A New Approach to Physics of Nuclei

E. G. Drukarev, M. G. Ryskin, V. A. Sadovnikova
Petersburg Nuclear Physics Institute,
Gatchina, St. Petersburg 188300, Russia

We employ the QCD sum rules method for description of nucleons in nuclear matter. We show that this approach provides a consistent formalism for solving various problems of nuclear physics. Such nucleon characteristics as the Dirac effective mass and the vector self-energy are expressed in terms of the in-medium values of QCD condensates. The values of these parameters at saturation density and the dependence on the baryon density and on the neutron-to-proton density ratio is in agreement with the results, obtained by conventional nuclear physics method. The contributions to and are related to observables and do not require phenomenological parameters. The scalar interaction is shown to be determined by the pion–nucleon -term. The nonlinear behavior of the scalar condensate may appear to provide a possible mechanism of the saturation. The approach provided reasonable results for renormalization of the axial coupling constant, for the contribution of the strong interactions to the neutron–proton mass difference and for the behavior of the structure functions of the in-medium nucleon. The approach enables to solve the problems which are difficult or unaccessible for conventional nuclear physics methods. The method provides guide-lines for building the nuclear forces. The three-body interactions emerge within the method in a natural way. There rigorous calculation will be possible in framework of self-consistent calculation in nuclear matter of the scalar condensate and of the nucleon effective mass .

PACS numbers 21.65.-f, 21.65.Mn, 24.85.+p

C o n t e n t s

  1. Introduction

  2. Nucleon QCD sum rules in vacuum

    2.1.   General ideas
    2.2.   Explicit form of the SR equations
    2.3. Inclusion of radiative corrections

  3. QCD sum rules in nuclear matter

    3.1.   Choice of the variables
    3.2.   Operator product expansion
    3.3.   Model of the spectrum

  4. Nucleon self-energies in the lowest orders of OPE

    4.1.   General equations
    4.2.   Left-hand sides of the sum rules
    4.3.   Approximate solution
    4.4.   Gas approximation
    4.5.   Asymmetric matter
    4.6.   Possible mechanism of saturation

  5. Other characteristics of the in-medium nucleons

    5.1.   Axial coupling constant
    5.2.   Charge-symmetry breaking forces
    5.3.   Nucleon deep inelastic structure functions

  6. Intermediate summary

    6.1.   Reasons for optimism
    6.2.   Reasons for scepticism
    6.3.   Response to the sceptical remarks

  7. Four-quark condensates

    7.1.   General equations for contribution of the four-quark condensates
    7.2.   Approximations for the four-quark condensates
    7.3.   Perturbative Chiral Quark Model
    7.4.   Four-quark condensates in the PCQM

  8. Contribution of the higher order terms

    8.1.   Symmetric matter with the four-quark condensates
    8.2.   Radiative corrections
    8.3.   Asymmetric matter
    8.4.   Many-body interactions

  9. Self-consistent scenario

  10. Summary

  11. Epilogue

1 Introduction

In the present paper we review our approach to description of a nucleon, placed into nuclear matter. The in-medium characteristics of nucleon can tell us much about the medium itself. On the other hand, the introduction of nuclear matter enables to separate the problems of nucleon interactions from those, connected with individual features of specific nuclei. Thus investigation of nucleon characteristics in nuclear matter is an important step for studies of physics of nuclei.

Until mid 70-th the studies of nuclear matter were based on the nonrelativistic approach. Since the publication of the paper [1] description of the in-medium nucleon was based on Dirac phenomenology. It was successful in describing most of characteristics of nucleons both in nuclear matter and in finite nuclei [2, 3]. In the meson exchange picture the vector and scalar fields correspond to exchange by the vector and scalar mesons between a nucleon and the nucleon of the matter. This picture is called Quantum Hadrodynamics (QHD). In the simplest version (QHD-I) only vector and scalar mesons are involved. In more complicated versions some other mesons are included [2, 4]. While the studies in framework of the nonrelativistic approach are going on, i.e. the applications of the Nuclear Density Functional Method provided very accurate description of the data [5, 6].

