Thermal Schwinger pair production at arbitrary coupling
TThermal Schwinger pair production at arbitrary coupling
Oliver Gould ∗ and Arttu Rajantie † Department of Physics, Imperial College London, SW7 2AZ, UK (Dated: June 12, 2017)We calculate the rate of thermal Schwinger pair production at arbitrary coupling in weak externalfields. Our calculations are valid independently of many properties of the charged particles produced,in particular their spin and whether they are electric or magnetic. Using the worldline formalism,we calculate the logarithm of the rate to leading order in the weak external field and to all ordersin virtual photon exchange, taking us beyond the perturbative expansion about the leading order,weak coupling result.
PACS numbers: 11.10.Wx, 11.15.Kc, 11.27.+d, 12.20.Ds, 14.80.Hv
I. INTRODUCTION
In the presence of an electric field, empty space is un-stable to the production of electron-positron pairs, calledSchwinger pair production [1]. The usual perturbativevacuum is not the true vacuum, the lowest energy state,and hence it decays. At finite temperature, the energyavailable from the thermal bath enhances the rate of de-cay.For weak coupling much has been done to gener-alise Schwinger’s original result, including the effect oftemporal and spatial variation in the external field [2–9]; the presence of an additional high energy photonor other particle [10–14]; a finite temperature [15–24];higher loops [25–29] and back reaction [30–34]. However,at stronger coupling, where perturbation theory breaksdown, much less is known.In this paper we calculate the rate of Schwinger pairproduction from a thermal bath, making no assumptionsabout the strength of the coupling. We do though restrictourselves to weak external fields. Our results are arguedto be valid for the full range from zero to infinite coupling[27, 35] (see, however, [36–38]).There are many applications of this calculation but ourinterest stems from the wish to better understand thepair production of magnetic monopoles [39–41]. Theo-retical understanding of this is poor and there is a press-ing need for concrete calculations of rates and cross sec-tions for ongoing experimental searches such as MoEDAL[42] at CERN. The Dirac quantisation condition im-plies that if magnetic monopoles exist, they are neces-sarily strongly coupled. Their charge, g , must satisfy g = 2 πn/e ≈ . n , where n ∈ Z and e is the charge ofthe positron. As a result of this, calculations are difficultand cross sections for their pair production are extremelypoorly understood.In the collisions of elementary particles it has been ar-gued that the pair production of ’t Hooft-Polyakov mag-netic monopoles [40, 41] is exponentially suppressed by ∗ [email protected] † [email protected] [43] e − π/e ≈ − , (1)even at arbitrarily high energies. The suppression can beseen as due to its large, coherent structure and the smalloverlap of this state with the initial, perturbative state.An analogous exponential suppression has been explicitlydemonstrated for soliton pair production in a particularscalar theory [44–46]. Note though that this argumentdoes not apply to pointlike, Dirac monopoles. For them,the lack of any small parameter has meant that therehave been no estimates of pair production cross sectionswhich have been derived from first principles. Conse-quently accelerator searches for magnetic monopoles canonly yield upper bounds on the production cross section[47], they cannot constrain the mass, spin or charge ofmagnetic monopoles.In this paper, we consider a different process, relevantto physical situations where there are strong magneticfields and high temperatures. In such situations pairproduction proceeds via the magnetic dual of Schwingerpair production, as first considered at zero temperatureby Affleck, Manton and Alvarez [35, 48]. For this pro-cess the arguments of Ref. [43] do not apply. Furtherwe can calculate the rate of pair production from firstprinciples, the result being valid for both Dirac and ’tHooft-Polyakov monopoles (see appendix B).In particular, strong magnetic fields and high temper-atures arise in heavy ion collisions [49, 50], though notin elementary particle collisions. The difference is crucialand our results suggest that, for sufficiently light mag-netic monopoles, pair production in heavy ion collisionsis not exponentially suppressed as in Eq. (1). Like thecase of ( B + L ) violation [51–56], we suggest that this isbecause, in the initial thermal state, the energy is spreadacross many degrees of freedom.We can study magnetic monopole Schwinger pair pro-duction via its electromagnetic dual as we do not con-sider electric and magnetic charges simultaneously. Inthis case the duality amounts simply to a relabelling ofelectric degrees of freedom and charges as magnetic. Wewill, in the bulk of the paper, refer to pair productionof particles with charge g in an external field E , whether a r X i v : . [ h e p - t h ] A ug electric or magnetic. The mass of the charged particles isdenoted by m . As our calculation reduces to a semiclas-sical one, we only rely on the classical electromagneticduality.Our calculation is also of relevance to Schwinger pairproduction of atomic nuclei, especially those with charges Ze (cid:38) Z (cid:38)
3) where the usual weak coupling ap-proaches break down, and to pair production of quarksin QCD, in the Abelian dominance approximation [57–61]. Further it gives an all-orders correction to the knownweak coupling results. This will be of interest to currentand future experimental studies of Schwinger pair pro-duction (see for example [62] for a discussion), as well asto multi-loop and asymptotic analyses of the QED per-turbation series [36–38].In Sec. II we set up our calculation using the world-line formalism, deriving an expression for the thermalSchwinger rate at arbitrary coupling. In Sec. III weexplain a key approximation that we make, the dilute in-stanton gas approximation. In Sec. IV we derive analyticresults in various limits and in Sec. V we extend beyondthese limits via numerical calculations. We also discussthe general form of the rate in terms of a phase diagram.In Sec. VI we conclude and suggest further work.Throughout we use natural units, c = (cid:126) = k B = 1. II. WORLDLINE EXPRESSIONA. Zero temperature rate
Physically, we consider at an initial time a state, suchas a thermal state. We choose the state such that, in theabsence of an external field, there are no net productionor annihilation rates. If we then adiabatically turn onan external field, the initial state becomes unstable to anet production of charged particles. We wish to calculatethis rate.We denote by | Ω (cid:105) the zero temperature state in theabsence of the external field, the so called false vacuum.The probability of the decay of this state is given by P = 1 − |(cid:104) Ω | ˆ S | Ω (cid:105)| = 1 − e −V Γ , (2)where ˆ S is the S-matrix including the external field. Asboth the false vacuum state and the external field arehomogeneous, the quantity of interest is the probabilityper unit spacetime volume, or the rate per unit volume,Γ. The rate of pair production is given by twice theimaginary part of the energy density of the initial state.Γ = 2 V Im( − i log (cid:104) Ω | ˆ S | Ω (cid:105) )= 2Im( E ) , (3)where V is the volume of spacetime and E is the energydensity of the false vacuum. Note that for this to makesense we should do the calculation in a finite volume andtake the volume to infinity at the end. We can analytically continue Eq. (3) to Euclideantime, as the energy density can equally well be calcu-lated in Euclidean time. It then becomes [63]Γ = 2 V Im( − log (cid:104) Ω | ˆ S E | Ω (cid:105) ) , (4)where ˆ S E refers to the “S-matrix” corresponding to Eu-clidean time evolution. The generalisation of this resultto non-zero temperatures can then easily be made. Us-ing the Matsubara formalism, finite temperatures simplycorrespond to finite Euclidean time extents and periodicboundary conditions [64].We consider quantum electrodynamics (QED) andscalar quantum electrodynamics (SQED) in 4D flatspacetime. In the worldline formalism these two theo-ries are related, the only difference being the presence ofa spin factor in the QED worldline path integral. In Ap-pendix A we show that the spin factor does not turn upin the leading term for weak external fields. As we willmake this approximation below, we restrict our atten-tion to SQED, the final results also being valid for QED.SQED is the model of a photon, A µ , interacting with amassive charged scalar particle, φ , with charge g . Theintroduction of the external field, A ext µ , is achieved byshifting the gauge field, A µ → A µ + A ext µ , in the covari-ant derivative of φ . The Euclidean Lagrangian is then L SQED := 14 F µν F µν + D µ φ ( D µ φ ) ∗ + m φφ ∗ , (5)where F µν = ∂ µ A ν − ∂ ν A µ is the field strength; D µ = ∂ µ + igA ext µ + igA µ is the covariant derivative and m isthe mass of the charged particle. We assume the scalarself-coupling, i.e. λ ( φφ ∗ ) /
4, is sufficiently small that wemay ignore it, at least in the range of energies considered.Note that for QED no such term would arise.We write the false vacuum transition amplitude as apath integral and note that we may integrate out thecharged particle, as it enters quadratically. (cid:104) Ω | ˆ S E | Ω (cid:105) = (cid:90) D A µ D φ e − (cid:82) x L SQED = (cid:90) D A µ det( − D + m ) − e − (cid:82) x F µν F µν = (cid:90) D A µ e − Tr log( − D + m ) − (cid:82) x F µν F µν , (6)where (cid:82) x := (cid:82) d x and the functional integrations arenormalised such that the amplitude is 1 for zero exter-nal field. The normalisation drops out once we take theimaginary part of the logarithm to find the rate, as inEq. (4).We can now use Schwinger’s trick (i.e. Frullani’s inte-gral) to express the logarithm as a proper time integral[1] log( A ) = − (cid:90) ∞ d ss (e − As − e − s ) , (7)and drop the second term as it is field independent andwill not contribute an imaginary part. The UV diver-gences of the theory will then turn up as divergences atsmall s which can be renormalised using the heat kernelexpansion. Introducing the proper time integral leads tothe expression Tr(e − ( − D + m ) s ), which we express as apath integral over closed worldlines, [65–69]Tr(e − ( − D + m ) s ) = (cid:90) D x µ e − S [ x µ ,A ext µ + A µ ; s ] , (8)where the action is given by S [ x µ , a µ ; s ] := m s + 14 (cid:90) s d τ ˙ x µ ˙ x µ − ig (cid:73) a µ d x µ , (9)and ˙ x µ := d x µ / d τ . This is the worldline path integralfor a charged scalar particle with the reparameterisation invariance fixed such that the einbein (also called thevierbein or tetrad by analogy to 4 dimensions) is equalto 2 (see for example chapter 1 of [70]). The false vacuumtransition amplitude is nowΓ = − V Im log (cid:90) D A µ e − (cid:82) x F µν F µν (cid:40) ∞ (cid:88) n =1 n ! (cid:18) n (cid:89) j =1 (cid:90) ∞ d s j s j (cid:90) D x µj e − S [ x µj ,A ext µ + A µ ; s j ] (cid:19)(cid:41) . (10)At each order in n the integration over the photon isnow Gaussian and can be done exactly, resulting in aneffective non-local worldline action. If we denote the freephoton propagator by G µν ( x j , x k ), we can write this asΓ = − V Im log (cid:40) ∞ (cid:88) n =1 n ! (cid:18) n (cid:89) j =1 (cid:90) ∞ d s j s j (cid:90) D x µj (cid:19) e − (cid:80) nk =1 ( S [ x µk ,A ext µ ; s k ] − g (cid:80) nl =1 (cid:72) (cid:72) d x µk d x νl G µν ( x k ,x l )) (cid:41) . (11)Integrating out the photon has left us with a non-local,long range interaction. At this point we have made no ap-proximations regarding the strength of the external field,or of the coupling. The relatively simple exponential formof Eq. (11) only obtains for Abelian gauge fields (see forexample [71]).At weak coupling, g (cid:28)
