Adiabatic regularization for Dirac fields in time-varying electric backgrounds

The adiabatic regularization method was originally proposed by Parker and Fulling to renormalize the energy-momentum tensor of scalar fields in expanding universes. It can be extended to renormalize the electric current induced by quantized scalar fields in a time-varying electric background. In this work we further extend the method to deal with Dirac fields. This requires a self-consistent ansatz for the adiabatic expansion, in presence of a prescribed time-dependent electric field, which is different from the conventional WKB-type expansion used for scalar fields. Our proposal is consistent, in the massless limit, with the conformal anomaly. We give the renormalized expression of the electric current and analyze, using numerical and analytical tools, the pair production induced by a Sauter-type electric pulse. We also analyze the scaling properties of the current for a large field strength.


I. INTRODUCTION
The landmark work of Heisenberg and Euler [1], motivated by earlier work of Sauter [2], stablished the instability of the quantum vacuum under the influence of a prescribed (slowly varying) electric field. If the field is sufficiently strong, real electron-positron pairs can be created. This result was re-obtained by Schwinger in the modern language of quantum electrodynamics by finding a positive imaginary contribution to the effective action W . The quantity e −2Im W represents then the probability that no actual pair creation occurs during the history of the field [3].
The quantum mechanism driving the spontaneous creation of particles by a gravitational field was discovered by Parker in the early sixties by studying quantized fields in an expanding universe. The crucial fact is as follows [4]: creation and annihilation operators evolve, under the influence of the expansion of the universe (or a generic time-varying gravitational field), into a superposition of creation and annihilation operators. During a cosmic period when the expansion factor is almost constant one can interpret the effect of the gravitational field on the particle number and unambiguously establish the spontaneous creation of real particles by the evolving gravitational field. Major applications of this remarkable phenomena occurs in the very early universe [5,6] and in the vicinity of a collapsing star forming a black hole [7]. These pioneer works on particle creation lunched the theory of quantum fields in curved spacetime, as a first step to merge gravity and quantum mechanics within a self-consistent and successful framework [8][9][10][11]. The underlying machinery was also employed to study time-varying electromagnetic fields [12,13]. In the limit of a slowly varying electric field the Schwinger result can be recovered.
In the gravitational scenario, the most relevant physical observable is the energy-momentum tensor. Its vacuum expectation value T µν possesses ultraviolet (UV) divergences and has to be regularized and renormalized. In the seventies many methods were proposed to this end, as explained in the monographs [8,10,11]. For homogeneous, time-dependent spacetimes a generic expression for T µν was obtained for scalar fields within the so-called adiabatic regularization scheme [14]. The adiabatic method uses a mode by mode subtraction process, naturally suggested by the definition of a single-particle state in an expanding universe, and in such a way that preserves the basic symmetries of the theory. This method is widely used in cosmology and it turns out to be very efficient to implement numerical computations. It has been recently extended to spin-1/2 fields [15,16].
As mentioned above, the analysis of particle creation by time-varying electric fields can be carried out using the techniques first proposed to treat curved backgrounds. The electromagnetic field is considered as an external, unquan-tized background, while the created particles are excitations of the quantized matter field. From the experimental side, this particle production effect is also of special interest since it may not be far from being experimentally detected in high intensity lasers [17], and in beam-beam collisions [18]. This effect is also very important in astrophysical [19,20] and cosmological scenarios [21][22][23], and in non-equilibrium processes induced by strong fields [24]. In this context, the most important physical local expectation value is the electric current j µ , which also possesses ultraviolet divergences and has to be renormalized in a proper way. Recent discussions on foundational issues related to the particle number density of the created particles, adiabatic invariance, and unitary evolution can be seen in [25][26][27].
Due to the similarities with the gravitational case, it is a good strategy to re-adapt the adiabatic regularization scheme to the case in which the external background is an electric field. This proposal was successfully done for fourdimensional scalar fields and for fermionic fields in two dimensions [27][28][29][30][31]. Other renormalization methods have been generalized to incorporate an electromagnetic background, as for instance the Hadamard point-splitting method for complex scalar fields [32,33]. The aim of this work is to further extend the adiabatic regularization/renormalization method to Dirac fields in presence of an electric field background in four spacetime dimensions. This extension requires a self-consistent ansatz for the adiabatic expansion of the field modes. We give a proper ansatz, which cannot be fitted within the conventional WKB-type expansion used for scalar fields [8,10,11]. This extension of the adiabatic method is in agreement with the trace anomaly. We carry out the adiabatic renormalization and provide a general expression for the renormalized electric current. We illustrate the power of the method by studying with detail a Sauter-type electric pulse.
The paper is organized as follows. In Section II we introduce the basic ingredients of our ansatz to construct the adiabatic expansion of the four-dimensional fermionic modes coupled to a prescribed time-dependent electric field. Section III is devoted to explain the details of the adiabatic renormalization procedure in this context. In particular, we give a generic and explicit expression of the renormalized electric current. We also test the consistency of the method and discuss some intrinsic renormalization ambiguities. In Section IV we study the particular case in which the background field is a Sauter-type electric pulse. We analyze the particle production phenomena in terms of the renormalized electric current. We also discuss the scaling properties of the created current. In Section V we state our main conclusions. Our work is complemented with a series of appendices where we give technical details. We also discuss in Appendix C the connection between the adiabatic method and the Hadamard renormalization scheme for charged scalar fields.

