Interplay between Nucleon Interaction
and Nucleon Mass in Dense Baryonic Matter
[0.8cm] WonGi Paeng^{*}^{*}*email:
Department of Physics, Hanyang University, Seoul 133791, Korea
Hyun Kyu Lee^{†}^{†}†email:
Department of Physics, Hanyang University, Seoul 133791, Korea
Mannque Rho^{‡}^{‡}‡email:
Institut de Physique Théorique, CEA Saclay, 91191 GifsurYvette cédex, France &
Department of Physics, Hanyang University, Seoul 133791, Korea
Chihiro Sasaki^{§}^{§}§email:
Frankfurt Institute for Advanced Studies, D60438 Frankfurt am Main, Germany
February 14, 2021
Abstract
The dilatonlimit fixed point and the scaling properties of hadrons in the close vicinity of the fixed point in dense baryonic matter uncovered in hidden local symmetry implemented with spontaneously broken scale symmetry are shown to reveal a surprisingly intricate interplay, hitherto unsuspected, between the origin of the bulk of proton mass and the renormalizationgroup flow of the nuclear interactions. This rends a theoretical support to the previous (phenomenologically) observed correlation between the dropping nucleon mass and the behavior of the nuclear interactions in dense matter described in terms of half skyrmions that appear at a density denoted in skyrmion crystals. The role of the meson degree of freedom in the source for nucleon mass observed in this paper is highly reminiscent of its important role in the skyrmion description of nucleon mass in hidden local symmetric theory. One of the most notable novel results found in this paper is that the nucleon mass in dense baryonic medium undergoes a drop roughly linear in density up to a density (denoted ) slightly above nuclear matter density () and then stays more or less constant up to the dilaton limit fixed point. The possibility that we entertain is that coincides with or at least close to . We note that this feature can be economically captured by the paritydoublet model for nucleons with the chiralinvariant mass . It is found in oneloop renormalizationgroup analysis with the Lagrangian adopted that while the NN coupling “runs” in density, the NN coupling does not scale: it will scale at twoloop or higherloop order, but at a slower pace, so it is more appropriate to say it “walks” rather than runs. The former implies a drastic change in the nuclear tensor forces, affecting, among others, the nuclear symmetry energy and the latter generating the stiffness of the EoS at density higher than that of normal nuclear matter.
1 Introduction and Conclusion
If hidden local symmetry (HLS for short) is assumed to hold in the vicinity of chiral restoration in dense baryonic matter and if symmetry is a good flavor symmetry for the vector mesons and in medium as it is in free space, it follows from RG (renormalizationgroup) considerations that the hidden gauge coupling constant and the “effective” V (vectormeson)nucleon coupling constant will scale in density as the quark condensate scales, and consequently as in the chiral limit, both the nucleon and nucleon couplings as well as their masses will go to zero [1]. This implies two dramatic effects in nuclear processes even slightly above nuclear matter density: (1) The suppression of the nuclear coupling will remove the exchange tensor force and hence strongly affect the nuclear symmetry energy [2]; (2) the suppression of the nucleon coupling will soften shortrange repulsion in nuclear interactions and hence make the baryonic matter collapse at moderate density [3]. The first effect, when treated appropriately, turns out to be consistent with the EoS of compactstar matter that involves central densities of (where is the normal nuclear matter density), in fact playing a crucial role for explaining, within the HLS framework, the recently discovered 2solar mass neutron star [4]. On the other hand, the second effect, if unmodified, would be disastrous for the stability of baryonic matter in the density regime relevant to compact stars.
It is the purpose of this paper to suggest how to avoid the disaster due to the second (negative) effect without affecting the first (positive) effect. The key observation is that the symmetry, seemingly good in the matterfree vacuum, must break down in medium, and hence the properties of the isovector and isoscalar vector mesons behave markedly differently as density is increased. The key element in this is the origin of the bulk of proton mass that appears to have no direct link to chiral symmetry, its RG flow and its unsuspected association with the property of the NN interaction in dense medium. Our conclusion is that while the and masses tend to zero (in the chiral limit), perhaps not in the same way, as density is increased, the effective nuclear coupling “walks” in contrast to the effective nuclear coupling that “runs.” This feature was indicated in the phenomenology of compactstar matter studied in [4], and we show in this paper how that feature can be understood in the framework of HLS, e.g., vector manifestation and dilatonlimit fixed point.
