Pion structure in the nuclear medium
Using the light-front pion wave function based on a Bethe-Salpeter amplitude model, we study the properties of the pion in symmetric nuclear matter. The pion model we adopt is well constrained by previous studies to explain the pion properties in vacuum. In order to consistently incorporate the constituent up and down quarks of the pion immersed in symmetric nuclear matter, we use the quark-meson coupling model, which has been widely applied to various hadronic and nuclear phenomena in a nuclear medium with success. We predict the in-medium modifications of the pion electromagnetic form factor, charge radius and weak decay constant in symmetric nuclear matter.
pacs:21.65.Jk, 14.40.Be, 13.40.Gp
One of the most exciting challenges in hadronic physics is to investigate the changes in hadron properties in a nuclear medium as well as in a nucleus (1); (2). In particular, partial restoration of chiral symmetry in a dense nuclear medium has not yet been confirmed by experiment, although it is generally accepted to occur based on quantum chromodynamics (QCD). In spite of the notorious complex number or sign problem of the fermion determinant, one hopes that lattice QCD simulation will eventually be able to study hadron properties in a nuclear medium with relatively high baryon densities (3); (4); (5); (6).
Chiral symmetry is one of the most important symmetries in QCD. Therefore, it is very interesting to study the pion properties, since it is the Nambu-Goldstone boson of the theory which is realized in nature due to the spontaneous breaking of chiral symmetry. Thus, it is natural to ask how the pion properties would be modified in a nuclear medium, where chiral symmetry is expected to be (partially) restored (see Ref. (7) for a review concerning this question).
There exist several works on pion properties in a nuclear medium, e.g. using the Nambu-Jona-Lasinio (NJL) model (8), studies were made for the dynamical mass (of the pion-like mode) (9), the pion structure function (10), and the mass and decay constant (11); (12); (13). Other studies dealt with the pion decay constant in a composite-operator approach (14), pion cloud effects on the Drell-Yan scattering (15), mass shifts via in-medium chiral perturbation theory (16); (17); (18), masses and decay constants within a Dyson-Schwinger and Bethe-Salpeter equation ansatz (19). Furthermore, the pion in finite density has been studied with QCD sum-rule approaches (20), using virial expansions (21), a non-local chiral quark model (22), a relativistic mean field approach (23), and by exploiting operator relations in QCD (24). However, only one of them (23) examined the in-medium modification of the pion elastic form factor (in asymmetric nuclear matter) based on the mixing mechanism at the hadronic level. In the present work, we investigate the pion structural properties in symmetric nuclear matter based on quark degrees of freedom.
In order to do so, it is necessary to have a reliable pion model which is well constrained and successful in describing its properties in vacuum. However, because of the Nambu-Goldstone boson nature of the pion with an anomalously small mass, its description in vacuum itself is not straightforward either, and a special treatment is necessary. Furthermore, even if such a successful model is available, one requires a proper description of the system’s state as well as a consistent current operator to perform a meaningful comparison with experimental data. To properly define the state of a relativistic system, three space-time hyper-surfaces were identified by Dirac (25). These hyper-surfaces correspond to different forms of relativistic Hamiltonian dynamics, namely instant form, front form and point form.
In the present study, we apply the front-form dynamics incorporating consistency between the current operator and the state from a field theoretical point of view with a fixed number of particles. In particular, although the state has an infinite number of components in the Fock-space (26), only the lowest Fock component or valence component is usually modelled and used for calculating electroweak form factors. In principle, the infinite set of coupled eigenvalue equations for the Hamiltonian operator in the Fock space can be replaced by an effective squared-mass operator or an effective Hamiltonian acting on the valence sector. At the same time, it is also possible to express systematically the higher Fock-state components of the wave function as functionals of the lower ones (26); (27); (28). The effective electroweak current operator for the valence component of the state can also be consistently derived within the field theoretical framework of the Bethe-Salpeter equation projected on the equal light-front time (29); (30). However, the major advance in the extraction of the form factors from the front-form wave function is the realization that in the Drell-Yan frame the pair production does not contribute unlike in the frame (31).
