Schwinger Effect in 4D de Sitter Space and Constraints on Magnetogenesis in the Early Universe
SSchwinger Effect in 4D de Sitter Space andConstraints on Magnetogenesis in the Early Universe
Takeshi Kobayashi (cid:63), † and Niayesh Afshordi † , ‡ (cid:63) Canadian Institute for Theoretical Astrophysics, University of Toronto,60 St. George Street, Toronto, Ontario M5S 3H8, Canada † Perimeter Institute for Theoretical Physics,31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada ‡ Department of Physics and Astronomy, University of Waterloo,200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
We investigate pair creation by an electric field in four-dimensional de Sitter space. The expectationvalue of the induced current is computed, using the method of adiabatic regularization. Understrong electric fields the behavior of the current is similar to that in flat space, while under weakelectric fields the current becomes inversely proportional to the mass squared of the charged field.Thus we find that the de Sitter space obtains a large conductivity under weak electric fields in thepresence of a charged field with a tiny mass. We then apply the results to constrain electromagneticfields in the early universe. In particular, we study cosmological scenarios for generating large-scalemagnetic fields during the inflationary era. Electric fields generated along with the magnetic fieldscan induce sufficiently large conductivity to terminate the phase of magnetogenesis. For inflationarymagnetogenesis models with a modified Maxwell kinetic term, the generated magnetic fields cannotexceed 10 − G on Mpc scales in the present epoch, when a charged field carrying an elementarycharge with mass of order the Hubble scale or smaller exists in the Lagrangian. Similar constraintsfrom the Schwinger effect apply for other magnetogenesis mechanisms. [email protected] [email protected] a r X i v : . [ h e p - t h ] N ov ontents | eE | (cid:29) H . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Weak Electric Force : | eE | (cid:28) H . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Comments on Very Light or Massless Scalars . . . . . . . . . . . . . . . . . . 15 Particle creation by a time dependent background happens in various situations. The well-knownexample is the production of charged particles under strong electric fields [1, 2, 3, 4, 5], arising from atime dependent vector potential, as studied by Schwinger. Similar phenomena are also seen in curvedspacetimes, where time dependent gravitational backgrounds produce particles. Such gravitationaleffects are particularly important for cosmology, as the large-scale structure in the universe can beseeded by the accelerated expansion during the inflationary epoch (see e.g. [6] for a review). Inaddition to the cosmic structures, the magnetic fields in our universe may also have a cosmologicalorigin. The possibility of electromagnetic fields existing in the early universe motivates us to lookinto effects induced by electromagnetic fields in curved spacetimes. Recently, the Schwinger effect intwo-dimensional de Sitter (dS) space was studied in [7] (see also [8]). The authors found behaviorsquite different from those in flat space; for instance, a large current is induced under weak electricfields, when the mass of the charged particle is much smaller than the Hubble scale, a phenomenondubbed as “hyperconductivity.”With cosmological applications in mind, in this paper we explore the Schwinger process in four-dimensional dS space. Considering a charged scalar, both the electric and gravitational backgroundfields give rise to the production of the scalar particles. Our strategy is to study the expectationvalue of the induced current, as in [7, 9, 10, 11, 12, 13, 14]. This allows us to analyze cases whereeven the adiabatic vacuum does not exist in the asymptotic future, in other words, regimes where thescalar mass and electric force are much smaller than the Hubble scale and thus the scalar excitationsare not well described as particles. Upon computing the expectation value of the current whose1ormal expression has ultraviolet divergences, we use the method of adiabatic subtraction [15, 16,17, 18, 19, 20] in order to remove the infinities. Under strong electric fields, i.e. | eE | (cid:29) H , theinduced current J is obtained as J ∝ e E H e − πm | eE | , (1.1)where H is the Hubble rate, E the electric field amplitude, e the scalar charge, and m is thescalar mass. Such a behavior of the current is analogous to that from the Schwinger process in flatspace [13, 14]. On the other hand, with weak electric fields, i.e. | eE | (cid:28) H , we find that the currentdepends linearly on E , J ∝ e EH m . (1.2)Thus we confirm that for small mass, a four-dimensional dS also induces large current from weakelectric fields. However, unlike in the two-dimensional case [7] where the current under weak electricfields is exponentially suppressed for massive scalars, in four-dimensions the scaling (1.2) holds forarbitrary masses. Therefore charged massive scalars can also give rise to non-negligible conductivityin a dS universe under weak electric fields.After analyzing the Schwinger effect in de Sitter space, we move on to apply the results to con-strain electromagnetic fields in the early universe. We particularly focus on cosmological scenarios forgenerating large-scale magnetic fields during the inflationary epoch [21, 22]. Inflationary magneto-genesis is generically accompanied by the generation of large electric fields as well [23, 24, 25, 26, 27],which gives rise to a current via the Schwinger process. When the induced current becomes large, itsbackreaction to the Maxwell fields becomes non-negligible and can prevent any further generation ofthe magnetic fields. Such considerations allow us to constrain models of inflationary magnetogenesisfrom the Schwinger effect. Focusing on models where the electromagnetic fields are generated bya time dependent coupling on the Maxwell kinetic term (of the form I ( t ) F µν F µµ [22]), we findthat the Schwinger effect presents a serious obstacle to generating primordial magnetic fields dur-ing inflation. For example, having a field in the action that carries an electric charge of order theelementary charge and mass of order the Hubble scale or smaller, the Schwinger effect prohibitsinflationary magnetogenesis from producing magnetic fields larger than 10 − G on Mpc scales inthe current universe. The bound depends on the charges and masses of the fields in the action,however the Schwinger effect is shown to pose a major challenge for generating magnetic fields aslarge as 10 − G, which is the lower bound on the extragalactic magnetic fields suggested by therecent gamma ray observations [28, 29, 30, 31, 32, 33, 34].This paper is organized as follows: We investigate the Schwinger effect in a four-dimensional dSspace in Section 2. After explaining the setup, we carry out the usual Bogoliubov calculations inSubsection 2.1, limiting ourselves to the regime of | eE | , m (cid:29) H so that the adiabatic vacuum existsin the asymptotic future. The reader interested in the induced current/conductivity or constraints onmagnetogenesis can skip this subsection, as the results obtained from the Bogoliubov calculation willonly be used upon making semiclassical estimates in later discussions. In Subsection 2.2, we computethe expectation value of the current, using the method of adiabatic regularization. The behavior ofthe induced current is studied in various limits, including regimes where the Hubble scale is muchlarger than the electric force and the scalar mass. We then apply the results to constrain inflationarymagnetogenesis in Section 3. This section can also be considered as providing discussions on the2ssue of backreaction to the background electric field, in the context of magnetogenesis scenarios.Finally, we conclude in Section 4.Throughout this paper, we take the principal values − π ≤ arg (cid:36) ≤ π for the phase of complexnumbers (cid:36) . In order to study the Schwinger process in a four-dimensional dS space, we analyze QED coupledto a charged complex scalar: S = (cid:90) d x √− g (cid:26) − g µν ( ∂ µ − ieA µ ) ϕ ∗ ( ∂ ν + ieA ν ) ϕ − m ϕ ∗ ϕ − F µν F µν (cid:27) , (2.1)where F µν = ∂ µ A ν − ∂ ν A µ . The background spacetime is fixed to dS, ds = a ( τ ) (cid:0) − dτ + dx + dy + dz (cid:1) , (2.2)where the conformal time τ is expressed in terms of the constant Hubble parameter as τ = − aH < , H = daa dτ = const . (2.3)Here we have taken τ → − to denote the asymptotic future. We use Greek letters for the spacetimeindices µ, ν = τ, x, y, z , and Latin letters for spatial indices i, j = x, y, z .In order to describe a constant and uniform electric field, we consider a vector potential of theform A µ = EH τ δ zµ , E = const . (2.4)Then a comoving observer with 4-velocity u µ ( u i = 0, u µ u µ = −
