DOI: 10.22661/AAPPSBL.2020.30.2.76
QCD Vacuum and Dense Matter
YOUNGMAN KIM* RARE ISOTOPE SCIENCE PROJECT, INSTITUTE FOR BASIC SCIENCE, DAEJEON 305811, KOREA
^{*}Electronic address: ykim@ibs.re.kr
communicated by Yongseok Oh
Quantum Chromodynamics (QCD) is widely accepted as the fundamental theory for the strong interaction and is also notorious for its nonperturbative nature at low energies relevant for nuclear and hadron physics. It has been known that the QCD vacuum has a rich structure that is responsible for chiral symmetry breaking and/or confinement of QCD. In this short article, we propose a method to study possible roles of the QCD vacuum in nuclear and hadron physics in the framework of a QCD effective theory with quarks, gluons and Goldstone bosons.
INTRODUCTION
One of the ultimate goals in nuclear physics is to achieve a rigorous understanding of the properties of hadronic matter and finite nuclei in the frame work of the underlying theory of the strong interaction, Quantum Chromodynamics (QCD).
Ab initio or microscopic approaches with two and threenucleon chiral interactions have been developed to study nuclear matter and finite nuclei. Since chiral symmetry is one of the crucial properties of QCD, these approaches are contemporarily most apt for studying hadronic matter and nuclei at low energies in the context of effective theories of QCD.
In this short article, we suggest a way to connect some intrinsic QCD properties with hadron and nuclear physics using an effective model of QCD with quark and gluons. The QCD vacuum has a rich structure that is responsible for chiral symmetry breaking and/or confinement of QCD: quarkantiquark condensate, monopoles, instantons, dyons, Copenhagen (spaghetti) vacuum, etc. The influence of such rich QCD vacuums on the properties of nuclear matter and nuclei should be an interesting problem to be pursued. For example, the role of the quarkantiquark condensate, which is responsible for the spontaneous symmetry breaking of chiral symmetry, in nuclear and hadron physics has been extensively studied. The QCD vacuum is often characterized by background gluon fields or by a vacuum condensate of gluon fields. There can be several methods to dig into the effect of the QCD vacuum on the properties of nuclear matter and finite nuclei.
In this work we use an effective model of QCD with quarks, gluons and pions [1], which is defined between the chiral symmetry breaking scale (~ 1 GeV) and the confinement scale ΛQCD , to study how the nontrivial QCD vacuum featured by the gluon fields affects the physics of dense matter or, in general, hadron and nuclear physics. As a specific realization of such a vacuum condensate of gluon fields, as a first attempt we adopt the Savvidy vacuum [2], which is extended to the Copenhagen (spaghetti) picture of the QCD vacuum [36].
In Sec. 2, as an interesting example of investigating an intrinsic QCD property in dense matter, we briefly introduce a parity doublet model and review some recent work using the parity doublet model in dense matter. After a short introduction to the chiral quark model and the Savvidy vacuum, with a brief summary of the Copenhagen vacuum, we make our proposal for a path from QCD vacuum to dense matter in Sec. 3.
PARITY DOUBLET MODEL
QCD vacuum can be parameterized by the quark condensate, which breaks chiral symmetry. In this section, as an example of effective theories of QCD with hadronic degrees of freedom, investigating the role of QCD vacuum in dense matter we discuss two distinctive pictures on the origin of nucleon mass in the chiral limit.
As it is wellknown, in the linear sigma model the nucleon mass in the chiral limit is given by
 (1)

where g_{𝜋} is the pionnucleon coupling constant and σ_{0} is the vacuum expectation value of the sigma field. Here σ is the chiral partner of the pion. . It can be easily seen from Eq. (1) that the nucleon mass in the chiral limit is solely a result of spontaneous chiral symmetry breaking and will be zero when σ_{0} = 0, i.e., chiral symmetry restoration.
A characteristic feature of the parity doublet model is that the mass of the nucleon in this model contains a so called chiral invariant mass in addition to the contribution from spontaneous chiral symmetry breaking. In the parity doublet model [7], two nucleon fields transform in a mirrorlike way under the chiral SU(2)_{L }× SU(2)_{R} transformation
 (2)

where R ∈ SU(2)_{R} and L ∈ SU(2)_{L}. With this mirror assignment, it is easy to see that a mass term m_{0}(_{}) is invariant under chiral transformation. Here, m_{0} is called the chiral invariant nucleon mass. Then, the nucleon part of the parity doublet model Lagrangian [7] reads
 (3)

To obtain the mass of the nucleon N(938) and its parity partner N(1535), we diagonalize the kinetic and mass terms
 (4)

