Fate of the topological susceptibility in two-color dense QCD

In Quantum Chromodynamics (QCD), the $U(1)_{A}$ anomaly, i.e., the non-conservation of the $U(1)_{A}$ axial current caused by the gluonic quantum corrections, plays crucial roles in the low-energy physics governed by the spontaneous breaking of the chiral symmetry. For instance, the $U(1)_{A}$ anomaly affects the hadron mass spectrum to yield the heavy $\eta^{\prime}$ meson [1] and the order of the chiral phase transition in QCD matter [2]. In addition to these low-energy aspects, the $U(1)_{A}$ anomaly is also closely related with topological vacuum structures of QCD [3], which is described by the anomalous gluonic operator tagged with the $\theta$ parameter. The characteristics of the $\theta$-dependent QCD vacuum is captured by the topological susceptibility: the curvature of the QCD effective potential with respect to $\theta$.

The symmetry breakings in QCD are reflected in meson susceptibility functions defined by two-point functions of quark composite operators in the low-energy limit. At the hadronic level, the so-called chiral-partner structure would be an indication of this property [4]. That is, at high temperature and/or density where chiral symmetry tends to be restored, masses of the mesons related by the chiral transformation become degenerate, and so do the corresponding meson susceptibility functions. This implies that, indeed, the meson susceptibility functions can be regarded as alternative probes to measure the strength of chiral symmetry breaking and restoration. In a similar way, the effective restoration of $U(1)_{A}$ symmetry can be quantified by the degeneracies of the meson susceptibility functions connected by the $U(1)_{A}$ transformation.

Making use of the Ward-Takahashi identity (WTI) associated with chiral symmetry, one can show that the topological susceptibility is also correlated with the chiral- and $U(1)_{A}$-partner structures in the meson susceptibility functions [5, 6, 7, 8, 9]. Thereby, the topological susceptibility can also be referred to as the indicator for the breaking strength of $U(1)_{A}$ symmetry through the chiral phase transition. In fact, lattice QCD simulations at the physical quark masses support that the magnitude of the topological susceptibility smoothly decreases at high temperatures [10, 11, 12]. Furthermore, a strong correlation with the chiral restoration has also been studied through the meson susceptibility functions within $2$ flavor QCD [13, 14, 15, 16, 17] and $2+1$ flavor QCD [18, 19, 20].

Thus far, the susceptibilities in hot QCD matter have been explored by both lattice simulations [13, 14, 18, 19, 20, 15, 10, 11, 12, 16, 17] and effective model analyses [7, 21, 8] in order to gain deeper insights into the symmetry properties of QCD in the extreme environment. However, at finite quark chemical potential $\mu_{q}$ lattice QCD simulations with three colors suffer from the sign problem, and then the first-principle numerical computations cannot apply in baryonic matter straightforwardly [22]. For this reason, our understanding of QCD at low-temperature and high-density regime is still limited compared to that in hot medium.

In light of the difficulty of three-color QCD on lattice simulations with finite $\mu_{q}$, two-color QCD (${\rm QC_{2}D}$) with two flavors provides us with a valuable testing ground. This is because the sign problem is resolved in such QCD-like theory owing to its pseudo-real property [23]. Focusing on this fact, many efforts from lattice simulations are being devoted to understandings of, e.g., phase structures, thermodynamics quantities, electromagnetic responses, the hadron mass spectrum, and gluon propagators in cold and dense QC₂D matter [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. In association with such numerical examinations, theoretical investigations of QC₂D at finite $\mu_{q}$ based on effective models have been done [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65].

In QC₂D, diquarks composed of two quarks are treated as color-singlet baryons. In other words, baryons exhibit bosonic behavior similarly to mesons. Reflecting this fact in QC₂D, $SU(2)_{L}\times SU(2)_{R}$ chiral symmetry is extended to the so-called Pauli-Gursey $SU(4)$ symmetry, which allows us to describe diquark baryons and light mesons in the single multiplets. Accordingly, the spontaneously symmetry-breaking pattern caused by the chiral condensate is changed to $SU(4)\to Sp(4)$ [50, 51]. Despite such an extension of chiral symmetry, symmetry structures of the $U(1)_{A}$ axial anomaly induced by gluonic configurations essentially do not differ from those in ordinary three-color QCD.

To shed light on the role of the $U(1)_{A}$ axial anomaly in baryonic QCD matter, the $\mu_{q}$ dependence of the topological susceptibility has been numerically measured by lattice numerical simulations of QC₂D [30, 33, 41, 45]. The recent lattice result in Ref. [41] indicates that the effect of $\mu_{q}$ does not exert any influence on the behavior of the topological susceptibility in baryonic matter, resulting in an approximately constant value. In contrast, the other group shows that the topological susceptibility is suppressed in high-density regions [45]. Hence, there exist discrepancies among the lattice simulations at finite quark chemical potentials, and the fate of topological susceptibility at high-density regions is still controversial.