1) measures an electric field alongthe z -direction, E µ = u ν F µν = aEδ zµ , (2.5)with a constant field strength E µ E µ = E .The equation of motion of ϕ under the time dependent background is ϕ (cid:48)(cid:48) + 2 a (cid:48) a ϕ (cid:48) − ∂ i ∂ i ϕ − ieA z ∂ z ϕ + e A z ϕ + a m ϕ = 0 , (2.6)where the prime represents a τ -derivative, and the sum over repeated spatial indices is implied irre-spective of their positions. Upon quantizing the scalar field ϕ under the time dependent background,let us redefine the field as q = aϕ, (2.7)then the conjugate momenta are obtained from the action S = (cid:82) d x L in (2.1) asΠ = ∂ L ∂q (cid:48) = q (cid:48)∗ − a (cid:48) a q ∗ , Π ∗ = ∂ L ∂q (cid:48)∗ = q (cid:48) − a (cid:48) a q. (2.8)3e promote q , q ∗ , and their conjugate momenta into operators, q ( τ, x ) = 1(2 π ) (cid:90) d k (cid:110) a k q k ( τ ) e i k · x + b † k q ∗− k ( τ ) e − i k · x (cid:111) ,q † ( τ, x ) = 1(2 π ) (cid:90) d k (cid:110) a † k q ∗ k ( τ ) e − i k · x + b k q − k ( τ ) e i k · x (cid:111) , (2.9)and assign the commutation relations[ a k , a † p ] = [ b k , b † p ] = (2 π ) δ (3) ( k − p ) , [ a k , a p ] = [ b k , b p ] = [ a k , b p ] = [ a k , b † p ] = · · · = 0 , (2.10)as well as[ q ( τ, x ) , Π( τ, y )] = [ q † ( τ, x ) , Π † ( τ, y )] = iδ (3) ( x − y ) , [ q ( τ, x ) , q ( τ, y )] = [Π( τ, x ) , Π( τ, y )] = [ q ( τ, x ) , q † ( τ, y )] = [ q ( τ, x ) , Π † ( τ, y )] = · · · = 0 . (2.11)The relations (2.11) follow from (2.10) when the mode function q k satisfies the normalization con-dition: q k q (cid:48)∗ k − q ∗ k q (cid:48) k = i. (2.12)The mode functions obey the equation of motion (cf. (2.6)) taking the form of q (cid:48)(cid:48) k + ω k q k = 0 , (2.13)where the effective frequency squared ω k is ω k = ( k z + eA z ) + k x + k y + a m − a (cid:48)(cid:48) a (2.14)= 1 τ (cid:18) e E H + m H − (cid:19) + 2 τ k z eEH + k . (2.15)Here, k = ( k x + k y + k z ) / . In the asymptotic past τ → −∞ , the frequency is ω k (cid:39) k , and thus q k is a sum of plane waves. On the other hand, in the asymptotic future τ →
0, the frequencyapproaches ω k (cid:39) τ (cid:18) e E H + m H − (cid:19) , (2.16)whose rate of change is (cid:18) ω (cid:48) k ω k (cid:19) (cid:39) (cid:18) e E H + m H − (cid:19) − , ω (cid:48)(cid:48) k ω k (cid:39) (cid:18) e E H + m H − (cid:19) − . (2.17)Thus when e E /H + m /H is much larger than unity, then q k in the asymptotic future is wellapproximated by a WKB solution, in other words, there exists an adiabatic vacuum for ϕ .Let us now introduce the variables z ≡ kiτ, κ ≡ − i k z k eEH , µ ≡ − e E H − m H , (2.18)4here z and κ are purely imaginary, while µ is either real or purely imaginary. Then the equationof motion (2.13) is rewritten as d q k dz + (cid:26) z (cid:18) − µ (cid:19) + κz − (cid:27) q k = 0 . (2.19)Solutions of this equation are the Whittaker functions W κ,µ ( z ), M κ,µ ( z ), whose basic properties arelaid out in Appendix A. From the limiting form of W κ,µ ( z ) as | z | → ∞ shown in (A.7), we see thatthe function W κ,µ ( z ) represents the positive frequency solution in the asymptotic past. Thus wechoose the mode function as q k = e iκπ/ √ k W κ,µ ( z ) , (2.20)where the normalization is set from the condition (2.12), up to an arbitrary phase. Let us now evaluate the pair creation rate of the charged scalar particles. In this subsection we limitourselves to cases where e E H + m H (cid:29) , (2.21)so that there exists an adiabatic vacuum for the charged scalar in the asymptotic future. (Seediscussions around (2.17).) Then the scalar excitations can be interpreted as creation of particlesat some intermediate time, and the production rate can be obtained by computing the Bogoliubovcoefficients. Under (2.21), µ is purely imaginary, and we take arg µ = π/
2, i.e. µ = i | µ | , (2.22)throughout this subsection. In order to study the particle excitations at late times, let us now rewritethe mode function in terms of M κ,µ ( z ) (see also Appendix A, and note especially that 2 µ is not aninteger in this subsection), q k = e −| µ | π/ (cid:112) k | µ | { α k M κ,µ ( z ) + β k ( M κ,µ ( z )) ∗ } . (2.23)The coefficients α k and β k should satisfy | α k | − | β k | = 1 (2.24)from the normalization condition (2.12). Here we remark that, e −| µ | π/ (cid:112) k | µ | M κ,µ ( z ) (2.25)represents the positive frequency solution in the asymptotic future. From the limiting form of M κ,µ ( z )as z → τ → (cid:112) | ω k | exp (cid:26) − i (cid:90) τ dτ | ω k | (cid:27) (cid:39) (2 | µ | ) − / ( − τ ) i | µ | +1 / e i · const . . (2.26)5ere, upon obtaining the right hand side, we have used (2.16) and | µ | (cid:29) α k and β k are obtained from (2.20) and (2.23) by using the for-mula (A.5) as α k = (2 | µ | ) / e ( iκ + | µ | ) π/ Γ( − µ )Γ( − µ − κ ) , β k = − i (2 | µ | ) / e ( iκ −| µ | ) π/ Γ(2 µ )Γ( + µ − κ ) . (2.27)Choosing the vacuum | ¯0 (cid:105) in the asymptotic future by ¯ a k | ¯0 (cid:105) = ¯ b k | ¯0 (cid:105) = 0 for ∀ k , where¯ a k = α k a k + β ∗ k b †− k , ¯ b k = β ∗− k a †− k + α − k b k , (2.28)the number of created particles in the vacuum | ¯0 (cid:105) with charge ∓ e and comoving wave number ± k per comoving three-volume is (cid:104) ¯0 | a † k a k | ¯0 (cid:105) (2 π ) (cid:82) d x = (cid:104) ¯0 | b †− k b − k | ¯0 (cid:105) (2 π ) (cid:82) d x = | β k | (2 π ) = e iκπ + e − | µ | π π ) sinh(2 | µ | π ) . (2.29)Integrating this expression over all wave modes gives a divergent result:1(2 π ) (cid:90) d k | β k | = 1(2 π ) sinh(2 | µ | π ) (cid:26) H eE sinh (cid:18) πeEH (cid:19) + 2 πe − | µ | π (cid:27) (cid:90) ∞ dk k , (2.30)since it denotes the number of particle pairs produced from the infinite past to the infinite future.Instead of the sum over all times, we are rather interested in the produced number of pairs per unittime.Under the condition (2.21), the rate of change of the effective frequency ω k (2.15) is tiny in boththe asymptotic past and future, and thus there exist adiabatic vacua for the charged scalar. Here,let us estimate the time of particle creation by analyzing when the adiabaticity is violated, i.e., when ω k changes quickly, by studying how | ω (cid:48) k /ω k | grows in time. The quantity | ω (cid:48) k /ω k | vanishes in theasymptotic past, and it approaches the value (2.17) in the asymptotic future. Depending on theparameter values, the time evolution of | ω (cid:48) k /ω k | may or may not exhibit peaks in the intermediatetimes; e.g. for k z eE <
0, then | ω (cid:48) k /ω k | can just monotonically grow in time. In such cases where | ω (cid:48) k /ω k | does not exhibit peaks, we can instead focus on when | ω (cid:48) k /ω k | comes close to taking theasymptotic value (2.17). It can be checked that, for parameter sets that satisfy − µ (cid:29)
1, it isaround the time τ ∼ − k (cid:18) | µ | + 14 (cid:19) / (2.31)when the quantity | ω (cid:48) k /ω k | exhibits peaks, or approaches closely to its maximum value. Thus we make use of the rough estimate (2.31) and translate the k -integral in (2.30) into a timeintegral,1(2 π ) (cid:90) d k | β k | = ( | µ | + ) / (2 π ) sinh(2 | µ | π ) (cid:26) H eE sinh (cid:18) πeEH (cid:19) + 2 πe − | µ | π (cid:27) (cid:90) −∞ dτ ( aH ) . (2.32) The violation of the adiabaticity can also be studied in a different frame; by redefining the field and time as q k = a m χ k , dτ = a m ds , such that the form of the equation of motion (2.13) is preserved. The detailed behaviorof | ω (cid:48) k /ω k | (e.g., whether it peaks at a certain time, or monotonically grows) depends on the frame, however we remarkthat the order-of-magnitude estimate (2.31) of when | ω (cid:48) k /ω k | approaches its maximum value is independent of thechoice of the frame.
