On the Higher Loop Euler-Heisenberg Trans-Series Structure
OOn the Higher Loop Euler-Heisenberg Trans-Series Structure
Gerald V. Dunne and Zachary HarrisDepartment of Physics, University of Connecticut, Storrs, CT 06269-3046, USA
We show that the one-loop Euler-Heisenberg QED effective Lagrangian in a constant backgroundfield acquires a very different non-perturbative trans-series structure at two-loop and higher-looporder in the fine structure constant. Beyond one-loop, virtual particles interact, causing fluctua-tions about the instantons, whereby the simple poles of the one-loop Borel transform become branchpoints. We illustrate this in detail at two-loop order using Ritus’s seminal result for the renormal-ized two-loop effective Lagrangian as an exact double-integral representation, and propose a possiblenew approach to computations at higher loop order. Our methods yield remarkably accurate ex-trapolations from weak-field to strong-field, and from magnetic to electric background field, at bothone-loop and two-loop order, based on surprisingly little perturbative input.
I. INTRODUCTION
The exact renormalized one-loop QED effective Lagrangian in a uniform background electromagneticfield was computed long ago by Euler and Heisenberg [1–5]. This computation is made possible by theexistence of a simple exact integral representation of the electron propagator in a constant backgroundfield [6, 7]. When the constant background field is purely magnetic, the one-loop effective Lagrangianis real, but when the constant background field is electric, the one-loop effective Lagrangian has both areal and imaginary part. The non-perturbative imaginary part determines the rate of electron-positronpair-production from vacuum [1, 3]. The one-loop Euler-Heisenberg expression in [1] is an explicit Borel-Laplace integral representation, so the non-perturbative properties can be extracted straightforwardlyfrom the singularity structure of the exact Borel transform function, which has only pole singularities.This meromorphic property of the Borel transform no longer holds at higher loop order. The two-loopEuler-Heisenberg effective Lagrangian, which includes the new effect of photon exchange between thevirtual particles in the fermion loop (see Figure 1), was first calculated by Ritus [8–11], also based on theexact proper-time integral representation of the electron propagator. A new feature at two-loop orderis the necessity of mass renormalization, and Ritus found an exact two-parameter integral expressionincorporating both charge and mass renormalization. See also [12, 13]. Ritus’s two-loop expression isnot explicitly in the form of a Borel-Laplace integral, so the extraction of non-perturbative propertiesis less direct than at one-loop order. In this paper we discuss the extraction of this non-perturbativeinformation at two-loop, extending the analysis of [8–11, 14, 15]. The Borel transform is not meromorphicat two-loop, and the non-perturbative structure is different in several interesting ways. This distinctioncontinues at all higher loop orders. Since the constant background field fermion propagator is knownexactly, in principle one can express the l -loop Euler-Heisenberg effective Lagrangian as a 2( l − α may shed light on scattering amplitudes in strong backgroundfields, in particular those associated with high intensity lasers. For example, seminal work by Ritus andNarozhnyi has made predictions for the resulting structure at higher loop order for the special case wherethe background laser field is represented as a constant crossed field [25–27]. The physical impact of theseresults is an active area of investigation [28–31], seeking to build on the pioneering analysis of Ritus andNarozhnyi.The all-orders Euler-Heisenberg effective Lagrangian can be written as a series in α , the fine structure a r X i v : . [ h e p - t h ] J a n FIG. 1. The irreducible one-loop (left) and two-loop (right) diagrams contributing to the Euler-Heisenbergeffective Lagrangian, with the double solid lines representing the fully-dressed fermion propagator. constant L (cid:18) α, eFm (cid:19) ∼ ∞ (cid:88) l =1 (cid:16) απ (cid:17) l L ( l ) (cid:18) eFm (cid:19) (1)where F denotes the strength of the constant background field, which we consider here to be either magnetic or electric. At a fixed loop order l , corresponding to a given order in α , the weak fieldexpansion of the effective Lagrangian L ( l ) (cid:0) eFm (cid:1) is an asymptotic series with perturbative coefficients a ( l ) n : L ( l ) (cid:18) eFm (cid:19) ∼ π l − F ( l − (cid:18) eFm (cid:19) ∞ (cid:88) n =0 a ( l ) n (cid:18) eFm (cid:19) n , eF (cid:28) m (2)Here we have chosen a particular normalization of the expansion coefficients a ( l ) n , the motivation forwhich is explained below: see the discussion following (7).Thus, the full expansion (1) is a double series, an expansion both in α and in the field strength. We have separated a factor of (cid:0) απ (cid:1) l from our definition of L ( l ) (cid:0) eFm (cid:1) , to emphasize the fact that theloop expansion (1) is a perturbative expansion, which is also expected to be asymptotic [32]. If thebackground field is magnetic, of magnitude B , at l -loop order L ( l ) (cid:0) eBm (cid:1) is expected to be unambiguouslyBorel summable to a real expression. If the background field is electric, of magnitude E , at l -loop order L ( l ) (cid:0) eEm (cid:1) is expected to have both real and imaginary parts, also expressible as a well-defined Borelrepresentation. Furthermore, it is expected that at each loop order the electric background field resultcan be obtained by analytic continuation B → iE from the real magnetic field result. Here we analyzethese issues beyond the familiar one-loop result, using ideas and methods from Borel-´Ecalle summation[33–40].At one-loop order the non-perturbative imaginary part has a well-known polylogarithm expressionwhich can be expanded as a convergent weak-field instanton expansion:Im (cid:20) L (1) (cid:18) eEm (cid:19)(cid:21) = E π Li (cid:16) e − πm / ( eE ) (cid:17) (3)= E π (cid:26) e − πm / ( eE ) + 12 e − πm / ( eE ) + 13 e − πm / ( eE ) + . . . (cid:27) (4)Note that the instanton terms are not multiplied by fluctuation series, just numerical residue factors, sothis is a very simple example of a trans-series. By contrast, at two-loop order the weak-field instantonexpansion of the non-perturbative imaginary part is conjectured to be of the form [8–11]Im (cid:20) L (2) (cid:18) eEm (cid:19)(cid:21) ∼ πE (cid:40) e − πm / ( eE ) (cid:18) . . . (cid:19) + (cid:114) m eE ∞ (cid:88) k =2 e − kπm / ( eE ) (cid:32) − c k + (cid:114) eEm + . . . (cid:33)(cid:41) (5)where the numerical coefficient c k , for k ≥
