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In this section, we propose a method to study the effects of the QCD vacuum in nuclear and hadron physics. As a first attempt, our strategy for a way forward is to work with the constituent in the Copenhagen vacuum. Here, we assume that the Copenhagen vacuum is still valid in the presence of massive constituent quarks. We begin with a brief summary of the Copenhagen vacuum [5, 6].

2.1. A candidate of the QCD vacuum

There have been enormous amount of studies on the candidates of the true QCD vacuum, center vortices, instantons, monopoles, quark-antiquark condensate, etc, see [1–4] for a recent review. As a first attempt, we begin with a simple classical gluon field configuration, called the Savvidy vacuum. A constant chromomagnetic field is a non-trivial classical solution of the SU(2) Yang-Mills equation of motion. The real part of the one-loop vacuum energy in the constant chromomagnetic field is given by [19, 20] 1 11 gH 1 Re ǫ = H2 + g2H2 ln . 2 48π2  Λ2 − 2 This shows that the one-loop vacuum energy has a lower minimum energy at a non-zero value of H than the pertubative QCD vacuum, which implies that quantum fluctuations could generate the homogeneous chromomagnetic field. Right after this interesting result, a subsequent study [21] showed that the Savvidy vacuum is unstable due to the imaginary part in the one-loop vacuum energy, (gH)2 Im ǫ = . − 8π2 The origin of the instability is the tachyonic mode in the lowest Landau level [21]. For SU(2) Yang-Mills theory, with 3 a specific choice of the constant chromomagnetic field Gy = Hx, where the superscript denotes the color, the energy 1 2 eigenvalue of the gluon field, Wµ = (Gµ + Gµ)/√2 becomes

1 2 En = 2gH(n + )+ k 2gH . (1) r 2 3 ± Here, is due to the spin. Now, one can easily see that the energy eigenvalue can be negative for n = 0 with sign ± 2 − for spin, En = k gH. It is obvious that the constant magnetic field cannot be the true QCD vacuum because it 3 − is unstable andp breaks rotational symmetry and gauge invariance. For studies with SU(3) color, we refer to [22–26]. There have been many studies to remove the instability, for example, see Refs. [5, 6, 27–29] and references therein. It was shown in Ref. [5] that the chromomagnetic field has locally a domain-like structure and argued that a stable can be obtained by a superposition of such domains, which restores gauge and rotational invariance. This is called the Copenhagen vacuum. In Ref. [6], it was argued that the of the constant chromomagnetic field is zero everywhere in the vacuum which is the quantum liquid state [5]. Also, an interesting result was obtained that the coupling constant 2 that minimizes the vacuum energy of the Copenhagen vacuum is unexpectedly small αs = g /4π =0.37 [6].

2.2. Constituent quarks in Copenhagen vacuum

As a route to go from QCD vacuum to nuclear and hadron physics, we suggest to adopt the constituent quark model picture with a gluon background field. The starting Lagrangian for our low-energy effective model is the same with that of QCD except that the quark mass is not negligible compared to ΛQCD.

1 µν = ψi¯ Dψ mψψ¯ Fµν F + ..., L 6 − − 2 where Dµ = ∂µ + igGµ. Here, m represents the constituent quark mass whose origin is in general believed to be chiral . In the constituent quark model, the dominant contribution to the baryon mass is just the sum of the constituent quark masses. The gluon field is normalized as a a Gµ = GµT , 1 1 tr(T aT b)= δab,T a = λa , (2) 2 2 3 where λa denotes the Gell-Mann matrices. Fµν is the usual field-strength of the gluon field

