Energy-momentum tensor and effective Lagrangian of scalar QED with a nonminimal coupling in 2D de Sitter spacetime
aa r X i v : . [ h e p - t h ] F e b Energy-momentum tensor and effective Lagrangian of scalar QED with anonminimal coupling in 2D de Sitter spacetime
Marzieh Akbari Ahmadmahmoudi and Ehsan Bavarsad ∗ Department of Physics, University of Kashan, 8731753153, Kashan, Iran
We have studied the induced one-loop energy-momentum tensor of a massive complexscalar field within the framework of nonperturbative quantum electrodynamics (QED) witha uniform electric field background on the Poincar´e patch of the two-dimensional de Sitterspacetime (dS ). We also consider a direct coupling the scalar field to the Ricci scalar cur-vature which is parameterized by an arbitrary dimensionless nonminimal coupling constant.We evaluate the trace anomaly of the induced energy-momentum tensor. We show thatour results for the induced energy-momentum tensor in the zero electric field case, and thetrace anomaly are in agreement with the existing literature. Furthermore, we construct theone-loop effective Lagrangian from the induced energy-momentum tensor. PACS numbers: 04.62.+v,11.10.Kk,11.10.Gh
I. INTRODUCTION
The basic framework of quantum field theory in curved spacetime was originally introduced byParker [1] in the late of 1960s and followed by others. In the Parker’s pioneer work quantizationof fields was described and the average density of created particles in an expanding universe wasderived. 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 Hawkingradiation. With these discoveries the development of quantum field theory in curved spacetimereceived much further motivations; see, e.g., [3–5] for introduction. Indeed, a general curvedspacetime is not invariant under transformations of the Poincar´e group, as a consequence there isno a natural set of field modes which are invariant under Poincar´e transformations. This ambiguityleads to ambiguity in definition of particle concept. The fact that the field modes are defined onthe 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 moreadvantageous to construct locally-defined quantities; see [3] for a comprehensive review. One ∗ Electronic address: [email protected] such object of interest is the energy-momentum tensor which is constructed from fields and theirderivatives at the same point of spacetime. There are two important reasons for studying energy-momentum tensor [3, 4]. In Einstein’s equation the energy-momentum tensor appears as a sourceterm of the gravitational field, hence it can be used to investigate the backreaction effects of thematter on the dynamics of gravitational field. Also, it is a useful quantity to explore the physicalproperties of the quantum fields. Thus, studying the energy-momentum tensor of quantized fieldsget more interesting when the cosmological spacetimes have been considered.The regularized and renormalized energy-momentum tensor for different quantum fields incurved spacetime has been extensively studying using various methods. The renormalized energy-momentum tensor of a quantized neutral scalar field propagating in a spacetime of the type ofFriedmann-Lemaitre-Robertson-Walker (FLRW) universes has been analyzed in several cases ofinterest: (1) For a massive field with the arbitrary [6–12], minimal [13, 14], and conformal coupling[15–17] to the Ricci scalar curvature. (2) For a massless field with an arbitrary [18, 19], and confor-mal [20] coupling to the Ricci scalar curvature. In [21], the energy-momentum tensor and effectiveLagrangian of a massive neutral scalar field with the nonminimal coupling to the Ricci scalar cur-vature 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 ten-sor of a massive neutral scalar field with the nonminimal coupling to the Ricci scalar curvaturewas evaluated by computing the difference in energy-momentum between the in and out-vacuumstates. Then, it was realized that the energy-momentum tensor of the created particles describes aperfect 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 constatdecay due to the particle creation. In the work of [23] the energy-momentum tensor of a quan-tum 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 aphase of superacceleration in which the Hubble constant amplifies. With the aim of developingthe adiabatic expansion for the case of fermion fields, the average number of created particles andregularized energy-momentum tensor of a noninteracting, massive Dirac field in a spatially flatFLRW universe have been computed in Refs. [24–28].In order to make one step forward in the context of particle creation in curved spacetime, it seemsnatural to add an electromagnetic gauge field interacting with the quantum matter field. In fact astrong electromagnetic field background in the Minkowski spacetime can create pairs of particles[29–31] 