On the other hand, many efforts have been made to improve the QHD. There were many reasons for this. First of all, the QHD itself has several weak points. It is not clear, if the scalar meson does exist, the experimental data are controversial. It is rather an effective way of describing the two-pion exchange. The mass of this effective state is about 500 Mev. The mass of vector meson is about 780 MeV. Hence, the exchange by theses mesons takes place at the distances, where the nucleon can not be treated as a point -like particle. Thus the QHD inherited the problems of nonrelativistic phenomenology, connected with description of the interaction at small distances. The other weak points are discussed in [7, 8]. Also, it is desirable to match the QHD with the Quantum Chromodynamics (QCD ) which is believed to be a true theory of strong interactions. Another reason is that there are various problems in nuclear physics. It is desirable to have an approach, which would enable to calculate:

  • The nucleon single-particle potential energy , where in the density of the baryon quantum number. This enables one to find the saturation density and the single-particle binding energy .

  • Parameters of interaction with external fields. These are magnetic moments and the axial coupling constant . The latter is important for understanding of the chiral properties of the matter.

  • Neutron–proton mass splitting in isotope-symmetric matter. It was observed for a number of nuclei, being known as the Nolen–Schiffer anomaly.

  • Structure functions of the deep inelastic scattering. They describe the internal structure of nucleons. Investigation of the latter is important for construction of the quark models of nucleons, for studies of the confinement, etc. The reasons for some of the in-medium modifications of the structure functions are still obscure. They become a subject for discussions from time to time.

  • The single particle potential energy for hyperons in nuclear matter. Can a system of hyperons be stable? In other words, can a strange matter exist ?

As we said earlier, the first problem was solved in framework of Dirac phenomenology [1, 2] for the values of density close to the saturation value. The nucleon was considered as a relativistic particle, moving in superposition of vector and scalar fields and . In the rest frame of the matter . The dynamics of the nucleon is described by equation


with , . In nuclear matter the fields and depend only on the density , and do not depend on the space coordinates. The values of the fields are adjusted to reproduce either the data on nucleon–nucleon scattering or the nuclear data – see, e.g. [3]. However, each of the other listed problems requires additional improvements of the QHD. Turning to the other problems from the above list, note that the axial coupling constant changes due to polarization of medium by pions [9], with a crucial role of the delta-isobar excitations [10]. Thus, in order to solve the second problem one should introduce additional degrees of freedom into the QHD. Also, the third problem requires introduction of the nuclear forces, which break the isospin invariance [11]. Finally, the fourth problem is just unaccessible for traditional methods of nuclear physics.

There were several attempts to combine the QHD and the quark structure of nucleons. The nucleon-nucleon forces, constructed within such models enabled to reproduce semi-quantitatively or quantitatively the nucleon characteristics in nuclear matter. In the Quark–Meson Coupling (QMC) model [12] the nucleon was considered as a three-quark system in a bag. The quarks were directly coupled to and mesons. The values of the nucleon effective mass and of the in-medium coupling constant appeared to be somewhat smaller than in QHD [13]. This model is employed nowadays as well [14].

In another class of works the short-range interaction between the nucleons was treated as interaction between their quarks. The latter were described in framework of some QCD motivated models. The long-range NN interaction was described in terms of nucleons and pions. These works were reviewed in [15].

During the two latest decades much work was done on development of the Effective Field Theory (EFT). The starting point is the most general Lagrangian, which includes nucleons and pions as the degrees of freedom and respects all the symmetries of QCD, i.e. the Lorentz invariance, chiral symmetry, etc. [16]. The applications of the EFT is usually combined with the expansion in powers of the pion mass (Chiral Perturbation Theory). Expansion in powers of low momenta is also usually carried out. The works, based on the EFT are reviewed in [17], for some of recent developments - see [18]. The EFT approach does not touch the quark degrees of freedom of the nucleons.