Fµν ∂µGν ∂ν Gµ + ig[Gµ, Gν ] . ≡ − We now include the Copenhagen vacuum in the following way. In the present work we consider quarks with three colors, for instance, red, green and blue. Since our primary goal is to find out how the vacuum effect is imprinted on the baryon masses, we generalize the simple constant chromomagnetic field used in [21]. To incorporate the Copenhagen vacuum in the , we first define three mutually orthogonal unit vectors in a coordinate space

uˆk = (sin θ cos φ, sin θ sin φ, cos θ) , (3) uˆ(1) = ( cos θ cos φ, cos θ sin φ, sin θ) , (4) ⊥ − − uˆ(2) = (sin φ, cos φ, 0) . (5) ⊥ − (2) We chooseu ˆ⊥ as the spatial direction of the background gluon. Then,u ˆk denotes the direction of the constant chromomagnetic field as H~ = Huˆ = ~ G~ . In general, H~ can be defined as H~ a = ~ G~ a gǫabcG~ b G~ c. In the k ∇× ∇× − × present assumption, however, ǫabcG~ b G~ c might be zero since we will turn on the background gluon field only with a = 3. × ~ (1) (2) Now, we choose the background gluon field as G3 = Hx⊥ uˆ⊥ λ3/2; all the other gluon fields are purely quantum (1) (1) fields. Here, x⊥ =u ˆ⊥ ~x and λ3 is the third element of the Gell-Mann matrix, λ3 = diag(1, 1, 0). Then, it is easy to see that the quarks· with red and green colors couple to the background gluon field with opposite− sign (gH or gH) due to λ , while the blue quark does not. The eigenvectors of λ are given by − 3 3 1 0 0 R = 0 , G = 1 , B = 0 . (6) 0 0 1      

3. BARYON MASS

To show how our suggestion works in practice, we consider the effects of the Copenhagen vacuum on the baryon mass. In a usual (chiral) constituent quark model, with a confining potential we solve the Schr¨odinger equation or Dirac equation or Faddeev equation to describe the properties of the . It is widely recognized that the constituent quark mass contributes dominantly to the ground-state baryon mass. However, there exist some other contributions from (Goldstone ) or perturbative gluons. In the present work, the confinement is imposed by the background gluon fields (Copenhagen vacuum). Since the main goal of this work is to show how the QCD vacuum represented by the background gluon fields appears in physical quantities such as the baryon mass, we assume that the ground-state baryon mass is approximately the sum of three constituent quark masses (or ground state energies of three quarks). We then solve the equation of motion for the quarks in the presence of the constant chromomagnetic field considering the fact that the vacuum expectation value of the constant chromomagnetic field is zero everywhere in the Copenhagen vacuum [5, 6]. Now, we solve the Dirac equation with the constant chromomagnetic field, taking into account the fact that at each point one has rotational invariance in the Copenhagen vacuum. To fulfill this, we need to solve the Dirac equation ~ with the constant chromomagnetic field H = Huˆk, whereu ˆk is defined in Eq. (3), and take the average over θ and φ. We remark here that there have been an enormous number of works that solves the Dirac equations in a constant field, see Ref.[30] for a recent review. Therefore, we do not present how to solve the Dirac equation in our case. Interestingly, we find that the average over the angles, θ and φ, are trivial since the resulting Landau level of the quark is independent of the angles,

2 En = 2˜gH(n + γ 1)+ m , n =0, 1, 2, ... , (7) p − a where γ =1, 2 is for the spin up and down, respectively. Here,g ˜ = g/2 and the factor 1/2 comes from Gµ = Gµλa/2. To study the effects of the constant chromomagnetic field on the baryon masses, we assume that the dominant contribution to the baryon mass is the sum of the ground state energy of the constituent quarks and calculate the 4

TABLE I: Calculated baryon masses. m is the mass of u- and d-quarks and ms is for the s-quark. Baryon Quark contents Mass p uud 2m + pm2 + 2˜gH n udd 2m + pm2 + 2˜gH + 4 2 2 2 1 2 Σ uus 3 m + 3 ms + 3 pm + 2˜gH + 3 pms + 2˜gH 0 4 2 2 2 1 2 Σ uds 3 m + 3 ms + 3 pm + 2˜gH + 3 pms + 2˜gH − 4 2 2 2 1 2 Σ dds 3 m + 3 ms + 3 pm + 2˜gH + 3 pms + 2˜gH 0 2 4 1 2 2 2 Ξ ssu 3 m + 3 ms + 3 pm + 2˜gH + 3 pms + 2˜gH − 2 4 1 2 2 2 Ξ ssd 3 m + 3 ms + 3 pm + 2˜gH + 3 pms + 2˜gH 0 4 2 2 2 1 2 Λ uds 3 m + 3 ms + 3 pm + 2˜gH + 3 pms + 2˜gH

