Quantisation Across Bubble Walls and Friction

We quantise from first principles field theories living on the background of a bubble wall in the planar limit with particular focus on the case of spontaneous breaking of gauge symmetry. Using these tools, we compute the average momentum transfer from transition radiation: the soft emission of radiation by an energetic particle passing across the wall, with a particular focus on the longitudinal polarisation of vectors. We find these to be comparable to transverse polarisations in symmetry-breaking transitions with mild super-cooling, and dominant in broken to broken transitions with thin wall. Our results have phenomenological applications for the expansion of bubbles during first order phase transitions. Our general framework allows for the robust calculation of any particle processes of interest in such translation breaking backgrounds.

First order phase transitions (FOPTs) have long ago been proposed as having possibly occurred during the hot big-bang phase of the universe [1][2][3]. In the past, it was entertained that even within the known Standard Model (SM) of particle physics there might have been as many as two: chiral symmetry breaking/confinement in QCD at temperatures T ∼ 150 MeV and the spontaneous breaking of electroweak (EW) symmetry at T ∼ 160 GeV. Both are now understood to be smooth crossovers [4,5]. Infact, it is interesting to note that from the current laws of physics there is no conclusively established meta-stable vacuum for any temperature at zero chemical potential 1 . By contrast, FOPTs are ubiquitous in beyond the SM (BSM) theories. This is due firstly to a vast richness of important phenomenological consequences, among which baryogenesis [7][8][9][10][11][12][13][14][15][16][17][18], the production of heavy dark matter [19][20][21][22][23][24][25], primordial black holes [26][27][28][29][30] and gravitational waves (GW) [3,[31][32][33][34] to name a few. In particular, the EW phase transition is easily made first order in many BSM models [14,[35][36][37][38][39][40][41][42][43][44][45] and the consequent out-of-equilibrium dynamics (in conjunction with B violation in the SM) still make for an attractive theory of baryogenesis. Secondly, our currently most compelling picture of physics at the highest energy scales seems to suggest landscapes of countless meta-stable vacua. From this perspective FOPTs may even be expected during the post-inflationary era [46]. Finally, and perhaps most importantly from a phenomenological perspective, the advent of gravitational wave detectors has re-energised interest in these violent phenomena with the prospect of upcoming experiments possibly detecting a stochastic gravitational wave background relic [3,47,48]. Thus even FOPTs occurring in potential hidden sectors decoupled from the SM and its thermal history become of interest [49,50].
A FOPT proceeds through the nucleation and subsequent expansion of bubbles of new phase, pushed by the energy density difference ∆V between phases. If friction from the surrounding matter can be ignored, the bubble wall interpolating between the two phases will continue to expand with constant proper acceleration until they collide with each other, with most of the vacuum energy released thus going into kinetic energy of the walls. This scenario is known as runaway. If instead friction causes a pressure P which manages to equilibrate the driving force P ≃ ∆V , a constant subluminal terminal velocity is reached and energy is efficiently transferred to the medium. All phenomenological consequences listed above, for example the strength and spectral shape of the stochastic GW signal, depend crucially on the Figure 1: Diagrams corresponding to LO and NLO processes contributing to friction on a moving bubble wall. Emission of vectors with changing mass is generally the dominant 1 → 2 process for friction and the subject of this work. Tilde distinguishes objects in the new phase.
bubble velocity and which of the two regimes is realised. To this end, it becomes important to understand precisely the dynamics of an expanding domain wall in medium 2 . The analysis of bubble-medium interactions is a complicated problem which is largely still under investigation [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66]. Friction is expected in general to be an involved function of bubble velocity v w and the surrounding degrees of freedom (d.o.f.). A distinction can be made however between low v w , when a fluid description is most appropriate, and the ultra-relativistic regime, γ w ≡ 1/ 1 − v 2 w ≫ L w Γ int. , where L w is the wall thickness in its rest frame and Γ int. is the interaction rate between particles in medium [67]. We will focus on the latter regime in this work, where the wall can be said to be interacting with individual particles.
