Worldline Sphaleron for Thermal Schwinger Pair Production

IMPERIAL-TP-2018-OG-1 HIP-2018-17-TH Worldline sphaleron for thermal Schwinger pair production

Oliver Gould,1, 2, ∗ Arttu Rajantie,1, † and Cheng Xie1, ‡ 1Department of Physics, Imperial College London, SW7 2AZ, UK 2Helsinki Institute of Physics, University of Helsinki, FI-00014, Finland (Dated: August 24, 2018) With increasing temperatures, Schwinger pair production changes from a quantum tunnelling to a classical, thermal process, determined by a worldline sphaleron. We show this and calculate the corresponding rate of pair production for both spinor and scalar quantum electrodynamics, including the semiclassical prefactor. For electron-positron pair production from a thermal bath of photons and in the presence of an electric ﬁeld, the rate we derive is faster than both perturbative photon fusion and the zero temperature Schwinger process. We work to all-orders in the coupling and hence our results are also relevant to the pair production of (strongly coupled) magnetic monopoles in heavy-ion collisions.

I. INTRODUCTION Schwinger rate, Γ(E,T ), takes the form,

∞ In non-Abelian gauge theories, sphaleron processes, X Γ(E,T ) = c h(J · A)ni, (1) or thermal over-barrier transitions, have long been un- n derstood to dominate over quantum tunnelling transi- n=0 1 tions at high enough temperatures [1–4] . The same is where E is the magnitude of the electric field and T is true, for example, in gravitational theories [5]. On the the temperature. The leading term, c0, gives the one loop other hand, sphalerons have been conspicuously absent result, that of Schwinger [6], at zero temperature. There from the study of Abelian gauge theories. In this paper, has been some disagreement in the literature about the we make partial amends for this absence by finding a thermal corrections to c0. According to the worldline sphaleron in quantum electrodynamics (QED). We note instanton calculation, at low temperatures there are no that, unlike the corresponding quantum tunnelling tran- thermal corrections at one loop, so agreeing with Refs. sition at zero temperature, this sphaleron is not visible at [13–15] using alternative approaches. However at high any finite order in the loop expansion, which may explain enough temperatures, such that T > gE/(2m),2 where g why it has been missed in the past. is the charge coupling and m is the mass of the (light- In the presence of an electric field, empty space is non- est) charged particles, the worldline instanton calculation perturbatively unstable to decay into electron-positron suggests that the one loop calculation is no longer a con- pairs, called Schwinger pair production [6]. At zero tem- sistent truncation of the problem and thermal corrections perature, Schwinger pair production is determined by a are expected at leading order [8, 11, 12, 16]. See Refs. circular worldline instanton [7]. At nonzero temperatures [17–19] for alternative conclusions regarding the thermal, the rate of this process is enhanced by the energy of the one loop correction. The next order term, c1hJ · Ai, has thermal bath, and the worldline instanton is deformed also been calculated, both at zero [20–23] and at nonzero away from circular [8–12]. At sufficiently high tempera- but low temperature [10, 24]. Now, the presence of the tures the process becomes essentially thermal and is de- worldline sphaleron requires the dynamical photon inter- termined by a worldline sphaleron (which, in Ref. [12], action, J · A, in the exponent of the semiclassical expan- we referred to as the S instanton). In this paper we sion. Hence, it requires all orders in the loop expansion. briefly review the derivation of the worldline sphaleron Despite this, the worldline sphaleron itself is rather and then calculate the sphaleron rate, including the fluc-

arXiv:1806.02665v2 [hep-th] 9 Oct 2018 simple. The electric version consists of an electron and a tuation prefactor. positron a finite distance apart, such that the attractive At zero temperature, Schwinger pair production is Coulomb force between them is balanced by the force of a quantum tunnelling process visible at one loop. If the external electric field pushing them apart (see Fig. we denote symbolically the interaction between the dy- 1). namical (as opposed to external) photon field and the Following the work in Ref. [12], our calculations in charged particles as J · A, then the loop expansion of the this paper are carried out in both QED and scalar QED (SQED), making no assumptions as regards the magnitude of the charge coupling, g. We let g range from in- ∗ [email protected] finitesimal to infinite. In SQED we will assume however † [email protected] ‡ [email protected] 1 The word sphaleron was originally coined in Refs. [1, 2]. It denotes a static, localised, unstable field configuration, which is a minimum of the energy functional in all directions in function 2 Except where noted, we use the natural units common in high space except one, where it is a maximum. energy physics, where c = ~ = kB = 0 = 1. 2

We should add a quick note on what we mean by our initial thermal state. We consider temperatures, T , much less than the mass of any charged particles, m. In this case there are no charged particles in the initial state, it is a thermal bath of photons. Upon turning on the external field, charged particles are produced by the thermal Schwinger process. After an initial transitory period, FIG. 1. The simple worldline sphaleron, for the thermal which depends on how quickly the electric field is turned Schwinger process. The Coulomb attraction between electron, e−, and positron, e+, is balanced by the external electric on, and before the back-reaction of the charged parti- field, E, which pushes them apart. Pictured here in the Eu- cles becomes significant, there is a period during which clidean thermal approach, where the solution lives is R3 ×S1, charged particles are produced at a constant rate [25, 26]. the circumference of the circle being the inverse temperature, This rate is what we calculate. The process is relevant 1/T . only for the lightest charged particle (either the electron or the lightest magnetic monopole), because the lightest particles will be produced exponentially more quickly that the scalar self-coupling is weak, λ 13. We con- than heavier particles and, once produced, their presence sider circumstances when the calculation is semiclassical will cause Debye screening. and hence the rate is slow. In this regime the results are The chief result of this paper is the following, the largely independent of many properties of the charged sphaleron rate for thermal Schwinger pair production, particles.