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 gravitationalparticle creation phenomena in curved spacetime [34]. Thus, studying the Schwinger effect in thecosmological spacetimes would be interesting because it may amplify the gravitational particlecreation 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 motivatesthe study of Schwinger effect in the dS. Investigation of the Schwinger effect in the dS was initiatedby [37, 38]. The Schwinger effect and the rate of scalar pair creation process in the presence of auniform electric field background have been analyzed in a dS of two [38–42], four [43], and general[44] dimensions, by using the technique of Bogoliubov transformation that requires semiclassicalconditions. By using this technique, the influence of a uniform and conserved flux magnetic field onthe creation of scalar pairs by the Schwinger mechanism in a four dimensional de Sitter spacetime(dS ) has been explored [45, 46]; see also [47]. The authors of [45, 46] found that a strong magneticfield can intensify the Gibbons-Hawking radiation [48] of dS even when there is no an electric field.It is worth mentioning that the creation of Dirac pairs by Schwinger mechanism in a dS has alsobeen studied using semiclassical methods [49–53], with the main conclusion that a strong electricfield enhances significantly the gravitational pair creation. The Schwinger effect has been exploredin a flat FLRW [54], anti-de Sitter [40, 41, 55, 56], and a charged black hole [57–59] spacetimes;also, widely considered in studies relevant to the inflationary universe scenarios [60–71].The investigation of the Schwinger pair creation in a dS, by using the Bogoliubov transformationmethod, needs to define an adiabatic out-vacuum state at late times in addition to the adiabaticin-vacuum state at early times, which in turn requires to impose semiclassical conditions. In thesemiclassical conditions, either the mass of the particle or the eclectic potential energy across theHubble radius or both must be very larger than the energy scale determined by the curvature ofthe spacetime [39, 43]. On the contrary, the in-vacuum state of quantum fields in dS satisfies theadiabatic 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 widerranges of the related parameters. The regularized in-vacuum expectation value of the currentof a charged scalar field, caused by a uniform electric field background, has been computed intwo [39], three [44], and four [43] dimensional de Sitter spacetimes. The authors showed that inthe strong electric field regime, the induced current asymptotically approaches the semiclassicalcurrent. In particular, it was reported that for an essentially light scalar field in the weak electricfield regime, the induced current has an inversely proportional response to the electric field, whichis referred to as the infrared hyperconductivity phenomenon [39, 43, 44]. The derived resultsfor the induced current in dS [43] have been verified by applying an alternative regularizationthat is the point-splitting method [72], and also calculating the current by using the uniformasymptotic approximation method [60]. An investigation of the influence of a uniform magneticfield background on the current of created scalar pairs by a parallel uniform electric field backgroundin dS illustrates that there is a period of infrared hyperconductivity [45, 46]. In dS [51] and dS [73], the in-vacuum expectation value of the current of a Dirac field coupled to a uniform electricfield background has been analyzed. And the authors come to the conclusion that in the infraredregime the fermionic current is free of the hyperconductivity phenomenon, as opposed to the scalarcurrent. The negative current phenomenon is another remarkable feature of the regularized currentin dS , which is occurred for the scalar fields with essentially small masses [43, 72] and the Diracfields with any mass [73] in a certain range of the electric field strength when the current pointsin the opposite direction to the electric field background. By introducing a novel condition forrenormalization of the in-vacuum expectation values of the scalar and Dirac currents in dS , itwas 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 thebehaviours 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 amassive complex scalar field coupled to a uniform electric field background in the Poincar´e patchof dS . We also consider a direct coupling the scalar field to the Ricci scalar curvature of dS which is parameterized by an arbitrary dimensionless nonminimal coupling constant. To computethe expectation value, we will choose the in-vacuum state of the quantized scalar field, becauseit is an adiabatic and Hadamard state [38, 39]. We evaluate the expectation value to one-looporder, hence the ultraviolet divergences will naturally occur in our calculations. To remove