Theta vacuum effects on QCD phase diagram

Theta vacuum effects on the QCD phase structure in the \mu-T plane are studied by using the Polyakov-loop extended Nambu-Jona-Lasinio model and its extension, where \mu is the quark chemical potential and T is temperature, respectively. As the parameter \theta of the theta vacuum increases, the chiral transition becomes stronger. For large \theta, it eventually becomes first order even at zero \mu.


I. INTRODUCTION
Violations of parity (P ), charge conjugation (C) and charge-parity symmetries (CP ) are important subjects in particle and nuclear physics. For example, the strong CP problem is a longstanding puzzle; see for example Ref. [1] for a review of this problem. Lorentz and gauge invariance allow the Quantum Chromodynamics (QCD) action to have a term of the topological charge, where F a µν is the field strength of gluon. The parameter θ can take any arbitrary value between −π and π, where θ = −π is identical with θ = π. Nevertheless, experiment indicates |θ| < 3 × 10 −10 [2,3]. Since θ is P -odd (CP -odd), P (CP ) is then preserved for θ = 0 and ±π, but explicitly broken for other θ. Why is θ so small ? This is the so-called strong CP problem.
For T higher than the QCD scale Λ QCD , there is a possibility that a finite θ, depending on spacetime coordinates (t, x), is effectively induced, since sphalerons are so activated as to jump over the potential barrier between the different degenerate ground states [17]. If so, P and CP symmetries can be violated locally in high-energy heavy-ion collisions or the early universe at T ≈ Λ QCD . Actually, it is argued in Refs. [12,18] that θ may be of order one during the QCD phase transition in the early universe, while it vanishes at the present epoch [19][20][21][22][23]. This finite θ may be a new source of very large CP violation in the Universe and may be a crucial missing element for solving the puzzle of baryogenesis.
Furthermore, this effective θ(t, x) deviates the total number of particles plus antiparticles with right-handed helicity from that with left-handed helicity. The magnetic field, formed in the early stage of heavy-ion collision, will lift the degeneracy in spin depending on the charge of particle.
As a consequence of this fact, an electromagnetic current is generated along the magnetic field, since particles with right-handed helicity move opposite to antiparticles with right-handed helicity. This is the so-called chiral magnetic effect (CME) [18,[24][25][26]. CME may explain the charge separations observed in the recent STAR results [27]. Thus, the thermal system with nonzero θ is quite interesting.
In this letter, we study effects of the theta-vacuum on QCD phase diagram by using the twoflavor PNJL model [16,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] and its extension; see Ref. [16] and references therein for further information on the PNJL model. Particularly, our attention is focused on the two-flavor phase diagram in the µ-T plane, where µ is the quark chemical potential. In our previous work Ref. [16], as a theoretical interest, we investigated spontaneous P and C violations at finite θ and imaginary µ. As physical phenomena that may occur in the early universe or the high-energy heavy-ion collisions, we here examine the chiral and P symmetry restorations at finite T and θ and real µ.
This paper is organized as follows. In section II, the PNJL model and its extension are explained briefly. In section III, the numerical results are shown. Section IV is devoted to summary.

