# Isovector axial-vector form factors of octet baryons in QCD

###### Abstract

We compute the diagonal isovector axial-vector as well as induced pseudoscalar form factors of nucleon, and baryons by employing the light-cone QCD sum rules to leading order in QCD and including distribution amplitudes up to twist 6. Extrapolating our sum-rules results to low-momentum transfers, we make a comparison with experimental and lattice-QCD results where we can achieve a nice qualitative and quantitative agreement.

###### pacs:

13.75.-n, 14.20.-c, 12.38.-t## I Introduction

Form factors are important in hadron physics as they provide information about the structure, in particular the shape and the size, of the hadron. The baryon matrix elements of the axial-vector current are parameterized in terms of the axial () and the induced pseudoscalar () form factors as follows:

(1) |

where is the isovector axial-vector current, is the momentum transfer and is the baryon mass. Among all, the nucleon form factors have received much attention. The nucleon axial charge, which corresponds to the value of the form factor at zero-momentum transfer (), can be precisely determined from nuclear -decay (the modern value is Nakamura et al. (2010)). The dependence of the axial-vector form factor of the nucleon has been studied up to 1 GeV from antineutrino scattering Ahrens et al. (1988) and for GeV from pion electro-production on the proton Choi et al. (1993). In the high- region ( GeV), we have very small amount of relatively old data Baker et al. (1981). Our information about hyperon axial-vector form factors from experiment is also limited. However, both the low- ( GeV) and the high- ( GeV) regions will be accessible by higher-energy experiments such as Minera at Fermilab, which will give a complete understanding of form factors in a wide range of Drakoulakos et al. (2004). In these experiments, strangeness-production processes will be able to probe the hyperon form factors with precision. On the theoretical side, there exist some estimates for the axial charges of the hyperons from chiral perturbation theory (PT) Savage and Walden (1997); Jiang and Tiburzi (2008, 2009), large N limit Flores-Mendieta et al. (1998) of QCD and QCD sum rules (QCDSR) Chiu et al. (1985).

As for the induced pseudoscalar form factor, a recent result from muon-capture experiment predicts Andreev et al. (2007), where is the muon mass. There exist theoretical results from heavy-baryon PT as Bernard et al. (2002) in consistency with experiment. The prediction from manifestly-invariant PT is Schindler et al. (2007), where the first and the second errors are due to empirical quantities and truncation in the chiral expansion, respectively.

Concurrently, the lattice calculations provide a first-principles description of hadronic phenomena, which also serve as a valuable tool to determine the hadron couplings and form factors in a model-independent way. While systematic errors such as the finite lattice size and relatively heavy quark masses still exist, the developing technology of the lattice method shows promising advances in removing sources of these errors. Lattice QCD calculations of the axial charge and form factors of the nucleon have reached a mature level Edwards et al. (2006); Khan et al. (2006); Hagler et al. (2008); Alexandrou et al. (2007); Yamazaki et al. (2008). While it is difficult to measure hyperon properties experimentally due to their short lifetimes, the method of lattice QCD makes it possible to extract such information. Namely, there have been recent attempts to extract the hyperon axial charges and meson couplings using lattice QCD Lin and Orginos (2009); Sasaki and Yamazaki (2009); Erkol et al. (2009, 2010). Simulations with more realistic setups with smaller lattice spacing and larger lattice size employing much lighter quarks and a dynamical -quark are under way, which will also provide valuable information about hyperon form factors at high momentum transfers.

A complementary approach to lattice QCD is the method of QCD sum rules, which is a powerful tool to extract qualitative and quantitative information about hadron properties Shifman et al. (1979a, b); Reinders et al. (1985); Ioffe and Smilga (1984). In this approach, one starts with a correlation function that is constructed in terms of the interpolating fields, which are chosen with respect to the quantum numbers of the hadron in question. In the traditional method one proceeds with the calculation of the correlation function using the Operator Product Expansion (OPE), which is formulated with Wilson coefficients and local operators in terms of the nonperturbative structure of the QCD vacuum, in the deep Euclidian region. This correlation function is matched with an Ansatz that is introduced in terms of hadronic degrees of freedom on the phenomenological side. The matching provides a determination of hadronic parameters like baryon masses, magnetic moments, coupling constants of hadrons, and so on.