2, is c k = 12 √ k k − (cid:88) (cid:96) =1 (cid:112) (cid:96) ( k − (cid:96) ) , k ≥ We use natural units ( (cid:126) = c = ε = 1), with the fine-structure constant α = e / π . In keeping with the commonconvention for perturbative QED expansions, e.g. for the anomalous magnetic moment, we write the perturbative QEDexpansion parameter as α/π , and rescale the QED beta function coefficients correspondingly. In fact, in general it is a triple expansion since for each loop order l the weak field expansion (2) is itself a doubleseries expansion in the two Lorentz invariant combinations of the constant background field. Here, for simplicity, weconcentrate on a constant background field that is either magnetic or electric, but not both. For a constant field of either magnetic or electric nature, the relevant Lorentz invariant quantity is E − B . This conjectured two-loop structure is quite different from one-loop: there are (unspecified) fluctuationsabout the one-instanton term, and at higher instanton orders ( k ≥
2) there is a stronger weak-field pref-actor, (cid:113) m eE , followed also by fluctuations (only partially specified). Our goal here is to investigate thesefluctuation terms in the non-perturbative imaginary part for the two-loop effective Lagrangian in an elec-tric background field, by analytic continuation from the weak field expansion for a magnetic background.We focus on reconstructing non-perturbative information from finite-order perturbative information,rather than the standard method of seeking an exact closed-form (multi-)integral representation, whichcurrently appears prohibitively difficult even at three-loop order [17–19]. The motivation for this dif-ferent approach is as a proof-of-principle test for a new approach to Euler-Heisenberg computations athigher loop order.Note that while eF is a renormalization group invariant, both α and m are scale dependent. Atone-loop order, m is simply the bare mass and the charge renormalization is conventionally done at thephysical electron mass scale. At two-loop order, a field-dependent mass renormalization is required, and L (2) is again conventionally renormalized at the physical electron mass scale [8–10]. Being a doubly-perturbative expansion, it is not immediately obvious how to extract strong-field or short-distance ornon-perturbative information from the full all-orders result (1). Different expansions, for example atfixed order in α or in eF , are possible, and correlated or uniform limits may also be physically relevantin certain circumstances. For example, if we sum the leading weak field contributions to the imaginarypart of L ( l ) (cid:0) eEm (cid:1) for an electric background, this sum exponentiates to e απ times the leading one-loopresult [10, 18, 41]: Im (cid:20) L (cid:18) α, eEm (cid:19)(cid:21) ∼ αE π e απ e − πm / ( eE ) + . . . , eE (cid:28) m (7)This exponential factor can be computed from the world-line representation of the effective Lagrangian[41], or can be understood [10] as encoding the leading field-dependent mass shift at two-loop order: m → m − α/ ( eE ). By a Borel dispersion relation, this exponentiation can be translated into aconjecture for the leading large-order growth of the perturbative expansion coefficients a ( l ) n [15], whichmotivates the choice of overall normalization of these coefficients in (2).Here we propose a different approach to the higher-loop computations, based on expressing the per-turbative weak magnetic field expansion at a given loop order as an approximate Borel integral, whosestrong-field limit can be extracted with surprisingly high precision, and whose analytic continuation froma magnetic to an electric background can also be achieved with high precision, including exponentiallysuppressed non-perturbative information. Such an approach relies on efficient and near-optimal meth-ods to perform the necessary analytic continuations [38–40]. In this paper we test the feasibility of suchmethods applied to the Euler-Heisenberg effective Lagrangian at one-loop and two-loop, and we concludewith comments about the prospects for higher loop orders.
II. THE EXACT ONE-LOOP EULER-HEISENBERG EFFECTIVE LAGRANGIANA. Exact Results at One-Loop Order
We first review well-known properties of the one-loop Euler-Heisenberg QED effective Lagrangian ina constant background magnetic field, B , conventionally expressed as a proper-time integral [1, 5] L (1) (cid:18) eBm (cid:19) = − B (cid:90) ∞ d tt (cid:18) coth t − t − t (cid:19) e − m t/ ( eB ) (8)[Recall our notational convention that a factor of (cid:0) απ (cid:1) l is extracted at l -loop order]. The weak fieldexpansion of (8) is a prototypical effective field theory expansion, expressing the physics of the lightfields (photons) after integrating out the heavy fields (electrons/positrons) at the electron mass scale m : L (1) (cid:18) eBm (cid:19) ∼ B π (cid:18) eBm (cid:19) ∞ (cid:88) n =0 a (1) n (cid:18) eBm (cid:19) n , eB (cid:28) m (9)Here the factorially divergent one-loop expansion coefficients are known exactly: a (1) n = ( − n Γ(2 n + 2) π n +2 ζ (2 n + 4) (10)= ( − n Γ(2 n + 2) π n +2 (cid:18) · n +2 + 13 · n +2 + 14 · n +2 + . . . (cid:19) (11)The corrections to the leading factorial growth in (11) are exponential in n , which is directly relatedto the non-appearance of power-law fluctuation corrections in the instanton expansion (4) [5]. We havedeliberately written the Riemann zeta factor in the form in (11) to emphasize the correspondence betweenthe k factors multiplying the k -instanton terms in (4), and the residues of the singularities of the Boreltransform. See also Figure 7.The exact integral representation (8) can also be expanded in the strong field limit, yielding a conver-gent strong field expansion whose leading behavior is: L (1) (cid:18) eBm (cid:19) ∼ · B (cid:18) ln (cid:18) eBπm (cid:19) − γ + 6 π ζ (cid:48) (2) (cid:19) , eB (cid:29) m (12)where γ ≈ . ... is the Euler-Mascheroni constant, and ζ (cid:48) (2) = π ( −
12 log( A ) + γ + ln(2 π )) ≈− . A being the Glaisher-Kinkelin constant. The coefficient of the logarithmic term in(12) is the one-loop QED beta function coefficient β , associated with one-loop charge renormalization[2, 5]: β QED ( α ) = 2 α ∞ (cid:88) n =1 β n (cid:16) απ (cid:17) n = 2 α (cid:20) (cid:16) απ (cid:17) + 14 (cid:16) απ (cid:17) + . . . (cid:21) (13)The weak field expansion (9) is an asymptotic series [3, 5, 42–45], whose Borel sum is the one-loopEuler-Heisenberg integral representation (8), with one-loop Borel transform function: B (1) ( t ) := − π t (cid:18) coth( πt ) − πt − πt (cid:19) = 2 π ∞ (cid:88) n =0 ( − n ζ (2 n + 4) t n +1 (14)= 2 π ∞ (cid:88) k =1 tk ( t + k ) (15)Here we have chosen to make the convenient rescaling of the Borel variable by a factor of π , to absorbthe powers of 1 /π in (10)-(11), which has the effect of placing the Borel poles at integer multiples of i ,rather than at integer multiples of πi . Then the exact one-loop effective Lagrangian is recovered via theBorel-Laplace integral (note the extra factor of π in the exponent): L (1) (cid:18) eBm (cid:19) = πB (cid:90) ∞ dt e − m πt/ ( eB ) B (1) ( t ) (16)The small t expansion (14) of B (1) ( t ) generates the asymptotic weak magnetic field expansion (9), whilethe partial-fraction expansion in (15) exhibits the meromorphic nature of the one-loop Borel transformfunction B (1) ( t ), with an infinite line of integer-spaced simple pole singularities along the imaginaryBorel axis, at t = ± ik , for k = 1 , , , . . . . Therefore, under analytic continuation, B → iE , where E is a constant background electric field, these poles lead to the non-perturbative imaginary part of theeffective Lagrangian in (3), with weak-field expansion in (4). In the strong electric field limit the leadingbehavior can be obtained by analytic continuation from the strong magnetic field expansion in (12):Im (cid:20) L (1) (cid:18) eEm (cid:19)(cid:21) ∼ β (cid:16) π (cid:17) E , eE (cid:29) m (17)consistent with the strong field limit of the exact polylog expression in (3). This also follows from theexplicit representation of the one-loop effective Lagrangian in terms of the Barnes gamma function G [5]: L (1) (cid:18) eBm (cid:19) = 2 B m (cid:34) −
