The s-wave baryon nonleptonic decay amplitude in large-N_c chiral perturbation theory

The s-wave decay amplitude in the nonleptonic decay of baryons is analyzed within heavy baryon chiral perturbation theory in the large-N_c limit at one-loop order, where N_c is the number of color charges. Loop graphs with octet and decuplet intermediate states are systematically incorporated into the analysis and the effects of the decuplet-octet mass difference are accounted for. There are large-N_c cancellations between different one-loop graphs as a consequence of the large-N_c spin-flavor symmetry of QCD baryons. The predictions of large-N_c baryon chiral perturbation theory are in very good agreement both with the expectations from the 1/N_c expansion and with the experimental data.


I. INTRODUCTION
The remarkable success of the 1/N c expansion of QCD-where N c is the number of color charges [1, 2]-and its subsequent combination with heavy-baryon chiral perturbation theory [3] to describe several static properties of baryons has been evident over the past two decades.
Initially, a 1/N c expansion of the chiral Lagrangian was formulated in Ref. [3]; since then, the Lagrangian has been useful to evaluate nonanalytic meson-loop corrections to baryon amplitudes in the 1/N c expansion for finite N c . Specifically, the method was originally applied to compute flavor-27 baryon mass splittings at leading order in chiral perturbation theory [3]. Later, a number of additional baryon properties have also been successfully evaluated, namely, baryon axial-vector couplings [4,5], baryon magnetic moments [6,7], baryon vector couplings [8], and Dirac form factors [9] to name but a few.
The approach to compute nonanalytic meson-loop corrections in 1/N c baryon chiral perturbation theory at finite N c consists in identifying all the pertinent one-loop Feynman diagrams for the process under consideration. These diagrams are given by the product of a baryon operator with well-defined transformation properties under the spin-flavor symmetry times a loop integral, which depends nonanalytically on the light quark masses m q . In this way, the 1/N c and group theoretic structure of the loop corrections is manifest. Although theoretically the procedure is straightforward, in practice the reduction of the baryon operator becomes a rather involved task. With the advent of more powerful technical computing systems, the reduction is possible to an unprecedented level.
All the computations of baryon properties mentioned above generalize the formulas obtained previously in conventional baryon chiral perturbation theory (i.e., without a 1/N c expansion). An extra feature of the approach is that the 1/N c formulas exhibit the 1/N c and flavor-breaking structure of the one-loop corrections so various relations obtained in the limit of exact SU (3) flavor symmetry (for instance, the Coleman-Glashow relations for baryon magnetic moments [6,7]) can be better understood.
In the framework of baryon chiral perturbation theory, the analyses of sand p-wave amplitudes have been addressed in Refs. [10][11][12][13], each of which with some particular focus. References [11] and [13] evaluated the leading nonanalytical corrections including both octet and decuplet baryons as intermediate states, focusing on the |∆I| = 1/2 component of the decay amplitude (i.e., the so-called |∆I| = 1/2 rule was assumed to be valid). There are a few differences in some decay diagrams between these two analyses. Reference [12] also assumed the validity of the |∆I| = 1/2 rule, but included only octet baryons as intermediate states in the loops so the effects of the decuplet baryons were incorporated into the low-energy constants of the effective Lagrangian; to this purpose, all counterterms were included. While Refs. [11] and [13] conclude that good agreement with experiment cannot be simultaneously obtained using sand p-wave amplitudes at one-loop level, Ref. [12] claims the opposite.
In this paper, the applicability of the combined expansion in 1/N c and chiral corrections is extended to the analysis of decay amplitudes in the nonleptonic decays of baryons. Due to the enormous amount of algebraic calculations involved, it is more appropriate to present first the s-wave amplitude here and to leave the p-wave one for a further paper. To this end, one-loop graphs with intermediate spin-1/2 octet and spin-3/2 decuplet baryon states are analyzed including the full dependence on the decuplet-octet baryon mass difference, while at the same time including the cancellations that follow from the large-N c spinflavor symmetry of baryons.
The organization of the paper is as follows. In Sec. II the central ideas on the combined formalism are provided in order to introduce the notation and conventions. In Sec. III a theoretical description of baryon nonleptonic decays is presented, with emphasis on the calculation of tree-level s-wave amplitudes. In Sec. IV the one-loop contributions to the s wave amplitude are evaluated; partial operator reductions already performed in Ref. [8] are recognized to be present in the current analysis so they are borrowed and adapted to make up the new results. At this point, a direct comparison with conventional baryon chiral perturbation theory (i.e., without a 1/N c -expansion) is performed. The comparison is done by identifying the existing relations between the chiral coefficients and the operator coefficients that appear in the present analysis. Both analyses agree in full. In Sec. V a preliminary numerical comparison of the theoretical expressions with the available experimental information [14] through a least-squares fit is performed. The analysis is satisfactory. In Sec. VI some concluding remarks are addressed. The paper is complemented by two appendices. In Appendix A all the new operator reductions required are listed whereas in Appendix B all the coefficients that come along with the baryon operators in the several one-loop contributions are provided.