√ 2 3/2 3 3 2 2 (2s + 1) TWS (mT ) 2m g E/π gE g E T g T gT − T + T Γ(E,T ) ≈ e 1 + O 2 , 2 , , , , (2) (4π)3/2 sin πTWS sinh2 πT√WS m m m m E T 2T

where s is the spin of the charged particles (either zero where the rate is enhanced by some other ingredient. In or 1/2) and the expression applies for temperatures sat- particular, as we consider in this paper, a thermal bath isfying of photons in the initial state may significantly increase the rate of pair production, and hence one may observe 1/4 4gE3 pair production at lower field strengths. One way to ex- T > T ≈ . (3) WS π3m2 perimentally realise a thermal bath of photons would be to use a laser-heated hohlraum, as has been proposed to Eqs. (2) and (3) are valid to all orders in the coupling observe the Breit-Wheeler process, γγ → e+e− [31, 32]. g, though only to leading order in the five different di- Our initial interest in this calculation was for its rele- mensionless combinations of parameters within the big vance to magnetic monopole production in heavy-ion col- O symbol in Eq. (2). Hence all of these parameters must lisions. In the fireball of a heavy-ion collision there are be small for our approximations to be good. This is nat- strong magnetic fields and high temperatures. Hence, urally satisfied, for all but the last of these parameters, magnetic monopoles may be produced by the dual ther- if the charged particles are sufficiently heavy. mal Schwinger process [33]. There is much promise for Schwinger pair production of electrons and positrons magnetic monopole searches in current and future heavy- has yet to be experimentally observed, as inaccessibly ion collisions, in particular the MoEDAL experiment strong electric fields are required to generate an appre- [34, 35] at the LHC, so spurring this work. Here we ex- ciable rate. High intensity lasers have achieved electric tend the results of Ref. [12] by calculating the prefactor field strengths of O(0.01%) of the Schwinger critical field of the rate, an important quantity for making comparison 2 strength, Ec := πme/g [27] and the next generation of to experiment. high intensity lasers aim to be able to reach electric field Magnetic monopoles are strongly coupled to the pho- strengths of O(1%) of Ec [28–30]. Though this is not 2 ton. The minimum magnetic charge squared, gD = sufficient to observe the original Schwinger process, it (2π/e)2 ≈ 430, follows from the Dirac quantisation con- may be possible to observe induced Schwinger processes, dition [36]. Hence one must work to all orders in the coupling to derive results applicable to magnetic monopoles. If one does so and does not consider electric and mag- 3 Of course photon loops will generate this term. However, the netic charges simultaneously, one can study magnetic term is a pointlike interaction between scalar loops (given no monopole Schwinger pair production via its electromag- external legs) and, in the dilute instanton approximation that netic dual. In this case the duality amounts simply to a we will make, such loops are subdominant and are neglected. relabelling of electric degrees of freedom and charges as 3 magnetic. As our calculation reduces to a semiclassical dominated by a static field configuration, a sphaleron, one, we only rely on the classical electromagnetic duality. the rate is given by In the regime considered here, our results are valid both for elementary and ’t Hooft-Polyakov monopoles [37, 38]. −|ω−| Γ(E,T ) ≈ Im log(Z), (5) The paper is set out as follows. In Sec. II we de- πV fine the sphaleron rate and set up the calculation of it. In Sec. III we find the sphaleron and the spectrum of where V is the spatial volume, Z is the canonical par- fluctuations about it. In Sec. IV we note that the spec- tition function excluding the states containing the de- trum of fluctuations manifests the well-known self-force cay products and |ω−| is the rate of growth with time of instability. In Sec. V we specialise to a region of param- the unstable mode, responsible for the imaginary part of eter space where this instability does not arise. We then log(Z). It is assumed that the rate of decay is slow, so calculate the sphaleron rate, including the prefactor, the that the process is out of equilibrium. main result of this paper, which we extend to spin half We first consider the thermal Schwinger process in charged particles in Sec. VI. In Sec. VII we discuss the SQED, with the external field, E, pointing along the x3 implications of our results for the possible experimental direction. We write the finite temperature path integral observation of electron-positron pair production from a using the imaginary time formalism. The Euclidean La- purely photonic initial state. In Sec. VIII we discuss grangian for this theory is, the implications of our results for magnetic monopole 1 searches. In Sec. IX, we summarise our results. L := F µν F + D φ(Dµφ)∗ + m2φφ∗, (6) SQED 4 µν µ

where Fµν = ∂µAν −∂ν Aµ is the field strength and Dµ = II. GENERAL APPROACH ext ∂µ + igAµ + igAµ is the covariant derivative, showing a split of the photon field into external (or background) The thermal rate of decay of a metastable state has and dynamical parts. We assume the scalar self-coupling, been studied by many authors [39–42]. In regards to the i.e. λ(φφ∗)2/4, is sufficiently small that we may ignore it, thermal Schwinger process, in Ref. [12] we calculated the 3 −S at least in the range of energies considered . Note that logarithm of the rate, the factor S in Γ(E,T ) ∼ e . For for QED, which we consider later, no such term arises. the sphaleron process, we found The finite temperature partition function for this the- " r !# ory is, 2m g3E Γ(E,T ) ∼ exp − 1 − . (4) T 4πm2 Z R − LSQED Z = DAµDφ e x , (7) The difference from the expected Boltzmann suppression in the absence of an external field, 2m/T , can be under- where the Euclidean time direction has length 1/T and stood as a field dependent mass renormalisation. This the fields satisfy periodic boundary conditions in this di- accounts for the Coulomb corrections to the rest mass of rection. We leave the gauge fixing implicit. Eq. (7) can a particle-antiparticle pair at the separation where the be rewritten exactly using the worldline formalism, by external field and Coulomb force balance. The renor- carrying out purely formal manipulations [7, 12, 43]. Es- p 3 2 malised mass would be m∗ = m(1 − g E/(4πm )) so sentially a change of integration variables, the usefulness that the exponential suppression reads 2m∗/T . The same of the worldline representation is that it allows one to kind of mass renormalisation arises in Schwinger pair pro- circumvent the usual loop expansion, and obtain gauge duction at zero temperature [7, 20, 22]. invariant resummations of (infinite) classes of Feynman To go beyond this, we need an explicit expression for diagrams. In Ref. [12] we gave the following exact world- the rate, including the prefactor. As argued for in Refs. line representation of the thermal partition function in [39, 40], for high temperatures,4 where the process is SQED,

" ∞ n Z ∞ Z # 1 1 X 1 Y dsj µ − 1 S˜[x ;s ;κ,T˜] κ P HH dxµdxν G (x ,x ;T˜) Im log(Z) = Im log 1 + Dx e j j e k