ground-state baryon masses. As an illustrative example, we consider the mass in some detail. The of the proton in the spin-flavor space is given by 1 p = [ uud( + 2 )+ udu( + 2 ) | ↑i −√18 ↑↓↑ ↓↑↑ − ↑↑↓ ↑↑↓ ↓↑↑ − ↑↓↑ + duu( + 2 )] , (8) ↑↓↑ ↑↑↓ − ↓↑↑ where

uud( + 2 )= u( )u( )d( )+ u( )u( )d( ) 2u( )u( )d( ) , (9) ↑↓↑ ↓↑↑ − ↑↑↓ ↑ ↓ ↑ ↓ ↑ ↑ − ↑ ↑ ↓ and so on. The color wave function, which is common to all baryons, is given by

1 φcolor = (RGB RBG + BRG BGR + GBR GRB) . (10) r6 − − − The contribution from the first (second, etc) term in Eq. (8) with the first term in the color wave function (RGB) is 2 3m (m+2 m +2˜gH, etc). Here, m denotes the up and down constituent quark masses, mu = md = m = ms. After 6 2 consideringp all the possible combinations and averaging over seventy-two terms, we obtain mp↑ =2m + m +2˜gH. 2 We confirm mp↑ = mp↓, as it should be, and mp = mn =2m + m +2˜gH. p In Table I, we summarize our results of the ground-state baryonp masses. We first observe that the Gell-Mann- Okubo mass relation for the baryon octet [31], mΣ +3mΛ = 2(mN + mΞ), is satisfied. One may think this is trivial as we have assumed SU(2) isospin symmetry to set mu = md = m = ms. However, in our case the relation holds even with the contributions from gluon background fields. This is bec6 ause gluons are flavor-blind and therefore the contribution from the gluon background field is universal for up, down and strange quarks; for a up or it 1 2 1 2 is 3 m +2˜gH, while for a it is 3 ms +2˜gH. Asp manifest in Eq. (7), the spin projection alongp the direction of the constant chromomagnetic field is an important factor for the eigenvalue of the constituent quark, especially at the lowest Landau level (LLL). When n = 0 and γ = 1, the energy of the corresponding quark does not depend on H. Also, excited states depend on the QCD vacuum more strongly than the LLL, which implies that the QCD vacuum effect is more influential on the excited baryons than the ground-state ones. This observation may be attributed to a specific feature of the vacuum we chose that provides an external field to the system. Now, we estimate the value ofgH ˜ using the baryon masses as inputs. Since we have three inputs, 940 MeV for nucleons, 1130 MeV for mΛ and mΣ, 1320 MeV for mΞ, and three parameters to be determined, m, ms,gH ˜ , we may be able to fix the value ofgH ˜ . However, all the baryon masses in the table are not independent from each other due to the Gell-Mann-Okubo mass relation [31], and so we can at best find the relation among those three unknown parameters. In the figure 1, we plot the dependence of √2˜gH and ms on m. For example, if we take m = 300 MeV, then we obtain √2˜gH = 160 MeV. Though we cannot precisely determine the value of √2˜gH which characterizes the adopted QCD vacuum, we can at least see how the QCD vacuum affects the baryon masses. In Fig. 2, we plot the ratio m/ms as a function of √2˜gH to investigate what will be happening to SU(3) flavor symmetry as we increases H at the level of constituent quarks. We note again that we assumed mu = md. As it can be seen from the figure, it seems that flavor symmetry of the constituent quarks is becoming worse as H increases. Since the current quark mass for up and down quarks is roughly 5 MeV and about 100 MeV for the strange quark, 5 the ratio of the current quark is about 0.05. Therefore, the ratio approaches that of the current quarks H increases. This is because the contribution of the constituent quark mass to the ground-state baryon masses becomes smaller as H increases. We remark here that apart from the contribution from the gluon background field to the baryon mass, there can be a contribution from the dynamical gluons that are supposed to interact weakly in the constituent quark model. a If we take the analogy of the Cornell potential, V (r) = r + br, the contribution from the background gluon field may correspond to the confinement part of the potential,− while that from the dynamical gluons may correspond to the Coulomb part of the potential induced by one-gluon exchange. Since the main goal of this work is to propose a method to investigate how the QCD vacuum represented by the background gluon fields appears in physical quantities such as the baryon mass, we calculate only the dominant contributions to the baryon mass. We may dig into more precisely the ground-state baryon mass including decuplet baryons in a future study.