II. BARYON CHIRAL PERTURBATION THEORY IN THE 1/Nc EXPANSION
The aspects related to the 1/N c expansion for baryons have been discussed in detail in Refs. [3,15,16], so in this section a survey to introduce the notation and conventions used is provided. To start with, it should be recalled that in the large-N c limit the lowest-lying baryons 1 are given by the completely symmetric spin-flavor representation of N c quarks SU (2N f ) [15,17]. Under SU (2) × SU (N f ), this representation decomposes into a tower of baryon flavor representations with spins J = 1/2, 3/2, . . . , N c /2. Corrections to the large-N c limit are expressed in terms of 1/N c suppressed operators [15], which yields the 1/N c expansion of QCD.
The 1/N c expansion of a QCD m-body quark operator acting on a single baryon state can be written in the most general way as [16] O m-body where the O n (0 ≤ n ≤ N c ) constitute a complete set of linearly independent operator products which are of nth order in the baryon spin-flavor generators J k , T c and G kc , and the c n (1/N c ) are arbitrary unknown coefficients with an expansion in 1/N c beginning at order unity. Examples of 1/N c expansions for baryon operators include the 1/N c expansion of the baryon mass operator M and the baryon axial vector current A kc . The former is given by [16] where the coefficients m 0,1 n are a priori unknown parameters of order O(Λ χ ), and the superscripts attached to them indicate the spin-flavor representation they belong to. The first summand in Eq. (2) denotes the overall spin-independent mass of the baryon multiplet and the remaining terms, which are spin-dependent, constitute M hyperfine .
The 1/N c expansion of the baryon axial vector current, in turn, can be written for N c = 3 as [16] where the unknown coefficients a 1 , b n and c n also have expansions in 1/N c beginning at order unity and the leading operators that accompany them are given explicitly by Higher order operators are constructed from the previous ones by anticommuting them with J 2 . The chiral Lagrangian for baryons L baryon , formulated to understand the low-energy dynamics of baryons interacting with the pion nonet π, K, η, and η ′ in a combined expansion in 1/N c and chiral symmetry breaking, was given explicitly in Ref. [3]. In the baryon rest frame, L baryon reads where the covariant derivative reads The ellipsis in Eq. (7) refer to terms of higher order in the derivative and 1/N c expansions as well as terms involving explicit chiral symmetry breaking by the quark mass matrix. The Lagrangian depends on the meson field ξ(x) = exp[iΠ(x)/f π ] through the vector and axial-vector currents where Π(x) represents the nonet of Goldstone boson fields and f π ≈ 93 MeV/c 2 is the pion decay constant.