where := gE/m2 will act as the semiclassical param- µ 4 At lower temperatures the rate is given instead by a slightly eter, akin to ~. The path integrals over the xj are different expression, related by replacing |ω−| → 2πT in the over closed worldlines, with distance measured in units prefactor. of m/gE. In these units the effective coupling between 4 worldlines is κ := g3E/m2 and the effective temperature requires all orders in the loop expansion, so we do not is T˜ := mT/gE. We will refer to sj as Schwinger param- Taylor expand in κ. Rather, we note that for 1 the ˜ eters. The free thermal photon propagator is Gµν , where worldline action, S/, becomes large and a semiclassical µ, ν run over Euclidean indices 1, 2, 3, 4. The scaled ac- approximation should be valid. tion, S˜, for a single worldline, is5 In this context, one would expect the dominant contri- butions to Eq. (8) to come from configurations consisting Z 1 Z 1 s 1 µ 4 of a small number of worldlines. This is because configu- S˜[x; s; κ, T˜] := + dτx˙ x˙ µ + dτx3x˙ 2 2s 0 0 rations with more worldlines would be expected to have Z 1 Z 1 larger actions. Thus, as in Ref. [12], we perform a cluster κ 0 µ ν 0 0 − dτ dτ x˙ (τ)x ˙ (τ )Gµν (x(τ), x(τ ); T˜). expansion of Eq. (8), 2 0 0 (9) ∞ X Γ(E,T ) = Γ(n)(E,T ), (12) The interaction involving the photon propagator has a n=1 short distance divergence, corresponding to the electromagnetic contribution to the self-energy of the charged where Γ(n) is the contribution to Γ from clusters of n particles. In this paper we regularise this divergence fol- worldlines. Within the semiclassical approximation, the lowing Polyakov’s seminal work [44]. We add to the dis- cluster expansion is a dilute instanton expansion. The tance squared between points a short distance cut off, 2 leading contribution to the thermal Schwinger rate is a . Up to gauge fixing terms which vanish upon integra- given by the instanton with smallest action. This lead- tion around a closed worldline, the regularised thermal ing order term is approximately equal to the density of photon propagator is these instantons, and is exponentially small. Higher or- ∞ der terms in the cluster expansion are expected to be X n G (x , x ; T ; a) := G(x , x + e ; a)δ suppressed by powers of this density, or by subleading µν j k j k T 4 µν n=−∞ instanton densities. ∞ At low temperatures, T˜ 1, the dominant instanton −δ X µν is the circular worldline instanton [7, 49] (leftmost in Fig. = n 4π2 (x − x − e )2 + a2 (1) n=−∞ j k T 4 2), a saddle point of Γ . At higher temperatures, ther- q mal corrections deform this instanton, so increasing the T sinh 2πT r2 + a2 δ jk µν rate (second and third from left in Fig. 2). Above some = q q , 2 2 2 2 temperature, T˜CW (κ), where T˜CW (0) = 1/2, a second 4π r + a cos (2πT tjk) − cosh 2πT r + a jk jk instanton with different topology dominates, called a W (10) instanton in Ref. [12]. It consists of a charged particle and antiparticle oscillating back and forth, parallel to where e4 is the unit vector in the Euclidean time (2) 4 4 the external field, and is a saddle point of Γ (second direction and we have defined tjk := xj − xk and q from right in Fig. 2). At higher temperatures still, above 1 1 2 2 2 2 3 3 2 ˜ rjk := (xj − xk) + (xj − xk) + (xj − xk) . For TWS(κ), this W instanton ceases to exist and the domi- smooth worldlines without self-intersections the only di- nant instanton is the static worldline sphaleron solution vergence as a → 0 is the self-energy [44–47], which can (rightmost in Fig. 2), also a saddle point of Γ(2). The be absorbed by adding the following mass counterterm, instanton phase diagram outlined here has been estab- lished for 0 < κ ≤ 1 and may be subject to change at Z 1 larger values of κ. κ π p µ − 2 dτ x˙ x˙ µ. (11) 8π a 0

We may then take the limit a → 0 in the final re- III. THE SPHALERON AND FLUCTUATIONS sults. That Eq. (11) is a mass counterterm can be ABOUT IT seen by noting that the first two terms in Eq. (9) are a reparameterisation-fixed form of the same term [48]. µ Parametrically, the usual loop expansion, given by Eq. Eq. (9) gives the action for one worldline, x (τ), with Schwinger parameter s, and it gives the exponent of the (1), is here mapped to a Taylor expansion in κ. As men- (1) tioned in the introduction, to see the worldline sphaleron integrand of Γ , when divided by . In terms of the action for one worldline, the action for two worldlines, the scaled exponent of the integrand of Γ(2), is

˜ ˜ ˜ ˜ ˜ ˜ 5 There are two minor diﬀerences here with respect to Ref. [12]. S[x, y; sx, sy; κ, T ] := S[x; sx; κ, T ] + S[y; sy; κ, T ] We have rescaled s → s/2 and here we are treating negatively Z 1 Z 1 0 µ ν 0 0 charged particles as matter and positively charged particles as − κ dτ dτ x˙ (τ)y ˙ (τ )Gµν (x(τ), y(τ ); T˜). antimatter, which amounts to a plus rather than minus sign for 0 0 the third term on the right hand side. (13) 5

about the solution in a Fourier series,

µ µ ζ (τ) − ζ0 (τ) = ∞ √ √ µ X h µ µ i a0 + an 2 cos(2πnτ) + bn 2 sin(2πnτ) , n=1 µ µ ξ (τ) − ξ0 (τ) = ∞ √ √ µ X h µ µ i c0 + cn 2 cos(2πnτ) + dn 2 sin(2πnτ) . FIG. 2. Worldline instantons relevant for the thermal n=1 Schwinger process. Each column represents a different in- (17) stanton, each relevant for a given temperature. The rightmost instanton is the worldline sphaleron, which dominates the rate at the highest temperatures. The external field points The second order action is diagonal in these Fourier co- along the 3-direction and the 4-direction is the Euclidean time efficients. It can thus be expressed as direction. 1 1 S˜(2) = T˜(s − s )2 + T˜(s − s )2 2 x 0 2 y 0 Due to the double integral terms in the action, the cor- ∞ 4 1 X X responding equations of motion are integrodifferential + (αµaµaµ + γµcµcµ) 2 n n n n n n equations. n=0 µ=1 ∞ 4 The sphaleron is a static solution to these equations of 1 X X + (βµbµbµ + δµdµdµ) , motion. It consists of particle and antiparticle sitting a 2 n n n n n n fixed distance apart along the x3 axis (rightmost in Fig. n=1 µ=1 1 1 1 X 2). It is given by = T˜(s − s )2 + T˜(s − s )2 + λ χ χ , 2 x 0 2 y 0 2 i i i i (18) 1r κ 1 x(τ) = x0(τ) := 0, 0, , − (2τ − 1) , 2 4π 2T˜ where on the last line we have defined χi to run over 1r κ 1 y(τ) = y (τ) := 0, 0, − , (2τ − 1) , all the different fluctuations of the worldline, {χi} := 0 ˜ µ µ µ µ 2 4π 2T {an, bn, cn, dn}, and λi to run over all the corresponding µ µ µ µ 1 eigenvalues, {λi} := {αn, βn , γn , δn}. The eigenvalues sx = sy = s0 := (14) T˜ for n = 0 are