In Ref. (32), the effect of such pair-terms was studied in detail to evaluate the form factors of a system with two identical fermions, where effective constituent quark masses of the lowest Fock-space component were used to describe the pion properties in vacuum. The vertex function describes the momentum component of the coupling of the quarks to the pion bound state, where by momentum component we mean the light-front wave function obtained by integrating over the relative quark momentum, , after the separation of the instantaneous terms in the external quark legs. Namely, the momentum component is the light-front wave function which depends on the kinematic variables, and . In this model, the symmetric vertex function and the light-front valence wave function which are symmetric under the exchange of the fermion momenta are employed (32). (See also Ref. (33) for a nonsymmetric case.) Symmetry properties of a Bethe-Salpeter amplitude are derived from quantum field theory, where conceptual and phenomenological problems arise when a nonsymmetric vertex function is used to describe the pion (33); (31). For example, the form factor and weak decay constant cannot be reproduced simultaneously. In this study, we use the pion model of Ref. (32) which has all the desired properties discussed above.
The main purpose of the present article is to investigate the in-medium modifications of the pion properties, i.e. the electromagnetic form factor, radius and decay constant in symmetric nuclear matter, where the pion model (32) is adjusted so as to provide the best description of the electromagnetic form factor data in vacuum. Because the (symmetric) nuclear matter is translational and rotational invariant, usual Lorentz transformation can be applied. Thus, the input obtained in the nuclear matter rest frame by the nuclear matter model we employ, keeps the track of the Lorentz scalar and vector nature of mean fields, and thus our assumption on the pion vertex model enables us to extract the form factor in a covariant manner.
For the nuclear matter, we employ the quark-meson coupling (QMC) model (34); (35); (36) in order to include consistently the in-medium properties of the constituent up and down quarks in the pion in symmetric nuclear matter. The QMC model has been widely applied to various nuclear and hadronic phenomena in a nuclear medium with successes. Although our approach may be regarded as crude, this is a first exploratory study that treats both the constituent quarks forming nuclear matter in the bound nucleons and those in the pion immersed in this nuclear medium on the same footing. In particular, it is known that a treatment of nuclear matter based on quark degrees of freedom is very difficult and a description starting from first principles of QCD is far remote. In this sense, although phenomenological, this study may give some insight in the pion properties in a nuclear medium.
This article is organized as follows. In Section II we briefly describe the QMC model focusing on the properties of constituent up and down quarks and the pion vertex in symmetric nuclear matter. The expressions for the in-medium electromagnetic form factor of the pion are discussed in Section III, while the results for the in-medium pion properties, electromagnetic form factor, radius and weak decay constant are presented in Section IV. Finally, Section V is devoted to a summary and discussions.
Ii Quarks in nuclear matter
In order to study consistently the modifications of the pion properties in a nuclear medium, we need a reasonable model of nuclear matter based on the quark degrees of freedom, as well as a pion model which successfully describes the pion properties in vacuum. We first discuss the quark model description of nuclear matter with the QMC model, with presenting some results calculated for symmetric nuclear matter.
ii.1 Quark Model of Nuclear Matter: QMC Model
The QMC model was introduced in 1988 by Guichon (34) with the MIT bag model, and by Frederico et al. in 1989 (37) with help of a confining harmonic potential, both approaches to describe nuclear matter properties based on the quark degrees of freedom. The model has been successfully applied for various studies of finite (hyper)nuclei (35) as well as the hadron properties in a nuclear medium (see Ref. (36) for a comprehensive review). In the model the medium effects arise through the self-consistent coupling of phenomenological isoscalar-Lorentz-scalar (), isoscalar-Lorentz-vector () and isovector-Lorentz-vector () meson fields to the confined light-flavor and valence quarks — rather than to the nucleons. As a result the internal structure of the bound nucleon is modified by the surrounding nuclear medium with respect to the free nucleon case.
where , and are respectively the nucleon, Lorentz-scalar-isoscalar , and Lorentz-vector-isoscalar field operators, with
which defines the -field dependent coupling constant, , while is the nucleon- coupling constant. All the important effective nuclear many-body dynamics including 3-body nucleon force modeled at the quark level, will effectively be condensed in . Solving the Dirac equations for the up and down quarks in the nuclear medium with the same mean fields (mean values) and , which act on the bound nucleon self-consistently based on Eq. (1), we obtain the effective -dependent coupling at the nucleon level (34); (35); (36). The free meson Lagrangian density is given by,
where we have ignored the isospin-dependent Lorentz-vector-isovector -meson field, since we consider isospin-symmetric nuclear matter within the Hartree mean-field approximation. In this case the mean value of the -mean field becomes zero and there is no need to consider its possible contributions due to the -Fock (exchange) terms.
