The SU ( 3 ) Vector Currents in BChPT × 1 / N c

The SU(3) Vector Currents in BChPT×1/Nc I. P. Fernando 2, ∗ and J. L. Goity 2, † Department of Physics, Hampton University, Hampton, VA 23668, USA. Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA. Abstract Baryon Chiral Perturbation Theory (BChPT) combined with the 1/Nc expansion is applied to the SU(3) vector currents. In terms of the ξ power counting linking the low energy and 1/Nc expansions according to O(ξ) = O(p) = O(1/Nc), the study is carried out to next-to-next-toleading order, and it includes SU(3) breaking corrections to the |∆S |= 1 vector charges, charge radii, and magnetic moments and radii. The results are obtained for generic Nc, allowing for investigating the various scalings in Nc.


I. INTRODUCTION
Vector currents, being intimately related to the flavor SU (3) symmetry of QCD, represent a fundamental probe for hadron structure as well as for the breaking of SU (3) by quark masses. This is particularly interesting for baryons, where the electromagnetic current for nucleons, known empirically to remarkable accuracy [1], along with the magnetic moments  [2]. The reason is that β decay has a branching fraction of about 10 −3 , being dominated by the non-leptonic component. Fortunately, lattice QCD is producing results [3][4][5][6] which can be compared with the predictions of the approach in the present work. The experimental information on charge form factors is limited to the electric form factors of nucleons and the charge radius of the Σ − . This is however sufficient to predict the rest of the charge radii, whose SU (3) breaking effects are, at the order of the present calculation, finite nonanalytic in quark masses. The octet baryons' EM magnetic moments and nucleons' magnetic radii give an almost complete prediction for the rest of the currents but for one low energy constant (LEC) which requires knowledge of at least one weak magnetic moment of a ∆S = 1 current. In the approach followed here, results automatically extend to the vector current observables of the decuplet baryons and to EM transitions, e.g. the M 1 transition ∆ → N γ, most of which remain empirically unknown or poorly known. The study of electric currents in BChPT with inclusion of the spin 3/2 baryons dates back a quarter century [7,8], and numerous works have since been produced in various versions of that framework, among those close in spirit to the present one are found in Refs. [9][10][11][12][13][14][15], and works with additional constraints imposed by consistency with the 1/N c expansion are those of Refs. [16][17][18][19][20][21]. The present work formalizes the combination of BChPT and the 1/N c expansion [22] for the vector currents following the rigorous power counting scheme of the ξ expansion [23,24] based on the linking O(p) = O(1/N c ) = O(ξ). The combined framework was first applied to the SU (3) vector charges in Ref. [20], where the ξ expansion was not strictly implemented, however for the purpose of calculating the corrections of SU (3) breaking to the vector charges, restricted by the Ademollo-Gatto theorem (AGT), such omission has no very significant effect 1 . Here a complete study is presented to O(ξ 3 ) and O(ξ 4 ) (depending on the observable) of the SU (3) vector currents. The present work provides results for generic N c , permitting in this way to sort out in particular the large N c behavior of non-analytic terms in ξ stemming from one loop corrections, which gives additional understanding, as it has been shown for instance in the case of the Gell-Mann-Okubo relation and the σ terms discussed in Refs. [25,26]. The subject of magnetic moments has been addressed in the context of the 1/N c expansion in works limited to a tree level expansion in composite operators [27][28][29][30], and in works including one loop corrections in BChPT Refs. [17][18][19]21]. In addition to the BChPT, dispersive approaches implementing constraints of chiral dynamics have been implemented [31][32][33] and where in addition consistency with the 1/N c expansion has also been required [34][35][36][37][38]. Such works naturally give a range of applicability beyond the present one, which is limited up to the form factor radii.
This work is organized as follows: Section II presents the baryon chiral Lagrangians needed for the present work, Section III summarizes the one loop corrections to the vector currents, Section IV presents the analysis of the vector charges and radii, and Section V does the same for the magnetic moments and radii. A summary is presented in Section VI.
Several appendices are included for the benefit of readers intending to implement similar calculations.

