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Sphaleron transitions in a realistic heat bath

arXiv:hep-ph/9308307v2 21 Aug 1993Sphaleron transitions in a realistic heat bathA. Krasnitz∗IPS, ETH-Zentrum, CH8092 Zurich, SwitzerlandandR.

PottingUniversidade do AlgarveUnidade de Ciˆencias Exactas e Humanas,Campus de Gambelas, 8000 Faro, PortugalAugust 1993AbstractWe measure the diffusion rate of Chern-Simons number in the (1+1)-dimensionalAbelian Higgs model interacting with a realistic heat bath for temperatures be-tween 1/13 and 1/3 times the sphaleron energy. It is found that the measuredrate is close to that predicted by one-loop calculation at the lower end of thetemperature range considered but falls at least an order of magnitude short ofone-loop estimate at the upper end of that range.

We show numerically that thesphaleron approximation breaks down as soon as the gauge-invariant two-pointfunction yields correlation length close to the sphaleron size.IPS Research Report No. 93-10∗Supported by the Swiss National Science Foundation.1

Anomalous electroweak baryon-number violation may have played an important rolein setting the baryon number of the Universe to its present value [1, 2]. At temperaturesabove the gauge-boson mass scale electroweak baryon-number nonconservation is domi-nated by hopping over the finite-energy barriers separating topologically distinct vacuaof the bosonic sector.

Determination of the corresponding transition rate is a challengingnonperturbative problem, even in the range of validity of the classical approximation. Atthe lower temperature end of that range the barrier crossings are likely to occur in thevicinity of the saddle point (known as a sphaleron) of the energy functional.

The ratecan then be estimated using a field-theoretic extension of transition-state theory (TST)[17, 18]. At higher temperatures this analytic tool is no longer available, and direct mea-surement of the rate in real-time numerical simulations of a lattice gauge-Higgs systemis the only remaining possibility.Analytical saddle-point estimates of the rate in the Standard Model are complicatedby the fact that the corresponding sphaleron field configuration is not known exactly.At the same time, numerical real-time simulations of that system in its low-temperatureregime carry an enormous computational cost and are yet to be performed [1, 3].

In thissituation lower-dimensional models become a very useful test ground on which activation-theory predictions can be confronted by numerical experiments [4, 5, 6, 7, 8, 9, 10, 11, 12,13, 14, 15]. For this reason the (1+1)-dimensional Abelian Higgs model (AHM) studiednumerically in this work has attracted much attention recently [9, 4, 6, 11, 12].Determination of the transition rate should include its proper averaging over thecanonical ensemble in the phase space of a system in question.

One way to achieve thatwould be to generate the canonical ensemble of initial configurations, subject each ofthese configurations to the Hamitonian evolution, and average the transition rate overthe initial states of the system. Such procedure, while being perfectly valid, is very costlycomputationally.

To date, a single or a small number of initial configurations have beenused in Hamiltonian simulations of AHM [9, 12]. In addition, preparing initial configura-tions in case of a gauge theory presents a technical difficulty: if a standard importance-sampling method is used, resulting configurations will in general violate Gauss’ law.

Aspecial cooling procedure is required to eliminate static charge [9]. It is not clear thatsuch cooling does not cause the sample to deviate from the intended canonical ensemble.Another way to obtain the canonical ensemble average of the rate is to replace Hamil-tonian evolution by evolution in a heat bath.

The simplest form of the latter is imple-mented using phenomenological Langevin equations of motion. While Langevin approachguarantees thermalization of the system, it does so at the expense of introducing an arti-ficial viscosity parameter, thereby altering bulk dynamical properties of a field-theoreticsystem.

Numerical studies performed on different models show that transition rates in-deed strongly depend on viscosity [7, 8]. Recent analytical work [16] has also shown theimpact of heat-bath properties on quantities of transition-rate type.

In case of AHM,there also is a technical difficulty with the conventional Langevin approach: in orderto maintain gauge invariance, one is forced to use polar coordinates for the Higgs field;whenever the latter vanishes, the equations of motion are singular [4, 12].Recently we have proposed and tested a new method in which a field-theoretic system2

interacts with a heat bath at its boundaries [5]. The heat bath is constructed so as toimitate an infinite extension of the system beyond the boundaries.

