Soft contributions to the thermal Higgs width across an electroweak phase transition

We estimate the equilibration rate of a nearly homogeneous Higgs field, displaced from its ground state during the onset of an electroweak phase transition. The computation is carried out with Hard Thermal Loop resummed perturbation theory, and a significant part of the result originates from Bose-enhanced t-channel 2 ↔ 2 scatterings. The expression is shown to be IR finite and gauge independent. Possible applications to Langevin simulations of bubble nucleation are mentioned, and we also contrast with the friction affecting bubble growth.


Introduction
The recently approved LISA gravitational wave interferometer will offer unprecedented sensitivity to detect a possible weak-scale cosmological phase transition [1].Even though there is no such phase transition in the Standard Model of particle physics, many simple extensions do display first-order phase transitions.Therefore, it appears well-motivated to develop tools for a systematic study of their dynamics.
Many different momentum and energy scales (or length and time scales) play a role for the physics of a phase transition.The most macroscopic features of a phase transition, such as growing bubbles and the subsequent complicated fluid motion, are described by hydrodynamics.More microscopic features may be captured by kinetic theory, which should also permit for the computation of the transport coefficients appearing in the hydrodynamic equations.Some quantities, like the equation of state, are sensitive to the very smallest length scales, and need to be derived directly from the underlying quantum field theory.
In the intermediate domain of kinetic theory, which describes what is frequently referred to as "soft" physics, the relevant degrees of freedom are quasiparticles, whose properties get modified by thermal corrections.The modifications are generally divided into two types: dispersive and absorptive effects.Dispersive effects, such as thermal mass corrections, modify the relationship between the energy and momentum of a quasiparticle; we may say that they shift the real part of a pole appearing in a propagator.Absorptive effects describe a "width", i.e. an interaction or damping rate, and correspond to the imaginary part of the pole.
A most prominent quasiparticle is the Higgs boson.In vacuum, it has a large mass, m h,0 ≈ 125 GeV and, according to the Standard Model, a small width, Γ h,0 ≈ 4 MeV.
Both the Higgs mass and width change with the temperature.The mass changes through the evolution of the vacuum expectation value, originating from positive thermal corrections ∼ O(g 2 T 2 ) to the Higgs mass parameter [2], where g is a generic coupling constant.The width changes due to the Bose enhancement or Pauli blocking experienced by the decay products of 1 → 2 or 1 → 3 processes, but also due to new reactions, such as 2 → 1 "inverse decays", as well as 2 → 2 scatterings, involving additional particles in the initial state [3].In addition, 1 + n → 2 + n and 2 + n → 1 + n processes with n ≥ 1, may play a role.
When we go to very high temperatures, gT ∼ m h,0 , the positive thermal corrections to the Higgs mass parameter cancel against the negative vacuum term, and the electroweak symmetry is said to get restored.If the system underwent a second order transition, the Higgs mass could be made arbitrarily small by approaching the phase transition point.
In a more general situation, when the transition is a crossover or of first order, the Higgs mass does not become arbitrarily small.Let us consider the energy of scalar excitations having a vanishing spatial momentum, and adopt it as a definition of a thermal Higgs mass, denoted by m h ≡ m h,T .Different parametric regimes can be envisaged.Close to a crossover, perturbation theory breaks down, and we may say that the Higgs mass is of order m h ∼ O(g 2 T /π) [4].In contrast, in a first order transition, when the system interpolates between two stable phases, the Higgs mass could be larger in both of them (though it is tachyonic in between).For instance, assuming a partial cancellation between the negative vacuum term −m 2 h,0 /2 and the positive thermal correction ∼ O(g 2 T 2 ), we might assign the magnitude to the effective thermal Higgs mass squared [5,6].Organizing perturbative computations with this power counting has been argued to improve upon their convergence [7].
Under the assumption of eq.(1.1), the Higgs boson is lighter than most other particles, whose thermal masses are ∼ O(gT ).Then its 1 → 2 decays that dominate in vacuum are no longer kinematically allowed.However, the Higgs still has a thermal interaction rate.The purpose of this paper is to estimate this width, and show that the result is of order1 It is an interesting and non-trivial question which physical role the Higgs width of eq.(1.2) can play.We start this paper by briefly addressing this issue, in sec.2. Then we proceed to the explicit computation of Γ h , in secs.3 (for 2 ↔ 2 processes) and 4 (for 1 + n ↔ 2 + n processes).The results are illustrated numerically in sec.5, before we turn to an outlook in sec.6. Appendix A complements the body of the text, in which scatterings involving weak gauge bosons are considered, by estimating at leading-logarithmic level the Yukawa-mediated contribution involving top quarks and gluons.