In this paper, motivated by the above puzzle, we investigate the topological susceptibility in zero-temperature QC₂D at finite $\mu_{q}$ based on an effective-model approach. In particular, we employ the linear sigma model based on the approximate Pauli-Gursey $SU(4)$ symmetry invented in Ref. [65]. Notably, this model is capable of treating the $\eta$ meson which plays a significant role in describing the $U(1)_{A}$ anomaly structures consistently with other light mesons and diquark baryons. In QC₂D, since the diquarks obey the Bose-Einstein statistics, when the mass of the ground-state diquark becomes zero they begin to exhibit the Bose-Einstein condensates (BECs), leading to the emergence of the diquark condensed phase [50, 51]. This phase is also referred to as the baryon superfluid phase due to the violation of $U(1)_{B}$ baryon-number symmetry. Meanwhile, the stable phase with no such BECs connected to the vacuum, i.e., zero temperature and zero chemical potential, is called the hadronic phase. The former nontrivial phase triggers a rich hadron mass spectrum such as a mixing among hadrons sharing the identical quantum numbers except for the baryon number.

Within the linear sigma model, the influences of the $U(1)_{A}$ anomaly on hadrons are described by the so-called Kobayashi-Maskawa-’t Hooft (KMT)-type determinant interaction [66, 67, 68, 69], which only breaks $U(1)_{A}$ symmetry but preserves the Pauli-Gursey $SU(4)$ one. This interaction induces a mass difference between the pion and $\eta$ meson in the vacuum. Thus, in the present analysis we particularly focus on the strength of the KMT-type interaction, in other words, the mass difference between the pion and $\eta$ meson, in order to quantify roles of the $U(1)_{A}$ anomaly in the topological susceptibility. Besides, in lattice simulations source contributions with respect to the diquark condensate would be left sizable, so in this paper we also investigate the diquark source effects so as to facilitate the comparison with lattice data.

This paper is organized as follows. In Sec. 2 we present general properties associated with the topological susceptibility in QC₂D by focusing on the underlying QC₂D theory, and discuss symmetry partner structures of the meson susceptibility functions. In Sec. 3, the emergence of the Pauli-Gursey $SU(4)$ symmetry in QC₂D is briefly explained, and our linear sigma model regarded as a low-energy theory of QC₂D is introduced. In Sec. 4 we show how the topological susceptibility within the linear sigma model is evaluated by explicitly demonstrating the matching between underlying QC₂D and the linear sigma model. Based on it, in Sec. 5 we show our numerical results on the topological susceptibility at finite $\mu_{q}$. In order to facilitate the comparison with lattice simulations, in Sec. 6 we also exhibit the results in the presence of the diquark source contributions. Finally, in Sec. 7 we conclude our present study.

Our main aim of this paper is to reveal properties of the topological susceptibility in zero-temperature QC₂D with finite quark chemical potential $\mu_{q}$. In this section, we present an analytic formula of the topological susceptibility based on the underlying QC₂D Lagrangian [5, 6, 7, 8, 9], which is useful for the investigation within the effective-model framework of the linear sigma model.

The topological susceptibility is one of indicators to measure the magnitude of the $U(1)_{A}$ anomaly, which is related to nontrivial gluonic configurations such as the instantons [3]. Hence, we need to return to QCD Lagrangian where such microscopic degrees of freedom are treated manifestly. In two-flavor QC₂D, the Lagrangian including the so-called QCD $\theta$-term in Minkowski spacetime is of the form

$\displaystyle{\cal L}_{\rm QC_{2}D}=\bar{\psi}(i\gamma^{\mu}D_{\mu}-m_{l})\psi-\frac{1}{4}G_{\mu\nu}^{a}G^{\mu\nu,a}+\theta\frac{g^{2}}{64\pi^{2}}\epsilon^{\mu\nu\rho\sigma}G_{\mu\nu}^{a}G_{\rho\sigma}^{a}\ .$

(2.1)

