Electromagnetic form factors of singly heavy baryons in the self-consistent SU(3) chiral quark-soliton model
The self-consistent chiral quark-soliton model is a relativistic pion mean-field approach in the large limit, which describes both light and heavy baryons on an equal footing. In the limit of the infinitely heavy mass of the heavy quark, a heavy baryon can be regarded as valence quarks bound by the pion mean fields, leaving the heavy quark as a color static source. The structure of the heavy baryon in this scheme is mainly governed by the light-quark degrees of freedom. Based on this framework, we evaluate the electromagnetic form factors of the lowest-lying heavy baryons. The rotational and strange current quark mass corrections in linear order are considered. We discuss the electric charge and magnetic densities of heavy baryons in comparison with those of the nucleons. The results of the electric charge radii of the positive-charged heavy baryons show explicitly that the heavy baryon is a compact object. The electric form factors are presented. The form factor of is compared with that from a lattice QCD. We also discuss the results of the magnetic form factors. The magnetic moments of the baryon sextet with spin 1/2 and the magnetic radii are compared with other works and the lattice data.
A baryon can be viewed as valence quarks bound by the meson mean field Witten:1979kh (); Witten:1983tx () in the expansion within quantum chromodynamics (QCD), where denotes the number of colors. Witten showed explicitly in his seminal paper Witten:1979kh () that in two dimensional QCD the baryon can be considered as a bound state of valence quarks by the meson mean fields in the Hartree approximation. Since the mass of the nucleon is proportional to and the meson-loop fluctuations are suppressed by , such a mean field approach is valid in the large limit. The chiral quark-soliton model (QSM) Diakonov:1987ty (); Wakamatsu:1990ud (); Christov:1995vm (); Diakonov:1997sj () was developed based on this idea. In the large limit, the presence of the valence quarks produces the chiral mean fields coming from the polarization of a Dirac sea that in turn influences self-consistently the valence quarks. In this picture, the baryon arises as a soliton that consists of the valence quarks. A very important feature of the chiral mean field or the soliton is hedgehog symmetry. The QSM described successfully various properties of the baryon octet and decuplet such as the electromagnetic (EM) properties Kim:1995mr (); Kim:1995ha (); Wakamatsu:1996xm (); Kim:1997ip (); Silva:2013laa (), axial-vector form factors Silva:2005fa (), tensor charges and form factors Kim:1995bq (); Kim:1996vk (); Pobylitsa:1996rs (); Schweitzer:2001sr (); Ledwig:2010tu (); Ledwig:2010zq (), semileptonic decays Kim:1997ts (); Ledwig:2008ku (); Yang:2015era (), parton distributions Diakonov:1996sr (); Diakonov:1997vc (); Wakamatsu:1997en (), and so on.
Very recently, it was shown that singly heavy baryons can be considered as valence quarks bound by the pion mean field Yang:2016qdz (), being motivated by Diakonov Diakonov:2010tf (). In the limit of the infinitely heavy quark mass (), the spin of the heavy quark is conserved, which leads to the conservation of the spin of the light-quark degrees of freedom: Isgur:1989vq (); Georgi:1990um (). In this limit, the heavy quark inside a singly heavy baryon can be merely regarded as a static color source, which means that the heavy quark is required only to make the heavy baryon a color singlet. Thus, the flavor representations of the lowest-lying heavy baryons are given by , of which the baryon anti-triplet has and whereas the sextet has . The light quarks with being coupled to the spin of the heavy quark , the baryon sextet have spins and . So, there are two representations with spin and in the baryon sextet, which are degenerate in the limit of . The hyperfine spin-spin interaction will lift the degeneracy between these states with different spins Yang:2016qdz ().
In the QSM, we can apply basically the same formalism on the singly heavy baryons, which have been developed for the description of light baryons. Considering the heavy quark as a static color source, the heavy baryon can be regarded as a system of the valence quarks with the heavy quark stripped off from the valence level. In the case of the light baryons, the collective Hamiltonian is constrained by the right hypercharge imposed by the valence quarks, which selects the lowest allowed representations such as the octet () and the decuplet (). However, when it comes to the singly heavy baryons, this constraint should be changed to be because of the valence quarks inside a heavy baryon and yields the anti-triplet () and the sextet () as the lowest allowed representations. In addition, we need to modify the valence parts of all moments of inertia and quark densities for the calculation of any form factors. This extension of the QSM was rather successful in describing the masses of the lowest-lying singly heavy baryons in both the charmed and bottom sectors, and the mass of the was predicted Yang:2016qdz (); Kim:2018xlc (). Moreover, newly found resonances Aaij:2017nav () were well explained and classified. In particular, two of the resonances were interpreted as exotic baryons belonging to the anti-decapentaplet () with their narrow widths correctly reproduced Kim:2017jpx (); Kim:2017khv ().
