A comment on generalized Schwinger effect
arXiv:1712.06621v1 [gr-qc] 18 Dec 2017
Karthik Rajeev1∗, Sumanta Chakraborty2†and T. Padmanabhan1‡ 1 2
IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411007, India
Department of Theoretical Physics, Indian Association for the Cultivation of Science, Kolkata-700032, India
December 20, 2017
Abstract A spatially homogeneous, time-dependent, electric field can produce charged particle pairs from the vacuum. When the electric field is constant, the mean number of pairs which are produced depends on the electric field and the coupling constant in a non-analytic manner, showing that this result cannot be obtained from the standard perturbation theory of quantum electrodynamics. When the electric field varies with time and vanishes asymptotically, the result may depend on the coupling constant either analytically or non-analytically. We investigate the nature of this dependence in detail. We show that the dependence of particle production on coupling constant is non-analytic for a class of time-dependent electric fields which vanish asymptotically when a specific condition is satisfied. We also demonstrate that for another class of electric fields, which vary rapidly, the dependence of particle production on coupling constant is analytic.
1
Introduction
Production of particles by a non-trivial classical background source is a ubiquitous effect in quantum field theory (QFT). Hawking radiation from a black-hole, the production of particles in an expanding universe and the Schwinger effect are some of the most studied examples of this phenomenon [1–15]. Even though the physical mechanism of particle creation depends on the specific system we are considering, there are certain common features. For instance, when the classical background is spatially homogeneous, the analysis of particle production in many interesting scenarios reduces to the study of a time-dependent harmonic oscillator. Because of this similarity in the mathematical description, from the study of a specific QFT system, one can gain important insights about several other systems (for example, see [16]). In this paper, we will focus on the mechanism by which charged particle pairs can be produced from vacuum in a spatially homogeneous, time-dependent electric field [17–26], which could be called the generalized Schwinger effect. Besides its mathematical simplicity, there are several features of the Schwinger effect that makes it a promising model for understanding various aspects of QFT. An important such feature is its nonperturbative nature. The rate of particle production in a homogeneous time-independent electric field background depends on the field strength and coupling constant in a non-analytic manner (see [15] for ∗
[email protected] †
[email protected] ‡
[email protected]
1
a recent review). This shows that one cannot obtain this result from the standard perturbation theory of quantum electrodynamics (QED). While the predictions of perturbative approaches in QFT has been verified to very high precisions [27–30], the verification of non-perturbative results remains as a major challenge in experimental physics. Schwinger effect provides us such an opportunity to understand this relatively unexplored regime of QFT in a better (and possibly deeper) way. Therefore, recently there have been several studies regarding the experimental verification of Schwinger effect [31–34]. Even though the strength of the electric field that is required to test this phenomenon is beyond the current state-of-the-art, it will be accessible to some of the upcoming laser facilities like Extreme Light Infrastructure (ELI) [35] and European X-ray Free-Electron Laser (European XFEL) [36]. While the Schwinger mechanism is an exactly solvable problem, it is practically impossible to realize a spatially homogeneous, constant electric field in an experimental setup. In a laboratory setting, one can only produce electric fields that vanish asymptotically in time though it could be spatially homogeneous to a necessary level of approximation. A well studied configuration of electric field with this property in which the pair production can be analytically computed is the so-called Sauter type electric field given by E = (E0 sech 2 (αt), 0, 0), where α is a dimension-full constant [37–44]. The study of pair