2π 2π 4π The action of the sphaleron is α0 = {√ , √ , −√ , 0}, πκT˜ πκT˜ πκT˜

γ0 = {0, 0, 0, 0}. (19) 2 r κ S˜(κ, T˜) := S˜[x0, y0; s0, s0; κ, T˜] = 1 − . (15) T˜ 4π The four zero modes of γ0 correspond to translations of 4 the instanton. The ﬁfth, α0, corresponds to translation Expanding the action to second order about this solution in the parameter τ. The negative eigenvalue corresponds to increasing, or decreasing, the separation between the gives a surprisingly large number of terms, O(100), most ˜ of which are due to the nonlocal interactions. To proceed particles. It is negative for all κ and T . we deﬁne ζµ(τ) := xµ(τ) − yµ(τ) and ξµ(τ) := xµ(τ) + As regards the harmonic modes, by translational sym- yµ(τ). The solution given in Eq. (14) can then be written metry in the Euclidean time direction, one can see that as γn = βn, and δn = αn. Further, an nth harmonic at a given temperature, T˜, can be seen as n copies of an ˜ r n = 1 harmonic at the higher temperature nT . Hence, κ 1 the eigenvalues for n > 1 are given in terms of the n = 1 ζ(τ) = ζ0(τ) := 0, 0, , − (2τ − 1) , 4π T˜ eigenvalues by ξ(τ) = ξ0(τ) := 0, 0, 0, 0 . (16) αn(κ, T˜) = nα1(κ, nT˜), (20)

Due to the periodicity, we may expand the ﬂuctuations and likewise for the others. Explicitly the harmonic 6

λ eigenvalues are found to be, + 6000 ( √ + 1 2 ˜ 2 2 3 ˜2 π αn = (2πn) T − π κn T + √ 5000 + 2 3 κT˜ + 4000 √ 1 √ ˜ + + π πκn2T˜ + n + √ e− πκnT , ˜ + πκT 3000 + √ ++ ++ ++ ++ ++ 1 2 π + ++ ++ ++ 2 ˜ 2 3 ˜2 ++ ++ (2πn) T − π κn T + √ 2000 + +++ ++ +++ ++ 2 3 ˜ + +++ κT + ++ + + + √ √ 1000 + + ++ 2 1 − πκnT˜ + + ˜ + ++ + π πκn T + n + √ e , + ++ + + ˜ + + πκT + + + ++ n √ 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 1 2 2 π (2πn)2T˜ − π2κn3T˜2 − √ 2 3 κT˜ FIG. 3. For (κ, T˜) = (0.1, 1.2), the first 500 eigenvalues of √ fluctuations about the sphaleron, ordered by n, showing the 1 − πκnT˜ − 2π n + √ e , self-force instability at n = 30. πκT˜ ) 1 (2πn)2T˜ , (21) 2 carry out this procedure, we find a regime of parameters where one can separate the scales between the unstable UV modes and the rest of the fluctuations. ( √ 1 2 2 2 3 2 π The self-force instability is well known in classical elec- βn = (2πn) T˜ − π κn T˜ + √ 2 3 κT˜ trodynamics [54–56] and in various approximations to quantum electrodynamics [57–63]. Its existence in quan- √ 1 √ ˜ − π πκn2T˜ + n + √ e− πκnT , tum mechanical systems has been linked to the unbound- πκT˜ edness of the spectrum of the Hamiltonian [64–66]. √ 1 2 π In Refs. [59, 60] it was found that a nonrelativistic (2πn)2T˜ − π2κn3T˜2 + √ electron interacting with a quantised photon field does 2 3 κT˜ 2 not show the self-force instability, at least for e /(4π) . √ √ 2 1 − πκnT˜ 1. Starting from an extended charge distribution, with − π πκn T˜ + n + √ e , πκT˜ finite mass, they found that one could take the size of the √ 1 2 2 π charge distribution to zero and the result was free of the (2πn)2T˜ − π2κn3T˜2 − √ self-force instability. However, if they changed the orders 2 3 ˜ κT of the limits and first took the Compton wavelength to 1 √ ˜ zero (or the mass to infinity), the self-force instability was + 2π n + √ e− πκnT , πκT˜ present. Taking the mass to infinity amounts to dropping ) charged particle loop corrections. 1 (2πn)2T˜ . (22) In the dilute instanton gas approximation, extra 2 charged particle loops with S˜ ≥ O(1) are suppressed by the instanton density, as argued in Sec. II, and Ref. [12], and hence can justifiably be dropped. On the other IV. SELF-FORCE INSTABILITY hand, extra charged particle loops with vanishing action, as → 0, are not present in the semiclassical approx- These higher harmonics lead to an instability. For suffi- imation and are not necessarily suppressed. As → 0 2 2 ˜2 3 there are nontrivial such loops when there is a cancel- ciently large n, the negative self-force term, − 3 π κT n , dominates. Hence, there are an infinite number of neg- lation between the rest mass and Coulomb interaction ative eigenvalues (see Fig. 3). Within the context of terms. These are virtual particles, and they have a size the semiclassical approximation we make, this appears to ∼ κ/(8π), or g2/(8πm) in physical units. When there is be a serious problem. On general grounds, one expects a separation of scales between the virtual particle loop any instanton (or sphaleron) to only have a single neg- size, O(κ), and the instanton size, O(1/T˜), i.e. when ative eigenvalue [50]. However, following Refs. [51–53] κT˜ = g2T/m 1, the virtual particle loops should sim- we argue that the semiclassical configuration should be a ply renormalise the parameters of the theory [7], in par- saddle point of the effective action rather than the bare ticular the charge. This is assumed in our analysis. How- action, with fluctuations above some finite energy scale ever, when κT˜ = O(1), the saddle-point approximation µ already integrated out. Then, only fluctuations up to plus renormalisation is not expected to adequately take µ should be included in the fluctuation prefactor. Higher virtual charged particle loops into account. The self-force energy fluctuations contribute instead to the renormal- instability may be symptomatic of this. isation of parameters. Although we do not explicitly We define nSF such that the self-force turns those har- 7

