Schwinger Pair Production from Pad\'E-Borel Reconstruction

Schwinger pair production from Pad´e-Borelreconstruction

Adrien Florio1 1Institute of Physics, Laboratory of Particle Physics and Cosmology (LPPC), Ecole´ Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland. (Dated: December 27, 2019) In this work, we show how the knowledge of the first few terms of the Euler-Heisenberg La- grangian’s weak-field expansion in a magnetic field background is enough to reconstruct the pair- production rate in a strong electric field background. To this end, we study its associated truncated Borel sum using Padéapproximants, as advocated in a recent work by Costin and Dunne, J. Phys. A52, 445205 (2019).

INTRODUCTION Having in mind general quantum field theories, these are proofs of principle that a lot of non-perturbative in- In recent years, the program of ”resurgence” has formation might be at our hand, waiting to be extracted started to collect a number of successes in quantum me- from perturbative expansions. chanics and field theory. The idea behind it is that the This short note’s aim is to illustrate again the poten- typical asymptotic expansions that are to be dealt with, tial use of some of the ideas developed in these works in for example usual weak coupling expansions, are to be field theory, using one of the simplest ”non-perturbative” understood as being part of a transseries. In simple effects at hand, namely Schwinger pair production. In terms, transseries are sums of asymptotic series weighted particular, we will present two results. First, the knowl- by non-perturbative factors such as exponentials and log- edge of a few terms of the weak-field expansion of the arithms. A typical example is the semi-classical expan- Euler-Heisenberg effective Lagrangian in a background sion, which is a sum of perturbative/asymptotic expan- magnetic field is enough to reconstruct its strong-field be- sions around different saddle points. We refer the reader havior. Then, and perhaps more interestingly, the same to [1, 2] for pedagogical introductions to the topic. knowledge is enough to reconstruct the Euler-Heisenberg effective Lagrangian in a background electric field, for The very analytic structure of transseries implies weak and strong fields, including its imaginary part. This consistency relations between the different constituent means that this imaginary part, which gives the particle asymptotic series. In particular, large order coefficients production rate in a constant electric field, can be in- of a given expansion are known to be related to the small ferred from a few terms of a perturbative expansion. order coefficients of neighboring expansions. While being seemingly a mathematical curiosity, these relations have, for example, been used to predict the loop expan- SCHWINGER EFFECT, GENERALITIES sion around an instanton background for the quantum mechanical Sine-Gordon potential [3], predictions which Schwinger pair production is one of the most basics have been explicitly verified up to three loops using dia- field-theoretic non-perturbative effect, see [13] for an grammatic methods [4]. For other interesting examples extensive review. Its simplest realization is the vac- and reviews, we defer the reader to [5–9] and references uum emission of charged particles in the presence of therein. strong electric fields. A way to study it is to compute An immediate complaint against the potential practi- the one-loop fermionic effective action in a background cal usefulness of such approaches is that the knowledge electromagnetic field. Then, the phenomenon of pair- arXiv:1911.03489v2 [hep-th] 29 Jun 2020 of large orders terms of realistic quantum field theories productions is signaled by the appearance of an imagi- expansions is not necessarily available. In this spirit, ref- nary part in the effective action. For the sake of simplic- erence [10] started to investigate the amount of ”non- ity, we will hereafter restrict ourselves to the constant perturbative” information that can be extracted from background case. There, one can explicitly write down a finite number of terms of an asymptotic expansion. the effective Lagrangian [14]. For a purely magnetic field, Stunningly, using relatively few terms of the asymp- it admits the following closed-form [13] totic expansion of solutions to the PainlevéI equation (eB)2 m2 around real infinity, they were able to reconstruct the L (B) = ζ0 −1, (1) whole highly non-trivial analytic structure of this solu- eff 2π2 H 2eB tion throughout the whole complex plane. In a similar m2 m2 1 1 m2 2 spirit, the works [11, 12] successfully explored the phase + ζ −1, ln − + , H 2eB 2eB 12 4 2eB diagram of the λφ4 field theory by computing weak cou- 0 pling expansions up to nine loops and studying their as- with ζH (s, a) the Hurwitz zeta function and ζH (s, a) its sociated Borel sums. derivative with respect to s. The parameters m and e 2 are respectively the fermion’s mass and electric charge, a non-perturbative quantity; at any given order in (7), while B is the strength of the constant background mag- Γprod = 0. netic field. This expression is real and there is no pair- Actually, these weak-field expansions can be resummed production in a magnetic background, as there is apriori to (5) using Borel summation. In this language, again, no magnetically charged particle to be produced. The the imaginary part appears because of the presence of case of a pure electric field background is recovered by poles in the Laplace transform; (6) is ”Borel summable” analytically continuing B → ±iE [15] (note that in this while (7) is not. sense (1) can also be understood as the Euclidean space effective Lagrangian in an electric field background). Then, the effective Lagrangian does develop an imagi- STRONG-FIELD REGIME FROM WEAK-FIELD nary part, which can be written as [15] EXPANSION

