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SU(3)-flavour breaking in octet baryon masses and axial couplings

The lightest baryon octet is studied within a covariant and confining Nambu--Jona-Lasinio model. By solving the relativistic Faddeev equations including scalar and axialvector diquarks, we determine the masses and axial charges for \Delta S = 0 transitions. For the latter the degree of violation of SU(3) symmetry arising because of the strange spectator quark(s) is found to be up to 10%.


I. INTRODUCTION
In the quest to fully understand Quantum Chromodynamics (QCD) it is not sufficient to study baryons whose valence quark content consists only of the light u and d quarks. A solid understanding of all members of the baryon octet -that is, the nucleon, Λ, Σ and Ξ multiplets -remains a critical step. Early work on their structure centered on the constituent quark model [1,2] and the MIT bag model [3], later supplemented by chiral corrections associated with the cloud of virtual pions and kaons that surround a baryon [4][5][6][7][8][9][10]. Once their basic properties, such as masses, charge radii, magnetic moments and axial charges had been calculated, attention naturally turned to more complex properties, such as their parton distribution functions [11,12].
The empirical evidence concerning the structure of the hyperons is naturally far more limited than for nucleons. While we have fairly good data for their masses, magnetic moments and axial charges [13], little or nothing is known concerning their electromagnetic or axial form factors as a function of momentum transfer. Finding ways to explore these properties experimentally would be very valuable. On the other hand, over the last couple of decades lattice QCD has made steady progress in the calculation of octet baryon masses [14][15][16] and certain electroweak matrix elements [17]. These studies have been complemented by a judicious use of chiral effective field theory in order to extrapolate to the physical quark masses. Thus, we now have quite accurate determinations of the hyperon electric [18] and magnetic [19] form factors up to 1.4 GeV 2 , as well as low moments of their parton distribution functions [20]. It has even been possible recently to shed some light on the proton spin puzzle [21,22] by calculating the spin fractions carried by quarks across the octet [23].
On general grounds one would prefer to have models of octet baryon properties that are covariant as well as respecting the symmetries of QCD. The former is especially important if one wants to investigate parton distributions and form factors at high momentum transfer. The hope in building more sophisticated models is that through comparison with empirical data and lattice QCD studies one may develop a deeper understanding of how QCD works in the non-perturbative regime, including issues such as the importance and role of diquark correlations and chiral corrections [24], as well as the transition from non-perturbative to perturbative QCD [25].
In this work we investigate the masses and ∆S = 0 axial charges of the octet baryons within the framework of the covariant model of Nambu and Jona-Lasinio (NJL) [26][27][28][29][30], where confinement is simulated by employing propertime regularization [31][32][33]. Octet baryons are described by a Poincaré covariant Faddeev equation, where scalar and axialvector diquarks correlations are assumed to play a dominant role. Flavour breaking effects, introduced by a dressed strange quark that is approximately 50% heavier that the dressed light quarks, will also be studied.
The structure of the paper is as follows. Sect. II provides a brief introduction to the NJL model, including a discussion of the Bethe-Salpeter equation for mesons and diquarks. Sect. III introduces the Faddeev equation for octet baryons, discussing the solution for the Poincaré covariant Faddeev amplitude and octet masses. Finally, in Sect. V, the formalism is used to determine the axial charges associated with strangeness conserving beta decays. Sect. VI summarises our findings and presents some concluding remarks.

