University of Ljubljana Faculty of Mathematics and Physics Jernej Fesel Kamenik Role of Resonances in Heavy Meson Processes within Standard Model and Beyond Ph.D. Thesis Advisor: Prof. Svjetlana Fajfer Ljubljana, 2007 Univerza v Ljubljani Fakulteta za matematiko in fiziko Jernej Fesel Kamenik Vpliv resonanc na procese teˇzkih mezonov znotraj standardnega modela in njegovih razˇsiritev Disertacija Mentorica: prof. dr. Svjetlana Fajfer Ljubljana, 2007 MAJI For the conception and completion of this text I am much indebted to my advisor Svjetlana Fajfer, who has always managed to fine-tune and guide confidently my passage through the crevasses of studies and research. I am also especially grateful to Damir Be´cirevi´c, whose insight, energy and enthusiasm for the problems in the field continues to excite and inspire me. Not the least I thank him for his many insightful and valuable comments and suggestions he has given to the manuscript. I would also like to thank all the other collaborators, with whom parts of this work have been done, namely Nejc Koˇsnik, Jan O. Eeg and the sadly desist Paul Singer. With this I must not omit to mention the generosity of Doris Kim and Jim Wiss from the FOCUS collaboration, who have willingly shared parts of their experimental data with me. Thanks are also due to Miha Nemevˇsek and my father, Borut Kamenik, for carefully proof reading parts of the manuscript on a very short notice and calling my attention to numerous typos and other linguistic errors. Most of this work was done at the Department of Theoretical Physics at the Joˇzef Stefan Institute and I am indebted to the colleagues there for many enlightening discussions, especially to Miha Nemvˇsek, Nejc Koˇsnik, Jure Zupan, Borut Bajc and Jernej Mravlje. I am also grateful to the Laboratoire de Physique Th´eorique at Universit´e Paris Sud, Centre d’Orsay for the hospitality during spring 2005, where part of this work was done. However, many thanks also go to numerous excellent and welcoming hosts, devoted lecturers and stimulating student colleagues at numerous summer and winter schools which contributed enormously by deepening my understanding and broadening my horizons during the past four years. It seems to me impossible to thank enough to those that I hold dear, Maja, mine and her parents, and our closest friends. For they have expressed patience and understanding sometimes beyond to me reasonable amount for all my absences and hours spent at work, while completing this text. I would like to acknowledge that this work was supported in part by the Slovenian Research Agency. Support through the PAI project “Proteus” and the European Commission RTN network, Contract No. MRTN-CT-2006-035482 (FLAVIAnet) is also kindly acknowledged. Za zasnovo in neprecenljivo pomoˇc pri izvedbi priˇcujoˇcega dela sem hvaleˇzen mentorici Svjetlani Fajfer, ki je vedno znala izbrati in natanˇcno umeriti mojo pot mimo mnogih previs tekom ˇstudija in raziskav. Velika zahvala gre tudi Damirju Beˇcireviˇcu, ˇcigar vpogled, energija in zanesenjaˇstvo nad problemi na najinem skupnem podroˇcju me vedno znova navduˇsujejo in navdihujejo. Nenazadnje sem mu hvaleˇzen za njegove premnoge temeljite in obˇsirne komentarje ter predloge na predhodno verzijo tega teksta. Rad bi se zahvalil tudi preostalim sodelavcem na uspeˇsnih skupnih projektih, Nejcu Koˇsniku, Janu O. Eegu ter ˇzal prezgodaj preminulemu Paulu Singerju. Ob tem pa ne smem izpustiti omembe velikoduˇsnih Doris Kim in Jimu Wiss iz kolaboracije FOCUS, ki sta mi voljno ponudila del njihovih eksperimentalnih rezultatov v analizo. Hvaleˇzen sem tudi Mihi Nemevˇskuin mojemuoˇcetu, Borutu Kameniku, da sta v zelo kratkem roku skrbno prebrala dele besedila in me opozorila na mnoge tipkarske in druge jezikovne spodrsljaje. Priˇcujoˇce delo je v veliki meri nastalo na Odseku za teoretiˇcno fiziko Inˇstituta Joˇzef Stefan in zahvala gre vsem sodelavcem za premnoge diskusije, ˇse posebej pa bi ob tem rad izpostavil Miho Nemevˇska, Nejca Koˇsnika, Jureta Zupana, Boruta Bajca in Jerneja Mravljeta. Hvaleˇzen sem tudi Laboratoire de Physique Th´eorique na Universit´e Paris Sud, Centre d’Orsay, za gostoljubje spomladi leta 2005, kjer sem opravil del tukaj predstavljenih raziskav. Nenazadnje sem dolˇzan zahvale mnogim gostiteljem, predanim uˇciteljem in soˇstudentom na mnogih poletnih in zimskih ˇsolah, ki so mi omogoˇcili poglobiti moje razumevanje in razˇsiriti obzorja v zadnjih ˇstirih letih. Nikakor se ne morem dovolj zahvaliti mojim najdraˇzjim, Maji, mojim ter njenim starˇsem ter najinim najoˇzjim prijateljem, ki so vˇcasih izkazali ˇse preveliko mero potrpljenja in razumevanja ob mojih ˇstevilnih odsotnostih in predvsem urah porabljenih med pripravo priˇcujoˇcega teksta. Rad bi izpostavil, da je delo delno financirala Javna agencija za raziskovalno dejavnost Republike Slovenije. Stroˇski raziskav so bili delno kriti tudi iz projekta PAI “Proteus” ter s pomoˇcjo mreˇze RTN evropske komisije, pogodba ˇst. MRTNCT- 2006-035482 (FLAVIAnet). Great beauty seems invariably to portend some tragic fate. Michel Houellebecq, Les particules el´ementaires Abstract The effective theory based on combined chiral and heavy quark symmetry, the heavy meson chiral perturbation theory, is applied to studying the role of resonances in various processes of heavy mesons within and beyond the Standard Model. Chiral corrections including both positive and negative parity heavy meson doublets are calculated to the effective strong couplings featuring in the effective theory leading order interaction Lagrangian. Bare values of the chirally corrected couplings are extracted from the measured decay widths of charmed resonances. Chiral behavior of the couplings is studied in the leading logarithmic approximation. The mass splitting between heavy mesons of opposite parities spoils the chiral limit of the amplitudes. We restore a well behaved chiral limit by expanding the relevant loop integral expressions in inverse powers of the mass splitting. In semileptonic heavy to light decays we determine resonance contributions to the various form factors within an effective theory inspired model at zero recoil. We employ a form factor parameterization based on effective theory limits to extrapolate our results to the whole kinematical region in charm decays. We compare our results with experimental data and lattice calculations, and conclude that for a consistent description of the heavy to light semileptonic form factors, one needs to go beyond a single resonance pole approximation in the form factor parameterization. In semileptonic decays of B mesons to charm resonances we calculate chiral corrections to the relevant Isgur-Wise functions. We evaluate loop contributions of both positive and negative parity heavy mesons to the chiral running of the amplitudes. A well defined chiral limit is only restored after an appropriate loop integral expansion is performed. We calculate chiral loop corrections to the complete set of supersymmetric four-quark operators mediating heavy neutral meson mixing. The impact of heavy scalar meson contributions in the chiral loops on the chiral behavior of the bag parameters is studied and a well defined chiral extrapolation procedure is defined. Very rare nonleptonic decays of the Bc meson are studied within the Standard Model where they are mediated by box loop diagrams, and within a number of Standard Model extensions. Based on existing experimental searches for related B meson decays, limits are imposed on some of the models studied. The most promissing nonleptonic two-and three-body decay channels of the Bc meson in the search for such new physics contributions are identified. Key Words: heavy meson chiral perturbation theory, decays of charmed mesons, weak decays of heavy mesons, hadronic decays of heavy mesons, heavy neutral meson oscillations, new physics searches, lattice quantum chromodynamics PACS: 12.39.Fe, 13.20.Fc, 13.25.Ft, 13.25.Hw, 12.39.Hg, 12.38.Gc xiii Povzetek V doktorskem delu uporabimo efektivno teorijo, ki vkljuˇcuje tako kiralno simetrijo, kot simetrijo teˇzkih kvarkov, za ˇstudij vplivov resonanc na procese teˇzkih mezonov znotraj in izven standardnega modela. Vmoˇcnih razpadih teˇzkih mezonov izraˇcunamo kiralne popravke k efektivnim sklopitvenim konstantam, kjer upoˇstevamo prispevke teˇzkih mezonov tako pozitivne kot negativne parnosti. Gole vrednosti efektivnih sklopitev doloˇcimo iz razpadnih ˇsirin ˇcarobnih resonanc. Analiziramo tudi kiralno obnaˇsanje efektivnih sklopitev v pribliˇzku vodilnih logaritmov. Opazimo, da masna reˇza med teˇzkimi mezoni pozitivne in negativne parnosti pokvari kiralno limito amplitud. S pomoˇcjo razvoja zanˇcnih integralov po obratni vrednosti masne reˇze ponovno vzpostavimo dobro doloˇceno kiralno limito. Znotraj efektivnega modela prouˇcujemo prispevke resonanc k semileptonskim razpadom ˇcarobnih v lahke mezone. Napovedi v limiti niˇctega odboja ekstrapoliramo na celotno kinematsko podroˇcje s pomoˇcjo sploˇsne parametrizacije oblikovnih funkcij, ki temelji na limitah efektivnih teorij kvantne kromodinamike. Naˇse rezultate primerjamo z eksperimentalnimi podatki in izraˇcuni na mreˇzi. Zakljuˇcimo, da enostavni pribliˇzek enega pola ne more veˇc zadovoljivo opisati semileptonskih oblikovnih funkcij. V semileptonskih razpadih mezonov B vˇcarobne mezonske resonance izraˇcunamo kiralne popravke k funkcijam Isgur-Wise. Pri tem upoˇstevamo prispevke teˇzkih mezonov obeh parnosti h kiralnemu obnaˇsanju amplitud. Dobro definirano kiralno limito dobimo le po primernem razvoju zanˇcnih integralov. Izraˇcunamo kiralne popravke k celotnemu naboru kvarkovskih operatorjev, ki povzroˇcajo oscilacije teˇzkih nevtralnih mezonov. Obravnavamo prispevke teˇzkih skalarnih mezonov h kiralnemu obnaˇsanju parametrov “vreˇce” ter predpiˇsemo dobro definiran postopek njihove kiralne ektrapolacije. Zelo redke neleptonske razpade mezonov Bc obravnavamo znotraj standardnega modela, kjer potekajo le preko ˇskatlastih zank, ter znotraj nekaterih njegovih razˇsiritev. Na podlagi obstojeˇcih eksperimentalnih iskanj sorodnih razpadov mezona B, postavimo meje na parametre nekaterih obravnavanih modelov. Nato predlagamo najobetavnejˇse dvo-in trodelˇcne razpadne kanale mezona Bc za bodoˇca iskanja signalov nove fizike. Kljuˇcne besede: kiralna perturbacijska teorija s teˇzkimi mezoni, razpadi ˇcarobnih mezonov, ˇsibki razpadi teˇzkih mezonov, hadronski razpadi teˇzkih mezonov, oscilacije nevtralnih teˇzkih mezonov, signali nove fizike, izraˇcuni kvantne kromodinamike na mreˇzi Stvarni vrstilec -PACS: 12.39.Fe, 13.20.Fc, 13.25.Ft, 13.25.Hw, 12.39.Hg, 12.38.Gc xv Notation The characters from the middle fo the Greek alphabet µ, .,. . . in general run over space-time indices 0, 1, 2, 3, while the Latin indices i, j, k,. . . tun over spatial indices 1, 2, 3. The characters from the beginning of the Latin alphabet a, b,. . . in general run over light quark flavor indices 1, 2,. . . ,N in case of SU(N) chiral flavor theory. Spatial vector quantities are denoted with bold Latin letters e.g. p, while indices of Lorentz µ covariant quantities are writen explicitely e.g p. The metric used in the thesis is .µ. =diag(1,-1,-1,-1), where the indices run over 0, 1, 2, 3, with 0 the temporal index. The Levi-Civita tensor µ... is defined as a totally antisymmetric tensor with 0123 =1. The Einstein summation over repeated indices is assumed unless stated otherwise. The dot- product p · k denotes pµkµ. The Dirac matrices are defined so that .µ.. + ...µ =2.µ. .Also, .5 = i.0.1.2.3.The matrix .µ. i =[.µ,.. ]. The slash on a character denotes p = p µ.µ. The trace Tr runs over Dirac 2 matrix indices. The Hermitian adjoint of a vector, matrix or operator O is denoted O†.A bar on a Dirac bispinor u denotes —u = u†.0. The imaginary and real part of a complex number z are deonted (z)and (z) respectively. Natural units with . and the speed of light taken to be unity are used. The fine structure constant is thus .e.m. = e2/4. . 1/137. xvii Contents Contents xx List of Tables xxii List of Figures xxvi Povzetek doktorskega dela xxvii 1 Uvod...........................................xxvii 2 Efektivneteorijeteˇzkihinlahkihkvarkov ......................xxix 3 Hadronskeamplitude–pristopiinresonance ....................xxx 4 Moˇcnirazpaditeˇzkihmezonov ............................xxxii 5 Semileptonskirazpaditeˇzkihmezonov ........................xxxvii 5.1 Teˇzko–lahkiprehodi .............................xxxvii 5.2 Teˇzko–teˇzkiprehodi ............................. xlii 6 Meˇsanjeteˇzkihnevtralnihmezonov.......................... xlv 7 Redkihadronskirazpaditeˇzkihmezonov.......................xlviii 8 Zakljuˇcki ........................................ li 1 Introduction 1 2 Effective theories of heavy and light quarks 7 2.1 Whatisaneffectivefieldtheory?........................... 7 2.2 ExploringtheChiralsymmetryofQCD ....................... 8 2.2.1 Lightflavorsingletmixingandthe.. .................... 10 2.3 Symmetriesofheavyquarks.............................. 10 2.4 Combiningheavyquarkandchiralsymmetries ................... 12 3 Hadronic amplitudes 15 3.1 Operatorproductexpansion.............................. 15 3.2 Vacuumsaturationandresonancedominanceapproximations ........... 18 3.3 Parameterizationofhadronicamplitudes....................... 19 4 Strong decays of heavy mesons 25 4.1 Heavyquarkandchiralexpansion .......................... 26 4.2 Chiralcorrectionsincludingexcitedstates ...................... 27 4.2.1 Wave-functionrenormalization ........................ 27 4.2.2 Vertexcorrections ............................... 28 4.3 Extractionofphenomenologicalcouplings ...................... 30 xix 4.3.1 Renormalizationscaledependence,countertermcontributionsand1/mH corrections ................................... 32 4.4 ChiralextrapolationoflatticeQCDsimulations................... 34 4.4.1 Tamingresonancecontributions-thedecouplinglimit........... 36 4.4.2 Chiralextrapolationoftheeffectivemesoncouplings............ 37 5 Semileptonic decays of heavy mesons 41 5.1 Heavytolighttransitions ............................... 41 5.1.1 Semileptonicheavytolightmesonformfactors ............... 42 5.1.2 RelationsinHQETandSCETandFormFactorParameterization .... 44 5.1.3 HM.PTdescriptionincludingexcitedstates................. 48 5.1.4 Determinationofmodelparameters–comparisonwithexperiment.... 51 5.1.5 Summary .................................... 60 5.2 Heavytoheavytransitions .............................. 64 5.2.1 B › D(*) formfactors............................. 64 5.2.2 FrameworkandCalculationofChiralLoopCorrections .......... 65 5.2.3 ChiralExtrapolation.............................. 66 5.3 DiscussionandConclusion............................... 69 6 Heavy neutral meson mixing 71 6.1 .B =2operatorbasisandmixing .......................... 71 6.2 Chirallogarithmiccorrections............................. 73 + 6.3 Impactofthe1/2mesons .............................. 75 6.3.1 Decayconstants ................................ 76 6.3.2 Bagparameters................................. 77 6.4 RelevancetotheanalysisofthelatticeQCDdata.................. 79 6.5 Conclusion ....................................... 79 7 Rare hadronic decays of heavy mesons 81 7.1 Inclusive.S =2and.S =-1transitions ..................... 83 7.1.1 Operatorbasisandmixing .......................... 83 7.1.2 SM........................................ 83 7.1.3 BeyondSM................................... 84 7.2 NonleptonicdecaysofBc mesons........................... 88 7.2.1 Preliminaries .................................. 88 7.2.2 Amplitudes ................................... 93 7.3 Constrainingnewphysics ............................... 97 8 Concluding Remarks 101 AHM.PT Feynman rules 103 B One loop scalar and tensor functions, special cases 105 List of abbreviations 107 List of publications 109 Bibliography 110 List of Tables 1 Povzetek naˇsih rezultatov za efektivne sklopitve, kot je razloˇzeno vbesedilu. Vsevrednosti v redu ene zanke so dobljene ob zanemaritvi prispevkov kontraˇclenov na regularizacijski skali µ =1GeV. .....................................xxxiii 2 Napovedi naˇsega modela za vrednosti parametrov, ki nastopajo v formulah sploˇsne parametrizacije oblikovnih funkcij (17) za obravnavane razpadne kanale D › P .. Razpadni kanal D0 › .- oznaˇcen s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. . xl 3 Napovedi naˇsega modela za vrednosti parametrov, ki nastopajo v formulah sploˇsne parametrizacije oblikovnih funkcij (17) za obravnavane razpadne kanale D › P .. (b. =0za vse razpadne kanale). Razpadne kanale oznaˇcene s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. .................................... xl 4 Razvejitvena razmerja za semilptonske razpade D › P. Primerjava napovedi modela z eksperimentom. Razpadni kanal D0 › .- oznaˇcen s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. .............................. xlii 5 Razvejitvena razmerja ter razmerja delnih razpadnih ˇsirin za semilptonske razpade D › V. Primerjava napovedi modela z eksperimentom. Razpadne kanale oznaˇcene s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. ................... xlii 6 Razvejitvena razmerja razpadov .S =-1in .S =2mezona Bc - izraˇcunana znotraj modelov SM, MSSM, RPV in Z. . Za doloˇcitev neznanih kombinacij parametrov RPV (ˇcetrti stolpec) in Z. (peti stolpec) smo uporabili eksperimentalne gornje meje BR(B- › .-.-K+)< 1.8× 10-6 in BR(B- › K-K-.+)< 2.4× 10-6 . ............ li 1.1 Experimentally measured properties of the relevant charmed mesons and their dominant hadronic decay modes. The pseudoscalar ground states are listed for completeness. Unless indicated otherwise, the values are taken from PDG. ................... 4 4.1 Summary of our results for the effective couplings as explained in the text. The listed best-fit values for the one-loop calculated bare couplings were obtained by neglecting counterterms’ contributions at the regularization scale µ . 1GeV. .............. 32 4.2 Summary of probed input parameter ranges and coresponding fitted couplings’ variations as explained in the text. ................................. 34 5.1 The pole mesons and the flavor mixing constants KHP for the D › P semileptonic decays. 53 5.2 The pole mesons and the flavor mixing constants KHV for the D › V semileptonic decays. 56 5.3 Predictions of our model for the parameter values appearing in the general form factor formulas (5.39) for the various D › P .. decay channels considered. The D0 › .- decay channel marked with a dagger † has been used to fit the model parameters. .... 62 5.4 Predictions of our model for the parameter values appearing in the general form factor formulas (5.39) for the various D › V .. decay channels considered (b. =0for all decay modes as explained in the text).The decay channels marked with a dagger † have been used to fit the model parameters. ............................... 62 xxi 5.5 The branching ratios for the D › P semileptonic decays. Comparison of model predictions with experiment as explained in the text. The D0 › .- decay channel marked with a dagger † has been used to fit the model parameters. .................. 62 5.6 The branching ratios and partial decay width ratios for the D › V semileptonic decays. Predictions of our model and experimental results as explained in the text. The decay channels marked with a dagger † have been used to fit the model parameters. ...... 63 (*) 7.1 Numerical values of Bc › D(s) transition form factors at s =0by Kiselev. ....... 92 7.2 Pole masses used in Bc › D(*) transition form factors by Kiselev. ........... 92 (s) 7.3 B- › .-.-K+ and B- › .-.-K+ decay rates in various models and in terms of the relevant Wilson coefficients. ............................... 94 B- › D-D-D+ and B- › D-D- the relevant Wilson coefficients. ............................. 94 ss c s c D+ decay rates in various models and in terms of B- and B- › D- the relevant Wilson coefficients. ............................. 95 B- › K0D0.- and B- 0 s c c c c › D-.-K+ K-.+ decay rates in various models and in terms of D0K- decay rates in various models and in terms of the › K relevant Wilson coefficients. ............................... 95 B- and B- › D- 0 s c c › D-K0 K decay rates in various models and in terms of the relevant Wilson coefficients. ............................... 