Setup and motivation
A classic physics application for the electroweak phase transition is baryogenesis (for reviews see, e.g., refs.[9,10]).A key quantity in this context is the so-called sphaleron rate, proportional to the anomalous rate of baryon plus lepton number violation.The anomalous rate is generated by non-perturbative gauge field dynamics, at the momentum scale k ≡ |k| ∼ g 2 T /π.The gauge fields are the "slow" or "ultrasoft" modes of this problem, with a characteristic evolution rate ∼ g 4 ln(1/g)T /π 3 [11,12].However, the Higgs doublet also affects the sphaleron dynamics [13].Its evolution rate, from eq. (1.2), is just much faster.Therefore, in practical simulations [14], the Higgs doublet can be assumed to thermalize within a given gauge background, whereas the gauge fields evolve slowly via first-order (overdamped) gauge-covariant Langevin equations [12].
The very same framework that is used for determining the sphaleron rate, has also been applied to bubble nucleation [15,16].However, there is a qualitative difference between the sphaleron rate and the bubble nucleation rate.While the sphaleron rate is a smooth function of the temperature, and determined by its local value within a radius comparable to a magnetic screening length, ℓ M ∼ π/(g 2 T ), the length scale that is relevant for bubble nucleations is determined by macroscopic hydrodynamic boundary conditions, notably the Hubble rate.In classical nucleation theory, the radius of a critical bubble is where σ is the surface tension, L is the latent heat, T c is the critical temperature, and T n < T c is the nucleation temperature.As the Hubble rate is suppressed by T /m pl compared with the temperature, where m pl ≈ 1.22 × 10 19 GeV is the Planck mass, we proceed through the transition extremely slowly.Therefore, nucleations can happen close to T c , for instance at T n ∼ (0.99...0.999)T c .This may lead to a scenario with a bubble wall width ℓ w ∼ 10/T , bubble radius R ∼ 100/T , and inter-bubble distance ℓ B ∼ 10 10 /T .While the wall width is similar to the magnetic screening length, ℓ w ∼ ℓ M , the radius R could be much larger.
If we consider very large distances, R ≫ ℓ M , the relevant degrees of freedom are the hydrodynamics ones.At these scales, gauge field fluctuations are exponentially screened.Only gauge-invariant quantities, associated with conserved currents like the energy-momentum tensor, can display correlated long-distance fluctuations.
As a motivation for our investigation, we envisage a description of phase transition kinetics in which an effective low-energy scalar field is added to the usual hydrodynamic variables (temperature, T , and flow velocity, u µ ), to serve as an order parameter.The scalar field is gauge neutral; we do not consider the full Higgs doublet like in refs.[15,16].That said, the neutral component of the Higgs field is essential for the dynamics, as it can become tachyonic around the transition.Therefore, we heuristically postulate that the neutral Higgs component serves as an additional hydrodynamic variable. 2We denote its value by v.
When v is displaced from equilibrium, it relaxes towards it by dissipating its kinetic and potential energy into thermal entropy.Conversely, new fluctuations are constantly created by kicks from the medium.This physics is captured by a Langevin equation, where the metric convention (−+++) is assumed, 3 V is the Higgs potential, and ̺ is a stochastic noise term, with the autocorrelator The goal here is to estimate the value of the coefficient Υ.This should simultaneously fix Ω, through the fluctuation-dissipation relation Ω ≃ 2T Υ.The dots in the arguments of Υ and Ω in eqs.(2.1) and (2.2) stand for further gradients, such as u•∂, ∂ 2 , ... .The low-energy effective description is valid when the gradients are small.As this is the case for large bubbles, the Langevin description of eq.(2.1) can arguably be used to address nucleations around the electroweak phase transition [17][18][19][20].It has also been postulated to provide for a description of the subsequent bubble growth [21] (though then the noise is normally omitted).However, in the latter case, the gradients can be large within the bubble walls (∂ ∼ 1/ℓ w ), which draws the direct correspondence into question.
For a perturbative treatment, we write v in eq.(2.1) as a sum of a background value (v), and small fluctuations around it (δv ≡ h).We treat the background value as slowly varying, with V ′ (v, T ) ≃ 0.Then, eq.(2.1) turns into an equation for the fluctuations around the minimum.We denote m 2 h ≡ V ′′ (v, T ), remarking that this is only one among several possible definitions of a thermally modified Higgs mass. 4or the fluctuations, we go to momentum space, and to a local rest frame, Then the Green's function (propagator) corresponding to eq. (2.1) satisfies where the dots stand for higher powers of ω and k; the overall sign is a convention, corresponding to a Euclidean propagator; and the subscript K indicates that the frequency has been analytically continued to a Minkowskian one.Under the assumption we then obtain Υ from the inverse propagator as where the limit of small k should be understood in relation to mass scales like m h and gT .

