Phase diagram of quark-antiquark and diquark condensates at finite temperature and density in the 3-dimensional Gross Neveu model

# Phase diagram of quark-antiquark and diquark condensates at finite temperature and density in the 3-dimensional Gross Neveu model

Hiroaki Kohyama Department of Physics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, JAPAN
July 12, 2019
###### Abstract

We construct the phase diagrams of the quark-antiquark and diquark condensates at finite temperature and density in the 3D (dimensional) 2-flavor Gross Neveu model. We found that, in contrast to the case of the 4D Nambu Jona-Lasinio model, there is no region where the quark-antiquark and diquark condensates coexist. The phase diagrams obtained for some parameter region show similar structure with the 4D QCD phase diagram.

###### pacs:
12.38.Aw, 12.38Lg, 11.15.Pg, 11.10.Wx
preprint: OCU-PHYS 269

## I Introduction

The four-fermion interaction models are considered to be the effective theory for describing phase transitions. In dimensions (4D), the Nambu Jona-Lasinio (NJL) model can be regarded as an effective theory of QCDNJL (). The model successfully reproduces the chiral phase transition in quark matter (see, e.g. HatsuKuni (); onNJL ()). Furthermore, through recent studies of the diquark condensate, a variety of color superconducting phases are expected to be realizedAlford (). It has been shown that there is the region where the chiral(quark-antiquark) and diquark condensates coexist. (For a nice review, see, e.g. Mei ().)

The Gross-Neveu (GN) model proposed in 1974, is a model of Dirac fermions interacting via four-fermion interactionsGN (). The 2D GN model is a renormalizable quantum field theory, and the 3D GN model is renormalizable in the leading order. They are closely related to the Bardeen-Cooper-Schrieffer theory of superconductivityBCS (). Although the lower dimensional theory seems not to be realistic, the 2D GN type models are believed to be the effective models of 1 dimensional condensed matter systems such as conducting polymers like polyacetylenepolymer (). Since the GN model shares many properties with QCD, notably asymptotic freedom, chiral symmetry breaking in vacuum, the comparison of QCD and the GN model is an interesting subject.

A variety of works has been devoted to the study of the GN modelonGN (); Mal (); Kanemura (); Ulli (); Klimenko (); Kneur (); Repre (); Zhou (). By using bag-model boundary conditions, a closed formula for the effective renormalized coupling constant in the large-N limit, was derived in Mal (). In a constant curvature space, the phase structure of chiral symmetry breaking in the GN model at finite temperature and density is discussed in Kanemura (). The phase diagrams of the quark-antiquark condensate in the 2D GN model was obtained in Ulli (). In 3D, analyses so far have been done choosing 2-dimensional(2d) or 4-dimensional(4d) spinor representations for quarks. The case of the 2d representation is interesting itself because it is the nontrivial lowest-order representation. The phase diagram of the quark-antiquark condensate was constructed by employing the 2d representation in Klimenko (). On the other hand in the case of the 4d representation, there exists the (see, e.g. Appel ()) and the properties of the model bear resemblance to the NJL model in 4D. In this sense, this case is also interesting and the phase diagram of the quark-antiquark condensate was obtained in Kneur (). The relation between the 2d and 4d representations are discussed in Repre (). Furthermore, through analyzing the 3D GN with the 2d representation, the phase structure of the quark-antiquark() and diquark() condensates in vacuum (zero temperature and chemical potential()) was studied in Zhou () under the condition that the and condensates are much smaller than the cut-off scale of the model. It was found that the and condensates do not coexist at .

In this paper, we study the and condensates at finite temperature and density in the 3D GN model with the 2d representation. We obtain the phase diagrams, and discuss the similarities and differences between the 3D GN model and the 4D NJL model.

