Energy-momentum tensor and effective Lagrangian of scalar QED with a nonminimal coupling in 2D de Sitter spacetime

We have studied the induced one-loop energy-momentum tensor of a massive complex scalar field within the framework of nonperturbative quantum electrodynamics (QED) with a uniform electric field background on the Poincar\'e patch of the two-dimensional de Sitter spacetime ($\mathrm{dS_{2}}$). We also consider a direct coupling the scalar field to the Ricci scalar curvature which is parameterized by an arbitrary dimensionless nonminimal coupling constant. We evaluate the trace anomaly of the induced energy-momentum tensor. We show that our results for the induced energy-momentum tensor in the zero electric field case, and the trace anomaly are in agreement with the existing literature. Furthermore, we construct the one-loop effective Lagrangian from the induced energy-momentum tensor.


I. INTRODUCTION
The basic framework of quantum field theory in curved spacetime was originally introduced by Parker [1] in the late of 1960s and followed by others. In the Parker's pioneer work quantization of fields was described and the average density of created particles in an expanding universe was derived. Progress was also being made on this issue, when in the mid of 1970s Hawking discovered [2] that a black hole emits as a blackbody due to the particle creation which is known as Hawking radiation. With these discoveries the development of quantum field theory in curved spacetime received much further motivations; see, e.g., [3][4][5] for introduction. Indeed, a general curved spacetime is not invariant under transformations of the Poincaré group, as a consequence there is no a natural set of field modes which are invariant under Poincaré transformations. This ambiguity leads to ambiguity in definition of particle concept. The fact that the field modes are defined on the whole of at least a large patch of spacetime illustrates the global nature of the particle concept. This is in contrast with the at least quasi-local nature of physical detectors. Hence, it is more advantageous to construct locally-defined quantities; see [3] for a comprehensive review. One * Electronic address: bavarsad@kashanu.ac.ir such object of interest is the energy-momentum tensor which is constructed from fields and their derivatives at the same point of spacetime. There are two important reasons for studying energymomentum tensor [3,4]. In Einstein's equation the energy-momentum tensor appears as a source term of the gravitational field, hence it can be used to investigate the backreaction effects of the matter on the dynamics of gravitational field. Also, it is a useful quantity to explore the physical properties of the quantum fields. Thus, studying the energy-momentum tensor of quantized fields get more interesting when the cosmological spacetimes have been considered.
The regularized and renormalized energy-momentum tensor for different quantum fields in curved spacetime has been extensively studying using various methods. The renormalized energymomentum tensor of a quantized neutral scalar field propagating in a spacetime of the type of Friedmann-Lemaitre-Robertson-Walker (FLRW) universes has been analyzed in several cases of interest: (1) For a massive field with the arbitrary [6][7][8][9][10][11][12], minimal [13,14], and conformal coupling [15][16][17] to the Ricci scalar curvature. (2) For a massless field with an arbitrary [18,19], and conformal [20] coupling to the Ricci scalar curvature. In [21], the energy-momentum tensor and effective Lagrangian of a massive neutral scalar field with the nonminimal coupling to the Ricci scalar curvature in a de Sitter spacetime (dS) have been calculated by using the dimensional regularization.
In Ref. [22], to study the effects of the particle creation in a dS the finite energy-momentum tensor of a massive neutral scalar field with the nonminimal coupling to the Ricci scalar curvature was evaluated by computing the difference in energy-momentum between the in and out-vacuum states. Then, it was realized that the energy-momentum tensor of the created particles describes a perfect fluid with vacuum equation of state which vanishes for massless, conformally coupled field. Also, the author discovered that the invariant vacuum state and the effective cosmological constat decay due to the particle creation. In the work of [23] the energy-momentum tensor of a quantum noninteracting, massive, and nonminimally coupled scalar field in a dS has been investigated.
And, it was shown as a consequences of the quantum backreaction effects that there may exist a phase of superacceleration in which the Hubble constant amplifies. With the aim of developing the adiabatic expansion for the case of fermion fields, the average number of created particles and regularized energy-momentum tensor of a noninteracting, massive Dirac field in a spatially flat FLRW universe have been computed in Refs. [24][25][26][27][28].