4. SUMMARY AND DISCUSSION

To study the role of the QCD vacuum in nuclear and hadron physics, we have proposed to use the constituent quark model in the Copenhagen vacuum which provides a scenario for confinement [2]. As a first application of our proposal, we calculated the ground-state baryon masses. We found that the baryon mass depends on the quantity that manifests the Copenhagen vacuum, √gH˜ , and the Gell-Mann-Okubo mass relation for the baryon octet is satisfied. We also estimated the values of √2˜gH and ms as function of u and d quark mass m. When m = 300 MeV, we obtained √2˜gH = 160 MeV. We observed that the ratio m/ms approaches that of current quarks, which is because the contribution of the constituent quark mass to the ground-state baryon masses becomes smaller as H increases. In the constituent quark model, the constituent quark mass is from spontaneous , while the contribution to the baryon mass from the gluon background has nothing to do with chiral symmetry. In this sense, the baryon mass from the Copenhagen vacuum may be an origin of the chiral invariant mass defined in the parity doublet model [13], though we didn’t discuss the parity doubling structure in the baryon sector in our work. To be more realistic for the ground-state baryon properties such as the mass difference between octet and decuplet baryons, we need to include fluctuations such as one-gluon exchange and/or exchange. A promising set up for this and also for other observables in nuclear physics might be the chiral quark model (ChQM) proposed in [32, 33] with a suitable QCD vacuum such as the Copenhagen vacuum. If we cast our picture into the ChQM [33], the interpretation about the smallness of the gauge coupling may change slightly. In the ChQM, the large value of the gauge coupling at a scale ΛχSB 1 GeV leads chiral symmetry breaking and so the gauge coupling can be small in a vacuum with chiral symmetry breaking.∼ In our case, we may assume that the large value of the gauge coupling invokes chiral symmetry breaking and develops a background gluon field in the ChQM. As a future study, it will be interesting to study the effect of the QCD vacuum on other physical quantities such as the decay constant. It will be also informative to investigate if our approach can provide a unified platform to deal with quark matter and on the same footing.

1

0.8

0.6

0.4

√2˜gH 0.2 ms

0 0 0.1 0.2 0.3 m [GeV]

FIG. 1: Red solid line: √2˜gH, blue dashed line: ms. The numbers are given in unit of GeV. 6

Acknowledgments

The work of Y.K. was supported partly by the Rare Isotope Science Project of Institute for Basic Science funded by Ministry of Science, ICT and Future Planning, and NRF of Korea (2013M7A1A1075764). The work of I.A.M. was supported partly by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2018R1A5A1025563) and the Ministry of Science and Higher Education of the Russian Federation (project No. 0818-2020-0005) and with using resources of the Shared Services Center “Data Center of FEB RAS”. The work of M.H. was supported partly by JSPS KAKENHI Grant Number 20K03927.

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0.6

0.4 s m/m 0.2

0 0 0.2 0.4 0.6 0.8 1 √2˜gH [GeV]

FIG. 2: Ratio of the constituent quark masses.