production in the Sauter type field reveals an interesting feature of the generalized Schwinger effect. It turns out that the mean number of particles nk , produced with a given momentum k in the presence of a Sauter type field can depend on the coupling constant q and field strength E0 either analytically or non-analytically depending on whether the field varies rapidly or adiabatically in time, respectively. In particular, the expression for nk when the field is adiabatically varying can be shown to approach the corresponding expression for the Schwinger case. Therefore, one expects that the mean number of particles produced for a generic electric field configuration E(t) may depend on the coupling constant and field strength either analytically or non-analytically depending on certain conditions of the relevant parameters involved. Motivated by this, we seek for two general classes of electric field configurations such that the mean number of particles produced exhibit, either non-analytic dependence on the coupling constant (and field strength) when a specific condition is satisfied or, exhibit analytic dependence on the coupling constant (and field strength). The paper is organized as follows: In Section 2, we review the instability of the vacuum of QED in the presence of a Sauter type electric field. It turns out that the nature of dependence of the particle content on the coupling constant under Sauter type field depends on a parameter γ ≡ mα/(|qE0 |) where m and q are the mass and the charge of the field respectively. This parameter measures the degree of adiabaticity of the electric field in its time variation. We demonstrate that the mean number of produced particles nk as well as the probability of pair creation P depend analytically on |qE0 |, when γ 1 and are non-analytic functions of |qE0 |, when γ 1. In Section 3, we focus on a class of electric fields of the form E = (E0 f (αt), 0, 0), where f (s) vanishes in the limit |s| → ∞. We examine the conditions for the validity of perturbative analysis for this system. Further, we show explicitly that, when the aforementioned conditions are met, the mean number of particles produced is an analytic function of |qE0 |. For the electric field of the form E = (E0 f (αt), 0, 0), in Section 3.2, we consider the scenario in which perturbative analysis fails. We show explicitly that, in this non-perturbative regime, the mean number of particles produced has a factor which is non-analytic in |qE0 |, when the integral of f (s) has a certain asymptotic behaviour. Throughout the paper, we use a system of units in which c = ~ = 1. We work with (+, −, −, −) signature for the metric tensor.
2
2
Warm-up: Vacuum instability in Sauter type potential
Let us rapidly review the Schwinger effect in the Sauter type electric field to identify the two regimes in which the effect is perturbative or non-perturbative. Consider a spatially homogeneous electric field along the x direction, with the time dependence: E(t) = E0 sech 2 αt .
(1)
The associated vector potential could be taken as Aµ = (0, A(t), 0, 0) with Z A(t) = −
E(t)dt = −
E0 tanh αt . α
(2)
For α → ∞ the electric field would oscillate rapidly. In this case, the particle production rate is known to be analytic in |qE0 |. While, for α → 0, we have γ 1 and hence we can expand the vector potential A(t) in a Taylor series with the following leading order behaviour: A(t) ≈ −(E0 /α)αt = −E0 t, which mimics the standard Schwinger effect. Thus, in this limit, mean number of particles produced is expected to be non-analytic in |qE0 |. In order to study the vacuum instability in the presence of this field, one may compute the vacuum persistence probability P. It can be shown that, to the leading order, this probability is given by P = exp(−SE ) where SE is the Euclidean action evaluated for an instanton solution of the equation of motion dp/dt = qE, where p is the momentum [41, 42, 45–47]. This procedure works whenever, the solution exhibits periodicity in the Euclidean time, which happens for the Sauter potential. Let us now apply this procedure to obtain the vacuum persistence probability. We first need to determine the classical action for the charged particle. The equation of motion for a charged particle with charge q and mass m corresponds to dp/dt = qE, which can be integrated for the above electric field yielding, qE0 tanh αt = −qAx , (3) α while all the other components, e.g., py and pz are taken to be vanishing. The energy associated with the above trajectory is: q q m mα E = p2x + p2y + p2z + m2 = γ 2 + tanh2 αt; γ= . (4) γ |qE0 | px (t) =
