PHYSICAL REVIEW D 100, 125009 (2019)

Thermodynamics of accelerated fermion gases and their instability at the Unruh temperature

† ‡ George Y. Prokhorov ,1,* Oleg V. Teryaev,1,2,3, and Valentin I. Zakharov2,4,5, 1Joint Institute for Nuclear Research, 141980 Dubna, Russia 2Institute of Theoretical and Experimental Physics, NRC Kurchatov Institute, 117218 Moscow, Russia 3M. V. Lomonosov Moscow State University, 117234 Moscow, Russia 4School of Biomedicine, Far Eastern Federal University, 690950 Vladivostok, Russia 5Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia

(Received 6 July 2019; revised manuscript received 30 October 2019; published 9 December 2019)

We demonstrate that the energy density of an accelerated fermion gas evaluated within quantum- statistical approach in Minkowski space is related to a quantum correction to the vacuum expectation value of the energy-momentum tensor in a space with nontrivial metric and conical singularity. The key element of the derivation is the existence of a novel class of polynomial Sommerfeld integrals. The emerging duality of quantum-statistical and geometrical approaches is explicitly checked at temperatures T above or equal to the Unruh temperature TU. Treating the acceleration as an imaginary part of the chemical potential allows for an analytical continuation to temperatures T

DOI: 10.1103/PhysRevD.100.125009

I. INTRODUCTION In particular, the energy densities of accelerated gas of massless particles with spins 0 and 1=2 were evaluated Study of collective quantum properties of relativistic explicitly [9,10]. matter is crucial for the descriptions of media under As first noted in [9,11], the quantum-statistical approach— extreme conditions, in particular, of the quark-gluon rather unexpectedly—is sensitive to the Unruh temper- plasma produced in heavy-ion collisions. It led to the ature [12] discovery of new important chiral phenomena [1] such as chiral magnetic effect, as well as the influence of rotation a and magnetic fields on polarization [2–5] and the phase ¼ ð Þ TU 2π : 1:1 diagram [6–8]. A lot of efforts were made to improve our understanding of the effects associated with rotation and magnetic fields, while the role of acceleration a has been Let us remind that TU is the temperature of the radiation much less discussed in this context. seen by an accelerated observer. Within the quantum- Most recently, the situation has been changing due to statistical approach, the energy density changes its sign ¼ development of a novel approach to the quantum-statistical at T TU. physics based on the use of the Zubarev density operator. The Unruh effect is seen by an observer accelerated in There exists now a systematic way to include the accel- Minkowski vacuum. In this case, the relation (1.1) estab- eration a into the parameters characterizing equilibrium lishes a one-to-one correspondence between temperature and evaluate perturbative expansion in the ratio a=T. and acceleration. The quantum-statistical approach, on the other hand, treats T and a as independent parameters. We borrow the interpretation of these states at T>TU from *[email protected] quantum-field theory on the background of a space with † [email protected] horizon; see, e.g., [13,14]. Namely, the Euclidean version ‡ [email protected] of the Rindler space with a conical singularity provides an adequate image for a state with T and a being independent Published by the American Physical Society under the terms of parameters. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to As for the temperatures T