112 + ζ (cid:48) ( −
1) + 116 (cid:18) m eB (cid:19) + (cid:32) −
112 + m eB − (cid:18) m eB (cid:19) (cid:33) ln m eB − (cid:18) − m eB (cid:19) ln Γ (cid:18) m eB (cid:19) − ln G (cid:18) m eB (cid:19)(cid:35) (18)This Barnes representation of the one-loop effective Lagrangian is particularly convenient for analyticcontinuation of B , since the analytic properties of the Barnes G function are well known [46, 47]. B. Borel Analysis at One-Loop Order
In preparation for the two-loop analysis, where a simple Borel representation of the form in (8) is notavailable, and a closed-form expression such as (18) in terms of a special function like the Barnes functionis not known, we ask how we can recover accurate approximations to the various exact results listed inthe previous sub-section. We do this here at one-loop and then extend these methods to two-loop orderin Section III. Specifically, we begin with just a finite number of terms of the one-loop perturbative weakmagnetic field expansion in (9), and seek to recover:1. the strong magnetic field limit in (12) [this is a weak-field to strong-field extrapolation];2. the non-perturbative imaginary part (4) of the effective Lagrangian in an electric field background[this is analogous to a Euclidean to Minkowski analytic continuation].We develop a modified Pad´e-Borel approach that leads to remarkable precision with surprisingly littleinput information. Note that [48] has already showed that Pad´e-Borel summation achieves very accurateextrapolations of the weak magnetic field series for the one-loop Euler-Heisenberg effective Lagrangian.However, for the analysis at two-loop we require even higher precision; hence the new procedure describedbelow.Given only a finite number of terms of the weak magnetic field expansion (9), or equivalently only afinite number of terms in the small t expansion of the Borel transform in (14)-(16), the key to an accurateextrapolation to other regions of the complex B plane (recall that B < B (1) N ( t ) := 2 π N − (cid:88) n =0 a (1) n (2 n + 1)! ( πt ) n +1 (19)Since the one-loop Borel transform is meromorphic, the optimal approach [39, 40] is to use a Pad´eapproximation, which expresses B (1) N ( t ) as a ratio of polynomials: P [ L,M ] (cid:16) B (1) N ( t ) (cid:17) = P (1) L ( t ) Q (1) M ( t ) , where P (1) L ( t ) Q (1) M ( t ) = B (1) N ( t ) + O (cid:0) t L + M +1 (cid:1) (20)where L + M = 2 N + 1. The zeros of the denominator polynomial Q M ( t ) approximate the true singu-larities (see (15)) of B (1) ( t ), which lie at t = ± ik for k (cid:54) = 0 ∈ Z due to our rescaling t → πt . See Figure2. Tables of the one-loop Pad´e-Borel poles for N = 10 input terms and for N = 50 input terms areshown below in (21) and (22), in which we see poles stabilizing along the imaginary Borel axis at integermultiples of ± i . - ���� - ���� ���� ���� - �� - ���� - ���� - ���� ���� ���� - �� - ������ FIG. 2. Poles of the Pad´e approximation PB (1) N ( t ) in (24) for the truncated Borel transform, shown for both N = 10 (left) and N = 50 (right). Note that all the poles lie on the imaginary axis and tend towards integermultiples of ± i . This can be contrasted with two-loop case in Figure 8. While a Pad´e approximation could be applied directly to the truncated asymptotic weak magnetic field expansion (9),a significantly better extrapolation [39, 40] is achieved by the Pad´e-Borel method [49], making a Pad´e approximation inthe Borel t plane rather than a Pad´e approximation in the original physical variable eB/m . Pad´e poles from 10 input terms: ± { . i, . i, . i, . i, . i } (21)Pad´e poles from 50 input terms: ± { . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, . i, (22)18 . i, . i, . i, . i, . i, . i, . i, . i } We dramatically improve the quality of the extrapolation if we take advantage of known informationabout the opposite (strong field) limit, which corresponds to including information about the t → ∞ behavior of B (1) ( t ). Here we appeal to the fundamental physical interpretation of the logarithmic behaviorof the strong field expansion (12) in terms of charge renormalization and the conformal anomaly [1, 2,5, 8, 9], which translates into the requirement that B (1) ( t ) ∼ β πt + . . . , t → + ∞ (23)We therefore choose an off-diagonal Pad´e-Borel approximant with M = L + 1 = N + 1 PB (1) N ( t ) := P (1) N ( t ) Q (1) N +1 ( t ) (24)Note that we do not impose that the overall coefficient of πt · PB (1) N ( t ) in the limit t → + ∞ be equalto the physical value, β = . We simply impose the functional form that πt · PB (1) N ( t ) → constant, as t → + ∞ . Remarkably, the physical value, β , emerges in the large N limit, already at N ≈
10. SeeFigure 3.With this procedure, our Pad´e analytic continuation (24) of the truncated weak-field expansion (19)leads to an approximate Borel-Laplace integral representation for the one-loop effective Lagrangian asin (16): L (1) N (cid:18) eBm (cid:19) = πB (cid:90) ∞ d t e − m πt/ ( eB ) PB (1) N ( t ) (25)Figure 4 shows the extrapolation of this expression from the weak-field limit to the strong-field limit.Starting with just ten input coefficients of the weak field expansion, the modified Pad´e-Borel expressionin (25) extrapolates accurately over more than 8 orders of magnitude. This is a significantly farther-reaching extrapolation than in [48], due to our improved Pad´e-Borel transform in (24). This modifiedPad´e-Borel transform also explains why the pole structure shown in Figure 2, and in Equations (21)-(22), FIG. 3. N dependence of the limiting value lim (cid:104) πt · PB (1) N ( t ) (cid:105) t →∞ , for N ranging from 1 to 50, for the near-diagonal Pad´e approximant of the truncated one-loop Borel transform function in (24). The blue dots indicatethe original values obtained from expanding πt · PB (1) N ( t ) about t → ∞ , and which appear to be tending towards1 /