II. PNJL MODEL
Pioneering work on the parity violation and its restoration in the framework of the NJL model [13][14][15][45][46][47] was done by Fujihara, Inagaki and Kimura [13]. Boer and Boomsma studied this issue extensively [14,15]. In Ref. [16], we have extended their formalism based on the NJL model to that on the PNJL model. The two-flavor (N f = 2) PNJL Lagrangian with the θ-dependent anomaly term is where q = (u, d) denotes the two-flavor quark field, m does the current quark-mass matrix diag(m u , m d ), τ 0 is the 2 × 2 unit matrix, τ a (a = 1, 2, 3) are the Pauli matrices and 2 with the gauge field A ν a , the Gell-Mann matrix λ a and the gauge coupling g. In the NJL sector, G 1 denotes the coupling constant of the scalar and pseudoscalar-type four-quark interaction, and G 2 is the coupling constant of the Kobayashi-Maskawa-'t Hooft determinant interaction [48,49] the matrix indices of which run in the flavor space. The Polyakov potential U, defined later in (19), is a function of the Polyakov loop Φ and its Hermitian conjugate Φ * , where P is the path ordering and A 4 = iA 0 . In the chiral limit (m u = m d = 0), the Lagrangian density has the exact SU( is explicitly broken if G 2 = 0. The temporal component of the gauge field is diagonal in the flavor space, because the color and the flavor spaces are completely separated out in the present case. In the Polyakov gauge, L can be written in a diagonal form in the color space [29]: where loop Φ is an exact order parameter of the spontaneous Z 3 symmetry breaking in the pure gauge theory. Although the Z 3 symmetry is not an exact one in the system with dynamical quarks, it may be a good indicator of the deconfinement phase transition. Therefore, we use Φ to define the deconfinement phase transition. For simplicity, we assume below that m u = m d = m 0 .
We transform the quark field q to the new one q ′ with condensates η ′ , a ′ i and π ′ i . The Lagrangian density is then rewritten with q ′ as where m 0+ = m 0 cos θ 2 and m 0− = m 0 sin θ 2 . Making the mean field approximation and performing the path integral over the quark field, one can obtain the thermodynamic potential Ω (per volume) for finite T and µ: where In the right-hand side of (9), only the first term diverges. The term is then regularized by the three-dimensional momentum cutoff Λ [29,30]. Following Ref. [14,15], we introduce c as G 1 = (1 − c)G and G 2 = cG, where 0 ≤ c ≤ 0.5 and G > 0. Hence, the NJL sector has four parameters of m 0 , Λ, G and c. We put m 0 = 5.5MeV. The parameters Λ and G are so chosen as to reproduce the pion decay constant f π = 93MeV and the pion mass m π = 139MeV at vacuum. The remaining parameter c is a free parameter. Although the exact value of c is unknown, it is known from the analysis of the η-η ′ splitting in the three-flavor model that c ∼ 0.2 is favorable [50]. The value c = 0.2 has been also used in Refs. [14,15]. Therefore, we take c = 0.2 also here.
The PNJL model is reduced to the NJL model in the limit of T = 0, since in (9) the Polyakovloop potential U tends to zero and the logarithmic terms approach N c Θ(µ − E ± ± ), where Θ(x) = 1 for x > 0 and Θ(x) = 0 for x < 0, in the limit. Since P breaking at T = 0 and θ = π has already been studied in detail by the NJL model [13][14][15], we here concentrate ourselves on P restoration at finite T and θ = π.
The Polyakov potential U of Ref. [31] is fitted to LQCD data in the pure gauge theory at finite T [51,52]: The parameters included in U are summarized in Table I the full LQCD simulation [53][54][55][56] predicts. Therefore, we rescale T 0 to 212 MeV [36] so that the PNJL model can reproduce the critical temperature 173 MeV of the full LQCD simulation.  The variables X = Φ, Φ * , σ, π i , η and a i satisfy the stationary conditions, The solutions of the stationary conditions do not give the global minimum Ω necessarily. There is a possibility that they yield a local minimum or even a maximum. We then have checked that the solutions yield the global minimum when the solutions X(T, θ, µ) are inserted into (9). For θ = 0 and π, the thermodynamic potential Ω is invariant under P transformation, Since the four-quark coupling constant G contains effects of gluons, G may depend on Φ.
In fact, recent calculations [57][58][59] of the exact renormalization group equation (ERGE) [60] suggest that the higher-order mixing interaction is induced by ERGE. It is highly expected that the functional form and the strength of the entanglement vertex G(Φ) are determined in future by these theoretical approaches. In Ref. [42], we assumed the following form by respecting the chiral symmetry, P symmetry, C symmetry [16,38] and the extended Z 3 symmetry. This model is called the entanglement PNJL (EPNJL) model. The EPNJL model with the parameter set [42], α 1 = α 2 = 0.2 and T 0 = 190 MeV, can reproduce LQCD data at imaginary chemical potential [61][62][63][64][65][66][67][68][69][70] and real isospin chemical potential [71] as well as the results at zero chemical potential [53][54][55]. Recently, it was shown in Ref. [43] that, also under the strong magnetic field, the EPNJL model yields results consistent with the LQCD data [72]. The EPNJL model with this parameter set is also applied to the present case with real µ and finite θ.