A particle hitting the wall from the old phase can undergo many processes, which can be organised in terms of a perturbative expansion in couplings of the theory defined in the background of the wall profile, as sketched in fig. 1. The spontaneous breaking of translation symmetry means that momentum perpendicular to the wall is no longer conserved. The average momentum lost ⟨∆p⟩ times the flux of incoming particles is then the pressure opposing the bubble's expansion. It is most convenient to work in the rest frame of the wall. At leading order (LO) incoming particles either cross the wall or reflect. It is easy to show that, when reflections can be neglected 3 , where p 0 ∝ γ w is the (boosted) incoming particle's energy in the wall frame, ∆m ≡ √m 2 − m 2 the change in mass between phases, and n the number density in the plasma frame. This LO pressure is independent of γ w , scaling like ∝ T 2ṽ2 in the case of a thermal bath, whereṽ is the vev in the broken phase 4 [61,62].
However, later the same authors analyzed the next-to-LO (NLO) 1 → 2 processes in the same ultra-relativistic regime and found that, despite paying the price of the coupling, the emission of soft vector bosons that gain mass during the transition lead to a pressure/friction scaling like P NLO ∝ γ w [63], eventually dominating over the LO effect. This soft emission is known as transition radiation. While the original [63] focused on particles emitted forward into the wall (to the right in fig. 1), the authors of [66,69] considered also reflected emission (to the left in fig. 1) and argued it was larger by a factor of four.
However, all studies after [63] only considered the emission of transverse vector polarisations, ignoring the effects of longitudinal ones. The analysis of these modes is complicated by the rearrangement of particle degrees of freedom across a gauge symmetry breaking transition [70], which has naturally been the case of greatest interest. Moreover, it is well known that amplitudes involving NGBs can give spurious divergences without proper care. Recently it was shown that LO effects from longitudinal modes can have a large impact on pressure [68]. It thus becomes of interest to properly account for their contribution at NLO. In addition, a weakness of the treatments used so far is the frequent reliance on WKB approximations, which are known to break down for the soft momenta dominating the emission phase space.
In this paper, we approach the calculation of transition radiation by quantising field theories in the Lorentz-violating background of a domain wall from first principles. A complete orthonormal basis is constructed out of 'left' and 'right' mover energy eigenstates 5 , each wavemode having 'reflected' and 'transmitted' parts. We then carefully relate these to in and out 6 asymptotic eigenstates of 4-momentum.
In the case of vectors we show that the degrees of freedom across the wall are most conveniently described in terms of 'wall polarisations' τ and λ rather than the conventional transverse and longitudinals, as already pointed out in [70]. The two sets coincide only for zero perpendicular momentum ⃗ k ⊥ = 0. The advantage is that τ 1,2 and λ are not mixed between each other in the presence of the domain wall 7 . Moreover, in the case of gauge symmetry breaking, λ smoothly interpolates between a Higgs d.o.f. on the symmetric side and a third massive vector d.o.f. on the broken side. We show how to perform calculations using this basis consistently and avoid divergences which seem to appear in a naive analysis.
Although we explain how to (numerically) compute ⟨∆p⟩ for a general wall profile, we dedicate most of the work to approximating it in a way that is independent of the particular shape, with the rough scale L w playing the only role, while commenting on the sensitivity thereon. For the IR of the emitted spectrum k z ≲ L −1 w the wall appears effectively as a step function. This limit is particularly interesting theoretically (as well as phenomenologically important, as mentioned already) since everything can be computed analytically and relatively simply. For wavelengths longer than the wall width k z > L −1 w , the WKB approximation becomes applicable. The integral over the phase space thus splits into two contributions and the averaged 5 Throughout this paper, the reader should associate 'right-moving' with positive z−momentum and 'left moving' with negative z−momentum particles. 6 To be understood in the S matrix language. 7 Starting from ⃗ k ⊥ = 0, where the two sets coincide, the general τ and λ can be obtained by a general transverse Lorentz boost. Thus orthogonality is obvious. In general, they are also distinguished by whether in unitary gauge the z− component of the vector A µ is zero or not. See section 3.2.  Figure 2: Comparison between the averaged exchanged momentum from transition radiation due to the emission of τ and λ vector polarisations, in the limit of large incoming particle energy p 0 → ∞.
where the M are matrix elements for emission calculated using the respective approximations. As the incoming flux scales like γ w , we then have As a warm up, we study transition radiation in a theory with two scalars and observe some surprises. Though we find that the pressure from the emission of one scalar by the other always saturates at large velocities P NLO,scalars γw→∞ ∝ γ 0 w , we find also that there can be a intermediate regime of linear growth P NLO,scalars intermediate ∝ γ w . Interestingly, for scalars we find that the WKB contribution (second term in eq. (2)) dominates the momentum transfer in the asymptotic γ w limit.