˜ monics negative with n ≥ nSF . For T = O(1) and λi Multiplicity κ = O(1), we find that nSF = O(1). Above κ ≈ 3.0653 0 5 √ and for all T˜, the instability is at nSF = 1. For κT˜ 1, 4π √ ˜ 2 κ√T instead we find that 2 4π − √ 1 κT˜ 3 κ3/2 n = 1 − + O κ3 , (23) 1 (2πn)2T˜ 10 SF 3/2 2 √ κT˜ 9π 1 2 2 4π (2πn) T˜ + √ 4 2 κT˜ √ 1 2 4 4π ˜ (2πn) T˜ − √ 2 as can be seen in Fig. 3, where for (κ, T ) = (0.1, 1.2) 2 κT˜ we can see that nSF = 30. For κT˜ 1 the self-force problem is moved to parametrically high harmonics, or TABLE I. Table of eigenvalues of the fluctuations about the short distances, where the effects of virtual charged parti- worldline sphaleron assuming κT˜2 1. The lower single line cle pairs are significant and hence the naive semiclassical separates constant fluctuations from harmonic ones. approximation is expected to break down. For the pur- poses of the semiclassical calculation, one should cut off ˜ the higher harmonics below O(1/(κT )). The self-force λ0i Multiplicity instability is thereby moved into the ultraviolet, where it 0 8 can be considered separately, and should contribute only (2πn)2T˜ 16 to the renormalisation of couplings. 2 2 For electric charges the coupling is weak, g := e 1. TABLE II. Table of eigenvalues of the square of the corre- In this case κ 1, and the self-force problem is not sponding free particle path integral. The lower single line present at leading order in , which is the approxima- separates constant fluctuations from harmonic ones. tion we make. For magnetic charges, on the other hand, gM 1 and hence κ . Magnetic and electric charges, gM and e, are related inversely via the Dirac quantisa- ˜ tion condition, gM e = 2πj, where j ∈ Z. This relation- Above TWS, the rate is dominated by the sphaleron ship is expected to hold for the running couplings [67–69]. and hence by the second term in the cluster expansion, Hence, as one probes shorter distances the effective mag- Γ(E,T ) ≈ Γ(2)(E,T ), netic charge decreases; magnetic charge is anti-shielded. Z ∞ Z ∞ This has been argued to be an effective spreading-out of (2) |ω−| 1 dsx dsy 2 Γ (E,T ) = − Im magnetic charge over scales O(gM /(4πm)) [58]. Classi- πV 2! 0 sx 0 sy cally, a charge distribution spread out on these scales is Z Z µ µ − 1 S˜[x,y;s ,s ;κ,T˜] stable [59, 60], hence, for magnetic monopoles, one would Dx Dy e x y . (26) expect that the ultraviolet physics does not suffer from the self-force instability. We wish to evaluate this in the saddle point approximation about the sphaleron. In this approximation, it is given by V. THE SCALAR PREFACTOR Z ∞ |ω−| 2 − 1 S˜(κ,T˜) − T˜ (s −s )2 Γ(E,T ) ≈ − T˜ e Im ds e 2 x 0 πV x In this section we consider the regime for which the −∞ ∞ Z T˜ 2 Z 1 P self-force instability is not present. Thus we assume that − (sy −s0) − λiχiχi ˜ dsye 2 Dχi e 2 i , κT 1 as discussed in the previous section. We will −∞ further assume that κT˜2 1, in which case the eigen- 2T˜|ω | 1 Z 1 P − − S˜(κ,T˜) − λiχiχi values, λi, of Eqs. (19), (21) and (22), simplify consider- ≈ − e Im Dχi e 2 i , ably. They are given in Table I. We assume the running V (27) coupling is known at the relevant energy scale. The temperature, T˜ , above which the worldline WS where χi and λi are defined in Eq. (18). sphaleron dominates the thermal Schwinger process sat- We divide the integrations up into the constant fluc- ˜ isfies TWS & 0.5, at least for κ ≤ 1. For small κ, it grows tuations, which correspond simply to translations of the and is given approximately by worldlines, and the harmonic fluctuations, which corre- √ spond to sines and cosines. The final result involves a 2 T˜WS ≈ . (24) product of the contribution from the constant fluctua- π3/4κ1/4 tions, C, and the harmonic fluctuations, H, 2 Given that T˜ > T˜ and κT˜ 1, we must have that Z 1 P WS − λiχiχi κ 1 and T˜ must lie in the window Dχi e 2 i =: CH. (28) √ 2 1 We consider C first. From Table I, we can see that there < T˜ √ . (25) π3/4κ1/4 κ are eight constant fluctuations: five of these are zero 8 modes, one is a negative mode and the remaining two where the λ0i are defined by this equation, and are given are positive. Again, the result involves a product of the in Table II. The final result is the usual one for (eight contribution from these three groups, powers of) the free particle path integral in 1D quantum mechanics [71] with im/T → −T˜ / where T refers to C =: C C C . (29) Z N P time elapsed and the boundary conditions on the path To perform the integrations over the zero modes, we first integrals are periodic. In the second line we have factored put the worldlines in a large box with spatial sides of off the contribution from the constant modes. The 0 on length L˜ (in units of m/gE), and then in the result drop the summation symbol and in the integration measure terms subdominant in L˜. Pairing up the spatial zero denotes that only the harmonic modes are included. modes with the nonzero constant modes, we make use of In the integrations over the harmonic modes we the following elementary integral, must keep in mind that the change of variables, (xµ(τ), yµ(τ)) → (ζµ(τ) = xµ(τ) − yµ(τ), ξµ(τ) = Z L/˜ 2 Z L/˜ 2 µ µ − λ (x−y)2 x (τ) + y (τ)), was carried out. The Jacobian of the dx dy e 2 = transformation is 1/2 for each pair of degrees of free- −L/˜ 2 −L/˜ 2 √ Z ∞ √ dom or 1/ 2 for each degree of freedom. This can − λ ζ2 L˜ dζ e 2 1 + O √ . (30) be seen easily in the two dimensional transformation ˜ −∞ λL (x, y) → (ζ = x − y, ξ = x + y), Using this, and doing the trivial integrals over the two zero modes in the 4 direction, one can find that ∂x ∂x 1 1 1 |J| = ∂ζ ∂ξ = 2 2 = . (37) ∂y ∂y − 1 1 2 L˜3 m33V ∂ζ ∂ξ 2 2 CZ = = , (31) T˜2 T˜2 Multiplication by this Jacobian factor is equivalent to where V is the spatial volume in standard dimensionful multiplying the eigenvalues by 2. units. Multiplying by Eq. (36) and dividing by Eq. (35), H Defining the integration over the negative mode re- then takes the form quires an analytic continuation. This is done following ˜4 0 −1/2 T Y 2λi the classic work of Langer [70], resulting in an overall H = , (38) factor of 1/2 on top of the naive result, (2π)4 λ i 0i √ −1/2 ! 0 1 1/2 2 4π 1 1/4p where again the on the product symbol denotes that CN = (2π) − √ = ±i √ (πκ) T˜ . 2 κT˜ 2 2 only the eigenvalues corresponding to harmonic modes (32) are included. The infinite products of the ratios of these The sign ambiguity arises in the process of analytic con- eigenvalues are well defined and can be evaluated using tinuation and we must choose the negative sign. The the following identity, integrations over the two positive modes are elementary, ∞ −1 Y c2 πc 1 − = . (39) n2 sin (πc) √ !−2/2 n=1 2/2 4π √ CP = (2π) √ = πκT˜ . (33) κT˜ The result is Thus we arrive at T˜ H = √ √ . (40) 3/4 3/4 3 9/2 8 2π13/4κ3/44 sin 2π1/4 sinh2 π1/4 −iπ κ m V κ1/4T˜ κ1/4T˜ C = √ p . (34) 2 2 T˜ Putting it all together we find To perform the infinite integrations over the harmonic √ 3 3/2 − 2 (1− κ ) fluctuations requires regularisation or, equivalently, nor- 1 m (T˜ ) e T˜ 4π Im log(Z) ≈ − , malisation. We normalise the path integral measure with ˜ ˜ V 16π3/2 sin πTWS sinh2 π√TWS respect to the square of the equivalent free particle path T˜ 2T˜ integral, √ (41) ˜ 3/4 1/4 2 where we have reintroduced TWS ≈ 2/(π κ ) to Z T˜ R 1 2 Z 1 P µ − x˙ dτ − λ0iχiχi Dx e 2 0 = Dχie 2 i , simplify√ the expression. Note that it is consistent to keep the ∼ κ term in the exponent, though such terms were 6 L˜ Z 1 P0 dropped in the eigenvalues, as the correction is multi- 0 − λ0iχiχi = D χie 2 i , (35) T˜2 plicative rather than additive, as well as coming with a negative power of . L˜6 T˜4 ˜ ˜ = , (36) Eq. (41) is divergent at T = TWS, where the sphaleron T˜2 (2π)4 develops another zero mode. Below this temperature the 9