3. The red dots show a 4-th order Richardson extrapolation of this data. Observe that the physical value β = 1 / N ≈ - - FIG. 4. The blue curve is a log-log plot of the modified Pad´e-Borel sum of the truncated weak field expansionin (25), L (1) N , plotted here using only N = 10 perturbative input terms. This is indistinguishable from the exactclosed-form Barnes function expression in (18), plotted here as the translucent blue band. The gold curve showsthe weak field expansion (9), truncated at N = 10. The red curve shows the leading strong field behavior of L (1) in (12). Note that the truncated weak-field expansion fails even below the Schwinger limit eB ≈ m , whilethe modified Pad´e-Borel sum accurately interpolates over many orders of magnitude between the weak-field andstrong-field behavior. This plot was made using units in which e = m = 1. is much more accurate than that in [48], where the physical form of the large t behavior (23) was notimposed on the Pad´e-Borel transform.Similarly we can use the approximate Borel representation in (25) to achieve our second goal: ana-lytically continuing from a magnetic background to an electric background, to extract the exponentiallysmall non-perturbative imaginary part of L (1) in (3). Since the Borel singularities are all on the imaginaryaxis, we rotate the Borel contour to generate the imaginary part of the one-loop effective Lagrangian fora constant background electric field. Figure 5 shows (as blue dots) the result of this calculation, startingwith 10 terms of the weak magnetic field expansion. The red curve shows the exact result in (3), summedover all instanton orders, while the gold curve shows the leading one-instanton term. The agreement isexcellent in the weak-field limit, and also extrapolates accurately to much stronger fields.In Figure 6 we show that the precision of our extrapolation is sufficiently high that we can probe the exponentially small higher-instanton corrections to Im L (1) (cid:0) eEm (cid:1) , by dividing out the one-instanton factor, E / (2 π ) exp (cid:0) − πm / ( eE ) (cid:1) . Fitting this ratio with one-term, two-term and three-term exponential fits, - - - FIG. 5. A log-log plot of the imaginary part of the electric field effective Lagrangian at one-loop, calculated using N = 10 (blue dots). The gold curve shows the leading one-instanton contribution, ∝ exp (cid:0) − πm / ( eE ) (cid:1) , whichdisplays a small but noticeable deviation in the strong field limit. The red curve is the exact expression for theimaginary part in (3), including the sum over all instantons. See also Figure 6. This plot was made using unitsin which e = m = 1. FIG. 6. The ratio [blue dots] of the imaginary part of the electric field effective Lagrangian Im L (1) ( E ), dividedby the leading exponential term from (4), derived from our extrapolation using as input just N = 10 terms fromthe perturbative expression for a magnetic field background. The gold, red, and purple curves show a fit for thisratio, based on one, two, and three exponentially small correction terms, respectively. The fit coefficients aregiven in (26), from which it is clear that these sub-leading exponential corrections tend to the known form (3) ofthe instanton sum. This plot was made using units in which e = m = 1. we obtain the successively improving approximations:Im L (1) (cid:18) eEm (cid:19) ≈ E π (cid:16) e − πm / ( eE ) + 0 . e − πm / ( eE ) + . . . (cid:17) Im L (1) (cid:18) eEm (cid:19) ≈ E π (cid:16) e − πm / ( eE ) + 0 . e − πm / ( eE ) + 0 . e − πm / ( eE ) + . . . (cid:17) (26)Im L (1) (cid:18) eEm (cid:19) ≈ E π (cid:16) e − πm / ( eE ) + 0 . e − πm / ( eE ) + 0 . e − πm / ( eE ) +0 . e − πm / ( eE ) + . . . (cid:17) We see that the coefficients of this instanton expansion approach the exact k factors in (4). Thisdemonstrates that our extrapolation from magnetic to electric field is exponentially accurate: it recoversseveral orders of the exponentially suppressed corrections to the non-perturbative imaginary part ofthe one-loop effective Lagrangian for an electric background field, using as input only 10 terms of theperturbative weak magnetic field expansion. Another way to see this is to plot our improved Pad´e-Boreltransform (24), shifted slightly from the imaginary axis: Figure 7 shows the resulting poles at integerspacing along the imaginary axis, with residues following the exact 1 /k behavior in (15). FIG. 7. Singularity structure of the Borel transform PB (1) N ( t ), for N = 10, just offset from the imaginary axis, t → it + (cid:15) . The plot shows the real part (blue solid curve) of PB (1) N ( it + 1 / t = ik along the imaginary Borel axis, with residues falling off quadratically as 1 /k (dashed black curve). III. THE TWO-LOOP EULER-HEISENBERG EFFECTIVE LAGRANGIANA. Exact Results at Two-Loop Order
Whereas the one-loop Euler-Heisenberg effective Lagrangian in a constant magnetic field has a simpleBorel-Laplace integral representation (8), and can be expressed exactly in terms of the Barnes doublegamma function (18), no such closed-form expressions are known at two-loop for a constant magnetic orelectric field background. The most explicit representation for a magnetic background field is Ritus’sexact double-integral representation [8–11] L (2) (cid:18) eBm (cid:19) = B (cid:90) ∞ d tt e − t m / ( eB ) ( J + J + J ) (27)where J = 2 tm eB (cid:90) d ss (1 − s ) (cid:20) cosh( t s ) cosh( t (1 − s )) a − b ln ab − t coth t + 5 t s (1 − s ) (cid:21) (28) J = − (cid:90) d ss (1 − s ) (cid:20) c ( a − b ) ln ab − − b cosh( t (1 − s )) b ( a − b ) + b cosh t + 12 b − t s (1 − s ) (cid:21) (29) J = (cid:18) tm eB (cid:18) ln (cid:18) tm eB (cid:19) + γ − (cid:19)(cid:19)(cid:18) t coth t − − t (cid:19) (30)and the functions a , b and c are defined as a = sinh( t s ) sinh( t (1 − s )) t s (1 − s ) , b = sinh tt , c = 1 − a cosh( t (1 − s )) (31)Unlike the one-loop case (8), this two-loop expression (27)-(31) is not directly in Borel form. However,we can expand this as a perturbative weak-field series in eBm : L (2) (cid:18) eBm (cid:19) ∼ B (cid:18) eBm (cid:19) ∞ (cid:88) n =0 a (2) n (cid:18) eBm (cid:19) n , eB (cid:28) m (32)There is no simple closed-form expression for the coefficients a (2) n . However, the two-loop weak-fieldcoefficients a (2) n can be generated by a suitable expansion of the integral representation (27). In [14],15 terms of such a weak magnetic field expansion were obtained, which was enough perturbative datato argue that the leading large order growth of the two-loop expansion coefficients a (2) n has exactly thesame form as the leading large order growth (11) of the one-loop expansion coefficients a (1) n : a (2) n ∼ ( − n Γ(2 n + 2) π n +2 + corrections , n → ∞ (33)In this paper we have pushed the perturbative expansion (32) to much higher order, obtaining 50 terms ofthe two-loop weak magnetic field expansion. This expansion must be organized appropriately to respectthe various subtractions, in order to keep the s integrals finite. Our expansion strategy is described inAppendix V. This new weak-field perturbative data confirms the leading result (33), and furthermoreallows analysis of the subleading corrections: see Section III B below. The first 25 coefficients a (2) n arelisted in Appendix VI, and the first 50 coefficients are listed in an accompanying Supplementary file.This is the perturbative input data on which our subsequent Borel analyses are based.The leading strong magnetic field behavior at two-loop order is also known [8–10] L (2) (cid:18) eBm (cid:19) ∼ · B (cid:18) ln (cid:18) eBπm (cid:19) − γ −