In order to make one step forward in the context of particle creation in curved spacetime, it seems natural to add an electromagnetic gauge field interacting with the quantum matter field. In fact a strong electromagnetic field background in the Minkowski spacetime can create pairs of particles [29][30][31] which is known as the Schwinger effect; see [32,33] for a comprehensive review. Indeed, the physical mechanism underlying the Schwinger effect is analogous to that of the gravitational particle creation phenomena in curved spacetime [34]. Thus, studying the Schwinger effect in the cosmological spacetimes would be interesting because it may amplify the gravitational particle creation process; see [35] and references therein for a review. It is a well accepted paradigm [36] that strong electric and magnetic fields might be generated in the early universe which motivates the study of Schwinger effect in the dS. Investigation of the Schwinger effect in the dS was initiated by [37,38]. The Schwinger effect and the rate of scalar pair creation process in the presence of a uniform electric field background have been analyzed in a dS of two [38][39][40][41][42], four [43], and general [44] dimensions, by using the technique of Bogoliubov transformation that requires semiclassical conditions. By using this technique, the influence of a uniform and conserved flux magnetic field on the creation of scalar pairs by the Schwinger mechanism in a four dimensional de Sitter spacetime (dS 4 ) has been explored [45,46]; see also [47]. The authors of [45,46] found that a strong magnetic field can intensify the Gibbons-Hawking radiation [48] of dS 4 even when there is no an electric field.
The investigation of the Schwinger pair creation in a dS, by using the Bogoliubov transformation method, needs to define an adiabatic out-vacuum state at late times in addition to the adiabatic in-vacuum state at early times, which in turn requires to impose semiclassical conditions. In the semiclassical conditions, either the mass of the particle or the eclectic potential energy across the Hubble radius or both must be very larger than the energy scale determined by the curvature of the spacetime [39,43]. On the contrary, the in-vacuum state of quantum fields in dS satisfies the adiabatic conditions at all times. Hence, computation of expectation values of physical quantities, such as current and energy-momentum tensor, in the in-vacuum state enables us to probe wider ranges of the related parameters. The regularized in-vacuum expectation value of the current of a charged scalar field, caused by a uniform electric field background, has been computed in two [39], three [44], and four [43] dimensional de Sitter spacetimes. The authors showed that in the strong electric field regime, the induced current asymptotically approaches the semiclassical current. In particular, it was reported that for an essentially light scalar field in the weak electric field regime, the induced current has an inversely proportional response to the electric field, which is referred to as the infrared hyperconductivity phenomenon [39,43,44]. The derived results for the induced current in dS 4 [43] have been verified by applying an alternative regularization that is the point-splitting method [72], and also calculating the current by using the uniform asymptotic approximation method [60]. An investigation of the influence of a uniform magnetic field background on the current of created scalar pairs by a parallel uniform electric field background in dS 4 illustrates that there is a period of infrared hyperconductivity [45,46]. In dS 2 [51] and dS 4 [73], the in-vacuum expectation value of the current of a Dirac field coupled to a uniform electric field background has been analyzed. And the authors come to the conclusion that in the infrared regime the fermionic current is free of the hyperconductivity phenomenon, as opposed to the scalar current. The negative current phenomenon is another remarkable feature of the regularized current in dS 4 , which is occurred for the scalar fields with essentially small masses [43,72] and the Dirac fields with any mass [73] in a certain range of the electric field strength when the current points in the opposite direction to the electric field background. By introducing a novel condition for renormalization of the in-vacuum expectation values of the scalar and Dirac currents in dS 4 , it was shown that the infrared hyperconductivity period would be removed from the scalar current, however the negative current phase would still be present [74]. A satisfactory explanation for the behaviours of the current has been given in Ref. [75].
