Complex singularities around the QCD critical point at finite densities

Partition function zeros provide alternative approach to study phase structure of finite density QCD. The structure of the Lee-Yang edge singularities associated with the zeros in the complex chemical potential plane has a strong influence on the real axis of the chemical potential. In order to investigate what the singularities are like in a concrete form, we resort to an effective theory based on a mean field approach in the vicinity of the critical point. The crossover is identified as a real part of the singular point. We consider the complex effective potential and explicitly study the behavior of its extrema in the complex order parameter plane in order to see how the Stokes lines are associated with the singularity. Susceptibilities in the complex plane are also discussed.


I. INTRODUCTION
The critical point (CP) [1][2][3][4][5] of QCD at finite temperatures and densities is an important issue, and its existence and associated nature of the quark gluon plasma phase may be clarified by the heavy ion experiments in the near future [6]. This is one of the main targets lattice simulations are aiming at. The situation is, however, not conclusive due to the notorious sign problem. Various approaches such as the Taylor series [7][8][9][10] and the imaginary chemical potential [11][12][13][14][15][16][17] are adopted in order to circumvent the sign problem [18][19][20].
Their validity is controlled by the thermodynamic singularities in the complex chemical potential plane [21,22]. Such singularities are deeply associated with the partition function zeros. In [23], the QCD singularities have been investigated in the complex µ plane by using N f = 2 QCD with staggered quarks. In the present paper, we pursue this issue in terms of an effective theory, by focusing on the singularities in the vicinity of the CP in the complex chemical potential plane.
As pointed out by Lee-Yang [25], there is an analogy with two dimensional Coulomb gas.
In the infinite volume limit, the zeros accumulate on curves C with a "charge density" ξ(s), and the real part of the free energy Ω(λ) becomes Re Ω(λ) = −T C ds ξ(s) log |λ − λ(s)| . (1) Re Ω is continuous across C, while the "electric field" E = −∇Re (Ω) is discontinuous in the direction normal to the curve. The amount of the discontinuity is proportional to the charge density ξ. This curve named as the Stokes line is regarded as the location of a cut on a Riemann sheet of the analytic function ReΩ. It should be noted that the singularities occur at zeroes λ k for finite volume, while they appear only at branch points (not on the cut) in the infinite volume limit [21].
Such branch points are termed the Lee-Yang edge singularities. The Lee-Yang edge singularities have a strong influence, as the closest singularities to the real axis, on the behaviors of thermodynamic quantities for the real values of λ. They can also be regarded as critical points in the complex plane [37], and thermodynamic quantities become singular with the critical exponents characterized by the Lee-Yang edge singularities. In order to investigate what the edge singularities are like in the vicinity of the CP, we resort to an effective theory reflecting the phase structure of QCD [38]. This model is constructed based on the tricritical point (TCP) in the µ-T plane in the chiral limit, which has the upper critical dimension equal to 3, and thus its mean field description is expected to be valid up to a logarithmic correction. It provides some interesting physics in the vicinity of the tricritical point for vanishing quark mass m, and of the CP for small m. We make use of this model to investigate the nature in the complex µ plane. Thermodynamic singularities in the complex µ plane have been studied by Stephanov in terms of the random matrix theory [21].
In the present paper, we pay more attention to the influence of the singularities on the real µ axis.
When one introduces complex µ, the order parameter also becomes complex and so does the effective potential. Its µ dependence is quite intricate in the complex case. The above stated model, however, allows to analyze the complex potential. By focusing on the real part of the potential, we explicitly trace its extrema. In the vicinity of the singularity, their movements show different behaviors depending on where they pass. From its behaviors, we identify where the Stokes lines are located. We also see along the Stokes line that the critical exponent associated with the gap of Im Ω around the singular point differ from that on the real axis. The chiral susceptibility and quark number susceptibility in the complex µ plane are also discussed. By moving Re µ with fixed Im µ, the complex susceptibilities show a distinct behavior between Im µ < Im µ (s) and Im µ > Im µ (s) , where µ (s) denotes the singular point in the complex µ plane. We trace the peak of the complex susceptibilities. It turns out that the chiral susceptibility develops a peak at a location in agreement with the real part of the singular point in the vicinity of the CP. As a reminiscence of the singularity, the location of the peak of both the susceptibilities on the real axis shows the same temperature dependence as that of Re µ (s) .
In the following section, after a brief explanation of the model, we see how the effective potential looks above, on and below temperature of the CP for various values of real µ.
We then move to the complex µ plane, and investigate the singularities. Chiral and quark number susceptibilities are discussed in connection with the singularities. In section 3, we explicitly study the extrema of the complex effective potential and discuss the Stokes lines. The chiral and quark number susceptibilities in the complex plane are also discussed.
Summary is presented in section 4.