96 › D*- 0 B- and B- s c c › D*-K0 decay rates in various models and in terms of the K relevant Wilson coefficients. ............................... 96 B- and B- › D-*0 s c c › D-K*0 decay rates in various models and in terms of the K relevant Wilson coefficients. ............................... 97 › D*- *0 7.10 B- and B- s c c › D*-K*0 decay rates in various models and in terms of the K relevant Wilson coefficients. ............................... 98 7.11The branching ratios for the . c S =2decays of the B- lated within SM, MSSM, RPV and Z. models. The experimental upper bounds for the -6 -6 BR(B- › .-.-K+)< 1.8× 10and BR(B- › K-K-.+)< 2.4× 10have been used as an input parameters to fix the unknown combinations of the RPV terms (IV column) and the model with an additional Z. boson (V column). ............ 99 =-1and . meson calcu- S List of Figures 1 Enozanˇcni diagrami, ki prispevajo h kiralnim popravkom efektivnega moˇcnega vozliˇsˇca. .xxxiii 2 Renormalizacija sklopitve g vprocesu D*+ › D0.+. Primerjava kiralne ekstrapolacije v (I) SU(2)limiti z vodilnimi ˇcleni v razvoju zanˇcnih integralov (ˇcrna neprekinjena ˇcrta), (II) celotni logaritemski prispevki v SU(3)teoriji s teˇzko-lahkimi multipleti obeh parnosti (modra ˇcrtasto-pikˇcasta ˇcrta), (III) njihova degenerirana limita (siva ˇcrtasto-dvojno pikˇcasta ˇcrta), in (0) SU(3)logaritemski prispevki stanj negativne parnosti (rdeˇca ˇcrtasta ˇcrta), kot je razloˇzeno v besedilu. .............................xxxvi 3 Kiralna ekstrapolacija sklopitve h v razpadu D *+ › D0.+ . Primerjava kiralne ekstra 0 polacije z (I) razvojem zanˇcnih integralov v SU(2)limiti (ˇcrna neprekinjena ˇcrta), (II) celotni SU(3)logaritemski popravki (modra ˇcrtasto-pikˇcasta ˇcrta), in (III) njihova degenerirana limita (rdeˇca ˇcrtasta ˇcrta), kot je razloˇzeno v besedilu. .............xxxvi 4 Diagrama, ki prispevata k oblikovnim funkcijam H › P. .................xxxviii 5 Napovedi naˇsega modela (dva pola v ˇcrni neprekinjeni ˇcrti in en pol v rdeˇci prekinjeni ˇcrti) za porazdelitev suˇcnostne amplitude H2 +(s)v primerjavi s podatki kolaboracije FOCUS za semileptonski razpad D+ › K *0 . ............................. xl 6 Napovedi naˇsega modela (dva pola v ˇcrni neprekinjeni ˇcrti in en pol v rdeˇci prekinjeni ˇcrti) za porazdelitev suˇcnostne amplitude H2 - (s)v primerjavi s podatki kolaboracije FOCUS za semileptonski razpad D+ › K *0 . ............................. xli 7 Napovedi naˇsega modela (dva pola v ˇcrni neprekinjeni ˇcrti in en pol v rdeˇci prekinjeni ˇcrti) za porazdelitev suˇcnostne amplitude H2 0 (s)v primerjavi s podatki kolaboracije FOCUS za semileptonski razpad D+ › K *0 . ............................. xli 8 Diagram zanˇcnega popravka ˇsibkega vozliˇsˇca. .......................xliii 9 Kiralna ekstrapolacija naklona funkcije IW pri w =1(.(1)). Prispevki stanj negativne parnosti (ˇcrna crta) in domet moˇznih prispevkov stanj pozitivne parnosti, kadar razliko naklonov .(1)in .~(1)variiramo med 1(rdeˇca prekinjena ˇcrta) in -1(modra pikˇcastoprekinjena ˇcrta). .....................................xliv 10 Kiralna ekstrapolacija naklona funkcije .1/2 in njenega naklona pri w =1. Ekstrapolacija .1/2(1)skupaj s 1/.SH prispevki (ˇcrna neprekinjena ˇcrta), in domet moˇznih prispevkov k njenemu naklonu – .1. /2(1)– (sivo obmoˇcje) kadar variiramo razliko naklonov .(1), .~(1) in .1. /2(1)med 1(rdeˇca prekinjena ˇcrta) in -1(modra pikˇcasto-prekinjena ˇcrta). ....xliv 11 Diagrama, ki prispevata neniˇcelne kiralne popravke k psevdoskalarni razpadni konstanti teˇzko-lahkih mezonov. ..................................xlvi 12 Diagrami,kinastopajo v izraˇcunu kiralnih popravkov k operatorjem O1,2,4. ......xlvii 4.1 ”Sunrise topology” diagram contributing to heavy meson wave-function renormalization. The double line indicates the heavy-light meson and the dashed one the pseudo-Goldstone boson propagator. The full dot is proportional to the effective strong coupling. ...... 28 4.2 ”Sunrise road” topology diagram contributing to effective strong vertex correction. .... 29 4.3 Renormalization scale dependence of the fitted bare couplings as explained in the text. .. 33 xxiii 4.4 Effect of the mq and E. counterterms of the size order |.| on the solutions for the couplings g (top left ), |h| (top right) and g~(bottom) as explained in the text. ........ 33 22 4.5 Typical chiral logarithmic contributions -mlog(m/µ2)are shown for pion, kaon and . ii as a function of r =md/ms,with ms fixed to its physical value, and µ =1GeV. .... 35 4.6 The g coupling renormalization in D*+ › D0.+ . Comparison of chiral extrapolation in (I) SU(2)limit and loop integral expansion (black, solid), (II) complete SU(3)log contribution of both parity heavy multiplets (blue, dash-dotted), (III) its degenerate limit (gray, dash-double-dotted), and (0) SU(3)log contributions of negative parity states (red, dashed line) as explained in the text. ........................... 39 4.7 Chiral extrapolation of the h coupling renormalization in D *+ › D0.+.Comparison of 0 chiral extrapolation with (I) loop integral expansion in the SU(2)limit (black, solid), (II) complete SU(3)log contribution (blue, dash-dotted), and (III) its degenerate limit (red, dotted) as explained in the text. ............................. 39 5.1 A schematic view of the s worldsheet in heavy to light (H › P) semileptonic decays, with imaginary contributions to F+ form factor marked in red. Crossed circles indicate quasi-stable particle (resonance) poles, while the cut along the real axis represents the t-channel HP pair emission above threshold t0. The physical kinematical region in the s-channel is marked with blue. .............................. 45 5.2 Diagrams contributing to H › P form factors. The square stands for the weak current vertex. .......................................... 48 5.3 Comparison of D0 › .- transition F+ form factor s dependence of our model two poles extrapolation (solid (black) line), single pole extrapolation (dashed (red) line), lattice QCD fitted to two poles (dot-dashed (blue) line) and experimental two poles fits ((green) dotted and dash-double dotted lines). .............................. 53 5.4 Comparison of the D0 › K- transition F+ form factor s dependence of our model two poles extrapolation (solid (black) line), single pole extrapolation (dashed (red) line), lattice QCD fitted to two poles (dot-dashed (blue) line) and experimental two poles fits ((green) dotted and dash-double dotted lines). ........................... 54 5.5 Comparison of the D0 › K- transition F0 form factor s dependence of our model (solid (black) line), quark model of Melikhov & Stech (dashed (red) line) and lattice QCD fitted to a pole (dot-dashed (blue) line). ............................ 55 5.6 Comparison of the D0 › .- transition F0 form factor s dependence of our model (solid (black) line), quark model of Melikhov & Stech (dashed (red) line) and lattice QCD fitted to a pole (dot-dashed (blue) line). ............................ 55 5.7 Solutions of eq. (3.32) in the .2 × ..parameter plane for the various decay channels considered. ........................................ 57 5.8 Predictions of our model for the s dependence of the form factors V (s)(black solid line), A0(s)(red dashed line), A1(s)(blue dotted line) and A2(s)(green dash-dotted line) in D0 › K-* transition. .................................. 58 5.9 Predictions of our model for the s dependence of the form factors V (s)(black solid line), A0(s)(red dashed line), A1(s)(blue dotted line) and A2(s)(green dash-dotted line) in D0 › .- transition. ................................... 58 5.10Predictions of our model for the s dependence of the form factors V (s)(black solid line), A0(s)(red dashed line), A1(s)(blue dotted line) and A2(s)(green dash-dotted line) in Ds › . transition. .................................... 59 5.11 Predictions of our model (two poles in black solid line and single pole in red dashed line) for the s dependence of the helicity amplitude H2 (s)in comparison with scaled FOCUS + data on D+ › K *0 semileptonic decay. ......................... 59 5.12Predictions of our model (two poles in black solid line and single pole in red dashed line) for the s dependence of the helicity amplitude H2 (s)in comparison with scaled FOCUS - data on D+ › K *0 semileptonic decay. ......................... 60 5.13Predictions of our model (two poles in black solid line and single pole in red dashed line) for the s dependence of the helicity amplitude H2(s)in comparison with scaled FOCUS 0 data on D+ › K *0 semileptonic decay. ......................... 60 5.14Predictions of our model for the s dependence of the helicity amplitudes H2(s)for the i D+ › .0 semileptonic decay. Two poles’ predictions are rendered in thick (black) lines while single pole predictions are rendered in thin (red) lines: H+ (solid lines), H- (dashed lines) and H0 (dot-dashed lines). ............................. 61 5.15 Predictions of our model for the s dependence of the helicity amplitudes H2(s)for the i D+ › . semileptonic decay. Two poles’ predictions are rendered in thick (black) lines s while single pole predictions are rendered in thin (red) lines: H+ (solid lines), H- (dashed lines) and H0 (dot-dashed lines). ............................. 61 5.16Weak vertex correction diagram. ............................. 65 5.17Chiral extrapolation of the slope of the IW function at w =1(.(1)). Negative parity heavy states’ contributions (black line) and a range of possible positive parity heavy states’ contribution effects when the difference of slopes of .(1)and .(1)is varied between 1(red dashed line) and -1(blue dash-dotted line). ....................... 68 5.18Chiral extrapolation of the .1/2 function and its slope at w =1. .1/2(1)extrapolation including 1/.SH contributions (black solid line), and a range of possible extrapolation effects of its slope – .1. /2(1)– (gray shaded region) when the difference of slopes .(1), .(1)and .1. /2(1)is varied between 1(red dashed line) and -1(blue dash-dotted line). .. 68 5.19Example counterterm loop contributions yielding possible 1/.. and 1/...SH chiral corrections. The pseudo-Goldstone in the loop is emitted from a weak vertex counterterm. .69 6.1 The diagram which gives non-vanishing chiral logarithmic corrections to the pseudoscalar heavy-light meson decay constant. ............................ 74 6.2 The diagrams relevant to the chiral corrections to the SM bag parameter B1a.In the text we refer to the left one as “sunset”, and to the right one as “tadpole”. Only the tadpole diagram gives a non-vanishing contribution to the bag parameters B2,4a. ........ 75 6.3 In addition to the diagram shown in fig. 6.1, this diagram contributes the loop corrections + to the pseudoscalar meson decay constant after the 1/2mesons are included in HM.PT. 76 6.4 Additional diagrams which enter in the calculation of the chiral corrections to the operators O1,2,4. once positive parity heavy states are taken into account. .......... 78 — 7.1 Dominant contributions to the b › ssd (left) and b › dds—(right) transitions in the SM . Straight lines denote quarks while wavy lines represent W bosons. Filled dots stand for weak vertex insertion. .................................. 84 — 7.2 Dominant contributions to the b › ssd (left) and b › dds—(right) transitions in the MSSM. Dashed lines denote squarks while curly-straight lines represent gluinos. Filled dots stand for strong vertex insertion, while crosses denote off-diagonal squark mass insertions. ......................................... 85 — 7.3 Dominant contributions to the b › ssd (left) and b › dds—(right) transitions in the RPV model. Dashed lines denote sneutrinos, while filled dots stand for RPV vertex insertions. 86 — 7.4 Dominant contributions to the b › ssd (left) and b › dds—(right) transitions in the Z. model. Wavy lines denote Z. propagation, while filled dots stand for effective flavor violating Z-fermion vertex insertions. .......................... 87 xxvi 7.5 Diagrams contributing to the factorized matrtix elements of two body nonleptonic decays of B mesons. Double lines represent meson propagation, while crossed circles represent factorized weak current insertions. ............................ 7.6 Diagrams contributing to the factorized matrix elements of three body nonleptonic decays of B mesons. Dashed lines represent intermediate (resonant) state propagation while filled circles represent effective strong vertex insertions. ................. 89 90 Povzetek doktorskega dela 1Uvod Standardni model (SM) fizike osnovnih delcev je kvantna teorija umeritvenih polj, ki opisuje temeljne elektromagnetne, ˇsibke in moˇcne interakcije. Izoblikoval se je v ˇsestdesetih letih prejˇsnjega stoletja in je vse odtlej popolnoma obvladoval podroˇcje [1]. Osnovni gradniki SM so fermioni – leptoni in kvarki – ki so uvrˇsˇceni v tri druˇzine. Umeritvena grupa SM je SU(3)c×SU(2)L×U(1)Y , kjer SU(3)c zaznamuje umeritveno grupo kvantne kromodinamike (ang. quantum chromodynamics – QCD), SU(2)L je umeritvena grupa ˇsibkega izospina, medtem ko je U(1)Y umeritvena grupa ˇsibkega hipernaboja. Samo levoroˇcni kiralni ferminoi se transformirajo kot izospinski dubleti pod SU(2)L, medtem ko kvarki hkrati tvorijo fundamentalno tripletno reprezentacijo SU(3)c. Mase leptonov in kvarkov v SM generiramo s pomoˇcjo Higgsovega mehanizma – s spontanim zlomom simetrije, ko (kiralne) simetrije teorije njen vakuum ne spoˇstuje. V ta namen se v teorijo doda skalarni ˇsibko-izospinski dublet. Njegova vakuumska priˇcakovana vrednost zlomi umeritveno invarianco na podgrupo SU(3)c × U(1)EM in inducira mase ˇsibkim W± in Z umeritvenim bozonom. Kvarkovska polja v SU(2)L bazi v sploˇsnem niso lastna stanja mase. Zato jih obiˇcajno s pomoˇcjo unitarne matrike zavrtimo v masno bazo. Po konvenciji rotacijo izvedemo na poljih spodnjih kvarkov in rotacijsko matriko imenujemo Cabibbo-Kobayashi-Maskawa (CKM). V celoti jo lahko opiˇsemo s pomoˇcjo treh realnih kotov in ene kompleksne faze, ki krˇsi simetrijo CP. SM se lahko pohvali z mnogimi uspeˇsno prestanimi testi opisa osnovnih interakcij. Njegove napovedi so bile izdatno preverjene v pospeˇsevalniˇskih laboratorijih in se dobro ujemajo z meritvami do najviˇsjih energij dosegljivih do sedaj: precizni elektroˇsibki testi so v sploˇsnem v izjemnem ujemanju z napovedmi SM [2], medtem ko meritve krˇsitev simetrije CP vsistemih z mezoni K, D in B podpirajo CKM opis z eno univerzalno fazo [3, 4]. Zadnji osnovni gradnik, ki trenutno ˇse ˇcaka na svojo eksperimentalno odkritje je Higgsov bozon. Kljub velikim uspehom SM pa iz opazovanj ˇze vemo, da SM ne predstavlja popolne slike na najmanjˇsih prostorskih skalah. Tako na primer SM ne vsebuje gravitacije. Navkljub izrednim naporom, ki so jih v zadnjih desetljetjih teoretiˇcni fiziki namenili tej temi, je napredek poˇcasen in izsledki neprepriˇcljivi. Predvsem tudi zaradi skoraj popolne odsotnosti eksperimentalnih namigov na tem podroˇcju. Po drugi strani pa SM prav tako ne pojasni nedavno izmerjenih nevtrinskih oscilacij [5]. Te kaˇzejo na neniˇcelne mase nevtrinov, v nasprotju z opisom, ki ga ponuja SM.Hkrativse veˇc astrofizikalnih opazovanj nakazuje, da veˇcina materije v vesolju ni ne svetilna, ne barionske sestave [6]. Hkrati je relativno poˇcasna oziroma “hladna”. SM ne ponuja kandidatov za nebarionsko hladno temno snov. Nenazadnje naˇse trenutno razumevanje bariogeneze – tvorbe merjene asimetrije med barioni in anti-barioni – v zgodnjem vesolju zahteva mnogo veˇcje krˇsitve simetrije CP, kot so dovoljene znotraj SM [7]. Pravilna interpretacija eksperimentalnih podatkov in morebitna potrditev napovedi SM oziroma odkritje signalov nove fizike zahtevajo zanesljive izraˇcune relevantnih hadronskih procesov, xxvii xxviii temeljeˇc na fundamentalnem kvarkovskem opisu teorije. Neperturbativna narava QCD pri nizkih energijah, ki hkrati kvarke in gluone drˇzi ujete znotraj hadronov, nam pri tem povzroˇca obilico preglavic. Razvoj po sklopitveni konstanti v tem reˇzimu namreˇcniveˇc mogoˇc. Neposredni izraˇcuni opazljivk na podlagi osnovnih principov QCD so ˇse vedno mogoˇci s pomoˇcjo simulacij QCD na mreˇzi, vendar so te raˇcunsko izredno zahtevne [8]. Ena od moˇznosti, ki nam preostanejo je uporaba simetrij Lagrangevega operatorja, na podlagi katerih skonstruiramo efektivne teorije [9]. Neznane parametre v efektivni teoriji doloˇcimo iz eksperimentov ali, kadar je to mogoˇce, s pomoˇcjo neposredne primerjave z napovedmi polne teorije QCD. Takˇsne efektivne teorije lahko potem uporabimo neposredno za napovedi nekaterih eksperimentalnih procesov ali za oporo izraˇcunom QCD na mreˇzi pri pravilnem upoˇstevanju napak in aproksimacij. Ena pomembnih manifestacij moˇcne dinamike QCD pri nizkih energijah je pojav resonanc v spektru delcev. Zaznane so bile pred mnogimi leti v procesih pionov in kaonov, kjer so bile tudi podrobno raziskane [1]. Izkazale so se kot izredno vplivne v mnogih nizkoenergijskih procesih. Po eni strani omejujejo veljavnost doloˇcenih efektivnih teorij, ki resonanˇcnih pojavov niso sposobne zadovoljivo opisati. Hkrati je znano, da njihova prisotnost skoraj popolnoma zabriˇse prispevke redkih procesov znotraj SM oziroma nove fizike k oscilacijam mezona D in njegovim redkim razpadom [10]. Po drugi strani pa so fiziki dolgo predvidevali, da so zaradi relativno velikih mas kvarkov c in b prispevki resonanc teˇzkih mezonov v procesih teh dveh kvarkov manj pomembni. V zadnjih nekaj letih pa so mnogi eksperimenti poroˇcali o prvih opaˇzanjih resonanc v spektru ˇ ˇcarobnih mezonov [11, 12, 13, 14, 15, 16]. Studije osnovnih lastnosti teh novih stanj so bile ˇse posebej stimulirane zaradi dejstva, da mase resnanc v nasprotju s teoretiˇcnimi napovedi kvarkovskih modelov [17, 18] in izraˇcunov na mreˇzi [19, 20] niso leˇzale daleˇc nad masami osnovnih stanj. To hkrati namiguje na potencialno velik vpliv resonanc v procesih D in Ds mezonov in nam zastavlja naslednja vpraˇsanja: Ali lahko ocenimo pomembne vplive najniˇzje leˇzeˇcih resonanc teˇzkih mezonov v procesih osnovnih stanj teˇzkih mezonov? Ali lahko ohranimo nadzor nad temi efekti, ˇse posebej znotraj efektivnih teorij QCD? Ali nam lahko morda pomagajo razumeti nekatere vidike opaˇzenih in izmerjenih procesov osnovnih stanj teˇzkih mezonov? In konˇcno, katere zakljuˇcke pridobljene v ˇcarobnem sektorju lahko prenesemo in apliciramo v procesih mezonov B in Bs, katerih resonance so trenutno ˇse izven dosega eksperimentalnih laboratorijev. V tej disertaciji bomo raziskali mnogo aspektov resonanc v procesih teˇzkih mezonov [21, 22, 23, 24, 25, 26, 27, 28, 29]. Njihovi poglavitni prispevki bodo analizirani v relevantnem pristopu efektivnih teorij QCD. Znotraj tega ogrodja bomo izraˇcunali hadronske parametre, ki nastopajo v mnogih nizkornergijskih procesih in preuˇcili vpliv resonanc teˇzkih mezonov na opazljivke. Te vsebujejo moˇcne in semileptonske razpadne ˇsirine teˇzkih mezonov, kot tudi parametre meˇsanja nevtralnih teˇzkih mezonov. Moˇcni razpadni kanali, kadar so dovoljeni, ponavadi prevladujejo v izmerjenih razpadnih ˇsirinah, zato jih lahko uporabimo kot kriterije veljavnosti izbranega efektivno-teoretskega pristopa ter hkrati iz njih doloˇcimo osnovne parametre efektivnih teorij. Semileptonski razpadi, ki potekajo preko nabitih kvarkovskih in leptonskih tokov, SM opisuje ˇze v drevesnem redu. V teh procesih zato potrjeno prevladujejo prispevki SM. Poglobljene raziskave tega podroˇcja zato predvsem preverjajo konsistentnosti znotraj SM, kot so meritve razliˇcnih matriˇcnih elementov CKM ter testi unitarnosti matrike CKM. Po drugi strani pa meˇsanje teˇzkih nevtralnih mezonov znotraj SM poteka v redu ene ˇskatlaste zanke. Tako se v teh procesih odpira okno za iskanje prispevkov nove fizike, ki so lahko, ne pa nujno, obteˇzeni s faktorji zank. Znotraj naˇsega pristopa bomo obravnavali vse mogoˇce hadronske amplitude, ki nastopajo v meˇsanju teˇzkih nevtralnih mezonov znotraj SM in izven. Nazadnje bomo obravnavali tudi zelo redke hadronske razpade dvojno-teˇzkega mezona Bc, ki potekajo, tako kot mezonsko meˇsanje, znotraj SM ˇsele v redu ˇskatlaste zanke. Uporabili bomo nekaj pridobljenega znanja o vplivu resonanc na izraˇcune relevantnih hadronskih razpadnih amplitud. Tako bomo s pomoˇcjo obstojeˇcih meritev postavili nekatere nove meje na mnoge predloge nove fizike in hkrati predlagali 2. EFEKTIVNE TEORIJE TEZKIH IN LAHKIH KVARKOV xxix perspektivne nove smeri iskanja nove fizike. 