Singlet axial-vector coupling constant of the nucleon in QCD without instantons

We have analyzed axial-vector current-current correlation functions between one-nucleon states to calculate the singlet axial-vector coupling constant of the nucleon. The octet-octet and the octet-singlet current correlators, investigated in this work, do not require any use of instanton effects. The QCD and hadronic parameters used for the evaluation of correlators have been varied by (10 - 20)%. The value of the singlet axial-vector coupling constant of the nucleon obtained from this analysis is consistent with its current determination from experiments and QCD theory.

. Among the three flavor-diagonal coupling constants , the isovector coupling constant is the best understood and is measured from nuclear -decay. The eighth component is determined from the analysis of hyperon -decay in SU(3)f symmetry limit .
Indeed, in terms of SU(3)f parameters F and D, these two axial coupling constants are expressed as ! ! ,! ! and determined to be as [8,9] However, SU(3)f symmetry may be badly broken and an error in from 10% [10] to 20% [11] has been suggested. There is no direct way to measure . Theoretically, its calculation is challenging on account of its association with chiral anomaly. The first moment of spindependent structure function g1 of the nucleon can be related to the scale-invariant axial-vector coupling constants of the target nucleon. The experimental value of is obtained from measurement of g1 and combining its first moment integral with the measured values of and and theoretical calculation of the perturbative QCD Wilson coefficients.
Using SU(3)f symmetric value for and with no leading twist subtraction in the dispersion relation for polarized photon-nucleon scattering, COMPASS found [12] ! ! Several approaches have been used to calculate axial-vector coupling constants of the nucleon. Instantons, through axial anomaly relation, is believed to have an important role in the singlet axial-vector coupling constant of the nucleon [13]. Using numerical simulations of instanton liquid, Schaffer and Zetocha [14 ] have calculated axial-vector coupling constants of the nucleon. Though, they get a good result for , for the singlet case they get .
Using lattice QCD, Yang et al. have estimated the part of the proton spin carried by light quarks from anomalous Ward identities as =0.30(6) [15]. It hints to suggest that the culprit of the 'proton spin crisis' is the U(1)A anomaly. Chiral constituent quark model also gives a good result for and , but for the singlet case, it gives [16] . In a hybrid approach, where one takes into account one gluon exchange as well as effect of meson cloud, it has been possible to get a reasonably good result such as = 0.42 [17]. Similar result for quark spin contribution to the spin of the nucleon has been obtained using a spin-flavor based parametrization of QCD [18]. Three different approaches have been followed in QCD sum rule to calculate axial coupling constant of the nucleon. Ioffe and Oganesian [19] have used the standard QCD sum rule in external fields. Two-point correlation function of nucleon interpolating fields has been evaluated in the presence of a weak axial vector field. The limits on , the part of proton spin carried by light quarks, and , the derivative of the QCD susceptibility have been found from self-consistency of the sum rule. Belitsky and Teryaev [20] considered a threepoint function of nucleon interpolating fields and the divergence of singlet axial-vector current .
The form factor is related to vacuum condensates of quark-gluon composite operators through a double dispersion relation. In this approach, the extrapolation to involves large uncertainties. In the third approach by Nishikawa et al. [21,22], a two-point correlation function of axial-vector currents in one-nucleon state is evaluated. Here, the axial-vector coupling constants of the nucleon are expressed in terms of -N and K-N sigma terms and moments of parton distributions. The perturbative contribution is subtracted from the beginning and the continuum contribution can be reduced to a small value. The application of this method using singlet-singlet axial-vector current correlator for requires taking into account the chiral anomaly [21]. This gives appreciably high value of . The result was improved by the inclusion of instantons in the QCD evaluation of correlation function [22]. However, the result was extremely sensitive to critical instanton size and was not stabilized. Our own experience of working with singlet-singlet axial-vector current correlator, albeit in vacuum state [23,24 ], is that the sum rule does not work satisfactorily even on inclusion of instanton contribution. On the other hand, octet-octet and octet-singlet axial-vector current correlators work well. Instanton contribution is not needed in these last two sum rules. In view of this, in this work we will investigate octet-octet and octet-singlet axial-vector current correlators in one-nucleon states.
The results of the two sum rules can be combined to get and . The numerical evaluation of the sum rules requires use of several QCD and hadronic parameters. We have also studied consequences of variation of these parameters on sum rules.!

The sum rules!
Following [21,22 ], we consider the correlation functions of axial-vector currents in one-nucleon Actually, has two kinds of contributions: the connected and the disconnected terms.
Unlike the case of singlet-singlet correlator, the disconnected terms do not contribute to octetoctet as well as to octet-singlet correlators. Hence, the instanton contribution is not needed in our calculation. Eq.(1) can be written using Lehmann representation as !
where ! is the spectral function. We take Borel transform of even part in of both sides where . The nucleon matrix element of axial-vector current is given as!
Calling ! and realizing that has no singularity at We have calculated correlation functions using operator product expansion (OPE) by accounting for operators up to dimension 6. Our results for ! has some differences from those obtained in Ref. [21].
In above equations we have not taken into account continuum contribution about which we will comment later. It may be pointed out that the last terms in Eqs. (9a,b) arising from has not been considered earlier [21,22]. Our expression on the rhs of Eq.(9a) differs from the corresponding expression in Ref. [21 ] in other significant ways: the second and third terms differ in sign whereas fourth and fifth terms have somewhat different numerical coefficients.

Results and Discussion!
analyses. In addition to this, we also chose ! ! ! [28] , and ! from MSTW 2008 [25]. We observed that the sum rules were giving unphysical results for and for is obtained from experimental and phenomenological analyses. We believe this as a sign of robustness of our sum rules. In Table 1 we have listed some of those results for which !
, as a function of s, has minimum slopes in our designated interval s= (1.7-2.5) GeV 2 . Plots of some of these results are displayed in Figs. (1-6). We observe from the Table 1 that! ! was stuck to the lower end of the range of its variation and ! was confined to (300-325) MeV. The best results were obtained for y being in the range (0.16 -0.18).! As in any QCD sum rule calculation, our results have errors due to omission of contributions of higher dimensional operators and continuum contributions. From MSTW 2008 parametriz- The ratio of contributions of six-dimensional operators to that of four-dimensional operators is ~1/2 at s=2.5 GeV 2 , but the ratio of contribution of eight-dimensional operators to! that of fourdimensional operator is likely to be few percent, though their contribution to ! will get doubled on account of sign difference in the contributions of four-dimensional and sixdimensional operators. The continuum contribution comes from and states. A rough estimate shows that their contribution will be less than 1%. Thus we allow the error due to exclusion of contributions of higher dimensional operators and continuum contributions to be roughly 10%. Based on results given in Table 1 and the error estimates, we conclude !
where the first error is due to finite slope within the designated range of Borel mass parameter and the second one is due to omission of contributions of higher dimensional operators and continuum contribution. ! By choosing the correlator of singlet and octet axial-vector currents, we ensured that the disconnected diagrams do not contribute directly for determination of . However, the nonvalence components in the nucleon, such as strange quark-antiquarks and gluons, have an important role: they are directly responsible for the splitting of and . In QCD parton model, the axial coupling constants of a nucleon are related to polarized quark densities. Our results for !
We accept ! obtained in the entire designated interval of Borel mass parameter as our final result of sum rules. We also note an interesting point that the sign of spin-dependent parton density is decided by spin-independent quantities such as second moment of spinindependent parton distribution function of s-type and s-quark content of the nucleon. In conclusion, the present method of QCD sum rule, where correlation function of two axial-vector currents between one-nucleon states is studied, is capable of producing a result for singlet