# Pure sea-quark contributions to the magnetic form factors of baryons

###### Abstract

We propose the pure sea-quark contributions to the magnetic form factors of baryons, and , as priority observables for the examination of sea-quark contributions to baryon structure, both in present lattice QCD simulations and possible future experimental measurement. , the -quark contribution to the magnetic form factor of , and , the -quark contribution to the magnetic form factor of , are similar to the strange quark contribution to the magnetic form factor of the nucleon, but promise to be larger by an order of magnitude. We explore the size of this quantity within chiral effective field theory, including both octet and decuplet intermediate states. The finite range regularization approach is applied to deal with ultraviolet divergences. Drawing on an established connection between quenched and full QCD, this approach makes it possible to predict the sea quark contribution to the magnetic form factor purely from the meson loop. In the familiar convention where the quark charge is set to unity . We find a value of , which is about seven times larger than the strange magnetic moment of the nucleon found in the same approach. Including quark charge factors, the -quark contribution to the magnetic moment exceeds the strange quark contribution to the nucleon magnetic moment by a factor of 14.

^{†}

^{†}preprint: ADP-15-16/T918

It is well known that a complete characterization of baryon substructure must go beyond three valence quarks. Strange quark contributions to the properties of the nucleon have attracted a lot of interest since the originally puzzling EMC results concerning the proton spin EMC . While that particular motivation has faded Myhrer:2007cf ; Thomas:2008bd ; Thomas:2008ga , the role of the sea remains a central issue in QCD, especially with respect to lattice QCD. There such terms involve so-called “disconnected graphs”; that is, quark loops which are connected only by gluons to the valence quarks. Despite enormous effort Lewis:2002ix , only one direct lattice QCD calculation has produced a non-zero result Doi:2009sq , albeit with errors which mean that it is not statistically different from zero.

With very few exceptions, the form factor studies, which complement the recent experimental progress at facilities such as Jefferson Lab, deal with so called connected contributions, in which the external current acts on a quark line running directly from the hadronic source to sink. As discussed below, only a few studies have directly addressed the disconnected contributions. Perhaps the most famous example of a disconnected contribution is the strange quark contribution to the nucleon elastic form factors Leinweber:1995ie ; Leinweber:1999nf ; Leinweber:2004tc ; Leinweber:2005bz ; Leinweber:2006ug ; Wang:2009ta ; Thomas:2012tg ; Wang:2014nhf ; Shanahan:2014tja . Its fundamental importance is associated with the fact that it is directly analogous to the vacuum polarization contribution to the Lamb shift, the correct calculation of which confirmed the validity of Quantum Electrodynamics.

Parity-violating electron scattering (PVES) has proven to be a valuable tool for experimentally determining the strange quark contribution to the electromagnetic form factors of the proton. Under the assumption of charge symmetry, one can deduce the strange electric or magnetic form factor ( ) from measurements of the corresponding proton and neutron electromagnetic form factors and the neutral-weak vector form factor of the proton, through its contribution to PVES. While PVES measurements are very challenging, a number of groups have succeeded, starting with SAMPLE at Bates Spayde:2003nr and then A4 at Mainz a4 ; Maas:2004dh and G0 Armstrong:2005hs and HAPPEX Acha:2006my ; Aniol:2005zg ; Aniol:2004hp at Jefferson Lab. Up to now, the experiments have not provided an unambiguous confirmed answer to the sign of the strange form factors, although global analyses do tend to suggest that is favoured Young:2006jc ; Gonzalez-Jimenez:2014bia .

Though there exist predictions of the strange form factors from lattice with the help of charge symmetry Leinweber:2004tc ; Leinweber:2005bz ; Leinweber:2006ug ; Wang:2009ta ; Thomas:2012tg ; Wang:2014nhf ; Shanahan:2014tja , it is difficult to simulate this quantity directly because it is purely from the disconnected diagrams and is also quite small. Even an unambiguous determination of the sign of the strange form factor is an important step in the quest to understand the structure of nucleon. It is related to how the strange and anti-strange quarks are distributed in the nucleon. The sign of the strange form factor will shed light on whether the components in nucleon is dominated by colored di-quark configurations or by color singlet configurations Zou .

In this Letter, we propose that the quantity , the -quark contribution to the magnetic form factor of or similarly , the quark contribution to the magnetic form factor of , is equally important to . Because the light quark mass of the or quark governs the magnitude of the contribution, it is expected to be larger and less difficult to measure in lattice QCD. It is similar to the strange form factor in the sense that both of the quantities arise purely from “disconnected” sea-quark contributions. However, in an effective field theory framework, and are generated by a meson loop, which should be much larger than the strange form factor generated from a meson loop. They will serve as an ideal quantity for future lattice simulations and will shed light on the sea quark properties of baryons.

