Nucleation at finite temperature: a gauge-invariant, perturbative framework

We present a gauge-invariant framework for bubble nucleation in theories with radiative symmetry breaking at high temperature. As a procedure, this perturbative framework establishes a practical, gauge-invariant computation of the leading order nucleation rate, based on a consistent power counting in the high-temperature expansion. In model building and particle phenomenology, this framework has applications such as the computation of the bubble nucleation temperature and the rate for electroweak baryogenesis and gravitational wave signals from cosmic phase transitions.

Achieving a rigorous understanding of the thermal history of the electroweak (EW) symmetry breaking has been a long-standing challenge at the interface between particle physics and cosmology. The standard history of the Standard Model (SM) of particle physics contains no electroweak phase transition. Instead a smooth crossover between high and low temperature phases occurs [1,2]. A non-standard history with a first order phase transition is conceivable in theories beyond the SM (BSM) that are well motivated from the electroweak to TeV scale. BSM scenarios with extended scalar sectors can contain rich patterns of symmetry breaking [3] and potentially produce EW baryogenesis [4][5][6], and also rich phenomenology for high energy collider physics [7] and production of primordial gravitational waves [8]. These phenomena have been actively studied for decades and recently sparked increased interest in next-generation gravitational wave detector experiments [9][10][11] such as LISA [12].
One key ingredient for both EW baryogenesis and stochastic gravitational wave background production is a first order phase transition in the primordial plasma of particles that proceeds via nucleating bubbles of the low-temperature phase. The thermo-and bubble dynamics of such transitions can be reliably described nonperturbatively using lattice simulations [13][14][15]. However, exploring BSM scenarios and their thermodynamics invariably requires the use of perturbation theory, since fully comprehensive non-perturbative analyses of a multidimensional parameter space are computationally out of reach. A qualitative and quantitative assessment of the perturbative reliability requires robust theory computations when comparing to lattice data. In that respect, an unphysical gauge-dependence of the bubble nucleation rate has plagued particle physics phenomenology applications of the thermal phase transition literature for decades (c.f. [8,16] and references therein). While by now resolved at zero temperature [17][18][19][20][21][22], a similar, longstanding and open problem exists at finite temperature if the potential barrier between the phases is radiatively generated.
We resolve this long-standing problem by providing a practical, gauge-invariant framework for thermal bubble nucleation, intended for model-building and particle phenomenology applications. The reliability of the approach can be assessed by "benchmarking", against lattice analyses in the future. Such benchmarking is incompatible with the conventional approach since there the nucleation rate is gauge-dependent and an ill-defined unphysical quantity.
The bubble nucleation rate per unit volume, Γ, has the semiclassical approximation [23][24][25][26] Here, the exponent B = S 3 /T with S 3 being the threedimensional Euclidean effective action evaluated at the "bounce" solution that solves the classical Euclidean field equations [24,27]. The prefactor A is dimensionful and given by the characteristic mass scales of the theory, resulting from computing the functional determinants. The leading behaviour of the rate Γ is encoded in the exponent B.
The effective action is computable using the background field (φ) method. Therein, V eff (φ, T ) is the thermal effective potential describing the equilibrium free energy of the system and Z 1 2 (φ, T ) is the field renormalization factor. Both V eff (φ, T ) and Z(φ, T ) admit an expansion in the weak gauge coupling, denoted generically here as g. After the gradient expansion in powers of spatial derivatives (∂ i φ), the effective action reads [28,29] where the ellipsis contains terms of additional (∂ i φ)powers. While the gradient expansion in general does not converge [15,[30][31][32], one may obtain a self-consistent, gauge-invariant estimate of Γ by ensuring the two following conditions apply (A): Fields that generate the barrier must be parametrically heavier than the nucleating field. In this case leading orders can be correctly described by a derivative expansion, (B): Considering temperatures in the vicinity of the critical temperature, T c , such that a specific power counting in g applies, wherein the leading order potential has a radiatively generated barrier.
