Induced energy-momentum tensor of Dirac field in 2D de Sitter QED

We compute the expectation value of the energy-momentum tensor in the in-vacuum state of the quantized Dirac field coupled to a uniform electric field background on the Poincar$\rm\acute{e}$ path of the two dimensional de~Sitter spacetime ($\mathrm{dS}_{2}$). The adiabatic regularization scheme is applied to remove the ultraviolet divergencies from the expressions. We find, the off-diagonal components of the induced energy-momentum tensor vanishes and the absolute values of the diagonal components are increasing functions of the electric field which decrease as the Dirac field mass increases. We derive the trace anomaly of the induced energy-momentum tensor, which agrees precisely with the trace anomaly derived earlier in the literature. We have discusses the backreaction of the induced energy-momentum tensor on the gravitational field.


I. INTRODUCTION
The theory of renormalization of the quantum electrodynamics (QED) in Minkowski spacetime is well established. Generally, in a curved spacetime and especially in the case of the de Sitter spacetime (dS), there is no unique vacuum state to compute the expectation values of the quantities. Furthermore, the additional divergences are caused by the gravitational interactions. In order to establish the theory of renormalization in curved spacetime various methods have been developed; see [1,2] for introduction. Perhaps one of the most efficient methods to regularizing the expectation values of the quantities is the adiabatic subtraction method. This method is based on the approximation that the creation of a particle with the frequency much larger than the energy scale determined by the curvature of the spacetime can be neglected. Hence, the adiabatic expansion is a power series of the spacetime curvature. Specifically, in the case of a Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime with a spatially flat metric the number time derivatives of the scale factor determines the adiabatic order. Adiabatic subtraction process starts from considering a Wentzel-Kramers-Brillouin (WKB) type solution for the under consideration mode equation and proceeds to construction of the Fock space of the state vectors to compute the expectation values of the quantities; see [1,2] for details. Adiabatic regularization was introduced by Parker to obtain a finite expectation value of the particle number in an expanding universe [3]. After this original work, adiabatic regularization generalized and applied for various types of quantities and particles. Using adiabatic regularization scheme, the energy-momentum tensor of a scalar field has been computed in spatially flat [4][5][6][7][8][9][10][11][12][13] and closed [14] FLRW type universes, anisotropic spacetimes [15,16], and conformal coupling case [17]. The effective Lagrangian and the energy-momentum tensor of a scalar field in dS have been computed in Ref. [18] using the dimensional regularization method. Cosmological applications motivated authors of [19,20] to compute the energy-momentum tensor of the created scalar particles in a dS 4 . In [19], the finite energy-momentum tensor has been defined as difference of the expectation values in the in-vacuum and out-vacuum states; and the author found the decay of the effective cosmological constant due to the Hawking effect. In [20], a novel renormalization technique has been applied to compute the energy-momentum tensor, and the authors found that for a massive noninteracting scalar field quantum corrections may lead to a superacceleration phase where the Hubble constant increase. In all the works [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] the adiabatic regularization method developed and applied for a scalar field. The adiabatic expansion for the Dirac field has been introduced in [21] and further details provided in [22]. The energy-momentum tensor of the Dirac field in a spatially flat FLRW spacetime has been regularized using the adiabatic subtraction prescription [23]; see also [24][25][26]. As proved in [27] this method of the regularization gives the same result as the DeWitt-Schwinger point-splitting regularization, when applied to the energy-momentum tensor in a spatially flat FLRW universe.
In this paper, we purpose to go beyond considering only a scalar or fermion field in a curved spacetime background, by adding an electromagnetic gauge field to this picture. More specifically, we wish to consider a Dirac field coupled to a uniform electric field background in a dS. Creation of pairs in a strong electric field background in the flat spacetime is a well known nonperturbative phenomenon in quantum field theory which is referred to as the Schwinger effect [28][29][30]; see [31,32] for introduction. Possibility of existing strong electromagnetic fields in the early universe [33] motivates studying the Schwinger effect in the cosmological context [34]; see [35] for a recent review and references. Due to the cosmological event horizon dS emits particles [10,11,19,[36][37][38][39][40][41][42][43][44] which is known as the Gibbons-Hawking radiation [45,46]. In a dS, by using semiclassical approaches the Schwinger effect for scalar [47][48][49][50][51][52][53][54][55][56][57][58] and Dirac [59][60][61][62][63][64][65] fields has been investigated which may amplifies the Gibbons-Hawking radiation. Perturbative amplitude for production of the scalar [66][67][68][69][70] and Dirac [71][72][73] pairs in the presence of an electric field on the dS has been studied and the authors found that the pair production is significant only when the expansion parameter is large compered to the mass of the field. Recently, the effect of a magnetic field on the Schwinger effect in the dS 4 has been investigated [74]; see also [75][76][77][78][79][80][81], and it has been shown that the pair production is enhanced in the strong magnetic field regime. The Schwinger effect during inflation has been investigated using various approaches in Refs. [82][83][84][85][86][87][88][89][90][91]. To provide further evidence for the ER=EPR conjecture, the holographic Schwinger effect in dS was studied in Ref. [92].
