Bled Workshops in Physics Vol. 3, No. 3 Proceedins of the Mini-Workshop Quarks and hadrons (p. 67) Bled, Slovenia, July 7-14, 2002 Axial currents in electro-weak pion production at threshold and in the A-region * S. SircaQ'b, L. Amoreirac-d, M. Fiolhaisd-e, and B. Gollif'b Q Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia bJozef Stefan Institute, 1000 Ljubljana, Slovenia c Department of Physics, University of Beira Interior, 6201-001 Covilha, Portugal d Centre for Computational Physics, University of Coimbra, 3004-516 Coimbra, Portugal e Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal f Faculty of Education, University of Ljubljana, 1000 Ljubljana, Slovenia Abstract. We discuss electro-magnetic and weak production of pions on nucleons and show how results of experiments and their interpretation in terms of chiral quark models with explicit meson degrees of freedom combine to reveal the ground-state axial form factors and axial N-A transition amplitudes. 1 Introduction The study of electro-weak N-A transition amplitudes, together with an understanding of the corresponding pion electro-production process at low energies, provides information on the structure of the nucleon and its first excited state. For example, the electro-magnetic transition amplitudes for the processes y*p —> A+ —> pn0 and y*p —> A+ —> nn+ are sensitive to the deviation of the nucleon shape from spherical symmetry [1]. Below the A resonance (and in particular close to the pion-production threshold), the reaction y*p —> nn+ also yields information on the nucleon axial and induced pseudo-scalar form-factors. While the electro-production of pions at relatively high [2] and low [3,4] momentum transfers has been intensively investigated experimentally in the past years at modern electron accelerator facilities, very little data exist on the corresponding weak axial processes. 2 Nucleon axial form-factor In a phenomenological approach, the nucleon axial form-factor is one of the quantities needed to extract the weak axial amplitudes in the A region. There are basically two methods to determine this form-factor. One set of experimental data comes from measurements of quasi-elastic (anti)neutrino scattering on protons, deuterons, heavier nuclei, and composite targets (see [4] for a comprehensive list * Talk delivered by S. Sirca. 68 S. Sirca, L. Amoreira, M. Fiolhais, and B. Golli of references). In the quasi-elastic picture of (anti)neutrino-nucleus scattering, the vN —> |xN weak transition amplitude can be expressed in terms of the nucleon electro-magnetic form-factors and the axial form factor GA. The axial form-factor is extracted by fitting the Q2-dependence of the (anti)neutrino-nucleon cross section, dCT A(Q2) T B(Q2) (s -u) + C(Q2) (s -u)2 , (1) dQ2 in which GA(Q2) is contained in the A(Q2), B(Q2), and C(Q2) coefficients and is assumed to be the only unknown quantity. It can be parameterised in terms of an 'axial mass' ma as gA(q2) = gA(0)/(1 + q2/mA)2 . Another body of data comes from charged pion electro-production on protons (see [4] and references therein) slightly above the pion production threshold. As opposed to neutrino scattering, which is described by the Cabibbo-mixed V — A theory, the extraction of the axial form factor from electro-production requires a more involved theoretical picture [5,6]. The presently available most precise determination for MA from pion electro-production is MA = (1.077 ± 0.039) GeV (2) which is AMA = (0.051 ± 0.044) GeV larger than the axial mass MA = (1.026 ± 0.021) GeV known from neutrino scattering experiments. The weighted world-average estimate from electro-production data is MA = (1.069 ± 0.016) GeV, with an excess of AMA = (0.043 ± 0.026) GeV with respect to the weak probe. The ~ 5 % difference in MA can apparently be attributed to pion-loop corrections to the electro-production process [5]. 