As for the first term, $\psi=(u,d)^{T}$ denotes the two-flavor quark doublet and $D_{\mu}\psi=(\partial_{\mu}-i\mu_{q}\delta_{\mu 0}-igA_{\mu}^{a}T_{c}^{a})\psi$ is the covariant derivative incorporating effects from a quark chemical potential $\mu_{q}$ and interactions with a gluon field $A_{\mu}^{a}$. The $2\times 2$ matrix $T_{c}^{a}=\tau_{c}^{a}/2$ is the generator of $SU(2)_{c}$ color group with $\tau_{c}^{a}$ being the Pauli matrix. Besides, $g$ and $m_{l}$ are the QCD coupling constant and a current quark mass where the isospin symmetric limit is taken, $m_{u}=m_{d}\equiv m_{l}$. The second term in Eq. (2.1) is a gluon kinetic term where $G_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{b}$ is the field strength of gluons. The last ingredient of ${\rm QC_{2}D}$ Lagrangian in Eq. (2.1) is the $\theta$-term of QC₂D, which is described by a flavor-singlet topological operator $Q\equiv(g^{2}/64\pi^{2})\epsilon^{\mu\nu\rho\sigma}G_{\mu\nu}^{a}G_{\rho\sigma}^{a}$ tagged with the $\theta$-parameter. Our purpose in this subsection is to derive useful identities with respect to the topological susceptibility, so that only the $\theta$-dependent term in Eq. (2.1), which is gauge invariant, plays significant roles. For this reason, the gauge-fixing terms and the corresponding Faddeev-Popov determinant, which do not affect the following discussions, have been omitted in Eq. (2.1).

The generating functional of ${\cal L}_{\rm QC_{2}D}$ in the path-integral formulation is given by

$\displaystyle Z_{\rm QC_{2}D}=\int[d\bar{\psi}d\psi][dA]\exp\Biggl{[}i\int d^{4}x{\cal L}_{\rm QC_{2}D}\Biggl{]}\ ,$

(2.2)

and the $\theta$-dependent effective action of ${\rm QC_{2}D}$ is evaluated as

$\displaystyle\Gamma_{{\rm QC_{2}D}}=-i\ln Z_{{\rm QC_{2}D}}\ .$

(2.3)

The topological susceptibility $\chi_{\rm top}$ is defined by the curvature of $\Gamma_{{\rm QC_{2}D}}$, i.e., a second derivative with respect to $\theta$ at $\theta=0$:

$\displaystyle\chi_{\rm top}$

$\displaystyle=$

$\displaystyle-\int d^{4}x\frac{\delta^{2}\Gamma_{{\rm QC_{2}D}}}{\delta\theta(x)\delta\theta(0)}\Biggl{|}_{\theta=0}.$

(2.4)

Thus, from a straightforward calculation of Eq. (2.4) based on the QC₂D Lagrangian in Eq. (2.1), one can find that the topological susceptibility is described by a two-point correlation function of the topological operator $Q=(g^{2}/64\pi^{2})\epsilon^{\mu\nu\rho\sigma}G_{\mu\nu}^{a}G_{\rho\sigma}^{a}$:

$\displaystyle\chi_{\rm top}$

$\displaystyle=$

$\displaystyle-i\int d^{4}x\langle 0|TQ(x)Q(0)|0\rangle\ ,$

(2.5)

with $T$ denoting the time ordered product. It should be noted that contributions stemming from a product of $\langle Q\rangle$ have been omitted from Eq. (2.5) due to the parity conservation. The topological susceptibility in Eq. (2.5) is written in terms of the gluonic operator $Q$, which would not be a manageable expression since our task in this paper is to evaluate $\chi_{\rm top}$ from a low-energy effective model involving only hadronic degrees of freedom. Difficulties in matching the susceptibility from the effective models with that from underlying QC₂D are, however, remedied by utilizing the $U(1)_{A}$ axial rotation properly as demonstrated below.

Under the $U(1)_{A}$ rotation with a rotation angle $\alpha_{A}$, the quark doublet transforms as

$\displaystyle\psi\to\exp(i\alpha_{A}/2\,\gamma_{5})\psi\ .$

(2.6)

Meanwhile, within the path-integral formalism the gluonic quantum anomaly is generated by the fermionic measure $[d\bar{\psi}d\psi]$ according to the Fujikawa’s method [70], resulting in that the rotated generating functional reads

	$\displaystyle Z_{\rm QC_{2}D}$	$\displaystyle\to$	$\displaystyle\int[d\bar{\psi}d\psi][dA]\exp\Biggl{[}i\int d^{4}x\Biggl{(}\bar{\psi}i\gamma^{\mu}D_{\mu}\psi-m_{l}\bar{\psi}\exp(i\alpha_{A}\gamma_{5})\psi$		(2.7)
			$\displaystyle-\frac{1}{4}G_{\mu\nu}^{a}G^{\mu\nu,a}+(\theta-2\alpha_{A})\frac{g^{2}}{64\pi^{2}}\epsilon^{\mu\nu\rho\sigma}G_{\mu\nu}^{a}G_{\rho\sigma}^{a}\Biggl{)}\Biggl{]}\ .$

Thus, when choosing the rotation angle to be $\alpha_{A}=\theta/2$, the $\theta$-dependence of the QCD $\theta$-term is transferred into the quark mass term as

