Bled Workshops in Physics Vol. 16, No. 1 p. 16
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015

Analytic structure of nonperturbative quark propagators and meson processes*
Dalibor Kekeza and Dubravko Klabucarb
a Rugjer Boskovic Institute, Bijenicka c. 54,10000 Zagreb, Croatia b Physics Department, Faculty of Science, Zagreb University, Bijenicka c. 32, Zagreb 10000, Croatia
Abstract. The analytic structure of certain Ansätze for quark propagators in the nonperturbative regime of QCD is investigated. When choosing physically motivated parameterization of the momentum-dependent dressed quark mass function M(p2), with definite analytic structure, it is highly nontrivial to predict and control the analytic structure of the corresponding nonperturbative quark propagator. The issue of the Wick rotation relating the Minkowski-space and Euclidean-space formulations is also highly nontrivial in the nonperturbative case. A propagator form allowing the Wick rotation and enabling equivalent calculations in Minkowski and Euclidean spaces is achieved. In spite of its simplicity, this model yields good qualitative and semi-quantitative description of some pseudoscalar meson processes.
Lattice studies of QCD are complemented by the continuum QCD studies utilizing Dyson-Schwinger equations (DSE). Both ab initio DSE studies and DSE studies for models of QCD provide an important approach for the study of phenomena in hadronic physics both at zero and finite temperatures and densities -see, for example, Refs. [1,2]. Just like lattice QCD studies, the large majority of DSE calculations (including those of our group, e.g., [3]) are implemented in the Euclidean metric.
Nevertheless, solutions of the Bethe-Salpeter equation require analytic continuation of DSE solutions for dressed quark propagators Sq (p), into the complex p2-plane. Similar situation is with the processes that involve quark propagators (QP) and Bethe-Salpeter amplitudes: it is not enough to know propagators and the Bethe-Salpeter amplitude only in the spacelike region, for real and positive p2. It is important to know the analytic properties in the whole p2 complex plane.
Alkofer et äl. [4] have explored the analytic structure of the Landau gauge gluon and quark propagators. They have proposed some simple analytic Ansätze for these propagators. Based on their Ansätze, Jiang et äl. [5] provide an analytical approach to calculating the pion decay constant fn and and the pion mass Mn at finite density.
We want to investigate and further improve the analytic structure of the quark propagator S(p). It can be conventionally parameterized (in Minkowski
* Talk delivered by Dubravko Klabucar
Analytic structure of nonperturbative quark propagators ... 17
space) as
s(p) = -M-P2) P + (-p2) = z(-p2)-
and correspondingly in Euclidean space as
,2w ,	P + M(-p2)
jp2 - M2(-p2)
ÇM -, ,2^ (2, Z(p2)	^ ip + M(p2)
S (p) = ipCTv(p2) + CTs(p2) = . , , ,	= Z(p2)-
-ip + M(p2)	>2 + M2(p2) '
where M(x) is the dressed quark mass function and Z(x) is the wave function renormalization. Alkofer et al. [4] have explored the analytic structure of the quark (and gluon) propagator in the Landau gauge, using numerical solutions of the pertinent Dyson-Schwinger equations and fits to lattice data as inputs. Their Ansatze for Z and M (or crv and crs) include meromorphic functions (poles on the real axes or/and pairs of the complex conjugate poles) and functions with branch cut structures. Positivity violation in the spectral representation of the propagator shows the presence of the negative-norm contributions to the spectral function, i.e., the absence of asymptotic states from the physical part of the state space, which is sufficient (but not necessary) criterion for the confinement. While in the gluon propagator a clear evidence for positivity violation is found, the similar analysis shows that there is probably no such violation in the quark propagator [4]. The propagator with pairs of complex conjugate poles violates causality. It has been argued [6,7] that the corresponding S-matrix remains both causal and unitary (see also Ref. [1]).
Furthermore, complex conjugate poles can pose a problem for the analytic continuation from Minkowski to Euclidean space (Wick rotation) used by lattice gauge theory and functional methods. It has been also shown that complex conjugate poles in S(p) cause thermodynamical instabilities at nonvanishing temperature and density [8].