The aim of this paper is to study the expectation value of the energy-momentum tensor of a massive complex scalar field coupled to a uniform electric field background in the Poincaré patch of dS 2 . We also consider a direct coupling the scalar field to the Ricci scalar curvature of dS 2 which is parameterized by an arbitrary dimensionless nonminimal coupling constant. To compute the expectation value, we will choose the in-vacuum state of the quantized scalar field, because it is an adiabatic and Hadamard state [38,39]. We evaluate the expectation value to one-loop order, hence the ultraviolet divergences will naturally occur in our calculations. To remove these ultraviolet divergences, we will use the method of adiabatic regularization [12-14, 17, 76], because it is comparatively simpler than the other methods, such as, point-splitting regularization [7,19,20,77] and dimensional regularization [21,78]. It was verified [79] that the adiabatic and point-splitting regularization methods will lead to the equivalence result in spatially flat FLRW spacetimes. There has been several studies to investigate the energy-momentum tensor of created scalar and Dirac pairs by a uniform electric field in a dS. The energy-momentum tensor of created scalar pairs by a uniform electric field in a dS of general dimension was calculated by using the Bogoliubov coefficients in the two limiting regimes: the heavy scalar field [44], and the strong electric field [80]; which leads to a decay of the Hubble constant. An investigation of the gravitational consequences of scalar pair creation due to a uniform electric field background in the three [81] and four [82] dimensional dS has been made by calculating the regularized expectation value of the trace of energy-momentum tensor in the in-vacuum state. Recently, in [83] for a massive Dirac field coupled to a uniform electric field background in the Poincaré patch of dS 2 , the adiabatic regularized in-vacuum expectation value of the energy-momentum tensor has been evaluated. A common conclusion of [81][82][83] was that the sign of the trace can be either positive or negative, depending on the intensities of the parameters mass and electric field. Consequently, the Hubble constant decreases under the condition that the trace is positive, in contrast it increases when the trace is negative. The significant achievement of this paper is the construction of the effective Lagrangian from the regularized energy-momentum tensor.
The paper proceeds as follows. In the next section, we briefly introduce the elements of our analysis. In Sec. III, the expectation value of the energy-momentum tensor in the in-vacuum state, and the complete set of appropriate adiabatic counterterms are computed, we then obtain the regularized energy-momentum tensor. In Sec. IV, the regularized energy-momentum tensor is analyzed, then we use it to derive the trace anomaly and construct the effective Lagrangian.
Eventually, our conclusions are drawn in Sec. V. In the appendix, we include essential information which is needed to study of the paper.

II. THE QUANTUM SCALAR FIELD IN ELECTRIC AND DS BACKGROUNDS
In this section we will introduce the elements of model under consideration and setup our analysis. We imagine a massive charged scalar field which interacts with a uniform electric field background in the Poincaré patch of dS 2 . Hence, the scalar field is under the influence of two backgrounds, i.e., the electromagnetic and gravitational fields which are supposed to be unaffected by the dynamics of the scalar field. The classical action of a complex scalar field ϕ(x) of mass m and electric charge e which is coupled to an electromagnetic gauge field A µ in the dS 2 is where ξ is a dimensionless nonminimal coupling constant and R = 2H 2 , written in terms of the Hubble constant H, denotes the Ricci scalar curvature of the dS 2 . The metric g µν on the Poincaré patch of dS 2 can be read form the line element The coordinates conformal time τ and spatial coordinate x have ranges and cover half of dS 2 manifold. We consider a uniform electric field background with a constant energy density in the patch (2), which can be derived from the vector potential where E is a constant coefficient. Substituting the ingredients (2) and (4), the Klein-Gordon equation arising from the action (1) can be written as where the definitions of the dimensionless parameters are given by Since we ultimately wish to compute the expectation value of the energy-momentum tensor in the in-vacuum state, we only require that of the solutions of Eq. (5) which represent this vacuum sate.
Therefor, we impose the boundary condition that in the in region of the manifold as τ → −∞, the mode functions be plane waves of fixed comoving momentum k. The normalized positive U k (x), and negative V k (x), frequency mode functions that reduce to the plane wave form in the in region are found to be [see [38,39,44] for derivations] where the dimensionless physical momentum p and the parameter κ are expressed as In Eqs. (7) and (8), the factor W κ,γ denotes the Whittaker function; see, e.g., [84]. If the values of the parameters κ, γ, and the phase of the variable z satisfy conditions then the Whittaker function W κ,γ (z), with the help of gamma function Γ(z), can be represented by a convenient Mellin-Barnes integral The contour of integration is a straight line along the imaginary axis in the complex plane s from −i∞ to +i∞ that can be joined by a semicircle at the infinity to sort out the poles of Γ(1/2+ γ + s) and Γ(1/2 − γ + s) from the poles of Γ(−κ − s).