6y looking at the produced number of pairs within dτ , and dividing by a , we arrive at the pairproduction rate, i.e., the number of pairs produced per unit physical four-volume,Γ = H (2 π ) ( | µ | + ) / sinh(2 | µ | π ) (cid:26) H eE sinh (cid:18) πeEH (cid:19) + 2 πe − | µ | π (cid:27) . (2.33)Since the rate Γ is independent of time, the physical number density n of pairs at time τ is easilycomputed as n = 1 a ( τ ) (cid:90) τ −∞ d ˜ τ a (˜ τ ) Γ = Γ3
H . (2.34)The fact that n is a constant indicates that the Schwinger and gravitational particle creation balancesagainst the dilution of the number density due to the expansion of the universe. One sees that, whenthe mass and/or the electric field are large enough to satisfy the condition (2.21), the ϕ populationis always dominated by the particles created within a Hubble time.The vacuum persistence probability can also be computed in a similar fashion from |(cid:104) ¯0 | (cid:105)| = exp (cid:26) − (cid:82) d x (2 π ) (cid:90) d k ln (cid:0) | β k | (cid:1)(cid:27) . (2.35)Here we further assume m H ≥ in addition to (2.21), and use the formula for the dilogarithm, − (cid:90) z ds ln(1 − s ) s = ∞ (cid:88) j =1 z j j , for | z | ≤ , (2.36)for integrating over the angular direction k z /k . Then, converting the k -integral into the time integralusing (2.31), one can obtain the vacuum decay rate Υ vac : |(cid:104) ¯0 | (cid:105)| = exp (cid:26) − (cid:90) d x dτ a Υ vac (cid:27) , (2.37)as a series of the form,Υ vac = H (2 π ) (cid:18) | µ | + 14 (cid:19) / ∞ (cid:88) j =1 (cid:26) ( − j +1 j H eE e − j | µ | π sinh (cid:18) jπeEH (cid:19) + 2 πj e − j | µ | π (cid:27) . (2.38)Let us close this section by studying the limit where the Hubble parameter is much smaller thanthe mass and electric field strength. Taking H → H → Γ = ( eE ) (2 π ) exp (cid:18) − πm | eE | (cid:19) , (2.39)lim H → Υ vac = ∞ (cid:88) j =1 ( − j +1 j ( eE ) (2 π ) exp (cid:18) − jπm | eE | (cid:19) , (2.40)reproducing the familiar results for Schwinger pair production in Minkowski space [3, 4, 5] (seealso [13] for a recent analysis). 7 .2 Induced Current and Conductivity Once produced, the charged scalar particles move under the electric field and thus give rise to acurrent as well as a conductivity. The results obtained in Subsection 2.1 can be used to estimatethe induced current via | J | ∼ | env | , where v is the velocity of the particles. Such a semiclassicalapproach provides good approximations in some parameter regions (as we will see later), but not ingeneral. In particular, computing the contribution only from the created particles is not enough, assuch a naive picture violates local charge conservation [7]. Moreover, the analyses in the previoussubsection were limited to cases where the mass and/or the electric force are sufficiently larger thanthe Hubble scale, cf. (2.21).In this subsection we do not impose the condition (2.21), and directly compute the expectationvalue of the conserved current, J µ = ie (cid:110) ϕ † ( ∂ µ + ieA µ ) ϕ − ϕ ( ∂ µ − ieA µ ) ϕ † (cid:111) + h . c . (2.42)in the vacuum state | (cid:105) defined by a k | (cid:105) = b k | (cid:105) = 0 for ∀ k , cf. (2.9). Under the electric field alongthe z -direction, the expectation value vanishes except for its z -component, (cid:104) J z (cid:105) = − e (2 π ) a (cid:90) d k ( k z + eA z ) | q k | , (2.43)where the mode function is given in (2.20). However this expectation value diverges, as can be seenfrom the limiting form of q k as k → ∞ shown in (A.7). In order to explicitly see the divergence, letus first compute the integral by imposing a cutoff ζ on k , (cid:104) J z (cid:105) = − lim ζ →∞ e (2 π ) a (cid:90) ζ dk k (cid:90) − dr ( kr + eA z ) e iκπ k | W κ,µ ( z ) | , (2.44)where we have introduced r = k z k . (2.45)The integral is carried out in Appendix B, yielding (cid:104) J z (cid:105) = eaH (2 π ) lim ζ →∞ (cid:34) λ (cid:18) ζaH (cid:19) + λ (cid:18) ζaH (cid:19) − λ
36 + µ λ λ
15+ 45 + 4 π ( − λ + 2 µ )12 π µ cosh(2 πλ ) λ sin(2 πµ ) −
45 + 8 π ( − λ + µ )24 π µ sinh(2 πλ ) λ sin(2 πµ )+ Re (cid:26)(cid:90) − dr iλ
16 sin(2 πµ ) (cid:0) − µ + (7 + 12 λ − µ ) r − λ r (cid:1) × (cid:16)(cid:16) e πrλ + e πiµ (cid:17) ψ (cid:0) + µ + irλ (cid:1) − (cid:16) e πrλ + e − πiµ (cid:17) ψ (cid:0) − µ + irλ (cid:1)(cid:17)(cid:27)(cid:35) , (2.46) In terms of the current J µ (2.42), the Maxwell equation is written as ∇ ν F µν = J µ . (2.41) ψ ( z ) = Γ (cid:48) ( z ) / Γ( z ) is the digamma function, and λ is defined as λ = eEH . (2.47)We thus see that the expectation value of the current has quadratic and logarithmic divergences.Let us also remark that some of the terms in (2.46) blow up when µ = 0 , / , . . . . However theirsum does not necessarily diverge as µ approaches such values, and thus the finite part of (2.46) (i.e.terms without ζ ) is well-behaved.In order to regularize the divergences, we use the method of adiabatic subtraction [15, 16, 17,18, 19, 20]. The idea here is to compute quantities in the limit of slow variation of the background,then subtract their contributions from the formal expressions to obtain a finite result. (See alsoworks [9, 10, 11, 12] which applied adiabatic regularization to the analysis of Schwinger effect in flatspace.) Let us start by considering a mode function with a WKB form, q k ( τ ) = 1 (cid:112) W k ( τ ) exp (cid:26) − i (cid:90) τ d ˜ τ W k (˜ τ ) (cid:27) , (2.48)which is an exact solution of the equation of motion (2.13) if the function W k satisfies W k = ω k + 34 (cid:18) W (cid:48) k W k (cid:19) − W (cid:48)(cid:48) k W k . (2.49)Furthermore, when W k is real and positive, the normalization condition (2.12) is also satisfied. Here,recall from (2.14) that ω k takes the form of ω k = Ω k − a (cid:48)(cid:48) a , (2.50)with Ω k = (cid:8) ( k z + eA z ) + k x + k y + a m (cid:9) / . (2.51)Hereafter, let us assume the mass to be nonzero, i.e. m (cid:54) = 0, so that Ω k is positive definite. Inorder to parameterize the slowness of the evolution of the time dependent background, we assign anadiabatic parameter T − to each time derivative in (2.49) and (2.50); taking T → ∞ denotes thelimit of infinitely slow variation of the background. Then the function W k can be computed at eachorder in T − . The solution at leading order is simply W k = Ω k + O ( T − ) . (2.52)Higher order solutions can be obtained by recursively substituting the results into the right handside of (2.49); up to adiabatic order T − we obtain W k = ω k + 34 (cid:18) Ω (cid:48) k Ω k (cid:19) −
12 Ω (cid:48)(cid:48) k Ω k + O ( T − ) , (2.53)and so on. We note that our results are not altered by computing the adiabatic subtraction termsin a different frame, where q k and τ are redefined such that the equation of motion preserves theform of (2.13) (cf. Footnote 3). 9n the following we expand the current in terms of T − , then the lower order results will besubtracted from the formal expression (2.44). We will see that the adiabatic subtraction up toquadratic order is just enough to remove the divergences, as well as gives results that have thecorrect behavior in the Minkowski limit. We also remark that, since the formal expression of (cid:104) J (cid:105) vanishes, and also (cid:104) J i (cid:105) is homogeneous, it is clear that the adiabatic subtraction does not spoil thecurrent conservation. Detailed discussions on the method of adiabatic subtraction can be found in,e.g., [15, 16, 17, 18, 19, 20].For a real and positive W k (recall from (2.52) that we can have W k (cid:39) Ω k > − e (2 π ) a (cid:90) d k ( k z + eA z ) 12 W k . (2.54)Expanding this expression up to adiabatic order T − using (2.53) yields − e (2 π ) a (cid:90) d k ( k z + eA z ) 12Ω k (cid:34) k a (cid:48)(cid:48) a + 14Ω k (cid:8) ( eA (cid:48) z ) + ( k z + eA z ) eA (cid:48)(cid:48) z + (cid:0) a (cid:48) + aa (cid:48)(cid:48) (cid:1) m (cid:9) − k (cid:8) ( k z + eA z ) eA (cid:48) z + aa (cid:48) m (cid:9) + O ( T − ) (cid:35) . (2.55)We carry out the integration by imposing a cutoff ζ on k as in (2.44). After some algebra we obtainlim ζ →∞ e (2 π ) a (cid:34) − eA z ζ + 215 ( eA z ) + 13 eA z a m − eA (cid:48)(cid:48) z ln (cid:18) ζam (cid:19) + 16 a (cid:48) a eA (cid:48) z − a (cid:48)(cid:48) a eA z + 29 eA (cid:48)(cid:48) z + O ( T − ) (cid:35) . (2.56)Terms of adiabatic order T shown in the first line contains a quadratic divergence, while the termsof T − in the second line has a logarithmic divergence. Substituting the expressions for a (2.3) and A z (2.4), we find the adiabatic subtraction terms to belim ζ →∞ eaH (2 π ) (cid:34) λ (cid:18) ζaH (cid:19) − λ − λ m H + λ (cid:18) ζam (cid:19) + λ
18 + O ( T − ) (cid:35) , (2.57)where the first three terms arise from the order T expansion, and the other two terms from or-der T − . Comparing with the formal expression (2.46), the divergences of the expectation valueare seen to be removed by the adiabatic subtraction up to order T − . One can further expand upto order T − , which only gives finite terms. However, as we will see later, subtracting terms oforder T − spoils the behavior of the current in the flat space limit. Therefore, we subtract off terms10p to adiabatic order T − from (2.46) in order to obtain the regularized current, arriving at (cid:104) J z (cid:105) reg = eaH (2 π ) (cid:34) − λ
15 + λ (cid:16) mH (cid:17) + 45 + 4 π ( − λ + 2 µ )12 π µ cosh(2 πλ ) λ sin(2 πµ ) −
45 + 8 π ( − λ + µ )24 π µ sinh(2 πλ ) λ sin(2 πµ )+ Re (cid:26)(cid:90) − dr iλ
16 sin(2 πµ ) (cid:0) − µ + (7 + 12 λ − µ ) r − λ r (cid:1) × (cid:16)(cid:16) e πrλ + e πiµ (cid:17) ψ (cid:0) + µ + irλ (cid:1) − (cid:16) e πrλ + e − πiµ (cid:17) ψ (cid:0) − µ + irλ (cid:1)(cid:17)(cid:27)(cid:35) . (2.58)Comparing with the formal expression (2.46), the procedure of adiabatic subtraction has modifiedthe terms in the first line inside the parentheses.Here it is important to note that the hard cutoff ζ was introduced in the derivation only forcalculational convenience, so that the integrations of the formal expression (2.44) and the adiabaticexpansion (2.55) can be performed separately. Instead of using ζ , we could have subtracted theintegrand of (2.55) from (2.44) before carrying out the integral, then we would not see any infinitiesin the calculations. Such a procedure would be preferable for numerical studies.Before discussing the behavior of (2.58), let us parameterize the amplitude of the current as (cid:104) J z (cid:105) reg = aJ, (2.59)where J has mass dimension three. We also define the conductivity σ by σ = JE . (2.60)Then one sees from (2.58) that normalized quantities such as