One alternative to the traditional method as far as the hadron interactions at moderately large momentum transfers are concerned is the light-cone sum rules (LCSR) Braun and Filyanov (1989); Balitsky et al. (1989); Chernyak and Zhitnitsky (1990). In this technique, the light-cone kinematics at governs the asymptotic behavior of the correlation function. The singularity of the Wilson coefficients is determined by the twist of the corresponding operator. Then using the moments of the baryon distribution amplitudes (DAs), one can calculate the relevant hadron matrix elements.

LCSR have proved to be rather successful in extracting the values of the hadron form factors at high-momentum transfers. In Ref. Braun et al. (2006), the electromagnetic and the axial form factors of the nucleon have been calculated to leading order and with higher-twist corrections. It has been found that a light-cone formulation of the nucleon DAs gives a description of the experimental data rather well. This calculation has been generalized to isoscalar and induced pseudoscalar axial-vector form factors of the nucleon in Refs. Aliev and Savci (2007); Wang et al. (2006).

Our information about the DAs of the octet hyperons were scarce and as a result not much effort has been spent on these baryons. However, the DAs of octet hyperons have recently become available and their electromagnetic form factors have been calculated by Liu *et al.* Liu and Huang (2009a, b). Motivated by these advances in formulating the SU(3) sector in LCSR and ongoing simulations in lattice QCD to give a first-principles description of hadron interactions, in this work we study the axial-vector form factors of strange octet baryons using LCSR. Note that the axial-vector current is anomalous in QCD. Although this anomaly cancels in the isovector channel, it might have a significant contribution in the isoscalar channel. Since a study of the isoscalar axial-vector form factor would be unreliable without the inclusion of the anomaly effects, in this work we restrict our attention to the isovector form factors. To this end, we compute the diagonal isovector as well as the induced pseudoscalar form factors of nucleon, and baryons by employing their recently extracted DAs. Our paper is organized as follows: In the following section, we give the formulation of the baryon form factors on the light cone and derive our sum rules. In Section 3, we present our numerical results and in the last section, we conclude our work with a discussion on our results.

## Ii Formulation of baryon axial form factors

In the LCSR method one starts with the following two-point correlation function:

(2) |

where are the baryon interpolating fields for the , , . There are several local operators with the quantum numbers of spin-1/2 baryons one can choose from. Here we work with the general form of the interpolating fields parameterized as follows for the , and :

(3) |

with , and , which is an arbitrary parameter that fixes the mixing of two local operators. We would like to note that when the choice is made the interpolating fields above give what are known as Ioffe currents for baryons. Here , and denote the -, - and - quark fields, respectively, , , are the color indices and denotes charge conjugation.

The short-distance physics corresponding to high momenta and is calculated in terms of quark and gluon degrees of freedom. Inserting the interpolating fields in Eq. (3) into the correlation function in Eq. (2), we obtain

(4) |

where denote the quark fields and are constants which will be determined according to the baryon in question. represents the light-quark propagator

(5) |

Here the first term gives the hard-quark propagator. The second term represents the contributions from the nonperturbative structure of the QCD vacuum, namely, the quark and quark-gluon condensates. These contributions are removed by Borel transformations as will be explained below. We note that the hard-quark propagator receives corrections in the background gluon field, which are expected to give negligible contributions as they are related to four- and five-particle baryon distribution amplitudes Diehl et al. (1999). Following the common practice, in this work we shall not take into account such contributions, which leaves us with only the first term in Eq. (5) to consider.

The matrix elements of the local three-quark operator

( are real numbers denoting the coordinates of the valence quarks) can be expanded in terms of DAs using the Lorentz covariance, the spin and the parity of the baryon. Based on a conformal expansion using the approximate conformal invariance of the QCD Lagrangian up to 1-loop order, the DAs are then decomposed into local nonperturbative parameters, which can be estimated using QCD sum rules or fitted so as to reproduce experimental data. We refer the reader to Refs. Braun et al. (2006); Liu and Huang (2009a, b) for a detailed analysis on DAs of , , , which we employ in our work to extract the axial-vector form factors.