The mode functions (7) and (8) satisfy the conserved Wronskian conditions where we use a single dot above a symbol to denote the first conformal time derivative and two dots to denote the seconde conformal time derivative. To quantize the scalar field ϕ(x), we adopt the canonical procedure. Hence, we promote ϕ(x) to operator and expand it in terms of the compleat set of orthogonal mode functions (7) and (8) as where the annihilation a k , b k , and creation a † k , b † k , operators obey the commutation relations with all other commutators equal to zero. Then, we choose the in-vacuum state |in to be the state that is annihilated by a k and b k operators for all values of comoving momentum k.

III. COMPUTATION OF THE REGULARIZED ENERGY-MOMENTUM TENSOR
We are now ready to compute the expectation value of energy-momentum tensor of the scalar field in the in-vacuum state. In general, variation of the action δS with respect to the inverse metric δg µν defines the energy-momentum tensor as Vary g µν in the action (1) and use of definition (16) along with the Kline-Gordon equation of motion (5), yields the following symmetric expression for the energy-momentum tensor of the scalar field where Γ α µν is the Christoffel connection associated with the metric (2) whose nonzero components are or related to these by symmetry.
A. The evaluation of the expectation value in the in-vacuum state To evaluate the expectation value of the energy-momentum tensor in the in-vacuum state, we consider ϕ(x) as the scalar field operator and we would put Eq. (13) into the expression (17). Using the relations (14) and (15), we then obtain the integral expressions for the in-vacuum expectation values of the components of the energy-momentum tensor. Changing the integral variable from the comoving momentum k, to the dimensionless physical momentum p = −τ k, and imposing an ultraviolet cutoff Λ on p, the measure of integration can be written as Then, the in-vacuum expectation value of the timelike component can be expressed as where the coefficients I 1 , I 2 , . . . , I 7 denote the momentum integrals over the Whittaker functions, and are defined in Eqs. (A.1)-(A.7), respectively. Similarly, the in-vacuum expectation value of the spacelike component is expressed by By using Eq. (12), it can be verified that the in-vacuum expectation values of the off-diagonal components are equal to in T 01 in = in T 10 in = Ω 2 (τ ) H 2 π λΛ.
Substituting the expressions (A.8)-(A.14) into Eqs. (20) and (21), yields the unregularized invacuum expectation values of the timelike and spacelike components of the energy-momentum tensor, respectively. We find the unregularized timelike component where the notation log is used to denote the natural logarithm function and ψ denotes the digamma function which is given by the logarithmic derivative of the gamma function. Also, we find the unregularized spacelike component We see that the expectation values of the components of the energy-momentum tensor contain ultraviolet divergences. We will show below that these divergences will be subtracted by the adiabatic counterterms.

B. Adiabatic counterterms and regularization of the expectation values
In order to eliminate the divergent terms of the expressions (22)-(24), we employ the adiabatic regularization procedure. We return to the Kline-Gordon Eq. (5) and consider its positive frequency solution as Then the function h(τ ) satisfies the following field equation and it is convenient to rewrite the conformal time dependent squared frequency as where ω 0 (τ ) is given by where A 1 (τ ) is read from Eq. (4), and ∆(τ ) is given by To adjust the set of the required counterterms, following the usual prescription, we assume that the conformal scale factor Ω(τ ), and the electromagnetic vector potential A µ (τ ), to be of zero adiabatic order and the energy-momentum tensor T µν , to be of second adiabatic order in dS 2 . Therefore, ω 0 (τ ) is of zero adiabatic order and ∆(τ ) which can be rewritten as is of seconde adiabatic order. The Klein-Gordon Eq. (26) possesses a Wentzel-Kramers-Brillouin where the function W(τ ) solves the exact nonlinear second order differential equation Recall that the set of counterterms which are required to cancel the divergences from the expressions (22)-(24) must be constructed up to second adiabatic order. It is then necessary to find an adiabatic expansion up to second order for the function W. Thus, we write an appropriate series where the superscripts on the terms indicate their adiabatic orders. The iteration process begins by considering the zeroth adiabatic order. At this step, the adiabatic series (33) is truncated to W = W (0) . Substitution of this ansatz into Eq. (32) shows that the derivative terms on the righthand side of the equation are of second adiabatic order and since the ∆ term is of second adiabatic order too, all these