1, the non-local interactionterm in Eq. (11) can be dropped at leading order. Inthis case the sum exponentiates, leaving only one pathintegration which can be carried out exactly, leading toSchwinger’s result [1]Γ
Schwinger = m (cid:15) π ∞ (cid:88) n =0 ( − n +1 n e − π(cid:15) n . (12)In this paper we consider arbitrary coupling, g , for whichthe non-local interaction cannot be dropped. B. Finite temperature rate
The derivation thus far has been at zero temperature.At finite temperature, T = 1 /β , we make the followingreplacements (cid:104) Ω | ˆ S E | Ω (cid:105) → N − Tr e − ˆ Hβ , V → V T , where ˆ H is the Hamiltonian of the system in the presenceof the external field and the normalisation, N − , ensuresthe amplitude is 1 in the absence of the external field (see[20] for a physical discussion of thermal Schwinger pair creation). In the second line V T is equal to the spatialvolume, V , multiplied by the inverse temperature β .The rate is then given byΓ T = 2 V T Im (cid:26) − log (cid:16) Tr e − ˆ Hβ (cid:17) (cid:27) . (13)This transition to finite temperature can be madestraightforwardly using the Matsubara formalism, i.e. byenforcing periodicity in the Euclidean time coordinate, x ( τ ) = x ( τ ) + β , and including interactions betweenthe periodic copies.Including interactions between periodic copies is equiv-alent to replacing the photon propagator, G µν ( x j , x k ), byits thermal cousin, G µν ( x j , x k ; T ). In a general R ξ gaugethe ξ dependent term drops out when integrated arounda closed loop leaving just a term proportional to δ µν . Eq. (13) for the thermal rate has been advocated by Linde [72,73]. An analysis by Langer shows that a different expression for V T should be used, with the inverse temperature replaced bythe decay time of an intermediate state [55, 74, 75]. Though, aswe only work to exponential accuracy (i.e. the leading order ofthe logarithm) in this paper, the difference does not affect ourresults. This gauge independent part is G µν ( x j , x k ; T ) := ∞ (cid:88) n = −∞ G ( x j , x k + nT e ) δ µν = ∞ (cid:88) n = −∞ − δ µν π ( x j − x k − nT e ) = T sinh(2 πT r jk ) δ µν πr jk (cos (2 πT t jk ) − cosh(2 πT r jk )) , (14)where e is the unit vector in the Euclidean time di-rection and we have defined t jk := x j − x k and r jk := (cid:113) ( x j − x k ) + ( x j − x k ) + ( x j − x k ) . This is the Mat-subara thermal Green’s function in position space.To generalise Eq. (11), and get an expression for therate at finite temperature, one need only replace the zerotemperature Green’s function with that of Eq. (14), andimpose periodic boundary conditions in the Euclideantime direction, with period 1 /T . The aim of this paperis to calculate this thermal rate. C. Inclusive rate at fixed energy
We will also consider inclusive tunnelling rates at afixed energy E , i.e. rates from a microcanonical ensemble.In this case one makes the replacements (cid:104) Ω | ˆ S E | Ω (cid:105) → N − Tr (cid:16) δ ( E − ˆ H ) (cid:17) , V → V E , where the normalisation again ensures the amplitude is1 in the absence of the external field. In the second line V E is equal to the spatial volume, V , multiplied by sometimescale, which we expect to be O (1 / E ) on dimensionalgrounds. The exact form of V E will not concern us inthis paper as our final results are only to exponentialaccuracy. The rate, Γ E , is then given byΓ E = 2 V E Im (cid:26) − log (cid:16) Tr δ ( E − ˆ H ) (cid:17) (cid:27) , (15)The thermal density matrix is related to the microcanon-ical one by a sum over Boltzmann weights, or a Laplacetransform, e − ˆ Hβ = (cid:90) ∞ d E e −E β δ ( E − ˆ H ) . (16)Hence the inverse relation is via an inverse Laplace trans-form, δ ( E − ˆ H ) = lim B →∞ (cid:90) iB − iB d β πi e ( E− ˆ H ) β . (17)The microcanonical density operator is the projection op-erator onto the subspace of states with energy, E . These ...... FIG. 1. External photon legs denote couplings to the fixedexternal field whilst internal photon lines denote dynamicalvirtual photons. The Schwinger formula (Eq. (12)), validat weak coupling, accounts for the infinite set of diagramsrepresented in the first row. The quenched approximation alsoincludes all diagrams which include any number of internalphoton lines, with any topology. Some examples are shown inthe second line. Note that in SQED there are also four-pointinteractions involving two photons and two charged particles(not shown here though included in quenched approximation).In all the diagrams there is only one charged particle loop. rates and their relationship to thermal tunnelling rateshave been discussed by various authors [16, 75–78]. Forsufficiently slow rates one can expand the logarithms inEqs. (13) and (15) to derive the following approximaterelation, Γ T ∼ (cid:90) ∞ d E e −E β Γ E , (18)where we have ignored the ratio V E / V T as we will onlyuse the relation to exponential accuracy. III. THE DILUTE INSTANTON GAS
We wish to consider Schwinger pair production in QEDand SQED for arbitrary coupling, g . This requires goingbeyond perturbation theory in g . For a sufficiently weakexternal field, as we will show, an alternative set of ap-proximations are valid and allow us to proceed. Theseare the semiclassical and dilute instanton gas approxima-tions.Although Feynman diagrams will not be utilised in thiscalculation, they can illuminate the structure of the ap-proximations we will make. The rate, Eq. (11), con-tains only connected Feynman diagrams, due to the loga-rithm. The constituents of the contributing diagrams areinternal charged particles lines; external photon lines, for A ext µ ; internal, dynamical photon lines, for A µ , and ver-tices joining two charged particle lines and one or twophoton lines.At weak coupling, g (cid:28)
1, to leading order all depen-dence on the dynamical photon can be dropped. Thepath integrations in Eq. (11) are then uncoupled andthe sum exponentiates. The Feynman diagrams whichcontribute to this all contain one charged particle loopand an arbitrary number of external photon lines. Theseare the diagrams in the first row of Fig. 1. The sum ofthese diagrams at zero temperature is Schwinger’s origi-nal result, Eq. (12). At finite temperature the rate hasbeen calculated in Refs. [15, 17, 18, 79]. The inclusionof a single dynamical photon line (i.e. two loops) wascalculated first by Ritus at zero temperature [25–28] andby Gies at finite but low temperature [29]. In these cal-culations the approximation of weak external fields hasnot been made.At stronger coupling one must include the extra in-finitely many diagrams containing arbitrary numbers ofinternal, dynamical photon lines as well as arbitrary num-bers of charged particle loops. However, as we will ar-gue, for a sufficiently weak external field, diagrams with alarge number of charged particle loops will be suppressedand hence a loop expansion in charged particle loops ispossible. At each order one must sum the infinite setof diagrams containing a fixed, finite number of chargedparticle loops and an arbitrary number of both externaland dynamical photon lines. To first order this is thequenched approximation, which in this context was ar-gued to be valid (at zero temperature) in the Refs. [48]and [35] (see also [80]). Fig. 1 shows some examples ofdiagrams which contribute in the quenched approxima-tion.Following Refs. [48] and [35], we consider the situ-ation where the external field is weak and sufficientlyslowly varying to be considered constant. We choosethe external field to point in the 3 direction, F µν = − i ( δ µ δ ν − δ ν δ µ ) E (the factor of − i is present dueto the Wick rotation and the fact that E is the value ofthe Minkowskian field). As long as the worldline x µ ( τ )forms the boundary of some surface within the space,we can use Stokes’ theorem to re-express the interactionwith the external field, − ig (cid:73) A extµ d x µ = − ig (cid:90) (cid:90) F µν d x µ ∧ d x ν = − gE (cid:73) x d x , (19)which is simply the area enclosed by the worldline, pro-jected onto the 3-4 plane and multiplied by − gE . Now,we are in a position to set up the weak field approxima-tion to Eq. (11), which will amount to a semiclassicalapproximation. To see this it will be useful for us torescale the τ in the integrand of S , the parameters s j and the fields x µj ( τ ). We rescale them according to τ → τ /s j ,s j → s j /gE,x µj → x µj m/gE, (20)making all three dimensionless. The inverse temperaturemust be scaled in the same way as x µj ( τ ). We define thescaled temperature ˜ T := mT /gE and ˜ β := 1 / ˜ T .The full rate at finite temperature becomes, uponrescaling,Γ T = − V T Im log (cid:34) ∞ (cid:88) n =1 n ! n (cid:89) j =1 (cid:18)(cid:90) ∞ d s j s j (cid:90) D x µj e − (cid:15) ˜ S [ x j ; s j ; κ, ˜ T ] e κ(cid:15) (cid:80) k 1, the path integral is calcu-lable in the stationary phase, or semiclassical, approxi-mation. This is independent of the value of the coupling, g . The worldline configurations which dominate the path integral are those which satisfy the classical equations ofmotion. Of these, those which give a non-zero imaginarypart are those which are saddle points of the action withan odd number of negative modes in the spectrum offluctuations about the solution. The solutions relevantto tunnelling have just one negative eigenvalue and arecalled bounces or instantons.Note that the requirement that (cid:15) (cid:28) 1, which ensuressemiclassicality, entails that κ (cid:28) g , ˜ T (cid:29) Tm , (23)however, as we make no restrictions on g or T , κ and ˜ T are not thuswise constrained. This is key as we will onlycalculate to leading order in (cid:15) but to all orders in κ and˜ T .To proceed in calculating the rate, Eq. (21), we per-form a cluster expansion, as introduced by Ursell [81] and = + + + + . . . FIG. 2. The first three orders of the cluster expansion of therate. Each circle symbolises a closed worldline. The linesjoining them are interactions given by the two particle func-tion of Eq. (24). These diagrams are expressed algebraicallyin equations (25) and (26). Mayer [82]. We define the two-particle function f kl , for k (cid:54) = l , by f kl = exp (cid:26) κ(cid:15) (cid:73) (cid:73) d x µk d x νl G µν ( x k , x l ; ˜ T ) (cid:27) − . (24)The cluster expansion to Eq. (21) is then found by ex-panding in powers of f kl and grouping connected termsinto so-called clusters. Only connected terms contributeto Γ T . The expansion can be written asΓ T = ∞ (cid:88) n =1 γ n , (25)where γ n is the contribution to Γ T from clusters of n worldlines. The terms in the expansion can be mappedto connected graphs of increasing complexity, such as inFig. 2 (these are textbook results, see for example Ref.