In the present work, we want to investigate the EM properties of the lowest-lying singly heavy baryons with spin . Assuming that the mass of the heavy quark is infinitely heavy, we will show that the main features of the EM form factors are governed by the light quarks. Of course, the electric form factor requires a certain contribution from the heavy quark such that the charge of the heavy baryons should be correctly reproduced. Since the EM current is decomposed into the light and heavy parts, the heavy-quark contribution can be treated separately. Its effects on the electric form factors are just the constant ones within the present framework, which indicates that the heavy quark is considered to be a structureless particle. This is a natural consequence, because we deal with the heavy quark just as a static color source. Since any form factor in QCD should decrease rapidly as the square of the momentum transfer increases because of gluon exchanges between the quarks that constitutes a baryon, the present picture of the electric form factors may be put into question. However, keeping in mind that the QSM is the low-energy effective theory of the nucleon, we still can treat the heavy quark as a static one in the limit of , as far as the momentum transfer remains much smaller than the heavy-quark mass, i.e. . We will show that this approach indeed produces reasonable results for the electric form factors of the heavy baryons in comparison with the lattice data Can:2013tna (). In the case of the magnetic form factors, the situation is even better. In the limit of , the effects of the heavy quark vanishes, since the magnetic moment of the heavy quark is proportional to the inverse of the heavy-quark mass. Thus, the magnetic form factors of the singly heavy baryons are solely governed by the light quarks.
We sketch the structure of the present paper as follows: In Section II, we show how to compute the EM form factors of the heavy baryons within the QSM. In Section III, we present and discuss the numerical results of the EM form factors and the corresponding charge and magnetic radii. We also examine the effects of the symmetry breaking. The final Section is devoted to the summary and conclusions.
Ii General formalism
The EM current including the heavy quark (the charm quark or the bottom quark) is expressed as
where denotes the charge operator in , defined by
Here, and are the flavor SU(3) Gell-Mann matrices. The in the second part of the EM current in Eq. (1) stands for the heavy-quark charge, which is given as for the charm quark or as for the bottom quark. Since the magnetic form factor of a heavy quark is proportional to the inverse of the corresponding heavy-quark mass, i.e. , we can ignore the contribution from the heavy quark current in the limit of . However, we need to keep the second term in Eq. (1) when we compute the electric form factors. The EM form factors of the spin- baryons are defined by the matrix element of the EM current
where is the mass of the nucleon. The stands for the four-momentum transfer with . denotes the Dirac spinor with the momentum and the helicity . The Dirac and Pauli form factors and can be written in terms of the Sachs EM form factors, and
with . In the Breit frame, the Sachs form factors are related to the time and space components of the EM current, respectively
Thus, once we compute the matrix elements of the EM current, we can directly derive the EM form factors. Note that we consider the heavy-quark part separately.
The SU(3) QSM is characterized by the following low-energy effective partition function in Euclidean space
where and represent the quark and pseudo-Nambu-Goldstone boson fields, respectively. The is the effective chiral action
where stands for the generic trace operator running over space-time and all relevant internal spaces. The is the number of colors, and the Dirac differential operator defined by
where denotes the Euclidean time derivative. We assume isospin symmetry, i.e. . We define the average mass of the up and down quarks by . Then, the matrix of the current quark masses is written as . is written as
where and are the singlet and octet components of the current quark masses, expressed respectively as and . The SU(3) single-quark Hamiltonian is defined as
where represents the SU(3) chiral field. Since the hedghog symmetry constrains the form of the classical pion field as , where is the profile function of the soliton, the SU(2) chiral field is written as
with . We now embed the SU(2) soliton into SU(3) by Witten’s Ansatz Witten:1983tx ()
Since we consider the mean-field approximation, we can carry out the integration over in Eq. (8) around the saddle point (). This saddle point approximation yields the equation of motion that can be solved self-consistently. The solution provides the self-consistent profile function .
