piN scattering and electromagnetic corrections in the perturbative chiral quark model

We apply the perturbative chiral quark model to give predictions for the electromagnetic O(p^2) low-energy couplings of the ChPT effective Lagrangian that define the electromagnetic mass shifts of nucleons and first-order (e^2) radiative corrections to the piN scattering amplitude. We estimate the leading isospin-breaking correction to the strong energy shift of the pi(-)p atom in the 1s state, which is relevant for the experiment"Pionic Hydrogen"at PSI.

In [1,2] Weinberg and Tomozawa derived a model-independent expression for the S -wave πN scattering lengths using the current algebra relations and the PCAC assumption. To reproduce the result for the πN scattering lengths one can use the specific Lagrangian with the nucleon field N refered to as the Weinberg-Tomozawa (WT) term [3]- [7] which is part of the effective Weinberg Lagrangian. The effective Weinberg Lagrangian can be derived from the original σ-model [8] by performing a chiral-field dependent rotation on the nucleon field [3]. On the quark level the same exercise was done in the framework of the cloudy bag model [9,10]. The chiral transformation eliminates the nonderivative coupling of the chiral (pion) field with the nucleons/quarks and replaces it by a nonlinear derivative coupling (axial vector term + WT term + higher order terms in the chiral field). Note, that both realizations of chirally-symmetric Lagrangians (the original σ-model and the Weinberg type Lagrangian) shouldá priori give the same result for the πN S-wave scattering lengths. In ref. [11] in the framework of perturbative chiral quark model (PCQM) [12,13] we demonstrate that the equivalence between the two theories with nonderivative and derivative coupling of the chiral field to the quarks is also valid when including the photon field.
The purpose of this letter is to calculate first-order (e 2 ) radiative corrections to the nucleon mass and the pion-nucleon amplitude at threshold. We thereby predict the O(p 2 ) electromagnetic (e.m.) low-energy couplings (LECs) originally defined in the effective Lagrangian of Chiral Perturbation Theory (ChPT) [5,6]. Quantitative information about these constants is important for the ongoing experimental and theoretical analysis of decay properties of the π − p atom (for a detailed discussion see Ref. [14]). In particular, we give a prediction for the leading isospin-breaking correction to the strong energy shift of the π − p atom in the 1s state.
Following considerations are based on the perturbative chiral quark model (PCQM), a relativistic quark model suggested in [12] and extended in [13] for the study of low-energy properties of baryons. The model includes relativistic quark wave functions and confinement as well as the chiral symmetry requirements. The quarks move in a self-consistent field, represented by scalar S(r) and vector V (r) components of a static potential with r = | x| providing confinement. The interaction of quarks with Goldstone bosons is introduced on the basis of the nonlinear σ-model [8]. The PCQM is based on the effective, chirally invariant Lagrangian L inv [13] where ψ is the quark field, U = exp[iΦ/F ] is the chiral field and F = 88 MeV is the pion decay constant in the chiral limit [4,13]. In the following we restrict to the SU(2) flavor case, that isΦ →π = π τ . For small fluctuations of the mesons fields one can use the perturbation expansion in powers of the parameter 1/F . The PCQM was successfully applied to σ-term physics and extended to the study of electromagnetic properties of the nucleon [13]. Similar perturbative quark models have also been studied in Refs. [15].
The quark field ψ we expand in the basis of potential eigenstates as where the sets of quark {u α } and antiquark {v β } wave functions in orbits α and β are solutions of the Dirac equation with the static potential. The expansion coefficients b α and d † β are the corresponding single quark annihilation and antiquark creation operators.
The direct way to generate the WT term in the Lagrangian (1) is through introduction of a unitary transformation on the quark field ψ. The technique was, for example, performed in the context of the cloudy bag model [9]. With the unitary chiral rotation ψ → exp{−iγ 5Φ /(2F )}ψ the Lagrangian (1) transforms into a Weinberg-type form L W containing the axial-vector coupling and the WT term: where L W ;str I is the O(π 2 ) strong interaction Lagrangian, ✷ = ∂ µ ∂ µ and M π is the pion mass.