the result is valid for temperatures satisfying

3 3 21/4 T > TWS ≈ 4gE /π m . (44)

The prefactor is plotted in Fig. 4. VI. THE SPINOR PREFACTOR

For QED with (Dirac) spinor charged particles, just as for SQED, one can formally represent the exact partition function in terms of an infinite sum of integrals over worldlines. Compared to SQED each worldline path integral contains an additional spin-dependent factor. This can be seen by starting from the following identities FIG. 4. The sphaleron rate prefactor. The sharp rise at T = TWS is a divergence of the form c/(T − TWS ), where c is a det(D/ + m) = det(−D/ + m), constant. It is due to the presence of an extra zero mode. 2 1/2 = det −D/ + m2 , sphaleron is unstable due to the existence of another sad- i 1/2 = det −D2 + m2 − gΣµν (F ext(x) + F (x)) , dle point with lower action, the W instanton of Ref. [12]. 2 µν µν This may signal a parametric enhancement of the depen- (45) dence on the semiclassical parameter, [72]. This can be investigated by expanding the action to higher than where the Σµν are proportional to the generators of second order in the harmonic fluctuation which becomes Lorentz transformations in the spin 1/2 representation, a zero mode at T = TWS. Then one may perform the i.e. Σµν = [γµ, γν ]/2, where γµ are the Euclidean gamma non-Gaussian integral over this mode to find the correc- matrices (see Ref. [73] for a definition). Using the final tion to the prefactor (see for instance [9]). Whether or line of Eq. (45), one can see that the spin-1/2 functional not there are phenomenological consequences of this en- trace can be written in terms of the spin-0 trace and an hancement depends on precise values in a given situation, additional spin factor [74–76] though the exponential dependence may well dominate 2 2 i µν ext over such power law enhancements in the prefactor. The −(D/+m)s −(−D +m − gΣ (F (x)+Fµν (x))s Tr(e ) = Tr(e 2 µν ) higher harmonic divergences, at T˜ = T˜WS/n, are not Z µ ext relevant as they exist at lower temperatures where the µ −S0[x ,Aµ +Aµ;s] µ ext = Dx e Spin[x ,Aµ + Aµ; s], sphaleron does not dominate the rate. The last ingredient required to construct the sphaleron (46) rate is |ω−|, the rate of growth with time of the unstable µ ext mode. This is where S0[x ,Aµ + Aµ; s] is what one would get for a spin zero particle and the spin factor is given by 3 1/4 4gE µ ext i g R s dτΣµν (F ext(x)+F (x)) |ω | = 2πT ≈ 2π . (42) Spin[x ,A +A ; s] := T r P e 2 0 µν µν − WS π3m2 µ µ γ (47) where T r signifies the trace over spinorial indices and The rate of pair production of scalar charged particles γ P is the path ordering operator. is thus given by After scaling τ → τ/s, s → s/(gE) and x → (m/gE)x, √ 3 and performing the Gaussian integrations over the dy- 3/2 − 2m + g E/π TWS (mT ) e T T namical gauge field, one can show that the spin factor Γ(E,T ) ≈ (4π)3/2 sin πTWS sinh2 πT√WS is subleading in versus the spinless part of the action T 2T (see Appendix A of Ref. [12]). The spin factor does how- gE g3E T g2T gT 2 ever modify the prefactor at leading order in . To this 1 + O , , , , , (43) m2 m2 m m E order we need only to evaluate the spin factor on the worldline sphaleron. This gives, for each worldline, x, a where we have restored the dimensionful variables and multiplicative factor of 10

1 1 1 Z Z 0 0 34 0 µρ x0 0 ˜ ν x0 0 ˜ ν − Trγx Px exp s0Σx +κs0 dτ dτ Σx ∂ρ Gµν (x0, y0; T )y ˙0 + ∂ρ Gµν (x0, x0; T )x ˙0 2 0 0 1 34 1 1 = − Trγ Px exp Σ − , 2 x x 2T˜ 2T˜ = −2, (48)