All the traditional nuclear physics approaches face difficulties in attempts to describe the nucleon-nucleon (NN) interaction at small distances. On the other hand, the Quantum Chromodynamics (QCD) is believed to be a true theory of strong interactions. It has many unsolved problems at large distances. However, it becomes increasingly simple at small distances due to the asymptotic freedom. The interaction of quarks and gluons, which are the ingredients of QCD can be treated perturbatively. This is known as the asymptotic freedom [19]. It is tempting to use this feature of QCD in building the nucleon forces. One should take into account, however, that due to spontaneous breakdown of the chiral symmetry of the QCD, the vacuum expectations of some QCD operators (condensates) have nonzero values.

In medium with a nonzero value of the density of the baryon quantum number the condensates change their values. Also, some other condensates, which vanish in vacuum, obtain nonzero values.

The QCD sum rules (SR) method describes the vacuum hadron parameters basing on the quark dynamics at short distances, where the asymptotic freedom works. In other words, the dynamics of the quark system at the distances of the order of the confinement radius is described basing on that at small distances, where it is determined by the QCD condensates. Thus, the SR method enables to describe the hadron parameters in terms of the QCD condensates.

The approach was worked out in [20], where it was used for mesons. It is based on the dispersion relation for the function, describing the system which carries the quantum numbers of the hadron. The SR method was successfully applied for nucleons in vacuum [21], describing all their static and some of dynamical characteristics – see [22, 23] for a review. Thus it looks reasonable to try to apply the QCD SR method for the description of nucleons in nuclear matter. It was suggested in [24, 25, 26] that the parameters of nucleon in nuclear matter can be expressed in terms of the in-medium values of QCD condensates. In medium with a nonzero value of the density of the baryon quantum number the condensates change their values. Also, some other condensates, which vanish in vacuum, obtain nonzero values.

The generalization of the SR method for the case of finite densities was not straightforward. One of the main problems was the choice of variables, which enabled to separate the singularities connected with the in-medium nucleon from those connected with the medium itself. This was done in [24][26].

It was found also in these papers that the nucleon characteristics (effective mass and the vector self-energy ) can be presented in terms of the vector and scalar condensates


Here and are the density and vector of the ground state of the matter, is the quark field, the summation over flavors is carried out. The vector condensate is written in the rest frame of the matter. Due to conservation of the vector current the vector condensate is a linear function of


is just the number of valence quarks in a nucleon. The scalar condensate can be represented as [24, 25]


with caused by interaction of the nucleons of the matter. Since can be expressed in terms of the pion–nucleon term [27], while the latter is related to observables [28, 29], one can obtain the values of the nucleon parameters at least in the gas approximation. Since is small for the densities, close to the saturation point (see below), the values obtained in such approach are close to the physical ones. They appeared to be MeV, and MeV close to the saturation point. Thus, including only the condensates of the lowest dimension and neglecting the radiative corrections, we found that the QCD SR method reproduce the main features of the QHD [1].

However, the role of the condensates of higher dimension remained obscure. The contribution of the gluon condensate is rather small. However, an estimation for the value of the four-quark condensate , which suggests itself, destroys the agreement with the Walecka model. Also, the lowest order radiative corrections are numerically large. This took place for the vacuum as well. This caused doubts in possibility to expand the SR method for the case of finite densities.

The four-quark condensates are the most important among those of the higher dimension. Their calculation requires some model assumption on the quark structure of nucleon. We employed the Perturbative Chiral Quark Model (PCQM), suggested originally in [30]. We calculated these condensates [31] and found that previous naive estimation of its contribution was wrong, due to some cancelations which take place in any reasonable model.We demonstrated that the SR method provides the dependence of the nucleon characteristics and on the density and on the neutron-to-proton density ratio [32, 33], which is consistent with the results obtained by using traditional nuclear physics methods.