Hence, we focus on a temperature regime where the first two leading terms of the effective action are enhanced by powers of g and presentable schematically as Here a 0,1 are numerical coefficients that are computable in the gradient expansion at leading (LO) and next-toleading orders (NLO), respectively. ∆ encodes formally higher order corrections -that are O(1), up to potential logarithms of g [31] -that we presently do not compute. The bubble nucleation rate then reads We formally collected higher order effects encoded in ∆ in Eq. (3), to a yet unspecified prefactor A ′ ≡ Ae −∆ . The expression (4) results from an expansion based on the chain of scale hierarchies at high temperature where πT is the thermal scale of non-zero Matsubara modes, gT is the intermediate scale of zero modes that are thermally screened, and g 3 2 T is a characteristic scale of bubble nucleation, related to an effective mass of the nucleating field. For a systematic treatment of thermodynamics in the framework of three-dimensional, high temperature effective field theory (3d EFT) see [33,34]. This framework was recently extended [31] by presenting an approach for an effective description for bubble nucleation.
Here, we work at the leading high temperature expansion [35] that allows us to perform the computation without formally constructing a 3d EFT. Our goal is to compute LO and NLO terms in B and show their gauge invariance. We emphasize that this computation agrees with the generic EFT approach [31] at leading orders, and an accompanying article [32] revisits this computation in the EFT context, which allows to pursue improvement in terms of higher order corrections.
As in earlier literature both at zero [17,36] and high [30] temperature, we use the Abelian Higgs model as a concrete illustration. The Euclidean Lagrangian density for the Abelian Higgs model is with a U(1) gauge field B µ and a complex scalar field Φ. The covariant derivative for complex Higgs reads where g is the gauge coupling and the field strength tensor where φ ≡ φ(x) is a spatially dependent classical background field and H and χ are propagating quantum degrees of freedom. In perturbation theory the gauge can be fixed using R ξ -gauge [30,37] with Fadeev-Popov ghost (c,c) Lagrangian Although one need not relateφ directly to φ, it is convenient to identifyφ = φ/ √ 2 to eliminate mixing between the gauge field and Goldstone mode χ.
Equilibrium thermodynamics are described by the effective potential, whose minima should be separated by a barrier for a first order phase transition. While the Abelian Higgs model admits no tree-level barrier, the barrier arises radiatively through loop corrections. Similar to [17,30,35,38], we show that one-loop gauge field contributions can arise at the same order as the treelevel potential provided that model parameters assume the following power counting. First, we assume that which sets the size of the background field to the characteristic mass scale of the problem, the temperature. Second, we relate the size of the scalar self-coupling and gauge coupling by [35] λ ∼ g 3 .
By assuming that the quartic coupling is sufficiently small compared to gauge coupling, the cubic term induced by the gauge field will contribute at leading order, as seen below. Third, we assume which installs the high-temperature expansion of µ 2 ≪ T 2 . Finally, the effective thermally corrected mass of the scalar at leading order is Here, the first and third terms are individually of O(g 2 T 2 ). However, for µ 2 < 0, a cancellation between the first and remaining terms can render the overall sum in µ 2 eff smaller -parametrised by N > 0 -for some temperature range. Henceforth, we assume a temperature window where N = 1 and argue below that this occurs in the vicinity of the phase transition critical temperature [35,39].