Technically this meansthat the fields in the bulk of the system evolve according to the Hamiltonian equations ofmotion, while boundary fields are subject to Langevin evolution with a non-Markovianfriction kernel and colored noise.Our construction approximates a natural situationin which open systems are immersed in a similar environment. In this way, dynamicalevolution and canonical ensemble averaging occur at the same time while bulk dynamicalproperties of the system are intact.

Moreover, the new procedure does not suffer fromthe technical difficulties of the two old ones. In this work we apply the realistic heatbath (RHB) method to the study of sphaleron transitions in AHM.Earlier real-time simulations of AHM [9, 4] found that at low temperatures (about0.1 of the sphaleron energy) the temperature dependence of the rate qualitatively agreeswith that predicted by the 1-loop (TST) calculation of Ref.

[6]. The temperature rangeof our simulation is wider and includes somewhat higher temperatures, up to about 1/3the sphaleron energy, for which the TST result may no longer be reliable.

This allowsus to estimate the temperature at which TST loses validity. As will be demonstratedin the following, this is the temperature at which the scalar field correlation length fallsbelow the linear size of the sphaleron.

It is also interesting to determine the sign andmagnitude of the rate deviation from the TST prediction. If the rate we measure fallsconsiderably short of the latter, it might be indicative of the entropic rate suppressionwhich reflects the difficulty of creating a coherent configuration in high-temperatureplasma.

Our results, presented in the following, do indeed show dramatic slowdown ofthe rate growth.Our starting point is the (1+1)-dimensional lattice AHM Lagrangian which in suit-ably chosen units reads [4]L=a2Xj1ξ( ˙A1j,j+1 −A0j+1 −A0ja)2 + |(∂0 −iA0j)φj|2−a−2|φj+1 −expiaA1j,j+1φi|2 −12|φj|2 −12. (1)Here j labels sites of a chain whose lattice spacing is a.

The temporal component of avector potential, A0j, and a complex scalar field φj reside on sites of the chain, whereasthe spatial component of the vector potential, A1j,j+1, resides on links. Imposing theA0 = 0 condition one obtains a Hamiltonian (we shall drop the Lorentz index of A1 fromnow on)H = a2Xj ξEj,j+1a!2+ |πja |2 + |φj+1 −exp (iaAj,j+1) φj|2 + 12|φj|2 −12,(2)where πj and Ej,j+1 are canonically conjugate momenta of φj and Aj,j+1, respectively.The Hamiltonian equations of motion˙Aj,j+1=ξaEj,j+1,3

˙Ej,j+1=iφjexp(iaAj,j+1)φ∗j+1 −φ∗j+ h.c.,˙φj=1aπ∗j,˙πj=1aexp(iaAj,j+1)φ∗j+1 + exp(−iaAj−1,j)φ∗j−1 −2φ∗j−aφ∗j|φj|2 −1(3)are supplemented by the Gauss’ law constraint1a (Ej,j+1 −Ej−1,j) = Imπjφ∗j. (4)The same dynamics can be described in terms of gauge-invariant variables.

To this end,the Higgs field is rewritten in polar coordinates: φj = ρj exp(iαj). Defining bj,j+1 =αj+1 −αj −aAj,j+1, ǫj,j+1 = 1aEj,j+1 and introducing canonical momentum πρj for ρj wesee that (3) together with (4) is equivalent to˙ǫj,j+1=1aρjρj+1 sin bj,j+1,˙bj,j+1=1a ǫj+1,j+2 −ǫj,j+1ρ2j+1−ǫj,j+1 −ǫj−1,jρ2j!−aξǫj,j+1,˙ρj=1aπρj ,˙πρj=(ǫj,j+1 −ǫj−1,j)aρ3j+ 1a (ρj+1 cos bj,j+1 + ρj−1 cos bj−1,j −2ρj)−aρjρ2j −1.

(5)The equations of motion in this form involve only two pairs of real canonical variables,namely, ρj, πρj and ǫj,j+1, bj,j+1. It is easy to see that (5) follow from the HamiltonianH′=a2Xjξǫ2j,j+1 + ǫj,j+1 −ǫj−1,jaρj!2+ πρja!2+ 2a2ρ2j −ρjρj+1 cos bj,j+1+a4Xjρ2j −12(6)obtained from polar-coordinate form of (2) by substituting (4).In the following we shall use both presented forms of the equations of motion.

Onone hand, the Cartesian form (3) allows better numerical handling of sphaleronlike fieldconfigurations in which the Higgs field is close to zero at one or more sites. For thisreason we use it for real-time evolution in the bulk of an open gauge-Higgs system.