theseultraviolet divergences, we will use the method of adiabatic regularization [12–14, 17, 76], becauseit 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-splittingregularization methods will lead to the equivalence result in spatially flat FLRW spacetimes. Therehas been several studies to investigate the energy-momentum tensor of created scalar and Diracpairs by a uniform electric field in a dS. The energy-momentum tensor of created scalar pairsby a uniform electric field in a dS of general dimension was calculated by using the Bogoliubovcoefficients 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 consequencesof 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 ofenergy-momentum tensor in the in-vacuum state. Recently, in [83] for a massive Dirac field coupledto a uniform electric field background in the Poincar´e patch of dS , the adiabatic regularizedin-vacuum expectation value of the energy-momentum tensor has been evaluated. A commonconclusion of [81–83] was that the sign of the trace can be either positive or negative, dependingon the intensities of the parameters mass and electric field. Consequently, the Hubble constantdecreases under the condition that the trace is positive, in contrast it increases when the trace isnegative. The significant achievement of this paper is the construction of the effective Lagrangianfrom the regularized energy-momentum tensor.The paper proceeds as follows. In the next section, we briefly introduce the elements of ouranalysis. In Sec. III, the expectation value of the energy-momentum tensor in the in-vacuumstate, and the complete set of appropriate adiabatic counterterms are computed, we then obtainthe regularized energy-momentum tensor. In Sec. IV, the regularized energy-momentum tensoris 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 informationwhich 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 ouranalysis. We imagine a massive charged scalar field which interacts with a uniform electric fieldbackground in the Poincar´e patch of dS . Hence, the scalar field is under the influence of twobackgrounds, i.e., the electromagnetic and gravitational fields which are supposed to be unaffectedby 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 is S = Z d x √− g n g µν (cid:0) ∂ µ + ieA µ (cid:1) ϕ (cid:0) ∂ ν − ieA ν (cid:1) ϕ ∗ − (cid:0) m + ξR (cid:1) ϕϕ ∗ o , (1)where ξ is a dimensionless nonminimal coupling constant and R = 2 H , written in terms of theHubble constant H , denotes the Ricci scalar curvature of the dS . The metric g µν on the Poincar´epatch of dS can be read form the line element ds = Ω ( τ ) (cid:16) dτ − d x (cid:17) , Ω( τ ) = − Hτ . (2)The coordinates conformal time τ and spatial coordinate x have ranges τ ∈ (cid:0) − ∞ , (cid:1) , x ∈ R , (3)and cover half of dS manifold. We consider a uniform electric field background with a constantenergy density in the patch (2), which can be derived from the vector potential A µ ( τ ) = − EH τ δ µ , (4)where E is a constant coefficient. Substituting the ingredients (2) and (4), the Klein-Gordonequation arising from the action (1) can be written as (cid:20) ∂ ∂τ − ∂ ∂ x − iλτ ∂∂ x + 1 τ (cid:16) − γ (cid:17)(cid:21) ϕ ( τ, x) = 0 , (5)where the definitions of the dimensionless parameters are given by λ = − eEH , µ = mH , γ = r − λ − µ − ξ. (6)Since we ultimately wish to compute the expectation value of the energy-momentum tensor in thein-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 → −∞ , themode 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 regionare found to be [see [38, 39, 44] for derivations] U k ( x ) = (2 | k | ) − e iπκ e + ik x W κ,γ (cid:16) e − iπ | p | (cid:17) , (7) V k ( x ) = (2 | k | ) − e − iπκ e − ik x W κ,γ (cid:16) e + iπ | p | (cid:17) , (8)where the dimensionless physical momentum p and the parameter κ are expressed as p = − τ k, κ = − iλr, r = sgn( k ) . (9)In Eqs. (7) and (8), the factor W κ,γ denotes the Whittaker function; see, e.g., [84]. If the values ofthe parameters κ, γ , and the phase of the variable z satisfy conditions12 ± γ − κ = 0 , − , − , . . . , (cid:12)(cid:12) ph( z ) (cid:12)(cid:12) < π, (10)then the Whittaker function W κ,γ ( z ), with the help of gamma function Γ( z ), can be representedby a convenient Mellin-Barnes integral W κ,γ ( z ) = e − z Z + i ∞− i ∞ ds πi Γ (cid:0) + γ + s (cid:1) Γ (cid:0) − γ + s (cid:1) Γ (cid:0) − κ − s (cid:1) Γ (cid:0) + γ − κ (cid:1) Γ (cid:0) − γ − κ (cid:1) z − s . (11)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 / γ + s )and Γ(1 / − γ + s ) from the poles of Γ( − κ − s ).The mode functions (7) and (8) satisfy the conserved Wronskian conditions U k ˙ U ∗ k − U ∗ k ˙ U k = V ∗ k ˙ V k − V k ˙ V ∗ k = i, (12)where we use a single dot above a symbol to denote the first conformal time derivative and two dotsto denote the seconde conformal time derivative. To quantize the