where Trγx denotes a trace over spinor indices of the x redundant and the equation becomes worldline, Px is the path ordering operator along the x √ 3 worldline, Σµν := [γµ, γν ]/2, where γµ are the Euclidean 3/2 − 2me + e E/π 4TWS (meT ) e T T Dirac (gamma) matrices satisfying {γµ, γν } = 2δµν and Γ(E,T ) ≈ 0 0 (4π)3/2 sin πTWS sinh2 πT√WS denotes dependence on τ as opposed to τ. There is T 2T a second factor, for the other worldline, y, which differs 2 2 eE T eT by interchange of x and y. The two factors are equal to 1 + O e , 2 , , , (50) each other. Hence the worldline sphaleron rate for spinor me me E charged particles is where me is the mass of the electron and TWS ≈ 1/4 4eE3/π3m2 . Note that there are now only four di- Γ (E,T ) ≈ 4Γ (E,T ). (49) e spinor scalar mensionless combinations of parameters in the big O, rather than five. We describe as the region of validity of The 4 comes from a product of traces over gamma matri- this expression, where all of these dimensionless param- ces and hence can be seen as a factor of (2s + 1)2, where eters are small, and where T > T . s is the spin, to be compared with the single power of WS It is instructive to compare Eq. (50) to known results (2s + 1) which arises in Schwinger pair production at for E = 0 and for T = 0. In the former case, in a thermal zero temperature. Thus spin enters the rate of pair pro- bath of photons with zero electric field, to leading order, duction essentially trivially. The factor of (2s + 1)2 sim- electron-positron pair production proceeds via the colli- ply counts the possible spins of the produced particles: sion of pairs of photons, γγ → e+e−, the Breit-Wheeler up-up, up-down, down-up and down-down. This result process [31]. Above the kinematic threshold this process can be seen as a natural extension of the factor (2s + 1) has cross section approximately equal to the classical size which arises in Schwinger pair production. In that case, of the electron, σ ∼ e4/(16π2m2). Averaging this over in vacuum, the produced pairs must have opposite spins BW e the distribution of energies present in a thermal bath of so only up-down and down-up are possible. However, in photons one finds [77], our case, angular momentum can be exchanged with the thermal bath of photons. 4 3 e meT − 2me 2 T Γ (T ) ≈ e T 1 + O e , . (51) Note that the effects of spin and statistics should give BW 2(2π)4 m non-trivial corrections in stronger fields, at higher temperatures, or simply at later times, where the back reac- Adding a constant electric field does not change the rate tion of the produced charged particles cannot be ignored. of this process, as the additional photons have infinite However, in the physical situation we consider neither wavelength and hence zero energy. Pauli blocking nor Bose enhancement are relevant be- The exponential dependences of Eqs. (49) and (51) cause our initial state contains no charged particles and, are similar, though the prefactors differ markedly. In the in the presence of the weak external field, particles are presence of an electric field and a thermal bath of photons only produced exponentially slowly. with T > TWS, both processes are possible. In this case Combining this result with that of the previous section, the ratio of the worldline sphaleron rate, Γ(E,T ), and we arrive at the chief result of this paper, Eq. (2). the Breit-Wheeler rate is

−15/8 2 3/8 √ Γ(E,T ) π 4π m T e3E/π e T ≈ 2 F e , VII. ELECTRONS AND POSITRONS ΓBW(T ) 4 e eE TWS 3/8 √ 2 3 me e E/π 1754 e T 1, (52) So, what do our calculations imply for the pair pro- & E duction of the lightest electric particles in nature, the electron and the positron? In this case, the theory is where F (t) is deﬁned by this equation and the sec- weakly coupled, g2 := e2 1, where e is the charge of a ond line follows by evaluating F (t) at its minimum, at positron, and hence we may take the weak coupling limit t ≈ 1.37. Hence, in the domain of validity of both of Eq. (2), in which some of the approximations become results, the worldline sphaleron rate is faster than the 11

(purely thermal) Breit-Wheeler rate for all nonzero elec- these searches have led to upper bounds on cross sec- tric fields. In practice, the most easily physically real- tions for the pair production of magnetic monopoles. Un- isable regime, is probably the regime of weakest fields, fortunately, given the huge theoretical uncertainties on 2 4 1/3 E π(m T /4e) (equivalent to T TWS), where these cross sections, such experimental bounds cannot be the ratio of rates reduces to used to derive bounds on the properties of any possible monopoles, such as their mass, mM . 3/2 √ Γ(E,T ) 16π(mT ) e3E/π This problem does not extend to heavy ion collisions. ≈ e T 1. (53) 9/2 3/2 ΓBW(T ) e E In this case there are strong magnetic fields [81, 82] and high temperatures [83, 84] and thus magnetic monopoles In the regime of validity this amounts to a factor O(106). may be produced by the dual of the thermal Schwinger 2 The surprising result is that for weak fields, E me, the process. From a consideration of this, one can explicitly worldline sphaleron rate is parametrically faster than the calculate the cross section for magnetic monopole pair Breit-Wheeler rate, despite the fact one would expect the production. This has led to the first (lower) bounds on former to reduce to the latter as E → 0. However, one the mass of possible magnetic monopoles from any col- cannot take the E → 0 limit at fixed T while staying lider search [33]. These are also the strongest reliable in the region of validity of our calculation, due to the bounds on mM , though they are surprisingly weak, be- requirement E eT 2 (see Eq. (50)). ing only O(GeV). One can expect to see significant im- We can also compare the worldline sphaleron rate with provements in the near future, as the heavy-ion collisions that of the Schwinger rate at zero temperature [6], which gave rise to that bound [85] had a centre of mass energy per nucleon of only 8.7GeV, much lower than, for 2 πm2 (eE) − e 2 eE example, the last lead-lead run at the LHC in 2015, which Γ (E) ≈ e eE 1 + O e , . (54) Schwinger 4π3 m2 had a centre of mass energy per nucleon of 5020GeV. As briefly argued in the introduction, the rate of ther- In this case, in the regime T > TWS, and for weak fields, mal Schwinger pair production that we have calculated 2 E me, using the same reasoning as for the comparison in QED at arbitrary coupling, is directly applicable to with the Breit-Wheeler rate, we find, magnetic monopole production. This is due to classical electromagnetic duality, a symmetry of Maxwell’s equa- 1/8 √ 2 πm2 3 Γ(E,T ) m e − 2me + e E/π tions extended to include magnetic charges, & 5.81 e eE T T 1. ΓSchwinger(E) E µν µν µν µν (55) F → ∗F , ∗ F → −F , µ µ µ µ In the region of validity, the worldline sphaleron rate is jE → jM , jM → −jE, (56) exponentially faster than the zero temperature Schwinger µ rate. Numerically, one finds the enhancement to be in the where ∗ denotes the Hodge dual, jE denotes the elec- 3 region of 1010 . tric current and jM the magnetic current. In our result, Eq. (2), this amounts simply to a relabelling of electric quantities as magnetic. Under this duality our result VIII. MAGNETIC MONOPOLES is directly applicable to elementary (Dirac) monopoles. For ’t Hooft-Polyakov monopoles, which are composite, solitonic objects, our worldline calculation is also appli- As mentioned in the introduction, our initial inter- cable because their size is O(κ/4π) in our scaled units est in this calculation was for its relevance to magnetic and hence is parametrically smaller than the worldline monopole pair production, and hence to searches for sphaleron. Thus an effective description of ’t Hooft- magnetic monopoles. Due to the necessary strong cou- 2 2 2 Polyakov monopoles as magnetically-charged worldlines pling of magnetic monopoles, gM ≥ gD := (2π/e) 1, is applicable [12, 49, 86, 87]. perturbative techniques for computing their production Considering that magnetic monopoles are necessarily cross sections, σab→MM¯ , fail. As a consequence, these strongly coupled, we may take the strong coupling limit cross sections are largely unknown and, in fact, nonrig- of Eq. (2), in which some other of the approximations orous arguments for the order of magnitude of the cross become redundant and the equation becomes section differ by hundreds of orders of magnitude. This √ g3 B/π huge discrepancy is due to the presence or absence of an 2m M 2 2 3/2 − M + exponential suppression of the form exp(−16π/e ). For (2s + 1) TWS (mM T ) e T T Γ(B,T ) ≈ monopole production in collisions of “small” particles, (4π)3/2 sin πTWS sinh2 πT√WS such as pp → MM¯ or e+e− → MM¯ , this exponential T 2T suppression has been argued to be present for ’t Hooft- 1 g3 B g2 T g T 2 1 + O , M , M , M , Polyakov monopoles [78, 79] and may also be argued to 2 2 gM mM mM B be present for elementary (Dirac) monopoles [58]. (57) Experimental collider searches for magnetic monopoles have largely focused on collisions of “small” particles, where s is the spin of the magnetic monopole. The ex- such as protons or electrons [80]. The null results of pression applies for temperatures satisfying T > TWS ≈ 12