2 4 2 m eE − m π Following [10], we want to understand how much of the Im (L (E)) = Li e eE (2) eff 8π3 m2 2 full Lagrangian (1) we can reconstruct using a finite num- 2 ber of terms in (6). To this purpose, again as in [10], we 4 2 m eE h − m π i eE construct the corresponding truncated Borel sum. From = 3 2 e + ... , (3) 8π m it, we build Padéapproximants, which are then used to compute a resummed Lagrangian Lres through a Laplace with Li2 the second polylogarithm. From this expression, eff it is easy to see the famous exponential suppression to the transform. The idea behind this procedure is to try to production rate Γ , which by definition is [13] exploit the fact that, while the original expansion is only prod asymptotic, its Borel transform is convergent. Note also 0 Γprod = 2Im (Leff ) . (4) that very similar techniques were already used in the 80 s, see for example [18] for a thorough review on QCD strong Another representation of (1) that will be of use is the coupling expansion. following Laplace-type integral [14, 16] m2 To keep notations clear, we set x = 2eB and write the asymptotic expansion (6), truncated at order N, as e2B2 L (B) = − (5) eff 8π2 L (x, N) 1 1 eff ∼ (8) Z ∞ 2 4 6 4 dp 1 p − m p m 64π x coth p − − e eB . 2 N 2n 0 p p 3 X ζ(2n + 4) 1 (−1)n(2n + 1)! (2π)2n x From this representation, it is clear that the imaginary n=0 part in the electric case comes from the contribution to N 2n 1 1 X 1 the integral of the poles of the hyperbolic cotangent at = a , (9) 64π6 x4 2n x integer multiples of iπ. n=0 In the rest of this work, we will be concerned with the n ζ(2n+4) weak-field expansion of (1). It is given as [13, 17] with a2n = (−1) (2n+1)! (2π)2n . We also define a truncated Borel sum m4 N X a2n Leff (B) ∼ (6) BL (p, N) = p2n−1. (10) 4π2 eff (2n − 1)! ∞ 2n+4 n=1 X ζ(2n + 4) 2eB (−1)n(2n + 1)! , (2π)2n+4 m2 With these definitions, we construct a Padéapproxi- n=0 mant of (10). Padéapproximants are rational functions with ζ(x) the Riemann zeta function. For the electric constructed to match a given series at specific points. field, the expansion reads They are typically used to try to reproduce the ana- lytical structure of a function by extrapolating it away from some regions. Their rational nature allows for the m4 Leff (E) ∼ (7) emergence of poles and branch cuts, which appear as ac- 4π2 cumulations of poles. They can be found in a variety of ∞ 2n+4 X ζ(2n + 4) 2eE places in the physics literature. As a specific example, we (2n + 1)! . (2π)2n+4 m2 can mention attempts to analytically continue Euclidean n=0 lattice data to Minkowski space through Padéapproxi- Both series are asymptotic because of the factorial mants, see [19] and references therein. growth of their coefficients. They are also both real To have easy access to the poles of our Padéfunction to all orders. It is in this sense that the rate (4) is and have good control over the numerical Laplace trans- 3

Magnetic case, strong fields from weak fields Pad´e-Borelreconstruction improves as N increases. This boils down to the fact that the Borel sum (10) is convergent; every new order contributes making the result 101 Weak-field exp. N = 2 more accurate. For example, only four terms of the weak- 100 ﬁeld expansion can be used to probe the strongly-coupled 1 N = 4 10− regime as far as x = 0.2.