The plan of the paper is as follows: In Sec.II we present the Lagrangian of the 3D GN model and introduce the mean-field approximation. In Sec.III we derive the thermodynamic potential. In Sec.IV we present the results of the numerical analyses. Among others, we show that there is no region where the and condensates coexist. In Sec.V we display the phase diagrams. We find that the structure of phase diagrams bear resemblance to the QCD phase diagram for some parameter region. Sec.VI is devoted to summary and conclusions. In Appendixes A and B we describe the intermediate calculation and the renormalization for the thermodynamic potential.

## Ii Gross Neveu Model

The general form of the Lagrangian density of the 3D 2 flavor massless Gross Neveu model with 2d representation reads

 L=¯qi⧸∂q +GS[(¯qq)2+(¯q→τq)2] +GD8∑a=1(¯q→τλaqC)(¯qC→τλaq). (1)

Here are the Pauli matrices in flavor space and is a Gell-Mann matrix in color space. and are the coupling constants of the and interactions, respectively, and is the charge conjugation matrix. For the Dirac matrices and , we use the forms as in Zhou (),

 γ0=(100−1),γ1=(0ii0),γ2=(01−10)=C. (2)

The charge conjugated field are defined by

 qC=C¯qT,¯qC=qTC. (3)

Following the reasonings described in Mei (), we reduce the Lagrangian density in Eq.(1) to

 L=¯qi⧸∂q +GS(¯qq)2 +∑a=2,5,7GD(¯qτ2λaqC)(¯qCτ2λaq). (4)

in Eq.(4) enjoys various symmetry properties, which are fully discussed in Zhou (). Here we choose a color direction for diquark condensate to blue, which is equivalent to select in Eq.(4) (see Mei ()). Then we finally arrive at the following Lagrangian density:

 L=¯qi⧸∂q +GS(¯qq)2 +GD(¯qτ2λ2qC)(¯qCτ2λ2q). (5)

Due to in Eq.(5), only two colors (red, green) participate in the condensate while all three colors (red, green, blue) do in the condensate.

Let us introduce the mean-field approximation and rewrite the Lagrangian density as follows:

 L= ¯qi⧸∂q−¯qσq−12Δ∗(¯qCτ2λ2q) −12Δ(¯qτ2λ2qC)−σ24GS−|Δ|24GD, (6)

where and are the order parameters for the and condensates:

 σ=−2GS⟨¯qq⟩andΔ=−2GD⟨¯qCτ2λ2q⟩. (7)

To deal with finite density system, we introduce a chemical potential being conjugate to the quark number . Then the Lagrangian density reads

 L= ¯qi⧸∂q+¯qμγ0q−¯qσq−12Δ∗(¯qCτ2λ2q) −12Δ(¯qτ2λ2qC)−σ24GS−|Δ|24GD. (8)

Introducing the Nambu-Gorkov basis Nambu ()

 Ψ=(qqC)and¯Ψ=(¯q¯qC),

and using the relation , we can write the Lagrangian density in a momentum space as

 L =12¯ΨG−1Ψ−σ24GS−|Δ|24GD, (9)

where

 G−1 =((⧸p−σ+μγ0)1f1c−τ2λ2Δ1s−τ2λ2Δ∗1s(⧸p−σ−μγ0)1f1c). (10)

and are the unit matrix in flavor, color and spinor space respectively.

## Iii The thermodynamic potential

### iii.1 Derivation of the thermodynamic potential

Following the standard method, we can evaluate the thermodynamic potential:

 Ω (σ,|Δ|)=σ24GS+|Δ|24GD −1βVln∫[dΨ]exp[12∑n,p¯Ψ(βG−1)Ψ], (11)

where is the inverse temperature and is the volume of the system. With the help of the formula

 ∫[dΨ]exp[12∑n,p¯Ψ(βG−1)Ψ]=Det1/2(βG−1), (12)

we can rewrite Eq.(11) as

 Ω (σ,|Δ|)=σ24GS+|Δ|24GD−1βVlnDet1/2(βG−1). (13)