In the case of spontaneous breaking of gauge symmetry we find that the total friction from vector emissions scales as ∝ γ w log gT sv for T /ṽ ≪ 1 in line with literature. We provide an updated fitted formula in eqs. (137) and (142). The logarithmic enhancement appears only for the τ polarisations, and is dominated by the step function contribution (the first term in the eq. (2)), however we also find that effects of the λ polarisations can lead to significant corrections for mild supercooling (ṽ T ∼ few). We compare the relative importance in fig. 2 (Left). The curves are only very weakly dependent on L w .
As a side application, we also compute the transition radiation when the bubble wall connects two vacua with broken gauge symmetry but different vevs v andṽ.
Again we highlight the sensitivity to wall width. For small enough L w we find an inter-relativistic regime with averaged exchange momentum growing linearly, which translates to pressure scaling like γ 2 w .
In this case, the contribution to friction from the longitudinal vector emission scales as P NLO,vectors γw→∞ ∝ γ w L −1 w (see fig. 2, Right) and can dominate over the transverse for thin wall.
Aside from the asymptotic p 0 → ∞ limit, we are also able to explore regimes with intermediate -though large -γ w . For symmetric → broken transition we find that the saturating value is reached at energies strongly dependent on the mass of the emitter particle, as shown in fig. 3 fig. 3). The paper is organised as follows: in the section 2 we work through a toy model with only scalars, introducing various elements of the calculation. In section 3 we quantise an Abelian Higgs model in the presence of a symmetry breaking domain wall, and present the results for transition radiation of vectors in section 4. We summarise in section 5.

Summary of notation:
In the rest of this paper, we will adopt the following conventions: 1. We treat the bubble wall in the planar limit where it is a domain wall centred around z = 0. We use a hybrid notation for four-vector Lorentz indices: µ = (n, z) ≡ (0, 1, 2, z). Coordinates are x µ = (x n , z) ≡ (t, ⃗ x ⊥ , z).

Similarly for mometa
3. We write the variation of mass across the wall ∆m ≡ √m 2 − m 2 .
4. L w is the thickness of the wall.
5. γ w (v w ) is the boost factor (the velocity) of the wall.
6. The momenta p, q, k will always be used as in fig. 1 and we define: 2 Simple example: scalars In this section, as a warm up for the more physically relevant case of the emission of gauge bosons, we go through the quantised theory of scalars fields in the presence of the domain wall and derive results for transition radiation for the case of one scalar emitting another. This toy example is sufficient to highlight many features of the calculation.
Consider two different scalar fields ϕ, ψ, the first of which feels the wall and has different mass depending on the phase, while the second for simplicity does not. The Lagrangian we consider is the following where m(z) interpolates between m 2 (z)| z→−∞ = m 2 = const and m 2 (z)| z→+∞ = m 2 = const. Similarly y(z) goes from y toỹ. The profiles change on the scale of the wall width L w around z = 0. The interactions in eq. (5) are not the most general, but are designed to mimic the vector case when y = const. The process that we will be studying is ψ → ψϕ, which would be forbidden by kinematics if it was not for the breaking of z−momentum.
Section summary: In section 2.1 and 2.2, we quantise the free theory, focusing on the ϕ field 8 , by defining a complete basis of solutions that solve its equations of motion. In section 2.3, we define a new basis which correspond to out-going eigenstates of momentum. Later, in section 2.4, we calculate the amplitude for the ψ → ψϕ transition in the step wall approximation, valid when k z L w ≲ 1. In section 2.5, we present the proper domain for the phase space integration over the final state. In section 2.6, we complete the emission spectrum discussing the calculation of the amplitude in the (opposite) WKB regime k z L w ≳ 1. In section 2.7, we summarise and present master formulae for the calculation of the averaged momentum transfer ⟨∆p⟩. We conclude discussing results for ⟨∆p⟩ and pressure P NLO in sections 2.8 and 2.9 respectively.