II. EDGE SINGULARITIES
We investigate the edge singularities of QCD in the complex µ plane. For this purpose, we resort to some effective theory describing the phase structure of QCD around the critical point. In the present paper, we adopt a model proposed by Hatta-Ikeda [38]. Although this model is based on the mean field, it is interesting in the sense that the inter-relationship between the TCP and the CP indicates characteristic behaviors of the phase structure.

A. Effective potential and CP
In this subsection, let us briefly explain the model which we deal with in the paper. We consider N f = 2 case. In the chiral limit, there exists a TCP at finite temperature and density. The TCP is connected to a critical point at µ = 0, which is in the same universality class as 3-d O(4) spin mode [40]. And a first order line goes down from the TCP toward lower temperature side. When quark mass m is introduced, the critical line is absent, and the surviving first order line terminates at a critical point (CP). This CP can be described by fluctuations of the sigma meson and is expected to share the same universality with 3-d Ising model [4]. Since the upper critical dimension of the tricritical point is equal to 3, the TCP in QCD phase diagram can be described by a mean field theory up to a logarithmic correction. As far as m is small, the universal behavior around the CP is also expected to be described in the mean field framework. Let us briefly explain the model [38] in the following.
Starting with the Landau-Ginzburg potential, which incorporates only the long wavelength contribution one expands it around the TCP (a = b = m = 0) assuming a and b as a linear function of µ and T , . Moreover, a first order line (a = 3b 2 /(16c)), existing in the region a > 0 and b < 0 and tangential to a = 0 at the origin in the a-b plane, is expected to be mapped into the regionμ 3 > 0 andt 3 < 0 and tangential to the straight line a = 0 at the TCP. This is realized when the followings hold where the left condition implies that the linear transformation from (a, b) to (μ 3 ,t 3 ) keeps the orientation.
By switching on m, the condition for the CP to exist at T = T E and µ = µ E leads to the coefficients and expectation value of σ It is noted that with the condition Eq. (4), T E < T 3 and µ E > µ 3 hold.
Expanding Ω(T, µ, σ) around Ω(T E , µ E , σ 0 ), we obtain thermodynamic potential around the CP given by whereσ = σ − σ 0 . This is the potential we use in the present paper.
The coefficients A i are given as follows as a function of T and µ.
The stability A 4 > 0 of the potential (10) gives It is checked if this stability condition is fulfilled in the following calculations.
Before discussing the properties of the model in the complex µ plane, it would be better to see what this model is like in the real µ case. This will also be helpful in order to discuss the Stokes line in the following section. Figures 2 and 3 indicate typical behaviors of Ω at temperature around the CP. Here we chose the following numerical values for simplicity, and the same values for these parameters are used for the calculations throughout the paper.
The left panel of Fig

B. Edge singularities
Let us now move to the complex µ plane. Using the potential in Eq. (10), the instability of the extrema occurs at such σ that are simultaneously satisfied. Namely, This occurs when the former cubic equation has vanishing discriminant. where Using the coefficients A i in Eq. (11), the discriminant Eq. (15) is solved as a function ofμ E andt E .
Fort E = 0.2 ( Fig. 3), for example, the discriminant Eq. (15) yields four roots such as The stability condition of the potential Eq. where Ω ′ = Ω " = 0 is fulfilled at σ = −0.07602 (×). Red line:μ E = −0.0222, which is equivalent to the real part of (iii) in Eq. (17).     E is shown in the right panel. The real part depends linearly ont E , while the imaginary one behaves ast βδ E with βδ = 3/2 as expected [21].
In comparison to the result in Ref. [21], we translate the behavior of the singularities in theμ E plane to that in the µ 2 plane. For this we need the location of the tricritical point, which is however unknown in this frame work. By putting some appropriate numbers for the tricritical point µ 3 , we could plot the singularities in the complex µ 2 plane. Re µ = Reμ E +µ 0 and Im µ = Imμ E . Here, µ 0 is given in terms of the coefficients C i 's and µ 3 ; The singularities deviate from the real µ 2 axis ast E increases from 0 in the same way as those in Fig. 5, since a mapping fromμ E to µ 2 is a conformal one.