The long-distance side of the correlation function is obtained using the analyticity of the correlation function, which allows us to write the correlation function in terms of a dispersion relation of the form

The ground-state hadron contribution is singled out by utilizing the zero-width approximation

and by expressing the correlation function as a sharp resonance plus continuum which starts above the continuum threshold, , i.e. for . The matrix element of the interpolating current between the vacuum and baryon state is defined as

where is the baryon overlap amplitude and is the baryon spinor.

The QCD sum rules are obtained by matching the short-distance calculation of the correlation function with the long-distance calculation. Using the most general decomposition of the matrix element (see Eq. (2.3) in Ref. Braun et al. (2000)) and taking the Fourier transformations we obtain

(6) |

for the axial-vector form factors at structure and

(7) |

for the induced pseudoscalar form factor at the structure . The explicit form of the functions that appear in the above sum rules are given in terms of DAs as follows:

We make the following replacements in order to obtain the sum rule for each baryon we consider:

Note that, in the final sum rules expression, the quarks do not appear explicitly but only implicitly through the DA’s, masses and the residues of the corresponding baryons. Thus these replacements simply instruct to use the DA’s, mass and residue of the corresponding baryon. They apply to both axial-vector and induced pseudoscalar form factors.

The Borel transformation is performed to eliminate the subtraction terms in the spectral representation of the correlation function. As a result of Borel transformation, contributions from excited and continuum states are also exponentially suppressed. The contributions of the higher states and the continuum are modeled using the quark-hadron duality and subtracted. Both of the Borel transformation and the subtraction of the higher states are carried out using the following substitution rules (see e.g. Braun et al. (2006)):

(8) |

where

is the Borel mass and is the solution of the quadratic equation for :

where is the continuum threshold.

Finally, we obtain the following sum rules for the axial-vector and induced pseudoscalar form factors, respectively:

(9) |

(10) |

To obtain a numerical prediction for the form factors, the residues, are also required. The residues can be obtained from the mass sum rules, and the residue of the is given by Aliev et al. (2010):

(11) | |||||

where , and

The residues for the nucleon and can be obtained from Eq. (11). can be obtained by setting and , and by the exchanges and . We use the following parameter values: GeV, GeV, , GeV, GeV, GeV, GeV and GeV.

## Iii Numerical Results and Discussion

In this section we give our numerical results for the axial-vector form factors of , and . For this purpose we need the numerical values of the baryon DAs. The DAs of the nucleon are given in Ref. Braun et al. (2006) as expressed in terms of some nonperturbative parameters which are calculated using QCDSR or phenomenological models (see also Ref. Lenz et al. (2009) for a comparison of nucleon DAs as determined on the lattice Gockeler et al. (2008) and with other approaches). In this work, we give our results using the parameter set known as Chernyak-Zhitnitsky-like model of the DAs (see Ref. Braun et al. (2006) for details). As for the DAs of and we use the parameter values as calculated recently by Liu *et al.* Liu and Huang (2009a, b). In Table 1 we list the values of the input parameters entering the DAs of each baryon.

Parameter | |||
---|---|---|---|

(GeV) | 0.005 | 0.0094 | 0.0099 |

(GeV) | -0.027 | -0.025 | -0.028 |

(GeV) | 0.054 | 0.044 | 0.052 |

0.23 | 0.38 | 0.40 | 0.22 | 0.07 |

The sum rules include several parameters that need to be determined. The continuum threshold value for the nucleon is pretty much fixed at 2.25 GeV in the literature also from a mass analysis. We choose the values 2.5 and 2.7 GeV, respectively for and . In order to see the dependence of the form factors on the continuum threshold, we vary the values of within a 10 region, which leads to a change of less than 10 in the final results.

The form factors should be independent of the Borel parameter . We consider the regions 1 GeV 2 GeV for the nucleon and 2 GeV 4 GeV for and . We observe that the sum rules are almost independent of in this region; a variation in this region leads to change of the order of 1 in the final results. Hence we give our numerical results at 2 GeV for nucleon and at 3 GeV for and .