2 Efektivne teorije teˇzkih in lahkih kvarkov Trdovraten problem fenomenoloˇskih raˇcunov v hadronski fiziki predstavlja neperturbativna narava moˇcne interakcije. Pristop efektivnih teorij se je v minulih desetletjih izkazal kot izredno koristno orodje v tovrstnih obravnavah. Kot je obiˇcaj v sodobni fiziki, tudi tu uporabljamo simetrije za poenostavitev zahtevnih problemov. Lagrangev operator QCD ima v limiti brezmasnih Nf kvarkov kiralno simetrijo SU(Nf )R × SU(Nf )L, za katero na podlagi mnogih eksperimentalnih in teoretiˇcnih argumentov predpostavimo, da je spontano zlomljena v vektorsko podgrupo SU(Nf )V .Posledica takˇsne sponatne zlomitve je pojav brezmasnih Goldstonovih bozonov, ki parametrizirajo faktorski prostor SU(Nf )R × SU(Nf )L/SU(Nf )V in so tudi edine prostostne stopnje v nizkoenergijskih procesih. Za najbolj pogost primer Nf = 3 Goldstonova polja zapiˇsemo v obliki matrike . . 11 . .8 + . .0 .+ K+ 62 .- 11 . .8 - . .0 K0 .= .. . .. . , (1) 6 2 K- 02 K - 3 .8 medtem ko v primeru Nf =2upoˇstevamo le pionska polja. Njihove efektivne interakcije ne vsebujejo prispevkov z manj kot dvema odvodoma kar omogoˇca razvoj po prenosih gibalnih koliˇcin p, kjer je npr. vsak odvod reda p. Lagrangev operator v vodilnem redu takˇsnega kiralnega razvoja je [1, 30] f2 .µ.ab.µ.† ba (mq)ab.ba +(mq )ab.† ba L(2) . = , (2) + .0 8 kjer je . = exp2i./f. Mase psevdo-Goldstonovih bozonov, ki so posledica mas kvarkov u, d in predvsem s, pogosto parametriziramo v obliki Gell-Mannovih formul [31] 28.0ms 28.0ms r+12 8.0ms r+2 m= r, m= ,m= , (3) .f2 Kf2 2 .8 f2 3 22 kjer je r = mu,d/ms in 8.0ms/f2 =2m- m.. K Nekoliko drugaˇcna je simetrija teˇzkih kvarkov, ki je posledica asimptotske svobode QCD. Pri dovolj velikih energijah, ki so povezane z masami teˇzkih kvarkov, je narava QCD perturbativna in v mnogih pogledih podobna QED. Hkrati v interakcijah teˇzkih kvarkov njihov spin prispeva le v obliki relativistiˇcnih kromomagnetnih uˇcinkov. V limiti neskonˇcno teˇzkih kvarkov ti uˇcinki izginejo in dobimo efektivno SU(2) spinsko simetrijo. Nenazadnje QCD loˇci med okusi kvarkov le po njihovih masah. V limiti, ko mase NQ teˇzkih kvarkov hkrati poˇsljemo proti neskonˇcnosti, postanejo ti efektivno nerazloˇcljivi in dobimo novo SU(NQ) okusnosimetrijo, stanjapa namestoponjihovi gibalni koliˇcini razlikujemo po hitrosti v. Efektivno teorijo, ki upoˇsteva omenjene simetrije teˇzkih kvarkov imenujemo efektivna teorija teˇzkih kvarkov (ang. heavy quark effective theory – HQET) Simetrije teˇzkih kvarkov so izredno uporabne tudi v kombinaciji s kiralno simetrijo lahkih kvarkov in sicer v opisu interakcij mezonov, ki vsebujejo par teˇzkega in lahkega kvarka. Spinska simetrija teˇzkih kvarkov tu zahteva, da so hadronska stanja neodvisna od spina teˇzkega kvarka, kar teˇzko-lahke mezone uredi v masno degenerirane pare glede na parnost in spin lahkih prostostnih stopenj znotraj hadrona. Osnovna takˇsna dubleta negativne in pozitivne parnosti sta Hv =(1+ v)/2[P * - Pv.5]in Sv =(1+ v)/2[P * .5 - P0v], ter vsebujeta v 1v *µ *µ osnovna psevdoskalarna (Pv), vektorska (P ), skalarna (P0v) in aksialna (P )stanja teˇzkih v 1v mezonov. Njihove interakcije doloˇcata kiralna simetrija in simetrije teˇzkih kvarkov. V prvem redu obeh razvojev zapiˇsemo Lagrangev operator takˇsne efektivne teorije teˇzkih mezonov in psevdo-Goldstonovih bozonov (ang. heavy meson chiral perturbation theory – HM.PT) [32, 33] (1) (1) (1) (1) = - + L + L L L mix, HM.PT + 12 12 (1) = -Tr (iv ·Dab -.ab.H )Hb + gTr L A Ha HbHa ab.5 , - 12 (1) =Tr (iv ·Dab -.ab.S)Sb + gTr L A Sa SbSa ab.5 , + 12 L(1) = hTr HbSaA+h.c.. (4) mix ab.5 Zh.c. smo oznaˇcili dodatni hermitsko konjugiran operator, Tr pa oznaˇcuje sled ˇcez Diracove . indekse. Uvedli smo ˇse operator . = . s katerim definiramo operatorja kiralnega vektorskega Vµ =(..µ.† + .†.µ.)/2 in aksialnega Aµ = i(.†.µ. -..µ.†)/2= i.†.µ..†/2 toka. Prvi nastopa v kovariantnem odvodu kinetiˇcnega ˇclena Lagrangevega operatorja Dµ = .ab.µ -Vµ ,drugi ab ab pa definira interakcije med pari teˇzko-lahkih mezonov in lihim ˇstevilom psevdo-Goldstonovih bozonov, ki jih parametrizirajo efektivne sklopitvene konstante g, h in g. Prosta masna parametra teˇzkih mezonov .H in .S bi lahko v primeru, da bi obravnavali le interakcije teˇzkih mezonov ene parnosti, postavili na niˇc s primerno redefinicijo hitrosti. V naˇsem primeru pa to ni veˇc mogoˇce in v izraˇcunih se nam pojavi nova invariantna koliˇcina –razlikaobeh ˇclenov, ki jo oznaˇcimo s .SH ..S -.H . Videli bomo, da ta koliˇcina pomembno vpliva na interpretacijo in veljavnost izraˇcunov znotraj HM.PT. V prvem redu razvoja v kiralni simetriji in simetriji teˇzkih kvarkov zapiˇsimo ˇse operator ˇsibkega toka (0)µ i. † i.† J= Tr[.µ(1 -.5)Hb].- Tr[.µ(1 -.5)Sb].+ O(1/mQ) , (5) (V -A)HM.PT ba ba 22 ki ga bomo potrebovali pri izraˇcunu ˇsibkih procesov teˇzkih mezonov. . in .. sta prosta parametra, ki ju lahko identificiramo z razpadnima konstantama teˇzkih mezonov lihe in sode parnosti. 3 Hadronske amplitude – efektivni pristopi in resonance Pri fenomenoloˇski obravnavi ˇsibkih interakcij v hadronskih sistemih pogosto uporabljamo nekatere standardne metode in matematiˇcne pripomoˇcke. Tako nam npr. metode razvoja v operatorsko vrsto omogoˇcajo razˇclenitev problema v perturbativen izraˇcun visokoenergijskih prispevkov z asimptotsko prostimi kvarki na eni strani, ter na temeljnem nivoju neperturbativen izraˇcun hadronskih matriˇcnih elementov operatorjev, ki pa vsebujejo le lahke prostostne stopnje QCD. Na kratko bomo oplazili nekatere sploˇsne lastnosti, pribliˇzke in relacije med takˇsnimi hadronskimi amplitudami. Osnovna ideja razvoja v operatorsko vrsto je razˇclenitev poljubnega nelokalnega produkta operatorjev v vsoto lokalnih operatorjev pomnoˇzenih z efektivnimi t.i. Wilsonovimi parametri CA1...Ak T{A1(x1)A2(x2) ...Ak(xk) ---› (x -x1,...,x -xk)On(x), (6) xi›xn n kjer T oznaˇcuje operator ˇcasovne ureditve. Moˇctakˇsne razˇclenitve je dvojna: prviˇcvelja na operatorski ravni, je neodvisna od zunanjih stanj, na katero jo apliciramo in nam zato sluˇzi za izgradnjo efektivnih Hamiltonovih operatorjev; drugiˇc pa nam omogoˇca razˇclenitev skal v problemih, kjer lahko izmenjavo virtualnih prostostnih stopenj pri visokih energijah zakodiramo v Wilsonove koeficiente, fiziko nizkih energij pa opiˇsemo z efektivnimi operatorji. Pogosto lahko – 3. HADRONSKE AMPLITUDE PRISTOPI IN RESONANCE xxxi tako vrednosti Wilsonovih koeficientov izraˇcunamo analitiˇcno oz. perturbativno. Preostane nam izvrednotenje matriˇcnih elementov efektivnih operatorjev med zunanjimi stanji (f| in |i), ki opisujejo verjetnostno amplitudo za proces Mfi Mfi = Ci f|Oi |i. . (7) i V tem izvrednotenju leˇzi srˇzvsehteˇzav povezanih z izraˇcunom ˇsibkih prehodov med hadronskimi stanji. Trenutno najboljˇsa metoda za takˇsne raˇcune so simulacije QCD na mreˇzi. Vendar je naloga tako teˇzavna, ˇse posebno v prehodih med teˇzkimi in lahkimi hadronskimi stanji, da morajo tudi “eksaktne” simulacije na mreˇzi uporabljati mnoge pribliˇzke. Eden takˇsnih fenomenolo ˇsko in teoretiˇcno motiviranih pribliˇzkov je zelo enostaven a izjemno uporaben pribliˇzek vakuumskega zasiˇcenja (ang. vacuum saturation approximation – VSA) oz. popolne faktorizacije. Formalno ga izrazimo tako, da med produkte operatorjev, ki jih lahko identificiramo s kvazi-stabilnimi hadronskimi stanji vstavimo celoten nabor kvantnih stanj, nato pa zavrˇzemo vsa razen vakuuma. Soroden je pribliˇzek zasiˇcenja z resonancami, kjer vmesna stanja modeliramo z izmenjavo resonanc znotraj nekega efektivnega pristopa oz. modela. Na primeru semileptonskih prehodov si oglejmo ˇse nekaj sploˇsnih lastnosti hadronskih matri ˇcnih elementov, ki so nam pogosto v pomoˇc pri analizi hadronskih procesov. Hadronski matriˇcni element, ki opisuje semileptonske prehode med (psevdo)skalarnimi mezoni (Pi › Pf) lahko v sploˇsnem parametriziramo s pomoˇcjo primernih Lorentzovih kovariant na podlagi gibalnih koliˇcin s katerima oznaˇcimo zaˇcetno in konˇcno stanje (pi in pf), pomnoˇzenih z oblikovnimi funkcijami – skalarnimi funkcijami kvadrata izmenjane gibalne koliˇcine s =(pi - pf)2.Matriˇcni element toka JV -A ima tako le dva ˇclena 22 m- m µPi Pf Pf(pf)| J|Pi(pi). = F+(s)(pi + pf)µ - (pi - pf)µ V -A s 22 m- m Pi Pf +F0(s)(pi - pf)µ , (8) s kjer sta F+,0 vektorska in skalarna oblikovna funkcija. Posebnost takˇsne izbire parametrizacije je, da v teˇznostnem sistemu zaˇcetnega stanja natanˇcno razloˇcuje prispevke stanj razliˇcnih vrtilnih koliˇcin k amplitudi. Kot nakazuje ze samo poimenovanje, F+ vsebuje le prispevke stanj z vrtilno koliˇcino 1 – vektorske prispevke, medtem ko F0 opisuje prispevke skalarnih stanj z vrtilno koliˇcino 0. Zahteva po konˇcni vrednosti amplitude pri s =0 nam da ˇse dodatno kinematsko omejitev F+(0) = F0(0) . (9) Podobno razˇclembo lahko naredimo tudi v primeru, ko imamo v konˇcnem stanju vektorski mezon (P › V ). Takrat lahko zapiˇsemo matriˇcna elementa aksialnega (JA)invektorskega (JV ) toka kot 2V (s) V (, pV )| JVµ |P (pP). = mP + mV .µ..ß.* pP.pVß, µ 2mV V (, pV )| J|P (pP). = -i. * · (pP - pV )(pP - pV )µA0(s) A s . * · (pP - pV ) . -i(mP + mV ) . *µ - (pP - pV )µ A1(s) s * 22 · (pP - pV ) m- m PV +i (pP + pV )µ - (pP - pV )µ A2(s), (mP + mV ) s (10) kjer smo z . oznaˇcili polarizacijo konˇcnega vektorskega stanja. Oblikovna funkcija V bo sedaj vsebovala vse prispevke vektorskih stanj, A1 in A2 opisujeta izmenjave aksialnih prostostnih stopenj, A0 pa psevdoskalarne prispevke. Tudi tokrat nam dodatna omejitev zagotavlja konˇcnost matriˇcnega elementa pri s =0 mP + mV mP - mV A0(0) - A1(0) + A2(0) = 0 . (11) 2mV 2mV V razpadih psevdoskalarnih v vektorske mezone je pogosto uporabna parametrizacija razpadne amplitude v obliki t.i. suˇcnostnih amplitud 2mP |pV (y)| 22 H±(y)=(mP + mV )A1(mP y) ± V (mP y), mP + mV H0(y)= mP + m. V [m 2 (1 - y) - m 2 ]A1(m 2 P y) - 2mP |pV (y)|2 . 2 A2(mP y), PV 2mP mV ymV (mP + mV )y (12) kjer je y = s/m2 in vektor gibalne koliˇcine konˇcnega stanja je podan kot P 22 [m(1 - y)+ m]2 PV 2 |pV (y)|2 = - mV . (13) 4m2 P 4Moˇcni razpadi teˇzkih mezonov Natanˇcno poznavanje efektivnih moˇcnih sklopitev v prvem redu HM.PT je bistveno za teoreti ˇcne izraˇcune ˇsibkih procesov teˇzkih mezonov znotraj HM.PT, saj te sklopitve nastopajo v vseh zanˇcnih kiralnih korekcijah k kateremu koli operatorju znotraj HM.PT. Trenutna najzaneslivej ˇsa metoda za oceno hadronskih matriˇcnih elementov so numeriˇcne simulacije QCD na mreˇzi. Zaradi raˇcunskih teˇzav ob pribliˇzevanju kiralni limiti, ˇstudije na mreˇzi uporavljajo velike vrednosti mas lahkih kvarkov. Fizikalne rezultate potem dobijo s pomoˇcjo kiralne ektrapolacije. Ta v postopek vnese nove sistematiˇcne napake, ki jih je izredno teˇzko nadzorovati. Z niˇzanjem mas kvarkov namreˇcpriˇcakujemo vse bolj izrazite uˇcinke spontanega zloma kiralne simetrije [34, 35, 36]. HM.PT nam omogoˇca vzpostaviti sistemtiˇcno kontrolo nad takˇsnimi efekti saj napoveduje kiralno obnaˇsanje hadronskih koliˇcin v procesih teˇzko-lahkih kvarkovskih sistemov. Njene napovedi lahko neposredno uporabimo kot vodilo pri kiralni ektrapolaciji rezultatov na mreˇzi. Znotraj HM.PT lahko izraˇcunamo kiralne logaritemske popravke (imenovane tudi ne-analitiˇcni ˇcleni). Priˇcakujemo, da bodo najbolj izraziti v limiti izredno majhnih energij ˇ oz. mas mq . .QCD. Ce je pogoj zagotovo izpolnjen za kvarke u in d, je situacija v primeru kvarka s precej manj jasna [37, 38]. Pravtako nejasna je velikost skale kiralne zlomitve ...Nekateri avtorji uporabljajo vrednosti okrog 4.f. . 1GeV [39], medtem kojodrugi raje enotijo z maso prve vektorske resonance m. =0.77 GeV [40, 30]. Obˇcasno se uporabljajo tudi ˇse manjˇse vrednosti. V sistemih teˇzko-lahkih mezonov postane situacija ˇse bolj zapletena. Prva orbitalno vzbujena stanja (jP =1/2+)namreˇcleˇzijo nenavadno blizu najniˇzje leˇzeˇcih stanj (jP =1/2-). Nedavna eksperimentalna odkritja mezonov D0s in D1s postavljajo velikost masne reˇze le na pribliˇzno .Ss . mD* - mDs = 350 MeV [13, 14, 15, 41]. Malce veˇcja je v primeru stanj brez 0s ˇcudnosti .Sq = 430(30) MeV [11, 12]. Hkrati raˇcuni QCD na mreˇzi v limiti statiˇcnih teˇzkih kvarkov [42] dajejo slutiti, da so masne reˇze majhne tudi v sektorju b kvarkov. Takoj opazimo, da sta tako .Ss kot .Sq manjˇsi od .., m. in celo mK, kar zahteva ponovni premislek o napovedih HM.PT. .j(q) .j(q) 4. MOCNI RAZPADI TEZKIH MEZONOV xxxiii Tabela 1: Povzetek naˇsih rezultatov za efektivne sklopitve, kot je razloˇzeno v besedilu. Vse vrednosti v redu ene zanke so dobljene ob zanemaritvi prispevkov kontraˇclenov na regularizacijski skali µ =1 GeV. Vnaˇsem izraˇcunu kiralnih popravkov k moˇcnim razpadom teˇzkih mezonov upoˇstevamo prispevke teˇzko-lahkih mezonov pozitivne in negativne parnosti. Uporabimo Lagrangev operator (4) in izpeljemo Feynmanova pravila za izraˇcun Feynmanovih zanˇcnih diagramov na sliki 1. Zapiˇsemo in razloˇcimo tudi vse potrebne kontraˇclene v redu O(mq), v katere lahko pospravimo neskonˇcne prispevke zanˇcnih izraˇcunov. Nato iz merjenih razpadnih ˇsirin ˇcarobnih resonanc izluˇsˇcimo gole vrednosti sklopitvenih konstant. Zaradi velikega ˇstevila kontraˇclenov v redu ene zanke, njihovih vrednosti ne moremo doloˇciti. V osnovni analizi, ki jo opravimo pri fiksni skali renormalizacije µ = 1 GeV, njihove prispevke zanemarimo [43]. V tabeli 1 med seboj primerjamo rezultate izraˇcunov v drevesnem redu, ter v redu ene zanke z in brez prispevkov teˇzko-lahkih mezonov pozitivne parnosti. Prispevke kontrˇclenov k izraˇcunom v redu ene zanke nato ocenimo posredno preko odvisnosti naˇsih rezultatov od skale renormalizacije, kot tudi neposredno s pomoˇcjo Monte-Carlo ˇzrebanja nakjuˇcnih vrednosti kontraˇclenov in njihovih prispevkov k analizi razpadnih ˇsirin. Dodatno preverimo tudi obˇcutljivost naˇsih rezultatov na vhodne podatke mas in predvsem masnih razlik teˇzko-lahkih mezonov. Nato obravnavamo prispevke dodatnih resonanc znotraj kiralnih zank h kiralni ekstrapolaciji, ki jo uporabljajo simulacije QCD na mreˇzi [45, 46]. Kaoni in mezoni . znotraj zank k takˇsni ekstrapolaciji praktiˇcno ne prispevajo, medtem ko poglavitna nelinearnost izhaja iz prispevkov pionov. Rezultati iz prejˇsnjega odstavka dopuˇsˇcajo moˇznost relativno velikih popravkov k renormalizaciji sklopitvenih konstant. Masna razlika med teˇzko-lahkimi mezoni obratnih parnosti .SH je namreˇc velika v primerjavi z masami pionov in povzroˇci, da imajo le-ti znotraj kiralnih zank veliko gibalno koliˇcino. To postavi veljavnost takˇsne razˇsirjene raˇcunske sheme pod vpraˇsaj, saj dajo navidezno najveˇcje popravke prav zanke v katerih nastopajo vzbujena stanja teˇzko-lahkih mezonov. Oglejmo si torej tipiˇcni zanˇcni integral v razˇsirjenishemi,ki bo Ha(v)Ha(v) Ha(v)Ha(v) Hb(v)Hb(v) Hb(v)Hb(v) .i(k).i(k) .i(k).i(k) Hc(v) Hc(v)Hc(v) Hd(v) .j(q).j(q) Slika 1: Enozanˇcni diagrami, ki prispevajo h kiralnim popravkom efektivnega moˇcnega vozliˇsˇca. Raˇcunska shema g |h| g Drevesni red 0.61 [44] 0.52 -0.15 Red ene zanke brez stanj pozitivne parnosti 0.53 Red ene zanke s stanji pozitivne parnosti 0.66 0.47 -0.06 xxxiv vsploˇsnem sedaj vseboval dve dimenzijski skali (m in .). Kiralna teorija dodatno zahteva, da morajo biti vse gibalne koliˇcine pseudo-Goldstonovih bozonov (pionov) majhne v primerjavi s skalo kiralne zlomitve ... Prvo skalo, ki nastopa znotraj zanˇcnih integralov indentificiramo z ˇ masami pseudo-Goldstonovih bozonov znotraj zank. Studije QCD na mreˇzi lahkotokoliˇcino spreminjajo in uporabljajo vrednosti vse do m ~ 1GeV, ˇceprav jo kiralna teorija varuje pred velikimi popravki in ima napovedno vrednost le za m . ... Po drugi strani pa lahko druga koliˇcina . v razˇsirjeni raˇcunski shemi vsebuje tudi masne razlike med teˇzko-lahkimi mezoni obratnih parnosti. V tem primeru vrednost . ni veˇc varovana ne s strani kiralne simetrije in ne simetrij teˇzkih kvarkov in lahko zavzame vrednosti reda tipiˇcne hadronske skale O(.QCD). Ko torej integriramo gibalne koliˇcine pionov znotraj zank tudi preko te skale, se napovedna moˇcin perturbativnost sheme poruˇsita. ˇ Ce smo lahko fenomenoloˇske sklopitve iz eksperimentalno merjenih razpadnih ˇsirin izluˇsˇcili ne ozirajoˇcse na takˇsne probleme (rezultati niso bili kritiˇcno odvisni od izbrane vrednosti .SH), pa je situacija v kiralnih ektrapolacijah popolnoma drugaˇcna. Tu namreˇcpriˇcakujemo, da bodo ne-analitiˇcni logaritemski prispevki prevladovali medtem ko lahko vse analitiˇcne prispevke enostavno priˇstejemo k relevantnim kontraˇclenom. V teoriji z le eno skalo (m) so tako prevladujoˇci popravki vedno oblike m2 log m2 in imajo dobro doloˇceno kiralno limito, ko gre m › 0. V naˇsi razˇsirjeni shemi pa dobimo med drugimi tudi nove prispevke oblike .2 log m2,ki v kiralni SH limiti divergirajo. Takoj razumemo, da bo situacija najslabˇsa prav v primeru pionov, katerih mase moramo iz vrednosti, simuliranih na mreˇzi, ekstrapolirati najdlje proti kiralni limiti. Teˇzavo poskusimo reˇsiti pri izvoru, zato se osredotoˇcimo na kiralno limito teorije in poskusimo narediti razvoj zanˇcnih integralov po majhnem parametru. V izbrani limiti so to ravno potence obratne vrednosti nove skale 1/.. Postopek bo legitimen ob predpostavki, da leˇzi relevantno obmoˇcje integracije stran od te skale, torej za majhne mase in gibalne koliˇcine pionov znotraj zank in dovolj velike vrednosti . ~ ... Tako pridelamo vsoto integralov oblike = .=large µ4-D µ. µ4-D µ. q-1 q · v q q q dD q dD q (1 + + ...), (14) (2.)D (q2 - m2)(v · q - .) (2.)D (q2 - m2). . kjer smo s tremi pikami oznˇcili viˇsje ˇclene v razvoju po 1/.. Postopek lahko razumemo tudi kot razvoj okrog limite, v kateri se vzbujena stanja razklopijo, njihovi prispevki k teoriji pa se preobrazijo v vrsto lokalnih operatorjev, duˇsenih s potencami 1/.. Interpretiramo jih kot nove prispevke h kontraˇclenom teorije brez dinamiˇcnih vzbujenih stanj. Kakrˇsna koli veˇcja odstopanja takˇsnega pristopa od napovedi teorije brez vzbujenih stanj s primerno zamaknjenimi parametri, bi signalizirala zlom razvoja. To bi pomenilo, da prispevkov dinamiˇcnih teˇzko-lahkih mezonov pozitivne parnosti v procesih osnovnih stanj ne moremo zanemariti. Priˇcakujemo, da bo razvoj dobro deloval na primeru kiralne teorije s simetrijsko grupo SU(2), ki vsebuje le dinamiˇcne pione kot pseudo-Goldstonove bozone, katerih mase so mnogo manjˇse od fenomenoloˇske vrednosti .SH. Za ilustracijo lahko skiciramo relevantne energijske skale v efektivni teoriji 2 2 m m . mu,d ~ < .SH . ms ~ K,. < .. . mQ. (15) .. .. Znotraj celotne SU(3) invariantne kiralne teorije s dinamiˇcnimi teˇzko-lahkimi mezoni obeh parnosti razvijamo po potencah {m.,K,.,.SH}/.. in {m.,K,.,.SH,..}/mQ,medtem ko v SU(2) kiralni teoriji z 1/.SH razvojem zanˇcnih integralov razvijamo po m./{..,.SH} in {m.,.SH,..}/mQ. Zgoraj opisan pristop uporabimo za ektrapolacijo efektivnih moˇcnih sklopitev g, h in g v redu ene zanke. Najprej zapiˇsemo vodilne logaritemske prispevke v SU(2) kiralni teoriji skupaj 4. MOCNI RAZPADI TEZKIH MEZONOV z vodilnimi 1/.SH popravki . .  22 1 mmg eff. 2 . 2 - . h2 g = g 1+ m log -4g 3+ , (16a) P*Pb.± . a (4.f)2 µ2 8.2 g SH . .  22 1 mmg eff. 2 .. gP*Pa.0 = g 1+ m. log -5g 2 - h2 3 - , (16b) a (4.f)2 µ2 8.2 g SH 1 m2 3 m2 heff. 2 . . . = h 1+ m log (2gg- 3g 2 - 3g2) - h2 , (16c) Pa0Pb.± (4.f)2 . µ2 4 2.2 SH 1 m2 3 m2 heff. 2 . . . = h 1+ m log (-2gg- 3g 2 - 3g2) - h2 , (16d) . Pa0Pa.0 (4.f)2 µ2 4 4.2 SH . .  22 1 mmg eff. 2 . 2 + . h2 . . = g 1+ m log -4g3+ , (16e) g  . Pa* 1Pb0.± (4.f)2 µ2 8.2 g SH . .  22 1 mmg eff. 2 .. gP*. .0 = g1+ m. log -5g2 + h2 3 - , (16f) a1Pa0 (4.f)2 µ2 8.2 g SH kjer loˇcimo med razpadi z nevtralnim ali nabitim pionom v konˇcnem stanju. Te formule bomo primerjali s celotnimi SU(3) kiralnimi prispevki tudi v teoriji z dinamiˇcnimistanjiobehparnosti. Vsledeˇci analizi bomo uporabili fenomenoloˇske vrednosti sklopitev iz tabele 1 in primerjali: (I) Razvoj zanˇcnih integralov v limiti SU(2). Vodilne prispevke da teorija brez vzbujenih stanj, mi pa bomo upoˇsetavali tudi vodilne poravke clenov reda 1/.2 SH. (II) Celotna SU(3) logaritemska ekstrapolacija s prispevki teˇzko-lahkih multipletov obeh parnosti. (III) Enako kot (II) vendar v degenerirani limiti .SH =0, (0) Kiralna SU(3) ektrapolacija brez 1/.2 prispevkov v enaˇcbah (4.17a-4.17f). SH Predpostavimo eksaktno SU(2) izospinsko limito in parametriziramo mase pseudo-Goldstonovih bozonov s pomoˇcjo Gell-Mannovih formul (3). Poslediˇcno v kiralni ektrapolaciji variiramo le razmerje r – med masama lahkih kvarkov in maso ˇcudnega kvarka, ki jo drˇzimo na njeni fizikalni vrednosti. Ker nas zanima le ne-analitiˇcna r-odvisnost naˇsih amplitud, lahko odˇsejemo skupno odvisnost od skale renormalizacije skupaj z vsemi prispevki, analitiˇcnimi v r.Naˇse rezultate normaliziramo na skupno vrednost v vseh primerih pri skali 8rab.0ms/f2 =.2 in jih odtu SH ekstrapoliramo proti kiralni limiti. Za primer si oglejmo primer efektivne sklopitve v procesu brez ˇcudnosti D*+ › D0.