II. NAMBU-JONA-LASINIO MODEL
The NJL model was formulated as a theory of elementary fermions which encapsulated dynamical chiral symmetry breaking in a transparent manner [26,27]. With the advent of QCD, it was reformulated with quarks as the fundamental degrees of freedom, such that the symmetries of QCD are respected. 1 In particular the NJL model exhibits dynamical chiral symmetry breaking, which, as implemented in this work, gives rise to approximately 95% of the nucleon mass.
The complete three-flavour NJL Lagrangian in theqq interaction channel -including only 4-fermion interactions -has the form [30] L =ψ i / ∂ −m ψ where λ represents the eight the Gell-Mann matrices andm = diag [m u , m d , m s ]. The NJL model does not include gluons as explicit degrees of freedom, as such the pointlike quark-quark interaction renders the NJL model non-renormalizable. We regularize the NJL model using the proper-time scheme, which mantains Lorentz and gauge invariance, it also removes unphysical thresholds for the decay of colour singlet bound states in their coloured constituents, thereby simulating quark confinement [31][32][33].
The dressed quark propagator in the NJL model is obtained from the gap equation illustrated in Fig. 1. The solution for a quark of flavour q = u, d, s has the form where, in the proper-time regularization scheme, the dressed quark masses each satisfy In this three-flavour NJL model, defined by Eq. (1), the gap equation does not introduce flavour mixing in the quark propagator, this is in contrast to the two-flavour case which in general has flavour mixing [30]. The quark-quark interaction needed for the two-body interaction kernel in the Faddeev equation (to be described shortly) can be obtained from Eq. (1) using Fierz transformations. Keeping only scalar and axialvector diquark correlations, the NJL interaction Lagrangian in the qq channel reads where C = iγ 2 γ 0 is the charge conjugation matrix and the couplings G s and G a give the strength of the scalar and Quark-antiquark and quark-quark bound states are obtained by solving the appropriate Bethe-Salpeter equation, which is illustrated in Fig. 2 for diquarks. The reduced t-matrices for scalar and axial-vector diquarks, with quark flavour content q 1 and q 2 , take the form 2 The bubble diagrams are given by where the flavour and colour traces have been taken, and the remaining trace is over Dirac indices only. The masses of the various diquarks are given by the poles in the corresponding t-matrix, e.g., the scalar diquarks masses are given by the pole condition For the octet baryon calculations we approximate the full diquark t-matrix by a contact + pole form, that is where the pole residues are given by

III. FADDEEV EQUATIONS FOR OCTET BARYONS
Octet baryons are constructed as solutions to a Poincaré covariant Faddeev equation, which is illustrated in Fig. 3, where the quark-diquark approximation used here has been made explicit [37]. A tractable solution to the Faddeev equation is obtained by employing the static approximation [38] to the quark exchange kernel, where the exchanged quark propagator becomes S q (k) → − 1 Mq . This approximation has been shown to yield excellent results for nucleon form factors [24] and quark distributions [39][40][41][42]. The Faddeev equation for each octet baryon then takes the form where B labels an octet baryon, Z B the corresponding quark exchange kernel and Π B (p) is a diagonal matrix containing the various combinations of quark and diquark propagator. Eq. (14) must be supplemented by a normalization condition, such that the normalized Faddeev vertex reads where Γ 0B (p, s) is the unnormalized vertex and the normalization condition that determines Z B will be discussed shortly.
For equal light quark masses the nucleon, Σ and Ξ Faddeev vertex functions contain one scalar diquark and two types of axial-vector diquark 3 , with a Dirac structure p = p Figure 3. Homogeneous Poincaré covariant Faddeev equation whose solution gives the mass and vertex function for each member of the baryon octet.
of the form (17) where b = [nucleon, Σ, Ξ] and Z b is the vertex normalization. The Faddeev vertex function for the Λ baryon, with equal u and d quark masses, contains two types of scalar diquark and an axial-vector diquark and therefore reads The quark exchange kernel for the nucleon, Σ and Ξ reads where, in each case, M 1 is the mass of the singly represented dressed quark and M 2 the mass of the doubly represented dressed quark. 4 The factor of 3 is obtained from projecting the kernel onto colour singlet states. For the Λ the quark exchange kernel is given by where M is the mass of the dressed light quark. The quark-diquark bubble diagram matrix for the nucleon, Σ and Ξ reads where for each baryon q 1 is the singly represented dressed quark and q 2 the doubly represented dressed quark. For the Λ the analogous quantity reads The quark-diquark bubble diagrams are given by Finally, the vertex normalization is given by Note, the value of p 2 which satisfies the Faddeev equation for each octet baryon defines its mass, M 2 B , and at that point the coefficients α i then define the octet baryon vertex function.