The next task is to determine the optimal mixing parameter . In the ideal case, the sum rules and hadron properties are independent of this parameter. In order to see if we can achieve such an independence, in Fig. 1 we plot the form factors as a function of , where we make a reparameterization using . We explicitly mark the point for Ioffe current, which corresponds to a choice . It is observed that a stability region with respect to a change in the mixing parameter can be found around 0. In further analysis, we concentrate on this stable region and compare the results with those obtained using Ioffe current.

In Fig. 2, we plot the of , and as a function of in the region 1 GeV ^{1}^{1}1The predictions of LCSR are not reliable at , but are reliable for larger then a few GeV., for the Ioffe current () and for the stable region of mixing parameter (). The qualitative behavior of the form factors agree with our expectations: The values of the axial-vector couplings fall off quickly as we increase the momentum transfer. While there is a considerable discrepancy between the Ioffe and the stable regions for nucleon form factors at low momentum transfers, the results for the form factors are very close to each other in the case of and . Particularly for form factor the two regions produce practically the same results.

For comparison, we also give the lattice-QCD results for , namely axial charges of the , and Erkol et al. (2010). It was found in Ref. Erkol et al. (2010) that the axial charges have rather weak quark-mass dependence and the breaking in SU(3)-flavor symmetry is small. Furthermore, the QCDSR results are not yet precise enough to resolve the small variation of axial charges as a function of quark mass in available lattice-QCD data. Therefore we show the values from SU(3)-flavor symmetric point only. We also note that regarding the signs of the form factors we adopt the convention used in Ref. Erkol et al. (2010).

is usually parameterized in terms of a dipole form

(12) |

A global average of the nucleon axial mass as determined from neutrino scattering by Budd et al. Budd et al. (2003), GeV, is in good agreement with the theoretically corrected value from pion electroproduction as GeV Bernard et al. (2002). A different prediction is made by the K2K Collaboration from quasielastic scattering as GeV Gran et al. (2006). To extrapolate the sum-rules results to low-momentum–transfer region, we have first tried a two-parameter fit to the dipole form. However this procedure fails to give a good description of data. Instead we fix to the experimental value and make one parameter fit from 2 GeV region. Inserting the experimental value for nucleon and fitting to the dipole form in Eq. (12), our sum rules in the stable region of produce GeV, a value larger than the experimental result. We make a similar analysis for and axial-vector form factors using the lattice-QCD values for and in the dipole form and find GeV and GeV. Our results show that axial masses of and are slightly larger than that of nucleon. Note that, in the VMD model, the pole of the form factors is given by the mass of the (axial) vector meson that couples to the current. The lightest axial vector meson has a mass of GeV Nakamura et al. (2010); hence our results also are larger from the predictions of the VMD model.

We have also tried to fit to an exponential form, viz.,

(13) |

which allows a plausible description of data with a two-parameter fit. In this case we have tried three fit regions, namely, GeV, GeV and GeV. Our results are shown in Fig. 2 and summarized in Table 2. The fits from around GeV region produce the empirical values of quite successfully in the case of and , while we obtain somewhat higher values of than that from lattice QCD for all fit regions. We also observe that the axial masses are very close to each other, which indicates a possibly small SU(3)-flavor symmetry breaking in consistency with lattice-QCD findings Erkol et al. (2010). It will be interesting to compare our sum-rules results to those from lattice QCD with more realistic setups when available in the near future.

Baryon | Fit Region (GeV) | (GeV) | |
---|---|---|---|

[1.0-10] | 1.68 | 1.20 | |

[1.5-10] | 1.24 | 1.33 | |

[2.0-10] | 0.97 | 1.42 | |

[1.0-10] | 1.11 | 1.32 | |

[1.5-10] | 0.92 | 1.40 | |

[2.0-10] | 0.77 | 1.48 | |

[1.0-10] | 0.46 | 1.25 | |

[1.5-10] | 0.41 | 1.29 | |

[2.0-10] | 0.35 | 1.35 |

In Fig. 3, we give similar plots for of and as a function of . The value of is negligibly small as compared to other form factors (consistent with zero as can also be seen in Fig. 1) therefore its figure is not shown. The results from Ioffe and the stable regions are very close to each other in the case of , while we observe some discrepancy for form factors. has a stronger dependence as compared to . Actually, has a pole around the pion mass and this can explain the difference in the behaviors of two form factors. We have, unfortunately, not been able to obtain a good fit of to either dipole or exponential functions. This is probably due to rapid increase of below 1 GeV, where we do not have reliable sum-rules data.