+ na sliki 2. Takoj opazimo, da vkljuˇcitev celotnih SU(3) logaritemskih prispevkov vzbujenih stanj v zankah vnese velika (. 30%) odstopanja od ekstrapolacije brez ˇ teh stanj. Ce pa namesto tega uporabimo razvoj zanˇcnih integralov, se odstopanja drastiˇcno zmanjˇsajo. Ekstrapolaciji znotraj SU(2) in SU(3) teorij brez vzbujenih stanj sta skoraj identiˇcni saj v obeh glavnino prispevajo pionske zanke. Vodilne prispevke izintegriranih vzbujenih stanj ocenimo s pomoˇcjo sivega podroˇcja med obema krivuljama SU(2) teorije v scenariu (I). Razlika nanese komaj 0.5%, kar kaˇze na dobro konvergenco razvoja. Za popolnost izriˇsemo ˇse diagram kiralne ektrapolacije efektivne sklopitve h v procesu D+ › D0.+ (ekstrapolacija sklopitve gpoteka v istem slogu kot g ob zamenjavi obeh sklopitev v ektrapolacijskih formulah). Tukaj scenarij (0) nima pomena saj v zunanjih stanjih nastopajo teˇzko-lahki mezoni obeh parnosti. Po drugi strani pa je razvoj zanˇcnih integralov v scenariju (I) ˇse vedno dobro definiran, ˇceprav je njegove fizikalna interpretacija sedaj manj jasna. Namreˇc iz teorije ne integriramo veˇc stanj posamezne parnosti temveˇcdejansko reˇzemo 0.05 0.050.050.1 0.10.10.15 0.150.150.2 0.20.20.25 0.250.250.3 0.30.3 r rr Slika 2: Renormalizacija sklopitve g vprocesu D*+ › D0.+ . Primerjava kiralne ekstrapolacije v (I) SU(2) limiti z vodilnimi ˇcleni v razvoju zanˇcnih integralov (ˇcrna neprekinjena ˇcrta), (II) celotni logaritemski prispevki v SU(3) teoriji s teˇzko-lahkimi multipleti obeh parnosti (modra ˇcrtasto-pikˇcasta ˇcrta), (III) njihova degenerirana limita (siva ˇcrtasto-dvojno pikˇcasta ˇcrta), in (0) SU(3) logaritemski prispevki stanj negativne parnosti (rdeˇca ˇcrtasta ˇcrta), kot je razloˇzeno v besedilu. 1 11 0.750.80.850.90.9510.750.80.850.90.9511-.g1-zankaD*+D0.+1-.g1-zankaD*+D0.+ Scen.0Scen.IScen.IIScen.III Scen.IScen.IIScen.III 1 11- -- .h .h.h1 11- -- zank zankzanka aa D DD* ** + ++ D DD0 00. ..+ ++ 0.95 0.950.95 0 00 0.9 0.90.9 0.85 0.850.85 0.8 0.80.8 0.05 0.050.050.1 0.10.10.15 0.150.150.2 0.20.20.25 0.250.250.3 0.30.3 r rr Slika 3: Kiralna ekstrapolacija sklopitve h v razpadu D *+ › D0.+ . Primerjava kiralne ekstrapolacije z 0 (I) razvojem zanˇcnih integralov v SU(2) limiti (ˇcrna neprekinjena ˇcrta), (II) celotni SU(3) logaritemski popravki (modra ˇcrtasto-pikˇcasta ˇcrta), in (III) njihova degenerirana limita (rdeˇca ˇcrtasta ˇcrta), kot je razloˇzeno v besedilu. 5. SEMILEPTONSKI RAZPADI TEZKIH MEZONOV xxxvii visokoenergijske prispevke posamiˇcnih zanˇcnih integralov, ki nastanejo kot posledica kinematike vmesnih in zunanjih teˇzko-lahkih stanj. Tudi tukaj so vplivi vodilnih popravkov takˇsnega razvoja reda velikosti 0.5%. Naˇsa analiza kiralne ektrapolacije sklopitve g je pokazala, da lahko prispevki vzbujenih teˇzkolahkih stanj pomembno vplivajo na naklon in krivino ektrapolacij v limiti m. › 0. Zagovarjamo tezo, da je to posledica velikih gibalnih koliˇcin pionovvzanˇcnih integralih, kjer nastopajo masne razlike med teˇzko-lahkimi multipleti razliˇcnih parnosti .SH, ki v kiralni limiti ne gredo proti ˇ niˇc. Ce pa uporabimo fizikalno motiviranih pribliˇzkov – takˇsne integrale razvijemo po potencah 1/.SH – se njihovi efekti drastiˇcno zmanjˇsajo in prispevajo k ektrapolaciji le ˇse reda 0.5%. Poslediˇcno lahko sklepamo o dobri konvergenci izbranega razvoja. Torej lahko zakljuˇcimo, da je mogoˇce drˇzati kiralne popravke zank v moˇcnih razpadih teˇzkih mezonov pod nadzorom v kolikor ekstrapolacije izvajamo pod skalo .SH , vsekakor pa ostajajo pomembni za natanˇcno doloˇcitev efektivnih moˇcnih sklopitev g, h in g. 5 Semileptonski razpadi teˇzkih mezonov Eden izmed trenutno najpomembnejˇsih programov v hadronski fiziki se ukvarja z doloˇcitvijo parametrov CKM iz ekskluzivnih razpadov. Bistvena sestavina tega pristopa je natanˇcno poznavanje oblikovnih funkcij v teˇzko-teˇzkih in teˇzko-lahkih ˇsibkih prehodih hadronov. Tradicionalno so najveˇc pozornosti poˇzeli razpadi bezonov B ter z njimi povezano izluˇsˇcenje faze CKM in absolutnih vrednosti elementov CKM Vub in Vcb. Hkrati pa v sektorju ˇcarobnih mezonov absolutne vrednoste Vcs in Vcd najnatanˇcneje doloˇca unitarnost CKM, medtem ko neposredne meritve v ekspereimentih zavira slabo teoretiˇcno poznavanje velikosti relevantnih oblikovnih funkcij. V obeh sektorjih lahko prisotnost blizu leˇzeˇcih resonanc vzbujenih teˇzko-lahkih mezonov trenutno sliko precej spremeni. 5.1 Teˇzko–lahki prehodi V tem razdelku bomo na kratko obnovili sploˇsno parametrizacijo oblikovnih funkcij v prehodih H › P ,ki sta jo prva zasnovala Be´cirevi´c in Kaidalov [47], ter izoblikovali podobno parametrizacijo tudi za vse oblikovne funkcije v prehodih H › V .Takˇsna parametrizacija mora upoˇstevati znana eksperimentalna dejstva o prisotnosti resonanc v relevantnih razpadnih kanalih ter tudi teoretiˇcne limite teorij HQET in teorije kolinearnih prostostnih stopenj (ang. soft colinear effective theory – SCET), ko so te relevantne za dan problem. Potem bomo analizirali prispevke nedavno odkritih ˇcarobnih resonanc k oblikovnim funkcijam prehodov H › P in H › V ,za kar bomo uporabili efektivni model na podlagi HM.PT v katerega bomo dodali nova polja, ter sploˇsnih parametrizacij oblikovnih funkcij. V naˇsi diskusiji se bomo omejili na prispevke v prvem redu kiralnega razvoja ter razvoja po obratnih vrednostih mas teˇzkih kvarkov 1/mQ. V limiti majhnega odboja, ko izhajajoˇci lahki mezon P vmasnem sistemu zaˇcetnega teˇzkega mezona H skorajda miruje HQET napoveduje dobro znana umeritvena pravila za vse oblikovne funkcije H › P in H › V [48]. Hkrati v obratni limiti, ko mezon P vmasnemsistemu zaˇcetnega teˇzkega mezona H izide z maksimalno energijo za oblikovne funkcije veljajo nekoliko drugaˇcna umeritvena pravila [49]. Naˇsa naloga je, da poiˇsemo primerno konsistentno parametrizacijo poteka oblikovnih funkcij, ki bo zvezno povezala obe limitni obmoˇcji. Pri tem nam nekoliko pomaga dejstvo, da ˇze poznamo prevladujoˇce prispevke k nekaterim oblikovnim funkcijam v limiti majhnega odboja. V bliˇzini tega kinemtatskega obmoˇcja leˇzijo namreˇc znane ˇcarobne resonance negativne parnosti, ki bodo prispevale bliˇznje pole v konfiguracijski ravnini. Le-te lahko v primeru vektorskih oblikovnih funkcij F+ v H › P ter V v H › V (kot tudi psevdoskalarne oblikovne funkcije A0) izoliramo in dobro doloˇcimo, saj pripadajo najniˇzje leˇzeˇcim Ha(v)Ha(v)Hb(v) .i(k).i(k) Ha(v)Ha(v)Hb(v) .i(k).i(k) xxxviii vektorskim (psevdoskalarnim) ˇcarobnim resonancam H* (H). Upoˇstevajoˇcˇse vse umeritvene limite ter kinematske relacije med oblikovnimi funkcijami, lahko njihovo energijsko odvisnost kompaktno parametriziramo kot F+(0) F+(s)= , (1 -x)(1 -ax) F0(0) F0(s)= , (1 -bx) V (0) V (s)= , (1 -x)(1 -ax) A0(0) A0(s)= , y) (1 -y)(1 -aA1(0) A1(s)= , 1 -bx A2(0) A2(s)= , (17) (1 -bx)(1 -bx) kjer smo oznaˇcili x = s/m2 in y = s/m2 in hkrati veljajo znane kinematske relacije med H* H. F+(0) = F0(0) ter V (0), A0(0), A1(0) in A2(0). Proste parametre gornje parametrizacije, kamor ˇstejemo tudi parametre polov a, a, b. in b. , doloˇcimo s pomoˇcjo efektivnega modela, ki temelji na HM.PT. Feynmanova pravila HM.PT veljajo v podroˇcju majhnega odboja in dajo prispevke k oblikovnim funkcijam H ›P vprvem redu razvoja na sliki 4. Opazimo, da desni diagram na sliki ˇze spominja na resonanˇcni prispevek. Vendar pa ob primerjavi s parametrizacijo vektorske oblikovne funkcije F+ opazimo, da lahko zdanimi 1/2+ in 1/2- polji HM.PT identificiramo le prvega izmed obeh polov, ki prispevata k oblikovni funkciji. Nerodnost razreˇsimo, tako da v efektivno teorijo vnesemo nov set jP = 1/2- polj H, ki predstavljajo radialno vzbujena stanja psevdoskalarnih in vektorskih mezonov. Potrebne spremembe Lagrangevega operatorja HM.PT so enostavne (1) (1) (1) L+= L- + Lmix, HM.PT Slika 4: Diagrama, ki prispevata k oblikovnim funkcijam H ›P . 12 L(1) = -Tr Ha )Hb (iv ·Dab -.ab. e H , - 12 L(1) = HbHaab.5 +h.c., (18) mix hTr A in (0)µ † Ja(V -A)HM.PT+= i.Tr[.µ(1 -.5)Hb].ba, (19) 2 kjer smo vnesli tudi tri nove parametre: . eje masni popravek novega multipleta, h je efektivna H sklopitev med osnovnimi in radialno vzbujenimi stanji negativne parnosti, .pa je efektivna ˇsibka sklopitev novih stanj, povezana z njihovo razpadno konstanto. Sedaj ˇstevilo polov v 5. SEMILEPTONSKI RAZPADI TEZKIH MEZONOV xxxix parametrizaciji oblikovnih funkcij H › P ustreza ˇstevilu resonanˇcnih prispevkov v HM.PT modelu. Podobno igro lahko poskusimo tudi v primeru prehodov H › V .Ker pa HM.PT ne vsebuje lahkih vektorskih mezonov, moramo teorijo spet razˇsiriti z fenomenoloˇskim modelom. Posluˇzimo se pogosto uporabljenega principa skrite simetrije [32], po katerem lahke vektorske mezone vpeljemo kot umeritvena polja neke razˇsirjene SU(3)V simetrije in jih opiˇsemo z operatorjem .^µ = i gV.µ,kjerje .µ matrika lahkih vektorskih mezonov . 2 . .1(.µ + .0 ) .+ K*+ . µµ µ 2 . .-.1(.µ - .0 ) K*0 . .µ = . µ µµ . . (20) 2 K*- *0 µ Kµ .µ Kinetiˇcni in masni Lagrangev operator takˇsnih polj sta potem 1 af2 µ 2 .)ba] - ab )(Vµ,ba - .^µ,ba), LV =[Fµ. (^.)abF µ. (^(Vµ - .^(21) ab 2g2 V kjer smo definirali Fµ. (^.)= .µ.^. - .. .^µ +[^.µ,.^. ]. V prvem redu razvoja 1/mH lahko potem zapiˇsemo tudi interakcijski Lagrangev operator med lahkimi vektorskimi mezoni in teˇzko-lahkimi mezoni [32, 50] Lint µ = -ißTr[Hbvµ.^Ha]+ i.Tr[Hb.µ. Fµ. (^.)baHa], (22) 1/2- ba Lint µ mix = -i.Tr[Hbvµ.^baSa]+h.c. +iµTr[Hb.µ. Fµ. (^.)baSa]+h.c., (23) Lint µ mix = -i.Tr[Hbvµ.^baHa]+h.c. +iµTr[Hb.µ. Fµ. (^.)baHa]+h.c., (24) ter (0)µµ J+= .1Tr[.5Hb.^]+ .2Tr[.µ.5Hbv..^. ]. (25) a(V -A)HM.PT baba S temi dodatnimi gradniki lahko sestavimo Feynmanove diagrame, ki prispevajo k prehodom H › V , in so topoloˇsko ekvivalentni tistim na sliki 4, z zamenjavo pionskih z lahko-vektorskimi linijami. Ti diagrami lepo reproducirajo strukturo polov v sploˇsni parametrizaciji oblikovnih funkcij s prispevki resonanc s primernimi kvantnimi ˇstevili. Izjema je oblikovna funkcija A2, katere parametrizacija vsebuje dva efektivna pola, medtem ko naˇs model napoveduje le prispevek ene (edine) aksialne resonance iz jP =1/2+ multipleta. Naˇspristop ˇzelimo primerjati z izmerjenimi razpadnimi ˇsirinami in kotnimi porazdelitvami semileptonskih razpadov ˇcarobnih mezonov v lahke psevdoskalarne in vektorske mezone. Zato moramo napovedi HM.PT modelov ektrapolirati ˇcez celotno kinematsko podroˇcje. Posluˇzimo se sploˇsnih parametrizacij oblikovnih funkcij (17) ter hkrati uporabiti ˇcimveˇcobstojeˇcih eks perimentalnih informacij. Efektivne parametre polov oblikovnih funkcij (a, a, b. in b)tako zasiˇcimo z merjenimi oz. teoretiˇcno napovedanimi masami ˇcarobnih resonanc, katerih prispevke smo identificirali v HM.PT modelu. Preostale parametre doloˇcimo s prilagajanjem napovedi za razpadne ˇsirine znotraj naˇsega pristopa z eksperimentalno izmerjenimi vrednostmi. Rezultati takˇsnega postopka so vrednosti parametrov v tabelah 2 in 3. Na podlagi tako dobljenih parametrov najprej primerjamo napovedi naˇsega pristopa z nedavno eksperimentalno analizo suˇcnostnih amplitud, ki jo je naredila kolaboracija FOCUS [51]. Primerjava porazdelitev posamiˇcnih suˇcnostnih amplitud je na slikah 5, 6 in 7. Opazimo, da je Razpad F+(0) F0(0) ab Tabela 2: Napovedi naˇsega modela za vrednosti parametrov, ki nastopajo v formulah sploˇsne parametrizacije oblikovnih funkcij (17) za obravnavane razpadne kanale D › P .. Razpadni kanal D0 › .- oznaˇcen s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. D0 › .-† D0 › K- D+ › .0 D+ › K0 Ds › . Ds › .. D+ › . D+ › .. Ds › K0 0.60 0.60 0.72 0.72 0.60 0.62 0.72 0.71 0.73 0.81 0.87 0.66 0.60 0.62 0.60 0.62 0.60 0.62 0.55 0.76 0.57 0.83 0.55 0.76 0.57 0.83 0.57 0.83 0.57 0.83 0.55 0.76 0.55 0.76 0.55 0.76 Razpad V(0) A0(0) A1(0) A2(0) a= ab. D0 › .-† D0 › K-*† D+ › .0† D+ › K0*† D+ › . › .† Ds › K0* Ds 1.05 1.32 0.61 0.31 0.99 1.12 0.62 0.31 1.05 1.32 0.61 0.31 0.99 1.12 0.62 0.31 1.05 1.32 0.61 0.31 1.10 1.02 0.61 0.32 1.16 1.19 0.60 0.33 0.55 0.76 0.57 0.83 0.55 0.76 0.57 0.83 0.55 0.76 0.57 0.83 0.55 0.76 2 H Tabela 3: Napovedi naˇsega modela za vrednosti parametrov, ki nastopajo v formulah sploˇsne parametrizacije oblikovnih funkcij (17) za obravnavane razpadne kanale D › P .. (b. =0 za vse razpadne kanale). Razpadne kanale oznaˇcene s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. 4 Napoved Napoved (en pol) 3 FOCUS [51] 2 1 0 [GeV2 ] 0.20.40.60.8x * 0 ( D + › K ) + Slika 5: Napovedi naˇsega modela (dva pola v ˇcrni neprekinjeni ˇcrti in en pol v rdeˇci prekinjeni ˇcrti) za porazdelitev suˇcnostne amplitude H2 (s) v primerjavi s podatki kolaboracije FOCUS za semileptonski + *0 razpad D+ › K . 2 ) [GeV2 10 8 NapovedFOCUS[51] Napoved(enp.) 0.2 0.4 0.6 0.8 x Slika 7: Napovedi naˇsega modela (dva pola v ˇcrni neprekinjeni ˇcrti in en pol v rdeˇci prekinjeni ˇcrti) za porazdelitev suˇcnostne amplitude H2(s) v primerjavi s podatki kolaboracije FOCUS za semileptonski 0 *0 razpad D+ › K . ujemanje napovedi naˇsega modela z eksperimentalnimi podakti dobro, ˇceprav pa so eksperimentalne napake ˇse velike. Poleg sploˇsne parametrizacije primerjamo tudi napovedi naˇsega modela z enostavnim nastavkom enega pola za vse oblikovne funkcije. Iz slik 5 in 6 je razvidno, da se takˇsen pristop ne ujema dobro z eksperimentalnimi podatki. Na koncu podamo ˇse napovedi za razvejtivena razmerja vseh relevantnih semileptonskih prehodov D › P in D › V in primerjamo naˇse napovedi z merjenimi vrednostmi iz PDG [52]. Rezultati so povzeti v tabelah 4 in 5. Za primerjavo smo v tabelo 4 vkljuˇcili tudi rezultate, dobljene z naˇsim pristopom a le enim polom v parametrizaciji oblikovne funkcije F+.Naˇs model ektrapoliran s sploˇsno parametrizacijo da v sploˇsnem rezultate zdruˇzljive s trenutnimi eksperimentalnimi rezultati, medtem ko ekstrapolacija z enim samim polom popolnoma odpove. V principu bi lahko naˇs pristop posploˇsili tudi na razpade mezonov B. Vendar pa so ti, zaradi 5. SEMILEPTONSKI RAZPADI TEZKIH MEZONOV 0.2 0.4 0.6 0.8 x Slika 6: Napovedi naˇsega modela (dva pola v ˇcrni neprekinjeni ˇcrti in en pol v rdeˇci prekinjeni ˇcrti) za porazdelitev suˇcnostne amplitude H2 (s) v primerjavi s podatki kolaboracije FOCUS za semileptonski - *0 razpad D+ › K . 10203040NapovedFOCUS[51]  H(D+›K*0) 0 2[GeV2] Napoved(enpol) H ( › *0 - 6 4 Tabela 4: Razvejitvena razmerja za semilptonske razpade D › P . Primerjava napovedi modela z eksperimentom. Razpadni kanal D0 › .- oznaˇcen s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. Razpad D0 › .-† D0 › K- D+ › .0 D+ › K 0 D+ › . s D+ › .. s D+ › . D+ › .. D+ › K0 s B(dva pola) [%] 0.36 3.8 0.46 9.7 2.6 0.86 0.11 0.016 0.33 B(en pol) [%] 0.36 0.43 0.51 1.1 0.38 0.03 0.006 0.0003 0.06 B(Eksp.)[%] 0.36 ± 0.06 3.43 ± 0.14 0.31 ± 0.15 6.8 ± 0.8 2.5 ± 0.7 0.89 ± 0.33 < 0.5 < 1.1 Razpad B [%] B (Eksp.) [%] .L/.T .L/.T (Eksp.) .+/.- .+/.- (Eksp.) D0 › .-† 0.20 0.194(41) [53] D0 › K-*† 2.22.15(35) [52] D+ › .0† 0.25 0.25(8) [52] D+ › K0*† 5.65.73(35) [52] Ds › .† 2.42.0(5) [52] D+ › . 0.25 0.17(6) [53] Ds › K0* 0.22 1.10 1.14 1.10 1.13 1.13(8) [52] 1.08 1.10 1.03 0.13 0.22 0.13 0.22 0.22(6) [52] 0.21 0.13 0.13 Tabela 5: Razvejitvena razmerja ter razmerja delnih razpadnih ˇsirin za semilptonske razpade D › V . Primerjava napovedi modela z eksperimentom. Razpadne kanale oznaˇcene s kriˇzcem (†) smo uporabili za prilagajanje novih parametrov. mnogo veˇcjega kinematskega podroˇcja mnogo bolj obˇcutljivi na vrednosti oblikovnih funkcij pri s . 0 in zato zahtevajo pristop, ki presega le upoˇstevanje najniˇzjih resonanc. 5.2 Teˇzko–teˇzki prehodi Na naˇsi misiji k natanˇcni doloˇcitvi matriˇcnega elementa CKM Vcb igrajo pomembno vlogo ˇstudije razpadov mezona B vˇcarobne resonance. Eksperimenti z namenom doloˇciti vrednost Vcb dejansko izluˇsˇcijo produkt |VcbF(1)|,kjer je F(1) hadronska oblikovna funkcija prehodov B › D ali B › D* pri niˇctem odboju. Pomankanje natanˇcnih informacij o obliki in velikosti teh oblikovnih funkcij je tako ˇse vedno glavni vir napak. V obravnavi hadronskih lastnosti s pomoˇcjo QCD na mreˇzi najveˇcje teˇzave nastanejo zaradi majhnih mas lahkih kvarkov. ˇ Studije na mreˇzi morajo uporabljati veˇcje mase in rezultate naknadno ekstrapolirati k njihovim fizikalnim vrednostim. V teh ˇstudijah je kiralno obnaˇasnje amplitud ˇse posebej pomembno. HM.PT je v tem pogledu zelo uporabna, saj nam omogoˇca nekaj kontrole nad napakami, ki se pojavijo ob pribliˇzevanju kiralni limiti. Ogledali si bomo torej popravke kiralnih zank znotraj HM.PT v semileptonskih razpadih B mezonov v ˇcarobne mezone obeh parnosti in doloˇcili njihov vpliv na kiralno ekstrapolacijo, kot jo uporabljajo ˇstudije QCD na mreˇzi pri obravnavi relevantnih oblikovnih funkcij. Ponovno uporabimo formalizem Lagrangevih operatorjev v efektivni kiralni teoriji teˇzkolahkih mezonov in pseudo-Goldstonovih bozonov. ˇ Sibki del Lagrangevega operatorja, ki opisuje .i(q) 5. SEMILEPTONSKI RAZPADI TEZKIH MEZONOV prehode med teˇzkimi kvarki lahko zapiˇsemo s ˇsibkimi tokovi teˇzkih mezonov v HM.PT [54, 55] cv. .bv › Ccb{-.(w)Tr Ha(v ).Ha(v) -.(w)Tr . Sa(v ).Sa(v) . -.1/2(w)Tr Ha(v ).Sa(v) +h.c.} (26) v prvem redu kiralnega razvoja in razvoja po obratnih vrednostih mas teˇzkih kvarkov 1/mQ. Ob tem smo oznaˇcili . = .µ(1 -.5)in w = v ·v.Simetrija teˇzkih kvarkov zapoveduje enakost .(1) = .(1) = 1, ki je imuna na vsakrˇsne kiralne korekcije. Po drugi strani pa vrednosti .1/2(w) niso tako omejene. Kiralne zanˇcne popravke k funkcijam Isgur-Wise .(.), .(.)in .1/2(.)izraˇcunamo na podlagi Feynmanovih diagramov oblike na sliki 8. Teˇzki mezoni v zaˇcetnem in konˇcnem stanju si lahko namreˇc izmenjajo en psevdo-Goldstonov bozon, medtem ko znotraj zanke teˇce ˇse par teˇzkolahkih mezonov pozitivne ali negativne parnosti. Prispevke vseh takˇsnih konfiguracij apliciramo na kiralne ekstrapolacije, ki jih uporabljajo ˇstudije QCD na mreˇzi, da prestavijo mase lahkih ˇ mezonov iz velikih vrednosti v simulacijah v bliˇzino kiralne limite [45, 46]. Ze v prejˇsnjih poglavjih smo opozorili na probleme s kiralno limito amplitud, ki vsebujejo masno reˇzo med teˇzko-lahkimi mezoni obeh parnosti .SH. Spet uporabimo razvoj po 1/.SH, s katerim umirimo logaritemske popravke majhnih mas pionov. Kot smo ˇze razloˇzili, takˇsen razvoj dobro deluje na teoriji SU(2), v kateri kaoni in ete, katerih mase bi konkurirale .SH, ne nastopajo v zankah. Zato zapiˇsimo le izraze zanˇcno popravljenih funkcij Isgur-Wise za zunanja stanja teˇzko-lahkih mezonov brez ˇcudnosti v teoriji SU(2): 2 3 2 m. .aa(w)= .(w) 1+ m. log g 22(r(w) -1) 32.2f2 µ2 -h2 m. 2 .(w) m. 2 .1/2(w) 1 -w -hg w(w -1) , (27) 4.2 .(w).2 .(w) in 2 3 2 m. 3 .1/2aa(w)= .1/2(w) 1+ m. log µ2 -gg(2r(w) -1) - (g 2 + g2) 32.2f2 2 22 2  +h2 m. m. .(w) m. .(w) (w -1) -hg w(1 + w)+ hgw(1 + w) . 4.2 2.2 .1/2(w) 2.2 .1/2(w) (28) Sedaj nariˇsemo kiralno ekstrapolacijo funkcij Isgur-Wise pod skalo .SH, na kateri funkcije tudi normiramo (sliki 9 in 10). Trenutno ne poznamo zanesljivih ocen velikosti .(1) in .1. /2(1), ki nastopata v formulah za kiralno ekstrapolacijo, ko upoˇstevamo prispevke stanj pozitivne parnosti. Ha(v) Hc(v) Hc(v) Hb(v) Slika 8: Diagram zanˇcnega popravka ˇsibkega vozliˇsˇca. 0 000.05 0.050.050.1 0.10.10.15 0.150.150.2 0.20.20.25 0.250.250.3 0.30.30.35 0.350.35 r rr Slika 9: Kiralna ekstrapolacija naklona funkcije IW pri w =1 (.(1)). Prispevki stanj negativne parnosti (ˇcrna crta) in domet moˇznih prispevkov stanj pozitivne parnosti, kadar razliko naklonov .(1) in .~(1) variiramo med 1 (rdeˇca prekinjena ˇcrta) in -1 (modra pikˇcasto-prekinjena ˇcrta). 0.9750.980.9850.990.99510.9750.980.9850.990.9951.(1)1zanka/.(1)drevesni.(1)1zanka/.(1)drevesni(1/2)-prispevki.(1)-.(1)=1.(1)-.(1)=-1 1.3 1.31.3 . ..( (( .  ) )) (1 (1(1zank zankzanka) a)a) ( (( .  )(dr )(dr)(dre eev vve ees ssn nni ii) )) /. /./. 1 11/ // 2 22 1 11 00.050.10.150.20.250.30.3500.050.10.150.20.250.30.351/12/2.(1zan.)/.1/2(drev.) 1/2.(1zan.)/.(drev.) 1/21/2(min) .(1zan.)/.(drev.) 1/21/2(max) r rr 1.2 1.21.2 1.1 1.11.1 Slika 10: Kiralna ekstrapolacija naklona funkcije .1/2 in njenega naklona pri w =1. Ekstrapolacija .1/2(1) skupaj s 1/.SH prispevki (ˇcrna neprekinjena ˇcrta), in domet moˇznih prispevkov k njenemu na. . ~ klonu – (1) – (sivo obmoˇcje) kadar variiramo razliko naklonov .(1), .(1) in .. (1) med 1(rdeˇca 1/21/2 prekinjena ˇcrta) in -1 (modra pikˇcasto-prekinjena ˇcrta). 6. MESANJE TEZKIH NEVTRALNIH MEZONOV Zato njune moˇzne prispevke ocenimo tako, da njuni vrednosti variiramo glede na vrednost .(1) med 1 in -1. Opazimo, da so prispevki stanj pozitivne parnosti h kiralni ekstrapolaciji funkcije Isgur-Wise .(1) pod skalo .SH majhni (okrog enega odstotka po naˇsi oceni). Podobno sploˇsno obnaˇsanje lahko pripiˇsemo funkciji .(1) ob zamenjavi g -g,.SH --.SH in .(1) -.(1). Tudi kiralna ekstrapolacija vrednosti .1/2(1) (skupaj z 1/.SH popravki) deluje zelo poloˇzno in nakazuje, da je linearna ekstrapolacija v tem primeru lahko dober pribliˇzek. Po drugi strani pa so potencialni prispevki h kiralni ekstrapolaciji odvoda .1. /2(1) po trenutnih grobih ocenah lahko precejˇsnji, do 30%. Naˇsi rezultati so ˇse posebej pomembni za izluˇsˇcenje oblikovnih funkcij s pomoˇcjo QCD na mreˇzi. Trenutne napake na parameter Vcb v ekskluzivnih kanalih so ˇze reda le nekaj odstotkov. To zahteva zelo natanˇcen nadzor nad teoretiˇcnimi napakami, ki lahko vplivajo na njegovo doloˇcitev. Razumevanje kiralnih popravkov je bistveno za zagotavljanje verodostojnosti izluˇsˇcenja oblikovnih funkcij in ocene napak na mreˇzi. Naˇse ocene vodilnih 1/.SH popravkov postavljajomejona natanˇcnost takˇsnih ekstrapolacij. 6Meˇsanje teˇzkih nevtralnih mezonov 0 Oscilacije v sistemih mezonov B0 -B posredujejo nevtralni tokovi, ki spreminjajo okus. d,s d,s Znotraj SM so prepovedane na drevesnem nivoju, zato nam njihove meritve omogoˇcajo dostop do delcev znotraj relevantnih zanˇcnih diagramov. Dandanes se natanˇcno merjene vrednosti .mBd =0.509(5)(3) ps-1 [56], in .mBs =17.31(33)(7) ps-1 [57] uporabljajo za omejitev oblike 17 unitarnostnega trikotnika CKM in tako doloˇcajo koliˇcino krˇsitev CP znotraj SM [3, 4]. Cilj nam oteˇzujejo teoretiˇcne napake v izraˇcunih vrednosti razpadnih konstant fBs,d in parametrov “vreˇce” (ang. bag parameters) BBs,d .Te koliˇcine lahko v principu izraˇcunamo na mreˇzi. Veliko oviro pri tem pa predstavlja majhna masa kvarka d, ki je v simulacijah ne moremo doseˇci neposredno, temveˇc le preko kiralne ekstrapolacije. Oglejmo si torej vplive teˇzko-lahkih mezonov pozitivne parnosti na izraˇcun kiralnih popravkov razpadnih konstant in parametrov vreˇce, ki nastopajo v 0 ˇstudijah prispevkov SM in supersimetriˇcnega SM k meˇsalnim amplitudam B0 -B d,s d,s. 0 Prispevki supersimetriˇcnega SM k meˇsalni amplitudi B0 -Bd,s se ponavadi obravnava v d,s tako imenovani supersimetriˇcni bazi .B = 2 operatorjev [58]: — i — j O1 = bi.µ(1 -.5)qbj.µ(1 -.5)q, j O2 =— bi(1 -.5)q i — bj(1 -.5)q, O3 =— bi(1 -.5)qj — bj(1 -.5)q i , (29) j O4 =— bi(1 -.5)q i — bj(1 + .5)q, — j — i O5 = bi(1 -.5)qbj(1 + .5)q, kjer sta i in j barvna indeksa. Znotraj SM k meˇsalni amplitudi pomembno prispeva le operator O1.Matriˇcni elementi gornjih operatorjev so ponavadi parametrizirani s pomoˇcjo parametrov vreˇce B1-5, ki predstavljajo merilo odstopanja od pribliˇzka VSA 0 B0 B |O1-5(.)| aa = B1-5(.) , (30) 0 Ba|O1-5(.)|Ba0VSA kjer smo z . oznaˇcili skalo renormalizacije, pri kateri loˇcimo nizko od visokoenergijskih prispevkov k amplitudam. xlvi Ha(v) Ha(v) Hb(v) Slika 11: Diagrama,kiprispevata neniˇcelne kiralne popravke k psevdoskalarni razpadni konstanti teˇzkolahkih mezonov. Za opis nizkoenergijskega obnaˇsanja matriˇcnih elementov operatorjev (29) uporabimo HM.PT. Preden se podamo v podrobnosti naj omenimo, da obstaja v eksaktni limiti statiˇcnih teˇzkih kvarkov (v tej limiti bomo operatorje oznaˇcevali s tildo) zveza med operatorji (29), in sicer 0 B |O3 + O2 + 1 O1|B0= 0, ki nam omogoˇca, da iz obravnave izloˇcimo enega izmed njih. V a2 a isti limiti definiramo tudi razpadne konstante teˇzko-lahkih mezonov f^namreˇckot 0|Aµ|Ba0(p)QCD 0|P|Ba0(p)QCD lim . = lim . = 0|A0|Ba0(v)HQET = if^avµ , (31) mb›. 2mBmb›. 2mB —— kjer smo z Aµ = b.µ.5q in P = b.5q oznaˇcili aksialni in psevdoskalarni tok. Nazadnje upoˇstevamo ˇse izsledke ˇstudije kiralnih popravkov v kaonski fiziki [59], po katerih se operatorja O4 in O5 razlikujeta le po barvnih indeksih – lokalni izmenjavi gluona, ki pa ne more vplivati na nizkoenergijske lastnosti amplitud. Ta dva operatorja morata torej utrpeti identiˇcne kiralne popravke. Tako nam za analizo kiralnih popravkov ostanejo le trije matriˇcni elementi operatorjev: O1, O2 in O4,ki jih s polji HM.PT zapiˇsemo kot . . . O1 = ß1XTr .†H.µ(1 -.5)X Tr .†H.µ(1 -.5)X aa X . . . +ß. Tr .†H.µ(1 -.5)X Tr .†S.µ(1 -.5)X 1X aa . . . +ß. Tr .†S.µ(1 -.5)X Tr .†S.µ(1 -.5)X, 1X aa . . . .†H.†H O2 = ß2XTr (1 -.5)X Tr (1 -.5)X aa X . . . +ß. Tr .†H (1 -.5)X Tr .†S (1 -.5)X 2X aa . . . +ß. Tr .†S (1 -.5)X Tr .†S (1 -.5)X, 2X aa .†H O4 = ß4XTr (1 -.5)X Tr [(.H)(1+ .5)X] a a X +ß. Tr .†H (1 -.5)X Tr [(.S)(1+ .5)X] 4Xa a +ß Tr .†S (1 -.5)X Tr [(.H)(1+ .5)X] 4Xa a +ß. 4XTr .†S (1 -.5)X Tr [(.S)a (1 + .5)X] , (32) a kjer smo oznaˇcili X .{1,.5,..,...5,...}. Najprej se osredotoˇcimo na kiralne popravke k (psevdoskalarnim) razpadnim konstantam teˇzkih mezonov. Prispevke dobimo iz zanˇcnih diagramov na sliki 11. Diagram na desni prispeva le v teoriji s teˇzko-lahkimi mezoni obeh parnosti, saj lahko znotraj zanke teˇcejo le (psevdo)skalarni mezoni. Izraˇcuni popravkov na podlagi teh diagramov ponovno pokaˇzejo, da prisotnost stanj pozitivne parnosti ne prizadane pionskih popravkov k razpadni konstanti, se pa popravki teh 6.MESANJETEZKIHNEVTRALNIHMEZONOV Ha(v) Hb(v) Hb(v) Ha(v) Ha(v) Ha(v) Ha(v) Hb(v) Ha(v) Ha(v) Hb(v) Ha(v) Slika 12: Diagrami, ki nastopajo v izraˇcunu kiralnih popravkov k operatorjem O1,2,4. stanj kosajo s kaonskimi in eta popravki. Relavantni kiralni logaritemski popravki tako ponovno prihajajo iz teorije s simetrijo SU(2)L . SU(2)R › SU(2)V in jih lahko zapiˇsemo kot 22 1+3g3 m ^2 . 2 fq = . 1 - m log + cf(µ)m, (33) .. 2(4.f)2 2 µ2 kjer smo z cf oznaˇcili relevantne kontraˇclene. Hkrati lahko enostavno preverimo, da so kiralni popravki k skalarni razpadni konstanti teˇzko-lahkih mezonov identiˇcni, z zamenjavo sklopitev g in g 22 1+3g3 m = .. 2 . 2 fq 1 - m log + cf(µ)m. (34) .. 2(4.f)2 2µ2 Kot smo ˇze pokazali v prejˇsnjih razdelkih, velja g2/g2 . 1, zato priˇcakujemo da bodo odstopanja od linearne ekstrapolacije za fq manjˇsa kot za f^q. Konˇcno se posvetimo meˇsalnim amplitudam, ki dobijo kiralne popravke iz diagramov na sliki 12. Ponovno diagrami v spodnjih dveh vrsticah prispevajo le ob upoˇstevanju teˇzko-lahkih stanj obeh parnosti. Osredotoˇcimo se na obmoˇcje m. . .SH in postopamo podobno kot doslej. Zanˇcne integrale namreˇc razvijemo po obratnih potencah masne reˇze med teˇzko-lahkimi mezoni obeh parnosti 1/.SH in ponovno pokaˇzemo, da ostanejo vodilni pionski logaritemski popravki neprizadeti in kiralna ektrapolacija znotraj SU(2) teorije dobro definirana. Zapiˇsemo 22 1 - 3gm Bdrevesni 2 . 2 B1q = 1 1 - m. log + cB1 (µ)m. , (35a) 2(4.f)2 µ2 2 . 3g2Y ± 1 m. Bdrevesni 2 . 2 B2,4q = 1+ m log + cB2,4 (µ)m, (35b) 2,4 .. 2(4.f)2 µ2 kjer smo oznaˇcili Y =(ß2* ,4/ß2,4), ß2 * = ß2.. + ß2...5 - 4ß2... in ß4 * = -ß4.. + ß4...5 . Morda je na tem mestu dobro povdariti, da je diskusija tega razdelka pomembna predvsem tudi za fenomenoloˇske pristope, ki oznaˇcujejo logaritemske popravke kaonov in et kot napovedi in hkrati doloˇcijo relevantne kontraˇclene iz limite velikega ˇstevila barv ali kakˇsnega drugega modela. Pokazali smo namreˇc, da so prispevki blizu leˇzeˇcih skalarnih resonanc po velikosti konkurenˇcni prispevkom kaonov in et, ter jih zato v takˇsnih diskusijah ne moremo zanemariti ali iz njih izloˇciti. Vendar pa je dejstvo, da bliˇznja skalarna stanja ne pokvarijo poglavitnih Ha(v) Hb(v) Ha(v) Ha(v) Hb(v) Ha(v) xlviii pionskih logaritemskih prispevkov, zelo dobrodoˇslo za ˇstudije QCD na mreˇzi, saj lahko te ˇse vedno uporabljajo formule HM.PT za ektrapolacijo njihovih rezultatov. Takˇsnipostopkipa morajo biti omejeni na teorijo SU(2) in pod skalo .SH. Izraˇcunali smo torej vodilne kiralne popravke k celotni bazi supersimetriˇcnih operatorjev, ki prispevajo k meˇsanju teˇzkih nevtralnih mezonov. Pokazali smo, da bliˇznje skalarne resonance ne vplivajo na vodilne pionske logaritemske prispevke, ki zato ostajajo zanesljivo vodilo pri kiralnih ektrapolacijah rezultatov simulacij na mreˇzi. Hkrati smo preverili, da se vodilni kiralni popravki k razpadnim konstantam teˇzko-lahkih mezonov sode in lihe parnosti ujemajo ob zamenjavi efektivnih sklopitvenih konstant g › g. 7 Redki hadronski razpadi .S =2in .S =-1mezonov Bc Redki razpadi mezonov B veljajozaeno najobetavnejˇsih podroˇcij zaodkritjenovefizike izven SM [60, 61, 62]. Priˇcakovati je namreˇc, da bodo novi virtualni delci vplivali na te procese. To ˇse posebej velja za razpade, ki potekajo preko nevtralnih tokov, ki spreminjajo okus, saj ti znotraj SM potekajo le preko zank. Ekstremni primer kaˇsnega pristopa je iskanje razpadov, ki so znotraj SM izredno redki, in ˇze samo opaˇzanje katerih bi pomenilo jasen signal nove fizike. Huitu, Lu, — Singer in Zhang so pred leti predlagali razpade b › ssd in b › dds—(v katerih se ˇcudnost spremeni za .S = -1oz. .S = 2) [63, 64] kot prototipne v takˇsnem iskanju. Njihov predlog temelji nadejstvu,daso takˇsni razpadi znotraj SM izredno redki, saj potekajo preko izmenjave gornjih kvarkov in bozonov W znotraj ˇskatlastih zank in imajo poslediˇcno razvejitvena razmerja reda 10-11 do 10-13 . Prihajajoˇci pospeˇsevalnik LHC bo med drugim izredno produktivna tovarna mezonov Bc, Priˇcakovanja za njihovo proizvodnjo se namreˇc gibljejo okoli 5 × 1010 dogodkov na leto ob luminoznosti 1034 cm-2s-1 [65]. ˇ Cetudi bi bila dejanska ˇstevilka nekaj redov velikosti manjˇsa, bo omogoˇcala ˇstudij redkih razpadov mezonov Bc, ki bo morda osvetlil prispevke fizike izven SM. — Posvetili se bomo torej izraˇcunom redkih prehodov b › ssd in b › dds—vdvo-in trodelˇcnih razpadnih kanalih mezona Bc znotraj SM ter nekaj najpopularnejˇsih okvirov nove fizike. Na podlagi znanih eksperimentalnih omejitev na prispevke obravnavanih modelov bomo podali napovedi za razvejitvena razmerja ter identificirali najperspektivnejˇse kanale za iskanje signalov nove fizike. — Efektivni ˇsibki Hamiltonov operator, ki zaobjema tudi procese b › ssd in b › dds—zapiˇsemo kot 5 . sn O sn sn + C sn O + C dn O dn dn + C dn O , (36) Heff. = C n=1 qi — operatorjev, ki prispevajo k procesom b › dds—(za q = s)in b › ssd (za q = d). Izberemo qi kjer C in oznaˇcujeta efektivne Wilsonove sklopitve, s katerimi pomnoˇzimo celotno bazo C O s 1 = iL — d .µb iL jR — d iL — d jR — d iL — d .µb iL jL — d .µs jL , jR , jL .µb .µs iR , s s O O .µs = = 2 3 (37) iR — d jL — d iR — d b jL — d s jR jL iL s iR s s O O b = = , , 4 5 skupaj z dodatnimi operatorji O. , ki jih dobimo s primernimi kiralnimi transformacijami s 1,2,3,4,5 gornjih (L - R), ter vse skupaj ˇse z obrnjenimi okusi kvarkov s in d. — Znotraj SM k procesu b › dds—(b › ssd) prispevata le operatorja Os(d).Glavne prispevke 3 7. REDKI HADRONSKI RAZPADI TEZKIH MEZONOV k Wilsonovim sklopitvam dajo top in ˇcarobni kvarki ter bozoni W znotraj ˇskatlastih zank (38a) G2 2 2 2 2 m m m m d,SM 3 2 W VtbVts * VtdVts* f + VcdVcs * F = m W W c c C g ,, 2 2 2 2 4.2 m m m m W W t t G2 2 2 2 2 m m m m s,SM 3 2 W VtbVtd * VtsVtd* f + Vcs V * cd F = m W W (38b) c c C g ,, 2 2 2 2 4.2 m m m m t W t W kjer je f(x)= 1 - 11x +4x2 4x(1 - x)2 - 3 2(1 - x)3 lnx, (39a) g(x, y)= 4x - 1 4(1 - x) + 8x - 4x2 - 1 4(1 - x)2 ln x - ln y. (39b) Z uporabo numeriˇ s,SM cnih vrednosti matriˇ d,SM cnih elementov CKM iz PDG [52] lahko postavimo mejo . 4 × 10-12 GeV-2 . 3 × 10-13 GeV-2 in C C . 3 3 Obravnavamo tudi prispevke nekaterih modelov fizike izven SM: minimalni supersimetriˇcni SM (MSSM) z in brez parnosti R (RPV) ter model z generiˇcnim dodatnim vektorskim bozonom Z. . Znotraj MSSM prispevata, podobno kot v SM, le operatorja Oq, medtem ko sta pripadajoˇca 3 Wilsonova koeficienta iz izmenjave parov gluinov (g) in spodnjih skvarkov ( d)[58] .2 .* s,MSSM S21.13 = 24xf6(x)+66 f6(x) (40a) - C , 3 216m2 de .2 .* S12.23 d,MSSM 24xf6(x)+66 f6(x) (40b) - C = , 3 216m2 de kjer sta 6(1 + 3x)ln x + x3 - 9x2 - 9x +17 f6(x)= , (41a) 6(x - 1)5 6x(1 + x)ln x - x3 - 9x2 +9x +1 f6(x)= 3(x - 1)5 2 in smo definirali x = m/m2 ege d , (41b) . Z uporabo obstojeˇcih mej na parametre MSSM (.ij, me, meg)iz d ˇstudij oscilacij mezono K, B in Bs ter drugih redkih razpadov mezonov K in B lahko spet ome- s,MSSM d,MSSM | . 5 × 10-12 GeV-2 da znotraj MSSM dopustimo interakcije tipa RPV, dobimo poglavitne prispevke ˇ . 2 × 10-12 GeV-2 jimo vrednosti |C in V primeru, C . 3 3 ze v dreve snem redu preko izmenjave snevtrinov (.). K efektivnemu Hamiltonovemu operatorju potem prispevajo predvsem operatorji Oq 4 3 in Oq 4 z Wilsonovimi sklopitvami 3 .. n31.* n12 .. n21.* n13 s,RP V 4 s,RP V C 4 = - = - C , , 2 2 m m .en .en n=1 3 n=1 3 .. n32.* n21 .. n12.* n23 d,RP V 4 d,RP V C 4 = - = - C , . 2 2 m m .en .en n=1 n=1 (42) Obstojeˇce ˇstudije ne omejujejo vseh parametrov (.. .), ki nastopajo v teh procesih, zato ijk, mebomo omejitve podali iz napovedi za merjene ekskluzivne razpadne kanale. Mnoge razˇsiritve SM vsebujejo dodatne nevtralne vektorske bozone Z. [66, 67]. Ti lahko prispevajo k efektivnemu Hamiltonovemu operatorju v drevesnem redu preko operatorjev O1q,3 Oq 1,3. ter tudi Pripadajoˇce Wilsonove koeficiente lahko zapiˇsemo kot s,Z. s,Z. BdR BdL 12 13 , C 1 s,Z. B dR dR B C =-4GFy.2BdL12BdR13,C1=-4GFy.2=-4GFy.2BdL12BdL13,Cs,Z 3=-4GFy.2=-4GFy.2BdL21BdR23,Cd,Z 1=-4GFy.2=-4GFy.BdL21BdL23,Cd,Z=-4GFy. 12 13 , BdR BdL 21 23 , 3 (43) d,Z. 1 C d,Z. 3 BdR BdR 21 23 , C 3 2 2 2 kjer je y =(g2/g1)2(.1 sin2 .+.2 cos2 .)in .i = m/m2 i cos2 .W .Z g1, g2, m1 in m2 smo oznaˇcili W umeritvene sklopitvene konstane in mase bozonov Z in Z,medtem ko . oznaˇcuje meˇsalni kot. Vizraˇcunu pogostosti razpadnih kanalov mezona Bc, ki potekajo preko kvarkovskih prehodov — b › dds—in b › ssd moramo izvrednotiti matriˇcne elemente efektivnih Hamiltonovih operatorjev med hadronskimi stanji. V prvem pribliˇzku uporabimo popolno faktorizacijo oz. VSA, ki v veˇcini primerov zadovoljivo opiˇse glavne lastnosti razpadnih kanalov, ki jih obravnavamo. Izjeme so kanali, v katerih razpadne amplitude v faktorizacijskem pribliˇzku izginejo. O takˇsnih razpadih naˇsa metoda molˇci; potrebni bi bili bolj natanˇcni pristopi. Obravnavamo dvodelˇcne razpadne *00 *00 › D*- , D*- kanale B- K K,D-K in D-K , kot tudi trodelˇcne kanale B- › D-K-.+ , csss s cs 00 D*-K-.+ , D-D*-D+ , D-D-D*+ , D-D-D+ , D0KK- K- in D*0K . Pri tem uporabimo s ss ss ss teoretiˇcne uvide iz hadronskih in semileptonskih ˇstudij teˇzkih mezonov iz prejˇsnjih razdelkov, kot tudi iz drugih virov. Predvsem se izkaˇze pomebna vloga vmesnih resonanc pri modeliranju hadronskih oblikovnih funkcij, ki nastopajo v faktorizacijskem pribliˇzku, pomagajo pa nam tudi natanˇcno doloˇcene vrednosti nekaterih sklopitvenih konstant znotraj HM.PT. Na podlagi izraˇcunanih amplitud za izbrane hadronske kanale, najprej omejimo proste parametre modelov RPV in Z,tako da naˇse napovedi primerjamo z obstojeˇcimi eksperimentalnimi mejami na razvejtiveni razmerji razpadov BR(B- › K-K-.+) < 2.4 × 10-6 in BR(B- › .-.-K+) < 4.5 × 10-6, ki so jih izmerili v eksperimentu Belle [68]. V primeru modela RPV normaliziramo mase snevtrinov na skupno masno skalo 100 GeV in dobljene meje zapiˇsemokot  3 2 100 GeV .. n31.* n21.* n12 + .. n13 < 9.5 × 10-5 , (44a) m.en n=1 3 2 100 GeV .. n32.* n21.* n21 + .. n13 < 9.5 × 10-5 . (44b) m.en n=1 Lahko pa predpostavimo, da poglavitni prispevki nove fizike prihajajo v obliki dodatnih bozonov Z. . V tem primeru dobimo meje na njihove sklopitve oblike y 2 |BsL BsR + BsR BsL | < 2.7 × 10-4 , (45a) 1213 1213 + BsR y 2 |BsL BsL BsR | < 5.6 × 10-4 , (45b) 1213 1213 in dL dR dR dL < 2.4 × 10-4 , (46a) 2 + B B B B y 21 23 21 32 dL dL dR dR 2 < 5.3 × 10-4 . (46b) + B B B B y 21 32 21 32 Razpad SM MSSM RPV Z. 1 × 10-21 5 × 10-20 7 × 10-9 9 × 10-10 D+ 4 × 10-19 5 × 10-19 1 × 10-8 1 × 10-9 › D-K+.- 2 × 10-16 5 × 10-15 4 × 10-7 2 × 10-6 › DsK-.+ 7 × 10-14 1 × 10-13 8 × 10-7 3 × 10-6 0 .-K0 4 × 10-20 2 × 10-18 2 × 10-8 1 × 10-9 › D 8. ZAKLJUCKI B- › D-D-D+ B- › D-D- B- B- B- B- B- B- › D- B- B- › D*- B- B- B- B- Tabela 6: Razvejitvena razmerja razpadov .S = -1 in .S =2 mezona B- izraˇcunana znotraj modelov c SM, MSSM, RPV in Z. . Za doloˇcitev neznanih kombinacij parametrov RPV (ˇcetrti stolpec) in Z. (peti stolpec) smo uporabili eksperimentalne gornje meje BR(B- › .-.-K+) < 1.8 × 10-6 in BR(B- › K-K-.+) < 2.4 × 10-6 . Meje (44a-46b) so zanimive, predvsem ker omejujejo kombinacije parametrov RPV oz. Z,v ortogonalni smeri od obstojeˇcih meritev oscilacij mezonov K, B, Bs, ter drugih redkih procesov. Na podlagi teh omejitev lahko konˇcno podamo tudi napovedi za razvejtivena razmerja mnogih moˇznih dvo-in trodelˇcnih razpadnih kanalov mezona Bc.Naˇsi rezultati so povzeti v s tabeli 6. Napovedi SM in MSSM so zanemerljivo majhne. Na podlagi omejitev na parametre s RPV iz razpadnih kanalov B- › .-.-K+ in B- › K-K-.+ dobimo v tem modelu najveˇcja s ss cccccccccccccc 00 K-K 4 × 10-18 7 × 10-18 9 × 10-9 6 × 10-10 › D › D-K0 4 × 10-17 2 × 10-15 4 × 10-8 3 × 10-7 K0 1 × 10-14 2 × 10-14 7 × 10-8 4 × 10-7 › D*-K0 4 × 10-17 2 × 10-15 4 × 10-8 3 × 10-7 K0 1 × 10-14 2 × 10-14 6 × 10-8 4 × 10-7 › D-K*0 8 × 10-17 3 × 10-15 3 × 10-9 5 × 10-7 * › D- s K 3 × 10-14 4 × 10-14 6 × 10-9 9 × 10-7 0 › D*-K*0 6 × 10-18 3 × 10-16 2 × 10-10 4 × 10-8 * › D*- s 2 × 10-15 3 × 10-15 4 × 10-10 5 × 10-8 K 0 s s c c c s razvejtivena razmerja za trodelˇcne razpadne kanale B- › D-K+.- in B- › D- 00 , B- › D- , B- › D*-K0 in B- › D*- K-.