Onthe other hand, the gauge-invariant form (5) involves less degrees of freedom and lendsitself easier to linearization. We therefore use it for the heat bath construction.As a heat bath we take AHM linearized in the vicinity of one of its gauge-equivalentvacua, which, as is well known, is a system of two free fields: the radial Higgs field ̺4

and the gauge field ε whose masses are√2 and √ξ, respectively. Those are coupled atthe boundary site of the AHM ((2) or (6)) each to its interacting counterpart, i.e.

̺ toρ, and ε to ǫ. Suppose for definiteness that the left boundary of the interacting systemseparating it from the linear heat bath is at j = 0 site of the chain.

The field equationsat the boundary are then modified compared to (5), namely, the second and the fourthequation of (5) are replaced by˙b−1,0=1a (ǫ0,1 −2 ∗ǫ−1,0) −aξǫ−1,0 + D(qξ, [ǫ−1,0]) + F(qξ, t),˙πρ0=1a (ρ1 cos b0,1 −2ρ0 + 1) −2a (ρ0 −1) + D(√2, [ρ0 −1]) + F(√2, t). (7)In going from (5) to (7) we linearized in the vicinity of ρ0 = 1, b−1,0 = 0 and, followingRef.

[5], introduced two terms describing interaction with the linear heat bath at theboundary. The D(m, [σ]) term represents the reaction to the motion of a boundary fieldσ from the heat bath.

The mass of the corresponding heat-bath field is m. Explicitly,D(m, [σ]) =Z t−∞σ(t′)χm(t −t′)dt′,(8)where the Fourier image of the causal response function χm(t) is˜χm(ω) = 2isign(ω)q(ω2 −m2)(1 + a2(m2 −ω2)/4)(9)for frequencies m < |ω|

In order forthe system to reach thermal equilibrium with the heat bath, the F(m, t) term describingthermal fluctuations of a heat-bath field at the boundary should be included. Accordingto fluctuation-disspation theorem, F(m, t) is a Gaussian random variable whose timeautocorrelation is related to χm(t):⟨F(m, t)F(m, t + τ)⟩= θZ τ0 χm(t′)dt′,(10)where θ is the temperature.

We solve numerically the system of equations (3) togetherwith (7) and its right-boundary analog. Numerical implementation of the boundary heatbath is explained in detail in Ref.

[5]. As has already been mentioned, the Cartesianform of equations of motion in the bulk is used due to its superior numerical properties.On the other hand, we use polar coordinates to evolve boundary fields.

In order to beable to transform from polar to Cartesian coordinates at the boundary, we need to keeptrack of the angular variable α of the boundary Higgs field. This is done with the helpof Gauss’ law which for the linearized system gives˙αj = 1a (Ej,j+1 −Ej−1,j) .

(11)5

We use second-order Runge-Kutta algorithm for numerical solution of (3,7,11). A num-ber of criteria were applied in evaluating the algorithm performance and in determiningthe value of the time step.

In particular, we tested the absorption properties of the simu-lated heat bath by dropping the noise term from the equations of motion and cooling aninitially hot system. The resulting cooling curve was then compared to that of the samesystem in a large real zero-temperature heat bath.

In a similar way we studied thermal-ization of an initially cold system in the heat bath. The temperature was measured byaveraging the kinetic energy of the radial Higgs field over the system and over the timehistory.

In all our simulations the temperature of a thermalized system was found to bewithin 3% of the assigned value.One special numerical issue to be dealt with in a real-time simulation of a gaugetheory is the accuracy of the Gauss’ constraint. The equations we solve are consistentwith the Gauss’ law.

However, numerical errors give rise to a small spurious static chargedensity which should be kept in check in order to have no impact on the quantities wemeasure, in particular, the sphaleron transition rate. Since the rate is exponentiallysensitive to the sphaleron energy, we computed the perturbative correction to the latterin presence of a small static charge distribution q(x), ∆Esph([q]).

The correspondingexpression is derived in the Appendix.We then used β∆Esph([q]) averaged over thesphaleron positions as a criterion of Gauss’ law violation (here and in the following βdenotes inverse temperature). In all our simulations the amount of the spurious chargewas far too small to have any measurable impact on the sphaleron transition rate.Our attention in this work is focused on the temperature dependence of the rate.For this reason we performed all our simulations at a fixed value of the gauge couplingξ = 10, with the exception of preliminary study at ξ = 0.5.