The basic assumption that we make is that local field theory can be applied to dense baryonic matter up to the density relevant to the EoS of compact stars. We will not, however, require that it be valid all the way to the chiral transition density denoted . In fact, we will not address what happens precisely at but consider approaching it from below. This means that we will not be able to properly account for the possibility of explicit quark degrees of freedom in discussing the EoS.
We consider HLS Lagrangian that contains as relevant degrees of freedom, the pions and the lowestlying vector mesons, and . It may very well be that for realistic treatment, as stressed recently [5], the infinite tower of hidden local symmetric vector mesons as indicated in holographic QCD models [6] need to be incorporated and the nucleons should be generated from such a generalized HLS Lagrangian. In this work, we will take the simplified Lagrangian in which the infinitetower is integrated out leaving only the lowest vector mesons – in addion to pions, add baryon fields coupled to the mesons à la HLS and implement a scalar dilaton field reflecting spontaneously broken scale symmetry linked to the QCD trace anomaly^{1}^{1}1The role of scalar fields in effective Lagrangians is highly problematic in general and it is not at all obvious how to do this also in our case. We will however be guided by phenomenology in lowenergy nuclear physics, namely, the EFT Lagrangian be treated at mean field with the parameters of the Lagrangian “sliding” with the density of the background in the spirit defined in [7]. This will be the guiding principle in what follows.. To be precise in definition, we shall call the dilatonimplemented HLS Lagrangian with baryon fields “dBHLS” while “ / BHLS” will stand for dilatonless baryon HLS Lagrangian. “HLS” will stand for the generic notion of hidden local symmetry as well as for mesononly theory (without dilaton).
Our objective is to access nuclear matter as well as denser baryonic matter with a single Lagrangian, i.e., dBHLS. At zero density, that is, in the matterfree space, / BHLS is “gauge equivalent” to baryon nonlinear sigma model and can be formulated, with the inclusion of chiral loops, to give a consistent chiral perturbation theory with baryons and vector mesons in a way parallel to HLS [1]. For this, scalar excitations, in principle, can be generated from BHLS by loops. The QCD scalar is highlying and hence does not figure in nonlinear sigma model in hadron dynamics at lowenergy. Scalar glueball excitations will figure for the QCD trace anomaly, but they are also too massive. Thus the role of the dialton in the vacuum is unclear and remains an unsolved problem. At tree order, however, the / BHLS encodes the current algebra, and the dilaton could be suitably interpreted in dBHLS as the lowest scalar excitation in lowenergy pion dynamics.
In going to nuclear matter density, one possible approach could be to do inmedium chiral perturbation theory with / BHLS as one does with nonlinear sigma model. However this requires highorder loop calculations, and this has not been done yet. What one can do instead is to do the mean field calculation with dBHLS as suggested in [8, 9]. The rationale there is that doing the meanfield with a chiral Lagrangian of the dBHLS type near the nuclear saturation density is equivalent to doing Landau Fermiliquid fixed point theory provided the parameters of the Lagrangian are suitably scaled [8, 9, 10]. What is required is that near the Fermiliquid fixed point, the scalar should be (predominantly) a chiral scalar with a mass around 600 MeV.
Now given the dBHLS Lagrangian with the parameters sliding with density, normalized at the nuclear saturation density , the question then is how to go to higher density close to the chiral transition point ? This question was raised and answered in [3]. It involves what is called “dilaton limit fixed point” (DLFP for short) – defined precisely later – introduced by Beane and van Kolck [11]. The basic idea is as follows: While the dilaton should be dominated by chiral singlet component at low density near in a complicated configuration consisting of multiquark and glueballs, as one approaches chiral restoration at , one should recover a GellMannLévy (GML)type linear sigma model with a scalar making up the fourth component of the chiral fourvector () [12].