After some manipulations which is given in Appendix A, the determinant becomes

 Det1/2(G−1) =[p20−E+2Δ]2[p20−E−2Δ]2[p20−E+2][p20−E−2], (14)

where and

 E±≡E±μ,E≡√→p2+σ2,→p2=p21+p22 E±Δ2≡E2+μ2+|Δ|2±2√E2μ2+σ2|Δ|2(≥0). (15)

Thus, we obtain

 Ω (σ,|Δ|)=σ24GS+|Δ|24GD −T∑±∑n∫d2p(2π)2[ln[β2(p20−E±2)]+2ln[β2(p20−E±Δ2)]]. (16)

The frequency summation may be performed in a standard mannerLeBellac ():

 ∑nln[β2(p20−E2)]=β[E+2Tln(1+e−βE)]. (17)

Then we finally obtain

 Ω(σ,|Δ|) =Ω0(σ,|Δ|)+ΩT(σ,|Δ|), (18) Ω0(σ,|Δ|) =σ24GS+|Δ|24GD−2∫d2p(2π)2[E+E+Δ+E−Δ], (19) ΩT(σ,|Δ|) =−2T∑±∫d2p(2π)2[ln(1+e−βE±)+2ln(1+e−βE±Δ)]. (20)

Here is independent contribution, which is ultraviolet divergent, while the temperature dependent part is finite. For the purpose of later use, we write , in Eq.(19), in the integral form in the 3D Euclidean momentum space,

 Ω0 (σ,|Δ|)=σ24GS+|Δ|24GD −∑±∫d3pE(2π)3[ln(p2E0+E±2p2E)+2ln(p2E0+E±Δ2p2E)]. (21)

The terms in the denominators in the second line of Eq.(21) are inserted so as to drop an irrelevant infinite constant.

### iii.2 Renormalized thermodynamic potential

As mentioned in the previous subsection, , part of is ultraviolet divergent. To eliminate the divergences, we introduce the counter Lagrangian as derived in Klimenko () (see also GN ()):

 LC =−12ZSσ2−ZD|Δ|2, (22) ZS =6π2Λ−34α,ZD=2π2Λ−14α, (23)

where is the D momentum cut-off and is the arbitrary renormalization scale. For completeness, the derivation of Eq.(23) is given in Appendix B.

Introducing the above counter Lagrangian, in Eq.(21) turns out to be finite and the renormalized becomes

 Ω0r −∑±∫d3pE(2π)3[ln(p2E0+E±2p2E)+2ln(p2E0+E±Δ2p2E)−3p2Eσ2−2p2E|Δ|2]. (24)

Note that the counterterms cancel the divergences and the integral becomes finite. Thus, after performing the renormalization, the renormalized thermodynamic potential () is finite and we carry out the numerical analyses on .

Before studying the and condensates, we rewrite by using the following parameters:

 σ0 ≡−2π3(14GS−38α), (25) Δ0 ≡−π(14GD−14α). (26)

Using and , we can write at as

 Ωr (σ,|Δ|)∣∣T=0=μ=−32πσ0σ2−1πΔ0|Δ|2 +13πσ3+13π(σ+|Δ|)3+13π|σ−Δ|3. (27)

Throughout in the following, we use the parameters and . By minimizing Eq.(27) for , one can easily verify that, when , corresponds to the condensate in vacuum.

## Iv Quark-antiquark and diquark condensates

We have obtained the thermodynamic potential in the previous section. Eqs.(24), (25) and (26) tell us that this model has two free parameters (). In this paper, we aim to study the system where the condensate always takes place in vacuum if , so we assume to be positive (see, e.g.Ulli ()). After fixing , there remains a free parameter and we introduce through

 r≡Δ0/σ0. (28)

There is no direct way of fixing the parameter , and we analyze for different ’s.