scalar field ϕ ( x ), we adopt thecanonical procedure. Hence, we promote ϕ ( x ) to operator and expand it in terms of the compleatset of orthogonal mode functions (7) and (8) as ϕ ( x ) = Z dk π h a k U k ( x ) + b † k V k ( x ) i , (13)where the annihilation a k , b k , and creation a † k , b † k , operators obey the commutation relations h a k , a † k ′ i = h b k , b † k ′ i = (2 π ) δ ( k − k ′ ) , (14)with all other commutators equal to zero. Then, we choose the in-vacuum state | in i to be the statethat is annihilated by a k and b k operators a k (cid:12)(cid:12) in (cid:11) = b k (cid:12)(cid:12) in (cid:11) = 0 , (15)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 scalarfield in the in-vacuum state. In general, variation of the action δS with respect to the inversemetric δg µν defines the energy-momentum tensor as T µν = + 2 √− g δSδg µν . (16)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 T µν = g µν (cid:20)(cid:0) ξ − (cid:1) g αβ (cid:16) ∂ α ϕ ∗ − ieA α ϕ ∗ (cid:17)(cid:16) ∂ β ϕ + ieA β ϕ (cid:17) − (cid:0) ξ − (cid:1) m ϕ ∗ ϕ − ξ Rϕ ∗ ϕ (cid:21) + (cid:0) − ξ (cid:1)(cid:16) ∂ µ ϕ ∗ ∂ ν ϕ + ∂ ν ϕ ∗ ∂ µ ϕ (cid:17) + ieA µ (cid:16) ϕ∂ ν ϕ ∗ − ϕ ∗ ∂ ν ϕ (cid:17) + ieA ν (cid:16) ϕ∂ µ ϕ ∗ − ϕ ∗ ∂ µ ϕ (cid:17) + 2 e A µ A ν ϕ ∗ ϕ + 2 ξ Γ αµν (cid:16) ϕ∂ α ϕ ∗ + ϕ ∗ ∂ α ϕ (cid:17) − ξ (cid:16) ϕ∂ µ ∂ ν ϕ ∗ + ϕ ∗ ∂ µ ∂ ν ϕ (cid:17) , (17)where Γ αµν is the Christoffel connection associated with the metric (2) whose nonzero componentsare Γ = Γ = Γ = ˙Ω( τ )Ω( τ ) , (18)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, weconsider ϕ ( x ) as the scalar field operator and we would put Eq. (13) into the expression (17). Usingthe relations (14) and (15), we then obtain the integral expressions for the in-vacuum expectationvalues of the components of the energy-momentum tensor. Changing the integral variable fromthe comoving momentum k , to the dimensionless physical momentum p = − τ k , and imposing anultraviolet cutoff Λ on p , the measure of integration can be written as Z + ∞−∞ dk (2 π ) = H Ω( τ ) X r = ± Z Λ0 dp (2 π ) . (19)Then, the in-vacuum expectation value of the timelike component can be expressed as (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) = Ω ( τ ) H (2 π ) X r = ± (cid:20) I − λr I + (cid:16) λ + 12 µ (cid:17) I + 12 I + I − ξ I + λr I (cid:21) , (20)where the coefficients I , I , . . . , I 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 thespacelike component is expressed by (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) = Ω ( τ ) H (2 π ) X r = ± (cid:20) I − λr I + (cid:16) λ − (cid:0) − ξ (cid:1) µ + 4 ξ (cid:17) I + 12 (cid:0) − ξ (cid:1) I + (cid:0) − ξ (cid:1) I − ξ I + (cid:0) − ξ (cid:1) λr I (cid:21) . (21)By using Eq. (12), it can be verified that the in-vacuum expectation values of the off-diagonalcomponents are equal to (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) = (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) = Ω ( τ ) H π λ Λ . (22)Substituting the expressions (A.8)-(A.14) into Eqs. (20) and (21), yields the unregularized in-vacuum expectation values of the timelike and spacelike components of the energy-momentumtensor, respectively. We find the unregularized timelike component (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) = Ω ( τ ) H (2 π ) (cid:20) Λ + µ log (cid:0) (cid:1) − ξ + µ λ − µ (cid:16) − i csc(2 πγ ) sinh (cid:0) πλ (cid:1)(cid:17) ψ (cid:16)
12 + γ + iλ (cid:17) − µ (cid:16) i csc(2 πγ ) sinh(2 πλ ) (cid:17) ψ (cid:16) − γ + iλ (cid:17) + λγ csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1)(cid:21) , (23)where the notation log is used to denote the natural logarithm function and ψ denotes the digammafunction which is given by the logarithmic derivative of the gamma function. Also, we find theunregularized spacelike component (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) = Ω ( τ ) H (2 π ) (cid:20) Λ − µ log (cid:0) (cid:1) + ξ + µ λ + µ (cid:16) − i csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1)(cid:17) ψ (cid:16)
12 + γ + iλ (cid:17) + µ (cid:16) i csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1)(cid:17) ψ (cid:16) − γ + iλ (cid:17) + λγ csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1)(cid:21) . (24)We see that the expectation values of the components of the energy-momentum tensor containultraviolet divergences. We will show below that these divergences will be subtracted by theadiabatic 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 adiabaticregularization procedure. We return to the Kline-Gordon Eq. (5) and consider its positive frequencysolution as f ( τ, x) = e + ik x h ( τ ) . (25)Then the function h ( τ ) satisfies the following field equation d h ( τ ) dτ + ω ( τ ) h ( τ ) = 0 , (26)and it is convenient to rewrite the conformal time dependent squared frequency as ω ( τ ) = ω ( τ ) + ∆( τ ) , (27)where ω ( τ ) is