Here, one can see that even if we assume chiral symmetry restoration, σ_{0} = 0, the masses of the nucleon and its parity partner remain finite and degenerate. In this parity doublet model, chiral symmetry restoration does not imply m_{N} = 0 in the chiral limit, but rather m_{N} ± = m_{0}. In [7] the value of the chiral invariant mass was determined as m_{0} = 270 MeV from the decay width of N^{★}(1535) → N + 𝜋. The model is extended by including ω mesons to study symmetric nuclear matter [8], where m_{0} ~ 800 MeV was used to reproduce nuclear matter saturation properties.
Now, we review some of recent attempts to estimate the value of the chiral invariant nucleon mass.
In [9], an extended parity doublet model with the additional potential term of σ and with hidden local symmetry was constructed to study asymmetric nuclear matter. It was shown that this extended parity doublet model reasonably reproduces the saturation properties of nuclear matter with the chiral invariant nucleon mass m_{0} in the range from 500 to 900 MeV. To find a physical quantity that is sensitive to the value of the chiral invariant mass, m_{0} dependence of the slope parameter L was also investigated, where L is defined by
Here, ρ_{0} denotes the saturation density and S is the nuclear symmetry energy. It was shown that the value of the slope parameter is insensitive to m_{0} [9].
In [10], the parity doublet model is further extended by including ∆ baryons in order to study ∆ matter and its role in partial chiral symmetry restoration. It was shown that stable ∆nucleon matter can be formed for ρ ≥ 1.5ρ_{0} and the emergence of ∆ matter enhances partial chiral symmetry restoration in dense matter.
To see if some properties of finite nuclei can constrain the value of the chiral invariant mass, a nuclear structure study based on the parity doublet model [9] was performed in [11]. For example, the nucleon density profile in ^{48}Ca was studied with different values of m_{0} (see Fig. 1).
Fig. 1: Calculated nucleon density profile in ^{48}Ca. The figure is taken from [11].
By comparing calculated binding energies of selected several nuclei with experimental values, the authors of [11] argue that m_{0} ~ 700 MeV may be preferred.
In [12], an extended parity doublet model including four light nucleons, N(939), N(1440), N(1535), and N(1650) was constructed. In addition to the nuclear matter saturation properties, such as saturation density, binding energy, incompressibility, and symmetry energy, the authors also considered the tidal deformability determined by the observation of gravitational waves from neutron star merger GW170817. They found that the chiral invariant masses are larger than about 600 MeV.
In [13], the extended parity doublet model developed in [9] was incorporated into a new transport code "DaeJeon BoltzmannUehlingUhlenbeck (DJBUU)" to see how observables in heavy ion collisions depend on the value of the chiral invariant nucleon mass.
Finally, it is interesting to note that the skyrmion description of compressed baryonic matter with a scalechiral symmetric Lagrangian predicts similar results for the value of the chiral invariant mass [14].
A WAY FROM QCD TO NUCLEAR MATTER
Understanding the role of rich QCD vacuum structures in hadron and nuclear physics is an interesting and important problem. A successful and promising way to address this issue is of course the lattice QCD. However, it is also desirable to have a theoretical tool to tackle the issue with less numerical calculation. Below, we propose a way to study in general the effects of the QCD vacuum in dense matter or in nuclear and hadron physics.
Chiral quark model
As a platform to go from the QCD vacuum to hadron and nuclear physics, we use a chiral quark model constructed in [1]. One of the main motivations of [1] was to understand why the nonrelativistic quark model works.
The chiral quark model is defined between confinement scale Λ_{QCD} ~ 200 MeV and chiral symmetry breaking scale Λ_{𝜒SB} ~ 1 GeV, and therefore it contains quarks, gluons and Goldstone bosons. The effective Lagrangian is given by [1]
 (5)

where
 (6)

The fieldstrength tensor of the gluon field is given by
In this model, the constituent quark mass m is assumed to come from the spontaneous breaking of chiral symmetry and the leading contribution to the baryon mass is just the sum of the constituent quark masses.
The Goldstone boson fields due to spontaneous SU(3)_{L} ×SU(3)_{R} chiral symmetry breaking are
where c = 1/√2 In terms of the matrix 𝜋, Σ and ξ are defined by
 (7)

where f ≃ 93 MeV.
One of the virtues of this model is that one can treat interactions involving pions perturbatively at low energies thanks to the wellestablished power counting scheme. In addition to this, interestingly, gluons are also perturbative. The value of the gauge coupling constant α_{s} was estimated from the ratio of color and electromagnetic hyperfine splittings of the baryon spectrum to be α_{s} ≃ 0.28.
For more details of the model we refer to see [1, 15].
Savvidy vacuum and beyond
Among many important and interesting candidates for the true QCD vacuum (see [16, 17] for a review), we choose the Savvidy vacuum [2] for a first attempt.
A constant chromomagnetic field is a nontrivial classical solution of the SU(2) YangMills equation of motion. The real part of the oneloop vacuum energy in a homogeneous chromomagnetic field is given by [2, 18]
which has a lower minimum energy at a nonzero value of H than the pertubative QCD vacuum. This, socalled Savvidy vacuum, implies that quantum fluctuations could generate the constant chromomagnetic field.
Soon after this interesting finding, a subsequent study showed that the Savvidy vacuum is unstable due to the imaginary part of the oneloop vacuum energy [3],
It is straightforward to identify the origin of the instability [3]. For SU(2) YangMills theory, with a specific choice for the constant chromomagnetic field _{}, the eigenvalue of the gluon field _{} becomes
 (8)