Of crucial importance is the following question: Is it possible to find an analytic Ansatz for the quark propagator solely with branching cut (or cuts) on the real timelike axes, with no additional structures (isolated singularities or cuts) in the complex plane? Such an Ansatz could be used for practical calculation of the processes involving quark loops.
Because of a complicated interplay between analytic structure of the functions Z and M on one side, and crs and crv on the other side, the approximation A = 1 has been applied. Then, the problem reduces to finding of appropriate functions M(x) and cr(x) = 1/(x + M2(x)). The most rigorous constraint is that the propagator S(p) —» 0 for all directions |p2| ^ oo in the complex p2 plane [9]. Furthermore, for large and positive values of x = p2 (spacelike momenta), function M(x) must be positive and approach to zero from above [4]. In the Euclidean regime, for real and positive values of x, the mass function should be fitted to match the form known from lattice and Dyson-Schwinger calculations.
Number of Ansatze for the quark mass function has been investigated. When choosing certain parametrization of the function M(x), with definite analytic structure, it is highly nontrivial to predict and control the analytic structure of the accompanying cr(x) function. Relevant mathematical theory and possibly related theorems (like Rouches theorem) are hardly applicable for this concrete problem.
18 Dalibor Kekez and Dubravko Klabucar
The best results were achieved with the Ansatz of the form M(x) = log(R(x)), where R is a rational function with certain good properties. The function M(x) has a few cuts on the real timelike axes, while the propagator dressing function ct(x) has both branch cuts and poles on the real timelike axes. No additional structure are present in the complex momentum plane. The quark propagator based on this Ansatz should allow for the Wick rotation and equivalent calculation in Minkowski and Euclidean spacetime.
Future work will include an improved fitting of the mass function M(x) and refinement of calculation with Z(x) = 1. Furthermore, we are planning to check whether our Ansatz satisfies the requirements of positivity violation.
The quark propagator obtained in this way, endowed with good analytic properties, should then be tried and adjusted so that it gives good results in various applications: the yy-transition and charge form factors of pions, ct and p form factors and decays, are just some of the interesting potential applications of the quark propagator Ansatz with good analytic structure. It is also necessary to investigate the related issue of the Bethe-Salpeter equation in Minkowski space. The quark loop contribution to various processes should also be studied using these improved quark propagators. Besides the processes like n,n,n' -;> YY that are described by an anomalous triangle diagram, there are interesting anomalous processes based on the pentagon diagram, like n,n' —^> 4n. (We could expect new results from high-statistics n' experiments like BES-III, ELSA, CB-at-MAMIC, CLAS at Jefferson Lab.) The non-anomalous processes n —> 3n is especially interesting because it is sensitive to the isospin violation. While the average u and d-quark mass, (mu + md)/2, is well known, there exists significant uncertainty in their mass difference, md—mu. The n —> 3n decay is particularly suitable for md — mu difference determination because of the suppressed electromagnetic contributions [10,11].
Since the microscopic understanding of strongly interacting matter (both in hadronic phase and in quark-gluon phase) is of great importance also for the physics of heavy ion collisions and compact stars, extending the quark propagator Ansatz with good analytic structure to finite densities and temperatures should also be investigated. This is necessary, for example (to name one concrete task), for extending our analyses of the n-n' complex [12,13] to finite densities and temperatures. Of particular interest is extending to finite density our analysis of the possible UA(1) symmetry restoration [14] in the n-n' complex.
Acknowledgement
This work has been supported in part by the Croatian Science Foundation under the project number 8799. The authors acknowledge the partial support of the COST Action MP1304 Exploring fundamental physics with compact stars (New-CompStar).