Most of our measurementswere done for a chain of length L = 100 and lattice spacing a = 0.5. Since our results areto be compared to the analytical prediction for AHM in the continuum, we checked theirdependence on both lattice cutoffs by performing additional simulations at a = 0.25 andat L = 50.

The Runge-Kutta time step was 0.01 for a = 0.5 and 0.004 for a = 0.25.For each set of a, L, and the temperature the simulation time was 105 time units. Toadd confidence to our measurements we also performed microcanonical simulations at anumber of parameter values.

The agreement with the canonical results was good exceptfor the β = 5, L = 100, a = 0.5 case for which the canonical simulation gives somewhathigher value of the rate.Following Ref. [4], we extracted the sphaleron transition rate from ∆CS(t), the time-averaged squared deviation of the Chern-Simons variable NCS ≡(2π)−1 R A(x)dx for alag t. For lags shorter than the average time between consecutive sphaleron transitions∆CS(t) is determined by fluctuations of NCS in the vicinity of one of its vacuum values.

Atlags much longer than the lifetime of a vacuum many uncorrelated sphaleron transitionswould have occurred, each changing the value of NCS by an integer, and a random-walkbehavior sets in:∆CS(t) = ΓLt,(12)where Γ is the sphaleron transition rate per unit length. Figure 1 illustrates the describedlag dependence of ∆CS(t).6

Figure 1: Lag dependence of the average squared deviation of the Chern-Simons variable.Once ∆CS(t) is known, it can be (at large enough values of t) fitted to a straightline through the origin to yield Γ. Since the values of ∆CS(t) at different t are stronglycorrelated, we found that, while the quality of fit remains high as more and more ∆CS(t)data points are included, the error on Γ is not reduced significantly by fitting with moredegrees of freedom.

We therefore simplified the procedure and extracted Γ from a singlevalue of ∆CS(t) at t = 1000. This choice of a lag is suitable since, on one hand, it ismuch shorter than our total simulation time (105), while on the other hand it is at leastseveral times longer than the average time between consecutive sphaleron transitions inthe temperature range considered.

We also verified that Γ remains constant in a widerange of lags including the chosen one.Summary of all our rate measurements is presented in Table 1, while for Figure 2 weselected the results that best reflect the important features of Γ dependence on the inversetemperature β, as well as on a. Obviously, no measurable dependence on L is observed.The absence of finite-size effects is to be expected of our heat-bath construction.

Namely,at low temperatures the linearized heat bath closely imitates the infinite extension of thenonlinear system beyond the boundaries.At high temperatures the system becomesless and less correlated in space and time, and, as a result, the boundary effects loseimportance. We also observe no dependence of the rate on a, except for the highesttemperature (β = 3) considered.

The virtual independence of Γ of the lattice spacing atlow temperatures has been found in earlier work [9, 4] and is confirmed by our results.As will be shown shortly, the difference in the rates between the a = 0.5 and a = 0.257

Figure 2: Temperature dependence of the transition rate Γ. The solid curves correspondto the TST prediction (13), with the value of ξ indicated near each curve.cases at β = 3 is consistent with other properties of the model at this temperature.

Ourrate measurements are to be compared to the TST prediction [6]Γ = ω−2π 6βEsph2π! 12 Γ(α + s + 1)Γ(α −s)Γ(α + 1)Γ(α)!

12exp (−βEsph) ,(13)where the sphaleron energy Esph = 2√2/3, α2 = s(s + 1) = 2ξ, and ω2−= s + 1 is thenegative squared eigenfrequency corresponding to the sphaleron instability.As Figure 2 clearly shows, the values of Γ at low temperature are close to those givenby (13). While the approach of measured Γ to the TST prediction is slow, the two nearlycoincide at the lowest temperature considered, β = 14.

This nice agreement shows onceagain that our heat bath construction works as intended. But the most notable feature ofour results is their dramatic departure from the TST-predicted values starting at aboutβ = 5.

For ξ = 10 the discrepancy is a factor of 5 already at β = 5, and grows at β = 3to a factor of 10 for a = 0.5 and a factor of 20 for a = 0.25. The situation is similar forξ = 0.5.

Moreover, in the latter two cases the rate practically does not grow betweenβ = 5 and β = 3. Note that the deviations are much larger than our measurement errorbars and are therefore statistically significant.It is natural to ask why this strong lagging of the measured rate behind the TST onebegins in the vicinity of β = 5.