For what follows, we need to generalize the GML sigma model to a paritydoubled model with nucleons in mirror assignment. In the original GML model (with nucleons in standard assignment), spontaneous breaking of chiral symmetry (in the chiral limit) entirely generates both the mass of the scalar and the mass of the nucleon. Thus in the limit that the symmetry is restored, both the and the nucleon should become massless. We will argue that while the mass could go to zero to join the triplet pions, the nucleon mass need not be zero at the symmetry restoration. This feature can be captured by introducing paritydoubling in the baryon configurations. We shall call the former “standard” baryons and the latter “paritydoubled” baryons. As will be elaborated in detail below, within the EFT framework, the nucleon mass cannot decrease much as density increases without getting into conflict with nature. To be specific, we write the nucleon mass parameter in the Lagrangian as
(1) 
where is a chirally invariant mass term, a constant independent of the chiral condensate , and is dynamically generated mass that tends to zero as . The mass term can appear in chiral Lagrangian without upsetting chiral symmetry provided paritydoublets are introduced [13]. A substantial , indicated in nuclear phenomenology [4], will be the key issue in this paper.
The dilaton limit fixed point (DLFP) arrived at in [3, 14] in the paritydoublet baryon model dBHLS can be summarized as follows. As shown in [3] – and will be recalled in detail below, the idea is to make a field reparametrization so that dBHLS Lagrangian with chiral symmetry in nonlinear realization can be linearized in the limit . If one assumes that symmetry holds for and , one arrives at the DLFP of the form
(2) 
where , is the axial vector coupling constant and is the “induced” nucleon coupling defined by the effective vectormesonnucleon coupling
(3) 
where is the hidden gauge coupling constant. As , we expect that, even for ,
(4) 
The “vector manifestation” (VM) of HLS (and / BHLS) corresponds to . We can see from (3) and (4) that the coupling can go to zero before the VM/HLS fixed point (identified with chiral restoration) is reached.
Several important consequences follow from the property (4).
If dBHLS were applied in the meanfield approximation to baryonic matter much denser than nuclear matter, then one would expect that the nuclear symmetry energy that figures importantly in the EoS for compactstar matter, proportional to , decreases and vanishes at the DLFP. One should however note that there is no solid argument why the meanfield approximation should hold at density much higher than the saturation density, i.e., Fermiliquid fixed point, where meanfield approximation is valid. In fact, would imply that the contribution to the nuclear tensor forces get suppressed, leaving the pion tensor force more effective. This feature turns out to lead to the stiffening – instead of the softening – of the symmetry energy, contrary to the meanfield expectation, as has been observed in effective field theory approach to nuclear dynamics. Indeed what was found in [4] provides a rather strong support for the prediction (4) for the meson. This suggests that the meanfield approximation breaks down at a density above nuclear matter. These matters are further discussed in Sec.5.1. As mentioned therein, this implies that associated with the skyrmionhalfskyrmion topological transition that plays an important role in the calculation of EoS in [4], changeover takes place from a Fermi liquid structure to a nonFermi liquid structure, resembling certain quantum critical phenomena in condensed matter.
The situation with the NN coupling is quite different. As noted in [3], (4) would imply the suppression of the principal mechanism in EFT involving exchanges for the short distance repulsion in nuclear interactions indispensable at high density. Since one cannot pinpoint the density at which the DLFP sets in, one cannot say that the predicted suppression of the exchange repulsion is inconsistent with nature. However it is generally believed in nuclear community that any significant reduction of the NN coupling would make the EoS of nuclear matter too soft, causing difficulty in getting correct saturation. In [4], it was observed that unless the nucleon mass is suitably reduced, thereby increasing repulsion, it would be difficult to reduce the NN coupling. This suggests a close correlation between the behavior of the nucleon mass and the NN coupling in dense medium. This issue will be further elaborated on in Sec. 5.2. One could investigate the interplay between the two phenomenologically in the EFT formalism employed in [4]. In this paper, we will present a theoretical reasoning as to how the interplay can take place.