$\displaystyle Z_{\rm QC_{2}D}$

$\displaystyle=$

$\displaystyle\int[d\bar{\psi}d\psi][dA]\exp\Biggl{[}i\int d^{4}x\Biggl{(}\bar{\psi}i\gamma^{\mu}D_{\mu}\psi-m_{l}\bar{\psi}\exp\left(i\theta/2\,\gamma_{5}\right)\psi-\frac{1}{4}G_{\mu\nu}^{a}G^{\mu\nu,a}\Biggl{)}\Biggl{]}\ .$

Following the procedure in Eqs. (2.3) and (2.4) with the rotated generating functional (LABEL:gene_func_mass_theta), the topological susceptibility $\chi_{\rm top}$ is now expressed by fermionic operators as

$\displaystyle\chi_{\rm top}=-\frac{1}{4}\left[m_{l}\langle\bar{\psi}\psi\rangle+im_{l}^{2}\chi_{\eta}\right]\ ,$

(2.9)

where $\langle\bar{\psi}\psi\rangle$ is the chiral condensate serving as an order parameter of the spontaneous chiral-symmetry breaking, and $\chi_{\eta}$ denotes an $\eta$-meson susceptibility function defined by

$\displaystyle\chi_{\eta}=\int d^{4}x\langle 0|T(i\bar{\psi}\gamma_{5}\psi)(x)(i\bar{\psi}\gamma_{5}\psi)(0)|0\rangle\ .$

(2.10)

It should be noted that, from the rotated generating functional in Eq. (2.7), a non-conservation law of the $U(1)_{A}$ axial current $j^{\mu}_{A}=\bar{\psi}\gamma^{\mu}\gamma_{5}\psi$ is also obtained as

$\displaystyle\partial_{\mu}j_{A}^{\mu}=2m_{l}\bar{\psi}i\gamma_{5}\psi+\frac{g^{2}}{16\pi^{2}}\epsilon^{\mu\nu\rho\sigma}G_{\mu\nu}^{a}G_{\rho\sigma}^{a}\ .$

(2.11)

The topological susceptibility (2.9) is further reduced to a handleable form. In fact, using the $SU(2)_{L}\times SU(2)_{R}$ chiral-partner relation shown in Appendix A, a chiral WTI with respect to the chiral condensate $\langle\bar{\psi}\psi\rangle$ is derived as in Appendix B, which reads

$\displaystyle\langle\bar{\psi}\psi\rangle=-im_{l}\chi_{\pi}\ .$

(2.12)

In this identity, $\chi_{\pi}$ is a pion susceptibility function defined by

$\displaystyle\chi_{\pi}\delta^{ab}=\int d^{4}x\langle 0|T(i\bar{\psi}\gamma_{5}\tau^{a}_{f}\psi)(x)(i\bar{\psi}\gamma_{5}\tau^{b}_{f}\psi)(0)|0\rangle\ ,$

(2.13)

with $\tau^{a}_{f}$ being the Pauli matrix in the flavor space. Therefore, inserting Eq. (2.12) into Eq. (2.9), the topological susceptibility is found to be determined in terms of a difference of $\chi_{\pi}$ and $\chi_{\eta}$ as

$\displaystyle\chi_{\rm top}=\frac{im_{l}^{2}}{4}(\chi_{\pi}-\chi_{\eta})\ .$

(2.14)

This expression is identical to the one obtained in ordinary three-color QCD through the WTI [5, 6, 7, 8, 9]. Here, to facilitate an understanding of the role of topological susceptibility, we insert the scalar meson susceptibilities $\chi_{\sigma}$ and $\chi_{a_{0}}$ in Eq. (2.14):

	$\displaystyle\chi_{\rm top}$	$\displaystyle=$	$\displaystyle\frac{im_{l}^{2}}{4}\left[(\chi_{\pi}-\chi_{\sigma})-(\chi_{\eta}-\chi_{\sigma})\right]\ ,$
	$\displaystyle\chi_{\rm top}$	$\displaystyle=$	$\displaystyle\frac{im_{l}^{2}}{4}\left[(\chi_{\pi}-\chi_{a_{0}})-(\chi_{\eta}-\chi_{a_{0}})\right]\ ,$		(2.15)

where $\chi_{\sigma}$ and $\chi_{a_{0}}$ are the susceptibility functions made of the composite operators $\bar{\psi}\psi$ and $\bar{\psi}\tau_{f}^{a}\psi$, respectively. Indeed, under the chiral $SU(2)_{L}\times SU(2)_{R}$ rotation and the $U(1)_{A}$ rotation, the meson susceptibility functions are transformed into each other:

as explicitly shown in Appendix A. With this transformation, one can realize that the topological susceptibility in Eq. (2.15) is described by the combinations of the chiral $SU(2)$ partner $\chi_{\pi}\leftrightarrow\chi_{\sigma}$ ($\chi_{a_{0}}\leftrightarrow\chi_{\eta}$) and the $U(1)$ axial partner $\chi_{\pi}\leftrightarrow\chi_{a_{0}}$ ($\chi_{\sigma}\leftrightarrow\chi_{\eta}$) . When chiral symmetry is restored and the order parameter of the spontaneous chiral symmetry breaking vanishes $\langle\bar{\psi}\psi\rangle\to 0$, the chiral partner becomes (approximately) degenerate:

$\displaystyle\mbox{$SU(2)_{L}\times SU(2)_{R}$ restoration limit}:$

$\displaystyle\begin{cases}\chi_{\pi}-\chi_{\sigma}\to 0\\ \chi_{a_{0}}-\chi_{\eta}\to 0\end{cases}.$

(2.16)

After the chiral restoration, the topological susceptibility is dominated by the $U(1)_{A}$ axial partner: $\chi_{\rm top}\sim\chi_{\eta}-\chi_{\sigma}$ ($\chi_{\rm top}\sim\chi_{\pi}-\chi_{a_{0}}$), so that $\chi_{\rm top}$ acts as the indicator for the breaking strength of $U(1)_{A}$ symmetry. It should be noted that $\chi_{\rm top}$ trivially vanishes in the chiral limit ($m_{l}=0$) as seen from Eq. (2.14). In this limit, the topological susceptibility is no longer regarded as the indicator. This can also be understood by the fact that when $m_{l}=0$, the $\theta$ dependence of the generating functional in Eq. (LABEL:gene_func_mass_theta) disappears, resulting in the vanishing topological susceptibility defined by a second derivative with respect to $\theta$.

When studying with a low-energy effective model, the analytical expression of (2.14) is useful for evaluating the topological susceptibility $\chi_{\rm top}$. Here, we show another expression of $\chi_{\rm top}$ so as to see contributions from the chiral condensate $\langle\bar{\psi}\psi\rangle$ clearly. That is, from the identity (2.12) one can rewrite Eq. (2.14) into

$\displaystyle\chi_{\rm top}=-\frac{m_{l}\langle\bar{\psi}\psi\rangle}{4}\delta_{m}\ .$

(2.17)

In this expression, the dimensionless quantity $\delta_{m}$ is defined by

$\displaystyle\delta_{m}\equiv 1-\frac{\chi_{\eta}}{\chi_{\pi}}\ ,$

(2.18)

which measures the variation of the susceptibility functions $\chi_{\pi}$ and $\chi_{\eta}$. Equation (2.17) indicates, indeed, that the topological susceptibility is proportional to the chiral condensate $\langle\bar{\psi}\psi\rangle$ and $\delta_{m}$; the explicit chiral-symmetry breaking is entangled with the $U(1)_{A}$ anomaly contribution captured by the quantity $\delta_{m}$ in $\chi_{\rm top}$. This structure plays an important role in determining the asymptotic behavior of $\chi_{\rm top}$ at sufficiently large $\mu_{q}$ where chiral symmetry is restored.

Furthermore, the Gell-Mann-Oakes-Renner (GOR) relation: $f_{\pi}^{2}m_{\pi}^{2}=-m_{l}\langle\bar{\psi}\psi\rangle/2$ [71], enables us to rewrite the topological susceptibility in Eq. (2.17) as

$\displaystyle\chi_{\rm top}=\frac{f_{\pi}^{2}m_{\pi}^{2}}{2}\delta_{m}\ ,$

(2.19)

where $f_{\pi}$ is the pion decay constant and $m_{\pi}$ is the pion mass. It is obvious from its derivation that Eq. (2.19) holds model-independently.^#1#1#1The GOR relation $f_{\pi}^{2}m_{\pi}^{2}=-m_{l}\langle\bar{\psi}\psi\rangle/2$ is derived model-independently but with an assumption that the two-point function of the pseudoscalar channel $D_{\pi}\delta^{ab}\equiv\int d^{4}x\langle 0|T(i\bar{\psi}\gamma_{5}\tau^{a}_{f}\psi)(x)(i\bar{\psi}\gamma_{5}\tau^{b}_{f}\psi)(0)|0\rangle{\rm e}^{-ip\cdot x}$ is dominated by the lightest pseudoscalar-meson pole : $D_{\pi}\propto i/(p^{2}-m_{\pi}^{2})$, as in the case of three-color QCD. Accordingly, the relation (2.19) also holds upon the pole dominance of the lightest pseudoscalar meson. Notably, the quantity $\delta_{m}$ in the vacuum is solely determined by the masses of pion and $\eta$ meson as