In the following we work in the nuclear matter rest frame. For symmetric nuclear matter in the mean-field approximation, the nucleon Fermi momentum (baryon density ) and the scalar density () associated with the -mean field can be related as,
where is the constant value of the effective nucleon mass at a given density and is calculated in the quark model. In the standard QMC approach (34); (35); (36), one uses the MIT bag model and the Dirac equations for the up and down quarks in symmetric nuclear matter are solved self-consistently with the and mean-field potentials. The Dirac equations for the quarks and antiquarks ( or , and or , quarks) in the bag of hadron in nuclear matter at the position ( bag radius) are given by (36),
where we have neglected the Coulomb force as usual, since the nuclear matter properties are due to the strong interaction, and we assume SU(2) symmetry for the light quarks, , and define . In symmetric nuclear matter, the isospin dependent -meson mean field in Hartree approximation yields in Eqs. (6) and (7), so we ignore it hereafter. The constant mean-field potentials in nuclear matter are defined by, and , with and being the corresponding quark-meson coupling constants, and the quantities inside the brackets stand for taking expectation values by the nuclear matter ground state (36). Note that, since the velocity averages to zero in the rest frame of nuclear matter, the mean vector source due to the quark fields as well, . Thus we may just keep the term proportional to in Eqs. (6) and (7).
The normalized, static solution for the ground state quarks or antiquarks with flavor in the hadron , may be written, , where and are the normalization factor and corresponding spin and spatial part of the wave function. The bag radius in medium for a hadron , , is determined through the stability condition for the mass of the hadron against the variation of the bag radius (36). The eigenenergies in units of are given by,
The hadron masses in a nuclear medium (free mass ), are calculated by
where , with , , and being the lowest bag eigenfrequencies. and are the quark (antiquark) numbers for the quark flavors and , respectively. The MIT bag quantities, , , , and are the parameters for the sum of the c.m. and gluon fluctuation effects, bag constant, lowest eigenvalues for the quarks or , respectively, and the corresponding current quark masses. and () are fixed by fitting the nucleon (the hadron) mass in free space. (See table 1 the nucleon case.)
For the nucleon case in the above, the lowest, positive bag eigenfunction is given by
with and the spin function and
where is the eigenvalue for the lowest mode, which satisfies the boundary condition at the bag surface, .
The same meson mean fields and for the quarks satisfy the following equations at the nucleon level self-consistently:
where is the constant value of the scalar density ratio (34); (35); (36). Because of the underlying quark structure of the nucleon used to calculate in the nuclear medium (see Eq. (10) with ), gets -dependence, whereas the usual point-like nucleon-based model yields unity, . It is this or that gives a novel saturation mechanism in the QMC model, and contains the important dynamics which originates in the quark structure of the nucleon. Without an explicit introduction of the nonlinear couplings of the meson fields in the Lagrangian density at the nucleon and meson level, the standard QMC model yields the nuclear incompressibility of MeV, which is in contrast to a naive version of quantum hadrodynamics (QHD) (38) (the point-like nucleon model of nuclear matter), results in the much larger value, MeV; the empirically extracted value falls in the range MeV. (See Ref. (39) for the updated discussions on the incompressibility.)
Once the self-consistency equation for the , Eq. (15), has been solved, one can evaluate the total energy per nucleon:
We then determine the coupling constants, and , so as to fit the binding energy of 15.7 MeV at the saturation density = 0.15 fm ( = 1.305 fm) for symmetric nuclear matter.