We analyzed the role of radiative corrections for the nucleon SR in vacuum and demonstrated that their influence on the value of the nucleon mass is small [34]. Also, in nuclear matter the radiative corrections do not change much the nucleon characteristics and [35].

We found that the nonlinear contribution to the scalar condensate is determined mainly by the pion contribution to the self-energy of the nucleon of the matter. Simple estimations show that this term may provide a saturation mechanism in our approach [36]. However, a rigorous treatment requires renormalization of the pion propagator by the particle-hole excitations. The renormalized pion propagator depends on the nucleon effective mass and on the in-medium value of the pion–nucleon coupling constant .The latter can be obtained by the SR method. This brings us to a self-consistent scenario. At the present level of our knowledge it requires some more phenomenological assumptions [37].

Note also, that the finite density SR method provided reasonable results for the axial coupling constant [38], for the neutron–proton mass splitting [39] and for the difference between the deep inelastic structure of nucleus and that of sum of those of free nucleons [40]. However, in these calculations only the condensates of the lowest dimensions have been included.

Now we give the details.

2 Nucleon QCD sum rules in vacuum

This approach is described in details in many publications. Earlier papers are reviewed in [41]. The nowadays state of art is presented in [22] and [23], see also the book [19]. However, to make the text self-consistent we recall the main points of the approach. We emphasize the points, which we shall need for the extension of the SR method for the case of finite density of the baryon quantum number.

2.1 General ideas

The nucleon QCD sum rules succeeded in describing of the nucleon characteristics in vacuum in terms of the vacuum expectation values of the products of quark or (and) gluon operators (QCD condensates) [7, 8]. This approach is based on the dispersion relation for the function

describing the propagation of the system which carries the quantum numbers of the proton, is the unit matrix. In the simplest form the dispersion relations are


As we shall see below, we do not need to worry about possible subtractions.

In quantum mechanics is just the proton propagator. In the field theory different degrees of freedom are important in different regions of the value of .

One can consider the proton as a system of three strongly interacting quarks. Due to asymptotic freedom of QCD
cite18a the description becomes increasingly simple at . This means that at the function can be presented as a power series of and of the QCD coupling constant . The coefficients of the expansion in powers of are the QCD condensates. Such presentation known as the Operator Product Expansion (OPE) [42] provides the perturbative expansion of the short distance effects, while the nonperturbative physics is contained in the condensates. In QCD SR approach the left-hand side (LHS) of Eq.(5) is considered at , and several lowest order terms of the OPE are included. In this and in the next Subsections we neglect the radiative corrections, i.e. we do not include interactions and self-interactions (self-energy insertions) of the quarks, putting .

Turning to the right-hand side (RHS) of Eq. (5) note that Im at with being the position of the lowest lying pole, i.e. is the proton mass. There are the other singularities at larger values of . These are the cuts corresponding to the systems “proton+pions”, the pole N(1440), etc. The next to leading singularity is the physical branching point , and one can write


with , ; – the residue at the pole. Now Eq. (5) takes the form


The upper index OPE means that several lowest OPE terms are included.

The SR approach is focused on studies of the lowest state. Thus we keep the first term on the RHS of Eq. (7) and try to write approximate expression for the contribution of the higher states. The detailed structure of the spectral function in the second term on the RHS of Eq. (7) cannot be obtained by the SR method. However, at


with denoting the discontinuity.

The standard ansatz consists in extrapolation of Eq. (8) to the lower values of , assuming that the cut starts at certain unknown point . In other words

and thus Eq. (5) takes the form


Recall that .

This “pole+continuum” presentation of the RHS makes sense only if its first term, treated exactly is larger than the second term, which approximates the higher states. The position of the lowest pole , its residue and the continuum threshold are the unknowns in Eqs. (9).

In the next step one usually applies the Borel transform. It is defined as


converting a function of into the Borel transformed function of the Borel mass . The reasons for applying the transform are

  • Since for any integer , it kills the polynomials of . Hence, it eliminates the divergent terms in the function . This explains, why we wrote the dispersion relation (5) without subtractions.