II. MODE EQUATIONS, ANSATZ AND ADIABATIC EXPANSION
Let us consider a massive 4-dimensional spinor field ψ interacting with a prescribed electric field. The corresponding Dirac equation reads where D µ ≡ ∂ µ − iqA µ and γ µ are the Dirac matrices satisfying the anticommutation relations {γ µ , γ ν } = 2η µν . We consider ψ as a quantized Dirac field, while the electromagnetic field is assumed to be a classical and spatially homogeneous field E(t) = (0, 0, E(t)). It is very convenient to choose a gauge such that only the z-component of the vector potential is nonvanishing: To prepare things to propose a consistent ansatz for the adiabatic expansion of the field modes it is very important to transform the Dirac field as ψ = U ψ, where U is the unitary operator U = 1 This transformation will allow us to express the Dirac field in terms of only two time-dependent functions (see (4)). The field ψ obeys the Dirac equation for the transformed matrices γ µ = U γ µ U † , namely: γ 0 = γ 3 γ 0 , γ 1 = −γ 3 γ 1 , γ 2 = −γ 3 γ 2 , γ 3 = −γ 3 . Substituting them in the Dirac equation we easily get Expanding the field in Fourier modes, ψ (t, x) = d 3 k (2π) 3/2 ψ k (t)e i k x , we obtain the following equation where k ≡ (k 1 , k 2 , k 3 ). The form of the above equation allows us to re-express the field in terms of two-component spinors as follows where η λ with λ = ±1 form an orthonormal basis of two-spinors (η † λ η λ = δ λ,λ ) verifying k1σ 1 +k2σ 2 +mσ 3 √ k 2 1 +k 2 2 +m 2 η λ = λη λ . Their explicit expressions are where κ ≡ k 2 1 + k 2 2 + m 2 . Substituting (4) in (3) and using the Dirac representation for the matrices γ µ , one obtains the following differential equations for the functions h I k and h II These equations are exactly the same as those obtained in the two-dimensional case [28], where κ plays here the role of the mass. With the solutions of these equations we can construct the u-type field modes (assumed to be of positive frequency at early times) as follows Similarly, one can construct the orthogonal v-type field modes (of negative frequency at early times) as The normalization conditions for this set of spinors, is the Dirac inner product, are ensured with the normalization condition which will be preserved on time. With this set of basic spinor solutions one can construct the Fourier expansion of the Dirac field operator where B kλ and D kλ are the annihilation operators for particles and antiparticles respectively. The normalization condition (10) guaranties the usual anticommutation relations for these operators: , and all other combinations are 0.