II. BARYON CHIRAL LAGRANGIAN
This section summarizes the pieces of the Baryon Chiral Lagrangian up to O(ξ 4 ) relevant to the calculations in this work. The details on the construction of the Lagrangians and the notations are given in Ref. [24]. In order to ensure the validity of the OZI rule for the quark mass dependency of baryon masses, namely, that the non-strange baryon mass dependence on m s be O(N 0 c ), the following combination of the source χ + is defined by [24]: Collecting the baryons in a spin flavor multiplet denoted by B, and using standard notation for the chiral building blocks (for details see [24]), the LO O(ξ) Lagrangian reads: where the hyperfine mass shifts are given by the second term, G ia are the spin-flavor generators (see Appendix A), and the axial coupling is at LOg A = 6 5 g A , being g A = 1.2732(23) the nucleon's axial coupling. The relevant terms in the O(ξ 2 ) Lagrangian are: where the flavor SU (3) electric and magnetic fields are denoted by E + and B + and given [24]. The term proportional to κ gives at LO the magnetic moments associated with all vector currents. The O(ξ 3 ) and O(ξ 4 ) Lagrangians needed for the one-loop renormalization of the vector currents are the following: The LECs g 1 and g 2 will be determined by charge radii, the term proportional to g 3 gives electric quadrupole moments for decuplet baryons and for transitions between decuplet to octet baryons, which will not be discussed here, and the term proportional to κ r gives a contribution to magnetic radii (D 2 B + ≡ D µ D µ B + being the covariant divergence of the magnetic field). The rest are quark mass and higher order in 1/N c corrections to the magnetic moments.
Throughout, spin-flavor operators in the Lagrangians are scaled by appropriate powers of 1/N c such that all LECs start at zeroth order in N c . Of course, LECs have themselves an expansion in 1/N c , kept implicit, which requires information for N c > 3 to be determined. In that sense each Lagrangian term has a leading power in 1/N c which is used to assign its order in the ξ power counting, followed by sub-leading terms in 1/N c due to the expansion of the corresponding LEC. In addition, each term in the Lagrangian is explicitly chiral invariant and its expansion in powers of the Goldstone Boson fields yields factors for each additional factor of a GB field.
For convenience the following definition is used: Note that δm gives rise to mass splittings between baryons which are the O(1/N c ) hyperfine term in Eqn. (2) and the O(p 2 ) quark mass term. The O(m q N c ) term inχ + becomes immaterial in the loop calculations as only differences of baryon masses appear for which such terms exactly cancel.

III. ONE LOOP CORRECTIONS TO CURRENTS
The one-loop corrections to the vector currents involve the two sets of gauge invariant diagrams A and B in Fig. 1, where the vertices are given in Appendix C. The explicit results are the following: where P n are projectors onto the corresponding baryon in the loop, p 0 is the residual energy of the initial baryon, q 0 is the incoming energy in the current, and Γ µa = g µ0 T a + i κ Λ 0µij f abc f cbd q i G jd contains both the electric charge and magnetic moment components. The one-loop wave function renormalization factor δẐ 1−loop can be found in [24], and the loop integrals I, K, K µ , K µν , H ij and H ijµ are given in Appendix B. Since the temporal component of the current can only connect baryons with the same spin, q 0 is equal to the SU (3) breaking mass difference between them plus the kinetic energy transferred by the current, which are all O(ξ 2 ) or higher and must therefore be neglected in this calculation.
In the evaluations one sets p 0 → δm in and p 0 + q 0 → δm out . In particular, for diagram A 1 , if it requires evaluation at q 0 = 0 such a limit must be taken in the end of the evaluation.
The U (1) baryon number current can used to check the calculation: only diagrams A 1+2 contribute, and as required they cancel each other. For a generic current vertex Γ, the combined UV divergent and polynomial piece of diagrams A 1+2 can be written as: where λ = 1 − γ + log 4π. The first term is proportional to quark masses through the GB mass-square matrix M 2 ab = m 0 δ ab + 1 2 d abc m c , and the second involves the baryon hyperfine mass splittings δm which are O(1/N c ) and, following the strict ξ power counting, the O(p 2 ) terms due to SU (3) breaking in δm are disregarded. The consistency with the 1/N c power counting can be readily checked. Diagrams A 3 and B 1,2 are separately consistent with the 1/N c power counting. Their polynomial contributions are the following: Reduction formulas that can be found in [25] are used to express the above in a base of irreducible operators, Eqns. (9) and (12) below.