III. BARYON NONLEPTONIC DECAYS
The dominant decays of baryons are the nonleptonic modes where B i and B f are the decaying and emitted baryons and π is the emitted pion, respectively, with momenta p i , p f and k. These ∆S = 1 processes are quite useful in understanding the electroweak interaction in hadrons. The decay amplitude for the nonleptonic decays of spin-1/2 baryons can be written as [14] where G F is the Fermi constant, m π + is the pion mass, and A and B are parity-violating s-wave and parity-conserving p-wave decay amplitudes. A and B are related to the amplitudes s and p by where M f , E f , and p f are the mass, energy, and three-momentum of the final baryon; the usual observables, e.g., the partial decay rate Γ and the decay asymmetry α, can therefore be expressed as Γ ∝ |s| 2 + |p| 2 and α ∝ (|s| 2 + |p| 2 ) −1 . As far as isospin is concerned, both the sand p-wave components consist of contributions describing ∆I = 1/2 and ∆I = 3/2 transitions. An unexpected experimental result is that the former transitions are more favored than the latter ones by nearly a factor of twenty to one. This enhancement is also seen in kaon nonleptonic decays. Thus the so-called ∆I = 1/2 rule seems to be a rather universal feature of nonleptonic decays and will be considered to be valid hereafter.
The present paper focuses only on the analysis of the s-wave decay amplitude within large-N c chiral perturbation theory. For definiteness, this amplitude can be obtained using a soft pion theorem as [16] where c is an explicit flavor index, Q c 5 is the axial charge, and H W is the weak Hamiltonian. The latter contains pieces which transform as where Q c is the vector charge [16]. Octet dominance assumption also implies that H W transforms as the (0, 6 + i7) component of a (0, 8) representation of the spin-flavor symmetry SU (2) × SU (3) [16]. The 1/N c expansion of a (0, 8) operator has been discussed in detail in Ref. [18]. A simple operator analysis reveals that only n-body operators with a single factor of either T c or G ic appear. The allowed 1and 2-body operators are and whereas higher-order operators are obtained as O c n = {J 2 , O c n−2 } for n ≥ 3. Thus, the 1/N c expansion for H W reads up to corrections of relative order O(1/N 2 c ). Here, h i are undetermined parameters with dimensions of mass. Hereafter, the flavor index u will stand for u = 6 + i7 so any operator of the form W u should be understood as W 6 + iW 7 . As in previous works (see for instance Ref. [8]), the naive estimate that matrix elements of T c and G kc are both of order N c , which is the largest they can be, is also implemented. The estimate is legitimate provided the analysis is restricted to the lowest-lying baryon states. Within this naive power counting, A kc and T c are both order N c and so is H W of Eq. (16).
The vector charge is given by Q c = T c to all orders in the 1/N c expansion [19]. Thus, the commutator [Q c , H W ] reduces to Substituting Eq. (17) into (13) yields the decay amplitude at tree level A (s) where the flavor index c will stand for c = 1 ∓ i2 and c = 3 for π ± and π 0 , respectively. For completeness, any operator of the form W c should be understood as (W 1 ∓ iW 2 )/ √ 2 and W 3 for c = 1 ∓ i2 and c = 3, respectively. For the observed processes the expressions read The s-wave amplitudes at tree level for the nonleptonic decays of octet baryons into octet baryons can be fully described only by two parameters h 1 and h 2 . Adding higher-order operators in the 1/N c expansion (16) results into redefinitions of the already existing parameters, e.g., h 1 → h 1 + h 3 /6 and so on.
The right-hand sides of Eqs. (19)-(25) can be straightforwardly compared to their counterparts, α (s) BiB f , obtained within heavy baryon chiral perturbation theory, which can be found in Eqs. (3.7) of Ref. [11]. The operator coefficients h 1 and h 2 are related to the chiral coefficients h D and h F , for N c = 3, by Isospin symmetry of the strong interactions implies three relations among the seven amplitudes, namely, so there are effectively four independent amplitudes; the preferred study cases are those with a charged pion in the final state [11], namely, A (s) 2 These amplitudes can be combined to eliminate h 1 and h 2 , leading to the celebrated Lee-Suwagara relation [20,21] which holds in the limit of exact SU (3) flavor symmetry. Equation (18) can also be used to compute the tree-level s-wave amplitude for the nonleptonic decays of decuplet baryons to decuplet baryons. Specifically, the Ω − baryon is the only member of the baryon decuplet that decays predominantly through the weak interaction. For the known processes the amplitudes read and The above expressions are related by isospin as The inclusion of the third operator coefficient h 3 is necessary in order to account for the third chiral coefficient h C introduced in heavy baryon chiral perturbation theory [11]. For N c = 3 they are related by