Fig. 1 plots the and condensates (normalized by ) at . In the case of (panel (a)), we see that the condensate disappears at , and there does not arise the condensate. Through numerical analyses, we have found that this is the case for . On the other hand for (panel (b)), at the condensate disappears and at the same time, the condensate arises. Similar results are obtained for and , where the transition densities are and , respectively. All the phase transitions in the panels (a)-(d) are of the first order. With increasing the ratio , the condensate becomes larger and eventually exceeds the condensate at . As seen from the panel (e) for , the condensate disappears and only the condensate exists for whole . More detailed analyses show that the condensate does not occur for . Thus the results are sensitive to the ratio . It should be emphasized that there is no region where the and condensates coexist. To clarify this fact, we show the close-up of the condensates near the phase transition point in Fig. 1(c).

Now we turn to the case. We display the results for (, ) (, ) and (, ) in Fig. 2. These panels show that the condensate for is at and at . The condensate for is at and at . Thus, as increases, the and condensates decrease. Note that the condensate appears at and does not appear at , which indicates that disappears at high temperature. More detailed analysis shows that the condensate at disappears for (see Sec.V). It should be noted that, within our numerical accuracy, there is no region where the and condensates coexist, also at finite temperature. The results for other values of are qualitatively the same. As the temperature increase, the condensates become smaller and completely disappear at the critical temperature. This is the signal of the phase transition from the condensate state to the normal state.

From Fig. 2, we see that the phase transition for is apparently of the first order and the case for , it is of the second order. Investigating the thermodynamic potential as a function of the condensate as in Ulli (), we have confirmed this fact. We will discuss the phase transition and its order in the next section in more detail.

## V The phase diagram

Through minimizing the thermodynamic potential, one obtains the phase diagram. In Fig. 3, we display the phase diagrams for various values of .

For , there appears the pure condensate phase at low temperature and density and no condensate phase appears. In the cases of , the phase diagrams bear resemblance to that of QCD. For with , the phase transition from the condensate phase to the condensate phase takes place at . For , as increases, the transition from the condensate phase to the normal phase takes place at , which applies also for . On the other hand, in the cases of and , the transition temperature for are and , respectively.

As increases, the region of condensate phase shrinks toward the axis and the region of the condensate phase increases toward the axis. For and , the condensate does not exist only and the condensate appears. Through numerical analysis, we have found that the condensate phase disappears completely at .

The points () shown in the panel (a), (b) and (c) in Fig. 3 represent the critical points from the first order phase transition to the second order. The phase transition below the critical temperature is the first order and above is the second order. On the other hand in the panel (d), there is no critical point and the phase transition from condensate to the condensate is always the first order. The more detailed analysis tells us that the critical point disappears for , and it always appears between the phase and the normal phase. With respect to the condensate, the phase transition from the to the normal phase is always of the second order.

## Vi Summary and conclusions

We have studied the and condensates in the 3D GN model with 2d spinor quarks, and obtained the phase diagram for various values of .

We have found that the behaviors of the and condensates at , in Fig. 1, bear resemblance to that of the 4D NJL model Mei (): With increasing , the condensate becomes more dominant and the condensate disappears completely for .

Fig. 1 and Fig. 2 show that, both for and , there is no region where the and condensates coexist. This is a characteristic feature in the 3D GN model which does not happen in the 4D NJL model.

From Fig. 3 (b) and (c), we see that for , there is a close resemblance between the phase diagrams for and that of QCD. The condensate phase here corresponds to hadronic phase in QCD, and the condensate phase corresponds to color superconducting phase. In the case for , there does not appear the condensate phase, and the phase diagram shows close similarity with the QCD phase diagram without color superconducting phase. However the diagrams for and are very different from the QCD case. Especially when and , there is no condensate and only the condensate exists.

The circles in Fig. 3 indicate the critical points with respect to the phase transition from first order to the second order. Note that the phase transition from the condensate to the condensate is always of the first order. As seen from Fig. 1 and 2, when condensate arises, the condensate disappears rapidly.