A. Adiabatic expansion
Armed with the above results we can determine a consistent adiabatic expansion of the four dimensional Dirac field modes interacting with the prescribed electric background. Based on the two dimensional expansion given in [28], and taking into account that the positive-frequency solution with vanishing electric field, in the representation associated to ψ , is given by with ω = k 2 3 + κ 2 , we propose the following ansatz for the field modes: where the complex functions F (t) and G(t) and the real function Ω(t) are expanded adiabatically Here, Ω (n) , F (n) and G (n) are functions of adiabatic order n. The adiabatic order of a given function will be determined by its dependence on the potential vector A(t) and its derivatives. In order to recover at leading order the exact solution with vanishing electric field A(t) = 0 we demand F (0) = G (0) = 1 and ω (0) = ω. With this condition we are implicitly fixing the adiabatic order of the potential vector A(t) to 1, hence,Ȧ(t) and A(t) 2 will be of order 2, A(t), A(t)Ȧ(t) and A(t) 3 of order three and so on. For a detailed discussion on the adiabatic order assignment see [29].
Plugging the ansatz (14) in the mode equations (6) and (7) and also in the normalization condition (10) we get a system of equations for the functions F (t), G(t) and Ω(t) In order to obtain the expressions of the adiabatic terms ω (n) , F (n) and G (n) , we introduce the expansion (15) into Eqs. (16), (17) and (18) and solve them recursively, order by order. Note that G(k 3 , qA) satisfies the same equations as F (−k 3 , −qA), hence we take G(k 3 , qA) = F (−k 3 , −qA). The system can be solved algebraically by iteration and the general solution is given by where we have parametrized F and G in terms of real functions as F = F x + iF y and G = G x + iG y . Note that there is an ambiguity in the imaginary part (21). However, it disappears when computing physical observables. Further discussions on this issue are given in [16]. For simplicity we choose With the initial conditions F y = 0 and ω (0) = ω and by fixing the ambiguity according to (22), the solutions for the adiabatic functions F (n) , G (n) and ω (n) are univocally determined. In Apendix A we give the four first terms of the adiabatic expansion.

III. ADIABATIC REGULARIZATION/RENORMALIZATION
In this section we will carry out the detailed renormalization of the vacuum expectation value of the electric current j µ = −q ψ γ µ ψ , which constitutes the most important physical quantity in the context of strong electrodynamics [34]. The only non-vanishing component of the electric current is the one parallel to the electric field. With the results of Section II A we can obtain the formal expression of the z-component of the mean electric current where k ⊥ = k 2 1 + k 2 2 . This expression is UV divergent and we have to renormalize it. The current has scaling dimension 3, meaning that the divergences could appear up to third adiabatic order, so we have to perform adiabatic subtractions until and including the third order (note that the energy-momentum tensor requires adiabatic subtractions of order 4) [8]. Therefore, the renormalized form of the electric current is with . These subtraction terms contain all the divergences of the electric current, giving us a finite and meaningful result for j 3 ren . The other components give a vanishing result. After computing the subtraction terms, we finally obtain

A. Conformal Anomaly
An important test of any proposed renormalization method is the necessary agreement with the conformal anomaly. Here we compute the trace anomaly with our proposed extended adiabatic method. The trace of the energy-momentum tensor is proportional to the mass of the field T µ µ = m ψ ψ . Although the two point function has to be renormalized until the third adiabatic order, the trace of the energy momentum tensor must be regularized up to fourth order, i.e.
In the massless limit the first term vanishes, so the anomaly should appear in the subtractions of adiabatic order 4, that is The vacuum expectation value of the two-point function ψ ψ is given by By using the adiabatic regularization method, one can find the 4th order subtraction terms. Hence, in the massless limit we get One can easily re-write this result in a covariant way, obtaining the result It fully agrees with the well-known result for the trace anomaly induced by an electromagnetic field for a Dirac field [35].