JeH , σe H = JeH λ (2.61)are uniquely fixed by the two parameters m/H and eE/H (or, equivalently, µ and λ ), representingthe mass and electric force relative to the Hubble scale. In particular, J/eH and σ/e H areindependent of time. Note also that J/eH and σ/e H are, respectively, odd and even under λ → − λ .In Figure 1 we plot J/eH and σ/e H as functions of λ , where each curve corresponds to adifferent choice of mass m/H . The solid lines are obtained from the regularized result (2.58). Wealso show dashed lines denoting semiclassical estimates of the current based on the computations inSubsection 2.1, which will be explained later.The plots show that when | eE | (cid:29) H , the conductivity monotonically grows with increasing | E | ,and becomes independent of the mass for a sufficiently large | eE | /H . On the other hand, theconductivity is independent of E under weak electric force | eE | (cid:28) H . In particular, for smallmass m (cid:28) H , the conductivity is strongly enhanced in the weak electric field regime. Let us nowstudy the behavior of J and σ in the limiting regimes of | eE | (cid:29) H and | eE | (cid:28) H , respectively.11 .001 0.1 10 1000 10 - - JeH eEH (a) current e H - - eEH (b) conductivity Figure 1: Induced current J and conductivity σ as a function of the electric field E . The displayedquantities are normalized by the Hubble parameter H and charge e . Each line is for a differentchoice of mass, m/H = 0 . . . | J | ∼ | e | n is shown as dashed lines, with colors representing the choiceof mass. 12 .2.1 Strong Electric Force : | eE | (cid:29) H In the limit of | λ | → ∞ , for a fixed m/H , the third and fourth lines of (2.58) approachRe (cid:26)(cid:90) dr · · · (cid:27) (cid:39) λ , (2.62)and thus largely cancels with the first term in (2.58). Consequently, the cosh term proportionalto λ (note that µ (cid:39) − λ ) dominates the current and yields JH (cid:39) sgn( λ ) eλ π = sgn( E ) 112 π | e | E H , (2.63) σH (cid:39) e | λ | π = 112 π | e E | H . (2.64)Thus one sees that when | eE | is sufficiently large relative to the Hubble rate H , the current J isquadratic in E , and thus σ is linear in E . These features are also seen in the plots, where all curvesconverge as | λ | → ∞ and the behavior becomes independent of the mass.Since the condition (2.21) is satisfied in the regime of | λ | (cid:29)
1, an adiabatic vacuum exists inthe asymptotic future for the ϕ fields. Hence let us try to estimate the current in this regime basedon the semiclassical picture of the created particles carrying the charge. Supposing the particles totravel with velocity v (cid:39)
1, then we can estimate the arising current by J (cid:39) sgn( E ) 2 | e | n = sgn( E ) 2 | e | Γ3 H . (2.65)Here n is the number density of the produced pairs, and we have used (2.34) upon moving to theright hand side. The pair production rate Γ was computed in (2.33). Taking the limit of H → m /eE →
0, one exactly reproduces the result (2.63).In Figure 1(a) the dashed lines show the semiclassical estimate (2.65) using the productionrate Γ (2.33). In the regime | λ | (cid:29)
1, the estimates agree well with the results obtained fromcomputing (cid:104) J z (cid:105) reg . For cases with m (cid:29) H , the suppression of the current at | eE | ∼ m correspondsto the mass suppression of Γ shown in (2.39). Thus we obtain an improved approximation for thecurrent (2.58) at | λ | (cid:29) J (cid:39) sgn( E ) 112 π | e | E H e − πm | eE | . (2.66)On the other hand, when | λ | (cid:28)
1, the semiclassical estimate is seen to break down. Particularly,for light masses m/H < /
2, the condition (2.21) does not hold and thus the expression (2.33) itselfcannot be extended to the | λ | (cid:28) m/H = 0 . m/H = 0 . | λ | (cid:29) J ∼ sgn( E ) | e | E ( t − t ) e − πm | eE | , (2.67)13here t is the initial time when the electric field is switched on. (The situation here is slightlydifferent from that in (2.66) where the electric field always exist. A finite t is introduced because,due to the absence of the Hubble dilution, J blows up in a Minkowski space if the electric fieldexisted from the infinite past t = −∞ . This is why the expression (2.66) diverges as H → T − produces a term that scales as λ , which gives a scalingbehavior of J ∝ E at | λ | (cid:29)
1, contrary to (2.67). Therefore we see that the adiabatic subtractionup to order T − not just removes the divergences, but also produces results with the correct behaviorin the Minkowski limit.Let us also remark that the scaling of J at | λ | (cid:29) J ∝ E , and thus theconductivity approaches a constant at large | E | [7]. As we will see in Section 3, σ being an increasingfunction of E in four-dimensions gives rise to stringent constraints on electromagnetic fields in theearly universe. | eE | (cid:28) H Taking λ → m/H , one can check that the current (2.58) becomes dominated by termslinear in λ , and approximated as JH (cid:39) e E π H (cid:26) ln (cid:18) m H (cid:19) + 16 π µ ( − µ )sin(2 πµ ) − ψ (cid:0) + µ (cid:1) − ψ (cid:0) − µ (cid:1)(cid:27) , (2.68)where µ = 94 − m H . (2.69)The expression gets further simplified when m (cid:29) H as JH (cid:39) π e Em . (2.70)Here the main contributions to the current are given by the terms λ ln( mH ) and Re { (cid:82) dr · · · } in (2.58),which originate from the adiabatic subtraction (2.57) and formal expression (2.46), respectively. Onthe other hand, for m (cid:28) H , the current is approximated by JH (cid:39) π e Em . (2.71)The current in this case mainly arises from the terms in (2.58) that involve cosh, sinh, and Re { (cid:82) dr · · · } ,which are all contributions originating from the formal expression (2.46).Therefore for arbitrary mass, the current and conductivity in the regime | λ | (cid:28) JH ∼ − × e Em , σH ∼ − × e H m . (2.72)Here we see that the conductivity is independent of E , and takes larger values for a smaller massratio m/H .In particular for light mass m (cid:28) H , the plot shows that as one decreases λ , the conductivitygrows until it approaches the constant value (2.72). The strong enhancement of the conductivity14or small mass scalars under weak electric fields are supported by the infrared contributions to thecurrent: This can be seen from the limiting behavior of the Whittaker function as z → W κ,µ ( z ) ∝ z −| Re( µ ) | . (2.73)(Here we note that this expression (2.73) is not valid for an arbitrary set of κ and µ , however we useit for the rough estimation in the following.) Hence one sees that the integrand of (2.44) for (cid:104) J z (cid:105) , inthe limit k →
0, scales as ∝ k −| Re( µ ) | ) . (2.74)The simple power counting estimate indicates that the current spectrum diverges in the infraredlimit when | Re( µ ) | >
1, i.e., e E H + m H < . (2.75)However, it should also be noted that a nonzero eE or m give | Re( µ ) | < , and thus integrating thespectrum (2.74) down to k = 0 yields a finite total current. We also remark that the spectrum ofthe adiabatic subtraction terms in (2.55) are finite in the infrared limit, since Ω k ≥ am > | λ | (cid:28) q k . The nature of the infrared modes being importantshould be contrasted to cases − µ (cid:29) k ∼ aH | µ | always dominate the ϕ population.The constancy of σ under weak electric fields, and its strong enhancement for small mass are alsoseen in a two-dimensional dS space [7]. However there is an important difference worth noting: Whilein four dimensions the conductivity under weak fields scales as σ ∝ m − for all masses, in a two-dimensional dS the scaling σ ∝ m − is only for light masses, and the conductivity is exponentiallysuppressed for m (cid:38) H . In Figure 1 we have plotted curves for scalar masses as low as m/H = 0 .
01. Cases for even smallermasses have similar behaviors under strong/weak electric fields; the conductivity takes the massindependent value (2.64) at | λ | (cid:29)
1, while at | λ | (cid:28) λ ata more or less similar rate, until it approaches the constant value (2.72) set by the mass. However,we remark that the valley of σ in the intermediate regime of λ becomes deeper for smaller m , andthe conductivity can even take negative values at | λ | ∼ m = 0, one sees that the regularized current (2.58) divergesdue to the ln( m/H ) term, which was introduced through the procedure of adiabatic subtraction atthe order T − . This divergence originates from Ω k (2.51) vanishing for m = 0 at( k z + eA z ) + k x + k y = 0 , (2.76)and thus blowing up the adiabatic subtraction terms. Note that (2.76) corresponds to the vanishingof the physical momentum ( p x , p y , p z ) = (cid:18) k x a , k y a , k z + eA z a (cid:19) . (2.77)15hese observations suggest that the adiabatic expansion taking T → ∞ is invalid for zero modesof massless fields, and thus the method of adiabatic regularization may not be applicable for themassless case. (See also discussions in [15, 16].) This issue may be resolved by imposing an infraredcutoff on the momentum integral, by considering that in the finite past the dS expansion started, orthe electric field was switched on.We should also mention that, even if a scalar has a tiny bare mass, a Hubble-induced mass canbe generated [35, 36]. See also discussions on effective mass in dS space in, e.g., [37, 38, 39, 40, 41].The charged scalars need to be protected from mass corrections in order for a dS universe to actuallypossess large conductivity under weak electric fields. In the previous section we studied the production of charged scalars in a fixed background of aconstant electric field and dS expansion. However the backreaction from the produced scalars maybecome relevant, especially when the induced conductivity is huge. Such considerations are im-portant upon discussing the aftermath of the Schwinger process, but can also be used to constrainelectromagnetic fields in a dS universe. In particular, the backreaction from the Schwinger pro-cess can impose severe constraints on cosmological models for generating primordial electromagneticfields during the inflationary dS phase.The possibility of the cosmological generation of magnetic fields during the inflationary epoch hasbeen studied by many authors, e.g. [21, 22, 23, 24, 25, 26, 27, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52],in order to explain the origin of the large-scale magnetic fields in our universe. In such inflationarymagnetogenesis scenarios, the magnetic fields are generically accompanied by the production of evenlarger electric fields. One of the guidelines towards constructing a consistent model has been to keepthe production of electric fields under control in order to avoid the electric fields from dominatingthe energy density of the universe [23, 24, 25, 26, 27]. However, we have seen in the previous sectionthat even if the electric fields do not dominate the universe, they can induce large conductivity inthe universe via the Schwinger process, which may have non-negligible backreaction on the Maxwellfields. Therefore, in this section we analyze the effect of the Schwinger process on magnetogenesisin an inflationary dS spacetime. We will see that, unless charged fields are absent in the action, orare very massive, the Schwinger effect can spoil the process of magnetogenesis. Through discussingmagnetogenesis, we will also see how the induced current backreacts on the background Maxwellfields.