IV. RESULTS FOR OCTET BARYONS MASSES
The NJL model employed in this work has the following parameters: two regularization parameters Λ IR and Λ U V ;  Table III. Results for the pole residues in the various meson and diquark t-matrices (c.f. Eqs. (10) and (11) ).
the dressed quark masses M u , M d and M s ; 5 two coupling constants, G π and G ρ , from theqq NJL Lagrangian, given in Eq. (1); and the two coupling constants G s and G a from the qq NJL Lagrangian. The infrared cutoff implements confinement and therefore should be of the order of Λ QCD and we choose Λ IR = 0.240 GeV; for the light-quark dressed masses we choose M u = M d = 0.4 GeV; the ultraviolet cutoff Λ U V and the couplings G π and G ρ are constrained by the empirical values for the pion decay constant, the pion mass and the ρ mass; the qq couplings are chosen to reproduce the physical nucleon mass and the nucleon axial coupling constant. Finally, the dressed s-quark mass is fixed to reproduce the empirical mass of the cascade baryon (Ξ). The resulting parameters are summarized in Tab. I. Results for the kaon mass together with the various diquark masses are given in Tab. II. The splitting between the various scalar diquarks and between the axialvector diquarks is the result of explicit SU (3) F breaking effects from the strange quark. The empirical light to strange current quark mass ratio in the MS regularization scheme is m s /m q = 27.5 ± 1.0 [13], while we find m s /m q = 21.7. For the analogous dressed quark mass ratio we obtain M s /M q 1.5, which illustrates that effects from DCSB are very much suppressed for the heavier strange quark. For completeness we give in Tab. III the pole residues for the meson and diquark t-matrices.
The octet baryon masses obtained by solving the appropriate Faddeev equation, as discussed in Sect. III, are given in Tab. IV. In the SU (3) F limit all octet baryon masses are degenerate and hence the mass splitting between octet baryons results solely from the heavier s quark