Complete basis
The quantisation of modes in the presence of a background profile arises in many corners of physics. A very similar task appears for example in the quantisation of field theory in black hole spacetimes. We found the treatment in [71] particularly useful. In the simple example of eq. (5) above, the discussion is relevant for ϕ, which satisfies with a z−dependent mass term. To perform second quantisation we need to first find a convenient basis of solutions of this equation. Far away from the wall the solution must become a plane wave. One in general can choose a complete basis of 'right' and 'left' moving solutions, which are defined as follows 9 with k 0 > m and with k 0 >m and we take k z ,k z to be strictly positive 10 . The factor k z /k z is included in eq. (8) to ensure appropriate normalisation (see below, eq. (14)). In the limit of no domain wall the r L , r R (t L , t R ) coefficients are zero (one) and ϕ L , ϕ R correspond simply to the plane waves with ∓k z momenta. The momentum along z is not conserved across the wall but asymptotically far from it becomes constant and fixed by the relations In general, we need to solve the equations of motion to find the expression of the coefficients r L,R , t L,R . Consequently, they will depend on the explicit form of the mass variation m 2 ϕ (z). However, here it will be sufficient to consider the step wall ansatz for the mass m 2 ϕ (z) using the Heaviside Theta function. The form of the coefficients for the scalar case under consideration can be obtained via matching at the origin z = 0, where the step wall lies, and take the form 9 Recall that the index n designates 0, 1, 2 and not the z direction. 10 As is well known, the basis formed by just ϕ R,k but allowing k z to take both signs (and k z = sign(k)( √ k 2 − ∆m 2 ) is also complete but not orthogonal and therefore less convenient. For example, the algebra of creation and annihilation operators would be more complicated.
These expressions are specific to the step-wall assumption, however the general treatment that we present here will hold for general r k , t k coefficients and could be easily adapted to a smooth wall case. Modes with m < k 0 <m decay exponentially on the right of the wall and are automatically included as right-movers. For these, k is purely imaginary with magnitude In a similar fashion, for the left moving solution we find and we explicitly note the condition k 0 >m, to avoid the inclusion of solutions growing exponentially at infinity. The left and right moving modes are orthonormal in the sense that Computing the integral in eq. (14) -among many other things -in the step function case requires the identity and its complex conjugate (which gives the integral from 0 to ∞). In eq. (14) the principle value (PV) pieces vanish as soon as we specify the relation betweenk z and k z , i.e.k 2 z = k 2 z + m 2 −m 2 . Notice these relations only hold because we can discard terms proportional to δ(k z + q z ) due to the strictly positive definition of k z , q z in our definition. If explicitly computing things like the Hamiltonian and operator algebra (see next subsection) it is also useful to know the other inner products: Finally, we would like to comment that in general a discrete number of bound states may appear in the spectrum, in addition to the scattering states studied above, if the function m 2 ϕ (z) is non-monotonic and has minima in the vicinity of the domain wall. These are of the form ϕ b ∝ e −ik n xn χ b (z), with χ b exponentially decaying for |z| → ∞ and should be included in the upcoming expansion eq. (17).

Quantisation
Now that we have a complete orthonormal basis of eigenstates in the presence of the wall, we can proceed to quantise the theory. The field ϕ can be expanded in the where d 3 k ≡ dk z d 2 k ⊥ , we defined E ≡ k 2 0 − k 2 ⊥ and k z is varied between [0, ∞). We choose to label states by their quantum numbers outside the wall 12 . Note we have trivially extended the definition of the left moving modes to the region E <m for convenience.
Using eqs. (14) and (16), one can show that where π ≡ ∂ t ϕ. Promoting Poisson brackets of ϕ and its conjugate momentum π to canonical commutation relations gives the familiar commutation algebra We can define two types of states which should be thought of as independent external states in any process. The space of physical states is thus the Fock space defined by arbitrary powers of a † R,k and a † L,k acting on the vacuum.

Out-going eigenstates of momenta
In the previous subsections we chose to quantise the orthonormal basis {ϕ R,k , ϕ L,k } and defined associated one-particle states |k R ⟩ and |k L ⟩. As we explain in more detail in appendix A by the use of wave-packets, these should be thought of as describing incoming particles with definite z−momenta k z and −k z respectively at t → −∞, but at t → +∞ they correspond to a superposition between a transmitted and reflected particle. As a consequence, the functions ϕ R,L are eigenstates of momenta only at t → −∞. They are well-suited for processes with asymptotic in-state ϕ particles.