References
1. C. D. Roberts and S. M. Schmidt, Prog.Part.Nucl.Phys. 45 (2000) S1-S103.
Analytic structure of nonperturbative quark propagators ... 19
2.	R. Alkofer and L. von Smekal, Phys.Rept. 353 (2001) 281.
3.	D. Kekez and D. Klabucar, Phys. Rev. D 71 (2005) 014004 [hep-ph/0307110], and our references therein.
4.	R. Alkofer, W. Detmold, C. Fischer, and P. Maris,Phys.Rev. D70 (2004) 014014.
5.	Y. Jiang et al, Phys.Rev. C78 (2008) 025214; Y. Jiang et al., Phys.Rev. D78 (2008) 116005.
6.	U. Habel, R. Konning, H. Reusch, M. Stingl, and S. Wigard, Z.Phys. A336 (1990) 435-447.
7.	U. Habel, R. Konning, H. Reusch, M. Stingl, and S. Wigard, Z.Phys. A336 (1990) 423-433.
8.	S. Benic, D. Blaschke and M. Buballa, Phys. Rev. D 86 (2012) 074002.
9.	R. Oehme and W. t. Xu, Phys. Lett. B 384 (1996) 269.
10.	D. Sutherland, Phys.Lett. 23 (1966) 384.
11.	J. Bell and D. Sutherland, Nucl.Phys. B4 (1968) 315-325.
12.	For analytic, closed form results for the masses and mixing in the r|-r|' complex, and for our earlier results on the t|-t|' complex, see [13].
13.	S. Benic, D. Horvatic, D. Kekez and D. Klabucar, Phys. Lett. B 738 (2014) 113.
14.	S. Benic, D. Horvatic, D. Kekez and D. Klabucar, Phys. Rev. D 84, 016006 (2011) [arXiv:1105.0356 [hep-ph]], and our references therein.
Bled Workshops in Physics Vol. 16, No. 1 p. 91
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015

Resonances in the Nambu-Jona-Lasinio model
Mitja Rosinaa'b
aFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, P.O. Box
2964,1001 Ljubljana, Slovenia
bJ. Stefan Institute, 1000 Ljubljana, Slovenia
Abstract. We have designed a soluble model similar to the Nambu-Jona-Lasino model, regularized in a box with periodic boundary conditions, in order to explore the properties of resonances when only discrete eigenvalues are available. The study might give a lesson to similar problems in Lattice QCD.
1 The quasispin NJL-like model
It is very instructive to understand the key features of a simplified model containing the spontaneous chiral symmetry breaking. Some time ago we have constructed a soluble version of the Nambu-Jona-Lasino model [1,2]. Now we explore what it tells about the sigma meson. We make the following simplifications:
1.	We assume a sharp 3-momentum cutoff 0 < Ipi < A;
2.	The space is restricted to a box of volume V with periodic boundary conditions. This gives a finite number of discrete momentum states, N = NhNcNfVA3/6n2 occupied by N quarks. (Nh,N c and Nf are the number of quark helicities, colours and flavours.)
3.	We take an average value of kinetic energy for all momentum states: Ipi —»
4.	While in the NJL model the interaction conserves the sum of momenta of both quarks we assume that each quark conserves its momentum and only switches from the Dirac level to Fermi level.
5.	Temporarily, we restrict to one flavour of quarks, Nf = 1.
Let us repeat the "Quasispin Hamiltonian" [1,2].
P = 4 A.
k=1 v 7
92 Mitja Rosina
Here y5 and |3 are Dirac matrices, mo is the bare quark mass and g = 4G/V where G is the interaction strength in the original (continuum) NJL.
We introduce the quasispin operators which obey the spin commutation relations
1 „ • 1 1
jx = 2 I , jy = 2 ilY5 , jz = 2 Y5 ,
n 1 + h(k) n 1 h(k)	n
Ra = Y_ -2-'La = -2-j*^ ' J* = + = j*^ '
k=1 k=1 k=1
The model Hamiltonian can then be written as
H = 2P(Rz - Lz)+ 2moJx - 2g(jX + jy) . The three model parameters
A = 648 MeV, G = 40.6 MeV fm3, m0 = 4.58 MeV have been fitted (in a Hartree-Fock + RPA approximation) to the observables
M = \l ( Eg(N) — Eg(N - 1)) - P2 = 335MeV
Q = < g|tMI9> = V( 9l L P(i)l9 ) = 1 ( 9lJxlg > = 2503 MeV3 i
mn = Ei(N) - Eg (N) = 138MeV.