IV. VECTOR CHARGES
In this section the SU(3) vector current charges and corresponding radii are analyzed.
The SU(3) breaking corrections to the charges already presented in [20] and [24] are discussed for completeness. At lowest order the charges are represented by the flavor generators T a .
The one-loop corrections are UV finite at Q 2 ≡ −q 2 = 0, and since up to O(ξ 3 ) the AGT is satisfied, the corrections to the charges are unambiguously given by UV finite one-loop contributions. Note that the AGT applies to the whole baryon spin-flavor multiplet. On the other hand, at finite Q 2 the one-loop correction has an UV divergent piece which is independent of quark masses and is renormalized via the terms g 1 and g 2 in L B , one of them removes the UV divergence (g 1 ) and the other one is a finite counterterm (g 2 ).
Combining the polynomial pieces in Eqns. (7) and (8) where f a 1 ≡ V 0a . The corrections to the | ∆S |= 1 charges, already discussed in [20], are evaluated using the physical valuesg A = 6 5 × 1.27 and F π = 92 MeV, however one needs to be aware that their values are effected by the NLO corrections, leading to a theoretical uncertainty. With the usual notation for those charges [20], evaluating the ratios δf 1 /f 1 in the large N c limit one finds that δf 1 /f 1 = O(1/N c ). However, this behavior sets in rather slowly at N c ∼ 20, emphasizing the fact that the non commutativity of the low energy and 1/N c expansions is very important at the physical N c = 3. The results are shown in Table I, where the errors are estimated from the above mentioned theoretical uncertainty. The agreement with recent LQCD calculations [4] is encouraging, and further improvement in the precision of those calculations would be very useful. For the charge radii the loop contributions are from diagrams A 3 and B 2 and the renormalization is provided by the LECs g 1 and g 2 in L B and L B respectively, of which only g 1 is required for canceling the loop UV divergence according to Eqn. (9) 2 . As is the case with form factors in ChPT, the charge radii depend logarithmically in the GB masses. They can be determined by fitting to the known electric charge radii of proton, neutron and Σ − , or simply fixed using the first two. If one wishes to study also the large N c limit, an assignment at generic N c of the quark electric charges has to be done. One such an assignment that respects all gauge and gauge-gravitational anomaly cancellations in the Standard Model is is given by [39] Using the three known charge radii, g 1,2 are determined modulo the main uncertainty stemming from the value used forg A . At the renormalization scale µ = m ρ , using the value ofg A ∼ 1 obtained by the analysis of the axial couplings [24], C HF ∼ 200 MeV, and with Λ = m ρ one finds g 1 1.33 and g 2 0.74. g 2 is sensitive to C HF , which is understood as a result that the non-analytic contributions to the neutron radius is very important, and thus sensitive to that parameter, while g 1 is not. One also observes that both LECs are crucial for obtaining a good description of the radii. For the used value of µ, the fraction of the loop contribution to r 2 of the proton is 15%, and for the neutron it is about 60%.
The short distance contributions are thus very important in both cases. The dominant non-analytic contributions to the radii are proportional to log m q , with other non-analytic terms involving the LEC C HF giving almost negligible contributions, making the results insensitive to it. Table II  to r 4 = 60 d 2 f 1 d(Q 2 ) 2 , is given by the one-loop non-analytic terms with contributions that are inversely proportional to quark masses. The curvature is nominally an effect O(ξ 4 ) in the form factor, which therefore receives contributions from terms O(ξ 6 ) in the Lagrangian, and only in the limit of sufficiently small quark masses will the non-analytic contributions obtained here be dominant. In the recent work of Ref. [38] the electric charge higher moments have been studied, where t-channel elastic unitarity has been implemented in the EFT along with the constraints of the 1/N c expansion [34][35][36][37][38]. In particular, for the curvature they find r 4 p = 0.735(35) fm 4 and r 4 n = −0.540(35) fm 4 , to be compared with the one-loop contributions found here, 0.032 and −0.021 fm 4 respectively, roughly a factor 25 smaller in magnitude in each case. Clearly the description of the curvature must be primarily given by higher order contact terms, and to the order of the expansion followed here, the failure to account for the curvature limits the present description of charge form factors to the expected range given by the radii, Q 2 0.05 GeV 2 .