C. Crossover
Since the singular points move away from the real µ axis for positive values oft E , substantial quantities like the chiral susceptibility show no singular behavior on the real axis, but the reminiscence of the singularity appears as crossover. At temperatures close to the CP (t E 0), it would then be natural to identify Reμ (s) E as the location of the crossover on the real µ axis [21]. In the example, Eq. (17), att E = 0.2, Ω forμ E = Reμ Thet E -dependence of χ σ as a function ofμ E is plotted in the left panel of Fig. 6. The peak of the curve shifts away fromμ E = 0 ast E increases, and its location are in agreement with the values of Reμ in the complex plane makes an effect on the locations of the crossover in a slightly different way. The quark number susceptibility is defined by whereσ is the value of σ at the global minimum of the potential in (10). Figure 7 shows χ q as a function ofμ E (t E =0 (black), 0.02 (red), 0.04 (green) and 0.06 (blue) ). At the CP (t E = 0), χ q diverges atμ E = 0, and away from the CP (t E > 0), χ q develops a finite amount of peak, whose hight becomes smaller ast E increases. Locations of the peak of χ q and χ σ are approximately the same, but deviate from each other as temperature increases from the CP temperature, reflecting the difference of explicit dependence of Ω ′ = ∂Ω/∂σ on m and µ, respectively; This will be discussed again in connection with the complex susceptibilities in III C.

III. ANALYTIC CONTINUATIONS
In this section, we consider the analytic continuation of the extrema of the effective potential. By tracing them in the complex µ plane, the Stokes lines are identified, which reflects the analytic structure around the branch points. It will also be seen that the crossover phenomenon of χ σ on the real µ axis reflects the characteristics of the singular behavior of χ σ in the complex plane. It is understood that although we focus on the behaviors in the complex upper-halfμ E plane, those also occur in the lower-half plane in the complex conjugate manner.
A. Analytic continuation of extrema of the effective potential The Stokes line is understood as a curve to which the Lee-Yang zeros accumulate as shown in Eq. (1). When the phase transition occurs, zeros accumulate onto the critical point by which two phases are separated on the real axis. As the parameter is analytically continued from the one phase to the other, the corresponding global minimum of the potential is also analytically continued. In the present system, fort E = 0, the critical point exists atμ E = 0, while fort E > 0, the singular point is shifted into the complex plane.
Let us considert E > 0, and moveμ E , as shown in the left panel of Fig. 8, from a point on the real axis (μ E > 0) to the other side atμ E < 0 by changing θ from 0 to π inμ E = ρe iθ for a fixed value of ρ. The singular point is located atμ where σ + (σ − ) is the location of the extremum analytically continued fromμ E > 0 (μ E < 0 ). In order to explicitly see how this occurs, we taket E = 0.1. At this temperature, the singular points are located atμ forμ E < 0 , respectively, while the extremum C is not associated with the phase transition.
For ρ = ρ (s) , the two minima A and B approach each other and meet together at σ (s) as shown in the middle panel of Fig.10. In this case, Re Ω of the two trajectories behaves as shown in the right panel of Fig. 9. A smooth encounter of the two trajectories occurs at θ 0 = 3.0384 (= 0.9672π), whose value is indeed in agreement with the location of the singularityμ (s) E = −0.0115 + i 0.0012. As ρ varies passing ρ (s) nearμ (s) E , the two trajectories make a rearrangement. In the leftmost panel of Fig.10 (ρ > ρ (s) ), three trajectories behave like those in the right panel of Fig. 8, i.e., A (B) is the global minimum forμ E > 0 (μ E < 0). In contrast with this, for ρ = 0.012 (< ρ (s) ), only a single extremum A is associated with analytic continuation from µ E > 0 toμ E < 0, as shown in the rightmost figure in Fig.10. That is, no encounter with the Stokes line is found in this case.
As another case of ρ > ρ (s) , let us comment a behavior when ρ is chosen to be a specific value ρ = 0.04322. In this case, two extrema B and C (instead of A and B) meet together at σ = −0.0795229 for θ = π (μ E = −0.04322), where the condition Ω ′ = Ω " = 0 is fulfilled discussed in Fig. 4 and the footnote.