3 3 2 1/4 4gM B /π mM . Note again that there are now only of e smaller than the critical temperature, Tc ∼ emM , four dimensionless combinations of parameters in the big at which the symmetry of the vacuum is restored. Thus, O, rather than five. if one were to perform a semiclassical calculation of the What do our results mean for magnetic monopole thermal Schwinger rate directly in the field theory, one searches? If the magnetic field and temperature in a should be able to calculate the rate up to T ∼ Tc ∼ emM , given heavy ion collision vary slowly on the time and so gaining a factor of 1/e on the worldline calculation. In length scales of the sphaleron, one may make the ap- this case one should be able to calculate the rate beyond proximation that the magnetic field and temperature are the semiclassical approximation following the approach of locally constant. That is, locally, the rate of pair produc- Refs. [88–90]. A classical, stochastic equation will gov- tion is approximated by the rate derived for a constant ern the real-time dynamics of the sphaleron, and hence field and temperature. In this case the cross section for determine the prefactor [91, 92]. pair production of magnetic monopoles in heavy-ion col- Above Tc in a grand unified theory one can no longer lisions, σMM¯ , is given by define a magnetic field and magnetic monopoles can no longer be thought of as particles. However, if such a sys- √ inel √ dσ ¯ ( s, b) dσ ( s, b) MM ≈ HI tem were to cool, magnetic monopoles would be produced db db via the freezing out of thermal fluctuations [93]. Z √ √ d4x Γ(B¯(x; s, b), T¯(x; s, b)),

(58) IX. CONCLUSIONS √ where s is the centre of mass energy, b is the im- inel The worldline sphaleron describes thermal Schwinger pact parameter, σHI is the total inelastic cross section ¯ √ pair production at sufficiently high temperature (T > for the√ specific heavy-ion collision and B(x; s, b) and T¯(x; s, b) are the event-averaged, local magnetic field TWS) in Abelian gauge theories, regardless of the strength of the coupling. The rate of this process, Eq. and temperature at a point√x in the fireball for events with centre of mass energy s and impact parameter b. (2), is the chief result of this paper. The integrals in Eq. (58) can all be done in the saddle In order to work to all orders in g, we were forced to point approximation. restrict our calculation to sufficiently heavy charged par- Due to the strong coupling of magnetic monopoles, ticles. In this regime the rate is exponentially suppressed, Eq. (57) is unfortunately valid only for parametrically by the Boltzmann factor, 2m∗/T ≈ 2m/T . For weakly 2 2 coupled electric particles, we must have m/T 1, which low temperatures, T/mM 1/gM ∼ e , and weak mag- 2 3 3 is not too restrictive and leads to the exciting possi- netic fields, B/mM 1/gM ∼ e . This is due to the requirement that the self-force instability be moved to bility of an experimentally observable rate of pair pro- much shorter length scales than those of the sphaleron, duction. Further, in its region of validity, the world- a hierarchy which breaks down when the rate, and con- line sphaleron rate turns out to be much faster than sequently the cross section, are still extremely exponen- both perturbative photon fusion and the zero temper- tially suppressed, ature Schwinger rate. However, for magnetic monopoles, the validity of our approximations require instead that σ ¯ 2 MM −c/e m/T g2 ≥ g2 ≈ 430, and hence the results derived in inel ∼ e , (59) D σHI this paper are unfortunately too suppressed to be directly for some c > 0, with c = O(1). This same factor has been applicable to monopole searches in heavy-ion collisions. 3 2 argued to suppress magnetic monopole pair production For larger values of T/m or g B/m , the exponential in high energy particle collisions [78, 79]. However, in our suppression of the rate, Eq. (4), is greatly reduced, lead- case the arguments of Refs. [78, 79] do not apply and we ing to predictions of a measurable rate of pair production. 3 2 have no reason to expect the same exponential suppres- It was a regime where g B/m = O(1) that led to the sion at higher temperatures. Rather Eq. (59) results due mass bounds in Ref. [33]. However, our calculation of the to the limitations of our approximation scheme as ap- prefactor breaks down in this case, due to the presence of plied to strongly coupled magnetic monopoles. Further the self-force instability. One might hope that the calcu- theoretical work is needed to go beyond this regime. lated exponential suppression still gives a good approx- 2 imation to the rate at O(1) values of g3B/m2, though At higher temperatures than O(e mM ), the inverse temperature becomes smaller than the classical radius of the corrections to this cannot be calculated within our 2 approach. a magnetic monopole, rc ∼ gM /(4πmM ), and in considering fluctuations of the sphaleron, the structure of the magnetic monopoles cannot be ignored. However, as the sphaleron is independent of the Euclidean time direction, ACKNOWLEDGEMENTS one might still expect the worldline sphaleron to capture the exponential dependence of the rate. OG would like to thank Sergey Sibiryakov and Toby For ’t Hooft-Polyakov monopoles this can be checked Wiseman for useful discussions. AR is supported by 2 explicitly. In this case, the temperature e mM is a factor STFC grant ST/P000762/1 and OG was supported first 13 by an STFC studentship and then by the Research Funds of the University of Helsinki.

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