On the other hand, in this work we will be interested in the momentum transfer to the wall, so it is more convenient to have a ϕ particle emitted as an asymptotic out-state with well-defined momentum at t → ∞. A complete orthonormal basis of such late-time eigenstates of momentum is given by where in the last equalities we related them to the basis of section 2.1. We emphasise again that in our notation k z ,k z > 0 always. ϕ out L,k and ϕ out R,k should be thought of as describing an outgoing final state particle with −k z and +k z momentum respectively. Remember that the function χ L,k vanishes for k z < ∆m and the corresponding Θ functions are implicit. At t → −∞ they are both superpositions of incoming particles from z = ±∞ and do not have well defined momentum. In practice we need to calculate the amplitudes with ζ L,R = χ * R,L wave functions. At the level of states we have where we explicitly remind ourselves that when 0 < k z < ∆m the left mover state does not exist. The different asymptotic states are illustrated in the fig. 4. We emphasise that which basis is used to quantise the theory is completely arbitrary and should be chosen according to the problem at hand. In our present paper we will consider only outgoing ϕ particles so that the basis {ϕ out R,k , ϕ out L,k } is actually more convenient. Either way the results will be the same. From now on we drop the label 'out' and we will refer to R (L) emission meaning using the mode functions {ζ R , ζ L }, if not stated otherwise.

Amplitudes
We now finally turn to computing the amplitude for the process ψ → ψϕ in the background of a domain wall. We have not discussed the quantisation of ψ since it does not feel the wall directly and there are no complications with respect to the standard theory. In the previous sections we argued that there are two processes we have to consider separately: the emission of a left and right moving ϕ particle, with respective wavefunctions ζ L (ζ R ). Having quantised the free theory, the treatment of perturbative interactions proceeds as standard, by defining an S-matrix in terms of the interaction Hamiltonian S = T exp −i d 4 xH Int where T here denotes time ordering, we have the amplitude of interest with I = L, R , Figure 4: Summary of asymptotic external states with definite 4−momentum, to be used in calculating any process in the background of a domain wall (or any localised z−dependent potential). The upper panels represent in-state particles incoming from z = −∞ and z = ∞. In the plane wave limit they correspond to states |k R ⟩ and |k L ⟩ with wavefunctions eqs. (7) and (8) respectively. The lower panels represent out-state particles travelling out towards z = −∞ and z = ∞. They correspond to states eqs. (24) and (25) with wavefunctions eqs. (22) and (23) respectively.
where p and q stand for the initial and final one particle states for the ψ field and their respective 4−momenta, and the last equality is up to leading order in perturbation theory (tree level). Notice we have defined the matrix element M as closely as possible to standard theory. Of course we cannot extract a z−momentum conserving delta function but rather M still contains the integral over z.
For the theory of scalars of eq. (5), we have H Int = −iyψ 2 (x)ϕ(x) and we can now proceed to explicitly computing amplitudes. In the case of a ζ L mode emission (where the emitted scalar has −k z momentum), the amplitude takes the form Instead, in the case of ζ R mode emission (where the emitted scalar has +k z momentum): where the square root factor comes from the normalisation condition in eq. (8).
We emphasise that to compute the total friction from ψ → ψϕ we must sum the contributions from both processes. Then we can compute the amplitude squared for the emission of a right/left mover, under the assumption that y = const., we obtain where for L emission we distinguished between the two branches corresponding tõ k z purely real and imaginary, and used eq. (12) to simplify in the latter case. The physical interpretation of left and right movers is straightforward. We can see the right movers as a particle emitted and then transmitted inside the bubble wall, while the left movers is a particle first emitted and then reflected by the wall outside.

Phase Space integration
We can now compute the averaged exchanged momentum, ⟨∆p⟩, times the amplitude for a ψ particle to emit a ϕ and integrate over the phase space of the final particles. ⟨∆p⟩(p), which is a priori a function of the four-momentum p of the incoming ψ, thus represents an average of the momentum exchanged over one interaction. Then, from ⟨∆p⟩ we will be able to compute the total friction thereof, by adding separately contributions from left and right movers with their respective slightly different allowed phase spaces and integrating over the incoming flux.