given by ω ( τ ) = + q k + 2 keA ( τ ) + e A ( τ ) + m Ω ( τ ) , (28)0where A ( τ ) is read from Eq. (4), and ∆( τ ) is given by∆( τ ) = 2 ξτ . (29)To adjust the set of the required counterterms, following the usual prescription, we assume that theconformal scale factor Ω( τ ), and the electromagnetic vector potential A µ ( τ ), to be of zero adiabaticorder and the energy-momentum tensor T µν , to be of second adiabatic order in dS . Therefore, ω ( τ ) is of zero adiabatic order and ∆( τ ) which can be rewritten as∆( τ ) = 2 ξ ˙Ω ( τ )Ω ( τ ) , (30)is of seconde adiabatic order. The Klein-Gordon Eq. (26) possesses a Wentzel-Kramers-Brillouin(WKB) form solution h ( τ ) = 1 p W ( τ ) exp (cid:18) − i Z τ dτ ′ W ( τ ′ ) (cid:19) , (31)where the function W ( τ ) solves the exact nonlinear second order differential equation W ( τ ) = ω ( τ ) + ∆( τ ) − ¨ W W + 3 ˙ W W . (32)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 adiabaticexpansion up to second order for the function W . Thus, we write an appropriate series W ( τ ) = W (0) ( τ ) + W (2) ( τ ) , (33)where the superscripts on the terms indicate their adiabatic orders. The iteration process beginsby 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 right-hand side of the equation are of second adiabatic order and since the ∆ term is of second adiabaticorder too, all these terms vanish. Therefore, we have W (0) ( τ ) = ω ( τ ) . (34)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 W (2) ( τ ) = 12 ω (cid:18) ∆ − ¨ ω ω + 3 ˙ ω ω (cid:19) . (35)1Thus, the adiabatic expansion of W ( τ ) up to second order is obtained from Eqs. (33)-(35) as W ( τ ) = ω ( τ ) + 12 ω (cid:18) ∆ − ¨ ω ω + 3 ˙ ω ω (cid:19) . (36)We need also the adiabatic expansion of W − ( τ ), which up to second order is given by1 W ( τ ) = 1 ω ( τ ) − ω (cid:18) ∆ − ¨ ω ω + 3 ˙ ω ω (cid:19) . (37)Putting together the pieces (31), (36), and (37) of Eq. (25) determines the adiabatic expansionof positive frequency mode function up to second order. Having these orthogonal adiabatic modefunctions f ( x ), we can perform similar steps which led from Eq. (7) to Eq. (15) and establish theadiabatic expansion of the quantum scalar field operator and the vacuum up to second order. Thecounterterms are then obtained by putting the adiabatic expansion of the scalar field operatorinto Eq. (17) and computing the expectation values of the resulting expressions in the adiabaticvacuum. After these remarks, we find the set of appropriate counterterms as T (adi)01 = T (adi)10 = Ω ( τ ) H π λ Λ , (38) T (adi)00 = Ω ( τ ) H (2 π ) (cid:20) Λ + µ log (cid:0) (cid:1) + 16 − ξ + µ λ + λ µ − µ log( µ ) (cid:21) , (39) T (adi)11 = Ω ( τ ) H (2 π ) (cid:20) Λ − µ log (cid:0) (cid:1) −
16 + 2 ξ + µ λ − λ µ + µ log( µ ) (cid:21) . (40)Subtraction of the counterterms (38)-(40) from the unregularized expressions (22)-(24), respec-tively, yields the regularized energy-momentum tensor, which is referred to as the induced energy-momentum tensor. We find that the off-diagonal components of the induced energy-momentumtensor vanish T = T = (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) − T (adi)01 = 0 . (41)The timelike component of the induced energy-momentum tensor is obtained T = (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) − T (adi)00 = Ω ( τ ) H (2 π ) (cid:20) ξ − − λ µ + µ log( µ ) − µ (cid:16) − i csc(2 πγ ) sinh (cid:0) πλ (cid:1)(cid:17) ψ (cid:16)
12 + γ + iλ (cid:17) − µ (cid:16) i csc(2 πγ ) sinh(2 πλ ) (cid:17) ψ (cid:16) − γ + iλ (cid:17) + λγ csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1)(cid:21) . (42)Eventually, the spacelike component of the induced energy-momentum tensor is given by T = (cid:10) in (cid:12)(cid:12) T (cid:12)(cid:12) in (cid:11) − T (adi)11 = − Ω ( τ ) H (2 π ) (cid:20) ξ − − λ µ + µ log( µ ) − µ (cid:16) − i csc(2 πγ ) sinh (cid:0) πλ (cid:1)(cid:17) ψ (cid:16)
12 + γ + iλ (cid:17) − µ (cid:16) i csc(2 πγ ) sinh(2 πλ ) (cid:17) ψ (cid:16) − γ + iλ (cid:17) − λγ csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1)(cid:21) . (43)2In the case of zero electric field, our result for the induced energy-momentum tensor can be com-pared to the energy-momentum tensor of a neutral scalar field in dS , which has been derived inRef. [7] using the covariant point-splitting technique. If we set λ = 0 in Eqs. (42) and (43), wethen find that the induced energy-momentum tensor can be written as T µν = H (2 π ) (cid:20) ξ −
16 + µ log( µ ) − µ ψ (cid:16)
12 + γ (cid:17) − µ ψ (cid:16) − γ (cid:17)(cid:21) g µν . (44)The result (44) differs from the corresponding result obtained in [7] only by a prefactor of 2, becausein [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 fieldobtained earlier. IV. IMPLICATIONS OF THE INDUCED ENERGY-MOMENTUM TENSOR
In this section we investigate the induced energy-momentum tensor and consider some of itsimplications.