Chiral effective field theory (EFT) is a useful tool with which to study hadron properties at low energy. There has been some work on strange form factors with heavy baryon chiral EFT Hemmert1 ; Hemmert2 . However, there is an unknown low energy constant appearing in the chiral Lagrangian, which has limited the capacity to calculate the strange magnetic form factor. In other words, the quantity one wishes to predict – the strangeness vector current matrix element – is the same quantity one needs to know in order to make a prediction Musolf ; Kubis . While this is the case in conventional chiral EFT, experience with finite-range-regularization (FRR), has shown that by varying the regulator parameter, one can model the shift in strength from the loop contributions into the core. This suggests that within FRR -EFT one might identify the core contribution with the tree level contribution and make the approximation that, for in the region of 0.8 GeV, the sea quark content of the core is negligible. In this way, full QCD results have been obtained rather successfully from quenched lattice data Leinweber:2004tc ; Leinweber:2005bz ; Leinweber:2006ug ; Wang:2009ta ; Wang:2008vb ; Wang:2012hj . We should emphasize that unquenching only works for the particular choice of regulator mass, around 0.8 GeV, because only then does one define a core contribution that is approximately invariant between quenched and full QCD.

We will apply heavy baryon chiral effective field theory with finite range regularization to study the pure sea-quark contribution to the magnetic form factors of baryons. In presenting the formalism, we choose to focus on the -quark contribution to form factors. This channel is very similar to the -quark contribution to the proton. In the standard convention where the quark charge is set to unity .

In heavy baryon chiral EFT, the lowest order chiral Lagrangian for the baryon-meson interaction which will be used in the calculation of the magnetic form factor, including the octet and decuplet baryons, is expressed as

(1) | |||||

where is the covariant spin-operator defined as

(2) |

Here, is the baryon four velocity (in the rest frame, we have ) and , and are the usual SU(3) coupling constants. The chiral covariant derivative, , is written as . The pseudoscalar meson octet couples to the baryon field through the vector and axial vector combinations

(3) |

where

(4) |

As explained above, following earlier successful studies of the connection between quenched and full QCD, our working hypothesis is that the quark contribution to the magnetic form factor of the comes purely from the meson loop diagram, which is shown in Fig.1. There are two types of diagram. Fig. 1a is the leading order contribution, where the external field couples to the meson. Fig. 1b is the next-to-leading order contribution, where the external field couples to the baryon. That the meson loop provides a very small contribution to the magnetic form factor was shown in the previous study of the strange magnetic form factor Wang:2014nhf . Here we consider the loop contribution. Both octet and decuplet intermediate states are included. The contribution from the process shown in Fig. 1a is expressed as

(5) |

where the respective terms correspond to the intermediate , and states. can be obtained as

(6) |

In the now standard notation, () is the regulator introduced in the finite range regularization with momentum (). () is the energy of a pion with momentum (). The charge of the quark has been set to unity, consistent with the universal convention when discussing the strange quark form factors of the proton.

The intermediate contribution in Fig. 1a has the following relationship with the

(7) |

For the decuplet part, the contribution is written as

(8) |

where is the mass difference between the and .

The next-to-leading order contribution of Fig. 1b is

(9) |

This includes octet, decuplet and octet-decuplet transition contributions in Fig. 1b. The octet contribution arising from the is written as

(10) |

corresponding to the state appearing in the configuration . The decuplet contribution from the is obtained as

(11) |

The transition contribution to the magnetic form factor is written as

(12) |

Here is the quark contribution to the magnetic moment of the baryon at tree level, i.e.

(13) |

For the last term in Eq. (9), the following transition moments is applied

(14) |

(GeV) | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|

LO | |||||

NLO | |||||

or |

In the numerical calculations, the parameters are chosen as and (). The coupling constant is chosen to be . The form of the regulator function, , could be chosen to be a monopole, dipole or Gaussian function, any of which would give similar results Young:2002ib . In our calculations, a dipole form is chosen because that is the empirical shape of the nucleon axial form factor Guichon:1982zk

(15) |

with GeV.

As we explained earlier, this choice has been widely applied in the extrapolation of lattice data for hadron mass, moments, form factors, radii, first moments of GPDs, etc. Young:2002ib ; Leinweber3 ; Wang:2014nhf ; Wang4 ; Wang5 ; Allton:2005fb ; Armour:2008ke ; Hall1 ; Hall2 . With this choice it has been shown that reasonable physical results can be obtained from the quenched lattice data at both leading and next leading order Young:2002ib ; Leinweber:2006ug ; Leinweber3 ; Leinweber:2004tc ; Leinweber:2005bz ; Wang:2009ta ; Wang:2008vb ; Wang:2012hj ; Wang:2014nhf . around 0.8 GeV is the value required to identify a core contribution that is invariant between quenched and full QCD. This invariance of the core is based upon the assumption that the 3-quark core of the contains no quark component.