[83]). The γ n are proportional to the imaginary partsof what might conventionally be called cluster integrals(commonly denoted b n ) for the ensemble of charged par-ticle worldlines, and so for brevity we will refer to themas cluster integrals.The first three are given by γ = − V T Im (cid:90) ∞ d s s (cid:90) D x µ e − (cid:15) ˜ S [ x ; s ; κ, ˜ T ] ,γ = − V T Im (cid:89) j =1 (cid:18)(cid:90) ∞ d s j s j (cid:90) D x µj e − (cid:15) ˜ S [ x j ; s j ; κ, ˜ T ] (cid:19) f ,γ = − V T Im (cid:89) j =1 (cid:18)(cid:90) ∞ d s j s j (cid:90) D x µj e − (cid:15) ˜ S [ x j ; s j ; κ, ˜ T ] (cid:19)(cid:26) f f + f f f (cid:27) . (26)Eq. (25) is still formally exact but, importantly, is nowexpressed in a form that we can directly approximate.We follow Refs. [84–90] in performing a dilute instantongas approximation. Essentially we will assume that theleading order behaviour of Γ T is captured by the lowestnon-zero term in the cluster expansion. This is a self-consistent approximation, the higher order cluster inte-grals being exponentially suppressed with respect to theleading term.First, suppose that there exists an instanton for γ ,so that γ (cid:54) = 0. The path integral is invariant undertranslations x µ ( τ ) → x µ ( τ ) + a µ . The instanton solution for γ will necessarily break the translation symmetryand hence fluctuations around the instanton will contain(at least) four zero modes. Integration over these canbe done using the collective coordinate method (see forexample [91]), resulting in an integral over the positionof the instanton.The higher order cluster integrals give the contribu-tions due to interactions between instantons. Approx-imate multi-instanton solutions can be constructed bycombinations of single instantons a large distance apart.The contribution of these approximate saddle points canbe found using the method of constrained instantons[87, 92]. The integrations over the collective coordinatesand constraints of these approximate instantons will takethe form of cluster integrals for a gas of classical pointparticles (rather than worldlines), with dipole interac-tions (as the worldlines are closed and hence have zeronet charge). In this way, the infinite number of degreesof freedom of each particle worldline are reduced to thefour degrees of freedom of a point in spacetime.From this perspective, the rate, Γ T , can be interpretedas the pressure of the instanton gas. Standard statisticalmechanical relations then give the density of instantons, n inst , as n inst = ∞ (cid:88) n =1 nγ n . (27)Combining equations (25) and (27), the rate, Γ T , can bewritten as an expansion in powers of the densityΓ T = n inst + B n + B n + O ( n ) . (28)This is the Virial expansion and the coefficients, B n , arethe Virial coefficients. For n ≥ B n = − γ . At weak coupling, this virial expansionhas been introduced previously in Refs. [27, 28].To leading order in the cluster expansion the instantondensity will be given simply by γ . The average separa-tion between instantons is then γ − / . The density of in-stantons can be considered small if this distance is muchlarger than the maximum size of the instantons, R .There is however a subtlety due to the long range in-teractions of the instantons which was also found in thedilute instanton gas expansion of QCD [85, 86]. The con-tribution to the action due to the interaction between apair of dipoles in four dimensions a distance | x | apart,decreases as 1 / | x | . This is such that, at zero tempera-ture, its integral over the volume of spacetime divergesproportionally to log( V T ). As a result there is such adivergence in the second Virial coefficient, B , and in all reducible diagrams, defined to be those diagrams that canbe split into two disconnected parts by cutting a singleline. On the other hand, at non-zero temperature, thereis no logarithmic divergence due to the finite extent ofthe Euclidean time direction.Hence, at finite temperature, for sufficiently small,non-zero γ , we expectΓ T = γ (cid:0) O ( γ L ) (cid:1) , (29)where L = Max( R, β ). This leading order approximationis equivalent to the quenched approximation. The semi-classical approximation of γ gives γ L ∼ e − ˜ S ( κ, ˜ T ) /(cid:15) ,where we have written ˜ S ( κ, ˜ T ) for the value of the scaledaction, ˜ S [ x ; s ; κ, ˜ T ], evaluated at the saddle point.On the other hand, if there does not exist an instantonsolution consisting of a single worldline, then γ = 0. Inthis case we must repeat the above arguments for thefirst non-zero cluster integral, γ n , say. In that case theparticles of the instanton gas would consist of groups of n worldlines and Eq. (29) would be replaced byΓ T = γ n (cid:0) O ( γ n L ) (cid:1) . (30)In this paper, we consider only the leading order termin the dilute instanton gas approximation, γ n . Further,we only calculate the exponential suppression of the lead-ing term. This is equivalent to saying that we calculatethe logarithm of the rate to leading order in the smallparameter (cid:15) . When n = 1, this islog(Γ T ) = − ˜ S ( κ, ˜ T ) (cid:15) + O (log( (cid:15) )) . (31)In semiclassically evaluating the terms γ n , the saddlepoint of the s j integrations can be easily found. For γ ,we find γ = − m (cid:15) ˜ V ˜ T √ π(cid:15) Im (cid:90) D x µ (cid:18) (cid:90) d τ ˙ x µ ˙ x µ (cid:19) − e − (cid:15) ˜ S [ x ; κ, ˜ T ] (32)where ˜ V ˜ T := V T m (cid:15) is the (dimensionless) scaled vol-ume and we have defined ˜ S [ x ; κ, ˜ T ] to be the scaled ac-tion evaluated at the saddle point of the s integration,˜ S [ x ; κ, ˜ T ] := L [ x ] − A [ x ] + κV [ x ; ˜ T ] , (33)written in this way to emphasise its geometric nature.The constituent terms are the (parameterisation fixed)length of the worldline L [ x ] := (cid:115)(cid:90) d τ ˙ x µ ˙ x µ , (34)the area projected onto the 3-4 plane A [ x ] := (cid:90) d τ x ˙ x , (35) and the interaction term V [ x ; ˜ T ] :=12 (cid:90) (cid:90) d τ d τ (cid:48) ˙ x µ ( τ ) ˙ x ν ( τ (cid:48) ) G µν ( x ( τ ) , x ( τ (cid:48) ); ˜ T ) . (36)The first term, L [ x ], is the only non-geometric term, inthe sense that it depends on the coordinates along theworldline. It is however equal to the length of the world-line when evaluated on-shell. Note that the action isinvariant under τ → τ + c , where c is a constant. Thecorresponding conserved charge is ˙ x ( τ ).In some cases there may be no instanton solution con-sisting of a single worldline. As we have argued, in thesecases one should next look for instanton solutions consist-ing of two and then more worldlines. The (scaled) actionfor n worldlines could be thought of as that for a singlediscontinuous worldline (where one does not take deriva-tives across the discontinuities), except that the kineticterm, Eq. (34), does not appear to be additive. How-ever, the kinetic term is, in fact, additive if each of thedisconnected worldlines have the same (parameterisationfixed) length, n (cid:115)(cid:90) d τ ˙ x µ ˙ x µ = (cid:115)(cid:90) /n d τ ˙ x µ ˙ x µ + · · · + (cid:90) n − /n d τ ˙ x µn ˙ x n µ . (37)For the instanton solutions relevant in this paper, thisallows us to always talk about a single (possibly discon-tinuous) worldline and to always use the action in Eq.(33). IV. INSTANTONSA. Finite temperature rate The problem of finding the rate of pair production dueto a weak external field at given g , E , m and T is nowreduced to a problem which depends only on two param-eters, κ and ˜ T . The general solution amounts to findingthe saddle point of ˜ S [ x ; κ, ˜ T ] with one negative mode,and the fluctuations about it.Integrations over fluctuations in the negative mode,via an analytic continuation, give the all important fac-tor of i [74, 93–95]. There are also zero modes due totranslation invariance. Integration over these degrees offreedom requires first introducing a constraint which fixesthe translation invariance and then integrating over thatconstraint. We choose to fix the centre of mass of theworldline to be at the origin, ¯ x µ = 0. Integration overthe constraint then gives a factor ˜ V T = m (cid:15) V ˜ T , can-celling the 1 / ˜ V ˜ T in Eq. (32). The remaining integrationsover positive mode fluctuations give a subleading prefac-tor.To calculate the logarithm of the rate to leading orderin (cid:15) , we need only find the instanton solution and calcu-late its action, ˜ S ( κ, ˜ T ). Even this is a difficult enoughproblem, made so by the non-local photon interaction in(33). In the following we consider the equations of mo-tion analytically in certain limits: κ (cid:28) 1, ˜ T (cid:28) T . Then for arbitrary κ and ˜ T we use numericalmethods. B. Inclusive rate at fixed energy From ˜ S ( κ, ˜ T ), we can calculate the inclusive rate atfixed energy. In the semiclassical approximation, Eq.(18) shows that the two rates are related by a Laplacetransform and hence the exponents of the rates are re-lated via a Legendre transform. In the thermodynamiclanguage ˜ S ( κ, ˜ T ) is the free energy divided by the tem-perature. The (scaled) energy of the solution ˜ E is˜ E = ∂ ˜ S∂ ˜ β , (38)corresponding to a physical energy E = m ˜ E . By furtherscaling the worldlines by x → x/β , taking the derivativewith respect to ˜ β , and then reversing the scaling, we findthe following useful result˜ β ˜ E = L [ x ] − A [ x ] , (39)which holds on-shell. The exponential suppression of therate of pair production at fixed energy isΣ = 1 (cid:15) ( ˜ S − ˜ E ˜ β )= 1 (cid:15) ˜Σ( κ, ˜ E ) . (40) C. Regularisation As we have mentioned the interaction term, V , di-verges at zero distance. For smooth worldlines, thisis the long known self-energy divergence of electromag-netism. Its appearance in the worldline formulation ofQED has been studied by many authors (see for example[67, 71, 96, 97]). The divergence, being due to the stronginteractions between nearby sections of a worldline, isproportional to its length.We first consider a well known regularisation schemedue to Polyakov [67]. At zero temperature this amountsto replacing the interaction term, V [ x ; 0], with V Polyakov [ x ; 0] := 18 π (cid:90) (cid:90) d τ d τ (cid:48) ˙ x µ ( τ ) ˙ x µ ( τ (cid:48) )( x ( τ ) − x ( τ (cid:48) )) + a − π πa (cid:90) (cid:112) ˙ x ( τ )d τ. (41) The second term in (41), proportional to the length ofthe worldline, is a counterterm which absorbs the shortdistance divergence of the first term. It is almost of thesame form as the term L [ x ] in the action (Eq. (34)),except without the reparameterisation fixing. On-shellthe two terms are equal, hence we can see it as a masscounterterm.This self-energy divergence has been shown to bethe only divergence for smooth loops with no intersec-tions [71]. Worldlines with discontinuous first deriva-tives (cusps) and intersections may arise when there aredelta function interactions in the action or when a ratioof scales is taken to zero. Such worldlines also generatelogarithmic divergences .Unfortunately the regularisation scheme of Eq. (41)leads to problems when trying to formulate the equationsof motion which prevent us taking the limit a → a , the counterterm gives a negative bare mass. Tobypass this problem we adopt an alternative regularisa-tion in our numerical calculations for which the bare massand the renormalised masses are equal, V R [ x ; 0] := 18 π (cid:90) (cid:90) d τ d τ (cid:48) ˙ x µ ( τ ) ˙ x µ ( τ (cid:48) )( x ( τ ) − x ( τ (cid:48) )) + a − π √ πa (cid:90) (cid:90) d τ d τ (cid:48) ˙ x µ ( τ ) ˙ x µ ( τ (cid:48) )e − ( x ( τ ) − x ( τ (cid:48) )) /a . (42)Equations (41) and (42) agree as a → (cid:15) andhence will not arise in the stationary phase approxima-tion we have made. Including these fluctuations shouldresult in the final rates depending on the renormalisedcharge as argued for in Ref. [35]. These logarithmic divergences can be interpreted as due toBrehmstrahlung radiation. They give the anomalous dimensionfor Wilson loops, and hence for the propagator of the chargedparticles. FIG. 3. Sphere of influence of point x , size O ( κ ), comparedto curvature of worldline on scale O (1). For the small worldlines of the short-lived virtual pairsto simply renormalise the charge, there must be a sepa-ration of scales between them and the larger worldlineswhich constitute the saddle point. We can make a simpleestimate for the scale of the virtual pairs by equating therest mass to the Coulomb attraction. This equality reads2 m = g / (4 πr ) and gives the distance between chargesas r = g / (8 πm ). In our dimensionless units, this trans-lates to a distance κ/ (8 π ), which