The pion model we adopt here (32) uses a vacuum constituent quark mass, MeV, in order to well reproduce the electromagnetic form factor data and decay constant. Therefore, to be consistent with this pion model, our nuclear matter is built with the same vacuum mass. The corresponding coupling constants and some calculated properties for symmetric nuclear matter at the saturation density, with the standard values of MeV and MeV, are listed in Table 1. For comparison, we also give the corresponding quantities calculated in the standard QMC model with a vacuum quark mass of MeV (see Ref. (36) for details). Thus we have obtained the necessary properties of the light-flavor constituent quarks in symmetric nuclear matter with the empirically accepted data for a vacuum mass of MeV; namely, the density dependence of the effective mass (scalar potential) and vector potential. The same in-medium constituent quark properties will be used as input to describe the pion immersed in symmetric nuclear matter.
In Figs. 1, 2 and 3, we respectively show our results for the negative of the binding energy per nucleon (), effective mass of the nucleon, , and effective mass of the constituent up and down quarks, , in symmetric nuclear matter.
As one can expect from the values of the incompressibility, MeV in Table 1, the result for with MeV shown in Fig. 1 varies slightly faster than that for MeV (36) with increasing density. As for the effective nucleon mass shown in Fig. 2 with MeV, also decreases faster than that for MeV (36) with increasing nuclear density.
Concerning the effective constituent quark mass shown in Fig. 3, a general comment in connection with the light-front model (32); (40) is in order: due to the pole structure of the propagators, the sum of the in-medium constituent quark masses must be larger than the effective mass of the pion, , namely . Moreover, the pion mass up to normal nuclear matter density is expected to be modified only slightly, where the modification at nuclear density fm averaged over the pion isospin states is estimated as MeV (1); (7); (17); (11). Therefore, we approximate the effective pion mass value to be the same as in vacuum, up to fm. In Fig. 3 we present the results for the calculated effective mass of the constituent quarks, , up to , focusing on the relevant region where is satisfied.
ii.2 Quark Propagator and Pion Vertex in Symmetric Nuclear Matter
In general, the quark self-energy in symmetric nuclear matter is modified by the Lorentz-scalar-isoscalar and Lorentz-vector-isoscalar potentials. In the Hartree mean field approximation discussed in section II, the modifications enter as the shift of the quark (antiquark) momentum via ; for quark(antiquark)) due to the vector potential, and in the Lorentz-scalar part through the the Lorentz-scalar potential as . Since the Lorentz transformation properties are retained in nuclear matter, these modifications can be implemented in the pion model (32) without difficulties. Then, the up or down quark propagator (containing the quark and antiquark components) in symmetric nuclear matter is given by,
while the in-medium pion vertex (32) is modified as,
where the normalization factor associated with is also modified by the medium effects. The regulator mass represents soft effects at short range, namely at about the 1 GeV scale, and may also be influenced by in-medium effects. However, since there exists no established way of estimating this effect on the regulator mass and we already approximate , we also employ . In addition, since it is correlated with the in-medium modified constant discussed in Section III, we use the vacuum regulator mass value to avoid introducing another source of uncertainty.
Iii The in-medium electromagnetic form factor model
The electromagnetic interaction of a pion, a spin-zero bound composite system in vacuum, is based on three ingredients (32); i) effective Lagrangian which models the coupling of the pion field to the quark fields, ii) a symmetric vertex function in momentum space, iii) effective constituent quark masses and the lowest Fock state. We follow the procedure in vacuum i) iii) with the in-medium constituent quark properties as input and calculate the in-medium pion properties using an effective Lagrangian density with a pseudoscalar coupling (41),
where is the coupling constant and is the in-medium vertex function. The coupling constant is given by the Goldberger-Treiman relation at the quark level, , with the in-medium pion decay constant . The constant in Eq. (19) is determined from the charge normalization for the spin-zero composite system and it is density dependent. The photon field is coupled the minimal way satisfying current conservation. The front-form coordinates are defined as, , and
The electromagnetic current associated with the is obtained from the corresponding Feynman triangle diagram;
where , and is the number of colors. The factor 2 stems from isospin algebra. (It is easy to prove that the Ward identity is satisfied in the Breit-frame: first one performs the trace in , and notices that the integrand of the resulting expression is odd in , and therefore .)
We choose the symmetric vertex function, Eq. (19), which also produces a symmetric light-front wave function under the interchange of the quark and antiquark momenta, which improves the physical description without the conceptual difficulties associated with the non-symmetric regulator (see also Refs. (31); (42)). The normalization constant in Eq. (19) is fixed by imposing the condition on the pion form factor.