Outline
In vacuum, the dominant Higgs decay channels are 1 → 2 processes into (almost) on-shell states, like h → b b or h → γγ, or into virtual states that decay further, like h → W + * W − * .At temperatures close to the electroweak transition, where according to eq. (1.1) the Higgs mass is smaller than the typical thermal masses, most of these channels close. 5t is possible, however, that the Higgs boson scatters off plasma particles before its decay, setting it off-shell ("brehmsstrahlung").This turns the would-be 1 → 2 decay into a 1 + n → 2 + n inelastic process, with n ≥ 1.Alternatively, we may pull one of the legs into the initial state, being then encountered with a 2 + n → 1 + n process.Simpler still, adding a single thermal particle to a 1 → 2 decay or 2 → 1 inverse decay, yields a 2 → 2 reaction (or 1 → 3 or 3 → 1, but these are phase-space constrained).It turns out that in an ultrarelativistic plasma, where all masses are small compared with the temperature, the 2 → 2 reactions often represent the leading channel, given that they are always kinematically allowed.We consider them in the present section, returning to the 1 + n ↔ 2 + n processes in sec.4. Specifically, we discuss here bosonic 2 → 2 reactions.The bosonic channels are important, because they can be enhanced by the Bose distribution.They may then develop an IR sensitivity to the Debye scale ∼ gT , turning the naive rate ∼ g 4 T /π 3 into ∼ (g 4 T /π 3 ) × (πT ) 2 /(gT ) 2 ∼ g 2 T /π.The corresponding fermionic channels, where the Bose enhancement is absent but the strong gauge coupling makes an appearance, are discussed in appendix A. We start by considering the confinement phase (T > T c ), where the Higgs expectation value v is set to zero (sec.3.2), and then turn to the Higgs phase (sec.3.3).

Confinement phase
Anticipating that the momentum transfers pertinent to t-channel 2 ↔ 2 scatterings will be soft (p, ǫ φ ≪ πT ), we carry out the computation with Hard Thermal Loop (HTL) resummed perturbation theory [22][23][24][25].The leading HTL diagram contributing to the scalar field selfenergy at temperatures above the electroweak phase transition (T > T c ) is shown in fig. 1.In the confinement phase, as this regime is often referred to, we denote the scalar field by φ rather than h, and its (thermally corrected) mass by m φ rather than m h .We assume m φ to be parametrically of the same order as m h , as given by eq.(1.1), in order to in principle permit for a comparison between the resulting widths in the two phases (cf.sec.5).
The HTL-resummed gauge propagator appearing in the diagrams takes the form where ξ is a gauge parameter; P = (p n , p) is an imaginary-time (Euclidean) four-momentum, with p n = 2πT n denoting a Matsubara frequency (n ∈ ); A refers to the particle species (gauge field); and the projectors read where we have denoted p ≡ |p|.The self-energy Π E P in eq.(3.1) turns out to contain an overall P 2 , and it is often convenient to factor it out, whereby we denote the dimensionless coefficient function as In order to carry out the Matsubara sums appearing in the loop, the propagators can be written in a spectral representation, where P = (p 0 , p) is a Minkowskian four-momentum.The spectral function is given by These representations hold if the Euclidean side is a decreasing function of p n , and the spectral function is a decreasing function of p 0 .
After the analytic continuation in eq.(3.5), the self-energies can be written as [26-28] Here the "2" stands for SU L (2) gauge bosons, and the thermal mass parametrizing their HTL self-energies reads where n S ≡ 1 is the number of Higgs doublets and n G ≡ 3 the number of fermion generations.The corresponding parameter for the U Y (1) gauge bosons reads The self-energies need to be evaluated both in a spacelike domain ("t-channel") and in a timelike domain ("s-channel").For |p 0 | < p, both self-energies have an imaginary part, originating from ln In contrast, the self-energies are real for |p 0 | > p. Then the imaginary part for eq.(3.5) originates from the pole induced by the tree-level frequency dependence, after recalling that where x pole satisfies f (x pole ) = 0.
The imaginary parts originating from eqs. (3.9) and (3.10) are frequently referred to as "cuts", and those from eq. (3.12) as "poles" [29].As there are two lines in a loop (cf.fig.1), we may then find "cut-cut", "pole-cut", and "pole-pole" contributions.Such a general situation is encountered in sec.3.3, and will be illustrated in fig. 3

