A few more comments on secularly growing loop corrections in strong electric fields

We extend the observations of our previous paper JHEP 1409, 071 (2014) [arXiv:1405.5285]. In particular, we show that the secular growth of the loop corrections to the two--point correlation functions is gauge independent: we observe the same growth in the case of the static gauge for the constant background electric field. Furthermore we solve the kinetic equation describing photon production from the background fields, which was derived in our previous paper and allows one to sum up leading secularly growing corrections from all loops. Finally, we show that in the constant electric field background the one--loop correction to the current of the produced pairs is not zero: it also grows with time and violates time translational and reversal invariance of QED on the constant electric field background.


I. INTRODUCTION
Schwinger's pair creation [1] is a well studied phenomenon. However, in our recent paper [2] we show that in QED on strong electric field backgrounds there are loop corrections which grow with time 1 . We use Schwinger-Keldysh diagrammatic technique and consider a constant electric field, E z = const, and electric pulse, E z (t) ∝ 1 cosh 2 (t/T ) . We show that after a long enough evolution in a constant field background (or as T → ∞ for the case of the pulse), loop corrections become of the order of the tree-level contribution. This effect is overlooked in the standard literature on the subject (see e.g. [4] - [31]). At the same time it is also a quite general phenomenon for other strong background fields: See [32], [33], [34], [35], [36], [37], [38], [39] for the same kind of effects in de Sitter space and [40] for a review. Furthermore, such a situation is quite a well known phenomenon in non-stationary condensed matter theory [41], [42], if one does not take into account the time variation of the level populations.
The growth of the loop corrections to the two-point correlation functions has bright physical consequences. In particular in [2], we observe that particle number density, α + µp α µ p , which is an element of the photon's Keldysh propagator, grows with time even if at the initial state it was zero. Thus, there is photon production together with charged particles from the background electric field. (If interactions between quantum charged and gauge fields are turned on, photons are produced by the background field together with the charged particles rather than by accelerating products of the pair creation, i.e. photons are produced even if the density of the charged pairs is zero.) This is true for the both types of electric backgrounds under consideration -constant and pulse. Furthermore, in [2] we show that the photon's retarded and advanced propagators and all propagator's of the charged particles do not receive such secularly growing corrections at the first loop. Also vertexes do not receive corrections that grow with time. These observations allowed us to simplify the system of the Dyson-Schwinger equations to take into account leading corrections. Along these lines we derive in [2] the kinetic equation for the photon production in the strong field backgrounds. The solution of this equation allows one to sum up leading secularly growing corrections from all loops.
All these observations in [2] have been made in the A µ = (0, 0, 0, A 3 = Et) gauge in the case of the constant field background. One of the goals of the present paper is to show that our result is gauge independent, i.e. we would like to repeat the calculation in the A µ = (A 0 = −Ez, 0, 0, 0) gauge. The point is that the observations of [2] are based on the fact that there is no energy conservation in time dependent backgrounds. Hence, it may seem unclear what the reason for the same phenomenon in the static gauge under consideration is, once there is energy conservation for the single particle problem. In this note we clarify this point.
Another goal of this note is to solve the aforementioned kinetic equation for photons. And finally we would like to see the impact of these effects on the current of the produced charged particles. The point is that at tree-level this current is zero in the constant electric field background [43], [44], because of the invariance of QED under the time translational and reversal invariance on the eternally and everywhere constant field background. We show that at the loop order these symmetries are broken and the current receives non-zero contributions that grow with time.

II. SETUP OF THE PROBLEM
We consider, here, a massive scalar field coupled to an electromagnetic field in (3+1) dimensions: where D µ = ∂ µ −ieA µ . We divide the full gauge potential into two pieces A µ = A cl µ +a µ -classical, A cl µ , and quantum, a µ , parts. Throughout this paper, we denote the external gauge-potential A cl µ as A µ . If not otherwise stated, in this note we study the constant field background in static gauge, where A 0 (z) = −Ez and A = 0. The quantization of the gauge field is straightforward. One just has to choose a convenient gauge for a µ . Below we choose Feynman gauge. For the charged scalars the situation is not so transparent because we use exact harmonics in the background field rather than plane waves. So we give here a few comments on how to quantize the theory in such a situation.
Introducing the following notationsk = (k 0 , k 1 , k 2 ), k ⊥ = (k 1 , k 2 ) and d 3k = dk 0 d 2 k ⊥ , we expand the charged scalar fields in harmonics as follows: The function f k ⊥ z − k 0 eE satisfies the following differential equation: Solutions of (2) are related via a Fourier transformation, which we give below, to those of: This equation defines harmonic functions in the temporal, A 3 = Et, gauge (see e.g. [2]). The Fourier relation in question can be seen after the change of variables k 0 − eEz = −eEZ and eET = k 3 + eEt. Then the solutions of (2) and (3) are related as follows: We use this Fourier relation throughout the paper and we give the explicit form of f k ⊥ below. From the commutation relations ak, the commutation relations between φ and its conjugate momentum π = (∂ t − ieEz) φ * takes the standard form: The last equality follows from the Fourier transformation (4). Also one has to use the conservation of the Wronskian for the solutions of (3). The free Hamiltonian for the charged scalars is diagonal: Here we have used the harmonic expansion of φ and the fact that harmonic functions obey the equation of motion, (2). From the obtained form of the free Hamiltonian (it is diagonal and time independent) we can see that in static gauge in the constant electric field background there is energy conservation for each single harmonic. But the energy is not bounded from below, because k 0 can have any sign. Because of the latter fact we will see that various particle creation processes will be allowed when the interaction with the quantum gauge field, a µ , will be turned on.