B. Stokes lines
In the case oft E > 0, therefore, the Stokes line runs for ρ > ρ (s) . The left panel of Fig. 11 indicates how the Stokes line emanates from the singular pointμ  Figure). It is seen that the Stokes line is alined on a straight line, which is tilted with angle ϕ = π/4 from an axis parallel to the negative axis of Reμ E . This is in agreement with analytic consideration shown in Appendix A 2 [43].
Ast E decreases to 0,μ (s) E approaches the origin, and the angle ϕ shows a increasing tendency. The middle panel of Fig. 11 shows the Stokes line emanating from the origin for t E = 0. This behavior is in agreement with ϕ = π/2 as shown in Appendix A 4. Fort E < 0, a first order phase transition is located on the positiveμ E axis. In the right panel of Fig. 11, we show the case fort E = −0.2, where a first order phase transition is located atμ E = 0.05553 (blue triangle). From this point, a Stokes line goes out upright with ϕ = π/2 (see Appendix A 3 [44]). In recent Monte Carlo study [41] of low temperature and high density QCD, the distribution of the Lee-Yang zeros have been calculated, and it looks similar to the behavior in the right panel of Fig. 11, suggesting a possible first order phase transition.
To close this subsection, we briefly discuss Re Ω and Im Ω along the Stokes line. On the Stokes line, two values of Re Ω continued from realμ E axis agree (Eq. (22)). In the left panel of Fig. 12, such Re Ω is plotted as a function of Imμ E fort E = 0.2, which increases linearly. In contrast to this, Im Ω shows a gap on the Stokes line (at the singular point, the gap vanishes). The behavior of the discontinuity ∆Im Ω ≡ Im Ω(σ + ) − Im Ω(σ − ) is shown in the right panel of Fig. 12. The gap grows as where Imμ E denotes the imaginary part of the points on the Stokes line (see the left panel in Fig. 11). The exponent 3/2 comes from the behavior of σ + near the singularity Re σ + ∼ (Imμ E − Imμ (s) E ) 1/2 , which reflects the fact that the critical exponents for the edge singularity are, in general, different from the usual ones on the real axis [26,37]. In the case oft E = 0, we have Re σ + ∼ (Imμ E ) 1/3 and ∆Im Ω ∼ (Imμ E ) 4/3 (see Appendix B).
This corresponds in the Ising case to the magnetization behaving like (h − h (s) ) 1/2 near the singular point h (s) on the imaginary magnetic field h axis for T > T c and ∼ h 1/δ for T = T c (δ = 3 for the mean field case). It is also noted that at a Lee-Yang zero for finite volume, the discontinuity is like ∆Im Ω = (2k + 1)π (k : integer) per volume.

C. Susceptibility in the complex plane
The crossover behaviors of the susceptibilities on the realμ E axis reflects the structure of the singularity in the complex plane. Let us then study the behaviors of the susceptibilities in the complex plane. We fix Imμ E and move Reμ E by separating the region into two, one is 0 ≤ Imμ E ≤ Imμ    E , Re Ω" shows a discontinuity (broken lines) when Reμ E crosses the Stokes line. Fig. 13 indicates Re Ω" as a function of Reμ E for fixed values of Imμ E (0 ≤ Imμ E ≤ Imμ (s) E ). Re Ω" develops a minimum, which, as Imμ E → Imμ    E (see Fig. 10). In contrast, the location of the minimum of the inverse real part of the quark number susceptibility (Re χ q ) −1 depend more onμ E than that of Re Ω" as shown in Fig. 14 (the deviation between Reμ (s) E and Reμ E of the minimum point on the real axis is around 10 % fort E = 0.2). It is stressed that the crossover phenomena on the real axis are originated from the same complex singularity, and that χ σ reflects the singularity in the complex plane more directly than χ q does.    Fig. 13 indicates the behaviors of Re Ω" as a function of Reμ E for fixed values of Imμ E . Figure 13 includes also those for 0 ≤ Imμ E ≤ Imμ