(The parameter γ is always positive, depending on only |qE0 |. But, throughout our discussion we will assume that qE0 is positive, so that the modulus will not be explicitly displayed hereafter.) To get the vacuum persistence amplitude (VPA) we need to evaluate the trajectory of the particle. Using the fact that E = m(dt/dτ ) we obtain, Z cosh αt dt p τ =γ . (5) 2 sinh αt + γ 2 cosh2 αt The above integral can be performed in a straight forward manner by substituting z = sinh αt, which ultimately results in, ! p γ 1 + γ2 sinh αt = p sinh ατ . (6) γ 1 + γ2
3
Further using the fact that px = m(dx/dτ ) and the above result connecting t and τ we obtain, √ 1+γ 2 ατ γ sinh γ dx 1 1 = tanh αt = s √ 2 , dτ γ γ 1+γ 2 2 1 + γ cosh ατ γ p which can be integrated by the substitution z = cosh( 1 + γ 2 ατ /γ) resulting into, ! p p 2 1 + γ sinh 1 + γ 2 αx = γ cosh ατ . γ
(7)
(8)
To obtain the VPA, we need to analytically continue the trajectory into Euclidean time. The corresponding trajectory in the Euclidean plane can be obtained by the transformation: t → itE and τ → iτE , giving, ! p 1 + γ2 γ ατE , (9) sin sin αtE = p γ 1 + γ2 ! p p 2 1 + γ sinh 1 + γ 2 αx = γ cos ατE . (10) γ Therefore in the Euclidean time τE we have a periodicity with the period τEp : τEp =
2πγ 2π p . ≡ 2 α ¯ α 1+γ
(11)
The action for the particle in the Lorentzian sector is given by Z Z S = −m dτ + q A(t)dx , Z = −m Z = −m
Z dτ − m
dτ Z
dτ − m
dx dτ
2 ,
sinh2 α ¯ τ dτ . 1 + γ 2 cosh2 α ¯τ
(12)
Transforming to the Euclidean sector and integrating over a complete period in the imaginary time, we obtain, Z cos2 (¯ ατE ) SE ≡ iS = m dτE . (13) γ2 2 1 − 1+γ sin (ατE ) 2 Integration yields, SE =
πm 2γ p . α 1 + 1 + γ2
4
(14)
We can compute the probability for vacuum persistence, P ≈ exp(−SE ) in the two limits: (a) γ 1 and (b) γ 1. As mentioned earlier, we expect P to be non-analytic in the field strength in the case γ 1, (which includes the case of a constant electric field), while it could be analytic for γ 1. For γ 1 case, from Eq. (14) it follows that the probability for pair production becomes, 2πm 2πm 2πqE0 −2 − − 2πm 1+ 1+ . (15) + O(γ ) ≈ e α P=e α αγ α2 Thus P is independent of the field strength to leading order in 1/γ. Hence in this limit the pair production probability is analytic in the field strength. On the other hand, when γ 1, the pair production probability takes the following form πm2 γ2 πm2 m2 α 2 3 P = exp − 1− +O γ ≈ exp − 1− . (16) qE0 4 qE0 4(qE0 )2 Thus it is clear that for small γ, the pair production probability is non-analytic, as anticipated. This is mainly due to the fact that γ 1 corresponds to α → 0 limit and hence is similar to a constant electric field.
2.1
Mean number of particles produced
It is useful to rework the above result more explicitly in terms of mode functions and Bogoliubov coefficients which will allow us to determine the mean number of particles produced in either limit. To do this we will start with the explicit solution for complex scalar field φ in a Sauter type potential. Given the spatial homogeneity, it is convenient to work in the Fourier space. The Fourier transform φk of a complex scalar field φ of charge q and mass m in the Sauter type electric field introduced in Section 2 satisfies the following differential equation " 2 # qE 0 2 2 φ¨k + m + |k⊥ | + kx − tanh αt φk = 0 . (17) α The solution to this equation which corresponds to the positive frequency modes at asymptotic past can be written in terms of hypergeometric functions as follows i
∗ ξin (t) = C1 e−iω− t (1 + e2αt ) 2α (ω− −ω+ ) F(a, b, c; y) ,
(18)
where, s 2 1 qE0 2 2 y = (1 + tanh αt); ω± = m + |k⊥ | + kx ∓ , 2 α i 2qE0 i 2qE0 a= + ω+ − ω− ; b=1+ − + ω+ − ω− ; 2α α 2α α
(19) c=1−
iω− , α
(20)
and C1 is an arbitrary normalization constant. Using the asymptotic behaviour of hypergeometric functions we can determine its form as t → ∞ to be: ∗ ξin (t) ≈ A∗ e−iω+ t + Beiω+ t ,
5
(21)
where, the mean number of particles produced at future infinity is given by [43, 44]: " r # 2 π 2 qE0 1 − 4 + sinh2 2α (ω+ − ω− ) cosh π α2 nk = |B|2 =
πω− sinh α
sinh
πω+ α
.