2 N = 6 This is our ﬁrst result. With the knowledge of only

) 10− N = 6

x the ﬁrst few terms of the weak-ﬁeld expansion (6), it is ( 4 3

m 10− possible to explore the regime of strong magnetic fields eff N = 4 L 4 by first constructing the corresponding truncated Borel 10− Padé - sum, Padéapproximating it and computing its Laplace 5 N = 2 10− Borel transform. 6 10− Padé-Borel N = 16 7 10− Closed form SCHWINGER EFFECT RECONSTRUCTED 1 0 10− 10 m2 Now, we will show that this method, using the same x = 2e B | | | | data, actually also gives access to the regime of strong electric fields. In particular, we will see that we can use Figure 1. Magnetic field effective Lagrangian. Closed-form it to recover the Schwinger pair production rate. (plain line), weak-field expansion (dotted lines) and Padé- To consider an electric field, we proceed with the ana- Borel reconstruction (dashed lines) for different truncation order N. The weak-field expansion has a typical asymptotic lytic continuation x → ∓ix. This leads us to study behavior; every order makes it break down faster. The Padé- Borel reconstruction takes advantage of the fact that the Borel Lres (∓ix, N) 1 1 sum is convergent; every order improves the answer. eff = (13) m4 64π6 x4 Z ∞ ±ipx 2N form, we use Padéapproximants of the type a0 + dpe P BLeff (p, N) . 0 N X cn P2N BL (p, N) = . (11) Technically, to compute this Laplace transform, we eff 1 + b p n=1 n consider all the different fractions of (11) separately. We then rotate the integration contour in the complex plane The coefficients cn, bn, which are in principle complex by some angle and take into account any poles we might numbers, are computed by matching this expression to have crossed in the process. (10) around p = 0, see [20] for an explicit algorithm. Let us first look at what we obtain for the real part Finally, we compute our resummed Lagrangian as fol- of the resummed electric field effective Lagrangian ob- lows tained through this analytic continuation, figure 2. As Lres (x, N) 1 1 in the magnetic case, few terms of the weak Lagrangian eff = (12) m4 64π6 x4 allows for a precise extrapolation up into the strong-field Z ∞ regime. In particular, the reconstruction is able to pre- −px 2N a0 + dpe P BLeff (p, N) . dict correctly non-trivial features such as the change of 0 signs which happens around x = 0.1 (note that we are Note in particular that without the Padéinterpolation, plotting the absolute value). we would have achieved nothing, as in this case (12) More interesting are the results for the imaginary part would literally be equal to (9). of the effective Lagrangian, i.e. the pair-production rate. We show the result of this procedure, which from now They are shown in figure 3. They behave in exactly the on we will refer to as Padé-Borelreconstruction, in fig- same way; few terms of the weak-field expansion still give ure 1. The plain black line is the closed-form (1). The a quantitatively correct prediction of the rate. As little as dotted lines are the truncated weak-field expansions, for the first two terms are required to reconstruct an imag- different truncation N. The dashed lines are the Padé- inary part which is qualitatively correct at weak-field. Borel reconstructed expressions for the same N. Note With only the first six terms one can make quantitative that x → ∞ resp. x → 0 corresponds to the weak resp. predictions up to strong fields. This has to be contrasted strong-field regime, the goal being to be able to extrap- again with the original asymptotic series, which uses the olate from the former to the latter. Being an asymptotic same data but is real to all orders. expansion, every order makes it break down for larger The perhaps surprising capability of the Padé-Borel values of x, i.e. for weaker fields. On the contrary, the reconstruction to recover the pair-production rate is due 4