B. Introduction of a mass scale and renormalization ambiguities
A crucial point in the adiabatic regularization method is to fix the leading order of the adiabatic expansion, namely ω (0) . It seems very natural to define it as ω (0) ≡ ω = k 2 + m 2 , as we did in Section II A. However, there exist an inherent ambiguity in the method [36]. It is possible to choose a slightly different expression for the leading term ω (0) ≡ ω µ = k 2 + µ 2 , where µ corresponds to an arbitrary mass scale. In order to obtain the new adiabatic subtractions with this new choice of the leading order, one has to rewrite the mode equations as where σ = κ − κ µ ≡ k 2 1 + k 2 2 + m 2 − k 2 1 + k 2 2 + µ 2 is assumed of adiabatic order 1. Note that we recover the original adiabatic subtraction method by choosing µ = m, and hence σ = 0.
In this context, the ansatz of the adiabatic expansion will take the form where the functions F µ (t), G µ (t) and Ω µ (t) are expanded adiabatically as in (15). In order to recover at order 0 the limit of vanishing electric field (and also the limit σ → 0, since σ is now assumed of adiabatic order 1) we demand as initial conditions F  (10), expanding the functions F µ (t), G µ (t) and Ω µ (t) adiabatically, and finally, solving them recursively, order by order. In Appendix B we give the details of the computation and also the expression of the adiabatic renormalization subtractions for the electric current. We remark that the introduction of a mass scale µ causes an unavoidable ambiguity in the renormalization procedure: it allows us to perform different adiabatic subtractions to render finite the physical observables, depending on the scale µ we choose. For instance, concerning the renormalized current ψ γ ν ψ one can compare it at two different scales. Using the results given in the Appendix B we easily obtain ψ γ ν ψ ren (µ) − ψ γ ν ψ ren (µ 0 ) = q 12π 2 ln This ambiguity can be absorbed in the renormalization of the coupling constant. To this end it is convenient to scale the field asÃ ν ≡ qA ν and rewrite the semi-classical Maxwell equations as The above relation for the current (33), re-expressed in terms ofF αβ , translates into the well-known shift: , obtained within perturbative QED using minimal subtraction in dimensional regularization [37]. The renormalized current given in (25) should be understood as defined at the natural scale of the problem, defined by the physical mass of the charged field, i.e., µ = m and hence q ≡ q(m).

IV. PHYSICAL APPLICATION: THE SAUTER ELECTRIC PULSE
As mentioned in the introduction, one of the main advantages of the adiabatic renormalization method is its proficiency to perform numerical computations and analytical approximations. We will devote this section to study the properties of the renormalized expression of the current (25) for the case of a pulsed electric field.
Let us consider the well-known Sauter-type pulse E(t) = E 0 cosh −2 (t/τ ) with τ > 0, and its corresponding potential A(t) = −E 0 τ tanh (t/τ ), which is bounded at early and late times, A(±∞) = ∓E 0 τ . This kind of pulse produces a number of particles, and then also a current, which tends to be constant when t → ∞. In Fig.1 we represent the evolution of the current induced by this pulse for different values of E 0 and τ . These figures have been obtained by solving numerically the differential equations for the modes and integrating the expression of the renormalized current (25). In figure (a) the field strength is fixed (E0 = 2Ec), where Ec = m 2 /q is the critical electric field (or Schwinger limit), that is the scale above which the electric field can produce particles. In figure (b) the width of the pulse is fixed (τ = 1/m). We have used dimensionless variables, in terms of the mass and the charge.