+,ter dvodelˇcne razpadne kanale B- › D-K0 .Poleg K K c teh, pa znotraj modela Z. dobimo velika razvejitvena razmerja ˇse v kanalih B- › D-K*0 in 0 B- KS oz. KL.Poslediˇcno bo v razpadnih kanalih, ki vsebujejo nevtralne psevdoskalarne kaone, zaradi prispevkov pingvinskih diagramov znotraj SM teˇzko zaznati vplive nove fizike [69]. V tem pogledu so bolj perspektivni razpadi v nabite kaone oziroma njihova vektorska stanja. Vnaˇsem izraˇcunu smo se naslanjali na pribliˇzek naivne faktorizacije, ki je kot prvi pribliˇzek c zadostna za opis grobih lastnosti prispevkov nove fizike. Tudi v primeru, da bi morebitni nefaktorizabilni prispevki znatno spremenili vrednosti hadronskih amplitud, je razkorak med napovedmi SM in nove fizike trenutno tolikˇsen, da v vsakem primeru ohranja relevantnost obravnavanih razpadnih kanalov v iskanju nove fizike, in to nemudoma, ko bodo na voljo veˇcje koliˇcine mezonov Bc. 8Zakljuˇcki Neperturbativna narava QCD je trdovraten problem raˇcunov v hadronski fiziki. Ena izmed njegovih manifestacij je pojav resonanc v hadronskem spektru. V procesih, kjer so izmenjane gibalne koliˇcine majhne v primerjavi s skalo zlomitve kiralne simetrije ~ 1 GeV, lahko uporabimo pristop efektivnih teorij, ki temelji na pribliˇzni kiralni simetriji lahkih kvarkov ter pribliˇzni › D*-K*0 . Eksperimenti namesto mezonov K0 ali K , dejansko izmerijo stanja mezonov simetriji okusov in spina teˇzkih kvarkov. V takˇsnem okviru lahko sistematiˇcno analiziramo vpliv najniˇzje leˇzeˇcih resonanc v procesih teˇzkih mezonov. HM.PT smo uporabili na primeru moˇcnih, semileptonskih in redkih procesih teˇzkih mezonov. V ogrodje efektivne teorije smo sistemtiˇcno vkljuˇcili najniˇzje leˇzeˇce spinske multiplete teˇzkih mezonov pozitivne in negativne parnosti. V prvem redu kiralnega razvoja smo pokazali, da lahko bliˇznje leˇzeˇca vzbujena stanja teˇzkih mezonov pomagajo razloˇziti nekatere lastnosti semileptonskih oblikovnih funkcij v razpadih teˇzkih v lahke mezone. Namreˇc, z uporabo omejene parametrizacije oblikovnih funkcij, ki temelji na pribliˇznih limitah efektivnih teorij QCD, smo uspeli zasiˇciti prispevke celotnega stolpa vmesnih stanj samo z najbliˇzje leˇzeˇcimi stanji primernih kvantnih ˇstevil. Parametrizacijo smo napeli na izrˇcun razpadne ˇsirine v kinematskem obmoˇcju majhnih izmenjav gibalne koliˇcine znotraj HM.PT. Takˇsen model je uspeˇsno reproduciral veˇcino oblikovnih funkcij H › P in H › V prehodov znotraj trenutnih eksperimentalnih napak in v ujemanju z obstojeˇcimi izraˇcuni QCD na mreˇzi. V drugih procesih, ki smo jih obravnavali, prispevajo vzbujena stanja teˇzkih mezonov ˇsele v drugem redu kiralnega razvoja – k tako imenovanim kiralnim popravkom. Obravnavali smo moˇcne razpade teˇzkih mezonov ter izraˇcunali efektivne moˇcne sklopitvene konstante med pari teˇzkih mezonov sode ali lihe parnosti ter lahkimi psevdoskalarnimi mezoni v drugem redu kiralnega razvoja. Iz merjenih razpadnih ˇsirin D* › D. in D. › D. smo izluˇsˇcili efektivne 0 sklopitvene konstante v prvem in drugem redu kiralnega razvoja. Vpliv velikega ˇstevila novih neznanih parametrov, ki nastopajo v izraˇcunih drugega reda, smo ocenili s pomoˇcjo variacij skale kiralne zlomitve ter otipanjem prostora parametrov ob prilagajanju izraˇcunov na eksperimentalne meritve. Nato smo ˇstudirali ekstrapolacijo sklopitvenih konstant v limiti, ko gredo mase lahkih psevdoskalarnih mezonov proti niˇc. Ugotovili smo, da dajo naivni izraˇcuni kiralnih popravkov z upoˇstevanjem vzbujenih teˇzkih stanj slabo definirano kiralno limito. Namesto tega lahko izvedemo razvoj v obratni vrednosti masnih razlik med osnovnimi in vzbujenimi stanji teˇzkih mezonov in tako reˇsimo kiralno limito izraˇcunov. Takˇsen razvoj je zanesljiv za majhne mase lahkih psevdoskalarnih mezonov, manjˇse od masnih razlik med osnovnimi in vzbujenimi stanji teˇzkih mezonov. Potem se prispevki vzbujenih stanj teˇzkih mezonov formalno izraˇzajo kot popravki viˇsjih redov v kiralnem razvoju teorije brez dinamiˇcnih vzbujenih stanj. Ti rezultati so ˇse posebej pomembni za ˇstudije QCD na mreˇzi, ki uporabljajo kiralno ektrapolacijo za do- sego fizikalne limite simuliranih mas lahkih kvarkov. Po naˇsih ugotovitvah je relevantna kiralna limita takih ektrapolacij SU(2) izpospinska limita, zanesljivo pa se lahko izvedejo le za pionske mase manjˇse od masnih razlik med osnovnimi in vzbujenimi stanji teˇzkih mezonov. Hkrati lahko ocenimo zanesljivost ektrapolacij takˇsne simetrije v redu vodilnih logaritmov z uporabo vodilnih prispevkov vzbujenih teˇzkih stanj viˇsjega reda. Razklopitev vzbujenih resonanc in njihove poglavitne prispevke smo preverili tudi na primeru semileptonskih oblikovnih funkcij v teˇzko-teˇzkih prehodih med mezoni sode in lihe parnosti, kjer smo izraˇcunali kiralne popravke k funkcijam Isgur-Wise. Za izluˇsˇcenje matriˇcnega elementa CKM Vbc namreˇc poleg izredno natanˇcne doloˇcitve razpadnih ˇsirin iz eksperimentov in oblikovnih funkcij iz izraˇcunov QCD na mreˇzi potrebujemo natanˇcno poznavanje kiralne limite. Ugotovili smo, da so efekti vzbujenih resonanc teˇzkih mezonov primerljivi s trenutnimi ocenami teoretiˇcnih napak in jih bo zato v prihodnjih ˇstudijah potrebno upoˇstevati. Redke procese teˇzkih mezonov analiziramo predvsem z namenom iskanja signalov nove fizike izven SM. Vendar pa lahko upamo na uspeh le ob dobrem poznavanju in nadzoru nad hadronskimi efekti. V ta namen smo obravnavali kiralno obnaˇsanje celotne supersimetriˇcne baze efektivnih operatorjev .B = 2, ki so odgovorni za oscilacije nevtralnih teˇzkih mezonov. Izraˇcunali smo popravke kiralnih zank v drugem redu kiralnega in prvem redu razvoja po masah teˇzkih kvarkov ter vkljuˇcili vplive teˇzkih mezonov sode parnosti. Potrdili smo razklopitev vzbu 8. ZAKLJUCKI liii jenih stanj in podali izraze za kiralno ekstrapolacijo celotne baze efektivnih operatorjev v redu vodilnih logaritmov. Naˇs rezultat bodo tako lahko uporabile prihodnje ˇstudije teh procesov s simulacijami QCD na mreˇzi. Kot pomoˇzni rezultat smo izraˇcunali vodilne kiralne logaritemske popravke k razpadnim konstantam teˇzkih mezonov sode parnosti. — Nazadnje smo izvrednotili zelo redke prehode b › ssd in b › dds—mezona Bc vpristopu efektivnih teorij. Hadronske razpadne amplitude smo ocenili s pomoˇcjo pribliˇzkov faktorizacije in zasiˇcenja z resonancami. Prehode smo analizirali znotraj veˇcih modelov nove fizike. Na podlagi obstojeˇcih eksperimentalnih mej na pogostosti razpadov B › KK. in B › ..K smo lahko omejili relevantne kombinacije parametrov nove fizike. Konˇcno smo na podlagi teh omejitev identificirali najobetavnejˇse dvo-in trodelˇcne neleptonske razpade mezona Bc,v katerih bi lahko s pomoˇcjo prihodnjih delˇcnih trkalnikov iskali signale teh redkih prehodov. Tekom raˇcuna smo razreˇsili tudi nekaj tehniˇcnih podrobnosti. Morali smo izluˇsˇciti celoten nabor kontraˇclenov v drugem redu kiralnega razvoja, ki prispevajo k moˇcnim prehodom med teˇzkimi mezoni sode in lihe parnosti, ter lahkimi psevdoskalarnimi mezoni. Vkljuˇcitev vzbujenih stanj teˇzkih mezonov je nato pokvarila kiralno limito izraˇcunov v redu vodilnih logaritmov. Problem smo razreˇsili s pomoˇcjo odrezanega razvoja zanˇcnih integralov po obratni vrednosti masnih razlik med osnovnimi in vzbujenimi stanji teˇzkih mezonov, na raˇcun zmanjˇsanja intervala zanesljivosti izraˇcunov znotraj HM.PT. V primeru semileptonskih prehodov med teˇzkimi in lahkimi mezoni smo morali pravilno reproducirati limiti HQET in SCET, da smo lahko dobili veljavno parametrizacijo oblikovnih funkcij. Hkrati smo morali med seboj pravilno napeti bazi oblikovnih funkcij znotraj QCD in HQET, ter identificirati prispevke izraˇcunov znotraj HM.PT k posamiˇcnim oblikovnim funkcijam. Ugotovili smo, da le takˇsno pravilno napenjanje oblikovnih funkcij verno reproducira prispevke resonanc pravilnih kvantnih ˇstevil k oblikovnim funkcijam. Nenazadnje smo morali zaradi strukture polov v parametrizaciji oblikovnih funkcij v naˇse HM.PT izraˇcune vkljuˇciti prispevke radialno vzbujenih stanj teˇzkih mezonov. V izraˇcunih kiralnih popravkov k meˇsanju teˇzkih nevtralnih mezonov smo morali predpisati pravilen postopek bozonizacije efektivnih operatorjev. Izkazalo se je, da je mogoˇce ogromen nabor vseh moˇznih struktur znotraj HM.PT s pomoˇcjo spinske simetrije teˇzkih kvarkov in identitet — matrik 4 × 4 znantno skrˇciti. Podobno smo morali na primeru prehodov b › ssd in b › dds— identificirati celotno bazo kvarkovskih operatorjev, kot tudi njihov tok in meˇsanje v prvem redu enaˇcb renormalizacijske grupe. Le tako smo lahko ohranili nadzor nad vodilnimi popravki QCD v visokoenergijskem reˇzimu. Nenazadnje smo za oceno mnogih hadronskih amplitud v dvo-in trodelˇcnih razpadih mezona Bc morali izvesti vhodne HM.PT izraˇcune ter po potrebi vkljuˇciti tudi prispevke lahkih vektorskih ter skalarnih resonanc. Ob tem smo smiselne fenomenoloˇske rezultate dobili le s pravilnimo predpisanimi postopki resonanˇcnega zasiˇcenja amplitud. v Chapter 1 Introduction The Standard Model (SM) of elementary particle physics is a quantum gauge-field theoretical description of fundamental electromagnetic, weak and strong interactions. It emerged in the 1960’s and has completely dominated the field ever since [1]. The building blocks of the SM are fermions – leptons and quarks – which come in three families. The SM gauge group is SU(3)c × SU(2)L × U(1)Y ,where the SU(3)c is the gauge group of Quantum Chromodynamics (QCD), SU(2)L is the gauge group of weak isospin, while U(1)Y is the gauge group of weak hypercharge. Only the left-handed chiral fermions transform as weak isospin doublets under the SU(2)L, while quarks also form the fundamental triplet representation of SU(3)c. The masses of leptons and quarks are generated via the Higgs mechanism – spontaneous symmetry breaking, where the (chiral) symmetry of the theory is not respected by the vacuum. For this purpose an additional scalar weak isospin doublet is introduced. Its vacuum expectation value also breaks gauge invariance of the theory to the subgroup SU(3)c × U(1)EM, inducing masses for the weak W± and Z gauge bosons. The quark fields in the SU(2)L basis are not the mass eigenstates in general. Therefore it is customary to rotate them to the mass eigenbasis by means of a unitary matrix. The rotation is conventionally conveyed to the down-quark fields and the rotation matrix is called the CabibboKobayashi- Maskawa (CKM) matrix. It can be fully described by three real mixing angles and a complex CP violating phase. The successes of the SM description are abundant. Its predictions have been extensively tested in accelerator facilities and agree well with the data measured up to energies available at present: the electroweak precision tests are generally in impressive agreement with SM predictions [2] while the CP violation experiments in K, D and B meson systems support the CKM description with one universal phase [3, 4]. The only elementary building block presently lacking experimental detection is the Higgs boson. However we also know from observations, that the SM cannot be the ultimate fundamental theory. For once, it does not include gravity. Although colossal theoretical efforts have been spent on the subject in the last few decades, the progress has been slow and the results inconclusive. Mainly also due to the lack of almost any experimental hints in the area. On the other hand, the SM also does not account for the recently measured neutrino oscillations [5]. Explanation of these requires non-zero neutrino masses, contrary to the SM prescription1.Thirdly, a growing number of astrophysical observations suggest that most of the matter in the universe is neither luminous nor baryonic [6]. In addition, most of it must be slowly moving or “cold”. 1In fact, the matter contents of the SM can easily be extended to include right-handed neutrinos and thus allowing for Dirac neutrino masses via the Higgs mechanism. However the observed smallness of the neutrino masses and the fact that right-handed neutrinos must be singlets under the SM gauge group seem to prefer alternative mechanisms which lie beyond the SM. 1 2 CHAPTER 1. INTRODUCTION The SM does not provide a candidate for nonbaryonic cold dark matter. Finally, our current understanding of baryogenessis – the generation of the measured baryon -antibaryon asymmetry – in the early universe requires levels of CP and baryon number violation much higher than allowed for in the SM [7]. There are also some conceptual and “aesthetic” problems with the SM. The running of the gauge couplings suggests a unification scale at 1014 - 1016 GeV although precise unification does not occur if one takes into account only SM fields [70]. Neither does the SM describe the dynamics of such unification. In fact, even the electroweak symmetry breaking has no dynamical explanation within the SM. It is imposed by construction and renders the masses of the elementary particles as free parameters. Compared to the large unification scale the . electroweak scale 1/GF ~ 250 GeV also appears to be very small. The large scale hierarchy manifests itself in form of large quantum corrections to the mass of the Higgs boson, which are quadratically divergent and thus sensitive to the UV completion of the theory2.Even more appealing is the “fine-tuning” required for the vacuum energy density when compared to the measured present critical density of the universe .c ~ 10-14 eV4 [6]. Its classical value, which is a free parameter in any quantum field theory (QFT), has to cancel corrections due to spontaneous symmetry breaking and the resulting vacuum condensates in the SM at energy scales from a few 100 MeV to a few 100 GeV. Another similar issue is related to the strong CP phase of the QCD vacuum. Its value, a free parameter of the SM, is severely constrained by the measurements of the electric dipole moment of the neutron [1]. All of these reasons call for physics beyond SM, and many proposals exist on what the physical reality ought to look like above the electroweak scale. For example, supersymmetric (SUSY) extensions of the SM attempt to resolve the Higgs hierarchy problem and provide suitable dark matter candidates [71, 72]. The simplest and most studied of these is the Minimal SUSY SM (MSSM) which adds a bosonic partner to each SM fermion and a fermionic counterpart to each boson, but also doubles the Higgs sector. Possible alternative proposals come in the form of (large) extra-dimensions in which some or all of the SM fields may propagate or extended SM symmetries. In order for any of these extensions to address the dynamics at the electroweak scale, new physical degrees of freedom have to appear at the TeV scale. On the other hand, many of these “low scale” new physics scenarios can be embedded into high scale unification theories, such as Grand Unified Theories (GUTs) attempting to describe the unification of the SM couplings at large scales in terms of incorporating the SM gauge group into larger symmetries [73]. Even more ambitious are the various String theories unified under the name of M-theory, which also attempt to address the quantization of gravity and the cosmological evolution of the very early universe [74, 75]. In order to get a handle on this plethora of high energy phenomenologies one must often rely on the so-called “bottom-up” approach to new physics. One constructs effective low energy theories by systematically parameterizing possible new physics contributions to low energy processes based on symmetry principles of the expected underlying theory. Within this framework, the SM itself is regarded as an effective low energy description of a grander theory, containing the SM particle content and gauge symmetry at low energies. A similar reasoning lies behind the MSSM, which is often regarded as the low energy effective theory of a high scale (and/or dimensional) GUT or String theory, containing a (slightly broken) SUSY SM particle content at energies close to the electroweak scale. Alternatively, one can focus on specific low energy aspects, common to various SM extensions. A common characteristic of various new physics models (including MSSM) is the appearance of a doubling of the Higgs sector, which can be put in the general form of a Two Higgs Doublet Model (THDM). On the other hand 2This has to be compared to the logarithmic divergences of fermionic fields due to chiral symmetries and gauge boson fields due to gauge invariance. many GUT and String theories also predict additional low energy U(1) gauge extensions to the SM – the appearance of additional Z. bosons. By focusing on such common aspects, one can extract important general signatures of various new physics proposals. The experimental challenge of finding new physics follows two main directions. In direct searches the idea is to produce the new particles and detect them directly (often through their decay products). This requires high enough energies at particle colliders such as the Tevatron, the upcoming LHC or the planned ILC. A complementary idea is to measure the effects of new particles in processes where they enter as intermediate virtual states. In this approach it is crucial to be able to disentangle the effects of new physics from those conveyed by the SM particles. One may then employ the “top-down” approach as prototyped by the Wilsonian Operator Product Expansion (OPE) [76]. Namely one may represent low energy Green’s functions or scattering amplitudes in terms of products of local operators, which are in term computed (matched to) the full original formulation of the SM and possible additional higher energy extensions. In this way SM and new physics contributions are clearly separated on an amplitude-by-amplitude basis. The task is then to compute low energy scattering cross sections and decay rates and compare them to precision measurements. This approach both tests SM predictions as well as probes possible new physics contributions. Experimentally it requires high statistics and precision measurements, such as those provided in the last years by the B and D meson factories at Belle, BaBar, CLEO-c and others. Among their successes are the by now established neutral meson oscillations in all neutral K - K [1], D - D [77, 78], B - B [56] and Bs - Bs [57, 79] meson systems, as well as ever tightening consistency constraints on the CKM unitarity and the CP violating phase. So far, no clear indications of new physics in these phenomena have been observed and several stringent experimental bounds on various new physics proposals have been imposed. In order to correctly interpret experimental results and justify the consistency with the SM or claim new physics signals, one first has to reliably calculate the relevant hadronic processes based on the quark picture of the OPE. Due to the nonperturbative and confining nature of low energy QCD, this turns out to be a daunting task. Namely, the expansion in the coupling constant is not applicable in this regime. Ab initio calculations, i.e., by starting with the QCD Lagrangian and finishing up with predictions for physical observables are still possible, through the use Lattice QCD techniques, but are computationally very challenging [8]. Lattice methods also have their own limitations. To get meaningful results, computations have to be done in Euclidean space-time, which makes calculations of processes with more than one hadron in the final state very difficult. Also, in order to make numerical difficulties tractable, a number of approximations have to be made, e.g., by working at relatively large pion masses. Another option that has been commonly used in the past, is to use symmetries of the QCD Lagrangian to construct effective theories [9]. Unknown parameters in the effective theory are fixed from experiments or, if possible, from perturbative comparison (matching) to full QCD. These effective theories may then be employed to either predict some experimental processes directly, or to assist nonperturbative Lattice QCD calculations making them more tractable and keeping control of the used approximations. One important manifestation of the strong QCD dynamics at low energies is the appearance of resonances in the particle spectrum. They have been detected long ago and studied extensively in the processes of pions and kaons [1]. Their effects proved to be critical in many low energy processes. On one hand they restrict the validity of effective theory approaches, which are not able to fully include their effects, e.g., in (resonant) .. scattering. Also, their dominant (long distance) effects are known to almost completely obscure contributions due to (short distance) SM OPE or possible new physics contributions in D meson oscillations and rare decays [10]. On the other hand, due to the relatively large c and b quark masses, heavy meson resonance effects Meson JP Mass [GeV] Width [GeV] Br. [%] (final states) — cd D+ 0- 1.869 ± 0.001 D*+ 1- 2.010 ± 0.001 (9.6 ± 2.2) × 10-5 67.7 ± 0.5(D0.+), 30.7 ± 0.5(D+.0) D *+ [12] 0+ 2.403 ± 0.014 ± 0.035 0.283 ± 0.024 ± 0.034 (D0.+)3 0 4 CHAPTER 1. INTRODUCTION Table 1.1: Experimentally measured properties of the relevant charmed mesons and their dominant hadronic decay modes. The pseudoscalar ground states are listed for completeness. Unless indicated otherwise, the values are taken from PDG. were long believed to be less significant in processes among hadrons involving these two quarks. In the last couple of years however, many experiments have reported first observations of resonances in the charm spectrum. In 2003, Belle [11] and FOCUS [12] experiments reported the observation of broad resonances D *+ and D*0, ca. 400 - 500 MeV higher above the usual 00 D states and with opposite parity. In the same year BaBar [13] announced a narrow meson DsJ (2317)+ . This was confirmed by FOCUS [14] and CLEO [15] which also noticed another narrow state, DsJ (2463)+ . Both states were also confirmed by Belle [16]. The basic properties of the relevant charmed mesons together with the dominating hadronic decay modes are listed in table 1.1. Studies of the basic properties of these states have been triggered particularly by the fact that the DsJ (2317)+ and DsJ (2463)+ states’ masses are below threshold for the decay into ground state charmed mesons and kaons, as suggested by quark model studies [17, 18] and lattice calculations [19, 20]. Their relative closeness to the ground state charmed mesons suggests possibly significant effects in the processes of the lowest D and Ds states and poses the following questions: Can we estimate the relevant effects of the lowest heavy meson resonances to the processes of heavy meson ground states? Can we keep their effects under control, especially within effective theories of QCD? Can they possibly help us to understand certain aspects of observed and measured ground state heavy meson processes? And finally what conclusions, drawn for the charm sector can we apply to the processes of B and Bs mesons, whose resonances are currently still beyond the reach of experimental facilities6 . In this thesis we will explore several aspects of resonances in the heavy meson processes [21, 3 Observed channel. 