## Iv Conclusions and Outlook

We have extracted the isovector axial-vector and induced pseudoscalar form factors of octet baryons by employing the LCSR method. These form factors provide information about the shape and the size of the baryons. The values of the hyperon DA’s were not known precisely and this prevented the studies on hyperon structure and form factors from QCD for a long time. However, the DA’s have been recently calculated up to twist six Liu and Huang (2009a, b), which allows us to give a description of form factors at high-momentum transfers. Unfortunately, there is no sufficient experimental data yet to compare our results with in this region. However, the new generation higher-energy neutrino experiments, such as Minera Drakoulakos et al. (2004) will span a wide region of momentum transfers and will probe baryon axial-form factors with high precision in the near future.

In the low-energy region, we have compared our results with those from experiment and two-flavor lattice QCD simulations Erkol et al. (2010). We have observed that there is a nice qualitative and quantitative agreement, which can be suitably reproduced by an exponential form. With the availability of the lattice-QCD data in the low-, as well as in the high- region, we will be able to give a more accurate comparison of these two complementary approaches. Work along this direction is still in progress, where it is aimed to extract baryon form factors in a wide range of momentum transfers with larger 2+1-flavor lattices of smaller lattice spacing and quark masses. We also aim to extract isoscalar form factors and extend our study to non-diagonal baryon transitions as well. Our work along this direction is also in progress.

###### Acknowledgements.

We gratefully acknowledge very useful discussions with M. Oka and T. T. Takahashi. This work has been supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) under project number 110T245. The work of A. O. is also partially supported by the European Union (HadronPhysics2 project Study of strongly interacting matter).## References