Taking the hint that the symmetry can be substantially broken in medium, what we propose is to depart from hidden local symmetry in and consider . Denoting gauge couplings by and and induced couplings by and , an analysis parallel to the case [3, 14] reveals that as , one approaches the DLFP
(5) 
which is what was found in [3, 14]. But there is a major difference for the coupling: There is no constraint for that coupling in going toward the DLFP. Furthermore as will be shown below, oneloop RG analysis shows that does not “run” in contrast to which drops rapidly to zero toward the DLFP. This leads to our conclusion that as density increases above , the symmetry must be broken down significantly, and the coupling does not drop as fast as does.
In short, the RG analysis with / BHLS and the mean field treatment with dBHLS, both given in this paper, the large analysis of dense skyrmion matter described in [15] and the phenomenological study of the EoS for compactstar matter of [4] all converge to the conclusion that the dropping of the inmedium nucleon mass stops at with the mass staying constant up to near chiral restoration, suggesting an in the paritydoublet model for baryons. This property is found to have an intimate connection with the role that the meson degree of freedom plays in the structure of nucleon and nuclear matter.
In what follows, we provide details to what are given in sketch above.
2 Hidden Local Symmetric ParityDoublet Model
In this section, we give a precise definition of the model we will study. We will first write down and discuss the hidden local symmetric parity doublet model without dilaton field, i.e., / BHLS [14]. We will be focusing on the case where the chiral invariant mass in (31) is nonvanishing, in fact, substantially big, but we will also discuss the case for , i.e., “standard” scenario.
We will limit our considerations to two flavors (). Motivated by the finding in [4], we relax the symmetry for and mesons. We assume that the symmetry of the Lagangian involved is , where is the global chiral symmetry and is the hidden local symmetry. We take as the gauge bosons of and as of . The basic quantities are the HLS gauge bosons, and ,
(6)  
(7) 
where and are the HLS gauge couplings that will be taken unequal for the local symmetry concerned. The vectors transform
(8)  
(9) 
with and in terms of and the two matrix valued variables and , combined in a specialunitary matrix representing the pion field
(10) 
transforming
(11) 
The variables ’s transform as
(12) 
They may be parameterized as
(13) 
Here denote the pseudoscalar NambuGoldstone (NG) bosons associated with the spontaneous breaking of chiral symmetry, and and are the NG bosons associated with the spontaneous symmetry breaking of and respectively. The s are absorbed into the HLS gauge bosons through the Higgs mechanism, giving rise to their HLS boson masses. , and are decay constants of the associated particles.
To construct hidden local symmetric Lagrangian, it is convenient to introduce the MaurerCartan 1forms
(14) 
with the covariant derivatives of
(15)  
(16) 
transforming homogeneously,
(17) 
and and .
With the above definitions, we can immediately write down the Lagrangian for the mesonic sector. To the leading order in derivative expansion (i.e., to ), it is
(18)  
with
(19) 
and the field strength is given by
(20)  
(21) 
We recover local symmetry if we set and in (18).
Now, we construct the Lagrangian with nucleons in a hidden local symmetric model.
When the nucleon transforms as
(22)  
(23) 
under chiral transformation in the GellMannLévytype linear sigma model, we assign the following transformation,
(24)  
(25) 
to the nucleon’s chiral partner in the mirror assignment[13], where . In hidden local symmetry model in the mirror assignment, the nucleon and its chiral partner are represented in the nonlinearized form, , under chiral transformation, which is given as a function of and ,
(26) 
with the nucleon doublet transforming as
(27) 
under hidden local transformation.