  • It emphasizes the contribution of the lowest state, since

  • It improves the convergence of the OPE series, since

The Borel transformed SR take the form


If both LHS and RHS of Eq. (11) were calculated exactly, the relation would be independent on . However, certain approximations are made on both sides. The analytical dependence of both sides on is quite different. The OPE on the LHS becomes increasingly true at large . The accuracy of the “pole+continuum” model used for the RHS increases at small . An important assumption is that there is an interval of the values of , where the two sides have a good overlap, approximating also the true functions .

Thus our task is to find the region of the values of , where the overlap can be achieved and to obtain the characteristics of the lowest state and and the value of the continuum threshold .

2.2 Explicit form of the SR equations

The general definition of the function , which is sometimes called “correlator” or the ”correlation function” is


where is the local operator with the proton quantum numbers. It is often refereed to as “current”. It was shown in [36] that there are three independent currents


Here are the color indices, is the charge conjugation matrix, while denotes the transpose in the Dirac space. It was shown in [43] that the operators and provide strong admixtures of the states with negative parity and of the states with spin 3/2 correspondingly. Thus, the calculations with the operator are most convincing. We shall assume in further studies.

Expansion of the matrix element on the RHS of Eq. (12) in powers of corresponds to expansion of its LHS in powers of . In the lowest orders of expansion the product on the RHS of Eq. (12) can be written in terms of those of two quark operators. The latter can be written, following the Wick theorem




is the free quark propagator, is the quark mass, are the basic matrices with the scalar, pseudoscalar, vector, pseudovector and tensor structures, i.e. and . For the fields which respect the chiral invariance all the expectation values on the RHS of Eq. (14) vanish. However in the QCD the expectation value () has a nonzero value. All the other condensates vanish due to invariance of vacuum. Thus


Now we present

with denoting the dimension of the condensate, contributing to the term or .

If all expectation values of the products of the quark fields are described by the free propagators (15), we find the leading OPE contribution to the structure , corresponding to the free three-quark loop. If one of the quark pairs is described by the second term on the RHS of Eq. (14), while the others are given by the first term, we find the leading contribution to the structure ). In this case two quarks form a free loop, while the current exchanges by a quark–antiquark pair with vacuum.

Direct calculation provides [21]


Here is the cutoff of the integral over in Eq. (12). Its value is not important, since the terms containing will be eliminated by the Borel transform.

Going beyond the single-particle presentation (14) one finds also the next to leading OPE terms. For example, the contribution is due to the lowest order interaction of the quark system with the gluon condensate

with and the color-electric and color-magnetic fields. This condensate also has a nonzero value only due to the violation of the chiral symmetry in the ground state of the QCD. The higher terms contain the four-quark condensate


Finally, the Borel-transformed sum rules can be written as




with . The factor is introduced in order to deal with the values of the order of unity (in GeV units). The contribution of continuum is moved to the LHS of Eqs. (19) (see Eq. (22) below). Following [21, 44] we can write


The terms and are the Borel transforms of the contributions and to the functions , with subtraction of the corresponding contributions of continuum. Recall that the lower index denotes the dimension of the condensate. The terms, proportional to condensates of higher dimension and do not contain the logarithmic loops and thus do not contribute to continuum. Actually the calculations are carried out in the chiral limit . The explicit form of the contributions is [21, 44]






The contributions are illustrated by Fig. 1. Note that we provided these equations mostly as illustration of the main ideas and did not include several numerically not very important terms.

The term presents the contribution of the four-quark condensates , which, generally speaking, obtain nonzero values for all structures . It is evaluated under the factorization approximation [6]

One can find numerical values of the main QCD condensates presented by Eq. (24). There is the well known Gell-Mann–Oakes–Renner relation (GMOR) for the scalar condensate [45]


Here and are the decay constant and the mass of the meson, and are the current masses of and quarks. Its numerical value is . The value of the gluon condensate was extracted from the analysis of leptonic decay of and mesons [46]. This data were supported by the QCD sum rules analysis of charmonium spectrum [20].