Having in mind 1 → 2 processes and the fact that we have been labelling the mode functions as a function of k z (> 0) we will parameterise the kinematic of the process and the following phase space integration in term of the variables (k z , k ⊥ ). Then wherek z is defined in eq. (9), q z = q 2 0 − k 2 ⊥ − m 2 ψ , while from the energy conservation we can write q 0 = p 0 − k 0 . Note we have used the cylindrical symmetry of the set up to make the first spatial component of the momentum zero. The averaged exchange of momentum ⟨∆p⟩ is given by the sum of the left and the right moving contributions, being independent external states in any process, where dP ψ→ψϕ I is the differential probability. We emphasise that the two phase spaces, respectively in the first and the second terms on the RHS of eq. (32), will be different in their lower limits on the k z integrals. The second term, the left-mover, contains also the contribution from modes which are exponentially decaying inside the wall. For the terms on the RHS of eq. (32), the right mover, we obtain where I = R, L and [...] q z =±q z k is intended to be the sum over q z = ±q z k . The contribution with q z = −q z k corresponds to reflection of the incoming particle ψ (see for details defined in Appendix A.2), and we believe it was missing in the previous literature. Of course in the ultra-relativistic regime this should be highly suppressed. However the contribution could be relevant for amplitudes which are sensitive to the width of the wall even in the regime m ≪ p 0 ≪ L −1 w . Notice that although the amplitude is suppressed, the sign change in ∆p z R means that for a given q 0 and a given interaction, the q z < 0 process contributes more to momentum exchange, as the exchange of momentum is larger.
The limits of integration of eq. (33) are different between R and L emission and are found ensuring the energy conservation alongside the reality of the q z momentum, obtaining for the (R) modes For what concern the limits of integration for the (L) modes, the only difference is in the lower limit of k z integration, where we need also to consider exponentially decaying modes inside the wall, that is Following this discussion in general there will be four contributions: We explicitly checked that in all the regimes we are interested the contributions with q z < 0 are largely subdominant and we will ignore them completely in the rest of this paper.

Emission in the WKB regime
So far we have been treating the bubble wall as a step function. This is a good approximation if the z momentum of the emitted particle is less than the inverse width of the wall, i.e. k z < L −1 w , where L w is the typical width of the wall i.e. the region where the mass varies significantly. How can we proceed if the particles momentum becomes comparable or larger than the width of the wall?
First of all, if we know the shape of the potential exactly we can solve for the left and right mover solutions as we have done above for the step-wall, and proceed with these functions in precisely the same way as before. In principle this can be done numerically, if the potential is known. However even in the case when we do not know the exact shape of the potential we can still obtain reliable results. Let us consider a particle hitting the wall with the momentum k z ≫ L −1 w much larger than the parametric wall width L w . At those large energies, the step wall approximation is not valid anymore, reflection will be suppressed and WKB approximation becomes applicable [63,65,66]. We can use approximate solutions to the χ R,L (ζ R,L ) basis functions using WKB approximation, and then proceed in the same way as in the section 2.4. In practice this means that we need to separate the phase space into two regions and apply two different approaches: In the WKB regime there reflection is suppressed and the amplitude can be schematically written as follows Calculation of this integral requires the knowledge of the functions m i (z). However in the background of the domain wall this functions are changing only in the vicinity of the wall and outside quickly reach the asymptotic constant values. This means we can split the amplitude into two pieces where the assumption is that domain wall is essentially present only between z ∈ [0, L w ]. In WKB regime all the momenta of the particles are much larger than inverse width of the wall p z L w ≫ 1 so the overall modification of momenta p(+∞) − p(−∞) ≪ p(∞) is much less than its absolute value (if the wall is not too thick m(z)L ≲ 1), this is why we have approximated p/p(z) → 1. Similarly ∆p(+∞)− ∆p(−∞) ≪ ∆p(∞), then from basic properties of Fourier transformations the amplitude with ∆p max The physics behind this relation is very simple: the wall of the width L w can lead to the momentum loss ∆p z at most L −1 w . This is expected since the processes with ∆p z ≫ L −1 w happen at distances ∼ ∆p −1 z much shorter than the typical wall width, however at such small distances we recover Lorentz symmetry along the z direction and transition radiation must be forbidden (we checked these statements for various wall ansatzes in appendix H). From these arguments we can see that independently of the wall ansatz the particle emission will be dominated by the region ∆p z L w ≲ 1. Then we can approximate the amplitude as follows Performing the z integrals for the first two terms is trivial and we get The last term scales very roughly as we can see it will be suppressed by the condition ∆pL w ≲ 1. Thus we arrive at the Bodeker-Moore formula [63] for reduced matrix element Now we can take this formula and perform the phase space integration. However we would like to emphasise simple but important detail about the calculations. Since we have ignored the part inside the wall there is no guarantee that the matrix element will be suppressed in the region with ∆p z L w ≫ 1. In all of our calculations we always: • impose ∆p z L w < 1 − Fourier decomposition properties • verify that M inside ≪ M outside − applicability of BM approximation (we anticipate here that satisfying this inequality turns out to be non-trivial for the longitudinal vector bosons).