must be smaller thanany scale present in the instanton for the charge renor-malisation effects to be independent. The modification ofthe photon-charged particle interaction at distances be-low g / (8 πm ) has been discussed in Refs. [98, 99], withregard to magnetic monopoles. D. Small κ expansion 1. A singular perturbation problem Throughout this paper, we make the approximationthat the external field is weak, i.e. that 0 < (cid:15) (cid:28) κ := g (cid:15) is proportional to (cid:15) . Hence,for not too large couplings, we will also have that 0 <κ (cid:28) 1. This is the case we will consider in this section.Parametrically large couplings, such that κ = O (1) willbe considered in sections IV E, IV F and V.For sufficiently small κ , one would naively expect thatwe could simply set κ = 0 in the scaled action ˜ S [ x µ ; κ, ˜ T ],so dropping the interaction term. The problem is thatthe interaction term diverges at short distance and hencecannot be ignored for arbitrarily small but positive κ .This signals that for small κ we are dealing with a sin-gular perturbation problem related to the existence ofwidely separated scales (see for example [100]).We seek distinguished limits of the action ˜ S [ x ; κ, ˜ T ],and its corresponding equations of motion, by consideringthe scalings x = κ α y and ˜ T = κ − α Θ. The aim is to findscalings such that there is a balance between two terms inthe action and that the leading order equations of motiongive non-trivial solutions. After the scaling the action is˜ S [ x ; κ, ˜ T ] = κ α L [ y ] − κ α A [ y ] + κV [ y, y (cid:48) ; Θ] . (43)There are three distinguished limits: α = 0 , / , 1. The first, the α = 0 scaling, corresponds to scales x = O (1),which we will refer to as the infrared (IR) problem. Thelast, the α = 1 scaling, corresponds to shorter scales x = O ( κ ), which we will refer to as the ultraviolet (UV)problem. The intermediate scaling, α = 1 / x = O ( κ / ), which we will refer to as the match-ing problem.For small κ , an approximate solution to the equationsof motion valid on all scales can be found by solvingthe leading order equations of motion in these three dis-tinguished limits and matching them smoothly together.The simplest of the three problems is the matching prob-lem, α = 1 / 2. The leading approximation amounts tosimply keeping the length term˜ S ( κ (cid:28) , ˜ T ) ≈ κ / L [ y ] . (44)The area and interaction terms are equally subdominanton these scales, both being of order κ . Solutions tothe minimisation of the length term are simply straightlines. Hence the IR and UV solutions must be matchedwith straight lines. The matching is done at some scale λ = O ( κ / ) which acts as a UV cut-off for the IR prob-lem and as an IR cut-off for the UV problem. The finalsolution should be independent of the specific choice of λ . The IR problem, the α = 0 scaling, in the leading ap-proximation amounts to simply dropping the interactionterm, i.e. to ˜ S ( κ (cid:28) , ˜ T ) ≈ L [ y ] − A [ y ] . (45)In terms of a Feynman diagram language, this approx-imation takes into account all external field photon ex-changes but no virtual photon exchanges. This is the toprow of Fig. 1, a one-loop approximation. Making this ac-tion stationary is the old problem of maximising the areaof a field given a fixed length of fencing. The solution atzero temperature is a circle of radius 1 in the 3-4 plane.At finite temperature the solution can be found using themethod of images.The UV problem, the α = 1 scaling, in the leadingapproximation amounts to dropping the area term, i.e.to ˜ S ( κ (cid:28) , ˜ T ) ≈ κ ( L [ y ] + V [ y, y (cid:48) ; Θ]) . (46)This equation determines the dynamics at scales y = κ − x = O (1). In terms of Feynman diagrams this ap-proximation takes into account all virtual photon loopsbut no external photon lines.Eq. (46) is the action of a massive charged particle, inthe absence of an external field. Hence we can immedi-ately find one solution, that of a straight line, the particlesitting still (or 4D rotations thereof). To find a solutionto the full equations of motion, valid at all scales we canstitch this straight line solution together with a solutionof Eq. (45). This is possible if for every point x on theworldline, we can draw a ball of size λ within which theworldline appears straight as κ → 0. As solutions to (45)0 FIG. 4. On scales x = O (1) there may appear an intersection,as in a). This must be resolved on shorter scales, x = O ( κ ),as b) or c). Due to the fact that the problem can be specifiedcompletely in the plane, these are the only possibilities. Thelong and short distance pictures are matched at some scale λ = O ( κ / ). are independent of κ , this will always be possible as longas the worldline has everywhere finite curvature and doesnot self-intersect. Otherwise, in the region of a cusp orself-intersection, the straight line solution to Eq. (46)cannot be used.Such cusps or self-intersections are not permissiblewhen all the parameters of the theory are finite as theygive new divergences which depend on the angle at thenon-analytic point, γ in Fig. 4. Due to this the di-vergences cannot be absorbed as counterterms in thefield theory. However, if a ratio of scales in the prob-lem goes to zero, as when κ → 0, such apparent cusps orself-intersections can appear at larger scales, O (1) and O ( κ / ) in our case. On the smallest scale, O ( κ ) inour case, these apparent non-analytic points can be re-solved as in Fig. 3. The separation of scales in processeswhich involve a large momentum transfer such as deepinelastic scattering, gives divergences for the same reason[101, 102].For apparent non-analytic points to be resolved onscales O ( κ ), there must exist solutions of (46) which havethe topology (in the plane) of b) and c) in Fig. 3. Notethat these two possibilities are related by a rotation by π/ 2. Hence all points of self-intersection can be resolvedif solutions with the topology of b) can be found for allangles π/ < γ ≤ π . These solutions must be stitchedtogether with the IR solutions at some scale λ . Thuswe must impose boundary conditions at λ such that thesolutions can be smoothly matched.The existence of such solutions can be made plausibleby noting that for π/ < γ ≤ π the scalar product be- tween the tangent vectors on the left and right hand sidesof b) is negative and hence the interaction term is repul-sive. The magnitude of the repulsion increases withoutbound as the worldlines approach each other, suggestingthat the worldlines should approach to some minimumdistance, | y − y (cid:48) | = O (1). The minimum distance isa function of the incoming angle γ and is independentof κ , as long as γ and π − γ are both O (1). Naivelyone might expect that the scaled action L [ y ] + V [ y, y (cid:48) ]would then be O (1) and hence the contribution to the˜ S = κ ( L [ y ] + V [ y, y (cid:48) ; Θ]) = O ( κ ). However the followingargument shows that this is not the case.The matching of the IR and UV solutions is carriedout at the scale λ . Due to the long range of the inter-action, the UV scaled action, L [ y ] + V [ y, y (cid:48) ; Θ], will getlarge as log( λ/κ ). This is the infrared divergence of theinteraction of two long straight worldlines which are notparallel. Upon matching the IR and UV solutions all de-pendence on λ must drop out. However we will be leftwith a contribution to the action of the order κ log( κ ).Overall we find that for solutions with apparent cuspsor intersections˜ S ( κ, ˜ T ) = c ( ˜ T ) + d ( ˜ T ) κ log( κ ) + O ( κ ) , (47)for some c ( ˜ T ) and d ( ˜ T ). For solutions without cusps orintersections the κ log( κ ) term is absent, i.e. d ( ˜ T ) = 0. 2. Small κ results At zero temperature the solution to the IR problemis a circle of radius 1. At every point, x , on the circle,a small ball of radius λ = O ( κ / ) can be drawn withinwhich the worldline looks approximately straight. Hence,the circle of radius 1 solves the equations of motion at allscales. The resulting action is˜ S ( κ, 0) = π − κ . (48)This result was first derived in [35, 48]. Due to the sym-metry of the problem this result is in fact exact for arbi-trary κ and hence applies even for parametrically strongcoupling. The prefactor is given by the determinant offluctuations about this solution. This can be computedat leading order in κ giving for the rateΓ ≈ (2 s + 1) m (cid:15) π e − (cid:15) ( π − κ ) , (49)where s is the spin of the charged particle. At κ = 0this reduces to Schwinger’s result for weak external fields(Eq. (12)).At finite temperatures, ˜ T , and small κ , the leading or-der solution to the IR problem is given by an infinitesequence of circles of radius 1 separated by a distance˜ β along the Euclidean time axis (see Fig. 5 b)). Fortemperatures such that ˜ T < / 2, these circles do not1overlap and, for sufficiently small κ , we are able to drawa small ball of radius O ( κ / ) within which there is a sin-gle worldline which looks approximately straight. Hencefor such temperatures, the sequence of circles solves theequations of motion at all scales and, to lowest order in κ ,the rate is the same as at zero temperature. This meansthat, at one-loop order, we find no corrections to the zerotemperature rate for ˜ T < / κ . We write the full action and it’ssolution as expansions in κ ,˜ S [ x ] = ˜ S [ x ] + κ ∆ ˜ S [ x ] ,x µ ( τ ) = x µ ( τ ) + κx µ ( τ ) + κ x µ ( τ ) + . . . (50)where ˜ S [ x ] = L [ x ] − A [ x ] and ∆ ˜ S [ x ] = V [ x ]. Firstorder perturbation theory requires us to simply evaluate∆ ˜ S [ x ]. We split this up into interactions between pairsof loops, κ ∆ ˜ S [ x ] = κ ∞ (cid:88) n = −∞ ∆ ˜ S n [ x ] , (51)where we have defined ∆ ˜ S n [ x ] to be the interaction be-tween the loop at the origin and that centred at Euclideantime n/ ˜ T , κ ∆ ˜ S n [ x ] = κ π (cid:73) (cid:73) d x d x (cid:48) ( x − x (cid:48) − n ˜ T e ) . (52)The x denotes the positions on the circle at the originwith respect to the origin. The x (cid:48) denotes the positionson the circle centred at Euclidean time n/ ˜ T with respectto its centre. The e is a unit vector in the Euclideantime direction. The result of the integration, for n (cid:54) = 0,is κ ∆ ˜ S n [ x ] = − κ (cid:32)(cid:114) − (cid:16) ˜ Tn (cid:17) − (cid:33) (cid:114) − (cid:16) ˜ Tn (cid:17) . (53)This was first derived in Ref. [13]. For n = 0, the integralis − κ/ T ≥ −∞ as ˜ T → / 2, i.e. where the zero temperatureinstantons touch. However, following the discussion ofsection IV D 1, the separation of scales breaks down whenneighbouring circles are only a distance O ( κ / ) apart,when 1 / − ˜ T = O ( κ / ).Unlike the zero temperature result, there are correc-tions at second order in κ due to the warping of the shapeof the circles. To calculate these, we must solve (cid:90) (cid:32) δ ˜ S δx µ ( τ ) δx ν ( τ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) x x ν ( τ (cid:48) ) (cid:33) d τ (cid:48) + δ ∆ ˜ Sδx µ ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) x = 0 . (54)The solution must lie in the 3 − x µ ( τ ) = (cid:15) ( τ )(0 , , cos(2 πτ ) , sin(2 πτ )) . (55)In terms of (cid:15) ( τ ), the terms in the action at second orderin κ are κ (cid:18)(cid:90) ˙ (cid:15) π d τ − π ¯ (cid:15) (cid:19) − κ ˜ T π (cid:90) (cid:16) ζ (4) + 24 ˜ T ζ (6)(1 − cos(4 πτ )) (cid:17) (cid:15) ( τ )d τ, (56)where ¯ (cid:15) denotes the square of the average of (cid:15) ( τ ) (notthe average of the square) and ζ denotes the Riemannzeta function . From Eq. (56) we find the equationsof motion for (cid:15) ( τ ), the solution of which can be foundstraightforwardly via a Fourier series expansion. The twoarbitrary parameters in the general solution are fixed bysatisfying