III. NUMERICAL RESULTS
The present PNJL model has eight condensates of quark-antiquark pair. However, a and π vanish [14][15][16], since m u = m d and the isospin chemical potential is not considered here. We can then concentrate ourselves on σ, η and Φ. We can also restrict θ in a period 0 ≤ θ ≤ 2π without loss of generality. Figure 1 shows T dependence of σ and Φ at µ = 0 and θ = 0; note that η = 0 in this case.
In both the PNJL and EPNJL models, σ (Φ) rapidly but continuously decreases (increases) as T increases. Therefore, the chiral restoration and the deconfinement transition are crossover. The crossover transitions occur more rapidly in the EPNJL model than in the PNJL model. Figure 2 presents T dependence of σ, η and Φ at µ = 0 and θ = π; note that this figure plots condensates σ and η before the transformation (6). For both the PNJL and EPNJL models, the eta condensation (η = 0) occurs at low T and hence P symmetry is spontaneously broken there. At high T , in contrast, η = 0 and hence P symmetry is restored. The critical temperature T P of P restoration is 202 (170) MeV in the PNJL (EPNJL) model.
The order of P restoration was reported to be of second order in the NJL model [14,15] but of first order in the linear sigma model [73]. In the EPNJL (PNJL) model, η is discontinuous in its zeroth (first) order. Thus, the PNJL model supports the second-order transition, but the EPNJL model does the first-order. The zeroth-order discontinuity (gap) of η in the EPNJL model is propagated to other quantities σ and Φ as the zeroth-order discontinuity (gap), according to the discontinuity theorem by Barducci, Casalbuoni, Pettini and Gatto [74]. This really takes place in Fig. 2, although the discontinuity (gap) is appreciable for Φ but very tiny for σ. The firstorder discontinuity (cusp) of η in the PNJL model is also propagated to σ and Φ as the first-order discontinuity (cusp) [37]. As mentioned above, the EPNJL model is more consistent with the LQCD data than the PNJL model. This means that the order of P restoration may be weak first order.
T dependence of σ, η and Φ is shown also for µ = 300MeV and θ = π in Fig. 3. In the case of large µ, P restoration is of first order in both the PNJL and EPNJL models. As µ increase with θ fixed at π, thus, P restoration changes from the second order to the first order in the PNJL model,    grams are mirror symmetric with respect to the µ-T plane at θ = π. At finite θ, σ 2 + η 2 is the order parameter of the chiral transition rather than σ itself [16]. In the PNJL model of panel (a), point A in the µ-T plane at θ = π is a TCP of P restoration and a critical endpoint (CEP) of chiral restoration at which the first-order (solid) line is connected to the crossover (dotted) line. Point C in the µ-T plane at θ = 0 is another CEP of the chiral transition [46,75]. The second-order (dashed) line from C to A is a trajectory of CEP with respect to increasing θ from 0 to π. Thus, the CEP (point C) at θ = 0 is a remnant of the TCP (point A) of P restoration at θ = π. In the EPNJL  QCD has the anomalous Ward identities among the chiral and eta condensates, σ and η, the gluon condensate g 2 64π 2 ǫ αβσρ F a µν F a σρ and the topological susceptibility χ t [12]: The identities are useful for checking the self-consistency of the proposed model [12]. The gluon condensate does not appear explicitly in the PNJL model, but the model satisfies The PNJL model also satisfies the second Ward identity (25) as shown below. In the PNJL model, the left-hand side of (25) is rewritten into with the inverse curvature matrix for the parameters φ = (σ ′ , η ′ , Φ, Φ * ). Note that C −1 ij is the susceptibility χ ij of order parameters φ i and φ j and the term including χ ij is of order O(m 2 0 ). Equation (27) turns out to be the second Ward identity (25) after simple algebraic calculations. For θ = 0, Eq. (27) is further rewritten into in the virtue of the θ-reflection symmetry, where Ω 0 N N = ∂ 2 Ω 0 ∂N 2 for the thermodynamic potential Ω 0 with no the meson potential U and no the Polyakov potential U.  O(m 2 0 ) coming from the second term of the right-hand side of (27) become significant.

IV. SUMMARY
In summary, we have investigated theta-vacuum effects on QCD phase diagram, using the PNJL and EPNJL models. For the both models, the chiral transition becomes strong as θ increases.
Particularly in the EPNJL model that is more reliable than the PNJL model, it becomes first order even at µ = 0 when θ is large. This is an important result. If the chiral transition becomes first order at zero µ, it will change the scenario of cosmological evolution. For example, the firstorder transition allows us to think the inhomogeneous Big-Bang nucleosynthesis model or a new scenario of baryogenesis. Our analyses are based on the two-flavor PNJL (EPNJL) model and effects of the strange quark are then neglected. It is very interesting to study the effects by using the three-flavor PNJL model [40].