Now one must find the set of parameters , which insure the most accurate approximation of by the functions and also the interval of the values of , where this can take place.The set of parameters , which minimize the function


will be referred to as a solution of Eqs. (11).

The appropriate interval


and the values of nucleon parameters


were found in [21, 44]. If we fix GeV, the SR provide


These values will be employed in the present paper.

It was shown also in [21] that the nucleon mass vanishes if there is no spontaneous breakdown of the chiral symmetry, e.g. if . Numerically [21, 44]


In the QCD SR approach the mass of the nucleon is formed due to exchange by quarks between the nucleon and vacuum.

The role of instantons in QCD SR was analyzed in [47], [48]. It was shown that for the current the contribution of instantons vanishes for the structure of the SR. In the scalar structure instantons form a factor, which change the LHS of SR by about . Following [22] we include the contribution into uncertainties of the numerical value of the vacuum expectation value . However, a rigorous analysis requires investigation of the dependence of the contribution.

Note that the solutions (28), (29) are not stable with respect to modification of the values of the condensates [49]. Even for the small changes the absolute minimum of the RHS of Eqs. (19) is provided by another solution, i.e. GeV, , [50]. We treat this solution as an unphysical one, since the contribution of the continuum exceeds more than twice that of the lowest pole. This contradicts the key assumption of the “pole+continuum” model for the spectrum – see Eq. (9).

The nucleon SR with another form of nuclear current were obtained in [51]. In [52] it was used also for the description of delta isobars. Further we shall mention some other applications.

2.3 Inclusion of radiative corrections

A typical radiative correction is shown in Fig. 2. In the analysis, carried out in [21, 44] the most important radiative corrections of the order have been included. These contributions were summed to all orders of . This is called the Leading Logarithmic Approximation. The LLA corrections are expressed in terms of the factor [53]


where MeV is the QCD scale, while is the normalization point, the standard choice is MeV.


Here the upper index indicates inclusion of the radiative corrections.

The radiative corrections to the OPE terms of the function have been calculated beyond the LLA in the lowest order of the expansion [54] (see also [55]). The results are


The running coupling constant in the one-loop approximation, with inclusion of three lightest quarks is


The calculations in [54, 55] have been carried out for . Since the linear momenta in the loops, corresponding to radiative corrections are of the order , it is reasonable to assume that in Eq. (34) , converting to GeV. Assuming (somewhat larger values are often used nowadays [56]) we find that the radiative correction to the contribution changes its value by about . This “uncomfortably large” correction was often claimed as the most weak point of the SR approach [57].

In [34] we investigated the influence of radiative corrections on the values of characteristics, obtained in framework of the Borel transformed nucleon SR in vacuum. We demonstrated that inclusion of the radiative corrections in various ways (the radiative corrections are totaly neglected, are included in LLA, are taken into account beyond the LLA in the lowest order) alter the value of nucleon mass by about . The radiative corrections modify mainly the value of the nucleon residue.

Note also that inclusion of the radiative corrections diminishes the role of the unphysical solution, mentioned above. Once they are included, minimization of the function defined by Eq. (26) is provided by the physical solution in a broader interval of the values of the condensates [49, 50].

3 QCD sum rules in nuclear matter

3.1 Choice of the variables

Now we shall try to use the SR approach for calculation of the nucleon parameters in nuclear matter. The propagation of the system which has a four-momentum and carries the quantum numbers of the proton is determined by the equation


with the ground state of the nuclear matter. It is just an analog of Eq. (12). We consider nuclear matter as a system of nucleons with momenta , introducing


In the rest frame of the matter .