the constraint, ¯ x µ = 0, which we are using tofix translation invariance. The solution is given by κ(cid:15) ( τ ) = − κ ˜ T π (cid:26) ζ (4) + 24 ζ (6) ˜ T + 6 ζ (6) ˜ T cos(4 πτ ) (cid:27) . (57)The constant terms reduce the radius of the circle and theterm proportional to cos(4 πτ ) makes the circle prolate(stretched in the x direction). Substituting this solutioninto Eq. (56) and putting it together with the zeroth andfirst order terms we arrive at˜ S ( κ, ˜ T ) = π − κ (cid:40) ∞ (cid:88) n =1 (cid:32)(cid:114) − (cid:16) ˜ Tn (cid:17) − (cid:33) (cid:114) − (cid:16) ˜ Tn (cid:17) (cid:41) + κ π (4 ζ (4) ˜ T + 48 ζ (4) ζ (6) ˜ T + 126 ζ (6) ˜ T ) + O ( κ ) . (58) Note that had the kinetic term been the actual length, ratherthan its reparameterisation fixed form, the only difference would be the replacement of ¯ (cid:15) with (cid:82) (cid:15) ( τ ) d τ . FIG. 5. Instanton solutions at small κ : a) at zero tempera-ture; b) at 0 < ˜ T < / 2; c) at ˜ T > / 2, the naive instanton ofoverlapping circles and d) at ˜ T > / 2, the lemon instanton. For the corresponding inclusive rate at fixed energy weconsider the Legendre transform of this sum. The energyis, from Eq. (38), ˜ E = − ˜ T ∂ ˜ S ( κ, ˜ T ) ∂ ˜ T . (59)The leading term on the right hand side takes the formof κ multiplying a function of ˜ T . A consideration ofthis function implies that if we wish to consider ener-gies much larger than κ , the corresponding temperaturemust be very close to 1 / 2. This is the region of param-eter space where the circular worldlines almost touch,the minimum distance d (cid:28) 1. The UV problem in thiscase is non-relativistic, with y being the “time” direc-tion. Solutions to this non-relativistic problem exist for d (cid:38) κ / . This implies that the Legendre transform ofEq. (58) is only valid for ˜ E (cid:46) κ / . For ˜ E (cid:46) κ takingthe Legendre transform analytically is made difficult bythe infinite sum. In the limited regime κ (cid:28) ˜ E (cid:28) κ / however we can fairly simply find the leading few terms.˜Σ( κ, ˜ E ) = π − E − κ / ˜ E / + κ (cid:26) − ∞ (cid:88) n =2 (cid:32)(cid:114) − n − (cid:33) (cid:18) − n (cid:19) − / (cid:27) − κ / ˜ E − / + ∞ (cid:88) n =2 κ / ˜ E − / n ( n − / + 351536 κ ˜ E − + O ( κ / ˜ E − / ) . (60)The leading enhancement, − E , has been long known inthe context of induced vacuum decay [103, 104].For ˜ T ≥ / κ as in Fig. 3. How-ever, once we include intersections a more general classof solutions to the full problem are possible: we can com-bine sections of circles with intersections. This is possi-ble as solutions to (45) must only locally be arcs of a circle with curvature 1. Of all possible solutions describ-ing pair production processes, that with minimum actionwill dominate the path integral and hence give the rateof pair production.The minimum action solution of this kind has beenfound by several authors [10, 16, 23, 105], though thereis some dispute about this [24]. It is given by a lemonshape, Fig. 5 d), and not by the overlapping circles, c).The angle of intersection (see Fig. 4) is given by γ ˜ T = 2 arcsin (cid:18) 12 ˜ T (cid:19) . (61)However, note that this worldline on IR scales is only asolution to the full problem if, for given γ ˜ T , the corre-sponding UV solution exists. At small but finite κ ournumerical calculations in Sec. V find instanton solutionswhich appear to approach the lemon instanton as κ → κ , is˜ S (0 , ˜ T ) = γ ˜ T + sin( γ ˜ T ) (62)where ˜ T > / 2. Below ˜ T = 1 / π ,at zeroth order in κ . The action and its first derivativeare continuous at ˜ T = 1 / 2, though the second derivativediverges as − / (cid:112) T − / T → / T = 1 / 2, for κ = 0. Below the phase transition thesolution has circular symmetry. This is broken above it.To get the leading order correction in κ it would seemwe must solve Eq. (46) in the region of the intersection(assuming such a solution exists). However, we can infact bypass this hard problem using perturbative renor-malisation. First, we evaluate the interaction term onthe leading order IR instanton, the lemon. This gives anunphysical logarithmic UV divergence from the intersec-tions V Lemon = κπ (cid:26)(cid:18) π − γ ˜ T (cid:19) cot( γ ˜ T )+1 (cid:27) log( λ )+finite terms , (63)where we have used λ for the short distance regulator,rather than a , as this should be of the order of the match-ing scale, as in section IV D 1. Though we cannot solvethe UV problem, we know that it must provide a com-pensating counterterm i.e. κπ (cid:26)(cid:18) π − γ ˜ T (cid:19) cot( γ ˜ T ) + 1 (cid:27) log (cid:18) κλ (cid:19) . (64)From the perspective of the short distance physics thisis an IR divergence, arising due to the matching scale λ being much larger than κ . The presence of κ in the loga-rithm is due to the scaling in the short distance problem.This meets our expectations, as explained at the end of3section IV D, leading to a contribution of order κ log( κ ),˜ S ( κ, ˜ T ) = γ ˜ T + sin( γ ˜ T )+ 1 π (cid:26)(cid:18) π − γ ˜ T (cid:19) cot( γ ˜ T ) + 1 (cid:27) κ log( κ ) + O ( κ ) . (65)where ˜ T > / 2. Note that the O ( κ log( κ )) term startsto dominate over the leading term when γ ˜ T = O ( κ ) and π − γ ˜ T = O ( κ ), or ˜ T = O ( κ − ) and ˜ T − / O ( κ ). Thissignals a breakdown of the separation of scales assumedin deriving Eq. (65) and a breakdown of the approximatesolution.We have perturbatively renormalised the problem.The subleading corrections at O ( κ ) depend on the so-lution to the short distance ( α = 1) problem. Our resultcould be nonperturbatively improved using the methodsof the renormalisation group.From Eq. (65) we can find the inclusive pair produc-tion rate at a fixed energy by Legendre transform,˜Σ( κ, ˜ E ) = π − γ ˜ E − sin( γ ˜ E )+ 1 π (cid:26)(cid:18) π − γ ˜ E (cid:19) cot( γ ˜ E ) + 1 (cid:27) κ log( κ ) + O ( κ ) , (66)where γ ˜ E := 2 arcsin( ˜ E / E [10] (or an off-shell photon[106]) or by a collision of particles with the same cen-tre of mass energy [107, 108]. These calculations involvethe same shaped instanton, though without the periodiccopies. The (scaled) exponential suppression in that case,including the O ( κ log( κ )) correction, is π − γ ˜ E − sin( γ ˜ E )+ 12 π (cid:26) − γ ˜ E cot( γ ˜ E ) + 1 (cid:27) κ log( κ ) + O ( κ ) . (67)Note that as 0 < κ < | γ ˜ E | < π , the correspondingrate is strictly lower than that given by the exponentia-tion of ˜Σ( κ, ˜ E ). This is as expected: the inclusive rate atenergy ˜ E is greater that the rate of the specific processat the same energy. E. Low temperature expansion Let us now consider the case of low temperatures,˜ T (cid:28) 1, but arbitrary coupling, κ . The effect of non-zero temperature is felt through the interaction poten-tial, coupling periodic copies of the circular worldline.Perturbation theory in ˜ T takes the form˜ S [ x ] = ˜ S [ x ] + ˜ T ˜ S [ x ] + ˜ T ˜ S [ x ] + ˜ T ˜ S [ x ] + . . . ,x µ ( τ ) = x µ ( τ ) + ˜ T x µ ( τ ) + ˜ T x µ ( τ ) + ˜ T x µ ( τ ) + . . . (68) The terms ˜ S n [ x ] are simply defined to be the coefficientsof ˜ T n in the full action. Note that: i) there is no lin-ear term in ˜ S [ x ]; ii) the coefficient of the quadratic termis zero due to the closure of the worldline loops and iii)the coefficients of all odd powers of ˜ T in ˜ S [ x ] vanish dueto cancellation between loops in the positive and nega-tive Euclidean time directions. These properties of the˜ S [ x ] expansion carry over to that of x µ ( τ ) by standardperturbation theory.First order perturbation theory gives˜ S ( κ, ˜ T ) = π − κ − ζ (4) κ ˜ T − ζ (6) κ ˜ T + O ( ˜ T ) . (69)These first two terms are the same as those coming fromthe expansion of Eq. (58). To calculate the coefficient ofthe O ( ˜ T ) term requires second order perturbation the-ory, which amounts to solving a somewhat complicatedintegrodifferential equation.As pointed out in Refs. [13, 27], expanding the expo-nential of Eq. (69) captures the two-loop corrections forweak fields and low temperatures (Eq. (76) in Ref. [25]and Eq. (65) in Ref. [29]).e − (cid:15) ( π − κ − ζ (4) κ ˜ T ) ≈ (cid:18) πα + 2 π αT m (cid:15) (cid:19) e − π/(cid:15) , (70)where α = g / (4 π ). At higher loop orders one would needto calculate also the semiclassical prefactor for compari-son.The Legendre transform of (69) gives the inclusive pairproduction rate at fixed, perturbatively low energies. Itis ˜Σ( κ, ˜ E ) = π − κ − ζ (4) / κ / (cid:32) ˜ E κ (cid:33) / − ζ (6) κ / ζ (4) / (cid:32) ˜ E κ (cid:33) / + O (cid:32) ˜ E κ (cid:33) / (71)where the requirement of low temperatures translatesinto the requirement that ( ˜ E /κ ) (cid:28) F. High temperature At sufficiently high temperatures the process of pairproduction becomes classical. The instanton is then in-dependent of the Euclidean time direction and is calleda sphaleron [53, 72, 73, 110]. The problem reduces fromfour to three dimensions. Further, due to the irrelevance The structure of this expansion is reminiscent of diagrammaticlow energy expansions about instantons, such as arose in thediscussion of electroweak baryon number conservation (see forexample [104, 109]). µ = 1 , 2) the problem be-comes one dimensional. The action gives the Boltzmannfactor.In our case the instanton consists of two worldlines: astationary charged particle, x µ ( τ ), and its antiparticle, y µ ( τ (cid:48) ), at a fixed distance, | x ( τ ) − y ( τ (cid:48) ) | = r . On sucha path the action reduces to˜ S Straight [ r ; κ, ˜ T ] = (cid:18) − r − κ πr (cid:19) T . (72)There is a stationary point of the action at r = (cid:112) κ/ π which gives the thermal instanton. The action is then˜ S Straight ( κ, ˜ T ) = 2 (cid:18) − (cid:114) κ π (cid:19) T , (73)which can also be written as ˜ S = ˜ E / ˜ T , where ˜ E is theenergy of the solution. Note that as ˜ T = (cid:15)T /m , thefactors of (cid:15) cancel in the exponent of the rate leavingjust the usual Boltzmann suppression in physical units.The instanton may give the rate of thermal pair pro-duction when it is the lowest action solution for givenparameters ( κ, ˜ T ). In a broad class of theories, thoughnot including SQED or QED, it has been shown that asolution must satisfy a further constraint for it to de-scribe tunnelling: the spectrum of linear perturbationsabout the solution must have one negative mode [111].Variation of r is one negative mode, present for all κ and˜ T .Due to the periodic boundary conditions, the com-ponents of both worldlines are each expressible as aFourier series. We define ζ µ ( τ ) := x µ ( τ ) − y µ ( τ ) and ξ µ ( τ ) := x µ ( τ ) + y µ ( τ ). The linearised eigenvalue equa-tions about the straight line solution inherit the non-locality of the full action. They are thus linear integrod-ifferential equations, the expressions being very long, sowe omit them here.For κ (cid:28) 1, the integrands of the non-local interac-tions become highly peaked, approaching delta functionswhich make the eigenvalue equations local. In this regimethe dynamics is non-relativistic and the eigenvalue equa-tions can be straightforwardly solved. In the spectrum ofeigenfunctions there are two sets of potentially unstablelinear fluctuations, one given by ζ ( τ ) = r + δ cos(2 πnτ )and the other by ξ ( τ ) = δ sin(2 πnτ ), where δ (cid:28) n ∈ N , and in each case all other components are zero.The eigenvalues are λ n ( κ, ˜ T ) ≈ 12 (2 πn ) ˜ T − π √ πκ ˜ T . (74)The lowest frequency mode is thus least stable, and isunstable when ˜ T < ˜ T λ =0 ( κ ) ≈ √ π − / κ − / . Thissignals the existence of another solution of lower actionwhich is continuously connected to the straight line solu-tion but which breaks time translation invariance. Hencethere is a second order phase transition in