As noted in [47,49] the in-vacuum state of the quantum fields in dS has the Hadamard form, i.e., the ultraviolet divergencies in the expectation values of the quantities in this state behaves similar to what is expected in the flat spacetime. Hence, it is preferred to study of the renormalization theory in dS. In the pioneer work [49] the Schwinger effect and the in-vacuum expectation value of the induced current of a scalar field in a dS 2 has been investigated. The authors found that in the strong electric field regime, i.e., when the electric field is sufficiently large compered to the Ricci scalar curvature and the scalar field mass, the current agrees with the semiclassical results [47].
A notable feature of the current was found in the weak electric field regime for a small enough scalar field mass compered to the Hubble constant, the current is inversely proportional to the electric field which leads to a phenomenon of infrared hyperconductivity. These remarkable results then motivate to study the induced current for both scalar and fermions in various dimensions of the dS. The Schwinger effect and the adiabatic regularized induced current of the scalar field in a dS 4 has been investigated in [50], and particularly the phenomenon of infrared hyperconductivity was reported. In order to compare the results, the vacuum expectation value of the induced current was renormalized using point-splitting method in [93] which completely agrees with the work [50]. In [56], those results for the current have been derived by using the uniform asymptotic approximation method. The Schwinger effect in a general D-dimensional dS, and as an example of odd dimensions, the induced current of the scalar field in the three dimensional dS have been investigated in [51] which a period of the infrared hyperconductivity phenomenon was observed.
Furthermore, the authors of [74] consider a uniform magnetic field with conserved flux parallel to a constant energy density electric field in the dS 4 and find that the density of states are proportional to the magnetic field, consequently the scalar Schwinger effect is enhanced in the sufficiently large magnetic field. They also show that in the sufficiently weak magnetic field the infrared hyperconductivity phenomenon occurs in the induced current of the scalar field. For the case of the Dirac field the Schwinger effect and the induced current of the created pairs have been investigated in dS 2 [62] and dS 4 [63]. Contrary to the scalar field case, no infrared hyperconductivity phenomenon has been observed in the induced fermionic current. Analysing the induced current for both cases the scalar [50,93] and Dirac [63] fields in dS 4 reveals a period of the negative current where the current flows in opposite direction of the electric field background weaker than a threshold value depending on the field mass. The negative current leads to the antiscreening effect which increases the magnitude of the electric field background. Obviously, the infrared hyperconductivity and the negative current phenomena are contrary to physical intuition. In attempting to address these peculiarities, a new renormalization condition defined for the scalar and Dirac induced currents in dS 4 which was named maximal subtraction [94]. The maximal subtraction scheme defines the asymptotic value of the renormalized vacuum expectation value equal to that of obtained from the semiclassical approaches. Following this scheme the infrared hyperconductivity period is removed, however a phase of the negative current would be present. Recently, in [95] all the contributions to the regularized induced current in dS 4 have been clarified. In all the works [50,51,62,63] adiabatic subtraction scheme applied to regularize the induced current. In de Sitter spacetime with any dimension, the energy-momentum tensor of the semiclassical Schwinger scalar has been computed in the heavy scalar field [51] and the strong electric field [96] regimes; the authors found a decay of the Hubble constant due to the pair creation. The adiabatic regularized in-vacuum state expectation value of the trace of the induced energy-momentum tensor for a charged scalar field coupled to a uniform electric field background in three [97] and four [98] dimensional dS has been studied. In [98], was shown that the gravitational backreaction effects of the trace of the induced energy-momentum tensor in the semiclassical regime leads to a decay of the Hubble constant, whereas in the infrared regime a superacceleration phenomenon occurs, i.e., the Hubble constant increases. The method of adiabatic subtraction for regularization of both cases scalar and spinor QED in an expanding universe has been extensively studied in Refs. [99][100][101][102]. In this paper our main goals are investigation of the induced energy-momentum tensor of the Dirac field in de Sitter QED and further exploring validity of the adiabatic subtraction regularization. Hence, we consider a massive charged Dirac field coupled to a uniform electric field background in a dS 2 . To regularize the expectation value of the energy-momentum tensor in the in-vacuum state we adopt the second order adiabatic expansion of the appropriate counterterms.
The paper is structured as follows. In Sec. II, we briefly introduce our model. In Sec. III, we compute the expectation value of the energy-momentum tensor and regularize it. We examine the regularized energy-momentum tensor and discuses its consequences in Sec. IV. Eventually, the conclusions are presented in Sec. V.