The spectrum of the function is much more complicated than that of the vacuum function . Our main task is to separate the singularities connected with the nucleon in the matter from those connected with the matter itself. Include in the first step only the two-nucleon interactions. The singularities connected with the matter manifest themselves as singularities in variable . Thus the separation can be done by considering and keeping . Thus we consider the dispersion relations


for the three structures of the function


Each contribution can be viewed as the sum of the vacuum term and that provided by the nucleons of the matter


This notation will be used also for the other functions.

We clarify the value of putting


with being the relativistic value of the nucleon energy on the Fermi surface. This insures that the nucleon pole on the RHS of Eq. (37) describes the nucleon, added to the Fermi surface. For the analysis, carried out in this section we can neglect the bound, thus putting


This was the choice of variables in our papers [9][11], [32, 33] and [35][40]. It was used also in [58], where the approach was used for calculation of the nucleon–nucleus scattering amplitude.

Note that the vacuum dispersion relation (5) can be viewed as a relativistic generalization of the nonrelativistic dispersion relation in time component (energy) , known as the Lehmann representation [59]. The latter is based on casuality. It converts into a dispersion relation in after being combined with the symmetric relation in negative values of . The reasoning does not work in medium, since the Lorentz invariance is lost. To prove the dispersion relation (37) we must be sure of the possibility of the contour integration in the complex plane. A strong argument in support of this possibility is the analytical continuation from the region of real . At these values the asymptotic freedom of QCD enables one to find an explicit expression for the integrand. The integral over the large circle may have a nonvanishing contribution. However, the latter contains only polynomials in which are killed by the Borel transform. Thus we consider dispersion relations in to be a reasonable choice.

On the contrary, dispersion relations in contain all possible excited states of the matter on its RHS. To illustrate the latter point, consider photon propagation in medium (see, i.e. [60]).

In vacuum the propagator of the photon which carries the energy and linear momentum is . It has a pole at . In medium it takes the form . Here is the dielectric function, related to the amplitude of the photon scattering on the ingredients of the medium. If the photon energies are small enough, dependence of on can be neglected (this is known as the dipole approximation). However, is a complicated function of . It depends on the eigen energies of the medium. The same refers to the function . However, the function still has a simple pole at the point , reflecting properties of the in-medium photon. Straightforward calculation of the value is a complicated problem. The same is true for the in-medium proton. The SR are expected to provide the in-medium position of the pole in some indirect way.

The approach, in which the dispersion relation in at fixed value of the three-dimensional momentum is the departure point was developed in [61]. It was used for description of nucleons [61, 62], delta-isobars [63] and hyperons [64] in nuclear matter. The results are reviewed in [65]. However, the possibility to separate the singularities connected with the nucleon in the matter from those connected with the matter itself in this approach looks to us to be obscure.

Thus we expect the Borel transformed dispersion relations in with fixed value of to be more reliable.

3.2 Operator product expansion

Note that the condition enables to use the OPE of the LHS of Eq. (37). Indeed, we find


and thus in the rest frame of the matter



The exponential factor on the RHS of Eq. (35) can be written as

with – the direction of momentum , . Hence, the integral in Eq. (37) is determined by , , and the function can be expanded in powers of , corresponding to expansion of in powers of . Note that condition (43) is the same as that for the validity of the OPE for the structure functions of the deep inelastic scattering [19].

In the case of vacuum the expansion of quark fields in powers of was indeed an expansion in powers of , corresponding to a power series of for the vacuum function . In medium the fields can be expanded also in powers of . This leads to expansion in powers of of the function , and thus, generally speaking, to infinite number of condensates in each OPE term. Fortunately, due to the logarithmic dependence of the quark loops, the leading OPE terms contain only finite number of condensates. We shall give details in the next Section.