One can readily write down, following [3, 14], the Lagrangian for paritydoublet nucleons coupled to HLS vectors
(28)  
where the covariant derivative of Q is
(29) 
and the are the Pauli matrices acting on the paritydoublet. , and are dimensionless parameters. To diagonalize the mass term in Eq. (28), we transform into a new field :
(30) 
where . We identify as parityeven and parityodd states respectively. The nucleon masses are found to be
(31)  
(32) 
Finally, we arrive at the Lagrangian in parity eigenstate as
(33)  
(34) 
It is convenient for the analysis of nucleon coupling to change slightly the Lagrangian (33). We define quantities belonging to the algebra of as
(35)  
(36) 
where
(37) 
and
(38) 
and are related to each other via
(39)  
(40) 
Then, the Lagrangians (18) and (33) take the form
(41)  
(42)  
where is replaced by and
(43)  
(44) 
Note that the meson couples to nucleon and NG , but there is no coupling to other mesons, i.e., and , at tree order. There can be treeorder and  couplings in the homogeneous WessZumino term in the anomalous part of the HLS Lagrangian that could give rise to a oneloop correction to nucleon coupling, but does not contribute at the order we are working with. We willl see in Sec. 4 that at the oneloop order, the coupling does not scale.
One can read off the vector meson mass and the coupling constant at tree level as
(45)  
(46)  
(47) 
The vector mesons couple to nucleons as
(48)  
(49) 
and the axial vectors coupling as
(50) 
where the subscripts stand for the parity of the nucleon doublet. When symmetry is restored, we will have and the nuclear coupling will vanish as in [3] in approaching the DLFP. In what follows, we will not assume symmetry.
3 Going Towards the DilatonLimit Fixed Point
In order to study what happens to the baryonic matter as density increases, we need to incorporate the dilaton field that represents spontaneously broken scale symmetry of QCD. The explicit scale symmetry breaking associated with the trace anomaly that is also presumably responsible for the spontaneous breaking [15] will not figure directly in our consideration.
We follow the standard trick of inserting the “conformal compensator” field into the Lagrangian (18) and (28) to obtain scale symmetric Lagrangian, with the scale invariance broken spontaneously. It is given by
(51)  
(52)  
(53)  
(54) 
where is the ColemanWeinbergtype dilaton potential that breaks scale symmetry spontaneously. We do not write down its explicit form since it is not needed for our purpose. Here, is the vacuum expectation value of at zero temperature and density.
To move towards a chiral symmetric GMLtype linear sigma model, we do the field reparametrizations – that also defines scalar – and
(55) 
or equivalently
(56)  
(57) 
With these reparametrized fields and going to parity eigenstates, one finds a complicated expression for (51) composed of a part that is regular, , and a part that is singular, , as , where is isospin index. The singular part that arises solely from the scale invariant part of the original Lagrangian (51) has the form
(58) 
where and
(59)  
(60) 
That be absent leads to the conditions that
(61) 
Using large sumrule arguments [11] and the RGE given in the next section , we infer^{2}^{2}2 It is most plausible that is reached only after is reached. This is because gives the constraint , but not . However the RGE given in the next section has the infrared fixed point , so the point at which could be very near the chiral restoration point at which is close to 1. In fact it is observed phenomenologically in GamowTeller transitions in heavy nuclei that .
(62) 
In the density regime where GMLtype linear sigma model is valid, the nucleon mass can be given as
(63) 
where is the vacuum expectation value of . As the chiral symmetry restoration point is approached, , so in the limit , we expect
(64) 
These are the constraints that lead to the dilaton limit as in [3] and announced above. It follows then that
(65) 
We thus find that in the dilaton limit, the meson decouples from the nucleon. In contrast, the limiting does not give any constraint on . The nucleon coupling remains nonvanishing in the Lagrangian which in unitary gauge (with ) and in terms of fluctuations and around their expectation values, takes the form
(66)  
which is the same as the Lagrangian given in [13] except for the nucleon interaction. This is just the nucleon part of the linear sigma model in which the is minimally coupled to the nucleon, applicable infinitesimally below the critical density with the mass replacing . What is significant with this result is that it shows that the suppression of the repulsion predicted with could be absent due to potentially significant symmetry breaking as is indicated in the phenomenology of EoS for dense baryonic matter. We will return to these matters in Secs. 5.2 and 6.