(22)
The two limits√discussed in the previous section corresponds to the following two limits in this√particular case: (a) α qE0 , which corresponds to γ 1 in the √ previous scenario, and (b) α (qE0 / m2 + k2 ), 2 2 2 equivalent to γ 1. Here k = kx + |k⊥ | . When α qE0 , it immediately follows that π(m2 + |k⊥ |2 ) nk = exp − , (23) qE0 which coincides with the pair production probability derived in the previous section associated with γ 1. As evident and anticipated the mean number of particles produced is non-analytic in the field strength. The probability P(k) that a particle with momentum k and an anti-particle with momentum −k is produced is given by P(k) = nk /(1 + nk ). When the electric field is weak (i.e., qE0 /m 1), so that nk 1, the probability of pair production becomes P(k) ≈ nk . Therefore, for kz = ky = 0 and the case of weak E, which was the case considered in Section 2, the probability of pair production to leading order in qE0 becomes P(k) = exp(−πm2 /qE0 ), which is in agreement with Eq. (16). √ The opposite limit, corresponding to α (qE0 / m2 + k2 ), can also be easily obtained from Eq. (22), leading to, nk ≈
sinh2
πqE0 kx α2 ε π α
√
2
k 2 + m2
,
(24)
where, ε2 = k 2 + m2 . Clearly, nk is analytic in qE0 in conformity with the result in the previous section. Again, we see that the particle production is non-analytic in the coupling constant for small α, while is analytic for large α. In the first case, the mean number of particles produced, to the leading order coincides with that of the standard Schwinger effect.
3
Analytic and non-analytic dependences in a general context
We will now generalise the previous results — which were obtained for a specific form of the electric field, viz. the Sauter field — to a more general configuration. For this analysis we will consider an electric field along the x-axis of the form [19] E(t) = E0 f (αt) ,
(25)
where, E0 is a constant. The vector potential can then be chosen to be (0, A, 0, 0), where A(t) = −
E0 F (αt) , α
(26)
where F is defined through dF (s)/ds = f (s). Since a physically realizable electric field vanishes as t → ∞, in Eq. (25), we impose the condition that f (αt) vanishes as we approach the asymptotic past and future 6
times. Therefore, the function F satisfies lim
|s|→∞
dF =0. ds
(27)
In this external background electric field, the Fourier transform φk of a complex scalar field of mass m and charge q satisfies the following time dependent harmonic oscillator equation: φ¨k + ωk2 φk = 0 ,
(28)
where the time dependent frequency ωk is given by ωk2 (t) = ε2 +
2kx mF (αt) m2 F (αt)2 . + γ γ2
(29)
and ε2 = k 2 + m2 . We will now study this equation and its solutions in the two appropriate limits.
3.1
Perturbative limit
In this subsection we will assume that the function F is bounded as follows: Condition 1 :
Fmin < F < Fmax .
(30)
Moreover, from Eq. (29), we see that the field strength dependent part of ωk2 can be treated as a perturbation whenever, F− < F (t) < F+ ,
(31)
p where, F± = γ(−kx ± kx2 + ε2 ) are the roots of the equation 2kx mF/γ + m2 F 2 /γ 2 = ε2 . Therefore, the field strength dependent term in Eq. (28), viz. (2kx mF/γ + m2 F 2 /γ 2 )φk can be always treated as a perturbation if the following conditions are satisfied, Condition 2:
Fmax F+ and |Fmin | |F− | .