3 N-A weak axial amplitudes The experiments using neutrino scattering on deuterium or hydrogen in the A region have been performed at Argonne, CERN, and Brookhaven [7-11]. (Additional experimental results exist in the quasi-elastic regime, from which MA has been extracted.) For pure A production, the matrix element has the familiar form M = (^A | vN) Gp cos 0c V2 ja (A | Va - Aa | N) where Gp is the Fermi's coupling constant, 0C is the Vud element of the CKM matrix, ja = U^ya(1 — y5)uv is the matrix element of the leptonic current, and the matrix element of the hadronic current Ja has been split into its vector and axial parts. Typically either the A++ or the A+ are excited in the process. The hadronic part for the latter can be expanded in terms of weak vector and axial form-factors [12] M = ^UAa(p Cv Cv Cv M + M2P|t + M52p + CA pA 3 i C4 I M + P|- M2 Y5F^a + CV jaY5 CA + CAja + m2 < u(p)f(w) Axial currents in electro-weak pion production 69 where = — üak(p') is the Rarita-Schwinger spinor describing the A state with four-vector p', and u(p) is the Dirac spinor for the (target) nucleón of mass M with four-vector p. (In the case of the A++ excitation, the expression on the RHS acquires an additional isospin factor of V3 since (A++ | Ja | p) = V/3(A+ | Ja | p) = v/3(A0 | Ja | p).) The function f(W) represents a Breit-Wigner dependence on the invariant mass W of the Nn system. The matrix element is assumed to be invariant under time reversal, hence all form-factors CV,A(Q2) are real. Usually the conserved vector current hypothesis (CVC) is also assumed to hold. The CVC connects the matrix elements of the strangeness-conserving hadronic weak vector current to the isovector component of the electro-magnetic current: (A++|Va|p) = V3 (A+|JgyT = 1)|p) , (A0 | Va | p) = (A+ | Jem(T = 1) I p) . The information on the weak vector transition form-factors CV is obtained from the analysis of photo- and electro-production multipole amplitudes. For A electro-excitation, the allowed multipoles are the dominant magnetic dipole Mi + and the electric and coulomb quadrupole amplitudes Ei+ and Si+, which are found to be much smaller than Mi + [2,3]. If we assume that Mi + dominates the electro-production amplitude, we have Cj = Cj = 0 and end up with only one independent vector form-factor cV = M cV C =— WC3 • It turns out that electro-production data can be fitted well with a dipole form for cV, Q2 2 CV(Q2) = 2.05 1 + 0.54 GeV2 An alternative parameterisation of Cj which accounts for a small observed deviation from the pure dipole form is Cj(Q2) = 2.05 Tl + 9V/QIj expl"-6.^V/Q2 The main interest therefore lies in the axial part of the hadronic weak current which is not well known. Extraction of CA(Q2) from data The key assumption in experimental analyses of the axial matrix element is the PCAC. It implies that the divergence of the axial current should vanish as m^ —> 0, which occurs if the induced pseudo-scalar term with C^ (the analogue of GP in the nucleon case) is dominated by the pion pole. In consequence, C^ can be expressed in terms of the strong nNA form-factor, CA(Q2) = f _ [2 G„nA M2 *V 3 2M Q2 + m2 70 S. Sirca, L. Amoreira, M. Fiolhais, and B. Golli while Of and Cf can be approximately connected through the off-diagonal Gold-berger-Treiman relation [13]. In a phenomenological analysis, Cf(Q2), Cf(Q2), and Cf(Q2) are taken as free parameters and are fitted to the data. The axial form-factors are also parameterised in "corrected" dipole forms cA(q2) = Ca (0) 1 + qiq2 bi + Q2 1 + Q. mAj In the simplest approach one takes at = bt = 0. Historically, the experimental data on weak pion production could be understood well enough in terms of a theory developed by Adler [14]. For lack of a better choice, Adler's values for CA(0) have conventionally been adopted to fix the fit-parameters at Q2 = 0, i. e. CA(0) = 0, CA(0) = -0.3 CA(0) = 1.2. (3) (4) (5) In such a situation, one ends up with MA as the only free fit-parameter. Several observables are used to fit the Q2-dependence of the form-factors. Most commonly used are the total cross-sections ct(Ev), and the angular distributions of the recoiling nucleon da da v^n Yoo T5 P 33 Y20 + 4 Trô P 31 ReY2i 4 Trô p 3-1 ReY22 where pmn are the density matrix elements and YLM are the spherical harmonics. Better than from the pmn coefficients, the Q2 dependence of the matrix