The cosmological enhancement of the electromagnetic fields are realized in inflationary magnetogen-esis scenarios by breaking the conformal symmetry of the Maxwell theory. To make our discussionsconcrete, we focus on a class of models where the conformal symmetry is broken by a time dependent16ffective coupling I on the Maxwell kinetic term, S = (cid:90) d x √− g (cid:26) − I F µν F µν − g µν ( ∂ µ − ieA µ ) ϕ ∗ ( ∂ ν + ieA ν ) ϕ − m ϕ ∗ ϕ (cid:27) . (3.1)The coupling I can be thought of as a function of other degrees of freedom, such as the inflatonfield. As in the previous section, we consider Schwinger process with a charged complex scalar andanalyze its effect on inflationary magnetogenesis. With the effective coupling, the Maxwell equation is modified to ∇ ν (cid:0) I F µν (cid:1) = J µ , (3.2)where the current J µ is shown in (2.42). In the following, we study the dynamics of the Maxwellfields in the Coulomb gauge, ∂ i A i = A τ = 0 . (3.3)Considering a dS background (2.2), and setting I to be homogeneous, the spatial component of themodified Maxwell equation reads A (cid:48)(cid:48) i − ∂ j ∂ j A i + 2 I (cid:48) I A (cid:48) i = a I J i . (3.4)We discuss electromagnetic fields defined in terms of A µ (instead of the normalized ˜ A µ = IA µ ), E µ = u ν F µν , B µ = 12 ε µνσ F νσ , (3.5)because it is A µ that the complex scalars couple to with charge e , and also since we consider thecoupling I to be fixed to unity in the present universe. Here u µ is the 4-velocity of the comovingobserver, and ε µνσ = η µνσγ u γ , where η µνσγ is a totally antisymmetric permutation tensor with η = −√− g . Thus ε ijk is totally antisymmetric with ε xyz = a . The time components B τ and E τ vanish, while the spatial components are E i = − a A (cid:48) i , B i = 1 a ε ijk ∂ j A k . (3.6)The magnitude of the fields are E ≡ E µ E µ = 1 a E i E i = 1 a A (cid:48) i A (cid:48) i , (3.7) B ≡ B µ B µ = 1 a B i B i = 1 a ( ∂ i A j ∂ i A j − ∂ i A j ∂ j A i ) . (3.8) The conformal symmetry can be broken alternatively by a mass term for the photon, m γ A µ A µ . In such models,significant magnetogenesis requires a tachyonic mass, i.e. m γ <
0, which for example can arise from non-minimalcouplings to gravity [21]. However, such theories have been pointed out to have problems including the appearance ofa ghost [24, 53, 54]. One could also imagine cases where the effective coupling I shows up not only in front of the Maxwell kinetic term,but in front of all the terms in the Lagrangian. In such cases, a time varying I can further induce ϕ production inaddition to the Schwinger and gravitational effects, and therefore the backreaction to the Maxwell fields are expectedto become stronger, resulting in even more stringent constraints on magnetogenesis than for (3.1). A τ cannot be taken to zero in the presence of charge, however, since we only use the equations including J µ forobtaining a rough criterion for the backreaction from J µ being non-negligible, we can approximately set A τ = 0 inorder to simplify the calculations. σ = J i E i , (3.9)(here we do not take the sum over i in the right hand side of (3.9)), the Maxwell equation (3.4) isrewritten as A (cid:48)(cid:48) i − ∂ j ∂ j A i + (cid:18) I (cid:48) I + aσI (cid:19) A (cid:48) i = 0 . (3.10) We shall first discuss the idealized situation where any charged fields are absent in the action. Thissubsection will also serve as a brief review of magnetogenesis in I F F scenarios.Let us focus on large-scale magnetic fields and neglect the spatial derivative term in the Maxwellequation (3.10), giving A (cid:48)(cid:48) i + 2 I (cid:48) I A (cid:48) i = 0 . (3.11)The general solution of this equation is A i = C + C (cid:90) dτI , (3.12)where C and C are constants. For instance, if the coupling I decreases in time as I ∝ a − s , with s > , (3.13)then the vector potential possesses a growing mode A i ∝ a s − , (3.14)and thus the electromagnetic fields are enhanced. Hereafter we suppose I to follow the scalingbehavior (3.13) during inflation, and then stays constant after inflation. Focusing on Maxwell fields with a certain wave number k , let us rewrite the magnetic fieldamplitude (3.8) as B ∼ k a A i A i . (3.15)Then, using A (cid:48) i /A i = (2 s − a (cid:48) /a which follows from (3.14), the ratio between the electric andmagnetic amplitudes with wave number k is obtained as (cid:12)(cid:12)(cid:12)(cid:12) EB (cid:12)(cid:12)(cid:12)(cid:12) = (2 s − aH inf k . (3.16)In this section we denote the (nearly) constant Hubble parameter during inflation by the sub-script “inf”. As can be seen from (3.15), the magnetic field after the magnetogenesis phase decays It could also be that the coupling I approaches a constant at some time τ during inflation, and thus magnetogenesisterminates before the end of inflation. In such cases, the constraint on magnetic fields we obtain in (3.29) is modifiedby I end → I ( τ ), and also obtains an additional factor of a ( τ ) /a end in the right hand side, which makes the boundmore stringent. This approximate expression of B is good enough for obtaining the E - B ratio (3.16). More detailed derivationsof (3.16) can be found in the references. B ∝ a − . Hence we can obtain a relation between the magnetic field strength in the presentuniverse and the electric field at the end of inflation as | B | = 12 s − ka H inf a end a | E end | . (3.17)Here the subscript “0” denotes quantities in the present epoch, and “end” at the end of inflation.We suppose the post-inflationary universe to be first dominated by an oscillating inflaton, and thuseffectively matter-dominated until reheating happens, (cid:18) H reh H inf (cid:19) = (cid:18) a end a reh (cid:19) , (3.18)where the subscript “reh” denotes quantities at reheating. After reheating, we consider the entropyto be conserved, i.e. s ∝ a − , and thus obtain a reh a ≈ × − (cid:18) M p H reh (cid:19) / . (3.20)The combination of (3.18) and (3.20) yields the expansion after inflation, a end a ≈ × − (cid:18) H reh H inf (cid:19) / (cid:18) M p H inf (cid:19) / , (3.21)which allows us to rewrite (3.17) as | B | M p ≈ × − s − ka M p (cid:18) H reh H inf (cid:19) / (cid:18) H inf M p (cid:19) / | E end | H . (3.22) Let us now study how the above picture of magnetogenesis is modified under the existence of chargedscalars in the action. From the Maxwell equation (3.10), one can read off the criterion for the inducedcurrent to be negligible as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ (cid:18) I (cid:48) aI (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) I . (3.23)In other words, when the ratio between σ and the rate of magnetogenesis is larger than ∼ I , theprocess of magnetogenesis can be strongly affected by the produced scalars. The σ term in (3.10)is seen to be a friction term for A (cid:48) i , thus a positive and large σ decays away the electric fields andprevents any further magnetogenesis. For the scaling (3.13) under consideration, the criterion (3.23)becomes | σ | H inf (cid:28) sI . (3.24) Upon computing the entropy density at reheating s reh = 2 π g s ∗ ( T reh ) (cid:18) π M p H g ∗ ( T reh ) (cid:19) / , (3.19)we have chosen the relativistic degrees of freedom to take the maximum value allowed in the MSSM, g ∗ = g s ∗ = 228 . g ∗ = g s ∗ = 10 .
75 gives a factor 7instead of 6 in the right hand side. I in the right hand sides can be understood from the fact that, when absorbingthe coupling by ˜ A µ = IA µ , the effective charge of ϕ becomes e/I . Hence a smaller I enhances thebackreaction of the produced ϕ on the Maxwell fields. This provides an explicit example of a problemthat arises when I is tiny, often referred to in the literature as the strong coupling problem [24, 43].However, even if I is never smaller than unity, we will see that the criterion (3.24) severely constrainsinflationary magnetogenesis.The criterion (3.24) can be used to set an upper bound on the magnetic fields that can begenerated during inflation. We evaluate the bound by using the approximate expression (2.66) forthe current under strong electric fields (the validity of using this approximation will shortly bediscussed). With the approximation, the conductivity is expressed as σH inf (cid:39) π | e E | H exp (cid:18) − πm | eE | (cid:19) , (3.25)which is solved for the electric field as | E | H (cid:39) π | e | σH inf exp (cid:26) W (cid:18) e π m H H inf σ (cid:19)(cid:27) . (3.26)Here, W ( x ) is the Lambert W -function which is the solution of W e W = x . For x ≥ W ( x ) is anon-negative and increasing function. Thus we note that | E | /H is an increasing function of σ/H inf when the other parameters are fixed. Moreover, since W ( x ) (cid:39) x for 0 ≤ x (cid:28)
1, the exponentialfactor in (3.26) approaches unity for small m . This limit can also be obtained directly by solvingthe approximation (2.64).From (3.26), the criterion (3.24) is translated into an upper bound on the electric field duringinflation, | E | H (cid:46) π sI | e | exp (cid:26) W (cid:18) e π sI m H (cid:19)(cid:27) , (3.27)which imposes a bound on the present magnetic field amplitude via (3.22), | B | M p (cid:46) − s s − ka M p (cid:18) H reh H inf (cid:19) / (cid:18) H inf M p (cid:19) / I | e | exp (cid:26) W (cid:18) e π sI m H (cid:19)(cid:27) . (3.28)Here, I end denotes the value of the effective coupling at the end of inflation. Note from (3.14) that s should be larger than (and not so close to) 1 / s s − should be of order unity for an efficient magnetogenesis. Further noting H reh ≤ H inf , then the bound can be rewritten as | B | (cid:46) − G (cid:18) ka Mpc (cid:19) (cid:18) H inf M p (cid:19) / (cid:32) √ πα | e | (cid:33) I Q , (3.29)where Q represents the mass dependence, Q = exp (cid:26) W (cid:18) e π sI m H (cid:19)(cid:27) (cid:39) exp (cid:26) W (cid:18) − e πα sI m H (cid:19)(cid:27) . (3.30)20ere we are using the Heaviside-Lorentz units, thus 1 G ≈ × − GeV , and the elementarycharge is √ πα ≈ .