terms vanish. Therefore, we have The next iteration is done by substituting the second order adiabatic series (33) into Eq. (32), using the result (34) and keeping only terms up to the second order. We then find Thus, the adiabatic expansion of W(τ ) up to second order is obtained from Eqs. (33)- (35) as We need also the adiabatic expansion of W −1 (τ ), which up to second order is given by Putting together the pieces (31), (36), and (37)  = Ω 2 (τ ) Subtraction of the counterterms (38)-(40) from the unregularized expressions (22)-(24), respectively, yields the regularized energy-momentum tensor, which is referred to as the induced energymomentum tensor. We find that the off-diagonal components of the induced energy-momentum tensor vanish The timelike component of the induced energy-momentum tensor is obtained Eventually, the spacelike component of the induced energy-momentum tensor is given by In the case of zero electric field, our result for the induced energy-momentum tensor can be compared to the energy-momentum tensor of a neutral scalar field in dS 2 , which has been derived in Ref. [7] using the covariant point-splitting technique. If we set λ = 0 in Eqs. (42) and (43), we then find that the induced energy-momentum tensor can be written as The result (44) differs from the corresponding result obtained in [7] only by a prefactor of 2, because in [7] a real scalar field has been considered, however here we have considered a complex scalar field ϕ(x), which has two real scalar field components. Thus, the induced energy-momentum tensor, in the zero electric field case, agrees with the energy-momentum tensor of a neutral scalar field obtained earlier.

IV. IMPLICATIONS OF THE INDUCED ENERGY-MOMENTUM TENSOR
In this section we investigate the induced energy-momentum tensor and consider some of its implications.

A. Analysis of the induced energy-momentum tensor
We begin our survey of the induced energy-momentum tensor by finding its qualitative behavior.  (42) and (43) are rather complicated, they have simple asymptotic forms in the limit λ → ∞, which are given by The asymptotic behaviors of the curves in Figs. 1 and 2, in the strong electric field regime, are well approximated by Eqs. (45) and (46), respectively. For the cases µ < 1, the second terms in both Eqs. (45) and (46), which depend on µ, dominate and as µ becomes smaller the magnitudes of T 00 and T 11 enhance by factor µ −2 . While, the first terms in both Eqs. (45) and (46), which become dominate for the cases µ 1, are independent of the value of µ and ξ.
We see clearly in Figs. 1 and 2 the characteristic decrease of the magnitudes of T 00 and T 11 at large mass parameter µ, and the increase at small µ. To find the asymptotic behavior of the induced energy-momentum tensor in the heavy scalar field regime that the condition µ ≫ max(1, λ, ξ) is valid, we can expand expressions (42) and (43) in Taylor series about µ = ∞. We then have where the coefficients c 1 and c 2 are given by In the heavy scalar field regime, the approximate expression (47) shows that the induced energymomentum tensor is suppressed as µ −2 instead of an exponentially suppression with a Boltzmann factor e −2πµ , which is derived by semiclassical approaches [44]. This behavior have been seen for the in-vacuum expectation value of the energy-momentum tensor of a Dirac field coupled to a uniform electric field in dS 2 [83]. Similar asymptotic behavior occurs in the in-vacuum expectation value of the current of a scalar field in four [43] and three [44] dimensional dS, and also the fermionic induced current in dS 4 [73]. Attempts have been made in Refs. [74,75] to address this observation.
Another feature of Figs. 1 and 2 is that some of the curves have a singularity. We remark that the graphs have been plotted on the logarithmic scales; hence the zero values of T 00 and T 11 are appeared as extremely sharp decrease in the graph. We stress that the components of the induced energy-momentum tensor, which are given by Eqs. (41)- (43), are continuous and analytic functions of the parameters mass, conformal coupling constant and electric field; as they must [85].

B. Trace anomaly
The trace of the induced energy-momentum tensor T , is contracted from the metric (2) and the components (41)-(43) as To calculate the trace anomaly, we take the combined lime of Eq. (49) as λ → 0, µ → 0, and ξ → 0.
We then find lim λ, µ, ξ→0 where in the last step we have used R = 2H 2 . The trace anomaly for a real scalar field has been obtained as (−R)/(24π) [86] in a general two-dimensional spacetime, where R is the Ricci scalar curvature of the spacetime. Here, note in particular that we have regarded a complex scalar field ϕ(x), which has two real scalar field components. Therefore, the result (50) is in agrement with the result obtained in the literature; see., e.g., [3,4] for a comprehensive review.