V. MAGNETIC MOMENTS
As mentioned earlier, at lowest order the magnetic moments of all vector currents are given in terms of the single LEC κ. In particular, using the EM current the LO value of κ Λ can be fixed from the proton's magnetic moment µ p in units of the nuclear magneton µ N , namely e κ 2Λ = µ p = 2.7928 µ N . Also, the M 1 radiative decay width of the ∆ at LO is given by: where ω is the photon energy. Using the above result for κ Λ gives Γ LO ∆→N γ = 0.38 MeV, to be compared with the experimental value 0.70 ± 0.06 MeV. In terms of the transition magnetic moment, the LO result is µ ∆ + p = 2  Table III. It is evident that there are significant SU (3) breaking effects, which together with the important spin-symmetry breaking observed. in particular in the ∆N M 1 amplitude indicate the relevance of the NNLO calculation. Note that all weak magnetic moments, i.e., magnetic moments associated with the ∆S = 1 currents are also fixed at LO, as they are empirically unknown. In the case of the neutron β decay the weak magnetic term is obtained from the isovector part of the EM magnetic moments of proton and neutron, which in this case, due to isospin symmetry, is quite accurate. On the other hand, in hyperon beta decay the effect of weak magnetism is too small to be at present experimentally accessible. Fortunately the advent of LQCD calculations of magnetic moments with increasing accuracy will allow the study of weak magnetism.
The one loop corrections to the magnetic moments are obtained from the spatial components of the vector currents depicted in Fig. 1  A and B 1 . Diagrams A 1,2 involve Γ ∝ G ia , which is similar to the axial currents already analyzed in Ref. [24]. The loop contributions to the Q 2 dependence of the magnetic form factors stem from diagram A 3 .
The UV divergencies of the one loop diagrams contributing to the magnetic moments after reduction of the corresponding expressions Eqns. (7) and (8) using a basis of spin-flavor operators read as follows: adding up to: The renormalization of the magnetic moments is provided by the Lagrangians with the LECs κ D,F,1,··· ,5 , and the magnetic radii receive only finite one-loop contributions and a finite renormalization by the term κ r . The β functions of the magnetic LECs resulting from Eqn. (12) are shown in Table IV.  The results of the fits are shown in Table V. Since the input magnetic moments have errors (much) smaller than the theoretical error of the present calculation estimated to be of the order of NNNLO corrections or about 5%, the χ 2 has been normalized for estimating the LECs' errors. Important correlation is found between the following pairs of LECs: κ 4 − κ 5 ,  As mentioned earlier, the ∆N γ amplitude at LO is too small by roughly 30%, a manifestation of an important spin-symmetry breaking effect. The effect receives a small non-analytic contribution (at µ = m ρ ), and the contributions from the contact terms are as follows: One of the early tests of the magnetic moments in SU (3) was provided by the Coleman-  Table VI. The rest of the radii are then predictions which can hopefully be tested in the future with LQCD calculations. Note that the lion share of the magnetic radii is from the short distance terms proportional to κ r with the loop contribution from diagram A 3 in Fig. 1 giving up to 20% for proton, neutron and Σ − and less than 10% for the rest.  Finally, a calculation of the curvature of the EM magnetic moments yields: r 4 p = 0.38 fm 4 and r 4 n = 0.54 fm 4 to be compared with those obtained in Ref. [38], which are which turn out to be very small, and therefore requiring for their description an extension of the present work to higher order.
where R represents the SU (3) multiplet of the baryon, and γ indicates the possible recouplings in SU (3). The matrix elements of interest are then given by: where the reduced matrix elements are (here p = 2S, q = 1 2 (N c − 2S)):

Appendix B: Loop integrals
The one-loop integrals needed in this work are provided here. The definition d d k ≡ d d k/(2π) d is used.
The Feynman parametrizations needed when heavy propagators are in the loop are as follows: where the A i are heavy particle static propagators denominators, and the B i are relativistic ones.

Appendix C: Interaction and vector current vertices needed in loop calculations
The interaction and currents vertices needed in the one-loop calculations are given for completeness.
gA q is incoming, and Γ µa = g µ0 T a + i κ Λ 0µij f abc f cbd q i G jd .