3. We also note 1 Mpc ≈ × eV − . The mass dependent factor Q is agrowing function of m/H inf , with lim m → Q = 1. Moreover, Q is of order unity when the argumentof W is smaller than ∼
1. In other words, the bound (3.29) is independent of the scalar mass if themass ratio satisfies m H (cid:46) παe sI . (3.31)When fixing all the parameters except for H inf , then the magnetic upper bound (3.29) is an increasingfunction of H inf while (3.31) is satisfied, scaling as H / . On the other hand, when (3.31) is violated,the Q factor becomes important and the upper bound turns into a decreasing function of H inf .In Figure 2 we plot the upper bound (3.29) as a function of H inf , for a fixed set of parameters k/a = 1 Mpc, e = 4 πα , m = 0 . I end = 1. The scaling factor s is taken to be of orderunity (its explicit value is unimportant here as varying s by order unity makes little difference in theplot). It is clearly seen that the upper bound switches from a decreasing to an increasing functionof H inf , as the ratio m/H inf decreases and starts to satisfy (3.31). - - - - - - - - - -
10 10 - - H inf [GeV] B [ G ] m/H inf Figure 2: Upper bound (3.29) on the present amplitude of magnetic fields with correlation length k/a = 1 Mpc − from the Schwinger effect constraint. A scalar with charge e = 4 πα and mass m = 0 . I end = 1. The horizontal axes showthe inflation scale H inf (lower) and the ratio m/H inf (upper). In the region where m/H inf is tiny,further constraints may arise due to the strong enhancement of the conductivity at the early stageof magnetogenesis (see the main text for details).As we have seen, the magnetic field bound (3.29) derive from the Schwinger effect constraint onthe electric field (3.27), which can be recast in the form of | eE end | H (cid:46) παe sI Q . (3.32)Hence for e ∼ πα and I end ∼
1, it is evident that the restrictions on magnetogenesis arise fromthe Schwinger effect in the large | λ | regime, as was assumed upon using the approximation (2.66).21ere we remark that a large charge e (cid:29) πα can push the constrained region into the small | λ | regime and thus invalidate the usage of (2.66). However, the conductivity in the | λ | (cid:28) | λ | (cid:29) | λ | , the conductivity σ eventually deviates from the exponential fall so that it cansmoothly connect to the plateau at | λ | (cid:28)
1. Such behavior also enhances σ relative to that predictedby (2.66), and thus gives a magnetic bound stronger than (3.29). The situation becomes more severefor cases with extremely light masses (i.e. m (cid:28) H inf ), where the conductivity is strongly enhancedat | λ | (cid:28)
1. The large conductivity under weak electric fields can affect magnetogenesis at its earlystage, long before the electromagnetic fields grow to values constrained by the bound (3.29) (thoughthe constraint would also depend on the value of I during inflation). Therefore the plot in Figure 2should be considered as a conservative bound for regions where the ratio m/H inf is tiny.We should also comment on the applicability of the results from the previous section, wherewe considered a constant and uniform electric field, on inflationary magnetogenesis where the elec-tromagnetic fields with finite correlation lengths are continuously being produced. Here it shouldbe noted that the typical time scale for the enhancement of the electromagnetic fields is, in thecase (3.14) under consideration, of order the Hubble time H − . Moreover, the electromagnetic fieldsare significantly enhanced after exiting the Hubble horizon, and thus we have given constraints onwave modes that are sufficiently larger than the horizon at the end of the magnetogenesis phase. Onthe other hand, the constraints are mostly due to the Schwinger process in the | eE | (cid:29) H regime,and thus the produced charged scalars typically have wave modes much smaller than the Hubbleradius, cf. (2.31). In this regime, it could also be checked that the ϕ population is always dominatedby the particles newly created within a Hubble time. Thus the length and time scales relevant tothe Schwinger process do not exceed those of the electric fields, validating our procedure of model-ing the electric fields produced during inflation as being constant and uniform upon evaluating theSchwinger effect constraints.In Figure 3 we plot the upper bound on the amplitude of the magnetic fields in the presentuniverse (3.29) as a function of the correlation length. The charge of the complex scalar is set tothe elementary charge e = 4 πα , and the coupling at the end of inflation to I end = 1. The scalingfactor s is chosen to be of order unity; the bound depends sufficiently weakly on s such that itsexplicit value is not important here. For the chosen sets of parameters in the figure, magnetogenesisis constrained by the Schwinger effect in the large | λ | regime where the approximation (2.66) is valid,and thus the magnetic bounds can be fully described by the expression (3.29).Each line represents the upper bound for a different set of the inflation scale H inf and the scalarmass m . The case of a high-scale inflation with H inf = 10 GeV (corresponding to a tensor-to-scalarratio of r (cid:39) .
2, as recently suggested in [55]) is shown as blue lines; the solid line is for m (cid:46) H inf ,and the dashed line for m = 10 H inf . In the former case, the mass dependent factor (3.30) is Q ∼ We could also say that when m (cid:28) H inf , independently of the magnetogenesis mechanism, the inflationary universecannot leave behind arbitrarily small electromagnetic fields; the strong enhancement of the conductivity under weakelectric fields forbids the electromagnetic fields from existing as a stable background. In this sense, a “lower bound”on the electromagnetic fields exists for inflationary magnetogenesis with light charged particles. Q ∼ and thus the bound is relaxed. - - - - - - - - - - a k [Mpc] B [ G ] H i n f = G e V m = . M e V H i n f = G e V H i n f = G e V -ray observations Figure 3: Upper bound on the present amplitude of magnetic fields produced during inflation fromSchwinger effect constraints. The blue and red lines are, respectively, for H inf = 10 GeV and H inf = 10 − GeV, with m (cid:46) H inf (solid lines) and m = 10 H inf (dashed lines). The green lineshows the case for the lowest possible inflation scale H inf = 10 − GeV, with m = 0 . e = 4 πα , and the effective coupling to I end = 1. Thecyan shaded region shows the magnitude of intergalactic magnetic fields suggested by gamma-rayobservations, whose lower bound is from [31].The bounds should be compared with results from recent gamma ray observations, that suggestthe existence of intergalactic magnetic fields of strength | B | (cid:38) − G , (3.33)when the correlation length is of Mpc scales or larger [28, 29, 30, 31, 32, 33, 34]. If the correlationlength λ B is much smaller than a Mpc, the lower bound improves as λ − / B . The suggested magneticfield strength is shown in the figure as the cyan shaded region, where the lower bound is takenfrom [31] (for the case of extended cascade emission). The observational bound has astrophysicaluncertainties (see e.g. [56, 57]) and more detailed work will be required to verify the claim, howeverwe already see from the plot that the Schwinger effect severely constrains the inflationary magneto-genesis scenario from producing such large-scale magnetic fields; the blue solid line shows an upperbound of | B | (cid:46) − G on Mpc scale and beyond. For other observational constraints on magneticfields, see the review [58]. 23ere we should note that, in order to generate magnetic power of (3.33) on Mpc scales fromthe magnetogenesis with the scaling behavior (3.13), then the inflation scale should actually satisfy H inf < − M p ∼ − GeV, otherwise the produced electric fields end up dominating the energydensity of the universe, as derived in [27]. (See also [23, 24, 26].) Thus in the figure we also plotthe magnetic bound for H inf = 10 − GeV as red lines, again with the solid line for m (cid:46) H inf and the dashed line for m = 10 H inf . With such a low inflation scale, the red solid line now gives | B | (cid:46) − G on Mpc scales.As long as m (cid:46) H inf , the bound scales as H / and thus becomes more severe for lower inflationscales. However, the situation is different when the mass is much larger than H inf . We have seenin Figure 2 that, when the mass is sufficiently large such that (3.31) is violated, then the upperbound (3.29) turns into a decreasing function of H inf . Thus we further plot the case of a chargedfield with an electron mass m = 0 . H inf = 10 − GeV,which is the lowest possible scale compatible with Big Bang Nucleosynthesis [59, 60], although itwould require instantaneous reheating and baryogenesis. This extreme case with m/H inf = 5 × is shown as the green line in the plot. Due to the very large mass-Hubble ratio, the conductivity issuppressed and thus relaxes the bound by Q ∼ . In