II. THE MODEL
We consider a massive Dirac field coupled to a uniform electric field with constant energy density on the Poincaré patch of dS 2 . We assume the electric and gravitational fields as classical backgrounds which are not affected by the presence the Dirac field. The half of dS 2 can be represented as a spatially flat FLRW spacetime where t is the proper time and H is the Hubble constant. By using the transformation the metric (2) can be expressed in a manifestly conformally flat form In order to have a uniform electric field with a constant energy density in the metric (3), we choose the electromagnetic vector potential to be where E is a constant.
The QED action for the Dirac field ψ(x) with mass m and charge e which is coupled to the electromagnetic vector potential (4) in the dS 2 with metric (3) can be written as where g is the determinant of the metric, and Γ µ are the Dirac gamma matrices in the dS 2 . The metric tensor of the spacetime g µν can be expressed in terms of the Minkowski spacetime metric η ab , by using the tetrad e a µ coefficients We choose the tetrad in the gauge then, the Dirac gamma matrices in the Weyl representation are given by where σ 1 , σ 2 , σ 3 are Pauli matrices. In this representation the adjoint Dirac field ψ(x) is defined The gauge covariant derivatives can be expressed as where the spin connection B µ is given by where we use the over dot to denote the derivative with respect to the conformal time τ .
The Dirac equation follows from the action (5), By using the fact that the Dirac Eq. (12) is invariant under spatial translations, we can write ψ(x) as decomposed two-component spinor Substituting the decomposed form of the spinor (13) into the Dirac Eq. (12), and by using Eqs. (4), (8) and (11) we arrive to a system of coupled deferential equations where we have defined The system of coupled first order linear deferential equations (14) can be converted into an equivalent two single second order linear deferential equations where the conformal time dependent frequency is given by The solutions of the Dirac Eqs. (16) with the desired asymptotic forms at early times τ → −∞ and late times τ → 0, which describe the in and out-vacuum states respectively, have been obtained in Ref. [62]. Since, we wish to compute the expectation value of the energy-momentum tensor in the Hadamard in-vacuum state, we take only the solutions of Eqs. (16) with the desired asymptotic behaviour at early times. Hence, the components of the normalized positive frequency mode spinor are given by [62] and, the components of the normalized negative frequency mode spinor are obtained [62] V where W is the Whittaker function [103], and the normalized step function has been defined as We have defined the dimensionless variables The normalized mode spinor components which are given by Eqs. (18)- (21), may be represented in the doublet form Then, it can be shown that these doublets satisfy the relation where I denotes the 2 × 2 dimensional unit matrix.
The set of complete orthonormal mode spinors (24) determines the in-vacuum state |0 which is annihilated by acting the operators a k for a fermion and b k for an antifermion with the comoving momentum k, as Then, the Fock space of state vectors is built by successive acting the creation operators a † k and b † k for fermions and antifermions, respectively. To quantize the Dirac field ψ(x) we impose the anticommutation relations Then, the quantized Dirac field operator ψ(x) can be written as