Electric case from magnetic case, real part Padé-Borel poles

1 8 10 N = 6 6 100 Padé-Borel 4

1 10−

) N = 4 2 ix

( 2 4 10− ) p m N = 2 π eff 0 L 3 Im( − 10 2

Re 4 − 10− 4 − 5 10− 6 Padé-Borel N = 16 − N = 2 N = 4 6 10− Closed form 8 N = 6 N = 16 − 1 0 0.3 0.2 0.1 0 0.1 0.2 0.3 10− 10 − − − m2 Re(p) x = 2e E π | | | |

Figure 2. Real part of electric field effective La- Figure 4. Poles of the Padé-Borelreconstruction in the Borel grangian. Closed-form (plain line) and Padé-Borelrecon- plane, for different truncation order N. Dotted lines are mul- struction (dashed lines) for different truncation order N. The tiples of 2πi, where poles accumulate as N is taken larger. Padé-Borelreconstruction leads to correct and convergent re- This is the correct analytic structure of the actual Borel sum, sults even after analytic continuation. which has single poles at non-zero multiples of 2πi. Note that the Padé-Borelapproximation requires more than a single pole per multiple of 2πi to reproduce the correct functional dependence.

Pair-production reconstructed to the fact that the Padéapproximants of the truncated 101 Borel sums are able to reproduce the correct analytic N = 6 100 Padé-Borel structure of the Borel sum. In terms of our variable x, 1 the actual Borel sum (5) is a meromorphic function with 10− 2 N = 4 single poles at x = 2πin for n ∈ Z, n 6= 0. As already 10− 3 mentioned, the imaginary part (3) can be understood as 10− 4 coming from the contribution of every single pole. It is 10− 2 dominated by the lowest-lying ones at x = ±2πi prod 5 Γ 10− Padé N = 16 N = 2 6 lead. 10− Closed form Γprod 1 −2πx 7 = e , (14) 10− 2 32π3x2 8 10− 9 1 2πx which we also show in figure 3. 10− 0.1 3 2 e− × 32π x 10 As the Padé-Borelapproximants are constructed only 10− from an asymptotic expansion around the real axis it is, 1 0 10− 10 however, a non-trivial fact that they are able to mimic m2 correctly this analytic structure. We show it occurring in x = 2e E | | | | figure 4, where we display the poles of our Padéapprox- imants. As the truncation order N is taken to be larger, Figure 3. Pair-production rate in a background elec- they accumulate around x = 2πin. Note that to approxi- tric field (imaginary part of the electric field effective La- grangian). Closed-form (plain line) and Padé-Borelrecon- mate the correct prefactors, a single pole is replaced by a struction (dashed lines) for different truncation order N. combination of different ones centered around x = 2πin. The dotted line is the leading exponential suppression to The leading poles at ±2πi are first reproduced accurately the Schwinger rate (shifted for readability). While N = 2 by the truncation order N = 6, which is consistent with gives an imaginary part which is only qualitatively correct the behavior of the results presented in figure 3. for weak-field, N = 4 and larger leads to a quantitatively This is our second and most important result. The correct prediction of the Schwinger rate for a whole range of knowledge of a few terms of the weak-field expansion of field-strengths. the effective Euler-Heisenberg Lagrangian in a magnetic field background is enough to reconstruct the particle 5 production rate in a strong electric field. (Taylor & Francis, 2008). [2] M. Mariño, Fortsch. Phys. 62, 455 (2014), arXiv:1206.6272 [hep-th]. CONCLUSION [3] G. V. Dunne and M. Unsal, Phys. Rev. D89, 105009 (2014), arXiv:1401.5202 [hep-th]. [4] M. A. Escobar-Ruiz, E. Shuryak, and A. V. Tur- This work can be summarized as follows: using only biner, Phys. Rev. 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Ulybyshev and L. Von Smekal, Comput. Phys. Commun. 237, 129 (2019) lating conversations and its invitation to BNL, where the doi:10.1016/j.cpc.2018.11.012, arXiv:1801.10348 [hep- aforementioned talk was given, and M. Shaposhnikov for ph]. feedback on this work. The author is supported by the [20] C. Brezinski and M. Redivo-Zaglia, Journal of Computa- Swiss National Science Foundation. tional and Applied Mathematics 284, 69 (2015), ortho- Quad 2014 .

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