A. Analysis of the induced energy-momentum tensor
We begin our survey of the induced energy-momentum tensor by finding its qualitative behavior.Figures 1 and 2 show graphs of the magnitudes of the timelike | T | [see Eq. (42)] and spacelike | T | [see Eq. (43)] components of the induced energy-momentum tensor versus the electric fieldparameter λ , respectively. In these figures note especially that the both scales are logarithmicto cover several orders of magnitude. Several features are clear from these figures. For the cases µ &
1, in the strong electric field regime that the condition λ ≫ max(1 , µ, ξ ) is valid, | T | and | T | are independent of the values of the parameters µ and ξ ; hence all the curves asymptoticallyapproach one another at the right end of the figures. Although the expressions (42) and (43) arerather complicated, they have simple asymptotic forms in the limit λ → ∞ , which are given by T ≃ Ω ( τ ) H (2 π ) λ (cid:18) − µ (cid:19) , (45) T ≃ Ω ( τ ) H (2 π ) λ (cid:18) µ (cid:19) . (46)The asymptotic behaviors of the curves in Figs. 1 and 2, in the strong electric field regime, are wellapproximated by Eqs. (45) and (46), respectively. For the cases µ <
1, the second terms in both3 - (cid:1) | T | (cid:2) μ = μ = μ = μ = μ = (cid:4) = (cid:0) = FIG. 1: The normalized magnitude of the timelike component of the induced energy-momentum tensor | T | τ , versus the normalized electric field λ = − eE/H , that both scales are logarithmic. The curvescorrespond to different values of the mass parameter µ = m/H , and the conformal coupling constant ξ . Eqs. (45) and (46), which depend on µ , dominate and as µ becomes smaller the magnitudes of T and T enhance by factor µ − . While, the first terms in both Eqs. (45) and (46), which becomedominate 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 and T atlarge mass parameter µ , and the increase at small µ . To find the asymptotic behavior of the inducedenergy-momentum tensor in the heavy scalar field regime that the condition µ ≫ max(1 , λ, ξ ) isvalid, we can expand expressions (42) and (43) in Taylor series about µ = ∞ . We then have T ≃ − T ≃ Ω ( τ ) H π (cid:18) c µ + c µ + O (cid:0) µ − (cid:1)(cid:19) , (47)where the coefficients c and c are given by c = 160 − ξ ξ ,c = λ − λ ξ − ξ
15 + ξ − ξ . (48)In the heavy scalar field regime, the approximate expression (47) shows that the induced energy-momentum tensor is suppressed as µ − instead of an exponentially suppression with a Boltzmannfactor e − πµ , which is derived by semiclassical approaches [44]. This behavior have been seen for thein-vacuum expectation value of the energy-momentum tensor of a Dirac field coupled to a uniform4 - (cid:3) | T | τ μ = μ = μ = μ = μ = ξ = ξ = FIG. 2: The normalized magnitude of the spacelike component of the induced energy-momentum tensor | T | τ , versus the normalized electric field λ = − eE/H , that both scales are logarithmic. The curvescorrespond to different values of the mass parameter µ = m/H , and the conformal coupling constant ξ . electric field in dS [83]. Similar asymptotic behavior occurs in the in-vacuum expectation valueof the current of a scalar field in four [43] and three [44] dimensional dS, and also the fermionicinduced current in dS [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 thatthe graphs have been plotted on the logarithmic scales; hence the zero values of T and T areappeared as extremely sharp decrease in the graph. We stress that the components of the inducedenergy-momentum tensor, which are given by Eqs. (41)-(43), are continuous and analytic functionsof 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 thecomponents (41)-(43) as T = g µν T µν = H π (cid:20) ξ − − λ µ + µ log( µ ) − µ (cid:16) − i csc(2 πγ ) sinh (cid:0) πλ (cid:1)(cid:17) ψ (cid:16)
12 + γ + iλ (cid:17) − µ (cid:16) i csc(2 πγ ) sinh(2 πλ ) (cid:17) ψ (cid:16) − γ + iλ (cid:17)(cid:21) . (49)5To calculate the trace anomaly, we take the combined lime of Eq. (49) as λ → µ →
0, and ξ → λ, µ, ξ → T = − H π = − R π , (50)where in the last step we have used R = 2 H . The trace anomaly for a real scalar field has beenobtained as ( − R ) / (24 π ) [86] in a general two-dimensional spacetime, where R is the Ricci scalarcurvature 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 withthe 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 thedefinition (16) and construct the effective action S eff such that its functional derivatives reproducethe expressions (41)-(43), then we can identify the effective Lagrangian L eff . We begin by intro-ducing the induced current J µ , which is the regularized in-vacuum expectation value of the currentof the scalar field ϕ ( x ), whose dynamics is described by the action (1). The induced current hasbeen computed in Ref. [39], and is given by J µ = Ω( τ ) Hπ eγ csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1) δ µ . (51)Thus, the effective electromagnetic potential A.J , can be constructed by combining Eqs. (4) and(51) as
A.J = g µν A µ J ν = H π λγ csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1) . (52)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 T = 12 Ω ( τ ) (cid:16) T + A.J (cid:17) , (53) T = −