element can be determined from the differential cross-section da/dQ2. In particular, since the dependence on CA and CA is anticipated to be weak at Q2 ~ 0, then da dQ2 (Q2 = 0) « ( CA(0) )2 The refinements of this crude approach are dictated by several observations. If the target is a nucleus (for example, the deuteron which is needed to access specific charge channels), nuclear effects need to be estimated. Another important correction arises due to the finite energy width of the A. In addition, the non-zero mass of the scattered muon may play a role at low Q2. All these effects have been addressed carefully in [15]. The sensitivity of the differential cross-section to different nucleon-nucleon potentials was seen to be smaller than 10 % even at Q2 < 0.1 GeV2. In the range above that value, this allows one to interpret inelastic data on the deuteron as if they were data obtained on the free nucleon. The effect of non-zero muon mass is even less pronounced: it does not exceed 5 % in the region of Q2 - 0.05 GeV2. The energy dependence of the width of the A resonance was observed to have a negligible effect on the cross-section. The final value based on the analysis of Argonne data [9] is CA(0) = 1.22 ± 0.06. (6) 2 a Axial currents in electro-weak pion production 71 At present, this is the best estimate for C^(0), although a number of phe-nomenological predictions also exist [16]. We adopt this value for the purpose of comparison to our calculations. There is also some scarce, but direct experimental evidence from a free fit to the data that C^(0) is indeed small and CA(0) is close to the Adler's value of -0.3 (see Figure 1). We use CA(0) = -0.3 in our comparisons in the next section. Fig. 1. One- and three-standard deviation limits on CA(0) and CA(0) as extracted from measurements of v^p —> |oTA++. The square denotes the model predictions by Adler [14]. (Figure adapted after [7].) 4 Interpretation of CA(Q2) in the linear a-model The axial N-A transition amplitudes can be interpreted in an illustrative way in quark models involving chiral fields like the linear a-model (LSM), which may reveal the importance of non-quark degrees of freedom in baryons. Due to difficulties in consistent incorporation of the pion field, the model predictions for these amplitudes are very scarce [17]. The present work [18,19] was partly also motivated by the experience gained in the successful phenomenological description of the quadrupole electro-excitation of the A within the LSM, in which the pion cloud was shown to play a major role [20]. 4.1 Two-radial mode approach We have realised that by treating the nucleon and the A in the LSM in a simpler, one-radial mode ansatz, the off-diagonal Goldberger-Treiman relation can not be satisfied. For the calculation of the amplitudes in the LSM, we have therefore used the two-radial mode ansatz for the physical baryon states which allows for different pion clouds around the bare baryons. The physical baryons are obtained 72 S. Sirca, L. Amoreira, M. Fiolhais, and B. Golli from the superposition of bare quark cores and coherent states of mesons by the Peierls-Yoccoz angular projection. For the nucleon we have the ansatz |N) = NnP2 [®N|Nq) + ®NA|Aq)] , (7) where Nn is the normalisation factor. Here and ®na stand for hedgehog coherent states describing the pion cloud around the bare nucleon and bare A, respectively, and P1 is the projection operator on the subspace with isospin and angular momentum 1. Only one profile for the ct field is assumed. For the A we assume a slightly different ansatz to ensure the proper asymptotic behaviour. We take |A) = Na{P3®A|Aq) +J dkn(k)[a|mt(k) | N) ]22 j , (8) where NA is the normalisation factor, |N) is the ground state and [ ] 2 3 denotes the pion-nucleon state with isospin | and spin 3. We have interpreted the localised model states as wave-packets with definite linear momentum, as elaborated in [13]. 4.2 Calculation of helicity amplitudes We use the kinematics and notation of [13]. For the quark contribution to the two transverse (À = 1 ) and longitudinal (À = 0) helicity amplitudes we obtain /A (q) s A A -