2470-0010=2019=100(12)=125009(10) 125009-1 Published by the American Physical Society PROKHOROV, TERYAEV, and ZAKHAROV PHYS. REV. D 100, 125009 (2019) acceleration providing an imaginary part of the chemical We also discuss the application of instability at Unruh potential [9,15]. To this end, we need a nonperturbative temperature to the description of heavy-ion collisions. The representation for the energy density for the fermion gas at pioneering attempts to relate the thermalization and the nonvanishing (T;a). We did suggest such a representation universality of the hadronization temperature to the Unruh to be valid in our previous paper [9]. This integral effect were made in Refs. [19–21]. The observation of the representation was fitted to reproduce the first three terms instability existence allows us to introduce the picture of of the perturbative expansion in a=T and here we associate hadronization which proceeds through the stage of for- it with a novel type of polynomial Sommerfeld integrals mation of a state with high acceleration and temperature which demonstrate, in the particular case of the energy lower than TU. The instability is then responsible for the density, the absence of perturbative terms beyond the first decay of this state into final hadrons. three terms known explicitly. The paper has the following structure. Section II dis- To summarize, at T>TU,wehavetwodual represen- cusses perturbative results for the energy density of an tations for the energy density of the fermion gas. One is accelerated massless fermion gas, obtained in the frame- provided by the integral representation, see (3.1), derived work of the quantum-statistical approach. Section III dem- within the quantum-statistical approach. The other one is onstrates the possibility of representing a perturbative result given by the quantum correction to the vacuum expectation in terms of Sommerfeld integrals of a new type and shows value of the T00 component of the energy-momentum by integration in the complex plane that these integrals are tensor in a space with horizon and conical singularity. polynomial. Section IV is devoted to quantum-field theory The two representations, indeed, turn identical upon the with a conic singularity and shows that the results of proper identification of the corresponding parameters. With this approach exactly coincide with the quantum-statistical a stretch of imagination, we can say that starting with the approach. Section V discusses analytic continuation into thermodynamics of the accelerated gas we get the horizon region TTU. There is a discontinuity at the point T ¼ TU [9,15]. Analytic continu- II. ENERGY DENSITY OF ACCELERATED ation allows to associate this instability at T ¼ TU with the crossing of the pole of the Fermi distribution in the complex FERMION GAS plane. This pole is a nonperturbative manifestation of the The properties of a medium in a state of global observation that acceleration appears as an imaginary chemi- thermodynamic equilibrium can be described by the cal potential [15]. However, the transition from T>TU to quantum-statistical Zubarev density operator of the form Tenergies (compensated by occupation of the corresponding operator, Jˆ are the generators of the Lorentz transforma- levels with negative energy) and Minkowskian vacuum. x tions displaced to the point x, and ϖμν is a tensor of thermal An analogy to this process provided, for example, by the vorticity. Acceleration effects are contained in the term superradiance from the ergosphere of a rotating black hole, ϖ ˆ μν where negative energy levels also exist [16]. μνJx , because On the other hand, in the framework of the approach with ˆ μν ˆ μ ˆ μ ϖμνJx ¼ −2αμKx − 2wμJx; ð2:2Þ space with a boundary, when T ¼ TU the conical singu- larity disappears and the cone turns into a plane. So, we see α ¼ ˆ μ that this phenomenon is echoed by a quantum instability where μ aμ=T is the thermal acceleration, Kx is the which is arising at the same point. It is amusing that a boost operator, wμ ¼ ωμ=T is the pseudovector of thermal ˆ μ similar picture arises [17,18] in the context of vacuum vorticity, Jx is the angular momentum operator. It is stability in external fields. important to note that, as followed from (2.1), from the