In ref. [11] we demonstrate explicitly for the πN amplitude up to order (1/F 2 ) that the two effective theories, the original one involving the pseudoscalar coupling and the Weinberg type, are formally equivalent, both on the level of the Lagrangians and for the matrix elements. This equivalence is based on the unitary transformation of the quark fields, where, in addition, the quarks remain on their energy shell. The same relation also holds in a fully covariant formalism, when quarks/baryons are on their mass shell. Particularly, we show that the Weinberg-Tomozawa result can be reproduced with the use of the original Lagrangian (1) if: i) we use the expansion of the chiral field up to quadratic terms and ii) we employ the full quark propagator including the antiquark components. The two forms of the Lagrangian also yield the same results when including the photon field. For the equivalence to hold it is essential that the photons are introduced consistently in both formalisms, that is by minimal substitution. One can prove that both Lagrangians yield the same results for radiative corrections to the πN scattering amplitude at threshold.
In this letter we apply the developed formalism to study e.m. corrections of nucleon properties, such as the mass and the πN scattering amplitude. We perform all calculations using the technically more convenient Lagrangian (3). Introduction of the e.m. field A µ is accomplished by minimal substitution into Eq. (3): where Q is the quark charge matrix.
Following the Gell-Mann and Low theorem [16] the e.m. mass shift ∆m em N of the nucleon with respect to the three-quark ground state |φ 0 > N is to order e 2 in the e.m. interaction. Subscript "c" in Eq. (5) refers to contributions from connected graphs only. Superscript "N" indicates that the matrix elements have to be projected onto the respective nucleon states. These nucleon states are conventionally set up by the product of single quark SU (6) spin-flavor and SU(3) c color w.f. (see details in [13]), where the nonrelativistic single quark spin wave function is replaced by the relativistic ground state solution. With the quark-photon interaction defined by the Lagrangian the e.m. mass shift ∆m em N is generated by two diagrams: one-body (Fig.1a) and two-body diagram (Fig.1b).
The leading e.m. corrections (up to order e 2 /F 2 ) to the πN scattering amplitude at threshold are generated by the interaction Lagrangian where L W ;str I is given in Eq. (3) and the additional e.m. part L W ;em I is given by The πN amplitude in the presence of O(e 2 ) radiative corrections is given by The diagrams for O(e 2 /F 2 ) radiative corrections to the πN amplitude at threshold are shown in Fig.2. To evaluate the diagrams in Figs.1 and 2 we use the photon propagator D µν in the Coulomb gauge 1 to separate the contributions from Coulomb and transverse photons.
First, we analyze the e.m. mass shift of the nucleon. The contributions of diagrams Fig.1a and Fig.1b are given by where iG ψ (x, y) =< 0|T {ψ(x)ψ(y)}|0 > is the quark propagator in a binding potential. In the following we truncate the expansion of the quark propagator to the ground state eigen mode: where |N is the SU(6) spin-flavor w.f. of the nucleon. Here we introduce the proton charge (G p E ) and magnetic (G p M ) form factors (f.f.) calculated at zeroth order [13] (meson cloud corrections are not taken into account) with where χ Ns is the nucleon spin w.f. and σ N is the nucleon spin operator. Note that the contributions of Coulomb and transverse photons to the e.m. mass shifts (see Eqs. (12)) are related to the nucleon charge and magnetic f.f., respectively. The sum is equivalent to the charge matrix of nucleons (Q N being the nucleon charge). In the limit m N → ∞ (when we neglect the contribution of G p M in Eqs. (12)) we obtain for the e.m. mass shifts consistent with the result (Eq. (12.4)) of Ref. [17]. Hence, the e.m. mass shift of the neutron vanishes in the heavy nucleon limit.
In the numerical analysis we use the variational Gaussian ansatz [13] for the quark ground state wave function with the following analytical form: where N = [π 3/2 R 3 (1 + 3ρ 2 /2)] −1/2 is a constant fixed by the normalization condition d 3 x u † 0 (x) u 0 (x) ≡ 1; χ s , χ f , χ c are the spin, flavor and color quark wave functions, respectively. Our Gaussian ansatz contains two model parameters: the dimensional parameter R and the dimensionless parameter ρ. The parameter ρ can be related to the axial coupling constant g A calculated in zeroth-order (or the three quark-core) approximation: Therefore, ρ can be replaced by the axial charge g A by means of the matching condition (17). The parameter R can be physically understood as the mean radius of the three-quark core and is related to the charge radius of the proton in the leading-order approximation as In our calculations we use the value g A =1.25 obtained in ChPT [4]. Therefore, we have only one free parameter, that is R. In the numerical studies [13] R is varied in the region from 0.55 fm to 0.65 fm, which corresponds to a change of r 2 E P LO from 0.5 to 0.7 fm 2 . The exact Gaussian ansatz (16) restricts the potentials S(r) and V (r) to a form proportional to r 2 . They are expressed in terms of the parameters R and ρ (for details see Ref. [13]).