Here the superscript denotes color and ± is due to the spin. Now, one can easily see that the energy eigenvalue can be negative for n = 0, _{}. The calculations above were extended to SU(3) and SU(4) in Ref. [4]. It is obvious that the constant magnetic field cannot be the true QCD vacuum because it is unstable and breaks rotational symmetry and gauge invariance. It was argued that the instability can be removed by the formation of domains.
It was shown in Ref. [5] that the chromomagnetic field has locally a domainlike structure: the field has different orientations in different domains. At long distance the orientations of domains is random so that both gauge and rotational symmetries of the vacuum can be restored in the infrared regime.
In Ref. [6], an interesting observation was made that the coupling constant that minimizes the vacuum energy of the random color magnetic quantum liquid is unexpectedly small, α_{s} = g^{2} /4𝜋 = 0.37.
Effective action
As a first step, we consider a simple choice for the constant chromomagnetic field, _{}, in two color QCD. Here "3" denotes the color. With this simple choice of the constant chromomagnetic field, we will be able to benefit from previous works with a constant U(1) electric or magnetic field. A difference is that in our case the gluon field carries color indices, and also we have more than one flavor. The interaction term between quarks and gluons is
 (9)

where τ^{a} is the Pauli matrix of color SU(2). Since the eigenvalue of τ^{3} is ±1, we have to consider two constant chromomagnetic fields, _{}.
With this difference in mind, we evaluate the oneloop effective action with constant chromomagnetic field at finite baryon number density. The fermion propagator in an external field is given by [19]
 (10)

Here, ± or ∓ in the offdiagonal terms are due to the eigenvalue of τ^{3}. We note here that the diagonal terms are independent of the eigenvalue of τ^{3}. The corresponding thermal fermion propagator is given by [20]
 (11)

where f_{F}(ω) is the FermiDirac distribution. Now, we evaluate the oneloop effective action at finite density using the thermal propagator [20, 21],
Since the diagonal part of the fermion propagator is independent of the eigenvalue of τ^{3}, the contributions from _{} to the effective action are the same. As the FermiDirac distribution becomes the heaviside step function at zero temperature, it is relatively easy to evaluate the effective action in cold dense matter. The effective action is nothing but the grand canonical potential, L _{e f f} = Ω, and so we can study the various effects of the QCD vacuum characterized by the constant chromomagnetic in dense matter. The results for the role of the Savvidy vacuum in dense matter, including the effective quark mass, to address the issue of the chiral invariant mass will be reported elsewhere in near future.
A proposal
So far we have discussed the chiral quark model, the Savvidy vacuum and the Copenhagen vacuum (spaghetti vacuum).
The chiral quark model itself is, by definition, describing deconfined and chiral symmetry broken phase. Then, what is happening when we combine the Savvidy vacuum or the Copenhagen vacuum with the chiral quark model? As an immediate example of the application of the chiral quark model in the Savvidy vacuum, we are currently studying how to estimate the chiral invariant nucleon mass as a function of the constant chromomagnetic field H, both in free space and in dense matter. Once this task is successfully completed, we may be able to connect indirectly a QCD vacuum specified by H with physical quantities through the parity doublet model.
Eventually, once we combine the chiral quark model with the Copenhagen vacuum or the center vortex picture, this combined model may describe confined phase with chiral symmetry breaking as the center vortices are responsible for quark confinement [22].
In a nutshell, the proposal is that a QCD vacuum can be represented by nontrivial gluon field configurations in the chiral quark model and that we can study the role of the QCD vacuum in nuclear and hadron physics using the chiral quark model with the nontrivial gluon field configuration.
SUMMARY
It is important to develop a tool to seriously investigate how QCD is reflected in nuclear and hadron physics, especially in the properties of nuclear matter and finite nuclei. As an example of such endeavors, we reviewed recent studies based on the parity doublet model in which the nucleon mass has a component called the chiral invariant nucleon mass, in addition to the contribution from spontaneous chiral symmetry breaking by the quarkantiquark condensate.
We then proposed a way to study possible roles of the QCD vacuum in nuclear and hadron physics in the chiral quark model. As a first attempt, we used the Savvidy vacuum which supports constant chromomagnetic field. Though the chiral quark model in the Savvidy vacuum is at best a toy model because it breaks rotational symmetry and gauge invariance and because the vacuum is unstable, we expect that this combined model may shed light on indepth studies to reveal the remnants of QCD vacuum in nuclear matter or in finite nuclei. Some of the problems associated with the Savvidy vacuum can be resolved by using an improved vacuum such as the Copenhagen vacuum. We also expect that the chiral quark model in a nontrivial QCD vacuum, which supports confinement, can provide a convenient means for treating quark matter and nuclear matter on the same footing.
Acknowlegments: The author expresses his sincere gratitude to Masayasu Harada, Sangyong Jeon, Myungkuk Kim, YoungMin Kim, ChangHwan Lee, Yuichi Motohiro, WonGi Paeng, Ik Jae Shin, and Yusuke Takeda for collaborations.
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