125009-2 THERMODYNAMICS OF ACCELERATED FERMION GASES AND … PHYS. REV. D 100, 125009 (2019) point of view of quantum-statistical mechanics, the effects The most unusual property of the Eq. (3.1) is the of acceleration can be described in space with the usual appearance of a bosonic type contribution in it. This Minkowski metric by adding a term with a boost to the contribution corresponds formally to a gas of massless density operator. bosons with 4 degrees of freedom (d.o.f.) and in limit In [10], a perturbation theory in ϖ was developed at a T → 0 it is the only nonvanishing contribution finite temperature. This perturbation theory was used in [9] Z to calculate the mean value of the energy-momentum tensor d3p jpj ¼ 0 μ ¼ 0 ¼ 0 ρðT → 0Þ¼4 : ð3:2Þ of the accelerated fermion gas when wμ ; ;m . ð2πÞ3 2πjpj a − 1 The following expression was obtained for the energy e density: The appearance of such a term can be qualitatively related 7π2T4 T2jaj2 17jaj4 to the equivalence principle [26]. Somewhat similar coun- ρ ¼ þ − þ Oðjaj6Þ; ð : Þ 60 24 960π2 2 3 terterm was also introduced in [11,27], while its bosonic pffiffiffiffiffiffiffiffiffiffiffiffiffi nature can be attributed to imaginary chemical potential, μ where jaj¼ −a aμ, in what follows we will denote connected with acceleration. Note also, that our counter- jaj¼a. term is positive, while in [11] a similar counterterm is It is easy to see that (2.3) satisfies the condition negative.   From a mathematical point of view, integrals of the form (3.1) are a new type of Sommerfeld integrals (look, e.g., ρ ¼ a ¼ 0 ð Þ T 2π ; 2:4 [27]). Similar Sommerfeld integrals have already been discussed in the literature in various contexts; however, which is an indication of the Unruh effect from the point (3.1) differs by the presence of an imaginary term in the of view of the quantum-statistical approach with the exponent. Zubarev density operator [9,11,24]. A remarkable property of the integrals (3.1) is their polynomiality. Here we present a simple method that allows III. NOVEL CLASS OF POLYNOMIAL us to show this polynomiality and better understand its ’ – SOMMERFELD INTEGRALS source. We will use the Blankenbecler s method [28 30], originally used in nuclear physics. We generalize this In this section, we discuss an interesting property related method to the case of antisymmetric weight function and to the solution (2.3) and the possibility of representing it in imaginary chemical potential. Equation (3.1) can be con- the form of a new type of Sommerfeld integrals and show verted to that these integrals are polynomial. Z Z  In [9], an integral representation was proposed for (2.3) 4 4 ∞ 3 ∞ 3 ρ ¼ a þ T x dx þ x dx Z   2 2 xþiy x−iy 3 jpjþ jpj − 120π π 0 e þ 1 0 e þ 1 d p ia ia   ρ ¼ 2 þ Z 2 Z 2 ð2πÞ3 jpjþ ia jpj− ia ∞ ∞ 1 þ T 2T 1 þ T 2T x dx x dx e e þ 2iy − ; ð3:3Þ Z xþiy þ 1 x−iy þ 1 d3p jpj 0 e 0 e þ 4 : ð3:1Þ ð2πÞ3 2πjpj a − 1 e ¼ a where y 2T and we substituted the value of a Bose It was shown in [9], that for T>TU Eq. (3.1) exactly integral right away, as its polynomiality is obvious in coincides with the perturbative result (2.3) (with advance. Let us consider a more general case of integrals of Oðjaj6Þ¼0). the type (3.3), for almost arbitrary weight function. Two Equation (3.1) also receives further support from the types of integrals are possible, consideration of the Wigner function [25] on the basis of Z Z which, in particular, it is also possible to show the addition ∞ ð Þ ∞ ð Þ ¼ f x dx þ f x dx ia Is1 xþiy x−iy ; of an imaginary term with acceleration 2 to the chemical 0 e þ 1 0 e þ 1 potential [9]. Equation (3.1) is remarkable in that it Z Z ∞ gðxÞdx ∞ gðxÞdx automatically leads to the condition (2.4) since in this case I 2 ¼ − ; ð3:4Þ s xþiy þ 1 x−iy þ 1 the bosonic “counterterm” in (3.1) turns out to be exactly 0 e 0 e equal to the first integral with the opposite sign. As discussed in [9], this fact is a manifestation of the where fðxÞ¼−fð−xÞ is an odd function, and gðxÞ¼ 3 Unruh effect within quantum-statistical mechanics [11]. gð−xÞ is an even function. In the case of (3.3), fðxÞ¼x 2 Indeed, the energy density of the Minkowskian vacuum is and gðxÞ¼x . First, we calculate the integral Is1 and normalized to zero and (2.4) demonstrates that the energy assume that 0