12 Ω ( τ ) (cid:16) T − A.J (cid:17) . (54)Variation of S eff = − Z d x √− g n T + A.J o , (55)with respect to the inverse metric g µν gives δS eff = 12 Z d x √− g (cid:26) (cid:16) T + A.J (cid:17) g µν − A µ J ν (cid:27) δg µν , (56)6then definition (16), leads to Eqs. (41), (53), and (54). Therefore, Eq. (55) is the desired one-loopeffective action of scalar QED in dS , and the corresponding effective Lagrangian reads L eff = − √− g (cid:16) T + A.J (cid:17) . (57)Substitution of Eqs. (49) and (52) into Eq. (57) yields the explicit form of the effective Lagrangianas L eff = √− g (cid:16) H π (cid:17)(cid:20) − ξ + λ µ − µ log( µ ) + µ (cid:16) − i csc(2 πγ ) sinh (cid:0) πλ (cid:1)(cid:17) ψ (cid:16)
12 + γ + iλ (cid:17) + µ (cid:16) i csc(2 πγ ) sinh(2 πλ ) (cid:17) ψ (cid:16) − γ + iλ (cid:17) − λγ csc (cid:0) πγ (cid:1) sinh (cid:0) πλ (cid:1)(cid:21) . (58)The scalar QED effective action in two-dimensional de Sitter and anti-de Sitter spacetimes hasbeen obtained in Ref. [40], by employing the in-out formalism which is introduced by Schwingerand DeWitt; see, e.g., [87] for a review. In the in-out formalism, the effective action is relatedto 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 awell-defined out-vacuum state to calculate the Bogoliubov coefficients, it is necessary to adopt thesemiclassical approximation. In the semiclassical regime, the parameters λ, µ , and ξ are constrainedas [39, 43, 44] λ + µ + 2 ξ ≫ . (59)However, the approach that we adopt in this paper involves only the in-vacuum state. Hence, we donot 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 ,compared with those effective Lagrangians which are derived under the semiclassical condition. V. CONCLUSIONS
This paper has investigated the one-loop induced energy-momentum tensor of a complex scalarfield in the context of scalar QED in a two-dimensional de Sitter spacetime. The dynamics of thescalar field is described by the action presented in Eq. (1). We have assumed that the scalar fieldpropagates in a uniform electric field background in the Poincar´e patch of dS . The metric of thespacetime can be read from Eq. (2), and the electric field background is described by the vectorpotential (4). Since, a systematic treatment of ultraviolet divergences in expectation values whichare computed in an adiabatic and Hadamard state is relatively simple and straightforward, we7calculate the expectation value of energy-momentum tensor in the in-vacuum state. The resultsfor the expectation values of the energy-momentum tensor components in the in-vacuum state aregiven by Eqs. (22)-(24). We have used adiabatic subtraction method to regularize the expectationvalues, the complete set of the appropriate counterterms is obtained in Eqs. (38)-(40). Then,each of the expressions (22)-(24) is regularized by subtracting its counterterm. This procedureremoves all the ultraviolet divergences and brings us to the induced energy-momentum tensorwhose components are given by Eqs. (41)-(43). The components of the induced energy-momentumtensor are continuous and analytic functions of the parameters mass µ , conformal coupling constant ξ , and electric field λ . We showed that, in the zero electric field case, the induced energy-momentumtensor takes the form (44), and agrees with the energy-momentum tensor of a neutral scalar fieldobtained earlier in the literature.We observe that the off-diagonal components of the induced energy-momentum tensor vanish.Figures 1 and 2 reveals the behaviors of the magnitudes of timelike T and spacelike T com-ponents of the induced energy-momentum tensor, respectively. For fixed values of µ and ξ in thestrong electric field regime λ ≫ max(1 , µ, ξ ), the magnitudes of T and T increase significantlywith increasing λ ; except in the close neighborhoods of the zero values of T and T . Recall thatin the figures the both scales are logarithmic; hence near the zero values of T and T a singularbehavior for the curves is seen. In the strong electric field regime, T and T can be well approx-imated by the expressions (45) and (46), respectively. For fixed values of λ and ξ , the magnitudesof T and T decrease with increasing µ . In the heavy scalar field regime µ ≫ max(1 , λ, ξ ), theapproximate expressions for T and T are given by Eq. (47).The trace of the induced energy-momentum tensor has been obtained in Eq. (49), which yieldsthe trace anomaly (50). In the discussion below Eq. (50), we have pointed out that our result forthe trace anomaly is in agrement with the trace anomaly of a massless conformally coupled realscalar 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) fromthe induced energy-momentum tensor. More precisely, the expression (58) is the nonperturbativeone-loop effective Lagrangian for a scalar field coupled to a uniform electric field background inthe Poincar´e patch of dS . In the derivation of the effective Lagrangian (58), we do not imposeany semiclassical condition such as (59). Consequently, our result for the effective Lagrangiancan be examined in larger domains of the parameters λ, µ , and ξ , compared with those effectiveLagrangians which are derived by using semiclassical approaches.8 Acknowledgments