A. Late times behavior of the electric current
We can obtain an expression of the current at late times for an electric background that vanishes at early and late times. Let us consider a pulse such that in the early and late time limits the potential is bounded as A(−∞) = −A 0 , A(∞) = A 0 , and its derivatives vanish. From equations (6) and (7), one can see that at late times t → +∞ the modes behave as [27] h where ω in/out = (k 3 ∓ qA 0 ) 2 + κ 2 , and α k and β k are the usual Bogoliubov coefficients satisfying the relation |α k | 2 + |β k | 2 = 1, that ensures the normalization condition (10). The coefficient |β k | 2 gives the density number of created particles at any value of k.
The renormalized electric current at late times induced by an electric pulse in terms of the coefficient |β k | 2 can be obtained by introducing the expression of the modes at late times (35) in the expression of the current (25). We obtain, for large t, In Appendix D we prove that the second integral of this expression vanishes, so the current at late times is given by the simple expression As expected, the final current is related to the number density of particles. The analytic expression of |β k | 2 depends on the form of the background.

B. Scaling behavior for large field strength
It is interesting to study the behavior of the current in the limit of large field strength. To this end, we consider again the example of the Sauter pulse, for which the coefficient |β k | 2 is given by (see [27] for more details) Plugging it into (37) we can obtain the current at late times induced by the pulse. As a test, one can compare the results given by (37) with the ones given by the exact expression (25) for large t, which are represented in Fig.1.
For this pulse, assuming qE 0 > 0, the large field strength limit corresponds to qE 0 0. A numerical analysis of the expression (38) shows that the relevant values of κ and k 3 are of the order of √ qE 0 and qE 0 τ , respectively. Therefore, in order to study properly the limit of large E 0 , it is convenient to introduce the following set of dimensionless variables and study the limit x → ∞ maintainingk 3 andκ constant. Then, we rewrite |β k | 2 as whereω in/out (x) = x 2 (k 3 ± 1) 2 + xκ 2 . In the limit x → ∞ the above expression for |β k | 2 is independent of x, and it is given by Substituting the expression (41) into (37) and taking into account that k3−qE0τ ωout ∼ −1 for large E 0 , we obtain the behavior of the current at late times created by a high intensity pulse Assuming now that qE 0 >> m 2 , the above integral (42) can be done exactly and we finally obtain which is the predicted expression of the current in the limit of large field strength E 0 . We can also obtain the total number density of created quanta for the Sauter pulse in this limit It is interesting to compare the result (43) with the one obtained for a scalar field. The coefficient |β k | 2 in this case has a different expression, but it tends to the same limit for large E 0 (41). Therefore the scaling behavior of the current at late times ( j 3 scalar ren ∼ 1 3π 3 q 3 E 2 0 τ ) will be the same as in the fermionic case, except for the factor 2, on account of the absence of the spin degree of freedom.
For completeness, it is worth to see how the above results can also serve to describe the Schwinger limit, i.e, a constant electric field. Note that the expression (41) has been obtained for the limit E 0 τ 2 >> 0, so it should also be valid for the limit of large τ , keeping E 0 constant, which describes a pulse with a large width. Bringing this limit to the extreme case τ → ∞, we get |β k | 2 ∼ exp (−π k 2 ⊥ +m 2 qE0 ), which is the well-known expression for the beta coefficients of a constant electric field [12] leading to the Schwinger formula for the vacuum persistence amplitude.