The matrix elements of the EM current (3) can be computed within the QSM by representing them in terms of the functional integral in Euclidean space
where the baryon states and are respectively defined by
The heavy baryon current can be constructed from the valence quarks
where represent spin-flavor indices and color indices. The matrices are taken to consider the quantum numbers of the soliton. The creation operator can be constructed in a similar way. Integrating over the quark fields, we find that the matrix element of the EM current can be decomposed into two separate parts, i.e. the valence part and the sea part
where denotes the trace over the Dirac spin, flavor, and color spaces.
When we carry out the integral over the field, we have to take into account the zero-mode fluctuations. Since the zero modes are not small, we need to perform the integral over the collective coordinates corresponding to the zero modes. These zero modes are relevant to continuous symmetries of the present problem. In fact, there are three translational and seven rotational zero modes. Since we are interested in the evaluation of the EM form factors, we need to deal with the translational zero modes properly, since the soliton is not invariant under translation and its translational invariance is restored only after integrating over the translational zero modes. In the course of treating the translational zero modes, the Fourier transform of the EM densities appears naturally as the expressions of the EM form factors. On the other hand, the rotational zero modes determine the quantum numbers of the lowest-lying heavy baryons.
The zero modes can be considered by a slowly rotating and translating hedgehog:
where designates a unitary time-dependent SU(3) collective orientation matrix and represents the time-dependent translation of the center of mass of the soliton in coordinate space. Then the Dirac operator in Eq. (10) is changed into the following form
where stands for the translational unitary operator, denotes the angular velocity of the soliton defined by
and the velocity of the translational motion is defined by
Since we assume that soliton moves and rotates slowly, we can deal with and perturbatively. The strange current quark mass is also considered to be small, we expand the Dirac operator with respect to perturbatively. Then, the effetive chiral action can be written as
and stand for the collective Lagrangians for the rotational corrections and symmetry breaking part. denotes the Euclidean time . Thus, the relevant collective Hamiltonian can be written as
and are the soliton moments of inertia. The parameters , , and for heavy baryons are defined by
where that the three parameters , , and are expressed in terms of the moments of inertia and . The valence parts of them are different from those in the light baryon sector by the color factor in place of . The expression of is similar to the sigma term again except for the factor: . The detailed expressions for the moments of inertia and are found in Ref. Kim:2018xlc ().
Because of the symmetry-breaking part of the collective Hamiltonian , the baryon wave functions are no more pure states but are mixed ones with those in higher SU(3) representations. Thus the wave functions for the baryon anti-triplet () and the sextet () are derived respectively as Kim:2018xlc ()
with the mixing coefficients
respectively, in the basis for the anti-triplet and for the sextets. The parameters , , and are given by
Having obtained the mixming parameters, we are able to express explicitly the wave function of a state with flavor and spin in the representation in terms of a tensor with two indices, i.e. , one running over the states in the representation and the other one over the states in the representation . Here, represents the complex conjugate of the , and the complex conjugate of is given as . Since a singly heavy baryon consists of light valence quarks, the constraint imposed on the right hypercharge should be modified from to . Thus, the collective wave function for the soliton with valence quarks is written as
where represents the dimension of the representation and a charge corresponding to the baryon state , i.e. .
The complete wave function for a heavy baryon can be constructed by coupling the soliton wave function to the heavy quark
where denote the Pauli spinors and the Clebsch-Gordan coefficients.
The final expression for the electric form factor of a heavy baryon can be written as
where the first part of Eq. (39) is the light-quark contribution to the electric form factor whereas the second part corresponds to the point-like heavy quark. The electric charge density of a light-quark part can be expressed as
where the explicit expressions for the electric densities can be found in Appendix A. The indices and are dummy ones running over and , respectively. In the present mean-field approach, the heavy-quark contribution to the electric form factor is just the constant charge of the corresponding heavy quark ( or ), because the heavy quark is assumed to be a static color source and a point-like particle. Of course, this is a rather crude approximation but it is still a reasonable one as far as we consider the electric form factors in low regions. Thus, we set in the present work.