The effect of the vector potential in the loop integral cancels identically due to the choice of the pion vertex. Therefore, only the mass shift of the quarks is relevant in the loop integral. In the four-momentum integration, we apply the transformation, , so that no trace of the vector potential remains in the expressions of the pion decay constant and electromagnetic form factor. It is worthwhile to observe that, the current Eq. (21) means to be covariant, while the quark propagator and pion vertex are computed in the rest frame of nuclear matter. However, both the shift of the time component of the quark momentum and effective mass in the medium allow us to recover a boost invariant form for the form factor as the shift in the virtual quark energy can be absorbed by a variable change in the loop momentum, since we have assumed that the vertex parameter is unchanged by the medium effects, which may be justified by the fact that it corresponds to a short-range scale deep inside the pion wave function, which is much smaller than the nucleon size.
We work in the Breit-frame, where the momentum transfer, , , lies in the plane with , and (the angle in Ref. (31) corresponds to ). The initial and final momenta of the composite spin zero bound state with mass are: , and . The Drell-Yan condition is recovered with , while the condition (43) in the Breit-frame follows from . We here use .
In general, the pion form factor in medium can be extracted from the covariant expression:
If covariance and current conservation are satisfied in the calculation, one can obviously compute the electromagnetic form factor in any frame and from any nonvanishing component of the current.
In the following, to compute the pion elastic form factor, we use the pseudo-scalar Lagrangian density of Eq. (20), the current defined by Eq. (21), and the symmetric vertex function given by Eq. (19) with the plus component of the current, , associated with Eq. (21) in the Breit-frame (with in the plane). Only two nonvanishing contributions in Eq. (21) contribute to the form factor (44); (33); (31); (45):
With the replacement , the two contributions to the form factor obtained from , which correspond to the two diagrams shown in Fig. 4, are given by the following expressions:
with the “in-medium on-the-energy shell” values of the individual momentum given by
For the trace of the operator in Eq. (26),
The detailed forms of and in vacuum after integration over can be found in Appendices A and B of Ref. (32).
The sum of the contributions and in vacuum was already shown to yield the covariant result (32). The different directions of in the Breit-frame can only change the absolute values of and , but not the sum. For example, with we have , and thus alone yields the covariant result (32).
The in-medium quark Dirac propagator after the variable shift, , can be decomposed using the front-form momenta (26),
where . The second term on the right-hand side of Eq. (31) is an instantaneous term in the light-front time. The instantaneous term contributes to both, and , due to the analytic structure of the symmetric vertex function of Eq. (19). These contributions are of nonvalence nature, as they are not reducible to the impulse approximation within the valence wave function.
Iv Valence Light-front wave function
The valence component of the light-front wave function in vacuum was obtained in Ref. (32). The external two-fermion space-time coordinates of the Bethe-Salpeter amplitude are constrained to equal light-front time after dropping the instantaneous terms of the external Dirac propagators (30). However, the effect of the instantaneous terms in a Bethe-Salpeter approach is included in the effective operators, together with the valence wave function (30). In the present treatment, the Bethe-Salpeter amplitude with the in-medium pion vertex of Eq. (19) can be written as (32),
Separating out the instantaneous terms in the quark propagators as well as the remaining spinor operator part in the numerator of Eq. (32) and the and factors in Eq. (32), the momentum part (the part depends on the plus and transverse momenta) of the valence component of the light-front wave function with is given by,
where is a normalization factor,
Performing the integration in Eq. (33), one has
where , with ; , and the square of the mass is . Since the momentum part of the wave function is symmetric under the exchange of the fermion momenta, we have a second term in Eq. (34), which is different from Ref. (33).
Once the normalization constant is obtained from the condition (see Eq. (23)), the probability of the valence component for the pion in medium can be calculated by setting .
For convenience, we introduce the transverse momentum probability density,
and integration of leads to the in-medium probability of the valence component in the pion:
The in-medium pion decay constant, , is defined as the matrix element of the partially conserved axial vector current in symmetric nuclear matter, with the ground state :
Using and the interaction Lagrangian density, Eq. (20), for the pion-