4Average of Belle [11] and FOCUS [12] values from [80]. 5Average of Belle [11] and CLEO [81] values from [80]. 6During the final stages of preparation of this thesis D0 collaboration has reported the first observation of axial resonances in the B spectrum [82]. Their properties and interpretation are yet to be analyzed in detail. D0 0- D*0 1- D*0 0+ 0 0 D 1+ 1 1.865 ± 0.001 2.007 ± 0.001 2.350 ± 0.0274 2.438 ± 0.0305 cu— < 0.002 0.262 ± 0.0514 0.329 ± 0.0845 61.9 ± 2.9(D0.0) (D+.-)3 (D*+.-)3 0- Ds D* 1- s DsJ (2317)+ 0+ DsJ (2463)+ 1+ 1.968 ± 0.001 2.112 ± 0.001 2.317 ± 0.001 2.459 ± 0.001 cs— < 1.9 × 10-3 < 0.005 < 0.006 5.8 ± 2.5(Ds.0) (D0.+)3 s (D*0.+)3 s 22, 23, 24, 25, 26, 27, 28, 29]. Their leading order contributions, either at tree level or at one loop, will be analyzed in the relevant effective theory approach to QCD. Within this framework we will calculate hadronic parameters entering various low energy processes and study the impact of heavy meson resonances on observables. These include strong, semileptonic decay rates of heavy mesons as well as neutral heavy meson mixing parameters. Since strong decay channels, if open, usually dominate the measured decay widths, one may use these as benchmarks on the validity of the chosen effective theory approach and also determine from them basic parameters of the effective theory. Semileptonic decays, mediated by quark and charged lepton weak currents proceed at tree level in the SM and are confirmed to be dominated by SM contributions. Their detailed study may therefore produce important consistency checks within the SM, such as the determination of the various CKM matrix elements and testing its unitarity, provided the relevant hadronic effects are well understood. Heavy neutral meson mixing, on the other hand, is mediated by box diagrams in the SM. This makes it an important arena for studying possible new physics contributions, which may or may not be suppressed by loop factors. Within our approach we will analyze all possible hadronic amplitudes entering heavy neutral meson mixing within the SM or beyond. Finally, we will also analyze very rare hadronic decays of the doubly heavy Bc meson, which are, like the neutral meson mixing, mediated by box diagrams in the SM. There we will make use of some of the knowledge on the impact of resonances in the calculation of the relevant hadronic decay amplitudes in order to constrain various new physics proposals based on existing experimental searches and also propose prospecting new search directions. The outline of the thesis is as follows. In the first two chapters we introduce the prerequisites for the phenomenological studies in the subsequent chapters. In chapter 2 we introduce the concept of effective field theories with a focus on the effective theory approaches to QCD in the limits of small and large quark masses. In chapter 3 we review some commonly used tools in hadronic calculations, such as the OPE, general hadronic matrix element parameterizations, and some of their approximations. In chapter 4 we analyze strong decays of heavy mesons within an effective theory approach, including loop contributions of excited heavy meson resonances. We attempt to extract the relevant effective strong meson couplings from the measured decay rates and study the impact of the resonances on the coupling extraction from Lattice QCD calculations. In chapter 5 we analyze the leading contributions of the heavy meson resonances to semileptonic decays. Both heavy to light as well as heavy to heavy meson transitions are analyzed. While in the former, heavy resonances may contribute already at tree level, in the latter their contributions are loop suppressed. Similar, loop suppressed contributions to heavy neutral meson mixing hadronic amplitudes are studied in chapter 6. Finally, chapter 7 contains our analysis of the very rare hadronic decays of the Bc meson within the SM and some of its extensions. The conclusions are gathered in Chapter 6, while some further technicalities of our calculations as well as brief descriptions of studied SM extensions are relegated to the appendices. 6 R . N Chapter 2 Effective theories of heavy and light quarks 2.1 What is an effective field theory? The content of quantum theory is encoded in its Green’s functions, which in general depend in a complicated way on the properties (e.g. particle momenta) of the initial and final states. In particular they exhibit nonanalytic behavior such as cuts and poles in the configuration variables, which arise when the kinematics allow for physical intermediate states. Conversely, when the kinematics are far from being able to produce a certain propagating intermediate state, the contribution of that state to the Green’s function of interest will be relatively simple, well approximated by the first few terms in a Taylor expansion (e.g. of the incoming momenta of the scattering problem). Instead of Taylor expanding each amplitude it turns out to be much more profitable to expand the Lagrangian in local operators that only involve the light degrees of freedom, where the expansion is in the powers of the generalized momenta of the light fields (appearing as derivatives in the Lagrangian) divided by the scale of heavy physics. Such a Lagrangian is called an effective field theory. Although the heavy modes do not appear explicitly anymore, their contributions are encoded through the parameters of the effective theory1.There are many situations in which effective field theories are of utility [83]: • They allow one to compute low energy scattering amplitudes without having a detailed understanding of the short distance physics, or to avoid wasting effort calculating tiny effects from known short distance physics (such is the OPE and the effective weak Hamiltonian). • In nonperturbative theories (such as low energy QCD) one can construct a predictive effective field theory for low energy phenomena by combining power counting of operators with symmetry constraints of the underlying theory (such as the .PT and HM.PT). • By regarding theories of known physics as effective field theory descriptions of more fundamental underlying physics, one can work bottom up, extrapolating from observed rare processes to a more complete theory of short distance physics (this approach is taken in many studies of BSM physics, such as MFV or grand unification). At present, the general approach of effective field theory is followed in many contexts of the SM and even in more speculative theories like grand unification, supergravity, extra dimensions or superstrings. 1This aspect of effective theories is not unique to quantum phenomena. Integrating out certain regions or scales of the phase space in order to simplify the description of certain phenomena has also been found to be of high value in other fields such as (classical) statistical mechanics or (classical) field theories. 7 8CHAPTER 2. EFFECTIVE THEORIES OF HEAVY AND LIGHT QUARKS 2.2 Exploring the Chiral symmetry of QCD One of the earliest and also one of the most successful examples of effective theories is the chiral perturbation theory (.PT) which builds upon the approximate chiral symmetry of QCD at low energies. We will briefly review it in this section. The QCD Lagrangian with Nf (Nf =2, 3) massless quarks q(n) =(u, d,...) Nf (n)i L0 = q QCD Dq(n) + Lgauge + Lheavy quarks n=1 Nf (n)(n)(n)(n) = qi + qi + Lgauge + Lheavy quarks, (2.1) Dq Dq LL RR n=1 where  D = .µDµ is the QCD covariant derivative and qR,L =(1±.5)q/2, has a global symmetry SU(Nf )R × SU(Nf )L ×U(1)V × U(1)A . (2.2) chiral group G At the effective hadronic level, the quark number symmetry U(1)V is realized as baryon number. The axial U(1)A is anomalous and is broken by nonperturbative effects. Theoretical and phenomenological evidence suggests that the chiral group G on the other hand is spontaneously broken to the vector subgroup H = SU(Nf )V . The axial generators of G are realized non-linearly and associated with them are the N2 -1 massless pseudoscalar Goldstone bosons .(x)= .i.i(x) f parameterizing the G/H right coset space. Here .i are the broken generators of G and .i(x) are the Goldstone fields. For the Nf = 3 case, the . can be written as . . . (2.3) . , .8 while in the Nf = 2 only the pion fields remain. To preserve all the symmetries of the fundamental theory in the effective Lagrangian, it is essential to construct it out of the Goldstone field functions which transform linearly under G (see e.g. [84] for details). A customary choice is . = exp2i.(x)/f, which transforms as . › R.L†,where R and L are the corresponding generators SU(Nf )R and SU(Nf )L respectively. f is an undetermined constant of energy dimension one, which can be identified with the Goldstone boson decay constant. We continue by factoring out the broken generators of G from the quark fields q = .(.)q,where .(.) transforms under G as .(.) › .(.)U(x). Here .(x) is the transformed Goldstone matrix and we demand that U(x)be anelement of H. In general it will also be a function of .(x). Consequently, qtransforms as q› U(x)qand we have to modify its covariant derivative to account for the co . ordinate dependence . q =(D + V). D q where the vector field Vµ =(..µ.† + .†.µ.)/2and . = transforming as . › L.U† = U.R† . It can be easily checked that Vµ transforms under G as Vµ › UVµU† + U.µU† . There exists another operator which can be built up of .(x), has the properties of an axial vector field Aµ = i(.†.µ. - ..µ.†)/2= i.†.µ..†/2 and is transforming under G as Aµ › UAµU† . Its role will become apparent later. The Lagrangian of the SM is not chiral invariant. The chiral symmetry of the strong interactions is broken by the electroweak interactions generating in particular non-zero quark masses. The basic assumption of .PT is that the chiral limit constitutes a realistic starting point for .=.... 1.6.8+1.2.0.+K+ .-1.6.8-1.2.0K0K-K0-23 2.2. EXPLORING THE CHIRAL SYMMETRY OF QCD a systematic expansion in chiral symmetry breaking interactions. Namely we extend the chiral invariant QCD Lagrangian (2.1) by coupling the quarks to external hermitian matrix fields vµ = rµ + lµ, aµ = rµ - lµ, s, p[9]: Nf L = L0(m)((n) - q(n) qv(m,n) + a(m,n).5)q(m)(s(m,n) - ip(m,n).5)q. (2.4) QCD + m,n=1 Here vµ and aµ will contain external photons and weak gauge bosons so that Green’s functions for electromagnetic and semileptonic weak currents can be obtained as functional derivatives of the generating functional Z[v, a, s, p] with respect to external photon and weak boson fields. The scalar and pseudoscalar fields s, p on the other hand give rise to Green’s functions of (pseudo)scalar quark currents, as well as providing a very convenient way of incorporating explicit chiral symmetry breaking through the quark masses. To preserve the manifest chiral symmetry of the effective Lagrangian, we promote it to a local symmetry and treat the external fields as spurions with the transformation properties rµ › RrµR†+iR.µR† , lµ › LlµL†+iL.µL† and s + ip › R(s + ip)L† . Accordingly we have to introduce the covariant derivatives for pion fields Dµ.= .µ. - irµ.+ i.lµ (as well as the appropriate external field stress-energy tensors). The physically interesting Green’s functions are then functional derivatives of the generating functional Z[v, a, s, p] at chosen values of the spurion fields. In particular for the quark masses we use s = mq . diag(mu,md,...). Even more generally, any effective quark operator (e.g. from the OPE of the effective weak Hamiltonian ) can be incorporated into the effective chiral Lagrangian by coupling it to the appropriate external chiral spurion field and then constructing the corresponding source terms out of the Goldstone fields. The effective chiral Lagrangian is usually organized in a derivative expansion based on the chiral power counting rules. One prescribes chiral powers p to all the constituent field operators and then builds the Lagrangian out of them by constructing all the terms adherent to the symmetries up to a given chiral power. In general, this procedure counts the powers of derivatives on the Goldstone fields as well as the number of external field insertions which at the level of Green’s functions translates to counting the powers of pseudo-Goldstone masses and exchanged momenta. Using the simplest choice for the quark mass chiral counting s ~ p2 one arrives at the lowest O(p2) order Lagrangian describing the low energy strong interactions of light pseudoscalar mesons [1, 30] f2 L(2) . = 8 .µ.ab.µ.† ba + .0 (mq)ab.ba +(mq)ab.ba † . (2.5) A trace is taken over the repeated light quark flavor indices. For the sake of clarity we have omitted all external currents (l = r = p = 0) except s in the second term, which induces masses 2 of the pseudo-Goldstone bosons m=4.0(ma + mb)/f 2 . While the normalization of the first ab term is canonical, the second term contains an unknown constant .0, which we can fit to the light pseudoscalar meson masses. Conversely, lattice QCD simulations often work in the exact SU(2) flavor isospin symmetry limit. There, one can parameterize the pseudo-Goldstone masses according to the Gell-Mann formulae as [31] 28.0ms 28.0ms r+12 8.0ms r+2 m= r, m= ,m= , (2.6) .f2 Kf2 2 .8 f2 3 22 where r = mu,d/ms and 8.0ms/f 2 =2m- m.. K Higher order terms in the chiral power counting can be constructed in this manner as well as terms involving any general external fields. The higher order terms in this expansion also serve a double role as the counterterms absorbing loop divergences from diagrams with insertions of lower order terms in the Lagrangian, thus keeping the theory renormalizable (in the general sense of the word). 10 CHAPTER 2. EFFECTIVE THEORIES OF HEAVY AND LIGHT QUARKS 2.2.1 Light flavor singlet mixing and the .. The eight SU(3) pseudo-Goldstone bosons: .+ , .- , .0 , K+ , K- , K0 , K0 and .8 have the —— same quantum numbers as the following quark-antiquark pairs: ud, du—, uu—- dd—, us—, su—, ds—, sd and uu—+ dd — - 2ss—. This suggests the existence of a ninth meson .0 that would correspond to the uu—+ dd —+ ss—singlet as the pseudo-Goldstone boson of the U(1)A axial symmetry of QCD. However, due to the axial anomaly, the mass of .0 is not protected and can be much larger than those of the other eight states. Nevertheless, for practical reasons .0 still has to be incorporated into the theory. Namely, its existence would entail mixing with the .8 state to form two distinct physical states (. and .) and this scenario needs to be taken into account. One of the common approaches is to pretend that there is no axial anomaly and add .0 to the matrix . (2.3) as an SU(3) singlet: . . 111 . .8 + . .0 + . .0 .+ K+ 632 .- 111 . .8 + . .0 - . .0 K0 .= .. . .. . . (2.7) 6 3 2 2 .8 + .1 K- K 0 - 3 .0 3 The mixing of .q and .0 to form physical states can in principle involve other states (e.g. .(1279) or even .c), can depend on the energy of the state or can be influenced by the axial anomaly. Consequently the mixing scheme can be very complicated. In this thesis, we use the approach developed by Feldman et al. [85]. There, the physical states . and .. can be written as linear combinations of .q and .s: . = .q cos . - .s sin., .. = .q sin . + .s cos .,where .q has . a(uu—+ dd—)/ 2 flavor structure and .s is an ss—state, while . is the mixing angle. The decay constants of . and .. follow the same pattern of state mixing: fq = fq cos ., f s = -fq sin ., .. f.q . = fq sin ., f.s . = fq cos ., (2.8) with the decay constants defined as .(p)| q.—µ.5q |0. = if.qpµ, .(p)| —. pµ, s.µ.5s |0. = if s .(p)| —.. pµ, .(p)| —.. pµ. (2.9) q.µ.5q |0. = if q s.µ.5s |0. = if s =1. first order of flavor symmetry breaking, it can be deduced that fq = f., fs =2f2 - f.2. K Therefore, if the .0 – .8 basis is used instead, two mixing angles rather than one are needed . = .8 cos .8 - .0 sin .0, .. = .8 sin .8 + .0 cos .0. The angles .0 and .8 are connected with ., fs .. and fd as .8 = . - arctan( 2fs/fq), .0 = . - arctan( 2fq/fs). The value of . can be obtained phenomenologically from the various measured processes involving . and .. states. The value that fits the data best is . =39.3. . 2.3 Symmetries of heavy quarks Since their early applications, symmetries of heavy quarks have been one of the key ingredients in the theoretical investigations of processes involving heavy quarks. They have been successfully applied to the heavy hadron spectroscopy, to the inclusive as well as a number of exclusive decays (for reviews of the heavy quark effective theory and related issues see [86, 33]). The important observation here is that for heavy enough quarks, the effective strong coupling of QCD, due to its renormalization group running and asymptotic freedom, will be small at the Due to the SU(3) flavor symmetry breaking effects and the axial anomaly fq /fs  In the 2.3. SYMMETRIES OF HEAVY QUARKS 11 mass scale of the heavy quark. This implies that on length scales, comparable to the compton wavelength .Q ~ 1/mQ, the strong interactions are perturbative and much like the electromagnetic interactions. Furthermore, heavy quark spin participates in the strong interactions only through relativistic chromomagnetic effects. Since these vanish in the limit of the infinite quark mass, the spin of the heavy quarks decouples as well. The resulting theory therefore contains an approximate SU(2) spin symmetry with the spin states of the heavy quark transforming in the fundamental representation. To arrive at this result formally, we start with the QCD Lagrangian for a single flavor Q of heavy quarks LQ = Q(i D - mQ)Q. (2.10) heavy quarks We separate the quark fields into their positive and negative frequency parts – i.e. into ”quark” and ”anti-quark” fields Q = Q(+) + Q(-),where Q(+) annihilates the Q quark while Q(-) creates the corresponding anti-quark. For an infinitely heavy (anti)quark field travelling with velocity v, it is useful to rescale the ground state energy of the effective theory Fock space relative to the mass of the heavy (anti)quark in the frame of reference. This is done by factoring out the dominant kinetic phase factor exp(±imQv · x) from the (anti)quark fields Q(±).Then we further project out the large components of the heavy (anti)quark spinors using a velocity dependent projection operators P± =(1 ±v)/2 to obtain the effective heavy (anti)quark fields h±(x)= P± exp(±imQv · x)Q(±)(x). They satisfy vh± = ±h±.We construct the HQET v vv (+) (-) Lagrangian from QCD by using the combined field hv = hv + hv while its orthogonal small (anti)quark spinor components h±(x)= P± exp(±imQv · x)Q(±)(x) can be integrated out using v their equations of motions [87] or more elegantly via direct Gaussian path integration of the generating functional Z[.v], where external sources .v only couple to the hv fields and none to hv.Consequently hv contribute only spin symmetry breaking corrections to the interactions among hv fields. They are proportional to the inverse powers of the heavy quark mass, yielding for the HQET Lagrangian LQ = hv(iv · D)hv + O (1/mQ)+ Lgauge + Llight quarks . (2.11) HQET The decoupling of heavy quark spin contributions in the leading term of eq. (2.11) is now intuitively manifest due to the absence of Dirac gamma matrices. Alternatively one can show, that it is invariant under the the generators of the SU(2) transformations Si = .5vi/2, where i =1, 2, 3, v · . = 0 and the heavy quark fields transform in the spinor representation D(S) (hv › D(S)hv, D(S)-1 = .0D(S)†.0). Also at leading order in the expansion, there are no quark-antiquark couplings as it would take an infinite amount of energy (twice) to pair- produce infinitely heavy quarks. We can generalize the above arguments to Nh flavors of heavy quarks (c, b,...). Since in QCD different quark flavors are only distinguished by their Lagrangian masses, for infinitely heavy quarks, QCD interactions become blind to the flavor of heavy quarks, exhibiting in total a U(2Nh) spin-flavor symmetry. The HQET Lagrangian then becomes Nh LHQET = h(n)(iv · D)h(n) + O (1/mQ)+ Lgauge + Llight quarks . (2.12) vv n=1 There are two important issues related to such HQET formulation. Firstly, the choice of the heavy quark velocity to be factored out of the fields is arbitrary and we can formally get a separate independent set of quark fields for each choice. The result is sometimes called velocity superselection rule, and related to it is the heavy quark velocity reparametrization invariance. It simply states, that any shift in the velocity of the heavy quark by v › v + /mQ,where . satisfies v · . = 0, can be accommodated by a corresponding redefinition of the heavy quark 12 CHAPTER 2. EFFECTIVE THEORIES OF HEAVY AND LIGHT QUARKS field hv ›exp(i. ·x)(1 + Secondly, apart from the kinetic term in eq. (2.12) whose /2mQ)hv. normalization is fixed via velocity reparametrization invariance, any effective operators involving heavy quark fields in HQET have to be properly matched to the corresponding operators in full QCD. Fortunately, due to the heavy mass scale, this matching can be performed perturbatively. As an example let us consider the heavy-light left-handed current operator (the case for the right-handed current proceeds identically) Jµ (V -A)QCD = qL.