the rate at thistemperature. The instability is exactly analogous to the Plateau-Rayleigh instability in fluid dynamics [112, 113];the Gregory-Laflamme instability in black strings [114];nuclear scission [115] and an instability in vacuum bub-bles at finite temperature [72, 73].These linear fluctuations remain eigenvectors of the fullintegrodifferential equations at larger values of κ . Theeigenvalue of the lowest frequency mode is then some-what more complicated, λ ( κ, ˜ T ) = 12 (2 π ) ˜ T − π κ ˜ T − π √ πκ ˜ T − π (cid:18) √ πκ ˜ T (cid:19) e −√ πκ ˜ T . (75)For the higher frequency modes, the eigenvalue is givenby λ n ( κ, ˜ T ) = nλ ( κ, n ˜ T ). The term, − π κ ˜ T , is dueto the interactions of each worldline with itself. It isthe much-discussed, electromagnetic self-force [116–118]and its contribution destabilises the straight worldlines,increasingly so at higher temperatures. This issue hasbeen discussed in a similar context in Ref. [98], where itwas argued that the photon-charged particle interactionis modified on sufficiently short scales, so diminishing theself-force.Setting Eq. (75) to zero defines the boundary betweenstability and instability to the lowest frequency perturba-tion. The boundary is described by a function, ˜ T λ =0 ( κ ),consisting of two branches which meet at κ ≈ . κ , thelower branch coincides with the long wavelength instabil-ity found in the non-relativistic analysis and the upperbranch is approximately given by ˜ T λ =0 ( κ ) ≈ /κ . Theinstability above the upper branch is due to the self-force.Higher frequency modes are more stable to the longwavelength instability but less stable to the self-force in-stability. As n increases the self-force term grows fastestso all sufficiently high harmonics are unstable. This in-stability is present for all κ and ˜ T . In fact, as this in-stability only depends on the shape of the worldlines atshort distances, O (1 / ( n ˜ T )), it is present for all smoothworldlines (and likely for more general worldlines too).Due to the translational symmetry in the Euclidean timedirection, the unstable harmonics of the sphaleron comein pairs, one a sine and the other a cosine.The self-force instability may be a sign of the break-ing down of the cluster expansion at larger values of κ .In support of this view, the self-force does not arise for κ (cid:28) κ . In the cluster expansion,only a small number of charged particle worldlines areincluded. Charged particle loops of size κ/ (8 π ) have asmall action, due to cancellation between the kinetic andinteraction terms. These loops make up the bubblingsea of virtual charged particle pairs, part of the quan-tum vacuum. Their presence modifies the photon-chargeinteraction on scales of order κ/ (8 π ), an effect which isnot included in the cluster expansion. A quantitative in-clusion of these effects is beyond the scope of this paperbut, as argued in Ref. [98], photon-charge interactions5 FIG. 6. Regions within which the straight line instanton isstable and unstable to the lowest frequency perturbations.The boundary is defined by setting Eq. (75) to zero anddefines a function, ˜ T λ =0 ( κ ), with two branches. should be weaker on scales of order κ ( g /m in dimen-sionful units) and below. This is interpreted as an effec-tive spreading-out of the charge, which prevents the self-force instability.For small deviations below the lower branch of˜ T λ =0 ( κ ), and for sufficiently small κ , the dynamics isnon-relativistic. In this case the equations of motion canbe solved, even beyond the linearised approximation, bystraightforward integration. The first integral of the mo-tion is the energy, ˜ E , − 14 ˙ r + U ( r ) = ˜ E , (76)where overdot signifies differentiation with respect to theEuclidean time coordinate, t , and U ( r ) = 2 − r − κ/ (4 πr ).Integrating this equation gives t ( r ), which can be in-verted to give r ( t ). The solutions are periodic, with pe-riod ˜ β ( κ, ˜ E ).In the non-relativistic regime the action is given by˜ S Nonrel ( κ, ˜ T ) = (cid:90) r R r L U ( r ) − ˜ E (cid:113) U ( r ) − ˜ E d r, (77)where ˜ E is treated as a function of ˜ β and κ and where r L and r R are the classical turning points on the leftand right. The non-relativistic approximation is validfor ∆ ˜ T := ˜ T λ =0 − ˜ T (cid:28) 1, hence we expand the integralthuswise˜ S Nonrel ( κ, ˜ T ) = ˜ S Straight ( κ, ˜ T ) − √ π / κ / ∆ ˜ T − π / κ / ∆ ˜ T + O (cid:16) ∆ ˜ T (cid:17) , (78)Note that the non-relativistic solution has a lower actionthan the straight lines, so it dominates the rate where it exists, ˜ S ( κ, ˜ T ) = (cid:40) ˜ S Straight ( κ, ˜ T ) , ∆ ˜ T ≤ , ˜ S Nonrel ( κ, ˜ T ) , < ∆ ˜ T (cid:28) . Also note that the difference between the two rates arisesat second order in ∆ ˜ T , showing that the transition be-tween the two solutions is a second order phase transition.The Legendre transform of these results give˜Σ( κ, ˜ E ) = (cid:40) ˜Σ Straight ( κ, ˜ E ) , ∆ ˜ E ≤ , ˜Σ Nonrel ( κ, ˜ E ) , < ∆ ˜ E (cid:28) . where ∆ ˜ E := ˜ E c − ˜ E and ˜ E c := 2(1 − (cid:112) κ/ π ), the thresh-old energy. The two functions are˜Σ Straight ( κ, ˜ E ) = 0 , (79)and ˜Σ Nonrel ( κ, ˜ E ) = π / κ / √ E + 3 π / √ κ / ∆ ˜ E − π / √ κ / ∆ ˜ E + O (cid:16) ∆ ˜ E (cid:17) . (80)The inclusive rate of pair production at a fixed energyis unsuppressed at the threshold energy. Just below thethreshold, ∆ ˜ E (cid:28) 1, the suppression is given by the non-relativistic result here. Note that the leading term in ∆ ˜ E can be written as ∆ ˜ E / ˜ T λ =0 . V. ARBITRARY TEMPERATURE AND κ For arbitrary temperature and κ there is no symme-try and no small parameter which can help us proceedanalytically. We adopt a numerical approach, in particular we dis-cretise the loop, representing it by a large number, N , ofpoints, x i , i = 0 , . . . , N − 1, and then write an approx-imation to the action where derivatives are replaced byfinite differences (see Appendix C for details). Note thatthis is not a lattice regularisation as the points are notconstrained to lie on a lattice but may lie anywhere in R , up to numerical accuracy.The number N must be chosen such that the distancebetween neighbouring points, | dx i | := | x i +1 − x i | , is muchsmaller than the smallest scale in the problem, the cut-off, a . Note that for a continuous worldline, the globalreparameterisation symmetry τ → τ + c means that ˙ x is constant. Thus, to leading order in 1 /N , | dx i | is in-dependent of i and hence equal to L [ x ] /N , where L [ x ] Note that solutions of the equations of motion are only instantonsif their actions are positive. At zero temperature this restrictsus to κ < π . The same condition holds at high temperaturesfor the straight line instanton. a must bechosen to be much smaller than any other scale in theproblem. In summary we require L [ x ] N (cid:28) a (cid:28) Min( κ, A − [ x ; i ]) , (81)where A [ x ; i ] is the proper acceleration of the worldlineat the point i . Note that the interactions between twodisconnected worldlines do not need regularisation, sowe may treat them exactly (up to discretisation errors).Computational constraints impose a maximum possible N ( ∼ in our case). This in turn imposes a mini-mum a and hence a minimum κ and a maximum properacceleration.The equations of motion are then 4 N (ignoring for themoment the symmetries and the zero modes) coupled,nonlinear algebraic equations which we solve numerically.Starting with an initial guess at the solution we itera-tively solve the linearised equations until converging ona solution of the nonlinear equations, i.e. the Newton-Raphson method. An accuracy of better than 10 − wasusually reached in about 3 iterations. Simpler gradientmethods cannot be used here as the solution is a saddlepoint, having one negative mode. A. Finite temperature results At low temperatures we can use the zero temperatureinstanton as an initial guess. Once the iterations haveconverged, we can then increase the temperature slightlyand repeat the procedure using the solution from thelast run as the initial guess for the next. In this waywe can find all solutions in the ( κ, ˜ T ) plane which arecontinuously connected to the low temperature solutions.These all have the topology of a circle and we refer tothem as C instantons.There are also instantons with the topology of railwaytracks: two infinitely long disconnected pieces. Over thewhole ( κ, ˜ T ) plane there exist such solutions consistingof two straight lines (see Sec. IV F) which we refer toas S instantons. Below the lower branch of ˜ T λ =0 (seeFig. 6), there exists another class of solutions with thistopology and with lower action. These consist of twowavy lines and we refer to them as W instantons. Forsmall κ and for temperatures just below the lower branchof ˜ T λ =0 , we can use the non-relativistic approximationof Sec. IV F as an initial guess. From there we can step in κ and ˜ T to find all the continuously connected solutions.As we approach the lower branch of ˜ T λ =0 from below,the W instantons become straighter and merge with theS instantons.As discussed in section IV C, we must regularise theinteraction potential. This introduces a third parameter, a , the short distance cut off. For each point in the ( κ, ˜ T )plane, we find the corresponding solution for a range of a and evaluate the action, ˜ S ( κ, ˜ T ; a ), and its Legendre FIG. 7. Examples of the three types of numerical solutions,all with κ = 0 . a = 0 . 02. From left to right: a) a Cinstanton with ˜ T = 0 . 5; b) a W instanton with ˜ T = 0 . 66; c)an S instanton with ˜ T = 0 . ≈ ˜ T λ =0 (lower branch). transform, ˜Σ( κ, ˜ T ; a ). For small enough a we should beable to fit these to a linear function˜ S ( κ, ˜ T ; a ) ≈ ˜ S ( κ, ˜ T ) + c ( κ, ˜ T ) a, (82)for some c ( κ, ˜ T ). To find ˜ S ( κ, ˜ T ) we then extrapolate to a → 0, ensuring that the a dependence is linear (see Fig.11).The Newton-Raphson method does not converge in thepresence of zero modes, essentially because the solutionis not unique. As described in section IV, we fix thetranslation zero modes by demanding that the centre ofmass of the worldline is at the origin, ¯ x µ = 0. Thereis also a fifth zero mode corresponding to the remainingglobal symmetry of reparameterisation invariance. Forworldlines with circular topology we fix this by demand-ing that x − x N/ − = 0. Given a suitable initial guessthis essentially fixes the point x to be at the bottomof the loop and x N/ − to be at the top. In the hightemperature case, where the instanton splits up into twoseparate worldlines, the global part of the reparameter-isation invariance must be fixed on each side separately.We do so by demanding that there is a turning point at x on the right hand side and at x N − on the left hand side,i.e. we fix the spatial derivative to be zero there. In allcases we use Lagrange multipliers to impose constraints.In the high temperature case we found that there isalso a quasi-zero mode associated with translating one ofthe halves forward in Euclidean time and the other down-wards. The presence of this quasi-zero mode slows theconvergence of the Newton-Raphson method. To preventthis slowing down we fixed ¯ x L = 0 and ¯ x R = 0, ratherthan simply (¯ x L + ¯ x R ) = 0 (subscripts L and R refer toleft and right hand sides). This over-constrains the prob-lem but the solutions thereby found are also solutions ofthe original problem. Further, from the parity symmetrywe expect solutions to satisfy this extra constraint.In this way we can start to fill in the ( κ, ˜ T ) plane withinstanton solutions, building up a contour plot of theaction and a phase diagram. Each of the three differentclasses of solutions (C, W and S