IV. ONE-LOOP CORRECTIONS TO THE S-WAVE AMPLITUDE IN BARYON NONLEPTONIC DECAYS
The most general one-loop graphs that contribute to the s-wave amplitudes in the nonleptonic decays of baryons are displayed in Fig. 2. The approach to evaluate one-loop corrections to a baryon operator from Feynman diagrams like the ones in Figs. 2(a,b,c) have been dealt with in Ref. [22]. The analysis, general enough to apply to any baryon operator transforming as a flavor octet, was first specialized to the baryon axial-vector current [4,5,22], later on to the baryon magnetic moment [6,7] and more recently to the baryon vector current [8]. With only minor adaptations, the very same approach can be implemented here to evaluate corrections to the s-wave amplitude of baryon nonleptonic decays. As for Figs. 2(d,e), they have weak vertices from the h π term in the chiral Lagrangian (2.10) of Ref. [11]. However, a detailed analysis performed in that references concludes that the s-waves are fairly insensitive to inclusion of the parameter h π . Therefore, Figs. 2(d,e) will not be treated here.
FIG. 2: One-loop graphs for s-wave decay amplitudes in baryon nonleptonic decays. A solid square represents a ∆S = 1 weak vertex, a solid circle represents a strong vertex, and a solid (dashed) line denotes a baryon (pion). Wave function renormalization graphs are not shown, but are taken into account in the analysis.
On the other hand, by retaining only the chiral logs in the loop integral (36), neglecting the pion mass and using the Gell-Mann-Okubo relation to express m 2 η as 4m 2 and Equations (42)-(44) and their reduced forms (47)-(49) can be compared with the heavy-baryon chiral perturbation results of Refs. [12] and [11,13], respectively. The chiral results agree among them but not with the ones obtained here. The difference can be traced back to the value of the pion wave function renormalization coefficient used in those calculations.
B. One-loop corrections from Fig. 2 The correction to the s-wave amplitude arising from Fig. 2(b,c) can be written as (cf. Eq. (14) of Ref. [5]) where A ja and A jb stand for the meson-baryon vertices, M is the baryon mass operator and Q ab (n) is a symmetric tensor which is written in terms of the corresponding loop integral I where b (m, 0, µ) stands for the degeneracy limit ∆ → 0 of the function I The function I b (m, ∆, µ), given in Eq. (A6) of Ref. [8]. Its first derivative is, for thhe sake of completeness, (55) therefore, in the ∆ → 0 limit, it reduces to The final expression for the correction to the decay amplitude from Fig. 2(b,c) can be organized as where all the contributions from flavor singlet, flavor 8 and flavor 27 representations, for N c = 3, can be cast into and where the ellipsis refer to operators that appear for N c > 3. It is understood that flavor singlet and flavor 8 contributions must be subtracted off Eq. and As in the previous case, the matrix elements of the operators in the operator bases can be easily obtained. In each case, only the leading ones are required because the rest are obtained in most cases by anticommuting with J 2 . Also, relations (40) can be used so the matrix elements can be found in Ref. [8].
C. Total correction from Fig. 2

(b,c)
Gathering together partial results, the final expressions for the correction to the s-wave amplitude in baryon nonleptonic decays from Fig. 2(b,c), for N c = 3, can be organized as and where I (1) b (m, ∆, µ) has been obtained from the Maclaurin series expansion The above expressions can be rewritten in terms of the chiral coefficients of Ref. [11], namely, and Again, by retaining only the chiral logs in the loop integral (36), neglecting the pion mass and using the Gell-Mann-Okubo relation leads to Equations (73)-(75) can be compared to their counterparts displayed in Eq. (3.6) of Ref. [11]. The expressions match identically for Although the above result is unexpected because in the chiral Lagrangian the coefficients are presumably independent, it could have been anticipated in the h 3 → 0 limit in Eq. (32). A deeper analysis of this asseveration is required.
To close this section, corrections to the Lee-Sugawara relation (28) can be readily be computed; they are made up from flavor or in terms of the chiral coefficients,

V. NUMERICAL ANALYSIS
At this stage, a least-squares fit can be readily performed to compare theory and experiment. The main aim of such an analysis is not to be definitive about the determination of the s-wave amplitudes, but rather to explore the working assumptions.
The available experimental data about baryon nonleptonic decays are given in the form of lifetimes, branching ratios, and decay asymmetries [14], which can be used to determine the sand p-wave amplitudes. The numerical analysis here will be performed using the data about those amplitudes not related by isospin, namely, A (s) . This information is listed in the second column of Table I. The use of these four amplitudes considerably limits the number of free parameters that can be determined from the fit itself. There are four operator coefficients from the 1/N c expansion of the baryon axial current Eq. (3), namely, a 1 , b 2 , b 3 , and c 3 , and three more from the 1/N c expansion of the weak Hamiltonian Eq. 16), namely, h 1 , h 2 , and h 3 . The first set of coefficients will be borrowed from the analysis of the baryon vector and axial vector currents of Ref. [8]; as for the second set, the limit h 3 → 0 will be assumed, which is equivalent to using the approximation (77). The expected error introduced with this assumption is order O(1/N 2 c ). For definiteness, the meson masses used are the experimental ones [14], the pion decay constant is f π = 93 MeV, the scale of dimensional regularization is µ = 1.0 GeV, and the decuplet-octet baryon mass difference is ∆ = 0.232 GeV. As for the loop integrals (36) and (54), only the nonanalytical terms will be retained. Unlike Ref. [12] where all counterterms up to order O(p 2 ) were accounted for, the computation of those counterterms is far beyond the scope of this work. Rather, their effects are estimated by adding in quadrature a nominal error of 0.3. This value is an estimation of first-order flavor symmetry breaking effects coming from those counterterms.
With all the above considerations, the best-fit parameters are which are given in units of G F m 2 π + √ 2f π and the quoted error is a consequence of the theoretical error added. For this constrained fit, χ 2 = 2.63 for 1 degree of freedom. Although the fit is not remarkably good, it does yield some interesting outputs.
The predicted amplitude A  Figs. 2(a) and 2(b,c), which correspond to A (s) 2a and A (s) 2b , respectively. Notice that there are large cancellations between loop-corrections so that the net contributions are roughly 1/N c of the tree-level ones, except for Ω − → Ξ * 0 + π − , where the loop correction turns out to be higher than expected. All this information is collected in Table I. The total amplitudes fairly reproduce the observed ones. This, however, is not a withdraw of the approach; instead, it is a consequence of the inputs used for the operator coefficients from the axial vector current. At present, the only calculation for the renormalization of the axial vector current in the context of large-N c chiral perturbation theory which accounts for the mass difference between octet and decuplet baryons available is the one of Ref. [8]. This calculation, however, does not include all baryon operators for N c = 3. This might be an inconvenient because it has been shown that loop corrections to the axial vector currents are exceptionally sensitive to deviations of the ratios of baryon-pion axial vector couplings from SU (6) values [22]. A major improvement in that calculation is desirable. This, however, represents a non negligible effort and will be attempted elsewhere.
Finally, the numerical evaluation of the Lee-Sugawara relation is a value which goes in the opposite direction to the experimental correction of approximately −0.24.