(32)
When this condition is satisfied, we can solve Eq. (28) perturbatively in powers of 1/γ. Therefore, if an electric field of the form given by Eq. (25) satisfies Condition 1 and 2, we can take the solution to Eq. (28) to be of the following form φk = φk(0) +
1 1 φk(1) + 2 φk(2) + · · · . γ γ
(33)
The O(1/γ) solution which approaches a positive frequency mode at t → −∞ can be easily found to be Z 0 2mkx t it φk = e − dt0 sin[ε(t − t0 )]F (αt0 )eiεt + O(γ −2 ) . (34) γε −∞ Due to the appearance of sin[ε(t − t0 )], we see that a positive frequency mode in the asymptotic past evolves into a linear combination of both positive and negative frequency modes. That is ( eiεt , t → −∞ φk ≈ (35) Aeiεt + B ∗ e−iεt , t → ∞
7
where, A = 1 + O(1/γ) and ∗
B =
πkx qE0 iαε2
2ε f˜ + O(γ −2 ) , α
(36)
where, f˜(s) is the Fourier transform of f (τ ). Therefore, the mean number of particles produced is given by 2 |πkx (qE0 )f˜ 2 α | + O(γ −4 ) . (37) nk (∞) = α2 ε4 Note that this expression is a Taylor series in qE0 and hence analytic in the same. So we obtain a simple, closed form expression for the number of particles produced in this case. We will now provide an alternate derivation of Eq. (37). To do this, it is convenient to consider the following quantity. z=
B ∗ (t) 2iρ e , A∗ (t)
(38)
where, ρ˙ = ωk , along with A and B being the time dependent Bogoliubov coefficients. It can be shown that z satisfies the following differential equation z˙ + 2iωk z +
ω˙ k z2 − 1 = 0 . 2ωk
(39)
The time dependent particle number nk in terms of z is given by nk =
|z|2 . 1 − |z|2
(40)
The variable z was introduced in [48,49] for the analysis of particle production in an external background. It is worth mentioning that z is also related to the classicality of the φk and used extensively in the context of cosmology to understand the transition to classicality [50–53]. For γ 1, we have |z| 1 and ωk ≈ ε + mkx F/(εγ), so that the differential equation for z presented in Eq. (39) reduces, in the leading order to, d(ze2iεt ) kx (qE0 ) 2iεt = e f (αt) . dt 2ε
(41)
This implies that |z(t → ∞)| is given by πkx (qE0 ) ˜ 2 . |z(∞)| = f 2 αε α
(42)
We can easily see from this that the asymptotic value of mean number of particles produced is given by Eq. (37). Therefore, we have shown in this section that: if the electric field of the form given by Eq. (25) satisfies Conditions 1 and Condition 2, the asymptotic value of the mean number of particles produced nk is an analytic function of |qE0 |. As a consistency check, let us apply this to the case of Sauter type electric field that we discussed in Section 3. In this case the relevant f (τ ) is given by sech 2 (ατ ), whose Fourier transform evaluated at 8
2ε/α turns out to be α−1 ε csch(πα/ε). Therefore, from Eq. (37), the mean number of particles produced to leading order in γ is given by 2 nk =
πqE0 kx α2 ε
sinh2
π α
√
k2 + m2
+ O(γ −4 ) ,
(43)
which is in complete agreement with Eq. (24).
3.2
Non-Perturbative Limit
In this subsection we will consider electric field configurations of the form Eq. (25), which violates Condition 1 and 2, presented in Eq. (30) and Eq. (32) respectively. This corresponds to the case in which the field dependent part of ωk2 is much larger than ε2 in some range of time. We focus on the situation which satisfies the following conditions: Condition 3:
|F (t)| F+ , |F− | when |t| tc ,
(44)
where tc is some critical time. This implies that the perturbation theory cannot be used to find a solution to Eq. (28) valid for all times t. However, since we have assumed the field to vanish asymptotically, from Eq. (27) we see that the WKB approximation becomes valid at late and early times. The simplest example of this case is given by a constant field. To set the stage, we will briefly review this special case before doing a general analysis. The Fourier modes φk of a complex scalar field in a constant electric field is given by E = (E0 , 0, 0) satisfies the following harmonic oscillator equation, such that, φ¨k + ωk2 (t)φk = 0
(45)
where, for the choice of gauge Aµ = (0, −E0 t, 0, 0), ωk2 (t) = m2 + |k⊥ |2 + (qE0 t + kx )2 .