$\displaystyle\delta_{m}\to 1-\frac{m_{\pi}^{2}}{m_{\eta}^{2}}\ ,$

(2.20)

as long as we stick to the low-energy regime of QC₂D where $\chi_{\pi}\sim-i/m_{\pi}^{2}$ and $\chi_{\eta}\sim-i/m_{\eta}^{2}$ can apply. Therefore, Eq. (2.19) implies that the topological susceptibility in the vacuum is expressed by three basic observables in low-energy QC₂D: $f_{\pi}$, $m_{\pi}$ and $m_{\eta}$. We note that $\delta_{m}\to 1$ corresponds to the significantly large anomaly effects, while $\delta_{m}\to 0$ implies no such effects. We also note that Leutwyler and Smilga obtained the following form based on the Chiral Perturbation Theory (ChPT) in three-color QCD [72]:

$\displaystyle\chi^{\rm(LS)}_{\rm top}=-\frac{m_{l}\langle\bar{\psi}\psi\rangle}{4}=\frac{f_{\pi}^{2}m_{\pi}^{2}}{2}\ ,$

(2.21)

where the $\chi_{\eta}$ contributions are missing. Indeed, in Eq. (2.21), the large anomaly is accidentally taken into account: $\delta_{m}=1$. Even in the case of QC₂D, the Leutwyler-Smilga relation was also found in  [73].

In ${\rm QC_{2}D}$, diquarks (antidiquarks) carrying the quark number $+2$ ($-2$) are treated as color-singlet baryons, namely, baryons become bosonic similarly to mesons. Accordingly, the so-called Pauli-Gursey $SU(4)$ symmetry, which enables us to treat both the baryons and mesons in a consistent way, emerges [50, 51]. In this section, we briefly explain how the Pauli-Gursey $SU(4)$ symmetry manifests itself from QC₂D Lagrangian, and based on the symmetry we present the linear sigma model which describes couplings among the baryons and mesons.

Thanks to pseudoreal properties of the $SU(2)$ generators for color and Dirac spaces, $T_{c}^{a}=-\tau^{2}(T_{c}^{a})^{T}\tau^{2}$ and $\sigma^{i}=-\sigma^{2}(\sigma^{i})^{T}\sigma^{2}$ ($\sigma^{i}$ is the Pauli matrix in the Dirac space), one can show that the kinetc term of quarks coupling with gauge fields in QC₂D, i.e., the first piece in Eq. (2.1), is rewritten to

$\displaystyle{\cal L}_{\rm Q_{2}CD}^{(\rm kin)}=\Psi^{\dagger}i\sigma^{\mu}D_{\mu}\Psi\ ,$

(3.1)

with $\sigma^{\mu}=({\bm{1}},\sigma^{i})$, in the Weyl representation. In Eq. (3.1) the quark field $\Psi$ is given by a four-component column vector in the flavor space as

$\displaystyle\Psi=\begin{pmatrix}\psi_{R}\\ \tilde{\psi}_{L}\end{pmatrix}=\begin{pmatrix}u_{R}\\ d_{R}\\ \tilde{u}_{L}\\ \tilde{d}_{L}\end{pmatrix}\ ,$

(3.2)

where $\psi_{L(R)}=\frac{1\mp\gamma_{5}}{2}\psi$ denotes the left-handed (right-handed) quark field and $\tilde{\psi}_{L(R)}$ is the conjugate one:

$\displaystyle\tilde{\psi}_{L(R)}=\sigma^{2}\tau^{2}_{c}\psi_{L(R)}^{*}\ .$

(3.3)

Equation (3.1) implies that the quark kinetic term in QC₂D is invariant under an $SU(4)$ transformation for the quark field as

$\displaystyle\Psi\to g\Psi\ ,$

(3.4)

with $g\in SU(4)$. Thus, it is proven that $SU(2)_{L}\times SU(2)_{R}$ chiral symmetry in QC₂D is extended to the $SU(4)$ one which is often referred to as the Pauli-Gursey $SU(4)$ symmetry [50, 51].