(left).
Returning to the diagram in fig. 1, inserting the propagators, and carrying out the contractions, we find the inverse propagator where d 1 ≡ 1 and d 2 ≡ 3. It is important to note that the part of eq.(3.1) containing the gauge parameter has dropped out.The reason for this is that contributions originating from P µ P ν can be written as vanishes on−shell Here the first term is non-zero, but since it is independent of K, it has no imaginary part, and therefore gives no contribution to eq. (2.7).The second term drops out because of its antisymmetry, the third because the scalar self-energy is evaluated on-shell.
In the first term on the second line of eq.(3.13), the projector eliminates the dependence on P µ and P ν .From K µ and K ν , only the spatial part is left over.But as we are interested in the limit k → 0 (cf.eq.(2.7)), these drop out as well.
In the second term on the second line of eq.(3.13), which originates exclusively from δ µν /P 2 due to the argument in eq.(3.14), we may write In order to carry out the Matsubara sum in eq.(3.16), we insert eq.(3.4), and denote For later reference, we treat the Higgs propagator as if it had a general self-energy.Then where n B (x) ≡ 1/(e x/T − 1) is the Bose distribution.Subsequently we analytically continue k n → −i[ω + i0 + ], and take the imaginary part, For k = 0, the physical value is ω = m φ (cf.eq.(3.16)), but we keep the notation ω for the time being, because later on we apply the same formula for ω = m h .We note that the derivation also applies to 1/(1 + Π α P ), as long as this is a decreasing function of p n , yielding then a corresponding spectral function ρ α P .Returning to the specific case of eq.(3.16),where Π φ K−P = 0, we make use of the free spectral function , the result can then be expressed as with the graphs illustrating typical processes for the terms on each corresponding line.The wiggly line, connecting to the grey blob, denotes an HTL resummed spectral function ρ α P .The first case corresponds to t-channel exchange (|p 0 | < p), the second to an s-channel reaction (p 0 > p).The s-channel is realized if a "plasmon" goes on-shell; we return to this in sec.4.2.
Restricting to the t-channel part for now, eq.(3.16) becomes 6Im ∆ The corresponding kinematics is illustrated in fig.1(right).We note that |m φ − m 2 φ + p 2 | < p, so the latter factor is indeed in a spacelike domain.A limiting value of this finite integral will be evaluated in sec.5.2.

Higgs phase
We proceed to compute the Higgs width in the phase with a non-zero expectation value, v > 0. The corresponding diagrams are shown in fig. 2. The computation is technically more cumbersome than in the confinement phase, but it yields a non-trivial demonstration of the gauge independence of Υ, as well as of its partial continuity across the phase transition.
The HTL propagator of eq.(3.1) gets modified in the presence of electroweak symmetry breaking.If gv ∼ m E2 , i.e. v ∼ T , vacuum masses are of the same order as thermal corrections from HTL self-energies.Then the propagator of a W -boson in a general R ξ gauge can be written as The presentation is chosen so that there is only one pole in each part (no 1/P 2 ).The gauge boson vacuum mass is given by m W ≡ g 2 v/2, and similarly m Z ≡ g 2 1 + g 2 2 v/2.The gauge propagators are more complicated in the neutral sector (γ, Z 0 ).The reason is that the thermal self-energies do not "align" with the way that vacuum masses appear, so that the diagonalization of the 2 × 2 propagator matrix, and correspondingly the mixing angles, get modified in a p-dependent way (cf.appendix B of ref. [30]).
More precisely, eq.(3.23) contains three parts: the T-part, the E-part, and the gauge part.
The mixing angles change in different ways in the T and E-parts, because Π Ta and Π Ea are different (a = 1, 2).Moreover, the mixing angles are momentum-dependent, because Π Ta and Π Ea are so.The mixing angles do not get modified in the gauge part, because no thermal self-energy appears there.Similarly, the ghost and Goldstone sectors have no momentumdependent self-energies.In the following, we do not display the contributions of the neutral sector explicitly, but just refer to them collectively as the "Z-channel".
Proceeding with the computation, the first step is to check the cancellation of gauge dependence.Gauge-dependent pole locations, such as ξm 2 W in eq.(3.23), originate from the vector, ghost, and Goldstone propagators.Adding all terms and going on-shell (K 2 = −m 2 h = −2λv 2 ), all momentum-dependent terms cancel.Therefore the remainder has no imaginary part in the sense of eq.(2.7) (the momentum-independent gauge terms would be cancelled by further Feynman diagrams, not shown in fig.2).
Turning to the gauge independent terms, and taking already the limit k → 0, we find To identify the origin of the second line of eq.(3.24) in terms of the structures in eq.(3.23), we remark that the vertex ∼ hW + W − gives a factor ∼ m 2 W , and the E-part has a prefactor ∼ 1/m 4 W , from two appearances on the first line of eq.(3.23), which combine to give ∼ 1/m 2 W .The Matsubara sum in eq.(3.24) can be carried out with the help of eq.(3.19), once we insert eq.(3.4). 7In eq.(3.19), each propagator is represented by a corresponding spectral 7 As Matsubara frequencies now appear in the numerator, we are faced also with structures of the type Given that this does not decrease at large |p n |, the spectral representation from eqs. (3.4) and (3.5) is not literally applicable.However, we can add and subtract p 2 + Π α P in the numerator, so that the problematic part cancels against the denominator.The resulting "large" 1 yields no cut.By carrying out a tedious analysis of the other terms, it can be verified that the correct result can in fact be obtained with the naive replacement p n → −ip 0 .In the course of the proof, we need to note that in the domain |p 0 | < p, Incidentally, eq.(3.25) shows that In the left panel, we show a situation with ω = m h > 2ω pl , so that a pole-pole contribution exists.In the right panel, we assume ω = m h < ω pl , cf. eq.(1.1), and then it is absent.The integration domain of eq.(3.27) selects the lower pole-cut branch.The integration over the cut-cut domain can likewise be symmetrized, cf.eq.(4.13).
function.The spectral functions are non-vanishing for |p 0 | < p and |ω − p 0 | < p (cut parts), as well as |p 0 | > p and |ω − p 0 | > p (pole parts), respectively.In this section we focus on the pole-cut contributions (the kinematics is illustrated in fig.3).
Given that two same spectral functions appear, we can symmetrize the contributions, by picking up the cut parts from ρ α P and the pole parts from ρ α K−P .This fixes the integration domain of p 0 , yielding (3.27) Now, unlike in the confinement phase (cf.eq.(3.20)), the pole part cannot be given in closed form.However, making use of eqs.(3.9) and (3.10), and denoting by p 0 α (p) > 0 the location of a pole, with α ∈ {T,E}, eq.(3.12) allows us to write Im (pole) To find the values of p 0 α , which are functions of p, we need to determine numerically the zeros of the denominators, In eq.(3.27), the pole terms are evaluated with the four-momentum K − P = (ω − p 0 , −p).The Dirac-δ's in eqs.(3.28) and (3.29) therefore become δ(ω − p 0 − p 0 α ).This sets the integration variable p 0 in eq.(3.27) to p 0 = ω − p 0 α (p).Finally, we need to determine when the would-be pole crosses into the integration domain of eq.(3.27).For ω < ω pl , the minimal value of p, denoted by p α ≡ p α,min , is obtained from the condition p 0 = −p, cf.fig.3(right).Employing p 0 = ω − p 0 α , we see that ω + p α = p 0 α , i.e.
Having determined p α and p 0 α , eq. (3.27) can be estimated numerically.The value of the integrand at p ≫ m E2 can also be found analytically, and we return to this in sec.5.2.
The consideration of 1 + n ↔ 2 + n processes is, however, faced with a challenge.This is that the apparently leading reaction, 1 ↔ 2, is severely phase-space constrained, being open only if the masses of the participating particles (m 1 , m 2 , m 3 ) satisfy the usual inequalities Therefore, n = 1 can give a larger contribution than n = 0, even if it nominally involves more couplings.This indicates that the naive perturbative expansion breaks down, and a resummation over all values of n may be required.We will be confronted with this issue in sec.4.3.
Unfortunately, the resummation over n is well understood only in a particular kinematic domain, namely for hard momenta, p ≫ gT , where it goes under the name of Landau-Pomeranchuk-Migdal (LPM) resummation (cf., e.g., ref. [31] and references therein).In this domain, the problem can be mapped onto a dimensionally reduced effective theory [32], and NLO computations become accessible.In our situation, when all masses and momenta are soft, the lack of a supplementary scale hierarchy makes the resummation less transparent. 8n the present study, we address the 1 + n ↔ 2 + n processes without any additional resummation on top of the HTL one.In other words, we consider 1 + n ↔ 2 + n processes with either n = 0 or n = 1, whichever of these channels happens to be open.It turns out that in all cases considered, only one of them is open at a time.We note that the 2 ↔ 3 processes originate from the so-called "cut-cut contributions" [29] of HTL perturbation theory.

