The Nucleon Anapole Form Factor in Chiral Perturbation Theory to Sub-leading Order

The anapole form factor of the nucleon is calculated in chiral perturbation theory to sub-leading order. This is the lowest order in which the isovector anapole form factor does not vanish. The anapole moment depends on counterterms that reflect short-range dynamics, but the momentum dependence of the form factor is determined by pion loops in terms of parameters that could in principle be fixed from other processes. If these parameters are assumed to have natural size, the sub-leading corrections do not exceed ∼ 30% at momentum Q ∼ 300 MeV. [email protected] [email protected] [email protected] Parity-violating electron scattering has long played a role in understanding electroweak interactions, and has more recently been explored as a tool for the study of nucleon structure. The SAMPLE collaboration has carried out electron scattering measurements at a momentum transferred of Q = 0.1 MeV on both the proton [1] and the deuteron [2], for a simultaneous extraction of the strange magnetic (GM) and the axial form factor of the nucleon (GA). One quantity that contributes in electron scattering as GA is the anapole form factor, which is an extension for Q > 0 of the anapole moment. The anapole is a parityviolating electromagnetic moment of a charge particle with spin [3]. Recently the effect of the nuclear anapole moment in atomic parity violation was measured precisely in Cs transitions [4], and a discrepancy with theory found. Parity violation in this case is enhanced by nuclear medium effects. No such enhancement is present in parity-violating electron scattering off the proton and deuteron; however, the anapole form factor could still be visible. Using previous estimates of the anapole moment [5, 6], the proton data implies a positive value for GM [1], in disagreement with most theoretical predictions (for a summary, see Ref. [7]). Experiments of current interest [1, 2, 8, 9, 10] are performed at finite Q = −q2. For Q < MQCD, where MQCD ∼ 1 GeV is the characteristic QCD mass scale, we are deep in the non-perturbative regime of QCD, where currently the only possible systematic calculations are in terms of hadrons. At Q ∼ O(mπ) the photon can resolve the pion cloud around the non-relativistic nucleon, and calculations are possible in Chiral Pertubation Theory (ChPT), which involves pions, nucleons, and delta isobars, and which has been successfully applied to hadronic and nuclear systems [11, 12]. The first anapole calculations were limited to Q = 0 in leading [5, 6] and sub-leading orders [6, 13]. Recently, the full form factor of the nucleon was calculated in leading order [14, 15]. In this order the form factor comes entirely from the pion cloud and is purely isoscalar, while experiments are most sensitive to the isovector component [7]. Here we report results of sub-leading contributions to the nuclear anapole form factor, where the isovector part first appears. In the framework of ChPT, QCD symmetries are used as a guide to build the most general effective Lagrangian. The number of terms in the Lagrangian is not constrained by symmetries, which demands a power counting argument to order interactions according to the expected size of their contributions. In order to fullfill chiral symmetry requirements, pions couple derivatively in the chiral limit; this derivative coupling brings to the amplitude powers of pion momentum or powers of the delta-nucleon mass difference (comparable to the pion mass). Chiral symmetry breaking terms involve quark masses, so they bring into the amplitude powers of the pion mass. Thus one has a chiral index (∆) available to order the Lagrangian terms, L = ∑∆ L(∆). For strong interactions, the index counts powers of Q/MQCD, and it is given by ∆ = d + n/2 − 2, where n is the number of fermions fields and d counts the numbers of derivatives, powers of the pion mass, and of the delta-nucleon mass difference. In the presence of electromagnetic interactions, it is convenient to include in d powers of the charge e as well. Weak interactions, on the other hand, bring powers of a very small factor GFf 2 π , where GF is the Fermi constant and fπ the pion decay constant. Since we count these factors explicitly, negative indices appear. 1 Based on this power counting argument the interactions relevant to our problem are the following. The parity-conserving terms are well known [11]: L str/em = 1 2 (Dμπ) 2 − 1 2 mππ 2 + N̄ iv ·DN − gA fπ N̄ (τ · S ·Dπ)N + . . . (1) L str/em = 1 4mN N̄ [ (v ·D) −D ] N + i gA 2mNfπ N̄ {S ·D, τ · v ·Dπ}N − i 4mN N̄ [S, S ] [1 + κ + (1 + κ)τ3]NFμν + . . . (2) Here π denotes the pion field with fπ = 93 MeV the pion decay constant; N represents the heavy nucleon field of four-velocity v and spin S (in the nucleon rest frame v = (1,~0) and S = (0, ~σ/2)); Aμ is the photon field and Fμν is the photon strength field; Dμ = (∂μ−ieQAμ) is the covariant derivative, withQ ab = −iε3ab for a pion andQ = (1+τ3)/2 for a nucleon; and “...” stands for other interactions with more pions, nucleons and deltas. The pion-nucleon coupling gA and the magnetic photon-nucleon couplings κ (s) and κ are not determined from symmetry but expected to be O(1); indeed, one finds gA = 1.267, κ = −0.12, and κ = 5.62 [11]. The relevant parity-violating terms were discussed in Ref. [6]: L weak = − h (1) πNN √ 2 N̄(τ × π)3N + . . . (3) L weak = − 2 f 2 π N̄S {( h (1) A + h (2) A τ3 ) [ (π × ∂μπ)3 + eAμ (