Similarly to the kinetic part, the quark mass term, i.e., the second piece in Eq. (2.1), is also expressed in terms of the four-component field $\Psi$, which reads

$\displaystyle{\cal L}_{\rm QC_{2}D}^{(\rm mass)}=\frac{m_{l}}{2}\left(\Psi^{T}\sigma^{2}\tau_{c}^{2}E\Psi+\Psi^{\dagger}\sigma^{2}\tau_{c}^{2}E^{T}\Psi^{*}\right)\ .$

(3.5)

This term, however, breaks the Pauli-Gursey $SU(4)$ symmetry explicitly due to the presence of a symplectic matrix in the flavor space

$\displaystyle E=\left(\begin{array}[]{cc}0&{\bm{1}}\\ -{\bm{1}}&0\\ \end{array}\right)\ ,$

(3.8)

in between the two quark fields. For this reason, the systematic treatment based on the viewpoint of the $SU(4)$ symmetry is spoiled by the quark masses. To recover the systematics, we introduce a spurion field $\zeta_{\rm sp}$ which transforms as

$\displaystyle\zeta_{\rm sp}\to g\,\zeta_{\rm sp}\,g^{T}\ .$

(3.9)

To construct the $SU(4)$-invariant Lagrangian, the quark mass term is promoted to the spurion term

$\displaystyle{\cal L}_{\rm QC_{2}D}^{\rm(sp)}=-\Psi^{T}\sigma^{2}\tau^{2}_{c}\zeta^{\dagger}_{\rm sp}\Psi-\Psi^{\dagger}\sigma^{2}\tau^{2}_{c}\zeta_{\rm sp}\Psi^{*}\ .$

(3.10)

In fact, one can show that the quark mass term (3.5) is appropriately reproduced by taking the vacuum expectation value (VEV) of the spurion field as

$\displaystyle\langle\zeta_{\rm sp}\rangle=\frac{m_{l}}{2}E\ .$

(3.11)

In what follows, we construct the linear sigma model to describe hadrons at the low-energy regime of QC₂D, based on the symmetries explained above. The fundamental building block of the linear sigma model in QC₂D is a $4\times 4$ matrix field $\Sigma_{ij}$ whose symmetry properties are the same as those of a quark bilinear field $\Psi_{j}^{T}\sigma^{2}\tau_{c}^{2}\Psi_{i}$. That is, $\Sigma$ transforms as

$\displaystyle\Sigma\to g\Sigma g^{T}\ ,$

(3.12)

under the $SU(4)$ transformation. As explained in Ref. [65] in detail, the $\Sigma$ can be parameterized by low-lying hadrons in QC₂D as

$\displaystyle\Sigma=(S^{a}-iP^{a})X^{a}E+(B^{\prime i}-iB^{i})X^{i}E\ ,$

(3.13)

where $S^{a}$, $P^{a}$, $B^{i}$ and $B^{\prime i}$ represent scalar mesons, pseudoscalar mesons, positive-parity diquark baryons and negative-parity diquark baryons, respectively. The $4\times 4$ matrices $X^{a}$ and $X^{i}$ are generators of $U(4)$ defined by

	$\displaystyle X^{a}$	$\displaystyle=$	$\displaystyle\frac{1}{2\sqrt{2}}\begin{pmatrix}\tau^{a}_{f}&0\\ 0&(\tau^{a}_{f})^{T}\end{pmatrix}\;\;\;(a=0,1,2,3)\ ,$
	$\displaystyle X^{i}$	$\displaystyle=$	$\displaystyle\frac{1}{2\sqrt{2}}\begin{pmatrix}0&D_{f}^{i}\\ (D_{f}^{i})^{\dagger}&0\end{pmatrix}\;\;\;(i=4,5)\ ,$		(3.14)

where $\tau_{f}^{0}={\bm{1}}_{2\times 2}$ in the flavor space, and $D_{f}^{i}$ represent $D_{f}^{4}=\tau_{f}^{2}$ and $D_{f}^{5}=i\tau_{f}^{2}$. Following the parametrization given in Ref. [65], we employ the following hadron assignment for $\Sigma$:

$\displaystyle\Sigma=\frac{1}{2}\begin{pmatrix}0&-B^{\prime}+iB&\frac{\sigma-i\eta+a^{0}-i\pi^{0}}{\sqrt{2}}&a^{+}-i\pi^{+}\\ B^{\prime}-iB&0&a^{-}-i\pi^{-}&\frac{\sigma-i\eta-a^{0}+i\pi^{0}}{\sqrt{2}}\\ -\frac{\sigma-i\eta+a^{0}-i\pi^{0}}{\sqrt{2}}&-a^{-}+i\pi^{-}&0&-\bar{B}^{\prime}+i\bar{B}\\ -a^{+}+i\pi^{+}&-\frac{\sigma-i\eta-a^{0}+i\pi^{0}}{\sqrt{2}}&\bar{B}^{\prime}-i\bar{B}&0\end{pmatrix}\ ,$

(3.15)

where $\pi^{0}=P^{3}$ and $\pi^{\pm}=(P^{1}\mp iP^{2})/\sqrt{2}$ are the pions, $\eta=P^{0}$ is the $\eta$ meson, $\sigma=S^{0}$ is the iso-singlet scalar meson ($\sigma$ meson), $a_{0}^{0}=S^{3}$ and $a_{0}^{\pm}=(S^{1}\mp iS^{2})/\sqrt{2}$ are the iso-triplet scalar mesons ($a_{0}$ mesons), $B=(B^{5}-iB^{4})/\sqrt{2}$ [$\bar{B}=(B^{5}+iB^{4})/\sqrt{2}$] is the positive-parity diquark baryon (the antidiquark baryon), and $B^{\prime}=(B^{\prime 5}-iB^{\prime 4})/\sqrt{2}$ [$\bar{B}^{\prime}=(B^{\prime 5}+iB^{\prime 4})/\sqrt{2}$] is the negative-parity diquark baryon (the antidiquark baryon).