C. Effective Lagrangian
Now that we have obtained the induced energy-momentum tensor, it is possible to return the definition (16) and construct the effective action S eff such that its functional derivatives reproduce the expressions (41)-(43), then we can identify the effective Lagrangian L eff . We begin by introducing the induced current J µ , which is the regularized in-vacuum expectation value of the current of the scalar field ϕ(x), whose dynamics is described by the action (1). The induced current has been computed in Ref. [39], and is given by Thus, the effective electromagnetic potential A.J, can be constructed by combining Eqs. (4) and (51) as To reach our goal of deriving the effective action, it is convenient to rewrite the expressions (42) and (43) in terms of the trace (49) and the effective electromagnetic potential (52). We then obtain Variation of with respect to the inverse metric g µν gives then definition (16), leads to Eqs. (41), (53), and (54). Therefore, Eq. (55) is the desired one-loop effective action of scalar QED in dS 2 , and the corresponding effective Lagrangian reads Substitution of Eqs. (49) and (52) into Eq. (57) yields the explicit form of the effective Lagrangian as L eff = √ −g H 2 2π The scalar QED effective action in two-dimensional de Sitter and anti-de Sitter spacetimes has been obtained in Ref. [40], by employing the in-out formalism which is introduced by Schwinger and DeWitt; see, e.g., [87] for a review. In the in-out formalism, the effective action is related to the transition amplitude between in-vacuum and out-vacuum states of the quantum fields; hence it is required to use the Bogoliubov coefficients. In de Sitter spacetime, in order to have a well-defined out-vacuum state to calculate the Bogoliubov coefficients, it is necessary to adopt the semiclassical approximation. In the semiclassical regime, the parameters λ, µ, and ξ are constrained as [39,43,44] λ 2 + µ 2 + 2ξ ≫ 1.
However, the approach that we adopt in this paper involves only the in-vacuum state. Hence, we do not need to consider the out-vacuum sate which in turn requires the condition (59). Consequently, the effective Lagrangian (58) can be probed in larger domains of the parameters λ, µ, and ξ, ξ, and electric field λ. We showed that, in the zero electric field case, the induced energy-momentum tensor takes the form (44), and agrees with the energy-momentum tensor of a neutral scalar field obtained earlier in the literature.
We observe that the off-diagonal components of the induced energy-momentum tensor vanish. behavior for the curves is seen. In the strong electric field regime, T 00 and T 11 can be well approximated by the expressions (45) and (46), respectively. For fixed values of λ and ξ, the magnitudes of T 00 and T 11 decrease with increasing µ. In the heavy scalar field regime µ ≫ max(1, λ, ξ), the approximate expressions for T 00 and T 11 are given by Eq. (47).
The trace of the induced energy-momentum tensor has been obtained in Eq. (49), which yields the trace anomaly (50). In the discussion below Eq. (50), we have pointed out that our result for the trace anomaly is in agrement with the trace anomaly of a massless conformally coupled real scalar field in a general two-dimensional spacetime obtained earlier in the literature.
The major achievement of this research is the derivation of the effective Lagrangian (58) from the induced energy-momentum tensor. More precisely, the expression (58) is the nonperturbative one-loop effective Lagrangian for a scalar field coupled to a uniform electric field background in the Poincaré patch of dS 2 . In the derivation of the effective Lagrangian (58), we do not impose any semiclassical condition such as (59). Consequently, our result for the effective Lagrangian can be examined in larger domains of the parameters λ, µ, and ξ, compared with those effective Lagrangians which are derived by using semiclassical approaches. and I 2 = Λ + rλ log 2Λ − rλ − γ cot 2πγ − γ csc 2πγ e 2πλr − i 2 rλ csc 2πγ π sin 2πγ + e 2πλr + e −2πiγ ψ 1 2 − γ + iλr − e 2πλr + e 2πiγ ψ 1 2 + γ + iλr . (A.9) Also, integrals (A.3)-(A.7) have been computed in Ref. [88], by using the procedure explained in [43], and the following results have been obtained (A.14)