Finally, from this discussion it is clear that we should not worry about left emission in the WKB approximations since for left movers with k z > L −1 w the total loss of momenta ∆p z = p z − q z + k z > L −1 w , thus these processes must be strongly suppressed and we can safely ignore them.
Scalars example Let us apply this very generic discussion to the case of scalar radiation. Then the matrix element will be given by for the contribution outside of the wall. The contribution inside the wall (which we ignore) scales roughly as which is always less than one. We conclude that the neglected corrections coming from inside of the wall contributions are indeed negligible for scalars.

Procedure for the momentum transfer calculation: summary
In this section we summarise the previous discussion and results and give a concise prescription for the momentum transfer calculation. There are three types of contributions: where the first two correspond to the L, R emission in the step wall approximation and the last one to the R emission in WKB regime. These momentum transfers are given by the following phase space integrals: where the limits k z max , k 2 ⊥,max were defined in eq. (34) but we repeat them here for the reader's convenience, and also recall that ∆m 2 ≡m 2 − m 2 . M L , M R are the amplitudes for the process calculated using the step wall ansatz (see section 2.4) and M wkb red. is the amplitude in the WKB approximation, without the contribution inside the wall, calculated following the discussion in section 2.6. Note the presence of various Theta functions imposing cuts on phase space. For the ⟨∆p step L,R ⟩ cases these ensure that k z < L −1 w , i.e. step wall approximation is valid. For the WKB case the first step function Θ(k z − L −1 w ) comes from requirement of validity of WKB, i.e. large momentum compared to the wall width. The second step function Θ L −1 w − (p z − q z −k z ) is needed so that momentum transfer never surpasses the wall width (see discussion Table 1: We report here all phase space integration limits for each emission contribution after explicitly taking into account constraints imposed by Theta functions in eq. (46). The WKB regime is divided in two regions having imposed the constraint ∆p z L w < 1, as explained in section 2.7. All definitions Max & Min functions are included to also capture low p 0 or large L −1 w .
near eq. (39)). In practice the phase space integration was implemented as is shown in Table 1. All theta functions of the form Θ(±(L −1 w − k z )) are easily implemented cutting the k z integration appropriately for step and WKB regions. For the latter, the extra constraint p z − q z −k z < L −1 w amounts to cutting also the k 2 ⊥ integration as follows where To derive this, note that if p z −k z − L −1 w < 0, the positivity of q z means we are done with no extra condition. If instead p z −k z − L −1 w > 0, squaring the constraint and solving for k 2 ⊥ gives eq. (49).

Momentum transfer from scalar emission
Following the procedure outlined in the previous section 2.7 we can start the calculation of the momentum transfer from scalar emission ψ → ψϕ in our toy model eq. (5). As explained above, the total averaged momentum transfer is to be computed in three separate contributions ⟨∆p step L,R ⟩, ⟨∆p wkb R ⟩. Numerical integration is relatively straightforward and representative results are shown in fig. 5. Approximate analytical expressions can be derived, with some details given in appendix E.1, and are presented when useful. There are several parameters in the problem so that which of the three contributions dominate is a function of different hierarchies. Generically we find that at large energies p 0 → ∞ the WKB contribution always dominates and falls off as 1/p 0 . Computing the phase space integrals in this asymptotic limit, we obtain a very good approximation On the other hand, at low and intermediate relativistic energies, the step function contributions typically dominate. This behaviour is amplified in two independent regimes. For very thin wall L w → 0 the WKB contribution naturally only turns on at higher energies p 0 ≫ L −1 w , with ⟨∆p step R ⟩ temporarily dominating in its place (see bottom-right panel of fig. 5). Nonetheless, the trend is still reasonably approximated by interpolating backwards the asymptotic result of eq. (51).