instantons) has a regionof existence and a region within which it has the lowestEuclidean action (the actions denoted respectively by ˜ S C ,˜ S W and ˜ S S ). If two solutions exist at a given point in the7 π κ ˜ C instantonsW instantonsS instantons FIG. 8. Contour plot of the action, ˜ S ( κ, ˜ T ), as calculatednumerically. The solid red and blue lines are our numericalresults and the solid green lines are given by Eq. (73). Theregion in the top right, bounded by the green dotted line, isthe region within which the S instanton is the only known so-lution. The blank region between the solid red and blue linesand for small κ is where we could not maintain the hierarchiesof Eq. (81) with N = 2 points. The dashed blue lines arelinear extrapolations from the contours found numerically tothe same value of the action at κ = 0 (Eq. (62)). plane, that with lower Euclidean action determines therate, and hence defines the phase. Only the regions withpositive action can describe tunnelling processes. Fig. 8is a contour plot of the Euclidean action as calculatednumerically.The phase diagram that emerges is quite interesting.The S instantons exist over the whole ( κ, ˜ T ) plane. TheC and W instantons do not. Where we have found theW instantons to exist, they have lower action than the Sinstantons. It also seems to be the case for the C instan-tons. It is the case at κ = 0 and we can give an argumentfor it at κ = 4 π . The action of the C instantons goes tozero at (4 π, 0) whereas that of the S instantons goes tozero at (4 π, /π ). Further, if we can assume that ˜ S C and˜ S S decrease with increasing temperature (i.e. the solu-tions have positive energy), then, where the C instantonsexist for κ = 4 π and ˜ T > 0, they must have lower action.We have not found numerically a region within whichboth the C and W instantons exist. It may be that theyexist in disjoint regions, or it may be that they coexistnear their phase boundary where our numerical calcula-tions fail.From Fig. 8 we can see the existence of two lines ofphase transitions: ˜ T CW ( κ ) separating the C instantonsfrom the W instantons and ˜ T W S ( κ ) separating the Winstantons from the S instantons. From our numericalresults, within the range of parameters explored, the linedefined by ˜ T = ˜ T W S ( κ ) appears to coincide with thelower branch of ˜ T λ =0 ( κ ). This line is a line of secondorder phase transitions, as discussed in section IV F. Theorder of the phase transitions at ˜ T = ˜ T CW ( κ ) is notclear, except at κ = 0 where it is of second order, asdiscussed in section IV D 2. At κ = 0 we also have that˜ T CW (0) = 1 / 2. Above this we can say nothing preciseas, in the region around ˜ T = ˜ T CW , we have not been able C instantonsW instantonsS instantonsLemon instantons (κ= ) FIG. 9. A slice through the ( κ, ˜ T ) plane at fixed κ = 0 . κ = 0. The expressionfor their action is π for ˜ T < / T > / 2. The transition temperature, ˜ T CW , lies somewhere inthe grey shaded region. The difficulty in maintaining thehierarchies of Eq. (81) have prevented us from calculating itmore accurately. maintain the hierarchies of Eq. (81). However it appearsthat ˜ T CW ( κ ) ≈ / 2, at least for κ ≤ κ, ˜ T ) outside the region spanned by our numer-ical calculations (see Fig. 8), there is little we can sayabout the form of the phase diagram. The two lines ofphase transitions may cross at some point ˜ T CW ( κ ∗ ) =˜ T W S ( κ ∗ ), which we denote by ( κ ∗ , ˜ T ∗ ), or even at mul-tiple points. Alternatively the line of phase transitionsbetween C and W instantons may remain forever belowthat of W and S instantons, i.e. ˜ T CW ( κ ) < ˜ T W S ( κ ) for all κ . More work is needed to better understand the phasediagram for larger κ and ˜ T .For comparison with the analytic results enumeratedin sections IV D, IV E and IV F, in Fig. 9 we also givea plot comparing the action as a function of ˜ T , for fixed κ = 0 . B. Fixed energy results We also calculate the Legendre transform of these re-sults. To calculate the energy of a solution we use Eq.(39). Fig. 10 is a contour plot in the ( κ, ˜ E ) plane of theexponential suppression, ˜Σ.At κ = 0 the relevant instanton is the lemon shapedone discussed in section IV D 2. The corresponding sup-pression is given by Eq. (66) and there is no phase tran-sition for any 0 < ˜ E < ˜ E c . On the other hand for κ > κ = 0(Eq. (66)). The extrapolations for both C and W in-stantons look good. How these instantons match ontothe lemon instantons at κ = 0, and where the phasetransition between them lies, is not clear. Note that forsmall, non-zero κ and small ˜ E the leading terms for both8 κ ˜ C instantonsW instantonsS instantons FIG. 10. Contour plot of ˜Σ( κ, ˜ E ), the exponential suppressionof the inclusive rate of Schwinger pair production at fixed en-ergy. The blank region in the top right, bounded by the greenline, is the region within which the exponential suppressionof the S instanton is negative. The blank region between thesolid red and blue lines and for small κ is where we could notmaintain the hierarchies of Eq. (81) with N = 2 points.The dashed blue lines are linear extrapolations from the con-tours found numerically to the same value of ˜Σ at κ = 0 (Eq.(66)). C and W instantons agree (equations (60) and (66)). C. Numerical errors For a selection of our numerical solutions we performedvarious checks. For the C and W instantons we computedthe lowest few eigenvalues of perturbations about the so-lutions and always found that there was one negativemode, as required for the solution to be interpreted as atunnelling solution. We also computed the spectrum ofeigenvalues about some S instantons, finding one nega-tive eigenvalue for temperatures above the lower branchof ˜ T λ =0 and more than one below this temperature. Theapparent absence of the self-force instability due to higherharmonic fluctuations may be due the cut-off, a , and dueto the discretisation of the worldlines. The conservationof ˙ x was accurate to about 1 part in 10 or better. Thesolutions were found to be symmetric under a rotationby π in the 3-4 plane, to numerical accuracy.The dominant errors in our numerical calculation aredue to the difficulty of maintaining the hierarchies of Eq.(81). We have rejected solutions for which L [ x ] / ( N a ) > . 15 or for which a/ Min( κ, A − [ x ; i ]) > . 2. The errorsdue to the finiteness of these quantities manifests in theextrapolation a → a , the dependence ofthe action on a is strongly nonlinear. This is due to a becoming comparable with the distance between points, L [ x ] /N , and hence the discreteness of the representationof the worldline becomes significant. We implementedan algorithm to fit to only the linear part of the plot.For each point in the ( κ, ˜ T ) plane we assemble the data { a, ˜ S ( κ, ˜ T ; a ) } in an array, ordered by the value of a . Wethen fit straight lines to all subsets of at least four con- + + + ++ + ++ + + + + FIG. 11. Plot of extrapolation of data to remove cut-off, a → 0. Here are plotted data points for ˜ S ( κ, ˜ T ; a ) for W instantonsat ( κ, ˜ T ) = (0 . , . a is expected to be due tothe effect of the discreteness scale, here L [ x ] /N ≈ . secutive data points, ensuring that this covers a rangeof a such that the maximum value is at least twice theminimum value. For each fit we calculate the standarderror in the result. For our final result, ˜ S ( κ, ˜ T ), we takethat with least standard error. We also throw away re-sults for which the standard error due to the fit is greaterthan 0 . 01, though in most cases it is much smaller.For ˜ T = 0 we have both approximate, numerical re-sults and an exact, analytic expression, Eq. (48). Thedifference between the two is found to increase with κ upto about 0 . 01 at κ = 1, using N = 2 points. This errorscales with the number of points as 1 /N . We also havean exact, analytic expression for large temperatures, Eq.(73). Unfortunately though, for the corresponding in-stanton, the S instanton, due to the enhanced symmetrythe zero modes corresponding to time translation and toreparameterisation invariance cannot be fixed as for theother instantons, preventing convergence of the Newton-Raphson method.As we approach the phase transition between C and Winstantons, ˜ T CW ( κ ), it becomes more difficult to main-tain the hierarchies of Eq. (81). Hence we expect errorsthere to be greater. VI. CONCLUSIONS In this paper, we have extended previous results onSchwinger pair production to arbitrary couplings and ar-bitrary temperatures. To achieve this we restricted our-selves to weak, constant external fields. This restrictionwas shown to result in a semiclassical approximation andwithin this approximation we have calculated the leadingbehaviour of the logarithm of the rate. As a by-product,we were also able to obtain inclusive pair production ratesat fixed energy.We adopted the worldline description. In this frame-work the problem reduced to one of solving the instantonequations of motion for a self-interacting worldline, an in-9teresting geometric problem.For weak couplings, like in QED, our results comple-ment the extensive literature on the subject, providing analternative approach which holds at all temperatures andin which some issues are clearer. In this case κ < (cid:15) andour approach, which includes all orders in κ but just theleading order in (cid:15) , does not seem necessary. However, aswe have discussed, the singular nature of the small κ per-turbation means that one aught not to simply set κ = 0from the outset. Doing this may lead to the incorrectinstanton and hence to incorrect results at leading orderin (cid:15) .In this weak coupling regime, and at temperatures˜ T < / 2, our results give small corrections to the lead-ing order results. When expanded they capture the two-loop, thermal correction for weak fields. We also find nothermal correction at one-loop in this temperature range.Though we have calculated analytically the correction tothe exponent to leading and next to leading order in κ ,a full calculation of higher order loop corrections wouldrequire also the thermal corrections to the prefactor.At higher temperatures, ˜ T > / 2, the singular natureof the weak coupling expansion gives non-trivial correc-tions to the naive κ → κ appear to approach the well known lemon shapedinstanton as κ → 0. At leading order, this solution gives anon-zero thermal enhancement to the rate. Note howeverthat this enhancement is not present in the one-loop ap-proximation, which breaks down due to the singular na-ture of the weak coupling expansion. The lemon shapedinstanton also shows an enhanced correction to the expo-nential suppression of order κ log( κ ) /(cid:15) . This dominatesover order the (cid:15) correction for sufficiently small (cid:15) .At intermediate and strong couplings our results opennew avenues. Using them one can make reliable estimatesfor the pair production rate of strongly charged parti-cles via the thermal Schwinger process. In particular onecould apply these results to the pair production of mag-netic monopoles. Sufficiently light magnetic monopoleswould be produced amply in the strong magnetic fieldsand high temperatures present in heavy ion collisions, inneutron stars and in the early universe.In this paper we have only calculated the exponentialsuppression of the rate. For direct phenomenological ap-plication, one should also calculate the prefactor. TheRefs. [105, 119, 120] would be an apt place to start.They all find similar instantons to ours, though in theo-ries without dynamical long-range forces.The appearance of the self-force instability in the semi-classical evaluation of the path integral raises some in-triguing