III. REGULARIZATION OF ENERGY-MOMENTUM TENSOR
The energy-momentum tensor of the Dirac field is defined by variation of the action (5) with respect to variation of the inverse metric δg µν , or more precisely After some algebra definition (29) yields the expression for the energy-momentum tensor of the Dirac field, coupled to the uniform electric field background in the dS 2 , as A. In-vacuum state expectation values By considering the Dirac field ψ(x) as the quantum operator which is given by Eq. (28) and using the relations (26) and (27), we arrive at the following expressions for the in-vacuum state expectation values of the energy-momentum tensor components where ℜ denotes the real part, for the spacelike component and the symmetric off-diagonal component obtained To regularize the ultraviolet divergencies in the momentum integrals of Eqs. (31)-(33), we have considered cutoff K for the comoving momentum k. Accordingly, cutoff Λ = −Kτ is defined for the dimensionless physical momentum p; see definitions in Eq. (23).
To evaluate the integrals over momentum involving the Whittaker functions in Eqs. (31) and (32), we follow the procedure that introduced in Refs. [49,50] for computing the induced current of the scalar field in a dS 2 and dS 4 , respectively. By using this method, the induced current of the Dirac field in a dS 2 [62] and dS 4 [63] has been computed. The appropriate Mellin-Barnes integral representation [103] of the Whittaker function is given by which is vialed for arbitrary values of the phase of the argument in the domain |ph(z)| < 3π/2.
The condition is that the contour of integration must be arranged in a way that the chain of poles of Γ( 1 2 + γ + s)Γ( 1 2 − γ + s) lies a part from the chain of poles Γ(−κ − s). By means of Eq. (34) and the theorem of residues, we then obtain the final results and where log denotes the natural logarithm function, and the digamma function ψ which is defined by the first derivative of the logarithm of the Gamma function; see, e.g., [103].
To remove the ultraviolet divergencies from the expectation values given by Eqs. (33), (35) and (36) we apply adiabatic subtraction scheme. This method has been used to regularize the induced fermionic current in dS 2 [62] and dS 4 [63]. To find the appropriate counterterms, we perform the same procedure as used in [62]. As usual in the literature, we assume that the electromagnetic vector potential and energy-momentum tensor in the two dimensional spacetime to be of adiabatic order zero and two, respectively. We begin by considering a WKB type solution for Dirac equations (16) as where a = 1, 2 is the spinor index, N a is a normalization coefficient and the real functions X a , Y a are required to satisfy equations Inserting the function Y a from Eq. (38) into Eq. (37), we find that U a can be expressed as Then, substituting U a from Eq. (40) into decoupled Dirac Eqs. (16) yields an algebraic equation for the function X a , which exactly is The remaining undetermined factor in the adiabatic mode spinor (37) is the normalization constant N a . We will show below that the counterterms are independent of N a . We constraint the components of the adiabatic mode spinor (37) to obey the normalization condition Following adiabatic regularization method, the set of counterterms are constructed from the expectation values of the energy-momentum tensor components (30) in the adiabatic vacuum sate which is described by the solutions (37). Such expressions can be written as and, for the spacelike component where K is an ultraviolet momentum cutoff. The off-diagonal component finally is obtained Defining the ratio of the spinor components one can rewrite Eqs. (43) and (44) as The expressions for U 1 and U 2 can be read from Eq. (40) with a = 1 and a = 2, respectively.
Substituting these expressions into the coupled Dirac Eqs. (14), we then obtain where X 1 satisfies Eq. (39) with a = 1. By virtue of Eqs. (43)-(49) the counterterms are independent of the normalization constants N a , as mentioned above.
We now return to Eq. (41), to find a solution for the function X 1 . Obviously, terms appearing on the right side of Eq. (41) are of two higher adiabatic order than those on the left side. By using the fact that the conformal time dependent frequency ω [see Eq. (17)] is of zero adiabatic order, at the lowest adiabatic order the function X a is obtained where the superscripts denote the adiabatic order. Then, up to the second order the adiabatic expansion of X a is constructed iteratively and is given by (51) with a = 1, respectively. Then, substituting the second adiabatic order expansion of the expression on the right side of Eq. (49) into Eqs. (47) and (48), we finally obtain Subtracting the second adiabatic order counterterms, given by Eqs. (45), (52) and (53) from the corresponding original expressions for the in-vacuum expectation values which are given by Eqs. (33), (35) and (36), leads to the regularized induced energy-momentum tensor and = Ω 2 (τ ) H 2 4π The important result is that the off-diagonal component of the regularized induced energymomentum tensor vanishes

IV. EXAMINATION OF THE INDUCED ENERGY-MOMENTUM TENSOR
In this section we examine the regularized in-vacuum expectation value of the energy-momentum tensor which is given by Eqs.

A. Asymptotic behaviors
We now analyze the induced energy-momentum tensor in the limiting values of the electric field and the Dirac filed mass, to find the explicit asymptotic behaviors shown in Figs. 1 and 2.

Strong electric field
In the strong electric field regime, the asymptotic behavior of the induced energy-momentum tensor is obtained by expanding T 00 (54) and T 11 (55) in a Taylor series about λ = ∞ with fixed µ. We then obtain the leading order terms as

Weak electric field
In the weak electric field regime, the asymptotic behavior of the induced energy-momentum tensor is obtained by expanding T 00 (54) and T 11 (55)  of light Dirac field µ ≪ 1, we obtain the leading order terms where γ E = 0.577 · · · is the Euler's constant. And for the case of heavy Dirac field µ ≫ 1, the leading order terms are given by