3.3 Model of the spectrum

Now we turn to the RHS of Eq. (37). The function defined by Eq. (35) can be written as


with standing for the system with the baryon number . Here is the ground state, summation over all states with is assumed. This equation is illustrated by Fig. 3

The matrix element contains the term , which adds the nucleon (the “probe nucleon”) to the Fermi surface of the state , while the rest nucleons are spectators. If interactions of this nucleon with the matter are neglected, this contribution to has a pole at . Interactions of the probe nucleon with the matter shift the position of the pole. In the mean field approximation shown in Fig. 4a the shift does not depend on . Going beyond the mean field approximation we find the Hartree self-energy diagrams (Fig.4b), which have singularities in . Due to condition (41) they do not add new singularities in variable to the function . The exchange (Fock) self-energy diagram is shown in Fig. 4c.

The matrix element contains also the terms with standing for the system, containing the nucleon and mesons. The current creates the nucleon and a meson with the mass , which is absorbed by the nucleons of the matter – see Fig. 5. This contribution has a cut in complex plane, at . It has also a cut in , which is not important for us, since is fixed.

The matrix element contains the terms , with describing a system with the baryon quantum number equal to zero. The other nucleons are spectators. The system can be a set of mesons, meson, etc. These contributions depend on the variable , providing singularities at , with the mass of state – see Fig. 6. Thus the lowest singularity in corresponds to the branching point , corresponding to the real two-pion state in the channel. A single-particle meson state with the mass generates a pole at . The diagram, shown in Fig. 4c. is also one of the contributions, which has singularities in the channel.

Note that the antinucleon state corresponding to generates the pole , shifted far to the right from the lowest lying state.

Thus the spectrum of the function consists of the pole at , a set of higher laying poles, generated by the channel and a set of branching points. The lowest lying branching point is separated from the position of the pole by a much smaller distance than in the case of vacuum ( in the latter case). Note, however, that at the very threshold the contribution is quenched since the vertices contain linear moments of intermediate pions. Thus the higher singularities can be considered as separated from the pole .

The situation becomes more complicated if we include the interaction of the probe nucleon with nucleons of the matter. The corresponding amplitudes depend on the variables . This causes the cuts, running to the left from the point . Its contribution thus is not quenched by the Borel transform. However, as we shall see in Sec. 8, such multinucleon interactions require inclusion of the condensates of high dimensions on the LHS of (37). Such contributions do not have logarithmic loops, thus contributing only to the pole terms on the RHS of Eq. (37). Hence, this singularities should be disregarded in our approach, and the three-body forces are included in the mean-field approximation in our approach.

To summarize the results of this Section, we use Borel transformed dispersion relations of the function at fixed . The OPE is used on the LHS. The ”pole+ continuum” model is used for the RHS. Now we have three SR equations


Here and are the position of the nucleon pole in medium and the value of its residue, is the in-medium value of the effective threshold. The meaning of the parameters will be clarified in next Section.

4 Nucleon self-energies in the lowest orders of OPE

In the Subsections we consider the symmetric nuclear matter, with equal number of protons and neutrons. Also, due to isotope invariance


4.1 General equations

Start with description of the nucleon pole. We can write for the inverse nucleon propagator in medium


with being the propagator of the free nucleon, while


is the general expression for the self-energy in the nuclear matter. In the kinematics, determined by Eq. (41) we find




For the new position of the nucleon pole we find




Thus in Eq. (45)


Note that Eqs. (50) correspond to those for the vector self-energy and for the effective mass in nuclear physics – see, e.g. [66]. Also in accordance with definition accepted in nuclear physics the scalar self-energy is


In the nonrelativistic limit the proton dynamics is determined by the potential energy


Keeping the only terms, which are linear in the density we find a simple expression for the shift of the position of the nucleon pole


Instead of Eqs. (19) we have now


Here are the Borel transforms of the LHS of Eq. (37), multiplied by – see the previous Section, while


where is the effective value of the in-medium value of the residue, i.e. , following Eqs. (49) and (52).

Employing Eqs. (57) and (58) one finds


4.2 Left-hand sides of the sum rules

4.2.1 Contribution of the condensates of lowest dimension

The calculation is based on the presentation of the single-quark propagator in nuclear matter