To this end recall the underlying assumption of (13):8

ξaLβΓξaLβΓRHB100.51002.890.09 ± 0.01100.5508.87(37 ± 4) × 10−4100.51003.910.068 ± 0.007100.25502.990.046 ± 0.004100.51004.950.034 ± 0.003100.25504.990.037 ± 0.004100.51006.890.015 ± 0.002100.25507.020.016 ± 0.002100.51008.90(37 ± 4) × 10−4100.25508.98(33 ± 5) × 10−4100.510011.85(32 ± 3) × 10−50.50.51002.97(78 ± 8) × 10−4100.510013.86(60 ± 7) × 10−60.50.51004.97(67 ± 7) × 10−4100.5504.960.033 ± 0.0050.50.51007.00(25 ± 3) × 10−4100.5506.880.016 ± 0.0020.50.51008.95(54 ± 7) × 10−5Microcanonical100.51004.940.055 ± 0.007100.25503.000.040 ± 0.004100.51008.95(40 ± 5) × 10−4Table 1: Summary of transition rate measurements. The inverse temperature β is givenas deduced from the average kinetic energy of the radial Higgs field.Chern-Simons number diffusion is dominated by evolution of configurations resemblingthe vacuum into those resembling the zero-temperature sphaleron.

It is clear that theNCS diffusion can only be described in these terms as long as the Higgs field is correlatedon a length scale larger than the sphaleron size (2√2). The corresponding correlationlength λ can be found by measuring a gauge-invariant two-point function [19]Cjl = φ∗jφl exp−ial−1Xk=jAk,k+1= ρjρl exp−il−1Xk=jbk,k+1.

(14)A rough estimate of λ at low temperatures is obtained by averaging Cjl over the thermalensemble with H′ of (6) replaced by its linearized version. Performing Gaussian integra-tion over the b variables one finds λ = 2β.

This is, in fact, an overestimate of λ, sincethermal fluctuations of the radial Higgs field are not taken into account. Figure 3 showsthe values of λ obtained by fitting ⟨Cjl⟩to const × exp(−|j −l|/λ).

As expected, thebreakdown of the saddle-point approximation for the rate occurs as λ becomes smallerthan the sphaleron size. Note that this is true for both values of ξ considered.

At β = 3λ = 1.47 for ξ = 10, only about 3 times larger than the lattice spacing a = 0.5. Hence thefield strongly fluctuates at length scales comparable to the lattice spacing.

It is thereforenot surprising that we find the a dependence of the rate at this temperature.At this point it is unclear what causes the sharp slowdown of the rate growth atthe high-temperature end of our measurement range. We cannot exclude a possibilitythat at temperatures in question crossing the NCS = half −integer separatrix in theconfiguration space of the model [1] in close vicinity of the sphaleron saddle point isstill strongly preferred energetically, but is already suppressed entropically.

It is usuallyassumed [1, 3, 13, 15] that at temperatures above the sphaleron energy the rate growslike a power of the temperature. It could be that what we observe at β ≤5 is a crossover9

Figure 3: Temperature dependence of the correlation length deduced from the gauge-invariant two-point function (14). The sphaleron size is shown by the dashed line forcomparison.from the exponential to power-law behavior of the rate.

It is not clear, however, thatat such a crossover the rate should stop growing as it does for ξ = 10, a = 0.25 and forξ = 0.5, a = 0.5. The only way to resolve this puzzling situation is by rate measurementsat still higher temperatures, as well as smaller lattice spacings.

We plan to do so in thefuture.To summarize, we performed an accurate measurement of the sphaleron transitionrate in AHM averaged over the canonical ensemble. The latter was obtained by im-mersing the system in a realistic heat bath.

The ergodicity of real-time evolution wasthus achieved without having to introduce an artificial viscosity parameter. Our ratemeasurements approach the corresponding TST estimate at low temperature.

This is inagreement with Ref. [9] where similar measurements were performed microcanonically.The highest temperature studied in that work was 0.103Esph, well within the range ofapplicability of TST.

In going beyond that range, we found dramatic slowdown of therate growth, with suppression factor as large as 20 relative to TST at the highest tem-perature considered. Our measurements show that the breakdown of TST occurs as soonas the correlation length deduced from Cjl (14)) becomes comparable to the sphaleronsize.