(46)
The exact solutions of Eq. (45) are known in terms of parabolic cylinder functions. But, the exact number of particles produced at t = +∞ can be found using the WKB solutions of Eq. (45) in the limit |t| → ∞. Let us denote by ξk(in) , the ‘in-modes’, which are solutions to Eq. (45) that behave as positive frequency modes at t → −∞. Similarity, we define the ‘out-modes’ ξk(out) , which are the positive frequency solutions of Eq. (45) at t → ∞. From Eq. (46), we see that (|ω˙ k |/ωk2 ) 1 as |t| → ∞. Hence, in the asymptotic past and future the WKB approximation is valid. In these regions, we have the following approximation for ξk(in) and ξk(out) Z 1 iqE0 t2 As t → −∞: ξk(in) ≈ p exp i |ωk |dt ≈ (−qE0 t)−iλ/2−1/2 exp − , (47) 2 |ωk | Z 1 iqE0 t2 iλ/2−1/2 As t → ∞: ξk(out) ≈ p exp i |ωk |dt ≈ (qE0 t) exp , (48) 2 |ωk | ∗ where, λ = (|k⊥ |2 + m2 )/(qE0 ). Let us consider the evolution of ξk(in) from t → −∞ to t → ∞. Since ∗ {ξk(out) , ξk(out) } is a complete set of solutions of Eq. (45), we can write ∗ ∗ ξk(in) = Aξk(out) + B ∗ ξk(out)
9
(49)
where, A and B are the standard Bogoliubov coefficients. The mean number of particles produced nk is then given by |B|2 . One can use the asymptotic expansions of the parabolic cylinder functions to compute B. But there is simpler and more elegant procedure (originally due to Landau), which can be used to ∗ obtain B and hence, nk . To follow this procedure, we start with the WKB approximation for ξk(in) given by (−qE0 t)iλ/2−1/2 exp iqE0 t2 , t → −∞ ∗ 2 (50) = ξk(in) 2 2 A(qE0 t)−iλ/2−1/2 exp − iqE0 t + B ∗ (qE0 t)iλ/2−1/2 exp iqE0 t , t → ∞ 2 2 ∗ Now, in the asymptotic expression for ξk(in) at t → −∞, if we treat t as a complex variable and rotate t in the complex plane from arg[t] = 0 to arg[t] = π, the solution nicely gets mapped to the asymptotic expression for ξk(out) at t → ∞. Therefore the Bogoliubov coefficient B is just:
B = e−iπ/2 e−πλ/2 .
(51)
Hence, the mean number of particles produced is given by nk = e−πλ , which is the standard result. To generalise this result we note that the time dependent frequency ωk has the following series expansion near |t| → ∞ : ω02 ω04 ω(t) ≈ κ|t| 1 + 2 2 − 4 4 + ... (52) 2κ |t| 8κ |t| =
1 X
Cn (κ)|t|2n−1 ,
(53)
n=−∞
where, κ = (qE0 ) and ω02 = (m2 + |k⊥ |2 ). Note that there are only odd powers of |t| in the expansion of ω(t) near |t| = ∞. We observe that the mean number of particles produced, found from rotating t in the complex plane is related to C0 as n = e−2πC0 ,
(54)
and the non analyticity comes from the fact that C0 ∝ 1/(qE0 ). We will now show that these features continue to hold even in a more general context. Motivated by the above observation, let us consider a class of electric fields satisfying Condition 3 as in Eq. (44), for which the mean number of particles produced is non-analytic in the coupling constant. This class is characterized by the following properties: • For, |τ | → ∞, f (τ ) is symmetric under τ → −τ . • F (τ ) diverges as a power series, as |τ | → ∞. The first condition implies that F (|τ |) has the following expansion near infinity F (|τ |) ≈
∞ X n=−∞
10
Cn |τ |2n−1 .