Since the integrations of the densities in Eq. (43) are given as
and , the electric form factor at turns out to be the charge of the corresponding heavy baryon.
The expression for the magnetic moment form factor of a baryon is written as
where the corresponding density of the magnetic form factors is given by
The indices and are the dummy indices running over . The explicit forms for the magnetic densities can be found in Appendix B. The matrix elements of the collective operators are explicitly given in Appendix D. The magnetic form factor at produces the magnetic moment of the corresponding baryon. So, it is convenient to express a collective operator for the magnetic moments Yang:2018uoj () as
Iii Results and discussion
We are now in a position to discuss the results from the present work. We first briefly mention how to fix the parameters of the model. We refer to Refs. Christov:1995vm (); Kim:1995mr () for a detailed explanation of numerical methods. The only free parameter of the QSM is the dynamical quark mass of which the numerical value was already fixed by computing various form factors of the nucleon. Its most preferable value is MeV. Nevertheless, we have checked whether the present results are sensitive to it with varied from 400 MeV to 450 MeV. All the form factors presented in this work are rather insensitive to the value of , so we choose the value for the best fit, i.e. MeV as in the light-baryon sector case Christov:1995vm (); Kim:1995mr (); Silva:2005fa (); Ledwig:2010tu (); Ledwig:2010zq (); Ledwig:2008ku (). Note that the same value of was selected also for the mass splitting of the heavy baryons Kim:2018xlc (). There are yet another parameters in the QSM: the average mass of the current up and down quarks , the strange current quark mass , and the cutoff parameter of the proper-time regularization. The value of was fixed to be MeV by reproducing the pion mass whereas the cutoff parameter is determined by reproducing the pion decay constant MeV.
The strange current quark mass can be in principle taken from its canonical value MeV which was obtained by reproducing the kaon mass in the model. However, MeV was used for the calculation of the form factors and other properties of the light baryons effectively. Very recently, the dependence of the mass splittings of heavy baryons on were examined within the same framework of the QSM Kim:2018xlc () and the best values of were obtained to be MeV and MeV for the mass splittings of the charmed baryons and the bottom baryons, respectively. Thus, instead of using the previous value 180 MeV, we will use the same values of as obtained in Ref. Kim:2018xlc () for consistency, regarding as an effective mass. However, the EM form factors of the heavy baryons show rather tiny dependence on the numerical value of , so the difference of the value does not affect the results at all.
iii.1 Electric form factors of the baryon anti-triplet and sextet with spin 1/2
The electric form factor of a baryon at is the same as its corresponding charge. Intergrating the electric charge density of the baryon given in Eq. (43) over three-dimensional space, one obtains the corresponding charge. In fact, the collective charge operator is found from Eq. (43):
where are the generators of group. Using the relations
Then we find the well-known Gell-Mann-Nishijima formula in
Sandwiching the charge operator between the collective baryon wave functions, we get the charge of the light-quark pair inside the corresponding baryon. In order to yield the correct charge of the baryon concerned, we have to introduce in addition the charge of a heavy quark inside it, as mentioned previously already. Thus, the dependence of the electric form factor of a heavy baryon in the present scheme is solely governed by the light quarks. The contribution of the point-like heavy quark is just its own constant charge as given in Eq. (39). Though this mean-field approximation may be a crude one, the dependence of the electric form factors will explain a certain characteristics of the electric structure of the heavy baryons.
In the present mean-field approach, the light-quark dynamics inside both a heavy baryon and a light baryon is treated on the same footing. Only difference lies in the different counting factor, as explained previously. Thus, it is of great interest to examine the electric charge and magnetic densities of the heavy baryons with those of the proton and the neutron, before we compute the EM form factors of the heavy baryons. In the left panel of Fig. 1, we draw the electric charge densities of the soliton for , which consists of the light-quark pair () with spin . Note that is a positive-charged member of the baryon anti-triplet. The heavy quark inside is assumed to be located at rest at the center of it. So, its charge density is just given by the delta function. The results are compared with those of the proton depicted in the right panel of Fig. 1. The general feature of the charge densities of the soliton of the light-quark pair inside is almost the same as the proton one. The electric charge of the light-quark pair inside is whereas the proton has . Thus, both the electric charge densities are positive definite over the whole region. The difference between these two electric charge densities is found only in the strength of the electric charge. Hence, the proton electric charge density turns out to be approximately three times larger than that of the soliton for . The sea-quark polarizations show marginal effects on both the electric charge densities.