4 RG Analysis of the Nucleon Coupling
We have arrived at the DLFP (2) by linearizing dBHLS Lagrangian (51) (which is gauge equivalent to nonlinear sigma model) in the meanfield approximation. Very near the fixed point, the resulting effective Lagrangian is a GMLtype linear sigma model, to which the meson is minimally coupled. The isovector vector meson is decoupled in the limit.
In this section, we interpret the DLFP in terms of RG (renormalizationgroup) flow. To do this, we take the hidden local symmetry Lagrangian with baryons but without the dilaton (i.e., / BHLS). As mentioned, in the chiral perturbation approach that we will be taking in this section, the role of scalar dilaton is problematic if introduced naively. We will first discuss the standard assignment for the nucleon, which gives a clearer picture of what’s going on, and then consider the mirror assignment with paritydoubling.
4.1 Renormalization of the vectornucleon coupling
In this subsection, we review shortly how to renormalize . The loop calculation is done in the background field gauge. When and are defined as the background fields of the and fields, the renormalization condition is given by
(67) 
where is given by the three point function of and presents the wavefuntion renormalization of () field. Expanding and using for the classical field (), one obtains the condition
(68) 
where the superscript (1) represents one loop. After taking the external momentum squared to be zero, the RGEs for are obtained by taking the derivative of both side of Eq. (68) with respect to the (loop) momentum cutoff . This is what is called in [16] “field theory approach” to Wilsonian renormalization group . As we see from Eq. (42), the coupling is distinguished from coupling in the Lagrangian when U(2) flavor symmetry is broken, and they are renormalized differently from each other as in Eq. (67) where . But, when U(2) flavor symmetry is unbroken, the coupling and the coupling carry one identical parameter and hence there will be only one RGE for the coupling to nucleon.
4.2 RGEs in the Standard Assignment
In the standard assignment with , we consider only the positiveparity nucleon (set ) in analyzing the RG properties of its coupling to mesons. The calculation is straightforward, the only difference from what was done in [14] being that we have local symmetry instead of . Omitting the details that are given in [14] – apart from terms involving the meson, we simply write down the RGEs at oneloop order,
(69) 
(70)  
(71) 
(72) 
The explicit expressions of and are given by
(73)  
(74)  
(75)  
(76)  
(77)  
(78)  
(79) 
To summarize the essential observations:

While coincides with the DLFP, does not “run” at oneloop order. There are two reasons for this. First, there are no contributions to RGE from oneloop vertex corrections to the NN coupling because all the divergences in NN 3point functions are canceled by those in the nucleon selfenergy diagrams figuring in wavefunction renormalization shown in Fig. 1^{3}^{3}3This result resembles the oneloop result in QED. In QED with gauge symmetry, all divergent terms of the photonelectronelectron three point function are canceled by the divergent terms in the electron self energy diagrams. This cancelation is due to gauge invariance. However this analogy does not extend to higher orders as mentioned in the text.. Second, there are no couplings to other mesons at tree order of the / BHLS Lagrangian, so there cannot be mesonloop contributions. However, as mentioned, at higher order this is no longer true. For instance there can be oneloop contribution to the nucleon coupling involving the homogeneous WessZumino (hWZ) term in the anomalous part of the Lagrangian. However the hWZ term goes as , so the oneloop contribution involving this vertex will correspond to normal twoloop order. Therefore we expect to run slowly, if at all. It may be more appropriate to characterize it as “walking.”

The RGE for nucleon mass, (69), has the fixed point . But, if at the fixed point, cannot be an infrared fixed point. Note that the nucleon coupling will affect the sign of . Since does not run at oneloop level, could become negative at the fixed point unless at that point. On the other hand, at twoloop or higher, could be “walking” to zero moving toward the DLFP, thereby making near the fixed point, in which case will become an infrared fixed point. This indicates an intricate interplay between the nucleon mass and the nuclear dynamics. This is an important point to which we will return in Section 6.

With the cutoff identified with a quantity related to density, as discussed in Appendix, we see that will tend to zero as density increases (or equivalently decreases). We consider the point where