E. B. is supported by the University of Kashan Grant No. 985904/1.
Appendix: momentum integrals over the Whittaker functions
In the appendix we present the definitions and explicit values of the coefficients I , I , . . . , I which are appeared in Eqs. (20) and (21). These coefficients are defined as I = e πλr Z Λ0 dpp (cid:12)(cid:12)(cid:12) W κ,γ (cid:0) − ip (cid:1)(cid:12)(cid:12)(cid:12) , (A.1) I = e πλr Z Λ0 dp (cid:12)(cid:12)(cid:12) W κ,γ (cid:0) − ip (cid:1)(cid:12)(cid:12)(cid:12) , (A.2) I = e πλr Z Λ0 dpp (cid:12)(cid:12)(cid:12) W κ,γ (cid:0) − ip (cid:1)(cid:12)(cid:12)(cid:12) , (A.3) I = e πλr Z Λ0 dpp (cid:12)(cid:12)(cid:12) W κ,γ (cid:0) − ip (cid:1)(cid:12)(cid:12)(cid:12) , (A.4) I = − e πλr ℑ Z Λ0 dpW κ,γ (cid:0) − ip (cid:1) W − κ,γ (cid:0) ip (cid:1) , (A.5) I = − e πλr ℜ Z Λ0 dpp W κ,γ (cid:0) − ip (cid:1) W − κ,γ (cid:0) ip (cid:1) , (A.6) I = e πλr ℑ Z Λ0 dpp W κ,γ (cid:0) − ip (cid:1) W − κ,γ (cid:0) ip (cid:1) , (A.7)where the operators ℑ and ℜ extract the imaginary and real parts of the expressions, respectively.Integrals (A.1)-(A.7) are seen to be of the same type as those momentum integrals which occurredin calculation of the induced current of a scalar field in dS [39] and dS [43]. For calculatingthese integrals, we consider the Mellin-Barnes integral representation of the Whittaker functionevaluated on a contour described just below Eq. (11) and accomplish the resulting integrals, asexplained in Ref. [43] with some routine modifications. We eventually find I = 12 Λ + rλ Λ + 12 (cid:16) γ + 3 λ − (cid:17) log (cid:0) (cid:1) + 516 − λ − γ − rλγ csc (cid:0) πγ (cid:1) e πλr − rλγ cot (cid:0) πγ (cid:1) − i (cid:16) γ + 3 λ − (cid:17) csc (cid:0) πγ (cid:1)(cid:20) π sin (cid:0) πγ (cid:1) + (cid:16) e πλr + e − πiγ (cid:17) × ψ (cid:16) − γ + iλr (cid:17) − (cid:16) e πλr + e πiγ (cid:17) ψ (cid:16)
12 + γ + iλr (cid:17)(cid:21) , (A.8)9and I = Λ + rλ log (cid:0) (cid:1) − rλ − γ cot (cid:0) πγ (cid:1) − γ csc (cid:0) πγ (cid:1) e πλr − i rλ csc (cid:0) πγ (cid:1)(cid:20) π sin (cid:0) πγ (cid:1) + (cid:16) e πλr + e − πiγ (cid:17) ψ (cid:16) − γ + iλr (cid:17) − (cid:16) e πλr + e πiγ (cid:17) ψ (cid:16)
12 + γ + iλr (cid:17)(cid:21) . (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 I = log (cid:0) (cid:1) − i π − i (cid:0) πγ (cid:1)(cid:20)(cid:16) e πλr + e − πiγ (cid:17) ψ (cid:16) − γ + iλr (cid:17) − (cid:16) e πλr + e πiγ (cid:17) × ψ (cid:16)
12 + γ + iλr (cid:17)(cid:21) , (A.10) I = 2Λ − rλ Λ + 12 (cid:16)
14 + λ − γ (cid:17) + rλγ cot (cid:0) πγ (cid:1) + rλγ csc (cid:0) πγ (cid:1) e πλr , (A.11) I = − Λ − (cid:16) γ + λ − (cid:17) log (cid:0) (cid:1) + 14 πrλ + 14 (cid:16) λ + γ − (cid:17) + 12 rλγ cot (cid:0) πγ (cid:1) + 12 rλγ csc (cid:0) πγ (cid:1) e πλr + i (cid:0) πγ (cid:1)(cid:16) γ + λ − − irλ (cid:17)(cid:20)(cid:16) e πλr + e − πiγ (cid:17) × ψ (cid:16) − γ + iλr (cid:17) − (cid:16) e πλr + e πiγ (cid:17) ψ (cid:16)
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