VI. CONCLUDING REMARKS
In this paper, the s-wave amplitudes in baryon nonleptonic decays were evaluated in heavy baryon chiral perturbation theory in the large-N c limit at one-loop order. All baryon operators present for N c = 3 were considered and the mass difference between decuplet and octet intermediate baryon states in the loop integrals was accounted for.
The calculation was performed following the lines of previous analyses in conventional heavy baryon chiral perturbation theory [11][12][13]. First, the validity of the ∆I = 1/2 rule was taken for granted, i.e., the ∆I = 3/2 component of the decay amplitude was neglected. Second, the assumption of octet dominance was also made, i.e., it was assumed that the 8 component dominates the 27 component in the weak Hamiltonian.
The method was simple. A baryon operator, gathering together tree and one-loop corrections, was constructed. The operator had a well-defined 1/N c expansion and correctly picked the octet component of the ∆S = 1 transitions under consideration. The matrix elements of that operator between SU (6) symmetric baryon states yielded the s-wave amplitudes. At tree-level, there were three unknown operator coefficients h i ; the first two went to octet-octet and the third one went to decuplet-decuplet transitions. These coefficients could be directly related to the chiral coefficients h D , h F , and h C . At one-loop order, two kinds of Feynman diagrams were evaluated. One of them was linear in the h i coefficients and the other one depended not only linearly on h i but also quadratically on the operator coefficients introduced in the 1/N c expansion of the baryon axial vector current.
From the theoretical point of view, working out all baryon operators for N c = 3 had several advantages. The most striking one was that it allowed a direct comparison with heavy baryon chiral perturbation theory results term by term. This comparison also revealed that the analysis of both octet-octet and decuplet-decuplet transitions could be described with only two parameters from the weak Hamiltonian, rather than the usual three, so that the third one was related by Eq. (77). Although it was at first an unexpected result, a similar one has already been obtained for the coefficients introduced in the chiral Lagrangian to first order in the quark mass matrix, Eq. (3.61) of Ref. [3].
A fit to data was also performed. The output fairly reproduced the observed s-wave decay amplitudes, due in part to the input coming from the analysis of the baryon axial vector current, which needs further improvement; once this improvement is achieved, a better fit in nonleptonic decays would be possible.
To close this paper, it is known for a fact that theory can lead to a good determination of either sor p-waves, but not both simultaneously. An intriguing question is whether the analysis of p-wave amplitudes, computed under the same footing as swave amplitudes, can yield a stable fit by using, among others, the above-mentioned two parameters from the weak Hamiltonian. This task, however, will be attempted in the near future.

Acknowledgments
This work has been partially supported by Consejo Nacional de Ciencia y Tecnología and Fondo de Apoyo a la Investigación (Universidad Autónoma de San Luis Potosí), Mexico.
Appendix A: Reduction of baryon operators Equation (51) contains n-body operators 3 , with n > N c , which are complicated commutators and/or anticommutators of the one-body operators J k , T c , and G kc . All these complicated operator structures should be reduced and rewritten as linear combinations of the operator bases (61)-(63), with n ≤ N c . The reduction, although lengthy and tedious in view of the considerable amount of group theory involved, is nevertheless doable because the operator bases are complete and independent. All the baryon operator reductions required for N c = 3 are listed here.