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¼ a FIG. 1. The slope of the integration contour y 2T is determined by the acceleration. Left: integration contour in (3.8) when 0 π. Z ∞ of existence of the imaginary chemical potential, and this I ¼ − FðxÞSðx þ iyÞdx s1 distinguishes our calculation from a similar one in [28–30]. Z0 ∞ The integrand in (3.8) has a second-order “Regge-like” − FðxÞSðx − iyÞdx; pole [31] in η ¼ −1, stemming from the Fermi distribution, 0 and a cut along the positive real semiaxis. To calculate the ∂ 1 exþiy Sðx þ iyÞ¼ ¼ − ; integral Is1, one has to close the contour in the complex ∂x exþiy þ 1 ðexþiy þ 1Þ2 Z plane at infinity as shown in the left plot in Fig. 1. x The integral I ¼ Iγ ¼ 0, since following [28,29] we ð Þ¼ ð Þ ð Þ G F x f x dx: 3:5 jDj < 1 I 0 assume . The integral along the whole contour C is then equal to After the change of variables and using the parity SðxÞ¼ ð− Þ ð Þ¼ ð− Þ S x and F x F x , we get IC ¼ Is1 þ I2: ð3:9Þ Z iyþ∞ 2π Is1 ¼ − Fðx − iyÞSðxÞdx: ð3:6Þ At the same time, due to the rotation angle equal to in −∞ iy the case of I2, Note that for obtaining (3.6), the oddness of function fðxÞ, ¼ − 2πiD ð Þ is crucial, leading to an even function FðxÞ. Also, the I2 e Is1: 3:10 presence of two integrals with þiy and −iy [which correspond to the appearance of contributions with þia Using the residue theorem, we get and −ia in (3.1)] is significant. Now let us present the Taylor expansion of the function ηD ¼ 2π ð− Þ ð Þ Fðx − iyÞ in the form of an exponent with the derivative (in IC iResη¼−1 ðη þ 1Þ2 F iy ; 3:11 other words, we use the translation operator) ∂ and after calculating the residue at the second-order pole, Fðx − iyÞ¼exDFð−iyÞ;D¼ : ð3:7Þ ∂ð−iyÞ we obtain

Making change of variables, η ¼ ex, we get πD Is1 ¼ Fð−iyÞð0

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πD a ¼ r−1; τ ¼ γrλ: ð4:2Þ I 2 ¼ Gð−iyÞð0 ; ð : Þ 60 24 960π2 2π 3 15 dimensional cone with an angular deficit 2π − a=T. The world line of a uniformly accelerated object corresponds to where the condition 0 2π. To summarize, the inverse acceleration for this world line, and the length of it is essential for polynomiality that Fermi distributions in the given circle determines the inverse proper temperature, (3.1) are taken with polynomial weights, and also that as it is shown on the left panel of Fig. 2. An essential symmetric combinations of integrals with ia appear (the property of the metric (4.3) is the presence of a conical odd weight function must correspond to the sum of the singularity at r ¼ 0. Note that, as in the Fig. 1, the angular integrals with ia, and even to their difference). If we use deficit of the cone is determined by the ratio of acceleration the physical interpretation of [9], then only the total con- to temperature, a=T. tribution of the modes with imaginary chemical potential The quantum-field theory in the space (4.3) can be þia and −ia is polynomial. constructed using the heat kernel method [13,14,32].Asa result, nonperturbative mean values for different operators can be obtained. In particular, in [13,14], the mean value for IV. DUALITY OF QUANTUM-STATISTICAL the energy density of Weyl spinor field was calculated as MECHANICS AND QUANTUM-FIELD follows: THEORY IN A SPACE WITH BOUNDARY   1 7π2 4 2 17 In this section, we show that the energy density of an ρ ¼ T þ T − ð Þ 2 2 4 : 4:4 accelerated gas can also be calculated in another way in 2 60 24r 960π r the framework of field theory in a space with a conical In this case, the last term, which is independent of temper- singularity [13,14]. Thus, we demonstrate duality between ature, is associated with the vacuum energy arising due to the quantum-statistical calculations and quantum-field theory Casimir effect in the space with a horizon [33]. in a space with a boundary. An amazing observation is that taking into account (4.2), Consider the Rindler metric in the form we get from (4.4)   2 2 2 2 2 2 4 2 2 4 ds ¼ −r dη þ dr þ dx⊥; ð4:1Þ 1 7π T T a 17a ρ ¼ þ − ; ð : Þ 2 60 24 960π2 4 5 where η ¼ γλ;x¼ r cosh η;t¼ r sinh η, and γ is a positive constant [in (4.1) for convenience, unlike the rest of the and thus, there is a complete agreement with the result (2.3) text, we consider definition of the metric such that g00 < 0]. (the difference in the coefficient 1=2 is associated with The world lines with r ¼ const; x⊥ ¼ const correspond to half the number of d.o.f. of the Weyl spinors in comparison uniformly accelerated motion. The relations between the with the Dirac spinors). This means that the energy density proper acceleration a and the proper time τ along these of accelerated matter can be calculated in two completely world lines, with the variables λ and r are determined by the different ways: either by means of the statistical Zubarev formulas density operator (2.1) and calculation of corrections in flat