V. AXIAL CHARGES
The axial charges of the baryons are important because they connect the strong and weak interactions. They are also related to the quark spin content of the baryons [43]. In fact, assuming SU (3)-flavour symmetry, all octet baryon decays can be parametrized by just three quantities: the Cabbibo angle, θ C , and the F and D couplings [44,45].
The flavour-triplet axial charge of an octet baryon, g B A , is given by the matrix element (28) where s µ =ū(p, λ) γ µ γ 5 u(p, λ) is the spin-vector of the octet baryon. The quark-spin fractions of the baryon are defined by where the u and d quark projection operators are given byP and the plus sign corresponds to the u quark. The various spin-fractions for the octet baryons under consideration are given by the sum of the six Feynman diagrams represented in Fig. 4 and have the structure The nomenclature for these Feynman diagrams is: a superscript Q implies that the operator acts directly on a quark (quark diagram) and a superscript D implies that the operator acts on (a quark inside) a diquark (diquark diagram); the notation q 1 [q 2 q 3 ] indicates a diagram with quark content q 1 q 2 q 3 contains only a scalar diquark of quark content q 2 q 3 . Similarly the notation q 1 {q 2 q 3 } indicates a diagram contains only an axialvector diquark of quark content q 2 q 3 ; and finally the notation q 1 {q 2 q 3 } ↔ q 1 [q 2 q 3 ] indicates the sum of the two diagrams where the operator induces a transition between scalar Figure 5. (Colour online) Feynman diagrams that represent the diquark axial current. The shaded circles are the diquark Bethe-Salpeter vertices and the shaded oval is the γ µ γ5 vertex.
and axialvector diquarks of flavour q 2 q 3 . The numerical coefficients arise from the flavour structure of the operator and the Faddeev amplitude.
In general the quark diagram with a scalar diquark spectator reads (37) and the analogous diagram with an axialvector diquark spectator is given by Similarly, the general form of the diquark diagram with a scalar diquark reads (39) and the analogous diagram with an axialvector diquark is given by where Λ µ [q2q3] and Λ µ αβ,{q2q3} represent, respectively, the interaction of a scalar and axialvector diquark, with an axialvector current, in the q → 0 and on-shell limits. Because the scalar diquark has spin-zero, we have Λ µ [q2q3] = 0, while for the axialvector diquark we have where α is the inital and β the final diquark polarization. The Feynman diagram for this contribution is illustrated in Fig. 5. For the various axialvector diquarks we find: g = −0.213 and g {ss} A = −0.194 6 . Note that in evaluating Eq. (41) we have used the on-shell condition, ε α (p) p α = 0, for both the inital and final axialvector diquark.
The final Feynman diagram represents the mixing between the scalar and axialvector diquarks induced by the axial current, this diagram reads where each contribution is given by 7 The diquark transition vertices are given by These vertices have the general form Evaluating these diagrams gives the spin-fractions which we summarize in Tab. VI. In addition to these body form factor contributions, the axial charge of the quark receives a finite renormalization. This renormalization is given by the inhomogeneous BSE illustrated in Fig. 6. The renormalized axial charge of the light quark is given by where Π (T ) The coupling G a1 is adjusted (G a1 = 1.0) to give M a1 = 1.26 GeV. The unrenormalized quark axial charge is unity, however for the renormalized axial charge we find g q A = 0.935. The value of the axial charge for octet baryon strangeness conserving transitions, for the bare case ("Bare") and for the case with a renormalized axial quark vertex ("BSE") are given in Tab. VII.
The axial charges for the octet baryons also receive corrections from the meson cloud and here we simply use the corrections determined using the Cloudy Bag Model (CBM) [46]. In practice this means that our results should be multiplied by 0.92 for g n A , 0.90 for g Σ A and 0.95 for g Ξ A . Our final results, including corrections from the quark vertex renormalization and the meson cloud are shown in Tab. VII. The only experimental value we have available is g n A . The values of the vector form factors f 1 (0) are 1, √ 2 and 1 for the nucleon, Σ, and Ξ, respectively. This comes from charge conservation when the electromagnetic form factors are computed in the octet.
We are now in a position to determine the size of SU(3)-flavour breaking effects for the axial charges of the octet baryons, using the SU(3)-flavour parametrization of Ref. [44], namely Within our model the values of F and D may be computed by choosing any pair of the previous relations. We call F Σ(Ξ) and D Σ(Ξ) the parameters calculated from (g A /f 1 ) Σ(Ξ) and (g A /f 1 ) n . From Tab. VII we obtain and The discrepancies suggest SU(3)-flavour symmetry breaking effects of around 10%, with F Σ /F Ξ = 0.89 and D Σ /D Ξ = 1.07. Since the strangeness conserving βdecays for the Σ − and Ξ − have not yet been measured, this result should be viewed as a prediction to be tested experimentally. We note that even larger SU(3) violation has been reported in the context of the proton spin problem [47]. In addition, a comparison of our results with the cloudy bag model computations performed in Ref. [48] shows that g Σ A is the same in both models, whereas g Ξ A is slightly smaller in magnitude in our work. The calculation of Ref. [48] includes one-gluon exchange and center of mass corrections, plus recoil effects and a rescaling factor to reproduce the experimental g n A . The other source of "data" with which our model might be compared are lattice QCD calculations. There has recently been good progress in the calculation of the electromagnetic form factors for the octet baryons [18,19], using chiral extrapolations of the lattice results. Clearly an extension of that work to weak form factors would provide a valuable test our model predictions.

VI. CONCLUSIONS
We have computed the masses and the ∆S = 0 axial charges of the baryon octet using a confining NJL model. The model results for the masses are in good agreement with the experimental values. While there are currently no measurements of the ∆S = 0 axial charges, other than for the neutron, we did find very close agreement between our results and those found within the cloudy bag model. Since there is currently considerable discussion concerning the degree of violation of SU(3) symmetry, the deviation of order 10% which we found is significant.
It will be important to extend the present investigation to calculate the chiral corrections explicitly within this model. Given the new lattice results for octet baryon electromagnetic form factors, we look forward to simulations of a similar quality for the axial form factors. Meantime, it would be very interesting to extend the present model to calculate the hyperon electromagnetic form factors.