V. CONCLUSIONS
In this work we have extended the adiabatic regularization method for 4-dimensional Dirac fields interacting with a time-varying electric background. Our proposal has required to introduce a non-trivial ansatz, Eq. (14), to generate a self-consistent adiabatic expansion of the fermionic modes. The given expansion turns out to be different from the WKB-type expansion used for scalar fields. With this extension we have obtained a well-defined prediction, Eq. (25), for the renormalized electric current induced by the created particles. Our proposal is consistent, in the massless limit, with the conformal anomaly. In parallel we have also explored the physical consequences of the introduction of an arbitrary mass scale on the adiabatic regularization scheme, finding consistency with the behavior of the effective scaling of the electric coupling constant. To illustrate the power of the method we have analyzed the pair production phenomenon in the particular case of a Sauter-type electric pulse. In particular, we have obtained the scaling behavior of the current in the strong field regime (Eq. (43)). In this appendix we give the explicit expressions of the adiabatic expansion of the fermionic field modes up to and including the fourth adiabatic order. We remind that G (n) (k 3 , qA) = F (n) (−k 3 , −qA).
Order 0 Order 1 Order 2 Order 4 where ω (n) /F (n) /G (n) (ω µ , κ µ , F µ , G µ ) are given by the expressions (19), (20) and (21) with the changes (ω, κ, F, G) → (ω µ , κ µ , F µ , G µ ). Note again that G µ (k 3 , qA) satisfies the same equations than F µ (−k 3 , −qA), and hence G (n) We also find an ambiguity in the imaginary part (B3). For simplicity we choose With the initial conditions (F µ ) The renormalized electric current for an arbitrary mass scale is given by with Appendix C: Matching Hadamard coefficients with scalar adiabatic regularization In this appendix, we relate the adiabatic regularization method with Hadarmard renormalization for charged scalar fields. To this end, we firstly review the adiabatic method for this case.
Adiabatic regularization for a charged scalar interacting with an electric field Let us consider a charged 4-dimensional scalar field interacting with a classical, homogeneous, time-dependent electric background E = (0, 0, E(t)) with potential vector A µ = (0, 0, 0, −A(t)). The Klein Gordon equation reads where D µ φ = (∂ µ + iqA µ )φ. Since the potential vector A µ is homogeneous, one can expand the scalar field in modes Once we have obtained the mode equation (C2), we can make an adiabatic expansion of the scalar field modes. To this end, we propose the usual WKB ansatz where W k can be expanded adiabatically as Ω k = ∞ n=0 ω (n) . The leading term (of zero adiabatic order) can be naturally taken as ω (0) = ω = k 2 + m 2 , corresponding to de limit of vanishing electric field. Next to leading order terms can be obtained recursively from (C2). The adiabatic expansion allows us to regularize the observables performing adiabatic subtractions. Since the first terms of the adiabatic expansion capture all potential divergences, one can subtract them, obtaining finite and meaningful results. With this method, we obtain the following vaccum expectation value of the two point function: . For the electric current, defined as j µ = iq[φ † D µ φ − (D µ φ) † φ], we get with j 3 (0−3) The choice of the leading term ω (0) is not completely fixed, and one can make a more general choice defining where µ is an arbitrary mass scale. With this new choice the adiabatic expansion can be recalculated, giving us slightly different subtraction terms. An exhaustive analysis of this ambiguity can be found in [36].
The ambiguity on the subtractions, leads to an ambiguity on the physical observables. For the two point function the ambiguity manifests as where α = 1 2(2π) 2 , and for the electric current we find j 3 ren (µ) = j 3 ren (µ 0 ) − Re-writing the equation above in a covariant way, we get j ν ren (µ) = j ν ren (µ 0 ) −

Matching with Hadamard renormalization
We can compare the results summarized above with the results given by Hadamard renormalization, particularizing for the case in which A µ = (0, 0, 0, −A(t)). Adopting the notation given in [32], the expectation value of the two point function can be expressed as and the electric current is given by property 1 −1 dx Pol a (x)P b (x) = 0 for a < b, where P ol a (x) is a polynomial of order a, we get that the integrals in (D4) vanish for n ≥ 3. This last property can be easily proven taking into account that P n (x) form a basis, and any function can be expanded as f (x) = ∞ b=0 c b P b (x) where c b = (b + 1/2) 1 −1 dx f (x)P b (x), and if the function is a polynomial f (x) = Pol a (x), for consistency c b = 0 for any b > a. Therefore, for all values of n involved in (D3) the integral vanishes, and then I = 0, as we wanted to prove.