III. ONE-LOOP CORRECTION
Because the free Hamiltonian, H 0 , is not bounded from below, the field theory under consideration is in the non-stationary situation. Hence, to calculate correlation functions one has to apply the Keldysh-Schwinger (KS) diagrammatic technique instead of the Feynman one [41], [42]. In such a formalism every particle is described by the matrix propagator, whose entries are the Keldysh propagator G K µν = 1 2 {a µ (x), a ν (y)} , and the retarded and advanced propagators G A,R µν = ∓θ(∓∆t) [a µ (x), a ν (y)] (and the same for the scalar fields, with a µ → φ). For our discussion it is instructive to see how the Keldysh propagators behave if the quantum average is done with the use of an arbitrary state |ψ . Performing the harmonic expansion of the quantum part, a µ (x), of the photon field we find that the photon's Kledysh propagator has the following form: Here n µν ( q, q ′ ) = ψ α † qµ α q ′ ν ψ , κ µν ( q, q ′ ) = ψ α qµ α q ′ ν ψ and q · x = |q|t − q · x. Furthermore, h.c. stands for the quantities containing ψ α qµ α † q ′ ν ψ = n µν ( q, q ′ ) − g µν δ (3) ( q − q ′ ) and κ * µν ( q, q ′ ) = ψ α † qµ α † q ′ ν ψ . Furthermore from (1) we find that scalar field's Keldysh propagator is as follows: Here n + k ,k ′ = ψ a † k ak′ ψ , κ + k ,k ′ = ψ |akbk′| ψ and h.c. stands for the expressions containing ψ aka † At the same time the form of the retarded and advanced propagators does not depend on the state |ψ . In [2] it was shown that there are no large (growing with time) loop corrections to the retarded and advanced propagators and also to the vertexes. This is a quite generic phenomenon: see e.g. [42] for the similar situations in different theories. It is straightforward to show that the same is true in the static gauge. Hence, we continue with the discussion of the Keldysh propagators. The reason why we present (6) and (7) here is that loop corrections contribute to n and κ in the Keldysh propagators of both fields.

A. Correction to the photon's Keldysh propagator
We start with the one-loop correction to the photon's Keldysh propagator in the limit t 1 +t 2 2 = t → ∞, when t 1 − t 2 = const. The initial state that we consider here is the one that is annihilated by all annihilation operators under consideration (a's, b's and α's). I.e. the tree-level Keldysh propagators G K and D K look as (6) and (7) with all n and κ equal to zero 2 .
Performing the same calculation as in [2] one can see that the one-loop correction to the propagator in question has the form of (6), where n µν ( q, q ′ , t) = δ (2) ( q ⊥ − q ′ ⊥ ) n µν (q 3 , q ′ 3 , q ⊥ , t) and κ µν ( q, q ′ , t) = δ (2) . The latter quantities are as follows: where t 0 is the moment of time after which we adiabatically turn on interactions between charged scalars, φ, and quantum gauge fields, a µ . In these expressions we neglect the difference between t 1,2 and t in the limit under consideration. This is mathematically rigorous if n µν and κ µν have a divergence as t → +∞ and if we would like to single out only the leading contributions. Otherwise we do such an approximation just to estimate the quantities under consideration. The physical meaning of such loop corrections is discussed in [2].
Let us consider n µν in (8). In order to estimate the expression in (8) we make the change of integration variables to: t ′ = t 3 +t 4 2 , τ = t 3 − t 4 . Then, we obtain the τ -integral in the range [t 0 − t, t − t 0 ], but its integrand is rapidly oscillating for large τ , as t → +∞ and t 0 → −∞. Hence, we can extend the upper and lower limits of the τ -integration to plus and minus infinity, respectively. Then, the integral over τ leads to the δ-function in the following expression: We further make the following change of integration variables Z = z 3 +z 4 2 and z = z 3 − z 4 . Also we change k 0 → k 0 − eEZ and k ′ 0 → k ′ 0 + eEZ. This change of integration variables allows us to simplify the integral over Z, which leads to a δ-function establishing that q 3 = q ′ 3 . As a result, To obtain this expression from (9) we have used that |q| = |q ′ | due to the presence of . Also we evaluate the integral over t ′ in (9). Finally, making the Fourier transformation (4), one can straightforwardly see that (10) coincides with the expression for n µν obtained in [2]. Thus, n µν is divergent as (t − t 0 ) → ∞. This divergence signals the presence of the photon production which starts right after the moment t 0 , when the interactions are turned on. It brakes the time reversal and translational invariance of QED on the constant field background. We discuss the physical meaning of all these observations in [2] in greater detail.
Let us continue now with the consideration of κ µν . In [2] we show that it does not receive growing contributions. (This, in particular, shows that the initial state for the photons is the appropriate vacuum state.) Now we are going to show, that in the static gauge, κ µν also does not grow with time. Similarly to the case of n µν here we also get that κ µν (q 3 , q ′ 3 , q ⊥ ) ∝ δ(q 3 +q ′ 3 ). Then, we take the limit t → ∞ and t 0 → −∞ in (8). This way we find that κ µν (q 3 , q ⊥ ) ∝ δ(k 0 +k ′ 0 +|q|)δ(k 0 +k ′ 0 −|q|). Hence, one can integrate out k ′ 0 to find that: The obtained expression contains only convergent integrals and, hence, is finite, if q = 0.