The assumptions of Eqs. (9)- (12) induce the chain of thermal scale hierarchies in Eq. (5). Non-zero Matsubara modes have masses ∼ πT . In the unbroken phase (φ = 0) zero Matsubara modes have the following scales: Higgs and Goldstone fields have masses ∼ µ eff ∼ g 3 2 T . At zero temperature it is sufficient to talk of transverse and longitudinal modes of the gauge field, but at high temperatures there is a further splitting of the three transverse modes into two spatial modes and one temporal. The gauge field temporal mode, B 0 , has a mass m B 0 ∼ gT ; while spatial gauge fields remain massless (m B = 0). In the broken phase (φ > 0), spatial and temporal gauge bosons, Goldstone and ghost fields have masses ∼ gφ ∼ gT , while the Higgs field that undergoes nucleation has parametrically lighter mass ∼ g 3 2 T . The full one-loop effective potential is found using the following background field dependent mass eigenvalues where masses for the zero Matsubara modes of the Higgs, Goldstone and temporal gauge field B 0 include resummed thermal corrections e.g. the T 2 -term in the third line describes Debye screening. The mass of the longitudinal component of the gauge field equals the ghost mass. Neither spatial gauge fields, nor ghost fields, develop thermal masses. Based on the power counting in g, the leading contribution to the effective potential is of O(g 3 T 4 ) and reads where the second line is the transverse gauge field contribution and the second term therein corresponds to the Debye mass (15) of the temporal mode. Near the phase transition, all terms in the potential should be approximately of the same order of magnitude which is assured by construction, given the assumed power counting in g in Eqs. (9)- (12). The leading-order potential of Eq. (17) at O(g 3 T 4 ) is gauge invariant. The one-loop contribution to the effective potential from the longitudinal gauge field, Goldstone field, and ghost fields is Note that ghosts contribute with a relative minus sign to other fields. These contributions give rise to an explicit ξ dependence, but these terms are of higher order compared to Eq. (17) due to a cancellation at leading order. Below we include the remaining O(g 4 T 4 ) terms at NLO in the effective potential which expands as where The last term on the second line originates from the oneloop Goldstone-ghost contribution in Eq. (18), and the last two lines correspond to the two-loop contributions of the B 0 field. The remaining terms result from two-loop diagrams with spatial gauge fields, ghosts and scalars (χ,H). The O(g 9 2 T 4 ) term in Eq. (19) arises at one-loop order from the Higgs field.
Consistent treatment of the effective action (2) at NLO also requires the inclusion of field renormalization: where The first term is from the two spatial modes and ghosts and the second term is from the temporal mode. Notably, at order O(g) the field renormalisation is independent of the gauge-fixing parameter. Higher order terms at O(g 3 2 ) arise from two-loop diagrams involving gauge, ghost and Goldstone fields, and one-loop diagrams with internal Higgs legs.
In the semiclassical approximation, the background field extremizes the leading-order action and can be found from the equation of motion for the leading-order potential where ∇ 2 ≡ ∂ i ∂ i is the three-dimensional Laplacian operator, and φ b is the "bounce solution" [40]. We expand the exponent of the nucleation rate as B = B 0 + B 1 : where β ≡ 1/T . The characteristic length scale R for nucleation is related to the typical bubble size, given by the inverse mass of the nucleating field R ∼ m −1 H ∼ g − 3 2 T −1 . This gives rise to the formal scaling d 3 x ∼ g − 9 2 T −3 . Together with the power counting for the effective potential and field renormalisation, this establishes the relative importance of the first two leading exponent terms of the nucleation rate: as already foreshadowed in Eqs. (3) and (4). Despite the 1/φ behaviour of Z NLO (φ b ), the contribution of the term [32] (c.f. also [31,41,42]).
Because the bounce solution has to be solved numerically from Eq. (23), the exponents B 0,1 are necessarily obtained by numerical integration. Nevertheless, their gauge-independence can still be proven analytically. The gauge-fixing parameter is absent at leading order since Z and V eff are both ξ-independent at this order, implying gauge-independence of both the bounce solution φ b and exponent B 0 . The gauge-invariance of B 1 is not immediately obvious since V eff NLO in Eq. (20) explicitly depends on ξ. To proceed, we utilise the Nielsen identities [37,43] in analogy to [17]. These identities have been discussed in the context of finite temperature in e.g. [30,36,44,45]. The Nielsen identity (in d-dimensional Euclidean space) relates the variation of the effective action with the gauge parameter to the corresponding Nielsen functional which admits a derivative expansion [30] C Together with the expansion of the effective action (2), this yields the Nielsen identities for the effective potential and field renormalization factor To employ these relations, we expand them in powers of g by first quoting the scaling of the Nielsen coefficients These coefficients are computed at leading order in [30,32]. We do not need the explicit expressions of D,D because the terms on the second line of Eq. (30) appear at O(g 2 ), and are hence suppressed relative to those on the first line at O(g). The leading order C LO ∼ gT is independent of the scalar background field φ at finite temperature. An explicit counting in powers of g in the identities (29) and (30) yields at O(g 4 T 4 ) in the first and O(gT ) in the second Nielsen identity. A combination of the explicit expressions (17), (20), (22), and (31) readily verifies both identities. In particular, the equality (34) holds since the NLO field renormalisation is ξ-independent and at LO C is φindependent.