particular, the upper bound on the Mpcscale is | B | (cid:46) − G, which is comparable to the value of the observational lower bound (3.33).In summary, under the existence of charged fields in the action, the Schwinger effect introducesa serious obstacle towards inflationary magnetogenesis. This is manifested in the form of an up-per bound (3.29) on the produced magnetic field, with possible corrections which typically makethe bound more severe, as discussed below (3.32). If for example the charged field has a massof order the Hubble scale or smaller, and carries charge of order the elementary charge, then theSchwinger constraint eliminates the possibility of the discussed inflationary magnetogenesis scenariobeing responsible for producing the extragalactic magnetic fields (3.33) suggested by gamma rayobservations. For cases with extremely light charged fields, i.e. m (cid:28) H inf , the phase of magne-togenesis may not be able to even start, as the induced conductivity under weak electric fields issubstantially enhanced. We stress that, as long as the mass of the charged field is not much largerthan the Hubble scale, the constraint is more severe for lower inflation scales. Hence lowering theenergy scale of inflation (as was considered in [51, 52] to circumvent previous constraints) does notimprove the situation.The Schwinger constraint can be relaxed if all charged fields have tiny charges, or if their massesare much larger than the inflationary Hubble scale. For instance, the running of the charge at highenergies may suppress the charges during inflation. Alternatively, if inflation is driven by the Higgsfield [61], then charged particles in the Standard Model may acquire very large masses while the Higgsfield takes large field values. Another possibility of relaxing the constraint is to have a large valuefor the effective coupling I at the end of the magnetogenesis phase, so that the backreaction from Combining (3.22) with the requirement that the electric field should not dominate the energy density of theuniverse, i.e., ρ E ∼ I E < M p H , (3.34)and further demanding I (cid:38) H inf (cid:46) − M p isnecessary for magnetic fields of | B | (cid:38) − G to be produced on scales of Mpc or larger. I here denotes the relative factor between theMaxwell kinetic term and the coupling term between the vector potential and the charged field. Thuswhen absorbing I into the definition of A µ , then a large I corresponds to a tiny effective charge.) If I is to approach unity in the present universe, then one could imagine a case where I takes a large valueat the end of inflation, and keeps decreasing after inflation. Such a possibility was investigated in [27],where magnetogenesis further continues in the post-inflationary epoch until reheating. Constraintsfrom the Schwinger effect can readily be applied for post-inflationary mechanisms of magnetogenesisas well; due to the low Hubble rate, the Schwinger process is expected to receive strong masssuppression, though we defer a careful study of Schwinger effect in post-inflationary scenarios to afuture work. We should also note that the Schwinger constraint is evaded if the charged fields areabsent from the action during inflation, e.g., it could be that the terms for the charged fields in theaction somehow emerge in the post-inflationary universe. In this work, we have analyzed particle creation by electric and gravitational fields in a four-dimensional dS space. In addition to the usual Bogoliubov computations, we calculated the expec-tation value of the induced current. By directly evaluating the current, we could investigate regimeswhere an adiabatic vacuum does not necessarily exist in the asymptotic future for the charged par-ticles. However, divergences had to be removed from the expectation value of the current operator.To this end, we applied the adiabatic regularization method. We saw that subtracting terms up toquadratic order in the adiabatic expansion removes all infinities, while also yields results that havethe correct behavior in the flat space limit.The expression for the regularized current is presented in (2.58). Under strong electric fields | eE | (cid:29) H , the limiting form of the current is shown in (2.63) (or (2.66)), whose behavior coincideswith that in flat space. In particular, the linear dependence of the conductivity on the electric field,i.e. σ ∝ E , which is inherent in a four-dimensional space, plays an important role upon constrainingelectric fields in the early universe. On the other hand, under weak electric fields | eE | (cid:28) H , theapproximate expression for the induced current is given in (2.72). The conductivity in this regimeis independent of the electric field, and moreover is inversely proportional to the mass squared, i.e. σ ∼ − × e H/m . Thus the dS space acquires a large conductivity under weak electric fieldsfor small masses, i.e. m (cid:28) H . This intriguing phenomenon with small masses is supported by theinfrared modes of the charged scalar, and was also seen to happen in a two-dimensional dS [7]. Formassive particles, the contribution to the current from each wave mode does not grow indefinitelytowards the infrared, however we note that the scaling σ ∝ m − holds for arbitrary mass. Thus evenmassive charged particles can give rise to a non-negligible conductivity under weak electric fields infour-dimensional dS space. This should be contrasted to the case of two-dimensional dS where σ isexponentially suppressed at large masses.We remark that the exactly massless case cannot be handled in the formalism presented in thispaper, as the adiabatic expansion breaks down for the zero modes of the massless field. It should alsobe noted that loop corrections may generate large masses to the charged scalars, and thus avoid thedS universe from obtaining an extremely large conductivity under weak electric fields. In this paper,25e have adopted the method of adiabatic regularization, however it would be important to computethe current with a different regularization or renormalization scheme and compare the results. Weleave this for future work.In the second half of the paper, we applied the above results to the early universe in orderto constrain cosmological scenarios for generating large-scale magnetic fields in our universe. Weshowed that the electric fields generated together with the magnetic fields can induce sufficiently largeconductivity to terminate the phase of magnetogenesis. We have especially focused on inflationarymagnetogenesis models with a modified Maxwell kinetic term I ( t ) F µν F µµ , whose coupling scalesas I ∝ a − s . The main constraints arise from the strong electric field regime | eE | (cid:29) H , wherethe behavior of the Schwinger process is similar to that in flat space. The upper bound from theSchwinger constraint on the produced magnetic amplitude is given in (3.29), and the bounds atvarious length scales are displayed in Figure 3. For instance, if the charged field has a mass of orderthe Hubble scale or smaller, and carries charge of order the elementary charge, then magnetic fieldswith correlation length of Mpc or larger is bounded as | B | (cid:46) − G for all possible inflation scales.Although the explicit bound depends on the masses and charges of the particles, we have shown thatunless charged fields are absent from the Lagrangian during inflation, the Schwinger effect makesit a formidable task for inflationary magnetogenesis to produce the extragalactic magnetic fields of ∼ − G suggested by gamma ray observations.In this paper we have focused on a certain class of inflationary magnetogenesis scenarios, howeverit would be interesting to systematically constrain inflationary magnetogenesis in general from theSchwinger effect. It is also important to constrain non-inflationary mechanisms, such as the scenarioin [27] which generates magnetic fields during the matter-dominated phase prior to reheating, and[62, 63, 64] during phase transitions.Through the discussions on magnetogenesis scenarios, we have seen that the Schwinger effectgives rise to strong constraints on electromagnetic fields under the existence of charged fields in theaction. This, in turn, suggests the exciting possibility of extracting information about the chargedfields in the Lagrangian, from the (non-)detection of primordial magnetic fields in our universe.
Acknowledgements
We would like to thank Jaume Garriga, Shinji Mukohyama, and Tanmay Vachaspati for usefulcomments on a draft. TK is also grateful to Marcelo Alvarez, Tony Chu, Chris Thompson, YukiWatanabe, and Aaron Zimmerman for very helpful discussions. This work was supported by the Nat-ural Science and Engineering Research Council of Canada, the University of Waterloo and PerimeterInstitute for Theoretical Physics. Research at the Perimeter Institute is supported by the Govern-ment of Canada through Industry Canada and by the Province of Ontario through the Ministry ofResearch & Innovation.