We have found that the phase structure drastically changes according to the value . With increasing , the condensate becomes larger. This means that the condensate becomes more dominant when increases. On the other hand, is related to the coupling constant through Eq.(26), and as increases, increases. In the same reason, when increases (decreases), becomes large (small), which causes to decrease (increase). With these observations in mind, we can conclude that when is small, the condensate is dominant over the condensate, and the condensate becomes larger when increases. This is the same phenomenon as seen in the 4D NJL model.

Finally it should be mentioned again that we have found the absence of the coexisting phase in the 3D GN model with 2d spinor quarks. As mentioned in Sec.I, the 3D GN model with 4d spinor quarks bears resemblance to the 4D NJL model. Then it is worth studying the phase diagram in the 3D GN model with 4d spinor representation to see whether the coexisting phase appears as in the 4D NJL model or does not appear as in the present case.

###### Acknowledgements.
I would like to express my sincere gratitude to A. Niegawa and M. Inui for useful discussions.

## Appendix A The derivation of Eq.(14)

The determinant of can be rewritten as

 DetG−1= Det ((⧸p−σ+μγ0)1f~1ciϵτ2Δ1siϵτ2Δ∗1s(⧸p−σ−μγ0)1f~1c) (29) × Det((⧸p−σ+μγ0)1f00(⧸p−σ−μγ0)1f),

where is the unit matrix and is the antisymmetric matrix in color (red and green) space, . For a block matrix with matrices , , and , we have the identity

 Det(ABCD)=Det(−CB+CAC−1D). (30)

Replacing , , and with corresponding elements in the first line of Eq.(29), we get

 Det((⧸p−σ+μγ0)1f~1ciϵτ2Δ1siϵτ2Δ∗1s(⧸p−σ−μγ0)1f~1c) (31) = Det(−|Δ|2+p20−→p2+σ2−μ2−2σ⧸p−μ⧸pγ0+μγ0⧸p)4 = (p20−E2−μ2−|Δ|2−2√E2μ2+σ2|Δ|2)4 ×(p20−E2−μ2−|Δ|2+2√E2μ2+σ2|Δ|2)4,

which leads to the first two factors in Eq.(14). After calculating the second line of Eq.(29), we finally obtain in Eq.(14).

## Appendix B Renormalization in vacuum

As mentioned in Sec.I, the standard 3D GN model is renormalizable in the leading order. This is also the case for the thermodynamic potential Eq.(16) in the present model with in the mean field approximation.

Following the procedure as in Klimenko () (see also GN ()), we carry out the renormalization and obtain the renormalized thermodynamic potential.

The divergent part of the potential is in Eq.(21) and the divergent integral is written as

 Ω0div (σ,|Δ|)=−3π2σ2Λ−2π2|Δ|2Λ (32)

It is to be noted that is independent of .

To eliminate this divergence, we introduce the counter Lagrangian , Eq.(22). For determining in , we consider the one-loop radiative correction to the propagator in the Minkowski momentum space as calculated in Klimenko (),

 Dσ(p2) =−i(1/2GS)+iΠ(p2), (33) Π(p2) =−NfNc∫d3k(2π)3Tr[⧸k(⧸k−⧸p)]k2(k−p)2, (34)

where and are the numbers of flavors and colors. shown above is ultraviolet divergent, and the counterterm is introduced so as to eliminate its divergent contribution. Here we employ the renormalization condition

 Dσ(p2)=−2iGS,atp20−→p2=−α2. (35)

This means that the counterterm should satisfy the following relation

 ZS=−iΠ(p2)∣∣p20−→p2=−α2. (36)

Going into the Euclidean space, and restrict the integration region with a sphere of radius . Then after evaluating integral Eq.(34) and using the condition Eq.(36), we obtain

 ZS=6π2Λ−34α. (37)

In the similar manner, one obtains the renormalization constant :

 ZD=2π2Λ−14α. (38)

Introduction of the counter Lagrangian eliminates , and we obtain the renormalized .

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