56 + 4 ζ (3) (cid:19) , eB (cid:29) m (34)where ζ (3) ≈ . β = 1 /
4, in the prefactor of the leading logarithmic factor of the strong field limit (34). However, exact closed-form expressions and Borel representations are known at two-loop for a constant self-dual field ,corresponding to the generating function of amplitudes for low-momentum external photons of fixed helicity [50–52]. - ���� - ���� ���� ���� - �� - ���� - ���� - ���� ���� ���� - �� - ������ FIG. 8. Poles of the Pad´e approximation PB (2) N ( t ) in (37) for the truncated Borel transform, shown for N = 10(left) and N = 50 (right). Note that the poles appear to be accumulating at ± i . Contrast with one-loop case inFigure 2 where the poles are integer-spaced along the imaginary Borel axis. Analytically continuing from a magnetic to an electric background, B → iE , the results in (33) and(34) yield the leading contributions to the non-perturbative imaginary part of the two-loop effectiveLagrangian L (2) (cid:0) eEm (cid:1) : Im (cid:20) L (2) (cid:18) eEm (cid:19)(cid:21) ∼ πE e − πm / ( eE ) , eE (cid:28) m β (cid:16) π (cid:17) E , eE (cid:29) m (35)These leading behaviors at two-loop are structurally identical to the leading behaviors at one-loop order(recall (4) and (17)). However, we show below that the subleading corrections at two-loop order are verydifferent from the corrections at one-loop order. B. Borel Analysis at Two-Loop Order
Based on the leading factorial growth in (33), and in analogy to the one-loop case analyzed in SectionII B, we define the two-loop (truncated) Borel transform B (2) N ( t ) := 2 N − (cid:88) n =0 a (2) n (2 n + 1)! ( πt ) n +1 (36)Note that we have adopted the same rescaling of t by a factor of π , as at one loop in (19), becausethe leading growth of the two-loop coefficients in (33) matches that at one-loop (11). Recall also thatat one-loop we obtained high-precision analytic continuations by using a modified Pad´e-Borel transform(24) that incorporated information about the strong magnetic field behavior of the one-loop effectiveLagrangian. Since the strong magnetic field limit in (34) has the same functional form as at one-loop,(12), we adopt the same strategy here. We analytically continue the truncated Borel transform (36) viaa near-diagonal Pad´e approximant, which encodes the logarithmic strong field behavior (34) at two-loop: PB (2) N ( t ) = P (2) N ( t ) Q (2) N +1 ( t ) (37)Figure 8 shows the singularities of this two-loop Pad´e-Borel transform, PB (2) N ( t ), based on 10 or 50 inputcoefficients (the first 25 coefficients are listed in Appendix VI, and the first 50 coefficients are listed inan accompanying Supplementary file). These plots confirm that the leading singularities are at t = ± i ,but they also show that the Borel plane singularity structure is much richer than in the one-loop case,1 - - FIG. 9. The blue curve is a log-log plot of the modified Pad´e-Borel sum of the truncated weak field expansionin (38), L (2) N , plotted here for N = 10. Compare with the one-loop result plotted in Figure 4. In contrast tothe situation at one loop, there is no exact expression for the Lagrangian at two loop against which to comparethe resummation. The gold curve shows the weak field expansion (32), truncated at N = 10. The red curveshows the leading strong field behavior in (34). Once again, we see that the truncated weak field expansionfails before the Schwinger critical field eB ≈ m , whereas the modified Pad´e-Borel sum interpolates over manyorders of magnitude between the weak-field and strong-field behavior. This plot was made using units in which e = m = 1. where the singularities are just isolated poles at integer multiples of the leading ones: recall Figure 2. Attwo-loop, Figure 8 suggests that the leading singularities, at t = ± i , appear to be branch points. Recallthat a Pad´e approximant represents branch cuts as lines of poles (interlaced by the Pad´e zeros) thataccumulate to the branch points [39, 40, 53]. This novel branch point structure is probed in more detailin Section III C below.Given the Pad´e approximation (37) to the Borel transform, the approximate two-loop effective La-grangian for the magnetic background is recovered by the Laplace transform L (2) N (cid:18) eBm (cid:19) = πB (cid:90) ∞ d t e − m πt/ ( eB ) PB (2) N ( t ) (38)The quality of this two-loop expression (38) as an accurate extrapolation of the two-loop effective La-grangian L (2) (cid:0) eBm (cid:1) from weak magnetic field to strong magnetic field, and from magnetic to electric field,relies on the quality of the analytic continuation of the Borel transform in the Borel plane, here providedby the Pad´e approximation (37).Using only 10 input perturbative coefficients a (2) n , in Figure 9 we plot the resulting Borel representation L (2) N (cid:0) eBm (cid:1) from (38). Similar to the one-loop result in Figure 4, we see that our modified Pad´e-Borelrepresentation at two-loop also extrapolates accurately over many orders of magnitude from the weak
10 20 30 40 500.10.20.30.40.5
FIG. 10. N dependence of the limiting value lim (cid:104) πt · PB (2) N ( t ) (cid:105) t →∞ , for N ranging from 1 to 50, for the near-diagonal Pad´e approximant of the truncated one-loop Borel transform function in (37). The blue dots indicate thevalues obtained from expanding πt · PB (1) N ( t ) about t → ∞ , and which tend towards the physical value β = 1 / - - FIG. 11. A log-log plot of the imaginary part (blue dots) of the two-loop effective Lagrangian in an electric field,calculated using N = 10 input perturbative terms. The gold curve shows the leading weak-field one-instantoncontribution in (35). This plot was made using units in which e = m = 1. magnetic field to the strong magnetic field regime. The excellent agreement of this extrapolation toasymptotically large magnetic field can be attributed to the fact that we have constructed our Pad´e-Borel transform in such a way that it incorporates the form of the known logarithmic behavior (34) ofthe two-loop effective Lagrangian: the Borel transform, which is generated from an expansion about t = 0, should be proportional to 1 / ( πt ) as t → + ∞ . Note that, (as at one-loop) we do not enforce thatthe coefficient of proportionality be equal to β . Remarkably, once again this fact emerges from our 50input coefficients, even though these coefficients were generated in the opposite limit near t = 0. SeeFigure 10.We can also use the approximate Borel expression (38) to achieve our second goal: analytic continuationfrom a magnetic to an electric background, producing an exponentially small non-perturbative imaginarypart of L (2) N (cid:0) eEm (cid:1) . The result is shown in Figure 11, again showing good agreement over many orders ofmagnitude of the external field strength. However, there is a clear deviation from this leading weak fieldcontribution even at eE ≈ m , coming from fluctuations about this one instanton term which were notpresent at one-loop in (4). This deviation is analyzed in the next Section. See Figure 13. C. Power Law Corrections at Two-Loop Order