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FIG. 2. Left: space with a conical singularity and the relationship of geometric characteristics, circumference, and distance to the cone point, with statistical parameters, acceleration, and temperature. Right: transformation of a cone into a plane with the temperature decreasing to T ¼ TU.

space, or by considering space with boundary, which At T ¼ TU, the cone turns into a plane as it is shown on the transforms to the space with the conical singularity (4.3) right panel of the Fig. 2 and the quantum-field approach in the framework of the heat kernel approach. with a conical singularity in its standard form does not The results obtained in the framework of this approach allow us to study the region TTU. pole η ¼ −1, as it is shown in Fig. 1. As will now be shown, Thus, the polynomiality of (2.3), which, as it was shown in in the region Tobservables is related to the quantum anomalies. 0 ¼ Is1 þ I2: ð5:2Þ It is well known that quantum anomalies can lead to the suppression of higher-order quantum corrections, and as the But I2 remains the same as in the region T>a=2π.Itis result, the exact expression of a physical quantity is described therefore easy to get by the first terms of the quantum-field perturbative series. π This statement is known as the Adler-Bardeen theorem [34] 2πiD D I 1 ¼ e Fð−iyÞ and is well known in quantum-field theory. s sinðπDÞ  Recently, it has been shown that anomalies play crucial 11ðπ Þ2 role in hydrodynamics. In various contexts, the relationship ¼ 1 þ 2 π − D − ðπ Þ3 i D 6 i D of quantum anomalies with the chiral vortical, magnetic,  and other chiral effects, as well as with the Hawking effect, 127ðπ Þ4 þ D þ Oððπ Þ5Þ ð− ÞðÞ was shown [27,35,36]. We would expect that quantum 360 D F iy 5:3 anomalies in hydrodynamics also should guarantee through a kind of the Adler-Bardeen theorem, the polynomiality in and the same formula for Is2 with Gð−iyÞ. Accordingly, for acceleration. However, finding the proof of this statement the energy density, we get remains an interesting unsolved problem. 127π2T4 11T2a2 17a4 Ta3 ρ ¼ − − − πT3a þ : ð5:4Þ V. INSTABILITY AT UNRUH TEMPERATURE 60 24 960π2 4π