While our calculation is motivated by chiral effective field theory with the same chiral Lagrangian, our calculation with FRR is at a physically motivated scale, where earlier work has suggested that the residual series of analytic terms best describes the three-quark core contributions. From the previous extrapolation of quenched lattice data, it is found that this preferred value of in the dipole regulator is around 0.8 GeV. The variation of from 0.6 to 1 GeV provides an estimate of the degree of model dependence of our result.

The contribution of the pure sea-quark contribution to the magnetic moment at leading and next-to-leading order is shown in Table I. The leading order diagram shown in Fig. 1(a) gives a negative contribution to the magnetic form factor. The contributions from the next-to-leading order diagrams are much smaller than the leading contribution. They depend on the parameter . Assuming SU(3) symmetry, one has Wang:2012hj ; Wang4 . In the previous extrapolation of nucleon magnetic form factors, we found equal and for full QCD and quenched QCD extrapolations, respectively Wang:2012hj ; Wang4 . Therefore, should be a good estimate. . In fact, this relation was applied in our previous investigation of nucleon magnetic form factors

In Fig. 2, we show the magnetic form factor versus at 0.6, 0.8 and 1.0 GeV. One can see that decreases in magnitude with the increasing . It is obvious that the magnetic form factor does not change sign for any of the choices of when increases. This is just like the strange magnetic form factor of the nucleon. However, the absolute value of is about one order of magnitude larger than . Since its absolute value decreases with the increasing , it would be preferable to attempt to measure the magnetic form factor at low . For example, when is less than 0.2 GeV, the absolute central value of is larger than 0.2 .

At , the quark contribution to the magnetic moment of the is . If we vary from 0.6 GeV to 1 GeV, will change from to . Numerical results show that remains negative over a large parameter range. Compared with the strange magnetic moment of the proton, the value of is about seven times larger Leinweber:2004tc ; Wang:2014nhf .

For unit charge sea-quarks, . Thus the magnitude of the sea-quark contribution further doubles in an experimental measurement of the contribution of the quark to the form factor of .

Motivated by the importance of establishing the properties of disconnected contributions to physical quantities in lattice QCD, we have shown that and have the practical advantage that their values are much larger than the strange magnetic form factor of the nucleon. Since the absolute value of is nearly one order of magnitude larger than the strange magnetic form factor of the nucleon, it would clearly be better to simulate this quantity in place of the strange form factor of the nucleon.

Since the lattice simulations will almost certainly be made over a range of light quark masses, we have investigated the pion mass dependence of . The results are shown in Fig. 3, where the upper, middle and lower lines are for 0.6, 0.8 and 1 GeV, respectively. From the figure, one can see that with increasing quark mass the absolute value of decreases. However, even at = 0.2 GeV, is still much larger than the strange magnetic moment of the nucleon at the physical pion mass.

An additional feature of the baryon is the presence of a strange quark in the two-point correlation function. In calculating the disconnected sea-quark contribution, one multiplies the disconnected loop by the standard two-point function in creating the full three-point function. The presence of a strange quark in the two-point function will assist in reducing statistical noise in the three-point correlation function for the pure sea-quark contribution.

Given that is dominated by the contribution of a meson loop and having strange quarks in the two-point correlation function is advantageous, one might also consider the quark contribution to the magnetic form factor of the or the the quark contribution to the magnetic form factor of the . These quantities are also determined by a meson loop. However, the coupling of and is much smaller resulting in a very small value of . Thus has unique advantages with respect to studies of the contributions to the structure of baryons through disconnected sea quark terms.

In summary, we have argued the importance of studying the pure sea-quark contributions to -baryon form factors, and . Because of the significant enhancement associated with the light or quarks, these observables have distinct quantitative advantages over the strange form factors of the nucleon. This enhancement arises because the pure light sea-quark contribution to the magnetic form factors of baryons is dominated by the -meson cloud contribution. This is much larger than the nucleon strange magnetic form factor which originates in the -meson cloud.

We calculated and within heavy baryon chiral effective field theory including both octet and decuplet intermediate states. The pure sea-quark contribution to the magnetic moment is , which is about seven times larger than the nucleon strange magnetic moment and 14 times larger for in experiment.

We also calculated the pion mass dependence of the pure sea-quark contributions. When the pion mass is about 300-400 MeV, the absolute value of is still around . It seems likely that future lattice simulations may be able to determine directly with more accuracy than the strange form factor of the nucleon, . The value or even the sign of would be very helpful in pinning down the size and origin of five-quark configurations in baryons.

## Acknowledgments

This work was supported in part by DFG and NSFC (CRC 110), by the National Natural Science Foundation of China (Grant No. 11475186) and by the Australian Research Council through grants FL0992247 (AWT), DP140103067 and DP150103164 (DBL) and through the ARC Centre of Excellence for Particle Physics at the Terascale.

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