On the real axis
Let us turn to behaviors on the realμ E axis. The right panel in Fig. 15 indicates thet E dependence of the location of the minimum of Re Ω" (square) and that of (Re χ q ) −1 (circle) on the realμ E axis. It is seen that both the locations depend linearly ont E coming from the linear dependence of Reμ  t E = 0.1. Right: Thet E dependence of the location of the minimum of Re Ω" (square) and that of (Re χ q ) −1 (circle) on the real axis (Imμ E = 0). A linear fit works well for both the quantities.
locations of the minimum of Re Ω" is invisibly small.
The susceptibility in the complex plane also behaves with the critical exponent characterized by the edge singularity. On the realμ E axis, it is given by 1/δ − 1 = −2/3, while in the complex plane, it changes to −1/2 (see Appendix B).

IV. CONCLUSION
We have discussed the thermo-dynamic singularities in the complex chemical plane in QCD at finite temperature and finite densities. For this purpose, we have adopted an effective theory incorporating fluctuations around the CP. Singularities in the complex chemical potential plane are identified as unstable points of the extrema of the complex effective potential. At CP temperature, the singularity is located on the real µ axis, and above CP temperature it moves away from the real axis leaving its reminiscence as a crossover. The location of the chiral susceptibility peak agrees with Reμ (s) E in the vicinity of the singularity. Simplicity of the model allows us to explicitly deal with the complex potential as a function of the complex order parameter and complex values of µ. We have had a close look at the behavior of the extrema of Re Ω(σ) in the complex order parameter plane. It is seen that two relevant extrema make a rearrangement at the singular point under the variation of µ around the singularity in the complex plane. It is also clearly seen that the Stokes line is located in different ways depending on above, on and below CP temperature, which provide information as to where the Lee-Yang zeros are located for finite volume. Along the Stokes line, Im Ω shows a gap, and the gap increases with the exponent characterized by the Lee-Yang edge singularity.
We have considered the chiral and quark number susceptibilities in the complex plane.
As a reminiscence of the singularity, the locations of the peaks of both the susceptibilities on the real axis exhibit a linear dependence ont E , reflecting thet E dependence of Reμ (iv) It may be worth while quantifying what is described here. In [23], the QCD singularities N f = 2 QCD with staggered quarks have been investigated by having a look at the effective potential with respect to the plaquette variable. In order to having a more contact with the present paper, some refinement of the computation would be necessary.
respectively. The exponent of Ω differs from that in the previous case (t 3 = 0), which causes the change of ϕ. The Stokes line is thus tilted with angle ϕ = π/4 in the vicinity of each critical point on the critical line.
3. Stokes lines fort 3 < 0 (m = 0) In the case oft 3 < 0, a first order phase transition occurs atμ 3 = µ c 3 , where In the vicinity of the µ c 3 , the global minimum and Ω behave as and respectively, whereσ 0 ,σ 1 and d 1 are coefficients, which depend intricately on C a etc. in Eq.
(3). The linear dependence of Ω in the broken phase yields ϕ = π/2. It suggests in finite volume that the Lee-Yang zeros at low temperatures and high densities are located parallel to the imaginary µ axis. This is in agreement with the recent Monte Carlo result obtained by utilizing the reduction formula of the reduced Dirac matrix [41].

Stokes line for the CP (t E = 0)
Att E = 0 and in the vicinity of the CP (m = 0), the global minimum and Ω behave as Here σ 0 is given in Eq. (7), andσ 1 and d 0 are coefficients, which depend on C a etc. in Eq.(3).

on the realμ E axis (t E = 0)
Here, the behavior of the chiral susceptibility around the CP is discussed. Since the singularity of the CP att E = 0 is located at the origin in the complexμ E plane, behaviors of the chiral susceptibility around the CP are given by fluctuations of σ at the global minimum of Ω in the vicinity ofμ (s) True vacuumσ which gives the global miminum of Ω(T, µ, σ) is obtained by solving ∂Ω ∂σ = 0, namely, This equation is cubic and it can be analytically solved. Ignoring higher-order terms inμ E around the CP, one obtains The curvature of Ω for true vacuum is calculated as follows: where the contributions of the first term (O(|μ E | 1 ) ) and second one (O(|μ E | 4/3 )) are neglected due to |μ E | ≪ 1.
The chiral susceptibility, thus, behaves as in the vicinity of the origin in the complexμ E plane.