Two sum rules involving leading vector form factors in hyperon semileptonic decays

When the weak and electromagnetic currents are considered to be in the same SU(3) octet, it is possible to derive two sum rules involving leading vector form factors in hyperon semileptonic decays in the limit of exact flavor SU(3) symmetry. According to the Ademollo-Gatto theorem, deviations of these sum rules from this limit are expected to occur due to second-order flavor SU(3) symmetry-breaking effects, which are evaluated within the 1/Nc expansion of QCD. Accordingly, one sum rule acquires no deviations whereas the other one obtains contributions coming from the 10+10¯ representation. The sum rules are validated using results obtained in heavy baryon chiral perturbation theory. Up to order O(p2) in the chiral expansion, the sum rules are fulfilled.


Introduction
Despite the enormous progress achieved in the understanding of the fundamental interactions with the Standard Model, first principles calculations of the properties of hadrons are not possible because the theory is strongly coupled at low energies. In order to resort this situation, several methods have been introduced to understand the low-energy QCD hadron dynamics and the 1/N c expansion [1] has played a crucial role. This method promotes QCD to a SU (N c ) non-Abelian gauge theory, where N c is the number of colors.
In this tenor, hyperon semileptonic decays (HSD) offer some challenges until today. In a recent paper [2], two sum rules involving the leading vector form factors f 1 were derived by considering that the weak currents and the electromagnetic current are members of the same SU (3) octet. According to the Ademollo-Gatto theorem [3], the leading vector form factors f 1 are protected against SU (3) symmetry breaking (SB) corrections to lowest order in m s −m, wherem denotes the mean mass of the up and down quarks. The purpose of the present paper is to provide some generalities about the method used to obtain those sum rules and to discuss some important implications.
The organization of this paper is as follows. In section II, the two sum rules involving leading vector form factors are derived by taking the matrix element of the vector current between eigenstates of the V 2 operator. The resulting expressions are valid in the exact flavor SU (3) symmetry limit. Subsequently, the two sum rules are modified to account for SB. In section III, a brief review about how Ademollo and Gatto came up with their result on the nonrenormalization of the electromagnetic and vector currents at first-order in SB is provided, followed by a survey on the 1/N c expansion in section IV. In section V, the expressions containing the operators that explicitly break symmetry at first-and second-order are introduced. Restrictions on the nonrenormalization of the electric charge serve as an important factor to understand which operator coefficients (since they are the ones responsible for SB effects) will be present in the expressions for f 1 . Collecting all partial findings, the operators responsible for SB in the sum rules are properly identified. Next, results obtained in the frame of chiral perturbation theory are used to validate the sum rules. Up to O(p 2 ) in the chiral expansion, the sum rules are satisfied. A validation beyond this chiral order is not currently possible because there are some partial results available. Finally, a numerical evaluation of the sum rules is also performed through a least-squares fit to data. Results are encouraging. Some closing remarks are given in section VI.

Sum rules for baryon vector form factors
The Cabibbo model for weak hadronic currents based on SU (3) symmetry has been a key approach to describe HSD [4]. Since flavor SU (3) symmetry is broken, its exact symmetry limit has been scrutinized using several methods in order to find discrepancies between theory and experiment and the leading weak form factors f 1 in HSD are the usual probes.
The starting point of the present analysis is the observation made by Pham about the matrix elements of the V = 1 V -spin multiplet [5]. He showed that these matrix elements can be related to each other in the limit of exact flavor SU (3) symmetry. From the two I-spin relations, and the rotated V -spin versions of the above relations become and where the bilinear forms q 1 Γ n q 2 are given in terms of quark fields q i and the matrices Γ n αβ are the usual ones that appear in applications of Dirac theory.
Decomposing the SU (3) baryon octet into eigenfunctions of V 2 yields Relations (2) are general enough to apply to matrix elements of any SU (3) octet ΔS = 1 operator. The Gell-Mann-Okubo mass formula is obtained from (2a) by relating B 2 | uIs |B 1 to f stands for the leading vector form factor at zero recoil in the limit of exact flavor SU (3) symmetry and M B i is the mass of the B i baryon. Applications to the axial-vector to vector form factor ratios g 1 /f 1 are obtained when Γ n = γ 5 γ μ [5].
The vector current can be straightforwardly analyzed for Γ n = γ μ . Consequently, there are two nontrivial expressions that relate the matrix elements of the vector current between baryon octet states, namely, Hence, in the limit of exact flavor SU (3) symmetry and neglecting isospin breaking, two relations between the leading vector form factors in HSD arise as a consequence of (7a) and (7b), namely, and When SB is present, relations (8) must be modified. After minor rearrangements, sum rules (8) thus read and 3 4 A few remarks are in order here. The coefficients on the right-hand-side of Eqs. (9) have been chosen to keep some resemblance to their counterpart in K l3 decays [6], namely, where will be referred to as a measure of SB hereafter. Similarly, δ SB k , which arise from SB, are formally of order O( 2 ) according to the Ademollo-Gatto theorem. Obtaining δ SB k requires and extra effort and it will be performed within the 1/N c expansion.