Heavy Dirac Field
For the case of heavy Dirac field, we can expand T 00 (54) and T 11 (55) in a Taylor series about µ = ∞ with fixed λ. We find that the expressions The trace of the induced energy-momentum tensor reads from Eqs. (54) and (55), whose magnitude has been plotted in Fig. 3. This figure shows that the magnitude of the trace is an increasing function of the electric field and decreases as the Dirac field mass increases. For the case of light Dirac field µ 1, the trace is negative T < 0. While, for the case of massive Dirac field µ 1, the trace T has a discontinuity at which its sign is changed. Let λ * be the value of λ at the discontinuity point of the trace T whose precise value is evaluated by solving the equation A numerical analysis illustrates that for λ < λ * the trace is positive T > 0, whereas when λ > λ * the trace becomes negative T < 0. To find the behaviour of λ * against the Dirac field mass µ at which the evaluation was made, we present Fig. 4 which shows that λ * increases as µ increases.
Before concluding, we would like to point out some consequences of the trace of the induced energy-momentum tensor, namely the trace anomaly and gravitational backreaction.

C. Trace anomaly
The trace anomaly of the induced energy-momentum tensor is derived by taking the limit of λ → 0 and µ → 0 in the expression (64), which yields where R = 2H 2 is the Ricci scalar curvature of dS 2 . This result agrees precisely with the well known trace anomaly [104] of a Dirac field in the two dimensional de Sitter spacetime.

D. Gravitational backreaction
In Ref. [105], the backreaction effect of the induced Dirac field current on the electromagnetic field background in dS 2 has been studied. The main conclusion was that the backreaction of the fermion pair creation does not amplifies the electric field background. In this section we wish to study the backreaction of the induced energy-momentum tensor on the gravitational field background. A natural way to do this is to use the Einstein gravitational field equations. However, as is well known, there is no Einstein gravity in two dimensional spacetimes. Perhaps the most natural analogues of the Einstein gravitational field equations in two dimensions is given by where R is the Ricci scalar curvature, Λ is the cosmological constant, G is Newton's gravitational constant in two dimensions and T is the trace of the energy-momentum tensor; see, e.g., [106,107].
In a dS 2 the cosmological constant vanishes, hence we set Λ = 0 in Eq. (67). In natural units, The pair creation begins from early times, τ → −∞, where the Ricci scalar curvature is given by where T is given by Eq. (64). Then, variation of both sides of Eq. (67) leads to the evolution equation for the Hubble parameterḢ Therefore, if T > 0 then the Hubble constant decays. This condition holds for the case of heavy Dirac fields µ 1 created by an electric filed background with the strength λ < λ * . Whereas, the Hubble constant amplifies and a superacceleration phenomenon withḢ > 0 occurs in the condition that T < 0. This is the case for light Dirac fields µ ≪ 1 created by an unbounded electric filed strength or creation of heavy quanta µ 1 in an electric filed with the strength λ > λ * .
In this paper we have computed the in-vacuum state expectation value of the energy-momentum tensor of a Dirac field in a uniform electric field background on the Poincaré path of dS 2 . As expected the expectation values in the Hadamard in-vacuum state acquire quadratic ultraviolet divergences; see Eqs. (33), (35) and (36). To obtain a finite result, we have applied the adiabatic regularization scheme. Since the energy-momentum tensor is of seconde adiabatic order in two dimensions, we have constructed the set of the appropriate counterterms up to the second order of the adiabatic expansions; see Eqs. (45), (52) and (53). Hence, we reach our goal of driving the induced energy-momentum tensor in Eqs. (54)- (56). We find that the off-diagonal components of the induced energy-momentum tensor vanish, and the magnitude of the diagonal components are increasing functions of the electric field and decrease as the Dirac field mass increases; see Figs. 1 and 2. More precisely, these behaviours have been determined in Eqs. (57)- (63).
The trace of the induced energy-momentum tensor is given by Eq. (64) and its magnitude is plotted in Fig. 3, that is an increasing function of the electric field and decreases as the Dirac field mass increases. The analytic behaviour of the trace in the limiting values of the electric field and mass can be read from Eqs. (57)- (63). We find that for the case of heavy Dirac field µ 1, the trace has a discontinuity at which the sign of the trace changes. The strength of the electric field at the discontinuity point, which is denoted by λ * , increases as the Dirac field mass increases; see In Sec. IV D, we have briefly discussed the evolution of the Hubble constant caused by the induced energy-momentum tensor; see the final result in Eq. (70). We find that creation of heavy Dirac fields µ 1 in an electric filed whose strength is bounded as λ < λ * leads to a decay of the Hubble constant due to the positive trace of the induced energy-momentum tensor. Whereas, creation of light Dirac fields µ ≪ 1 in an unbounded electric filed strength or heavy Dirac fields µ 1 in an electric filed with the strength of λ > λ * amplifies the Hubble constant and a superacceleration phenomenon withḢ > 0 occurs due to the negative trace of the induced energymomentum tensor.