A Some Properties of Whittaker Functions
In this appendix we lay out some of the properties of the Whittaker Functions that are useful forthe discussions in Section 2. For more details, see e.g. [65].26he Whittaker functions W κ,µ ( z ) = e − z z + µ U (cid:0) + µ − κ, µ, z (cid:1) ,M κ,µ ( z ) = e − z z + µ M (cid:0) + µ − κ, µ, z (cid:1) , (A.1)defined in terms of Kummer’s confluent hypergeometric functions U and M , are solutions of thedifferential equation d W dz + (cid:26) z (cid:18) − µ (cid:19) + κz − (cid:27) W = 0 . (A.2)Here, M κ,µ ( z ) does not exist when 2 µ = − , − , · · · . The fundamental pairs of solutions of (A.2) areformed by W κ,µ ( z ), W − κ,µ ( e πi z ) (for − π ≤ arg z ≤ π ), or M κ,µ ( z ), M κ, − µ ( z ) (for − π ≤ arg z ≤ π and 2 µ (cid:54) = 0 , ± , ± , · · · ).The functions have the properties( W κ,µ ( z )) ∗ = W κ ∗ ,µ ∗ ( z ∗ ) , ( M κ,µ ( z )) ∗ = M κ ∗ ,µ ∗ ( z ∗ ) , (A.3) W κ,µ ( z ) = W κ, − µ ( z ) , M κ,µ ( ze ± πi ) = ± ie ± µπi M − κ,µ ( z ) , (A.4)and are related through the formula (for 2 µ (cid:54) = 0 , ± , ± , · · · ): W κ,µ ( z ) = Γ( − µ )Γ( − µ − κ ) M κ,µ ( z ) + Γ(2 µ )Γ( + µ − κ ) M κ, − µ ( z ) . (A.5)The Wronskians are W κ,µ ( z ) dW − κ,µ ( e ± πi z ) dz − dW κ,µ ( z ) dz W − κ,µ ( e ± πi z ) = e ∓ κπi ,M κ,µ ( z ) dM κ, − µ ( z ) dz − dM κ,µ ( z ) dz M κ, − µ ( z ) = − µ. (A.6)As | z | → ∞ , the function W κ,µ ( z ) has a limiting form of W κ,µ ( z ) ∼ e − z/ z κ , for | arg z | < π. (A.7)As z →
0, the function M κ,µ ( z ) approaches M κ,µ ( z ) ∼ z µ +1 / . (A.8)We note that, in this paper we take the principal values − π ≤ arg (cid:36) ≤ π for the phase of complexnumbers (cid:36) . B Computation of the Current Before Regularization
In this appendix we perform the three-dimensional momentum integral in (2.44) for obtaining theexpectation value of the current before its divergences are regularized. We follow the calculationalprocedure in [7] for a one-dimensional momentum integral, with some modifications along the way.27he integral under consideration is (cid:90) d k ( k z + eA z ) e iκπ k | W κ,µ ( z ) | = 2 π (cid:90) ∞ dk k (cid:90) − dr (cid:18) kr + λτ (cid:19) e rλπ k | W − irλ,µ (2 kiτ ) | = − πτ lim ξ →∞ (cid:90) ξ dv v (cid:90) − dr ( rv − λ ) e rλπ | W − irλ,µ ( − iv ) | , (B.1)where we introduced real variables: r = k z k , λ = eEH , v = − kτ. (B.2)We have also put a cutoff ξ on the v -integral, which we will take to infinity at the end of thecalculation. Note from the definition (2.18) that µ is either real or purely imaginary, and that itsreal part lies in the range of 0 ≤ | Re( µ ) | ≤ . In the following analyses we further suppose µ (cid:54) = 0 , , , , (B.3)for calculational convenience. The excluded cases can be approached by taking the limits µ → , , , of the final result.Let us rewrite the Whittaker function using the Mellin–Barnes integral representation that isvalid for ± µ − κ (cid:54) = 0 , − , − , . . . , and | arg z | < π (recall that in this paper we take the principalvalues − π ≤ arg (cid:36) ≤ π for the phase of complex numbers): W κ,µ ( z ) = e − z/ πi (cid:90) i ∞− i ∞ Γ( + µ + s )Γ( − µ + s )Γ( − κ − s )Γ( + µ − κ )Γ( − µ − κ ) z − s ds, (B.4)where the contour of integration separates the poles of Γ( + µ + s )Γ( − µ + s ) from those ofΓ( − κ − s ). Also using ( W − irλ,µ ( − iv )) ∗ = W irλ,µ ∗ (2 iv ), then (B.1) can be rewritten as − πτ lim ξ →∞ (cid:90) ξ dv v (cid:90) − dr ( rv − λ ) e rλπ × (cid:90) i ∞− i ∞ ds πi Γ( + µ + s )Γ( − µ + s )Γ( irλ − s )Γ( + µ + irλ )Γ( − µ + irλ ) ( − iv ) − s × (cid:90) i ∞− i ∞ dt πi Γ( + µ ∗ + t )Γ( − µ ∗ + t )Γ( − irλ − t )Γ( + µ ∗ − irλ )Γ( − µ ∗ − irλ ) (2 iv ) − t . (B.5)Note that the µ ∗ ’s in the expression can be converted into µ ’s, since µ ∗ is equal to either µ or − µ .The integration contours of s and t are arbitrary, as long as they separate the poles as discussedbelow (B.4), and run from minus to plus infinity in the imaginary direction. Therefore, we choosethe contours to always satisfy Re( s ) , Re( t ) < . (B.6)Then the v -integral in (B.5) can be carried out as − πτ lim ξ →∞ (cid:90) − dr e rλπ Γ( + µ + irλ )Γ( − µ + irλ )Γ( + µ − irλ )Γ( − µ − irλ ) × (cid:90) i ∞− i ∞ ds πi Γ (cid:0) + µ + s (cid:1) Γ (cid:0) − µ + s (cid:1) Γ( irλ − s ) (cid:90) i ∞− i ∞ dt πi f r,s ( t ) , (B.7)28here f r,s ( t ) = Γ (cid:0) + µ + t (cid:1) Γ (cid:0) − µ + t (cid:1) Γ( − irλ − t ) × e i π ( s − t ) (2 ξ ) − s − t (cid:18) rξ − s − t − λ − s − t (cid:19) . (B.8)For a fixed set of r and s , the function f r,s ( t ) can have singularities at t = − ± µ − n (where n = 0 , , , · · · ), located on the left side of the integration contour of t , and t = − irλ + n, − s, − s ,on the right side of the contour.Upon integrating f r,s ( t ) over t , let us further specify the integration contour of s by requiring − < Re( s ) (B.9)to be always satisfied. We then carry out the t -integral by closing its contour in the right half-plane,without passing through any of the poles. The added integration contour of t does not contributeto the result, since an integral of f r,s ( t ) over a finite path along the real direction vanishes atIm( t ) → ±∞ , and also because any integral in the region Re( t ) > ξ → ∞ due to the condition (B.9). The residues of f r,s ( t ) inside the closed contour also vanish as ξ → ∞ ,except for those at the simple poles: t = − irλ, − irλ + 1 , − irλ + 2 , − irλ + 3 , − s, − s. (B.10)Among the six poles, the ones at t = − irλ, · · · , − irλ +3 give residues that have explicit ξ -dependence,while the residues at t = 2 − s, − s are independent of ξ . Instead of showing the full expression, inorder to reduce clutter we schematically writelim ξ →∞ (cid:90) dt πi f r,s ( t ) = lim ξ →∞ O ( ξ − s + irλ +3 , . . . , ξ − s + irλ ) + O ( ξ ) . (B.11)The s -integral in (B.7) with the ξ -dependent terms in (B.11) can be carried out similarly to the t -integral; closing the contour in the right half-plane, the only residues that survive as ξ → ∞ arethose at s = irλ, irλ + 1 , irλ + 2 , irλ + 3 . (B.12)These poles are not necessarily simple poles, and thus gives an expression that involves digammafunctions ψ ( z ) = Γ (cid:48) ( z ) / Γ( z ),lim ξ →∞ (cid:90) i ∞− i ∞ ds πi Γ (cid:0) + µ + s (cid:1) Γ (cid:0) − µ + s (cid:1) Γ( irλ − s ) O ( ξ − s + irλ +3 , . . . , ξ − s + irλ )= e − rλπ Γ (cid:0) + µ + irλ (cid:1) Γ (cid:0) − µ + irλ (cid:1) Γ (cid:0) + µ − irλ (cid:1) Γ (cid:0) − µ − irλ (cid:1) × lim ξ →∞ (cid:34) r ξ − λ − r ) ξ − r (cid:8) − µ + 4(2 − r ) λ (cid:9) ξ + λ (cid:8) − µ + ( − − λ + 12 µ ) r + 20 λ r (cid:9) × (cid:8) ln(2 ξ ) − ψ (cid:0) + irλ − µ (cid:1) − ψ (cid:0) + irλ + µ (cid:1)(cid:9) + · · · (cid:35) . (B.13)29n the final line, we have abbreviated terms that are independent of ξ by dots.On the other hand, the s -integral of the ξ -independent terms in (B.11) can be written as, (cid:90) i ∞− i ∞ ds πi Γ (cid:0) + µ + s (cid:1) Γ (cid:0) − µ + s (cid:1) Γ( irλ − s ) O ( ξ )= (cid:90) i ∞− i ∞ ds πi e iπs cos { π ( µ + s ) } cos { π ( − µ + s ) } sin { π ( s − irλ ) }× (cid:26) d r s − irλ + g r ( s ) − g r ( s − (cid:27) . (B.14)Here, g r ( s ) is a function of the form g r ( s ) = c r, − s − irλ − c r, − s − irλ − c r, s − irλ + c r, s + c r, s + c r, s , (B.15)and d r , c r, − , . . . , c r, are independent of s . In order to integrate the term with d r , let us temporarilyconsider integrating the modified function F p ( s ) = e iπs cos { π ( µ + s ) } cos { π ( − µ + s ) } sin { π ( s − irλ ) } s − irλ ) p , (B.16)where the power p is a constant that satisfies p >
1. Closing the contour path of s on the lefthalf-plane with a semicircle of infinite radius (here consider an arc that does not pass through anyof the poles, and taking the infinite radius limit in a discontinuous manner), one can check that theintegral of F p ( s ) along the arc vanishes due to p > s explained below (B.4), andat (B.6), (B.9), we further demand the contour to pass to the left of s = ± µ, ± µ . Such a pathalways exists for µ satisfying (B.3). Then the integral of F p ( s ) can be obtained by summing up itsinfinite series of residues at the simple poles: s = − ± µ − n, irλ − − n, (B.17)where n = 0 , , , . . . . Then we take the limit p → d r term in (B.14), which can be checked to take the formlim p → (cid:90) i ∞− i ∞ ds πi F p ( s ) = − γe − πrλ π cos { π ( µ + irλ ) } cos { π ( µ − irλ ) }− iπ sin(2 πµ ) (cid:34) e − iπµ ψ ( + µ + irλ )cos { π ( µ + irλ ) } − e iπµ ψ ( − µ + irλ )cos { π ( µ − irλ ) } (cid:35) . (B.18)Here, γ is Euler’s constant.As for integrating g r ( s ) − g r ( s −
1) in (B.14), we shift the variable for g r ( s −
1) by s → s + 1,giving (cid:18)(cid:90) i ∞− i ∞ − (cid:90) i ∞− − i ∞− (cid:19) ds πi e iπs g r ( s )cos { π ( µ + s ) } cos { π ( − µ + s ) } sin { π ( s − irλ ) } . (B.19)This can be evaluated by computing the residues of poles in the region sandwiched by the originalintegration contour and the shifted one (the choice of the contour was discussed above (B.17)), whichare simple poles at s = irλ − , − ± µ. (B.20)30umming up the contributions from (B.13), (B.18), (B.19), and integrating over r , we arrive atthe final result: (cid:90) d k ( k z + eA z ) e iκπ k | W κ,µ ( z ) | = πτ lim ξ →∞ (cid:34) λ ξ + λ ξ ) − λ − iπλ µ λ λ
15+ 45 + 4 π ( − λ + 2 µ )12 π µ cosh(2 πλ ) λ sin(2 πµ ) −
45 + 8 π ( − λ + µ )24 π µ sinh(2 πλ ) λ sin(2 πµ )+ (cid:90) − dr iλ