This suggests that a similar object might serve as a criterion for applicability of thesphaleron approximation in other theories, including the realistic 3+1-dimensional case.10

We gratefully acknowledge enlightening discussions with A. I. Bochkarev, Ph. de For-crand, E. G. Klepfish, A. Kovner, and especially A. Wipf.

Numerical simulations for thiswork were performed on the Cray YMP/464 supercomputer at ETH.AppendixIn this Appendix we give a perturbative estimate of the shift in the sphaleron energyin presence of a small static charge density q. For simplicity we use continuum, ratherthan lattice, formulation of AHM.

We shall also assume that the system has an infinitelength. Analogous to (2), the Hamiltonian depends on three pairs of canonical vari-ables A, E, ρ, πrho, α, πα.

Static configurations are obtained by minimizing the energywith respect to all the variables on a subspace constrained by Gauss’ law πα = E′ −q(prime means derivative with respect to the spatial variable x). Vacuum configurationscorrespond to the absolute minimum of the energy on that subspace, while sphaleronsminimize the energy among configurations with ρ = 0 at one point.

Two of the vari-ables, A and α, enter the Hamiltonian only in combination b = α′−A, thereby effectivelyreducing the number of variables by one. It is convenient to define a new complex fieldΦ(x) = ρ(x) expiZ x−∞b(x′)dx′.

(15)After eliminating πα with the help of Gauss’ law one obtainsH = 12Zdx"ξE2 + π2ρ + (E′ −q)2|Φ|2+ |Φ′|2 + 12(|Φ|2 −1)2#,(16)Extremization of energy leads, apart from the trivial condition πρ = 0, to a system ofcoupled equations for Φ and E:ξE −"E′ −q|Φ|2#′= 0, Φ′′ + (E′ −q)2Φ|Φ|4−Φ(|Φ|2 −1) = 0. (17)These equations are simplified if we introduce an electrostatic potential for the electricfield: E = Y ′.

If we require that q vanishes for |x| above certain value, E and |Φ| mustrespectively approach 0 and 1 as x →±∞. The electrostatic potential Y will thenapproach a constant value.

If that constant is chosen to be 0, (17) takes formY ′′ −ξ|Φ|2Y = q; Φ′′ + ξ2Y 2Φ −Φ(|Φ|2 −1) = 0. (18)We are interested in perturbative corrections to the lowest order in q to the vacuumY = 0, Φ = 1 and to the sphaleron configuration Y = 0, Φ = tanh(x/√2).

We willshow in the following that in both cases the correction to Y is first order in q. It is thenclear from the second equation of (18) that the correction to Φ is at best second order.Inspecting the Hamiltonian (16) and bearing in mind that the unperturbed solutionsextremize H for q = 0 we conclude that the correction to the energy is second order in11

q and comes solely from the first-order correction to Y . Our task therefore reduces tosolving the first equation of (18) in the unperturbed Φ background.

Writing Y (x) asR G(x, y)q(y)dy we obtain an equation for the Green’s function G(x, y):∂2x −ξ|Φ(x)|2G(x, y) = δ(x −y). (19)It is easy to verify thatGv(x, y) = −12√ξ exp−qξ|x −y|(20)for the vacuum (Φ(x) = 1).

For the sphaleron configuration (Φ(x) = tanh(x/√2)) thesolution of (19) is also straightforward but somewhat cumbersome. The result can beexpressed in terms of hypergeometric functions:Gs(x, y)=−C cosh−√2ξ x/√2cosh−√2ξ y/√2×Fα+, α−; γ;expx/√22 coshx/√2×Fα+, α−; γ;exp−y/√22 coshy/√2(21)if y ≥x, with x and y interchanged otherwise.

Hereα±=1 + √8ξ ± √1 + 8ξ2; γ = 1 + α+ + α−2;C=Γ(α+)Γ(α−)41+√2ξΓ(γ)Γ(√2ξ). (22)Substituting the correction to electric field into (16) we find the energy shift of a statedue to the static charge q:∆H = −ξ2ZdxdyG(x, y)q(x)q(y).

(23)This result is hardly surprising: G(x, y) is nothing but the Coulomb potential at x dueto a unit charge at y. Therefore to the lowest order in q the energy shift is equal to theCoulomb energy of external charge distribution.

The correction to the sphaleron energybarrier then follows immediately:∆Esph = −ξ2Zdxdy (Gs(x, y) −Gv(x, y)) q(x)q(y). (24)References[1] M. Shaposhnikov, Nucl.

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