With the matrix $\Sigma$ given in Eq. (3.15), our linear sigma model in QC₂D which respects the Pauli-Gursey $SU(4)$ symmetry is obtained as

$\displaystyle{\cal L}_{\rm LSM}$

$\displaystyle=$

$\displaystyle{\rm tr}[D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma]-V\ ,$

(3.16)

where the covariant derivative for $\Sigma$ is defined by

$\displaystyle D_{\mu}\Sigma=\partial_{\mu}\Sigma-i\mu_{q}\delta_{\mu 0}(J\Sigma+\Sigma J^{T})\;\;\;\mbox{with}\;\;\;J=\begin{pmatrix}{\bm{1}}&0\\ 0&-{\bm{1}}\end{pmatrix}.$

(3.17)

Here, the quark chemical potential $\mu_{q}$ is incorporated in the covariant derivative through gauging the $U(1)_{B}$ baryon-number symmetry. Besides, $V$ represents potential terms describing interactions among the hadrons, which is separated into three parts as

$\displaystyle V=V_{0}+V_{\rm sp}+V_{\rm anom}\ .$

(3.18)

$V_{0}$ represents an invariant part under the Pauli-Gursey $SU(4)$ symmetry. When we include contributions up to the fourth order of $\Sigma$ as widely done in the linear sigma model for three-color QCD, it takes the form of

$\displaystyle V_{0}=m_{0}^{2}{\rm tr}[\Sigma^{\dagger}\Sigma]+\lambda_{1}({\rm tr}[\Sigma^{\dagger}\Sigma])^{2}+\lambda_{2}{\rm tr}[(\Sigma^{\dagger}\Sigma)^{2}]\ ,$

(3.19)

where $m_{0}^{2}$ is a mass parameter, and $\lambda_{1}$ and $\lambda_{2}$ are coupling constants controlling the strength of four point interactions. The second piece of Eq. (3.18), $V_{\rm sp}$, is the spurion term corresponding to ${\cal L}_{\rm QC_{2}D}^{\rm(sp)}$ in Eq. (3.10), which is given by

$\displaystyle V_{\rm sp}=-\bar{c}\,{\rm tr}[\zeta^{\dagger}_{\rm sp}\Sigma+\Sigma^{\dagger}\zeta_{\rm sp}]\ ,$

(3.20)

where the parameter $\bar{c}$ is real, and has the mass dimension two. Although the $V_{\rm sp}$ is invariant under the $SU(4)$ transformation thanks to Eq. (3.9), the spurion field $\chi_{\rm sp}$ must be replaced by its VEV in Eq. (3.11) so as to incorporate the effect of the finite quark mass in a final step for evaluating physical observables.

The last piece in the potential (3.18), $V_{\rm anom.}$, includes $U(1)_{A}$ anomalous contributions which is responsible for the gluonic part in the non-conservation law of the axial current: the second term of the right-hand side (RHS) of Eq. (2.11). Within our present model, the $U(1)_{A}$ anomalous term is expressed by the Kobayashi-Maskawa-’t Hooft (KMT)-type interaction, [66, 67, 68, 69]

$\displaystyle V_{\rm anom}=-c({\rm det}\Sigma+\det\Sigma^{\dagger})\ .$

(3.21)

As demonstrated below, this anomalous term generates a mass difference between the pion and $\eta$ meson in the vacuum [65], and plays an important role in driving a finite topological susceptibility. It should be noted that the KMT-type interaction is described by four-incoming and four-outgoing quarks owing to the quark bilinear field $\Sigma_{ij}\sim\bar{\Psi}^{T}_{j}\sigma^{2}\tau_{c}^{2}\Psi_{i}$ based on the Pauli-Gursey $SU(4)$ symmetry. Thus, in hadronic-level diagrams, $V_{\rm anom}$ represents four-point interactions.

General expressions and characteristics of the topological susceptibility in QC₂D have been reviewed in Sec. 2, and the linear sigma model, which describes hadrons in the low-energy regime of QC₂D, has been invented in Sec. 3. In this section, we explain our strategy to evaluate the topological susceptibility within our linear sigma model through matching with the underlying QC₂D theory.