(55)
For later use, let us also note the asymptotic series for 1/F (|τ |), ∞ X 1 = C˜n |τ |2n−1 , F (|τ |) n=−∞
(56)
where, the coefficients Cn and C˜n satisfies, ∞ X
( Cn C˜l−n =
n=−∞
1, l = 1 0, ∀l ∈ Z − {1}
(57)
The Fourier modes in this electric field satisfies the equation of motion of a harmonic oscillator of unit mass and frequency is given by Eq. (29). For |t| → ∞, the frequency ωk is given by ωk (t) ≈
γ(|k⊥ |2 + m2 ) mF (αt) + kx + + O F (αt)−2 . γ 2mF (αt)
(58)
Proceeding exactly as in our analysis of the standard Schwinger effect, the WKB approximation for the ‘in’ and ‘out’ modes can be found to be Z Z 2 qE0 γ(k⊥ + m2 ) dt −1/2 As t → −∞: ξk(in) ≈ (|qE0 F (αt)|) exp i dtF (αt) + ikx t + (59) 2α 2m F (αt) Z Z 2 qE0 γ(k⊥ + m2 ) dt As t → ∞: ξk(out) ≈ (|qE0 F (αt)|)−1/2 exp −i dtF (αt) − ikx t − . 2α 2m F (αt) (60) ∗ ∗ Again, since ξk(in) can be written as a linear combination of ξk(out) and ξk(out) we have,
∗ ξk(in) =
( R 0 (|qE0 F (αt)|)−1/2 exp i qE dtF (αt) + ikx t + 2α ∗ Aξk(out)
γ(|k⊥ |2 +m2 ) 2m
∗
+ B ξk(out) , t → ∞
R
dt F (αt)
, t → −∞
(61)
where, we have to use Eq. (60) for the WKB approximation of ξk(out) near t → ∞. Let us now treat t as a complex variable and rotate t in the complex plane from arg[t] = 0 to arg[t] = π. As in the previous case, ∗ the WKB solution for ξk(in) at t → −∞ maps to the asymptotic expression for ξk(out) at t → ∞. The Bogoliubov coefficient B to leading order in qE0 is then given by ˜
B = e−iπ/2 e−πλC0 /2 ,
(62)
where, λ = (|k⊥ |2 + m2 )/(qE0 ). Therefore, we see that the mean number of particles produced has the following non perturbative part nk(non-pert) = e−
˜ π(m2 +|k⊥ |2 )C 0 qE
.
(63)
Therefore, we have shown that: For the class of electric field with the asymptotic properties mentioned above, the asymptotic value of the number of particle produced, nk , has a non-perturbative factor given by 2 ˜ nk(non-pert) = exp{−π(m2 + k⊥ )C0 /qE0 }.
11
As an example to this result and as a consistency check, consider the asymptotic expansion of the √ vector potential for the Sauter type potential in the limit α 1/ qE0 . This is given by: ASauter (t) ≈ E0 t + O[(αt)2 ] .
(64)
Therefore the corresponding expansion of 1/F for this case reduces to 1 1 ≈ , F (|τ |) |τ |
(65)
implying that, C0 = 1. So the particle number is given by nk(non-pert) ≈ e−
π(m2 +|k⊥ |2 ) qE0
,
(66)
which reproduces the result for mean number of particles produced of the standard Schwinger effect case, as expected.
4
Conclusion
The particle production in an external electric field serves as a toy model to understand various features of a generic QFT in an external classical background. An important feature of the particle production in the case of a constant electric field (Schwinger effect) is that it is a non-perturbative phenomenon. This is reflected in the fact that, the mean number of particles produced with a particular momentum is a nonanalytic function of the coupling constant and the field strength. However, when the external electric field is time-dependent, the mean number of particles produced could be either analytic or non-analytic in the coupling constant. A simple example which covers both the limits is provided by the Sauter type electric field in which case this behaviour is governed by a specific parameter in the potential. Motivated by this, we have investigated a more general class of field configurations to contrast analytic versus non-analytic dependence in the coupling constant. We have considered two fairly general classes of electric fields, such that, for one class the particle production is analytic while for the other it is non-analytic in qE0 . We have identified the conditions for this behaviour in a general setting. The results and the techniques have implications in other contexts, like for example, particle production in an expanding universe, which we hope to investigate in a future publication.
Acknowledgements Research of S.C. is supported by the SERB-NPDF grant (PDF/2016/001589) from SERB, Government of India. T.P’s research is partially supported by the J.C.Bose Research Grant of DST, India.