Using the above Nielsen identities, we demonstrate gauge independence of B 1 : Step (A) uses the Nielsen identity (33) and ξindependence of Z NLO ; (B) applies the equation of motion (23) and moves C LO outside the integrand due to its φ-independence; (C) applies Gauss's theorem; (D) follows from the asymptotic behaviour of the bounce solution at the boundary, This completes our proof of gauge invariance of the leading exponential of the nucleation rate. Figure 1 illustrates the quantitative impact of applying our framework as compared to the conventional gaugedependent approach. Fig. 1 (left) gives the λ-dependence of the strength of the transition, characterized by latent heat released in the transition, L, scaled to T 4 c . Fig. 1  (right) shows the T -dependence of S 3 /T . We fix g 2 = 0.42 and µ 2 eff = −M 2 /2, where M = 100, and where M and T are in arbitrary units of mass. Furthermore, we define λ and g at an initial renormalisation scale Λ 0 = 100 and run to a scale Λ = T using one-loop renormalization group evolution [32].
The dashed (solid) line in Fig. 1 (left) corresponds to the gauge-invariant result for L/T 4 c at LO (LO+NLO). For comparison, we also illustrate the conventional analysis based on direct minimization of the full one-loop effective potential (in the high-T expansion). The latter approach is gauge-dependent and inconsistent, as illustrated by the light grey band in which the gauge-fixing parameter ranges over ξ = (0, . . . , 5). The qualitative behaviour -the transition is stronger for smaller λ -agrees with the gauge-independent determination, but the quantitative result is ambiguous. Fig. 1 (right) shows the T -dependence of S 3 /T = B 0 + B 1 for fixed λ = 0.005 as the black solid curve. We have obtained this result using FindBounce [46] to find the bounce solution φ b of Eq. (23). For comparison, the light grey band significantly varies in S 3 /T when varying ξ = (0, . . . , 5) as manifestation of an inconsistent computation that is often encountered in the past literature (e.g. [8,47]). The latter result is based on computing Eq. (1) using Eq. (2), where the effective potential is directly computed at one-loop order in a fixed gauge. Therein, the bounce solution is found from the same gauge-dependent one-loop effective potential. This leftover gauge-dependence immediately signals an inconsistency, as the computation of a physical quantity should contain no ξ-dependence. For a discussion of other theoretical inconsistencies encountered in the "conventional" approach, such as the appearance of an imaginary part in S 3 and double counting of contributions from the nucleating field, see [31,48]. The renormalization scale-dependence of the results in Fig. 1, that can be used to monitor the accuracy of perturbation theory (c.f. [48,49]), is further discussed in [32].
The framework presented in this letter provides a way to obtain a gauge-independent, perturbative estimate of the thermal nucleation rate in the presence of radiative barriers. While we have worked in the Abelian Higgs model along the lines of previous literature [17,30], our framework readily generalises to more complicated gauge field theories, with radiatively generated barrier, and our practical approach can facilitate corresponding modelbuilding phenomenological studies. Ultimately, one must assess the quantitative and qualitative reliability of perturbative nucleation rate computations through comparison with non-perturbative calculations (see [15,50]). A meaningful comparison requires a well-defined, gaugeinvariant perturbative computation, which the framework presented herein provides.