- Nakamura et al. (2010) K. Nakamura et al. (Particle Data Group), J.Phys.G G37, 075021 (2010).
- Ahrens et al. (1988) L. A. Ahrens et al., Phys. Lett. B202, 284 (1988).
- Choi et al. (1993) S. Choi et al., Phys. Rev. Lett. 71, 3927 (1993).
- Baker et al. (1981) N. Baker, A. Cnops, P. Connolly, S. Kahn, H. Kirk, et al., Phys.Rev. D23, 2499 (1981).
- Drakoulakos et al. (2004) D. Drakoulakos et al. (Minerva Collaboration) (2004), spokespersons: J.G. Morfin, Kevin Scott McFarland, eprint hep-ex/0405002.
- Savage and Walden (1997) M. J. Savage and J. Walden, Phys. Rev. D55, 5376 (1997), eprint hep-ph/9611210.
- Jiang and Tiburzi (2008) F.-J. Jiang and B. C. Tiburzi, Phys. Rev. D77, 094506 (2008), eprint 0801.2535.
- Jiang and Tiburzi (2009) F.-J. Jiang and B. C. Tiburzi, Phys. Rev. D80, 077501 (2009), eprint 0905.0857.
- Flores-Mendieta et al. (1998) R. Flores-Mendieta, E. E. Jenkins, and A. V. Manohar, Phys. Rev. D58, 094028 (1998), eprint hep-ph/9805416.
- Chiu et al. (1985) C. B. Chiu, J. Pasupathy, and S. L. Wilson, Phys. Rev. D32, 1786 (1985).
- Andreev et al. (2007) V. Andreev et al. (MuCap Collaboration), Phys.Rev.Lett. 99, 032002 (2007), eprint 0704.2072.
- Bernard et al. (2002) V. Bernard, L. Elouadrhiri, and U. Meissner, J.Phys.G G28, R1 (2002), eprint hep-ph/0107088.
- Schindler et al. (2007) M. Schindler, T. Fuchs, J. Gegelia, and S. Scherer, Phys.Rev. C75, 025202 (2007), eprint nucl-th/0611083.
- Edwards et al. (2006) R. G. Edwards et al. (LHPC), Phys. Rev. Lett. 96, 052001 (2006), eprint hep-lat/0510062.
- Khan et al. (2006) A. A. Khan et al., Phys. Rev. D74, 094508 (2006), eprint hep-lat/0603028.
- Hagler et al. (2008) P. Hagler et al., Phys. Rev. D77, 094502 (2008).
- Alexandrou et al. (2007) C. Alexandrou, G. Koutsou, T. Leontiou, J. W. Negele, and A. Tsapalis, Phys. Rev. D76, 094511 (2007).
- Yamazaki et al. (2008) T. Yamazaki et al. (RBC+UKQCD), Phys. Rev. Lett. 100, 171602 (2008), eprint 0801.4016.
- Lin and Orginos (2009) H.-W. Lin and K. Orginos, Phys. Rev. D79, 034507 (2009), eprint 0712.1214.
- Sasaki and Yamazaki (2009) S. Sasaki and T. Yamazaki, Phys. Rev. D79, 074508 (2009), eprint 0811.1406.
- Erkol et al. (2009) G. Erkol, M. Oka, and T. T. Takahashi, Phys. Rev. D79, 074509 (2009), eprint 0805.3068.
- Erkol et al. (2010) G. Erkol, M. Oka, and T. T. Takahashi, Phys. Lett. B686, 36 (2010).
- Shifman et al. (1979a) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B147, 385 (1979a).
- Shifman et al. (1979b) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B147, 448 (1979b).
- Reinders et al. (1985) L. J. Reinders, H. Rubinstein, and S. Yazaki, Phys. Rept. 127, 1 (1985).
- Ioffe and Smilga (1984) B. L. Ioffe and A. V. Smilga, Nucl. Phys. B232, 109 (1984).
- Braun and Filyanov (1989) V. M. Braun and I. E. Filyanov, Z. Phys. C44, 157 (1989).
- Balitsky et al. (1989) I. I. Balitsky, V. M. Braun, and A. V. Kolesnichenko, Nucl. Phys. B312, 509 (1989).
- Chernyak and Zhitnitsky (1990) V. L. Chernyak and I. R. Zhitnitsky, Nucl. Phys. B345, 137 (1990).
- Braun et al. (2006) V. M. Braun, A. Lenz, and M. Wittmann, Phys. Rev. D73, 094019 (2006), eprint hep-ph/0604050.
- Aliev and Savci (2007) T. M. Aliev and M. Savci, Phys. Lett. B656, 56 (2007), eprint 0711.1757.
- Wang et al. (2006) Z.-G. Wang, S.-L. Wan, and W.-M. Yang, Eur. Phys. J. C47, 375 (2006), eprint hep-ph/0601060.
- Liu and Huang (2009a) Y.-L. Liu and M.-Q. Huang, Nucl. Phys. A821, 80 (2009a), eprint 0811.1812.
- Liu and Huang (2009b) Y.-L. Liu and M.-Q. Huang, Phys.Rev. D80, 055015 (2009b), eprint arXiv:0909.0372.
- Diehl et al. (1999) M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Eur.Phys.J. C8, 409 (1999), eprint hep-ph/9811253.
- Braun et al. (2000) V. Braun, R. J. Fries, N. Mahnke, and E. Stein, Nucl. Phys. B589, 381 (2000), eprint hep-ph/0007279.
- Aliev et al. (2010) T. Aliev, A. Ozpineci, M. Savci, and V. Zamiralov, Phys.Rev. D81, 056004 (2010), eprint 1002.2795.
- Lenz et al. (2009) A. Lenz, M. Gockeler, T. Kaltenbrunner, and N. Warkentin, Phys.Rev. D79, 093007 (2009), eprint 0903.1723.
- Gockeler et al. (2008) M. Gockeler, R. Horsley, T. Kaltenbrunner, Y. Nakamura, D. Pleiter, et al., Phys.Rev.Lett. 101, 112002 (2008), eprint 0804.1877.
- Budd et al. (2003) H. S. Budd, A. Bodek, and J. Arrington (2003), eprint hep-ex/0308005.
- Gran et al. (2006) R. Gran et al. (K2K Collaboration), Phys.Rev. D74, 052002 (2006), eprint hep-ex/0603034.