The values of our model parameters are very close to those of the full Nambu-Jona Lasinio model used by the Coimbra group [3] and by Buballa [4].
2 The spectrum of 0 and 0+ excitations -Emergence of the a meson
It is easy to evaluate the matrix elements of the quasispin Hamiltonian using the angular momentum algebra. If N is not too large the corresponding sparse matrix can be diagonalized using Mathematica.
Excited levels of the ground state band (R=L=N/4) in Fig. 1 are almost equidistant and are suggestive of n-pion states (in s-state). The level spacings AE are slightly decreasing with the assumed number of pions nn due to the attractive interaction between pions. Inbetween appear also "intruder states" which can be interpreted as sigma excitations. The interpretation as ct meson is further supported by the large value of the matrix element of Jx between the ground state and the "intruder state". (Odd "multipion states" have zero value and even ones have a rather small value.)
Resonances in the Nambu-Jona-Lasinio model
93
np	parity	E [MeV]	DE [MeV]	Intruder
8	+	866	63	
-		816		s(667)+p(136)+13 MeV
7 -		803	93	
6	+	710	99	
+		667		s(667)
5 -		611	108	
4	+	503	115	
3 -		388	123	
2	+	265	129	
1 -		136	136	
0	+	0	0	
Fig. 1. Levels of the ground state band (R=L=N/4), level spacing between opposite parity states, and the assumed number of pions nn pions
3 The width of the a meson
In the attempt to describe resonances when only discrete eigenvalues are available we get a discrete sigma resonance energy, but not its width. We are trying to get the complex pole. For that purpose, we explore the method of analytic continuation from the bound state [5]. For this purpose, we vary one of the model parameters, the bare quark mass m from the region where the ct meson would be bound (Ect < E2n) down to the physical value of m —» m0 (where ECT > E2n).
At m > 64 MeV there are two positive parity states between the first and second negative parity states (the one-pion and three-pion excitations); the lower one is the intruder (ct meson) and the upper one is the correlated two-pion state. At m = 64 MeV both positive parity states coincide - the threshold for ct —» 2n. When we decrease m further, the energy of the ct meson decreases slower than the 2n energy and it appears at higher multipion states. For the physical value m = m0 = 4.58 MeV ct is already the sixth excited state, next to the six-pion state. It is obviously in the continuum, prompt to decay into 2n, in a more complete choice of interaction.
The method consists of the following steps:
•	Determine the threshold value mth and calculate e = ECT — E2n as a function of m for m > mth.
•	Introduce a variable x = a/m — mth; calculate k(x) = i^f—e in the bound state region (Fig. 2).
•	Fit k(x) by a polynomial k(x) = i(c0 + cix + c2x2 + ... + c2Mx2M) .
94 Mitja Rosina
•	Construct a Pade approximant:
k( . _ . ao + aix + ... + aMxM . 1 + bix + ... + bMXM .
•	Analytically continue k(x) to the region m < mth (i.e. to imaginary x) where k(x) becomes complex.
•	Determine the position and the width of the resonance as analytic continuation in m (Fig. 3 and Fig. 4):
ETes _ Re (contm_>mok2), r _ -2Im (contm_>mok2) .
k [MeV1/2] 20
15
10 -
5 -
10
12
x [MeV1/2]
Fig. 2. The fit of k(x) with quadratic(lower middle) and quartic polynomial (upper middle) and with Pade approximants of order 1 (below) and 2 (above)
2
4
6
8
We notice that the results for ETes and r in Fig. 3 and 4 deviate strongly for first and second order Pade approximants. This is due to the large stretch for the analytic continuation so that convergence at higher orders cannot be expected. Nevertheless, it is rewarding that the physical values for ETes and r lie somewhere in the middle between both curves.
To conclude, the method of analytic continuation in this case is just a game, but it is instructive. Intentionally, we have plotted the energy and width of the d meson as a function of the corresponding pion mass rather than as a function of the model parameter m. This is reminiscent of the extrapolation of pion mass from about 500 Mev towards its physical value the way tha lattice people have to struggle.