Confinement phase
Starting with the confinement phase, and staying in the regime of eq.(1.1), a 1 ↔ 2 process can indeed be found.Concretely, it originates from the s-channel part of eq.(3.21), when this part is used in the result in eq.(3.16).Physically, this corresponds to the decay of a "plasmon" into two φ-particles, or its inverse process.
By definition, the s-channel corresponds to the regime p 0 > p. Then the self-energies in eq.(3.9) and (3.10) have no imaginary parts.Therefore a non-vanishing spectral function ρ α P can only emerge via eq.(3.12).If such a pole exists, we call it a plasmon.The plasmon dispersion relation, which we denote by p 0 α (p), can be found by looking for the zeros of eqs.(3.9) and (3.10) [26][27][28], similarly to eqs.(3.30) and (3.31).This can be done analytically only in specific limits, notably by expanding either in p/p 0 or (p 0 − p)/p, p ≪ p 0 : Equations ( 4.1) and (4.2) imply that the plasmon energy is m E2 / √ 3 for p ≪ m E2 .In the opposite limit p ≫ m E2 , it follows from eq. ( 4.3) that p 0 T2 ≈ p + m 2 E2 /(4p), and from eq. ( 4.4) that p 0 E2 is exponentially close to the lightcone, p 0 E2 ≈ p + 2p exp −2p 2 /m 2 E2 − 2 .We now observe from eq. (3.21) that the 2 → 1 process takes place if a solution can be found for the equality p 0 Ea (p)=m φ + ǫ φ , where ǫ φ ≡ p 2 + m 2 φ .For p → 0, the left-hand side reads m E2 / √ 3 and the right-hand side 2m φ , whereas for p → ∞, the left-hand side equals p and the right-hand side p+m φ > p.A solution exists (i.e. the curves cross) if 2m φ < m E2 / √ 3, as is indeed parametrically the case in the regime of eq.(1.1).
To be more precise, combining eqs.(3.12), (3.16) and (3.21), working out carefully the sign of the pole term, and noting that the result originates from the p 0 > 0 domain because of the energy conservation constraint, the s-channel contribution becomes In order to integrate over p 0 , we make use of a variant of eq.(3.29), where p 0 Ea is the solution of Subsequently, the momentum constraint can be expressed as where p Ea can be found by substituting the energy conservation constraint into eq.(4.7), The Jacobian is a bit intransparent, but it can be simplified by making use of the energy conservation constraint p 0 Ea = m φ + ǫ φ .The final result then takes the form whose numerical evaluation is deferred to sec.5.3.