Figure 2 compares the electric charge densities of the soliton inside the neutral member of the baryon anti-triplet with those of the neutron ones. Since the contains two down valence quarks, the charge distribution of the soliton inside becomes negative. Apart from the sign of the densities, the general behavior of the light-quark electric charge density of is very similar to that of the proton or . The sea-quark polarization inside is a stronger than those inside the or proton case. On the other hand, the neutron electric charge density is rather different from that of . In this case, the valence quarks govern the inner part of the neutron density, whereas the sea-quark polarization is dominant over its tail part. Thus even though the is the neutral baryon, its light-quark charge density behave very differently, compared with the neutron density. We will soon see that this difference will be clearly shown in the electric form factors of the neutral heavy baryons. All other charge densities of the positive-charged heavy baryons are very similar to that of , and those of the neutral ones to that of .
In Fig. 3, the electric form factors of the singly positive-charged heavy baryons with spin are drawn as functions of . They decrease monotonically as increases. This feature is very similar to that of the proton, which is already expected from the comparison of the charge densities in Fig. 1. So, it is also of great interest to compare the results of Fig. 3 with the proton electric form factor, as shown in Fig. 4. The electric form factor of the proton was obtained within the same framework with exactly the same parameters. The comparison exhibits a remarkable difference. The electric form factor of the proton falls off much faster than those of the singly positive-charged heavy baryons. It reveals a profound physical meaning: The heavy baryons are electrically compact objects, so that they are much smaller than the proton. This will be more clearly seen in the results of the charge radii which will be discussed later.
In Fig. 5, we show the results of the electric form factors of the neutral heavy baryons. They start to rise fast and then slow down as increases. The results are rather different from that of the neutron as discussed already in Fig. 2. The neutron electric form factor increases also as increases up to around and then starts to decrease very slowly Kim:1995mr (); Praszalowicz:1998jm (). The experimental data also confirm this behavior of the neutron form factor Sulkosky:2017prr (). As shown in Fig. 2, the neutron charge density is governed by the up quarks in the inner part of the neutron, whereas the negative-charged down quark dominates its tail part. On the other hand, the charge densities of the neutral heavy baryons are rather similar to those of the positive-charged ones except for the sign. Accordingly, the electric form factors of the neutral heavy baryons increase slowly and monotonically as increases. However, one should bear in mind that the present mean-field approach is only valid in the lower region, say, up to around or even lower values of .
In Fig. 6, we draw the numerical result of the electric form factor of , comparing it with those from lattice QCD Can:2013tna (). Figure 6 shows that the present result falls off faster than the lattice ones. However, we want to emphasize on the fact that the lattice data on the electric form factor of the proton with the unphysical value of the pion mass tend to decrease slower than the experimental data. For example, all the lattice calculations Capitani:2015sba (); Abdel-Rehim:2015jna (); Djukanovic:2015hnh (); Chambers:2017tuf () yield the results of the nucleon electric form factor, which fall off rather slowly in comparison with the experimental data. Even a very recent lattice calculation at the physical point Alexandrou:2017ypw () shows a similar feature. The same tendency was also found in the case of the tensor and anomalous tensor form factors of the nucleon Ledwig:2010tu (); Ledwig:2010zq (). The lattice results of these form factors Gockeler:2005cj (); Gockeler:2006zu () also fall off much more slowly than those results of the QSM.
In the present mean-field approach, there is in principle no difference between the electric form factors of the charmed baryons and those of the bottom baryons, since the same light quarks govern the dependence of the form factors. The charge of the bottom quark makes their electric form factors distinguished from those of the charmed baryons. In Fig. 7, we draw the electric form factors of the bottom baryons. In the upper left panel of Fig. 7, those of the neutral bottom baryons are depicted. Interestingly, they are all negative, which are different from those of the neutral charmed baryons. This can be understood from the differrent charges of the charm and bottom quarks. The upper right panel of Fig. 7 present the electric form factors of the negative-charged bottom baryons. That of is illustrated in the lower panel of Fig. 7. We find that it becomes negative at around , which is again due to the negative charge of the bottom quark.