B. Correction to the Keldysh propagator of the charged particles
The one-loop correction to the scalar Keldysh propagator, in the limit t = (t 1 + t 2 )/2 → ∞ and t 1 − t 2 = const, can also be expressed as (7) where n ± k ,k ′ , t = δ (2) k ⊥ − k ′ ⊥ n ± k 0 , k ′ 0 , k ⊥ , t and similarly for the case of κ ± . In this case, for example, There are similar expressions for n − and κ − .
In [2] we show that none of the n ± and κ ± receive corrections that grow with time. To make the same conclusion here we perform the same trick as at the end of the previous subsection. For example, let us consider n + and take t → +∞ and t 0 → −∞. Then, performing the same transformations as at the end of the previous subsection, we find: This expression contains only convergent integrals. Hence, n + cannot contain contributions that grow with time. Using the same line of arguments one can draw the same conclusion for the case of n − and κ ± .

IV. DISCUSSION
We would like to present here some additional physical consequences of the observations made above and in our previous paper.

A. Remarks on the loop correction to the current of the created particles
Since we have shown that the result of [2] is gauge independent, we prefer to use the temporal gauge, i.e. A µ = (0, 0, 0, −Et), because then the situation is easier to generalize to more physically natural situations such as the pulse background.
The fact that n ± do not grow with time does not necessarily mean that there is no charge particle production generated by loops. First, it is worth stressing here that the correct particle number in the temporal gauge is n ± k, t f k ⊥ ±t + k 3 eE 2 rather than n ± itself. Second, although n ± k, t → −∞ = 0, κ ± k, t → −∞ = 0 it is the case that n ± k, t → +∞ = n ± = 0, κ ± k, t → +∞ = κ ± = 0. This kind of behavior of n ± and κ ± is clearly another sign of the breaking of the time translational and reversal invariance of the theory, which is respected at tree-level.
What physical consequences should all this have? In e.g. [43], [44] it was shown that the tree-level current of the produced pairs, The physical meaning of (17) is transparent. The first term on the right hand side describes the photon production by the background field, while the second term accounts for the decay of the produced photons into charged pairs. These processes are allowed in the presence of the background field. The absence of other terms describing other processes is explained by their suppression by higher powers of e 2 [2]. The solution of (17) sums up leading corrections, i.e. unsuppressed powers of e 2 (t − t 0 ), from all loops. Here we would like to find/compare Γ 1 and Γ 2 and, hence, to solve this kinetic equation.
To find the relation between Γ 1 and Γ 2 , note that generic harmonic functions look like (see e.g. [43], [44]): Where A and B some constants. For example, for the in-harmonics B = 0. Then, one can see that f * k ⊥ t + k 3 eE is equal to f k ⊥ t + k 3 eE under the exchange of eE → −eE and k → − k. Using this relation and the change of k → q − k under the integrals in (18), one can show that Γ 1 = Γ 2 . The same is also true for the case of out-harmonics. As a result, for such a choice of the harmonic functions, the leading one-loop correction to n µν ( q, t) is exact and we have the linear growth in all loops. This means that the time translational and reversal invariance cannot be restored after summation of all loops.