Higgs phase
We now turn to the Higgs phase.In this case the diagrams are given by fig.2(c).The contribution of 2 ↔ 2 processes is given by eq.(3.27), and this get modified for the 1 ↔ 2 and 2 ↔ 3 processes considered in the present section.
As long as m W > 0, we can understand the kinematics of the problem from fig. 3. A 1 ↔ 2 "pole-pole" contribution would exist if the Higgs decayed into, or were generated from, two plasmons.The contribution can be written as The poles can be evaluated like in eqs.(3.28) and (3.29).The result involves the constraint δ p 0 − p 0 α (p) δ ω − p 0 − p 0 α (p) .Therefore the reaction is kinematically allowed only if If we are in the domain of eq.(1.1), this is not the case, and the pole-pole contribution is absent.
Instead, there is a cut-cut contribution where the gauge modes are in the t-channel, i.e. spacelike.Making use of the symmetry of the integrand, as well as eq.(3.25), it can be  [34].For reproducibility, more digits have been shown than are physically accurate.For the quark thermal mass m F , making an appearance in appendix A, we have employed the leading-order expression from eq. (A.1).
written as Here ω = m h .We return to a numerical evaluation in sec.5.3.

Outline
The integral representations in the confinement (cf.eqs.(3.22) and (4.11)) and Higgs phases (cf.eqs.(3.27) and (4.13)) depend on many scales (m φ , m h , m W , m Z , m E1 , m E2 , πT ).Consequently the functional forms are not transparent, and it is unclear if the results remain finite if hierarchies emerge between the scales.In order to clarify the situation, we consider the soft limit The soft regime is physically relevant, as it corresponds to being in the confinement phase (m W , m Z → 0) and going close to the phase transition, in principle even to a supercooled domain (m φ → 0).This is precisely the situation most relevant for bubble nucleation.We demonstrate that the physics originates from the exchange of soft momenta, p ∼ m Ea , and that eqs.(A.24), after inserting the couplings from table 1.

2 ↔ 2 processes
In the confinement phase expression of eq.(3.22), estimating ǫ φ ∼ p ≪ πT , the phase space distributions become Parts of Γ Ea (m φ −ǫ φ ,p) and the prefactor can be combined into For the friction coefficient from eq. (2.7) this yields Φ a (p) The integrand is peaked at p ∼ m Ea , where its value is ∼ O(1/m Ea ).A numerical evaluation is shown in fig. 4.
We now demonstrate where to find the same physics on the side of the Higgs phase.In eq.(3.27), we can set m W → 0 in the T-part, which then drops out.It would be tempting to put m W → 0 also in the E-part.However, as discussed below eq.(3.31), the limit needs to be taken carefully, in order to pick up the correct pole (otherwise we find a singular integral).Let us write the soft limit of the Higgs phase expression for Υ in a form analogous to eq. (5.5).If we add the Z-channel, for which 2g 2 2 → g 2 1 + g 2 2 in the limit m Z → 0, we get where the lower integration bound originates from eq. (3.33).For p ∼ p Ea ∼ m Ea , the full expression from eqs. (3.27) and (3.29) needs to be employed.The pole from eq. (3.31) can only be solved for numerically, and the same holds for p Ea .9 However, if we consider the contribution from momenta p ≫ m Ea , life simplifies.The self-energy Σ E2 P has the prefactor m 2 E2 /p 2 (cf.eq.(3.10)), and therefore represents a small correction in this domain.Then, from eq. (3.31), (5.7) The Jacobian becomes In the chosen integration domain, recalling that m h − p 0 > 0 appears as the energy variable in the pole part, we find p 0 ≈ m h − ǫ W .The factors in eq.(3.27) can be simplified as Altogether we find Φ a (p) (5.11) This agrees perfectly with the p ≫ m Ea ≫ m φ limit of eq.(5.5).
If we carry out the full integral, then the value of eq. ( 5.4) at small m φ and the value of eq.(5.6) at small m h , m W , m Z do not agree completely, with the Higgs phase value being ∼ 50% larger.At finite m h , the Higgs phase value also depends on the ratio m h /m W , i.e. on the scalar self-coupling λ.However, the limiting value at m h → 0 appears to be λ-independent, like in the confinement phase.