questions that require further work. So too doesthe instanton phase diagram, Fig. 8, for which furtherwork is needed to determine the form of the phase dia-gram at larger values of κ and ˜ T .The worldline description that we have developed herecould be used to calculate pair production rates for otherinduced Schwinger processes at arbitrary coupling. Forexample one could consider a non-constant external field. The numerical approach we have adopted would then di-rectly apply. ACKNOWLEDGMENTS The authors would like to thank Toby Wiseman,Sergey Sibiryakov, Ian Jubb, Edward Gillman, and LoisOvervoorde for illuminating discussions. OG would alsolike to thank Lois Overvoorde for help with the compu-tational work. OG is supported by an STFC studentshipand AR by STFC grant ST/L00044X/1. The computa-tional work was undertaken on the Cambridge COSMOSSMP system, part of the STFC DiRAC HPC Facilitysupported by BIS NeI capital grant ST/J005673/1 andSTFC grants ST/H008586/1, ST/K00333X/1. Appendix A: QED and SQED in weak external fields For sufficiently small scalar self-coupling, λ , the onlydifference between QED and SQED is the spin of thecharged particles. Following the manipulations of SQEDin Eq. (6) we see that the difference for a spin 1 / − ( − i /D − im ) s ) = (cid:90) D x µ e − S [ x µ ,A extµ + A µ ; s ] Spin [ x µ , A extµ + A µ ; s ] , (A1)where S [ x µ , A extµ + A µ ; s ] is given in Eq. (9) and thespin factor is given by Spin [ x µ , A extµ + A µ ; s ] := T r γ P e ig (cid:82) s d τ Σ µν ( F extµν ( x )+ F µν ( x )) (A2)where T r γ signifies the trace over spinorial indices, P isthe path ordering operator and Σ µν are the generators ofLorentz transformations in the spin 1 / µν = [ γ µ , γ ν ] / 2, where γ µ are the gamma matrices.The next step is to integrate over the gauge field A µ .Note that even with the inclusion of the spin factor theintegration over A µ is Gaussian and hence can be doneexactly. In the spin 0 case, the integration takes thefollowing form (cid:90) D A µ e − (cid:82) x (cid:82) y A µ ( x ) G − µν ( x,y ) A ν ( y )+ i (cid:82) x A µ ( x ) j µ ( x ) , (A3)where (cid:82) x := (cid:82) d x , and j µ ( x ) is given by j µ ( x ) = g (cid:90) s d τ ˙ x µ ( τ ) δ (4) ( x − x ( τ )) . (A4)0Performing the integration leads to the following expo-nentialexp (cid:26) − (cid:90) x (cid:90) y j µ ( x ) G µν ( x, y ) j ν ( y ) (cid:27) . (A5)The difference in the spin 1 / j µ ( x ) → j µ ( x ) + ig (cid:90) s d τ Σ µν ∂ ν δ (4) ( x − x ( τ )) , := j µ ( x ) + ξ µ ( x ) . (A6)Now, we scale all the dimensionful quantities as in sectionII, i.e. τ → τ /s , s → s/gE and x µ → x µ m/gE . Thisreduces all dependence on the parameters to dependenceon (cid:15) := gE/m (cid:28) κ := g E/m . We can nowwrite the interaction terms in the spin 1 / (cid:26) − κ (cid:15) (cid:90) x (cid:90) y j µ ( x ) G µν ( x, y ) j ν ( y ) − κ (cid:90) x (cid:90) y ξ µ ( x ) G µν ( x, y ) j ν ( y ) − κ(cid:15) (cid:90) x (cid:90) y ξ µ ( x ) G µν ( x, y ) ξ ν ( y ) (cid:27) , (A7)where j µ , ξ µ and G µν are now independent of g , E and m . In this paper, we have allowed κ to freely vary upto O (1) (that is because we only require κ (cid:28) g and weconsider strong coupling) however for the semiclassicalapproximation to be valid we require (cid:15) (cid:28) 1. Hence thespin dependent factors are subleading (as long as the di-mensionless parts are at most O (1)) and we can drop all ξ µ dependence.The net result of all this is that, to leading order in (cid:15) , the instanton describing pair production is the samefor both theories, as is the fluctuation prefactor aboutthe instanton, excluding an overall spin factor (2 s + 1).Charge renormalisation, which is not included to leadingorder in (cid:15) , is different in QED and SQED. In both caseswe expect the final results to depend on the renormalisedcharge, as discussed in section IV C. Appendix B: Worldline description of extendedparticles For elementary particles the geometric worldline de-scription arises naturally and can be derived by standardmethods from the field theoretic description, as shown insection II. For extended field configurations, such as soli-tons, no exact worldline description can exist. However,for circumstances where the extended field configurationis much smaller than all other scales, an effective world-line description can suffice [10, 48, 104]. This is analogousto the description of the motion of planets in the solarsystem in terms of the motion of points.In [48] just such an effective description was explicitlyderived for the ’t Hoof-Polyakov monopole [40, 41]. The worldline instanton that they found was a circle and theeffective worldline description was found to be valid whenthe radius of the circle was much larger than the size ofthe ’t Hooft-Polyakov monopole. We wish to generalisethis result for more general worldline curves.The ’t Hooft-Polyakov monopole is a static solution tothe field equations for the Georgi-Glashow model, andother similar theories. It is an extended solution andhence it cannot be said to be at a position, howeverit does have a well defined centre and core region, be-yond which all but the Abelian gauge field is exponen-tially damped. We can thus assign to the centre of themonopole solution a worldline, i.e. a map from the realline to the path in Minkowski space traced out by thecentre of the monopole solution. The static solution andLorentz transformations of it (which are also solutions)have straight, timelike worldlines. Static solutions of theEuclidean (Wick rotated) theory need not be timelike.Combinations of straight worldlines are no longer exactsolutions due to the interactions between them. However,in the limit that the monopoles are infinitely separatedthis should become an exact solution [90] . At finiteseparation, due to the long ranged interaction betweenmonopoles, the solution is no longer exact. However, wecan find an approximate solution following [48].We consider for example the Georgi-Glashow theory.To find an approximate solution to the full (Euclidean)field equations we first solve the equations of motion fora pointlike monopole in a given external magnetic field,at a certain temperature, including the self-interaction.These are the classical worldline calculations we have car-ried out in this paper. For simplicity, we restrict theworldline to the ( x , x ) plane. We use construct coordi-nates centred on the worldline and Fermi-Walker trans-ported along it (see for example [126]). We denote thecoordinate along the worldline as u and the normal co-ordinate in the plane as v , with ( x , x , v ) = (0 , , D i F ij + a ( u ) F vj + O ( a ( u ) v ) = [ D i φ, φ ] ,D i D i φ + a ( u ) D v φ + O ( a ( u ) v ) = λg ( | φ | − M W ) φ, (B1) In the Bogomolny-Prasad-Sommerfeld limit [121], i.e. when thescalar self-coupling is infinitesimally small and positive, such su-perpositions of static, like-charged monopoles are in fact exactsolutions. That is because in this limit the attraction due to theHiggs field is exactly cancelled by the repulsion due to the gaugefield [122]. At low energies the dynamics of such multi-monopolesolutions is given by geodesic motion of the collective coordinateson the configuration space of solutions [123–125]. D a = ∂ a + igA a ; i, j run over ( x , x , v ); λ is thefour point self-coupling of the Higgs particle; M W is themass of a W boson and a ( u ) is the magnitude of the accel-eration of the worldline. At zeroth order in the accelera-tion these equations are solved by the ’t Hooft-Polyakovmagnetic monopole, static along the worldline. Hence,at lowest order in the acceleration the full field theoreticcalculation reduces to the geometric, worldline one whichwe have pursued in this paper. This also requires thatthe radius of curvature of the worldline is large comparedwith the classical radius of the monopole solution. In ourdimensionless units this is κ (for the Georgi-Glashow the-ory) and hence we get the constraint κ (cid:28) a ( u ) − . Appendix C: Finite difference formulation In this appendix we give our discrete approximation tothe action in Eq. (33) and the corresponding equations ofmotion. We use a simple finite difference approximation˜ S [ x ] = (cid:115) N (cid:88) i ( x µi +1 − x µi ) − (cid:88) i x i ( x i +1 − x i ) − κ (cid:88) i,j ( x µi +1 − x µi )( x µj +1 − x µj ) G R ( x i , x j ; ˜ T , a )(C1)where i and j run over 0 , , ..., N − ∞ (cid:88) n = −∞ − π (( t + n/ ˜ T ) + r + a )) =˜ T sinh (cid:16) π ˜ T √ a + r (cid:17) π √ a + r (cid:16) cos (cid:16) π ˜ T t (cid:17) − cosh (cid:16) π ˜ T √ a + r (cid:17)(cid:17) , (C2)and likewise for the exponential counterterm ∞ (cid:88) n = −∞ √ π π a e − ( r +( t + n/ ˜ T ) ) /a =˜ T e − r /a ϑ (cid:16) π ˜ T t, e − π a ˜ T (cid:17) πa (C3)where t and r are the temporal and spatial differences asin section II B and ϑ is the Jacobi theta function of thethird kind. Due to the lack of well optimised numericallibraries for the Jacobi theta function, we in fact make adifferent choice of counterterm, which also reduces to the exponential regularisation for small ˜ T , G R ( x i , x j ; ˜ T , a ) =˜ T sinh (cid:16) π ˜ T √ a + r (cid:17) π √ a + r (cid:16) cos (cid:16) π ˜ T t (cid:17) − cosh (cid:16) π ˜ T √ a + r (cid:17)(cid:17) + √ π e − r /a e( cos ( π ˜ T t ) − ) / (2 π a ˜ T ) π a (C4)This regularisation is smooth and periodic in 1 / ˜ T , as wellas being relatively fast to numerically evaluate.One further point, also mentioned in section IV C, isthat there is no need to regularise the interactions be-tween disconnected parts of worldlines, one may use theunregularised propagator. This is useful as it removessome sources of error due to the finite cut off, a . In ourcalculations, we have used the unregularised propagatorfor the interaction between the left and right hand sidesof the W instantons. It could also be used, though wedidn’t, for the interaction between thermal copies for theC instantons.As discussed in section V, we fix the N zero and quasizero modes using Lagrange multipliers. Writing the con-straint equations as C a [ x ] = 0, where a runs over 1 , .., N ,we define a new action including the Lagrange multiplierterms ˜ S [ x, λ ] := ˜ S [ x ] + N (cid:88) a =1 λ a C a [ x ] . (C5)The λ a are the Lagrange multipliers.The equations of motion, which are simply 4 N + N coupled, nonlinear, algebraic equations are found by tak-ing partial derivatives of (C5), with respect to x ρk and λ b , ∂ ˜ S [ x ] ∂x ρk = 0 ,C b [ x ] = 0 . (C6)The Newton-Raphson equations derived from these, andan initial guess, are a system of linear equations, whichwe solve by LU decomposition, using the numerical li-brary Eigen 3 [127]. The 1 and 2 directions are trivial anddecouple, leaving 2 N + ˜ N equations, where ˜ N ( < N ) isthe number of zero modes in the reduced space. Appendix D: Numerical data Along with this paper, we have made available the nu-merical results presented in summary form in section V.They are in the file gould2017thermal results.csv . Thefirst line gives the column headings and all followinglines give the corresponding numerical data as commaseparated values. The meanings of the headings are asfollows.2pot: The nature of the interaction potential, seebelow.log2N: log ( N ), where N is the number of pointsin the worldline.kappa: the coupling, κ := g E/m .T: the temperature, ˜ T := mT /gE .a: the cut-off, a .S: the action, ˜ S := S/(cid:15) .E: the energy, ˜ E := E /m .len: the length of the worldline.kinetic: the gauge-fixed length of the worldline, ˜ L ,Eq. (34).i0: minus the area of the worldine, − ˜ A , Eq. (35).vr: the interaction, ˜ V R , Eq. 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