A. Analytical continuation to the region T < TU We note an interesting fact—according to (5.4) at T

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B. Instability intermediate unstable state had not been discussed, to our best knowledge. The analytic continuation to the region TTU and ρ ρ applied to hadronization, may lead to appearance of a sort of TTU is given by (3.15) and TTU T TU TTU U ∂ TTU U ∂T2 TTU U 30 of critical exponents and vacuum decay rates are possible. ∂2 73 As in [17], we show that violation of the classical ρ ðT → T Þ¼ a2; geometric constraint is accompanied by instability at the ∂T2 TT a : 5:6 ∂T U ∂T U between quantum-statistical and field-theoretical examples, Thus, we see that crossing of the pole leads to an instability there are certain differences in technical details. In par- in the energy density, which manifests itself in the ticular, in [17], quantum instability is due to imaginary part ∂2ρ to the effective action. In thermodynamics, on the other discontinuity of the second-order derivative 2 at the ∂T hand, energy density becomes negative, but remains real Unruh temperature. It is also easy to see from (5.5) that and the discontinuity manifests itself at jump of the second- near, but below TU, the energy density is negative [11,24], which also indicates instability in the system. order derivative (5.6). Also, energy density as the function ∂dρ of acceleration appears to be even for T>TU, but becomes The order of the derivative ∂Td, in which discontinuity occurs, turns out to be related to the pole order k in (3.8) odd below TU. (where k ¼ 2) and the order of polynomial weight n in Note, however, that the validity of the results based on integrals (3.4) as follows: the analytical continuation of the Eq. (3.1) deeply into the region T

125009-7 PROKHOROV, TERYAEV, and ZAKHAROV PHYS. REV. D 100, 125009 (2019) easily found by transforming them into contour integrals in Unruh temperature. To do this, let us consider the integral the complex plane. of the form Let us notice that polynomial Sommerfeld integrals exist Z Z  T4 ∞ xn ∞ xn for any integer dimension of the integrand. The lowest ¼ l þð−1Þnþ1 Is 2 y xþiy x−iy ; dimensional examples are known to be related to quantum π 0 e þ 1 0 e þ 1 anomalies [27]. In general case, Sommerfeld integrals ðA1Þ are expected to allow to obtain exact one-loop results. It is not ruled out that this is more general phenomenon, than ¼ a where y 2T. Equation (A1) is a special case of (3.4) for anomalies. polynomial weight functions. For l ¼ 0, n ¼ 3 and l ¼ 1, Further, we show that at Unruh temperature several n ¼ 2, one obtains integrals from (3.3). As described in a processes take place simultaneously. From the point of Sec. V,atT<2π, there is an additional contribution to this view of a space with a conical singularity, the angular integral due to the crossing of the pole. From (3.12),it deficit of the cone reaches its limiting value and the cone TT follows that ΔI ¼ I U − I U has the following form: turns into a plane. This behavior makes the analysis of s s s 4 þ1 region T

APPENDIX A: THE ORDER OF THE ∂d inþk−12n−k−1π2n−2kþl−1n! DERIVATIVE WITH DISCONTINUITY ΔI j ¼ ak−nþ2: ðA6Þ ∂Td s TU ðk − 1Þ! The purpose of this appendix is to derive Eq. (5.7) for the order of the derivative in which instability occurs at the One can easily derive jump (5.6) from (A6).

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APPENDIX B: INSTABILITIES ARISING FROM ðnþ1Þ ðnÞ iπð2nþ1ÞðD−1Þ −I 1 þ I 1 ¼ 2πiDe Fð−iyÞ: ðB1Þ REPEATED POLE CROSSINGS s s In this appendix, we show that when T

¼b1 þ a c Calculating now the energy density, and taking into account that n 2 4πT , we get         7π2T4 T2a2 17a4 πT3a Ta3 1 a T2a2 1 a 2 4πT3a 1 a 3 ρ ¼ þ − þ þ þ − þ 2π2T4 þ − þ 60 24 960π2 3 4π 2 4πT 2 2 4πT 3 2 4πT   1 a 4 þ 4π2T4 þ : ðB4Þ 2 4πT

Thus, we have reproduced Eq. (3.4) from [9], previously obtained on the basis of the properties of polylogarithms. It contains instabilities at T ¼ TU=ð2k þ 1Þ;k¼ 0; 1::, which lead at each point to the discontinuities of the second ∂2ρ derivative ∂T2. Equation (B4) allows formally to obtain the energy density of the accelerated gas for arbitrarily low temperatures: the corresponding plot was shown in [9] in Fig. 2. At the same time, as the temperature decreases, it turns out that all the coefficients at T4, T2a2;T3a… in (B4) begin to change (except for a4) and can become arbitrarily large in absolute value for big values of index n.

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