µQ. (2.13) At the tree level the HQET current can be written as Jµ (V -A)HQET, Tree = qL.µhv + O(1/mQ) . (2.14) Radiative corrections modify this result. The effective current operators present at the tree level are renormalized and additional operators are induced. Since in HQET the heavy quark velocity v is not a dynamical degree of freedom, the effective current operators can explicitly depend on it. The most general short-distance expansion of the vector current in the effective theory contains two operators of lowest dimension (three): µ J= C1(.)qL.µhv + C2(.)qLv µhv + O(1/mQ) , (2.15) (V -A)HQET where . is the regularization scale. After we have integrated out degrees of freedom when going from QCD to HQET, the C1,2 scale dependence reflects the non-trivial RG running of the effective theory operators (see section 3.1 for details). At the tree level the coefficients are C1 =1 and C2 = 0, and one recovers eq. (2.14). Explicit expressions for Ci(µ)athigher orders in .s are obtained from the comparison of the loop matrix elements of the currents in the full and in the effective theory. In addition to this, higher order power corrections in the 1/mQ expansion may be considered where operators of higher dimensions in HQET are taken into account in the matching procedure. 2.4 Combining heavy quark and chiral symmetries Heavy hadrons contain a heavy quark as well as light quarks and/or antiquarks and gluons (the heavy quark – antiquark pairs being suppressed in the mQ ›.limit of HQET). All the degrees of freedom other than the heavy quark are referred to as the light degrees of freedom . The total angular momentum J is a conserved operator with eigenvalues J2 = j(j +1). We have also seen that the spin of the heavy quark SQ is conserved in the mQ ›.limit (we define its eigenvalues sQ through S2 = sQ(sQ + 1)). Therefore, the spin of the light degrees Q of freedom S. defined by S. .J -SQ is also conserved in the heavy quark limit (eigenvalues S2 = s (s. + 1)). Heavy hadrons come in doublets (unless s. = 0) containing states with the total spin j± = s. ±1/2 obtained by combining the spin of the light degrees of freedom with the spin of the heavy quark sQ =1/2. These doublets are degenerate in the mQ ›.limit. Mesons containing a heavy quark Q are made up of a heavy quark and a light antiquark q— (plus gluons and qq—pairs). The ground state mesons are composed of a heavy quark with sQ =1/2 and light degrees of freedom with s. =1/2 forming a multiplet of hadrons with spin j =1/2 .1/2=0 .1 and negative parity, since quarks and antiquarks have opposite intrinsic * parity. These states are the D and D* mesons if Q is a charm quark, and the B and B mesons if Q is a b quark. The field operators which annihilate these heavy quark mesons with velocity (Q) *(Q) *(Q) v are denoted by Pv and Pvµ respectively, with Pv ·v = 0. Since these operators will mix under the heavy quark spin transformations, it is convenient to collect them into a single tensor 2.4. COMBINING HEAVY QUARK AND CHIRAL SYMMETRIES field operator transforming accordingly under the heavy quark spin and flavor symmetries of the HQET Lagrangian (2.12). Lorentz contractions with .µ and .5 convert vectors and pseudoscalars into bi-spinors so we can immediately identify the field Hv (Q) annihilating the ground state Qq— P *+ (or better h+q—)mesons as H+ = P+[-P+.5], where P+ serves to project out the large v vvv particle components of the heavy quark Q (h+) while the factoring out of the large momentum v +† phase factor is implicit. The expression for the creation field Hv + = .0Hv .0 follows from its bi-spinor transformation properties under CPT. An analogous procedure can be performed for P*- the mesons containing a heavy antiquark Q (h-) leading to H- = P-[ + P-.5], which v vvv creates the corresponding particles. The apparent difference in the relative sign between the pseudoscalar and vector components is conventional and defined so that the particle creation and anti-particle annihilation fields (and vice-versa) appear with the same relative sign between vector and pseudoscalar operators. The only further fields needed for the remainder of the thesis will be for the lowest lying mesons of positive parity (scalar P0 and axial-vector P* ), which can 1µ P*± ]2 be represented by S± = P±[.5 ±P± . v 1v 0v Due to the peculiar construction of the hadron fields, the normalization of states in HQET is different from that of full QCD. Namely the standard relativistic normalization of hadronic states of mass dimension -1 (possible spin labels are suppressed) H(p )|H(p)QCD =2Ep(2.)3.3(p -p) (2.16) is modified to factor out any dependence on the mass of the heavy quark, while the states are labelled by their four-velocity H(v )|H(v)HQET =2v 0(2.)3.vv. . (2.17) States normalized by using this HQET convention have mass dimension -3/2and the two normalizations differ by a factor . mH as well as possible power corrections . |H(p)QCD = mH |H(v)HQET + O(1/mQ) . (2.18) To take into considerations also the interactions with the pseudo-Goldstone bosons due to the chiral dynamics of the light antiquark (u, d,...) inside the heavy meson, these are factored out of the quark (and consequently hadron) fields. Under the chiral group G, Hv and Sv therefore transform as H(S)+ ›H(S)+U† and H(S)- ›UH(S)- . The most general effective vv vv Lagrangian containing positive and negative parity heavy mesons containing a heavy quark (we will drop the super-and subscripts ’+’ and v respectively and keep in mind, that an analogous Lagrangian can be written down for the heavy mesons containing a heavy antiquark, and that velocity reparametrization invariance connects different heavy quark velocity representations) to order O(p) in the chiral expansion and at leading order in the heavy quark mass expansion, that is invariant under heavy quark and chiral symmetries, and is a Lorentz scalar is [32, 33] (1) (1) (1) (1) L= L- + L+ L mix, HM.PT + 12 12 (1) = -Tr (iv ·Dab -.ab.H )Hb + gTr L A Ha HbHa ab.5 , - 12 (1) =Tr (iv ·Dab -.ab.S)Sb + gTr L A Sa SbSa ab.5 , + 12 L(1) = HbSaab.5 +h.c.. (2.19) mix hTr A For a general treatment of hadronic states with higher spins see e.g. [54, 88] 14 CHAPTER 2. EFFECTIVE THEORIES OF HEAVY AND LIGHT QUARKS µµ h.c. denotes an additional Hermitian conjugate term, D= .ab.µ -Vis the chiral covariant ab ab heavy meson derivative, while the trace Tr runs over Dirac indices. Chiral vector (V)and axial-vector (A) pseudo-Goldstone current operators have been defined in section 2.2, while g, h and g are three unknown effective couplings between heavy and light mesons. The .H and .S are the so-called residual masses of the H and S fields respectively. In a theory with only H or S fields, one is free to set .H =0 (.S = 0) since all loop divergences are cancelled by O(p2) (counter)terms at zero order in heavy quark expansion. However, once both fields are added to the theory, another dimensionful quantity .SH =.S -.H enters calculations and does not vanish in the chiral and heavy quark limits [89]. It is of the order O(p0)inthe standard chiral power counting and is usually it is accounted for via the appropriate pole offset in the heavy meson propagators. Although one is free to offset both positive and negative parity heavy meson poles, the end results of any calculation with well defines mass-shell conditions will only depend on the difference of both quantities. Alternatively, one could also boost the heavy mesons of different parities to different velocities so as to factor out this additional scale e.g. H(v)= exp(i.H v ·x)H(v)with v= v -.H /mQ,and S(v)= exp(i.Sv ·x)S(v)with v= v -.S/mQ. In this case however the splitting reappears in the form of a phase factor difference in L(1) in eq. (2.19). In terms of the momentum space Feynman rules this corresponds mix to modification of the momentum conservation at the HS. vertex inducing at leading order in 1/mH anew .SH /f coupling between positive and negative parity mesons and pseudo-Goldstone bosons (in addition to new 1/mH corrections). We see immediately that if .SH is comparable or larger than f, this leads to a strongly coupled theory. In addition it is not suppressed by powers of pseudo-Goldstone momenta (it has chiral power counting zero) and therefore spoils the chiral limit of the theory. However, the physical content of both theory representations is the same, meaning that at any perturbation order in both heavy quark expansion, chiral counting and .SH , both Feynman rule sets yield the same results for any Green’s function. The main difference is that in the first case we are actually able to re-sum all .SH contributions to all orders by solving the free field theory (including the extra .H and .S terms) exactly and obtaining the free heavy field propagators. This will prove to be of major importance in our calculations, there we adopt this approach throughout this work (for the list of derived Feynman rules used, see Appendix B). In the same way as sketched in the previous paragraph we must consider bosonization of HQET currents and more general operators that appear in electro-weak processes. Again we chose eq. (2.15) as an example. At the effective hadronic level of HM.PT we must construct all operators consistent with transformation properties (with respect to chiral, heavy quark and Lorentz symmetries) of the two effective HQET current operators up to the given order in the chiral and heavy quark expansions (formally this is done by inserting the same external spurion currents into generating functionals of both theories). At the O(p0) order in the chiral counting and at the leading order in the heavy quark expansion the effective current containing a single positive or negative parity heavy mesons simply reads (0)µ i. † i.† J= Tr[.µ(1 -.5)Hb].- Tr[.µ(1 -.5)Sb].+ O(1/mQ) , (2.20) (V -A)HM.PT ba ba 22 where . and .. are unknown parameters which can be matched to the heavy meson decay constants. Note that at this order there exists only one distinct operator for each parity because any insertions of the heavy meson velocity v can be reduced to this form by the use of the heavy meson velocity projection identities  = -H(S). In fact any general vH(S)= H(S)and H(S)v structure heavy-to-light current q—.Q,where . = 1,.5,.µ.5.µ,.µ. canbe translatedintothe effective bosonized form in the same manner [33, 55]. Chapter 3 Hadronic amplitudes – effective approaches and resonances In this chapter we will briefly review some standard methods used in the phenomenology of weak interactions of hadronic systems. Here we can consider as ”weak” all possible interactions apart from QCD, which may contribute significantly to quark dynamics at high enough energies but are almost completely swapped by strong interactions which confine quarks into hadrons at energies well below the electroweak scale. In addition to the prototype weak interactions of the electroweak SM, these may include contributions from possible new physics beyond the SM. The methods of OPE allow us to integrate out all degrees of freedom not directly associated with the external hadronic states and split the problem into a perturbative calculation of all short distance contributions using asymptotically free quarks on one hand, and an essentially nonperturbative calculation of hadronic matrix elements of operators, which however now contain only light degrees of freedom of QCD. We will also briefly touch upon some of the general properties, approximations and relations among these hadronic amplitudes. 3.1 Operator product expansion In this section we will briefly review the ideas behind OPE and its application to weak interactions. The original idea dates back to Wilson [76], who conjectured that the singular part (as x› y) of the product A(x)B(y) of two operators is given by a sum over other local operators CAB A(x)B(y) ---› (x- y)O n(y), (3.1) n x›y n where CAB(x- y) are singular c-number functions. Dimensional analysis suggests that CAB(x- nn y) behaves fox x› y like the power dOn - dA - dB of x- y,where dO is the dimensionality of the operator O in powers of mass or momentum. Since dO increases as we add more fields or derivatives to an operator O, the strength of the singularity of FAB decreases for operators O n n of increasing complexity, making their contributions to the sum (3.1) less and less relevant. The simple power counting argument is modified slightly by quantum effects in the renormalization group treatment, where anomalous dimensions of operators come into play. Another remarkable property of eq. (3.1) is that it is an operator relation: it holds regardless of what the states it acts on are. The OPE in general reads CA1...Ak T{ A1(x1)A2(x2) ...Ak(xk) ---› (x- x1,...,x- xk)O n(x), (3.2) xi›xn n 15 16 CHAPTER 3. HADRONIC AMPLITUDES with T being the time ordering operator. The application of the OPE to weak interactions comes from the observation that the distances at which weak interactions occur are set by the mass of the intermediate W and Z bosons, i.e., x -y ~1/mW . If one is interested in the processes at energy scales µ much smaller than the weak scale (µ mW ), or in other words, in the processes effectively occurring at typical distances 1/µ that are much larger than x -y ~ 1/mW , we can take the limit x › y (or equivalently mW ›.) and use the OPE. Formally we consider the generating functional of correlation functions and we focus on the relevant integration over the W and Z degrees of freedom. The charged current part of the action then contributes e.g. [90] R xLW ZW ~ [dW +][dW -]eid4, (3.3) where 1 !. . g2 ! 2 L= - .µW + -..W + .µW -. -..W -µ+ mW Wµ +W -µ + . Jµ +W +µ + Jµ -W -µ, W .µ 2 22 (3.4) with J+ =(—cstb)V -A +(—.µµ)V -A +(—=(J+)† . ud)V -A +(—)V -A +(—.ee)V -A +(—.. .)V -A,and J- µ µµ . = V CKM qi ij qj are the rotated weak states and g2 is the weak isospin coupling constant. We use 2 the unitary gauge for the W field and introduce Kµ. (x, y)= .(4)(x -y)[gµ. (.2 + m) -.µ.. ]. W After discarding a total derivative in the W kinetictermwehave RR d4xd4yWµ +(x)Kµ. (x,y)W. -(y)+i . d4x(Jµ +W +µ+Jµ -W -µ) ZW ~ [dW +][dW -]ei 2 g22 . (3.5) Performing a Gaussian functional integration over W ±(x) explicitly, we arrive at 2 R g -i d4xd4y(Jµ -(x).µ. (x-y)Jµ +(y)) ZW ~e82 , (3.6) where .µ. (x)is the W propagator in the unitary gauge. This result implies a nonlocal action functional for the quarks which we can expand in powers of 1/m2 to obtain a series of local W interaction operators of dimensions that increase with the order in 1/m2 To lowest order W . .µ. (x) ..(4)(x)gµ. /m2 and the effective action in eq. (3.6) becomes W 2 2 - gd4xJµ -J+µ , (3.7) 2 8mW corresponding to the usual effective charged current interaction Hamiltonian of the Fermi theory GF Heff = -. C2O2, (3.8) 2 . 22 where the definition of the Fermi constant GF / 2= g2/8mhas been used and a local four- W quark operator O2 = Jµ -J+µ with a Wilson coefficient C2 = 1 has been defined. Actually, the effective Hamiltonian (3.8) is valid only in the absence QCD interactions. Once these are taken J+µ into account, another four-quark operator appears in the OPE O1 = J- , where summation .ßµß. over the color indices ., ß of current quarks is understood. Formally we are integrating out all degrees of freedom at scales equal or larger than mW , including hard (energetic) gluons, while .. 3.1. OPERATOR PRODUCT EXPANSION practically, due to the asymptotic freedom of QCD, the corrections can be computed perturbatively by considering all the relevant correlation functions with free quarks in the asymptotic states. The effective weak Hamiltonian is then GF Heff = - (C1O1 + C2O2) , (3.9) 2 where the coefficient C1 is proportional to .s, while C2 ~ 1 is nonzero already at tree level in the perturbative QCD expansion as discussed above. A similar procedure can be employed to integrate out any possible heavy quarks in which case the current operators O1,2 in eq. (3.9) appear in the properly reduced form containing only the light quark fields. Finally, we may also consider neutral current weak processes including flavor changing neutral current (FCNC) processes in which case a number of new operators appears in the OPE corresponding to tree- level, penguin and box diagram contributions in the original theory. In the calculation of the Wilson coefficients typically expressions of the form .sln(µ/mW ) appear, where µ is a typical scale at which we want to study the processes. In the case of c and b quark decays, these are typically of the order of a few GeV. Thus, the ratio of scales in the argument of the logarithm can be very large, of order 100, and consequently the factor .sln(µ/mW ) of the order O(1). Even though the QCD coupling .s is not terribly large at the heavy quark scales and could be used as a perturbative expansion parameter, the appearance of large logarithms prevents the straightforward application of perturbation theory. All large logarithms of the form [.sln(µ/mW )]n have to be summed up using renormalization group equations. This is done by again considering correlation functions Oi. of operators appearing in the OPE both in the effective and in full theory. Oi. do not depend on the renormalization, whereas even after accounting for the renormalization of the quark fields in the original theory, due to the different UV structure of the effective theory, the OPE expressions have to be multiplicatively (0) (0) renormalized. In terms of the unrenormalized Wilson coefficients Cand operators O,of i i which neither depend on µ, the renormalization condition can be written as C(0) i O(0) i = Ci(µ)Z-1 ij (µ)Zji(µ)Oi(µ), (3.10) where the scale dependence of both the renormalized Wilson coefficients and the operators is fully determined by the renormalization matrix Zij(µ). In a compact form we can write dCi dZ = .ijCj,. = Z-1 , (3.11) dlnµdlnµ where we have introduced the anomalous dimension matrix ., which we determine be identifying the leftover singularities (or equivalently the logarithmic µ dependence as in eq. (3.11), but for the operators Oi) in the process of matching Green’s functions in both the original and the effective (OPE) theory. Using the evolution of the QCD coupling constant gS renormalization scheme) and their expansion 3 dgS (µ) g S = ß(gS)= -ß0 + ..., (3.12) d ln µ 16.2 .(0) .s .(.s)= + ..., (3.13) 4. 2 where .s = g/4. and ß0 =(11Nc - 2Nf )/3for Nf active flavors and Nc colors, the evolution S equation (3.11) can be solved to any given order. As an example we consider the leading order renormalization group (RG) evolution of a single Wilson coefficient (intheMS .0/2ß0 .s(mW ) C(µ)= C(mW ). (3.14) .s(µ) CHAPTER 3. HADRONIC AMPLITUDES The strong coupling constant .s(µ) appearing in (3.14) is at the one-loop order 4. .s(µ)= , (3.15) ß0 ln(µ2/.2 ) QCD where .QCD is the QCD scale (the value of which depends on the number of active flavors). Using the precisely measured value of the strong coupling constant at the Z boson mass, .s(mZ )= 0.1172 ±0.002 one arrives at .(5) = 216 ±25 MeV, while for µ