1 + n ↔ 2 + n processes
For the 1 + n ↔ 2 + n processes, we start with the confinement phase result from eq. (4.11), corresponding to n = 0.The transcendental constraint in eq.(4.9) cannot be solved analytically, but in the soft limit m φ ≪ m Ea , a "leading-logarithmic" solution works reasonably well, (5.12) This shows that p Ea ∼ m Ea ≫ m φ , and therefore ǫ φ ≈ p Ea ≫ m φ .Given that m Ea ≪ πT , it follows that .13)All in all this leads to Fixing p Ea from eq. (4.9), with the parameters from table 1, the result is plotted in fig. 4.
In the Higgs phase, eq. ( 4.13) may appear to vanish for m W → 0, due to the overall prefactor m 2 W . 10 However, the second term in eq.(4.13) becomes IR sensitive, as it contains the terms 1/P 4 and 1/(K − P) 4 .If we send m W → 0 + smoothly, keeping the ratio m h /m W fixed, the result remains finite.Its value at m W → 0 + depends strongly on m h /m W , i.e. on the scalar self-coupling λ.For the Standard Model value (cf.table 1), it is about a factor 3 larger than that following from eq. (5.14).In terms of the diagram in fig.2(c), this value originates from a double IR enhancement of soft scatterings, as (g 6 T /π 5 ) × (πT ) 4 /(gT ) 4 ∼ g 2 T /π.
However, we should be cautious about the numerical interpretation of the Higgs phase expression.The fact that the 1 ↔ 2 rate vanishes, but that the 2 ↔ 3 rate, in which the the participants of the 1 ↔ 2 process have been "dressed" by their finite interaction rates, is large, suggests a breakdown of the perturbative expansion.Alas, we are not aware of a clean procedure to resum the corresponding physics.

Conclusions and outlook
Even if the thermodynamic properties of an electroweak phase transition can be determined with high precision with modern tools (cf., e.g., ref. [35] and references therein), the understanding of its real-time dynamics, including nucleations and the subsequent bubble growth and collisions, remains partly on a more qualitative level.In fact, general doubts about the semi-classical picture have been raised recently (cf., e.g., refs.[36,37] and references therein).
To make progress, it may become necessary to head in the direction of numerical simulations of the full problem (cf., e.g., ref. [38] and references therein).A potential framework for numerical studies is offered by fluctuating hydrodynamics, with the normal degrees of freedom supplemented by a scalar order parameter, which satisfies a Langevin equation.The purpose of our study has been to estimate the parameters appearing in such a Langevin equation.We have done this predominantly on the side of the hightemperature phase, which becomes metastable as the temperature decreases below T c .
Concretely, we have estimated the width, or interaction rate, of a nearly homogeneous Higgs field, displaced slightly from its equilibrium value.Various contributions to this width are illustrated in fig. 4. In a concrete BSM model, required for actually generating a first-order transition, contributions from other particles should be added, however we do not expect them to change the qualitative features that we have observed.
We remark that the interaction rate we have found is quite large, of order g 2 T /π.A significant part originates from 2 → 2 scatterings, which would naively yield a thermally averaged rate ∼ g 4 T /π 3 .However, the t-channel exchange is IR-sensitive, regularized by the Debye scale ∼ g 2 T 2 (such a contribution originates from the longitudinal or "electric" E-part of the gauge propagator in eq.(3.1)).This boosts the naive rate by a factor ∼ (πT ) 2 /(gT ) 2 .Physically, these rapid soft scatterings decohere the system.The resulting classical physics should then lend itself to a Langevin description.
Even though we have studied quantitatively only the confinement phase (T > T c ), we have also inspected the formal expressions in the Higgs phase (T < T c ).This has permitted for a non-trivial crosscheck of the gauge independence of the equilibration rate, and the demonstration that in both phases it is the E-part polarization state that plays the decisive role.The results also demonstrate how 2 ↔ 2 scatterings are rather insensitive to particle masses, whereas 1 + n ↔ 2 + n scatterings depend strongly on them, particularly in the Higgs phase.This suggests the need for a yet-to-be-understood resummation, in order to obtain quantitative results also in the Higgs phase.
We end by recalling that a similar friction as appears in the Langevin equation, is also needed for studies of bubble growth.However, in that case gradients are large, and momentum exchange plays a role.In the traditional approach, the momentum exchange originates from the position dependence of quasiparticle masses across the wall, ∼ dm 2 (v(z))/dz [39].Barring IR divergences, this leads to a friction that vanishes in the high-temperature phase.Our friction has a non-zero value in the high-temperature phase, and an even a larger value in the Higgs phase, both of which we have treated as homogeneous (the field perturbations do have "on-shell" time dependence).In this spirit, we presume that our minimal result, depicted in fig.4, could represent a lower bound for the physical friction affecting bubble growth.