The Pad´e-Borel pole structure in Figure 8 suggests that the leading singularities at two-loop are branchpoints rather than poles. However, the leading large-order growth in (33) is the same as the one-loopleading large-order growth in (10), and this leading growth is associated with Borel poles. The resolutionof this apparent puzzle is that each of the symmetric leading Borel singularities, at t = ± i , is in fact a
10 20 30 40 50 - - - - - FIG. 12. The large order behavior of the modified coefficients ˜ a (2) n defined in (39). Different growth rates areshown here: f ( n ) = ( − n π n +2 / Γ (cid:0) n + (cid:1) (blue circles), f ( n ) = ( − n π n +2 / Γ (cid:0) n + + (cid:1) (gold squares),and f ( n ) = ( − n π n +2 / Γ (cid:0) n + − (cid:1) (red diamonds). The form involving Γ(2 n + ) is clearly favored. a (2) n ≡ a (2) n − ( − n Γ(2 n + 2) π n +2 (39)We now analyze the large order behavior of the modified coefficients ˜ a (2) n . Ratio tests indicate thefollowing leading growth of the ˜ a (2) n :˜ a (2) n ≈ ( − . × ( − n Γ (cid:0) n + (cid:1) π n +2 + . . . (40)See Figure 12, where we have adjusted the offset shift, finding the best agreement with the offset .After analytic continuation to an electric field, this corresponds to the following power-law correctionto the imaginary part of the two-loop effective Lagrangian:Im (cid:20) L (2) (cid:18) eEm (cid:19)(cid:21) ∼ πE e − πm / ( eE ) (cid:20) − . (cid:18) eEπm (cid:19) / + . . . (cid:21) (41)This form of the leading correction suggests a fluctuation expansion in powers of (cid:0) eEm (cid:1) / :Im (cid:20) L (2) (cid:18) eEm (cid:19)(cid:21) ∼ πE e − πm / ( eE ) (cid:20) d (cid:18) eEπm (cid:19) / + d (cid:18) eEπm (cid:19) / + d (cid:18) eEπm (cid:19) / + . . . (cid:21) (42)In Figure 13 we plot the result of fitting the imaginary part of the two-loop effective Lagrangian directlyfrom the integral representation. Using the fit interval eEm ∈ [10 − ,
1] we obtain fit parameters: d = − . d = 2 . d = − .
94. Figure 13 illustrates the improved agreement with the successive weak-field corrections. This form of the fluctuations about the one-instanton term fills in the first set of missingdots in Ritus’s conjectured expression (5). - - FIG. 13. The imaginary part of the electric field effective Lagrangian (blue dots), compared with the leadingweak-field one-instanton contribution (gold), and the fit including the additional weak-field power-law correctionsin (42) (red). This plot was made using units in which e = m = 1. D. Probing the Higher Instanton Terms
In fact, the power-law corrections discussed in the previous Section are not the whole story. Theweak-field expressions in (35) and (42) only include the effects of the leading Borel singularity at t = ± i :these are the “one-instanton” effects. But we also expect that there should be multi-instanton effectsassociated with Borel singularities at all integer multiples of the leading ones. These would appear asexponentially small corrections to the large-order growth of the perturbative expansion coefficients a (2) n .4 - - FIG. 14. Plot of the real part of the Pad´e-Borel transform, Re[ PB (2) N ( it + 1 / N = 50, showing thesingularity structure of the Pad´e-Borel transform along the imaginary axis. Note that without the conformal map,the accumulation of poles from the Pad´e-Borel approximation obscures the true singularity structure associatedwith the physical higher instanton terms. Compare with Figure 16 where the physical multi-instanton Borelsingularities at t = 2 i and t = 3 i are resolved. Therefore, the subleading corrections to the large order growth of the two-loop perturbative expansioncoefficients should have the following structural form: a (2) n ∼ ( − n Γ(2 n + 2) π n +2 (cid:8)(cid:0) (cid:1) + (cid:0) exponentially small corrections (cid:1) × (cid:0) power law corrections (cid:1)(cid:9) (43)Even though the leading large-order growth has the same form as at one-loop, compare (11) and (33),the structure of the corrections is very different: there are power-law corrections followed by muchsmaller exponentially suppressed corrections, which themselves have power-law corrections. This fact isdirectly responsible for the novel structure (5) of the non-perturbative imaginary part at two-loop order.Having studied the structure of the power-law corrections in the previous subsection, we now turn to theexponentially smaller corrections.At one-loop the first few exponentially small corrections can be resolved, see Figure 6, because thereare no power-law corrections (recall (11)), but at two-loop it is much more difficult because of theexistence of the (much larger) power-law corrections to the leading instanton term. These power-lawcorrections obscure the exponentially small corrections associated with multi-instantons. This problemcan be ameliorated by using more sophisticated Borel techniques, beyond Pad´e-Borel. Indeed, thisproblem is directly related to the fact that the Pad´e-Borel approximation represents the leading branchcut as a line of poles, which therefore obscures the existence of genuine multi-instanton singularities,which also lie on the imaginary axis, and are also expected to be branch points. See Figure 14, whichplots the Pad´e-Borel transform along the imaginary axis: the leading singularity at t = i can be seen, butbeyond that one sees coalescing Pad´e poles that are attempting to represent the branch cut t ∈ [ i, i ∞ ].This also explains why the multi-instanton Borel singularities are clear at one-loop from the Pad´e-Borelpole distribution in Figure 2 (because they are simple poles), but are not seen directly at two-loop orderfrom the Pad´e-Borel pole distribution in Figure 8 (because they are branch points). Fortunately, thereis a simple way to resolve this problem.The first step is to confirm that there are indeed integer-repeated Borel singularities, and to determineif they are in fact branch points. This problem can be resolved as follows [38–40]. We use a conformalmap [49, 54, 55] to map the doubly-cut Borel plane (based on the two symmetric leading branch pointsingularities at t = ± i ) into the unit disk in the conformal z plane. Specifically, the relevant conformalmap for this configuration is: t = 2 z − z , z = t √ t (44)A re-expansion inside the unit disk to