A
Perturbative Analysis of Schwinger effect
We have seen that one cannot apply perturbation theory to calculate the mean number of particles produced, if Condition 1 and 2 are not satisfied by the external electric field. The vector potential Aµ corresponding to a constant electric field along x-axis may be taken as Aµ = (0, −E0 t, 0, 0). One can identify that, in this case, F (αt) = αt and hence, Conditions 1 and 2 are clearly violated. Therefore, the 12
mean number of particles for the constant electric field case cannot be obtained from the perturbation analysis given in Section 3.1. However, just out of curiosity, one can ask: what does a naive perturbative analysis give for particle production in a constant electric field? To answer this, we start by noticing from Eq. (29) that the time dependent frequency ωk in for the Schwinger case has the following form ωk2 (t) = κ2 t2 + ω02 ,
(67)
where, κ = qE0 and ω02 = m2 + ky2 + kz2 and we have taken kx = 0. Let us expand the Bogoliubov coefficients in powers of ≡ κ2 /ω04 , as,
Figure 1: n(t) for ω0 = 1 and κ = 0.01. Brown graph is for n(t) calculated to leading order in and Blue graph is for the next to leading order calculation. A = 1 + A1 + A2 + ...
(68)
B = B1 + B2 + ...
(69)
where, An and Bn are proportional to n , for n = 1, 2, · · · . The following expressions, in this context, are also useful, ω˙k 2iρ e = C1 + C2 + ... 2ωk ω˙k −2iρ e = D1 + D2 + ... 2ωk
(70) (71)
where, κ2 te2itω0 ; 2ω02 κ2 te−2itω0 D1 = ; 2ω02 C1 =
iκ4 t3 e2itω0 (tω0 + 3i) ; ··· 6ω04 iκ4 t3 e−2itω0 (tω0 − 3i) D2 = − ; ··· 6ω04
C2 =
(72) (73)
The differential equation satisfied by the Bogoliubov co-efficients to O(2 ) can be written as, A˙ 1 = 0; B˙1 = 0;
A˙ 2 = C1 B1 B˙2 = D2 + D1 A1 13
(74) (75)
where the initial conditions are Ai (0) = Bi (0) = 0. Note that this is different from the initial conditions taken in the main body of this work. That is, the initial condition Ai (0) = Bi (0) = 0 implies that the quantum state at t = 0 is the instantaneous vacuum of the Hamiltonian of the system. On the other hand, we assumed the quantum state to be the vacuum at t = −∞ in the main body. We have chosen the initial conditions Eq. (74) and Eq. (75) just for convenience. The aim of this section is to demonstrate that naive perturbation theory fails when applied to the study particle production in constant electric field. Either of the initial conditions can be used to demonstrate the same hence, we have chosen the more convenient one here. The solution to second order is found to be κ4 8it3 ω03 + 6t2 ω02 + 6itω0 e2itω0 − 3e2itω0 + 3 (76) A1 = 0; A2 (t) = 192ω08 κ2 e−2itω0 −2itω0 + e2itω0 − 1 B1 (t) = − ; (77) 8ω04 κ4 e−2itω0 −4t4 ω04 + 20it3 ω03 + 30t2 ω02 − 30itω0 + 15e2itω0 − 15 (78) B2 (t) = − 48ω08 Therefore, the mean number of particles n(t) is given by κ4 κ4 8t8 ω08 + 80t6 ω06 − 90t4 ω04 + 225 − 36κ2 ω04 6t4 ω04 − 5t2 ω02 − 5 16 1152ω0 n κ4 h 2 4 + − 3 5κ + 2ω 2tω0 5κ2 3 − 2t2 ω02 + 6ω04 sin(2tω0 ) 0 16 1152ω0 o i + κ2 4t4 ω04 − 30t2 ω02 + 15 + 6ω04 cos(2tω0 )
n(t) =
+
κ4 36 2t2 ω010 + ω08 1152ω016
(79)
The mean number of particles produced clearly diverges, as can also be seen from Fig. 1. The diverging behaviour of the mean number of particles is due to the fact that the perturbation theory fails for this case. To be more specific, Condition 1 and 2 are not satisfied. Therefore, one has to employ other methods to compute the particle production on the constant electric field case. One such method is illustrated on Section 3.2.
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