References
1. B. T. Oblak and M. Rosina, Bled Workshops in Physics 7, No. 1 (2006) 92; 8, No. 1 (2007) 66; 9, No. 1 (2008) 98 ; also available at http://www-f1.ijs.si/BledPub.
[MeV] Ere
Resonances in the Nambu-Jona-Lasinio model
95
900
800
700
600
200
300
400
500
mn [MeV]
Fig. 3. The resonance energy ETes of the a meson as a function of the pion mass - extrapolation using Pade approximants of order 1 (below) and 2 (above)
T [MeV]
Fig. 4. The width F of the a meson as a function of the pion mass - extrapolation using Pade approximants of order 1 (below) and 2 (above)
2.	M.Rosina and B.T.Oblak, Few-Body Syst. 47 (2010) 117-123.
3.	M. Fiolhais, J. da Providencia, M. Rosina and C. A. de Sousa, Phys. Rev. C 56 (1997) 3311.
4.	M. Buballa, M.: Phys. Reports 407 (2005) 205.
5.	V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 69A (1978) 251, V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 96B (1980) 4, N. Tanaka et al. Phys. Rev. C59 (1999) 1391.
102 Povzetki v slovenščini
Analitična zgradba neperturbativnih kvarkovih propagatorjev in mezonskih procesov
Dalibor Kekeza in Dubravko Klabučarb a Rugjer Boškovič Institute, Bijenička č. 54,10000 Zagreb, Croatia
b Physics Department, Faculty of Science, Zagreb University, Bijenička č. 32, Zagreb 10000, Croatia
Raziskujemo analitično zgradbo nekaterih nastavkov za kvarkove propagatorje v neperturbativnem področju kromodinamike. Ce izberemo fizikalno motivirano parametrizacijo masne funkcije M(p2) oblečenih kvarkov, odvisne od gibalne količine in z določeno analitično zgradbo, je skrajno tezavno napovedati in obvladati analitično zgradbo ustreznega neperturbativnega kvarkovega propagatorja. Tudi problem Wickove rotacije, ki povezuje izrazavo v prostoru Minkowskega in Evklida, je skrajno tezaven v neperturbativnem območju. Izpeljemo obliko propagatorja, ki omogoča Wickovo rotacijo in dopusča enakovredne račune v prostoru Minkowskega in Evklida. Kljub preprostosti nudi ta model dober kvalitativen in semikvantitativen opis nekaterih pročesov z psevdoskalarnimi mezoni.
Primerjava med mezoni in resonancami WLWL pri energijah več TeV
Antonio Dobadoa, Rafael L. Delgadoa, Felipe J. Llanes-Estradaa and Domenec Espriub
a Dept. Fisica Teorica I, Univ. Complutense, 28040 Madrid
b Institut de Ciencies del Cosmos (ICCUB), Marti Franques 1, 08028 Barcelona, Spain.
Mikavni signali z Velikega hadronskega trkalnika (LHC) namigujejo, da morda obstajajo v področju zloma elektro-sibke simetrije resonance v območju več TeV. Spomnimo na nekajključnih resonanc mezon-mezon v območju GeV, ki bi utegnile imeti analogne resonance pri visokih energijah in nam sluzijo za primerjavo, hkrati z odgovarjujočo unitarizirano efektivno teorijo. Ceprav je podrobna dinamika lahko različna, pa zahteve po unitarnosti, kavzalnosti in globalnem zlomu simetrije (z uporabo metode inverzne amplitude) dovoljujejo prenos intuicije v večinoma neizmerjeno območje visokih energij. Ce bo povečano stevilo dogodkov na ATLASU okrog 2 TeV podprlo tako novo resonančo, to lahko pomeni anomalno sklopitev qqW.
Resonance v konstituentnem kvarkovem modelu.
R. Kleinhappel and W. Plessas
Theoretical Physics, Institute of Physics, University of Graz, A-8010 Graz, Austria
Na kratko poročamo o danasnjem opisu barionskih resonanc v realističnem modelu s konstituentnimi kvarki, v katerem običajno obravnavamo resonance kot