A. Contributions from quark-gluon scatterings
A.1.Outline In vacuum, the decay h → b b(g) belongs to the most important Higgs decay channels (here g stands for a gluon).At finite temperature, when m h is small (cf.eq.(1.1)), the 1 → 2 decay or inverse decay h ↔ b b is hindered by the quark thermal mass ∼ g 3 T , where g 2 3 ≡ 4πα s denotes the strong gauge coupling.However, the channel can be opened by adding one or two gluon legs.The corresponding diagrams are shown in fig. 5.
Even if the general structure of the diagrams in fig. 5 looks similar to those in fig.2, there is a major difference in the corresponding interaction rate.This becomes clear if we go to the soft limit, inspecting momenta p ∼ g 3 T ≪ πT .In contrast to the Bose enhancement O(ωT /p 2 ) seen in eq.(5.2) or (5.13), the corresponding fermionic expression is of O(ω/T ), cf.eq.(A.27).That is, compared with the bosonic one, we expect the quark-gluon contribution to be suppressed by O(p 2 /T 2 ) ∼ g 2 3 , supplemented by an associated factor ∼ N c /π 2 , where N c = 3 is the number of colours.However, as g 2 3 ∼ 1.0 at T ≃ 160 GeV (cf.table 1), this may not be a huge suppression numerically.
We compute the contributions of fig. 5 in the regime where the Higgs-induced fermion mass, m ψ ≡ h ψ v/ √ 2, is at most of the same order as the quark thermal mass, where C F is the quadratic Casimir coefficient of the fundamental representation.Here two frequently used thermal mass parametrizations have been displayed, m F and m ∞ .As will be reviewed below, the two masses originate via the approximate form of a dispersion relation valid for spatial momenta p ≪ g 3 T (m F ) or p ≫ g 3 T (m ∞ ), respectively.The numerical value used is given in table 1.It is appropriate to stress that the Fermi distributions are necessary for making the integral convergent, i.e. that momenta p ∼ πT play a role.Therefore the HTL computation we have presented, valid for momenta p ∼ m F , is not the full result.Nevertheless, it does represent the qualitative behaviour, as the sensitivity to the scales p ∼ πT is only logarithmic.We still need to estimate the contribution from the second term in eq.(A.10).We write the pole part as  To summarize, the order of magnitude of fermionic 2 ↔ 2 scatterings is given by eq.(A.20).This result is illustrated numerically in fig. 4.

A.3. 1 + n ↔ 2 + n processes
Finally we turn to 2 ↔ 3 processes, illustrated in fig.5(c).In the language of fig.3, they originate from cut-cut contributions.Making use of the symmetry of the integrand, we now restrict to the upper half of the cut-cut domain, like in eq.(4.13).From eqs. (A.7) and (A.9), we then obtain a variant of eq.(A.Here, apart from ∆ − P in eq.(A.17

Figure 1 :
Figure 1: (a) the imaginary part of a HTL-resummed Higgs self-energy contribution in the confinement ("symmetric") phase.Dashed lines denote scalar fields, wiggly lines gauge fields, and a blob HTL resummation.(b) a 2 → 1 contribution originating from process (a), which is kinematically allowed in the regime of eq.(1.1).(c) a 2 → 2 contribution originating from process (a), with the straight line standing for a generic particle species (scalar, fermion, gauge field).The kinematic variables correspond to those appearing in eq.(3.22), with ω = m φ and ǫ φ = p 2 + m 2 φ .

Figure 2 :
Figure 2: (a) imaginary parts of a HTL-resummed Higgs self-energy in the Higgs ("broken") phase.Dashed lines denote scalar fields, wiggly lines gauge fields, dotted lines ghosts, and a blob HTL resummation.(b) 2 ↔ 2 reaction originating from the processes in (a), with the solid line representing any particle species.(c) 3 ↔ 2 reaction originating from the processes in (a).

. 31 )
In eq.(3.31), the existence of m W > 0 plays an important role.Given that a pole lies in the domain |p 0 E2 | > p, so that p 2 − (p 0 E2 ) 2 < 0, we must have 1 + Σ E2 (p 0 E2 ,p) > 0. At the same time, Σ E2 (p 0 E2 ,p) < 0. Therefore a solution is found in a domain in which the absolute value | Σ E2 (p 0 E2 ,p) | is not too large.This leads to lim m W →0 + p 0 E2 = m E2 / √ 3, so that we are not on the light-cone, unlike what could be assumed by naively setting m W → 0 in eq.(3.31).

Figure 5 :
Figure 5: (a) the imaginary part of a HTL-resummed Higgs self-energy contribution, originating from Yukawa couplings.Dashed lines denote scalar fields, arrowed lines quarks, and a blob HTL resummation.(b) "pole-cut" contributions (or 2 ↔ 2 scatterings), originating from process (a).These are analyzed in sec.A.2. Thick lines without an arrow represent both quarks and antiquarks, i.e. an arrow in either direction, and curly lines stand for gluons.(c) "cut-cut" contributions, also originating from process (a).These are discussed in sec.A.3.

Table 1 :
Standard Model values for couplings and Debye masses at T ≃ 160 GeV