the original order, followed by a Pad´e approximation within theunit disk, separates subleading branch-points [38–40], as shown in Figure 15.The doubly-cut t plane is mapped to the interior of the unit disk, with the edges of the cuts mapped tosegments of the boundary, the unit circle. We expand the truncated Borel transform inside the conformal5 - - - - FIG. 15. Poles of the Pad´e-Borel approximation in the conformally mapped z plane. With N = 50 terms, thefirst three Borel singularities can be resolved as accumulation points of Pad´e poles located on the unit circle at θ = ± π , ± π , ± arctan (cid:16) √ (cid:17) , denoted by the green arrows. disk, and truncate at the same order as the t expansion: B (2) N (cid:18) z − z (cid:19) = N − (cid:88) n =0 b (2) n z n + O ( z N ) (45)This expansion uniquely defines the coefficients b (2) n . By construction, this expansion is convergent withinthe conformal disk. We then make a near-diagonal Pad´e approximation, and compute its poles. Thesesingularities are shown in Figure 15. The poles accumulating to z = ± i correspond to the leadingsingularities, since the conformal map (44) takes t = ± i to z = ± i . The next cluster of poles accumulateto z = ± e ± iπ/ , which are the conformal map images (on either side of the leading branch cuts) of thetwo-instanton singularities at t = ± i . The third cluster of z -plane poles in Figure 15 accumulate to theimages of the three-instanton singularities at t = ± i . Thus the conformal map reveals the existenceof integer-repeated higher instanton Borel singularities, and shows that they all have associated branchcuts. - - FIG. 16. Plot of the real part of the Pad´e-Conformal-Borel transform, Re[
PCB (2) N ( it + 1 / N = 50,showing the singularity structure of the conformally mapped Borel transform along the imaginary axis. The plotreveals the existence of higher Borel singularities at multiples of the leading singularity at t = i , correspondingto the multi-instanton expansion. It also confirms the branch cut nature of the singularity structure at twoloop. Compare with Figure 7 at one loop, where the Borel singularities are simple poles, and compare with theanalogous plot at two loop, but without the conformal map, in Figure 14. z -plane disk back to the Borel t using the inverse transformation in (44) [38, 40]. This produces thePad´e-Conformal-Borel approximation, and this is plotted along the imaginary t axis in Figure 16. We seethat the one-instanton, two-instanton and three-instanton singularities are all resolved. This should becontrasted with the result of the Pad´e-Borel approximation, without the conformal map, where nothingbeyond the one-instanton singularity can be clearly resolved: recall Figure 14. This failure of the Pad´e-Borel approximation to resolve higher instanton singularities is a direct consequence of the fact thatthe Pad´e approximation represents the leading branch cuts as sequences of poles accumulating to theleading branch points, as seen in Figure 14, and these poles obscure the existence of the genuine physicalhigher-instanton singularities, which are themselves branch points. The conformal mapping resolves thisproblem, as can be seen in Figure 16.To find the nature of the two-instanton Borel singularity at t = ± i , we study the approach to thepoint z = e iπ/ in the conformal disk. Writing z = r e iπ/ , we assume a power-law behavior (1 − r ) β forthe imaginary part of the Borel transform, and fit a good fit with β = − . See Figure 17. When mappedback to the Borel t plane, this corresponds to an imaginary part of the two-loop effective Lagrangian, atthe two-instanton level, of the formIm (cid:20) L (2) (cid:18) eEm (cid:19)(cid:21) two-instanton ∼ πE (cid:114) m eE e − πm / ( eE ) (46)which agrees with the form of the higher-instanton fluctuation prefactor conjectured by Ritus: recallEq. (5). It is quite remarkable that this doubly-exponentially-suppressed term can be deduced directlyfrom just the first 50 perturbative expansion coefficients of the effective Lagrangian in a magnetic fieldbackground. ���� ���� ���� ���� - ����� - ����� - ���������� FIG. 17. Blue: plot of the imaginary part of (1 − r ) / times the Pad´e-Conformal-Borel transform along the radialline z = r e iπ/ inside the conformal disk, approaching the z -plane image of the two-instanton Borel singularity.Red: approximate extrapolation of this plot to the singularity at z = e iπ/ . IV. CONCLUSIONS
We have shown that accurate extrapolations and analytic continuations of the two-loop Euler Heisen-berg effective Lagrangian can be recovered from a relatively modest number of terms of the perturbativeweak magnetic field expansion. These perturbative terms are generated from an expansion of Ritus’sseminal results for the renormalized two-loop effective Lagrangian in terms of two-parameter integrals[8–11]. The new physical effect at two-loop, compared to the well-known one-loop Euler-Heisenberg effec-tive Lagrangian, is that the Borel transform has branch point singularities, rather than just simple poles.These branch points reflect the interactions between virtual particles, and have the effect of producingfluctuation expansions multiplying the terms in the weak-field instanton expansion for the imaginary partof the effective Lagrangian in an electric field background. In order to probe these fluctuation correctionswe need high precision extrapolations, which we achieve using a combination of Pad´e approximationsand conformal maps to obtain a sufficiently accurate analytic continuation of the finite order truncationof the associated Borel transform [39, 40]. We have also incorporated the known physical informationabout the strong magnetic field limit, which is fixed by the QED beta function, and which determines the7
FIG. 18. The three diagrams which contribute to the Euler-Heisenberg effective Lagrangian at l = 3 loop order. functional form of the asymptotic limit of the Borel transform. In particular, with the input of just 10terms of the perturbative weak field expansion we find an accurate extrapolation from the weak magneticfield regime to the strong magnetic field regime, over many orders of magnitude. See Figures 4 and 9for the one-loop and two-loop results, respectively. Using 50 terms of the perturbative weak magneticfield expansion at two-loop order, we analytically continue to an electric field background and obtainnew information about the structure of the instanton expansion of the imaginary part of the effectiveLagrangian. We resolve the leading power law correction at the one-instanton level, and also identifythe exponentially further suppressed two-instanton term.Our analysis was motivated by the question of whether such extrapolations and associated non-perturbative information could be accessible at higher loop order (i.e. higher terms in the expansion (1)in the fine structure constant), starting not from a closed form multi-parameter integral representation,but from an explicit