Using (16) the proton f.f. at zeroth order are determined as [13]: With Eq. (19) the e.m. mass shift is finally given as where α = 1/137 is the fine structure coupling. For our set of parameters g The O(p 2 ) low-energy constants (LECs) f 1 , f 2 and f 3 contain the effect of the direct quark-photon interaction. Matching our results for the nucleon mass shifts to the predictions of ChPT [6] with we obtain following relations for the coupling constants f 1 , f 2 and f 3 : Our numerical result for f 2 = −8.7 ± 0.7 MeV is in good agreement with the value of f 2 = −8.3 ± 3.3 MeV [6,14] extracted from the analysis of the elastic electron scattering cross section using the Cottingham formula [17]. For f 1 +f 3 we get −1.5 ± 0.1 MeV.
We furthermore give a prediction for the separate values of f 1 , f 3 and the ratio f 1 /f 2 as deduced from our model analysis of e 2 corrections to the πN amplitude. We denote the corresponding matrix element associated with the nucleon flavor transition N 1 → N 2 by M (e 2 );ij In the Coulomb gauge only six diagrams (Fig.2a-2f) contribute to the radiative correction to the πN amplitude at threshold. The contribution of the other diagrams (Fig.2g-2o) vanishes. The contributions of the different diagrams of Fig.2 are as follow: Fig.2a and 2b where × T ij ψ 0 (x)ψ 0 (y)γ ν Qψ 0 (y)|φ 0 N for Fig.2c, Fig.2d and 2e, Fig.2f.
Truncating the quark propagator to the ground state mode the πN scattering amplitude at threshold including first-order radiative corrections is where The contribution of the Coulomb photons to the amplitude M e 2 πN inv is parametrized by the proton charge form factor (G E ), transverse photons are related to the proton magnetic (G M ) and axial nucleon (G A ) f.f. where the latter is given by [13] f 1 = −19.5 ± 1.6 MeV, Using these values of f 1 and f 2 we can estimate the isospin-breaking correction to the energy shift of the π − p atom in the 1s state. The strong energy-level shift ǫ 1s of the π − p atom is given by the model-independent formula [14]: where the leading order (LO) or isospin-symmetric contribution is ǫ LO 1s and the next-to-leading order (NLO) or isospin-breaking contribution is ǫ N LO 1s . The quantity ǫ LO 1s is expressed with the help of the wellknown Deser formula [18] in terms of the S-wave πN scattering lengths with ǫ LO 1s = −2α 3 µ 2 c A str and A str = (2a1 2 +a3 2 )/3. The reduced mass of the π − p atom is denoted by µ c = m p M π + /(m p + M π + ) and A str = (88.4 ± 1.9) × 10 −3 M −1 π + is the strong (isospin-invariant) regular part of the π − p scattering amplitude at threshold [19] (for the definitions of these quantities see Ref. [14]). In ChPT the quantity δ ǫ , the ratio of NLO to LO corrections, is expressed in terms of the LECs c 1 , f 1 and f 2 The quantity c 1 is the strong LEC from the ChPT Lagrangian [5,7] and F π = 92.4 MeV is the physical value of the pion decay constant [14]. In Ref. [13] we obtained c 1 = −1.16 ± 0.1 GeV −1 using the PCQM approach. Our prediction for c 1 is close to the value c 1 = −0.9m −1 N deduced from the πN partial wave analysis KA84 using Baryon Chiral Perturbation Theory [7]. Substituting the central values for our couplings f 1 = −19.5 MeV, f 2 = −8.7 MeV and c 1 = −1.16 GeV −1 into Eq. (32), we get δ ǫ = −2.8·10 −2 . Our estimate is comparable to a prediction based on a potential model for the πN scattering [19]: δ ǫ = −2.1 · 10 −2 .
In conclusion, we give predictions for the O(p 2 ) electromagnetic (e.m.) lowenergy couplings (LECs) f 1 , f 2 and f 3 as originally set up in the ChPT effective Lagrangian. The magnitude of f 2 and its relation to f 1 and f 3 are obtained from an analysis of the nucleon e.m. mass shift and the leading radiative corrections to the πN scattering amplitude at threshold. Using our values for f 1 and f 2 we also predict the isospin-breaking correction to the strong energy shift of the π − p atom in the 1s state. Latter prediction is extremely important for the ongoing experiment "Pionic Hydrogen" at PSI, which intends to measure the ground-state shift and width of pionic hydrogen (π − p-atom) at the 1% level [20].