Some generalities about the Ademollo-Gatto theorem
Following the assumption that the vector and electromagnetic currents are members of the same unitary octet and that the breaking of the unitary symmetry behaves as the eighth component of an octet, M. Ademollo and R. Gatto set up an important theorem on the nonrenormalization of the strangeness-violating vector currents [3].
In the Ademollo and Gatto formalism, the ath component of the current J a to first-order in SB is written as where B represents the baryon matrix, λ a stand for the Gell-Mann matrices and a 0 , b 0 , . . . , h are coupling constants. The parameter is a reminder of the order SB is accounted for. At the end can be set to one. The electromagnetic current is defined in the usual way as so the baryon charges can be obtained from (12) without any difficulties. For instance, the neutron charge, including first-order SB is Similar relations can be obtained for the remaining baryons. Therefore, there are eight equations (one for each baryon charge) and seven parameters to be found. However, the isospin relation leads to seven linear independent equations. Solving the system for the seven unknowns yields The solution of the system shows that the electromagnetic charge acquires no corrections to first-order SB (as it is expected since the electric charge remains unrenormalized to all orders in perturbation theory). As a direct consequence, the vector current is also protected against first-order SB. This is the original conclusion reached by Ademollo and Gatto [3].

Generalities on the 1/N c expansion
The formalism on the 1/N c expansion for baryons can be found in Ref. [8], so a few salient facts will be repeated. In the large-N c limit, the baryon sector has a contracted SU (2N f ) spin-flavor symmetry, where N f is the number of light quark flavors [7]. The 1/N c expansion is given in terms of 1/N c -suppressed operators with well-defined spin-flavor transformation properties. For N f = 3, the lowest-lying baryon states fall into a representation of the spin-flavor group SU (6).
The 1/N c expansion of any QCD operator transforming according to a given SU (2)⊗SU (N f ) spin-flavor representation is expressed in terms of n-body operators O n as [8] where the operator coefficients c (n) are undetermined and have power series expansions in 1/N c beginning at order one. The operators O n are polynomials in the spin-flavor generators J k , T c , and G kc , which are 1-body operators acting on the baryon states. They read Operators q † α and q α create and annihilate states in the fundamental representation of SU (6) and σ k and λ c are the Pauli spin and Gell-Mann flavor matrices, respectively.

The leading baryon vector form factor in the 1/N c expansion
In this section a brief description of the baryon vector current will be provided. The operator is constructed [2] in such a way that its matrix elements give the actual values of the leading vector form factor as is defined in HSD.
To start with, let V 0c denote the flavor octet baryon charge [9] where the subscript QCD indicates that the quark fields are QCD quark fields [9]. at zero recoil can be easily obtained by computing the appropriate matrix elements between baryon octet states for c = 4 ± i5.
In QCD, flavor SB is due to the strange quark mass m s and transforms as the eight component of an octet [8].
The expansion (19) has been truncated at the physical value N c = 3, so up to three-body operators are retained. Also, singlet and octet representations are explicitly subtracted off from the two-and three-body operators of the (0, 27) representation to keep only the truly (0, 27) contributions. Finally, operator coefficients c rep (n) accompany operators that belong to the exact flavor SU (3) limit whereas a rep (n) accompany n-body operators of the representation rep that explicitly breaks flavor symmetry. Higher-order operators are constructed by anticommuting J 2 with the already existing ones as O n+2 = J 2 , O n . These contributions are not included since they can be accounted for by redefining the operator coefficients.
The matrix elements of the operator V Q + δV Q between SU (6) symmetric baryon states give the actual values for the baryon charges Q B including first-order SB. For example, at the physical values N c = N f = 3, the proton charge reads Similar expressions are found for the other members of the baryon octet [2]. When analyzing the expressions for the full baryon charges, it is straightforward to notice that the coefficients of the (0, 27) representation are not independent and a new coefficient involving a 27 (2) and a 27 (3) can be introduced, namely, Since the new coefficient x 27 (2) appears in the eight expressions for the baryon charges this reduces the number of unknowns in the linear system by one. Solving the system of linear equations yields The above nicely reproduces the Ademollo and Gatto results using the 1/N c expansion formalism. Operator coefficients of (19) are related to the coupling constants introduced in (11) by Notice a matching between the various coefficients and the representations they belong to. The next step is to incorporate flavor SB at second-order into the 1/N c expansion for the The matrix elements of 2 δV Q between SU (6) symmetric baryon octet states give secondorder corrections to the baryon octet charges. For the proton the contribution reads Similar expressions for the remaining baryon octet charges are found [2]. Again, the operator coefficients of the (0, 27) representation are not independent and a new coefficient that includes b 27 (2) and b 27 (3) can be defined as y 27 Only eight out of twelve operator coefficients that appear in (24) make up the octet baryon charges. Again, using the argument that the electric charge remains unrenormalized to all orders in perturbation theory, it is possible to solve the resulting linear system for the eight unknowns, . (27d) Hence, following the above results, the vector current at second-order in flavor SB can be written in terms of eight operators and five operator coefficients. The matrix elements of V 4±i5 + 2 δV 4±i5 between SU (6) baryon states yields the actual expressions for the leading vector form factors at second-order in flavor SB. For the observed ΔS = 1 transitions, one has: where theẽ (n) coefficients are written in terms of those coefficients that come along with n-body octet operators. Substituting the obtained expressions for the leading vector form factors (28)-(31) in (9a) and (9b) yields