finite order perturbative expansion. This is because even at three-loop order (seeFigure 18) it has so far not been possible to find a parametric integral representation (a 4-fold parameterintegral at three loop) of the Euler Heisenberg effective Lagrangian, even though the exact propagatorsin a constant background field are known in a relatively simple integral representation form [17–19]. Ourresults suggest that an alternative strategy might be more practical: work instead with a perturbativeexpansion of the propagators, thereby generating a finite-order perturbative expansion of the l -loop ef-fective Lagrangian, from which extrapolations to other parametric regimes could be performed. To bepractical, such extrapolations must be achievable with a “reasonable” amount of perturbative input, andour results suggest that this may indeed be possible. To generate the perturbative expansion at higherloop order, one needs an efficient way to compute the renormalized effective Lagrangian, for exampleusing the background-field integration-by-parts methods developed in [16, 56].Certain structural facts are known about the Euler Heisenberg effective Lagrangian at higher looporders, and these could be used to constrain the higher-loop computations. The exponentiation e απ ofthe leading weak electric corrections to the imaginary part, as in (7), leads to a conjecture [15] that theleading large order behavior of the perturbative expansion coefficients has the same form for all looporders l : a ( l ) n ∼ ( − n Γ(2 n + 2) π n +2 , n → ∞ , ∀ l (47)Indeed, this conjectured behavior is the reason for the choice of the overall normalization of the pertur-bative expansion coefficients in (2): with this normalization choice we recover the exponential factor e απ in (7). This conjecture, along with the exponentiation in (7), would be interesting to confirm or disprovebeyond two-loop order. Physically, this correspondence is motivated by the interpretation of the mass m appearing in the exponential instanton factor, e − πm / ( eE ) , as the renormalized physical electron mass[10]. Already at two-loop order, this correspondence between the renormalized mass defined from thereal or imaginary part of the effective Lagrangian is sensitive to the finite mass renormalization. Thesituation at three-loop order is not yet clear [17–19], and we hope that the methods described here mightprovide an alternative approach to shed light on this open question.The leading strong magnetic field behavior at l -loop order (with l ≥
2) is also known, arising from theCallan-Symanzik equation in the strong field (or massless) limit [8–10]: L ( l ) (cid:18) eBm (cid:19) ∝ B (cid:18) ln (cid:18) eBπm (cid:19)(cid:19) l − + . . . , eB (cid:29) m (48)The overall coefficient is expressed in terms of the beta function coefficients up to order l . This leadingcontribution comes from the renormalon-like “ring diagram” with ( l −
1) fermion loops connected in asingle ring by ( l −
1) photon propagators. See Figure 19. This general fact could be used to constrainthe asymptotic behavior of the Borel transform at l -loop order. Deeper understanding of the strong It would be interesting to apply these perturbative Borel methods also to the reducible diagrams studied in [57, 58]. FIG. 19. Two equivalent views of the irreducible l -loop diagram giving the dominant strong-field behavior in(48) for the l -loop Euler-Heisenberg effective Lagrangian. There are ( l −
1) fermion loops, with the double linesdenoting fermion propagators in the constant background field, and one overall photon loop. field limit at higher order in the fine structure constant α may also shed light on the computation ofscattering amplitudes in strong background fields, in particular those associated with ultra-intense lasers[23, 24]. For example, seminal work by Ritus and Narozhnyi has made predictions for the resultingstructure at higher loop order for the special case where the background laser field is represented as aconstant crossed field [25–31]. V. APPENDIX: RITUS’S EXACT DOUBLE-INTEGRAL REPRESENTATION
The two-loop Euler-Heisenberg effective Lagrangian can be written as the double integral L (2) (cid:18) eBm (cid:19) = B (cid:90) ∞ d tt e − tm / ( eB ) ( J + J + J ) (49)with J = 2 tm eB (cid:90) d ss (1 − s ) (cid:20) cosh ts cosh t (1 − s ) a − b ln ab − t coth t + 5 t s (1 − s ) (cid:21) (50) J = − (cid:90) d ss (1 − s ) (cid:20) c ( a − b ) ln ab − − b cosh t (1 − s ) b ( a − b ) + b cosh t + 12 b − t s (1 − s ) (cid:21) (51) J = (cid:18) tm eB (cid:18) ln (cid:18) tm eB (cid:19) + γ − (cid:19)(cid:19)(cid:18) t coth t − − t (cid:19) (52)and a = sinh ts sinh t (1 − s ) t s (1 − s ) , b = sinh tt , c = 1 − a cosh t (1 − s ) (53)To generate a weak magnetic field expansion, for J and J we expand each term in the above expres-sions at small t to order O ( t n ). This is straightforward for most terms, withln ab = n (cid:88) k =0 k (cid:0) s k + (1 − s ) k − (cid:1) k (2 k )! t k (54)For the factors ( a − b ) − p , we first expand a − b = t n − (cid:88) k =0 (cid:20) k + 4)! 1 − (1 − s ) k +4 s (1 − s ) − k + 3)! (cid:21) t k = t n − (cid:88) k =0 A k t k (55)Then, the coefficients of this Taylor series raised to an arbitrary negative power can be generatedrecursively ( a − b ) − p = 1 t p n + p (cid:88) k =0 A ( − p ) k t k (56)9where (for A (cid:54) = 0) A ( − p )0 = 1 A p (57) A ( − p ) k = 1 kA k (cid:88) (cid:96) =1 [ (cid:96) (1 − p ) − k ] A (cid:96) A ( − p ) k − (cid:96) , k ≥ (cid:32) n (cid:88) k =0 a k t k (cid:33)(cid:32) n (cid:88) k =0 b k t k (cid:33) = n (cid:88) k =0 c k t k (59) c k = k (cid:88) (cid:96) =0 a (cid:96) b k − (cid:96) (60)Although certain terms in J and J look divergent with respect to the s integral, they are exactlycanceled by the expansion of other terms in the integrand. In addition, each of the remaining termscontains a factor of s (1 − s ), leaving well-defined integrals which result in polynomials in t . For J , thereis no s integral to perform, and so we can just expand the entire expression at small t to O ( t n ), yieldingpolynomials in t and polynomials multiplied by ln tm eB . Now, all the t integrals can be performed, wherewe can make use of the result (cid:90) ∞ d t e − m t/ ( eB ) t n ln (cid:18) tm eB (cid:19) = (cid:18) eBm (cid:19) n +1 Γ(2 n + 1) ψ (0) (2 n + 1) , n > −
12 (61)with ψ (0) ( x ) the digamma function. With this algorithm, we were able to generate fifty coefficients inthe weak magnetic field expansion of L (2) (cid:0) eBm (cid:1) , whereas previous analysis only obtained fifteen coef-ficients [14]. The first 25 weak magnetic field expansion coefficients of the two loop Euler-HeisenbergLagrangian can be found in Appendix VI, and the first 50 coefficients are contained in an accompanyingSupplementary file.0 VI. APPENDIX: COEFFICIENTS OF THE TWO-LOOP WEAK MAGNETIC FIELDEXPANSION n a (2) n − − − − − − − − − − − ACKNOWLEDGMENTS
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