More interesting is the case when the initial mass of ϕ is very light m ≪m, m ψ . Each contribution to momentum transfer becomes constant for an inter-relativistic plateau as can be observed in the leftmost panel of fig. 5. Moreover it is actually ⟨∆p step L ⟩ that dominates, with a value Further details can be found in appendix E.1.

Pressure from scalar emission
We now come to finally computing the pressure induced on the bubble wall from scalar emission calculated above. We will discover that in the case of m ≪m, this can be sizable. In order to find the pressure acting on the bubble wall we need to perform the integration over the flux of incoming particles We will focus on the cases displaying a constant behaviour for the exchange of momentum, that we call plateau for p 0 <mm ψ /m. If the average momentum transfer is a constant the integration is simple and we find in the ultrarelativistic case p z /p 0 → 1. To go from the second equality to the third, we used that in the wall frame, the plasma frame thermal distribution of ψ, n ψ (T ) is boosted. Summing all of the contributions and using eq. (52), we find At the highest scales the WKB contribution always dominates and falls as p −1 0 (while both step wall contributions fall as p −2 0 ). In the limit L w → 0, R emission also falls off as p −1 0 . Most interestingly, for large hierarchy m ψ /m, ⟨∆p⟩ is constant until p 0 ≲mm ψ /m.
We observe that this contribution becomes relevant and dominates over the LO contribution if y 2 8π 2 m ψ m > 1. We remind however the reader that this result for the pressure was obtained for a particular choice of interactions in the Lagrangian eq. (5) and parameters. We have chosen these couplings in order to mimic as much as possible the vector radiation from the current to be discussed in the next section. Whether the pressure contribution in eq. (55) can be phenomenologically relevant is left for further studies.

Spontaneously broken gauge theories
We now proceed to the phase transitions related to the spontaneous breaking of the gauge symmetries and the emission of the vector bosons. The procedure will in essence be exactly the same as what we presented for the case of the scalar emission. We will quantise the theory of a gauge field in the background of a domain wall interpolating between a symmetric and broken phase. As should be expected, the extra difficulty will involve dealing with gauge-fixing and the change of degrees of freedom due to the spatially-dependent rearrangement of the vacuum.
For simplicity, we will consider the Abelian Higgs model of a charged complex scalar H, whose potential V (H) is responsible for spontaneous breaking of the U (1) gauge symmetry. A second scalar field ψ charged under the same U (1) will play of the role of matter; its potential is trivial. The Lagrangian is where we are using the convention D µ ψ = (∂ µ + igA µ )ψ and A µ is the vector gauge field. We will in general not need commit to a specific potential but will simply assume that it has two minima at √ 2|H| = v,ṽ, where v = 0 corresponds to the symmetric phase. We will quantise the theory in the background of a domain wall √ 2⟨|H|⟩ = v(z). We will also be interested, both as a computational tool and as a phenomenological case in its own right, in imagining a distorted or more general class of potential with non-zero v <ṽ. We will call this scenario a broken to broken phase transition, in opposition to the usual case of symmetric to broken phase transitions.
To work with the theory described by eq. (56), one has to make two independent choices: what field coordinates to use for H, such as Cartesian or polar, and what gauge to impose. The value of each choice is determined by the particular application. Much of the following pages will be dedicated to arguing for the most convenient choices for our application.
If we are interested in studying the geometry of the vacuum manifold, polar coordinates H = 1 √ 2 (h + v)e i χ v are most convenient. The potential depends only on the modulus. In the symmetric phase however this coordinate choice is singular. On the other, Cartesian coordinates are well-defined everywhere where we have expanded around the background solution v = v(z). The Lagrangian for the Higgs and gauge fields becomes up to quadratic terms, where ∂ i ≡ ∂/∂h i , so that the last two terms are the zdependent mass terms of h 1,2 .
At a non-zero minimum of the potential, h 2 becomes massless and is the wouldbe Goldstone boson. As always for gauge theories, when v ̸ = 0 a mixing term appears between this goldstone boson and the gauge boson. However, in the context of a varying background, there is also an extra mixing proportional to ∂ z v. When v is a constant, the mixing can be eliminated completely while also gauge fixing by adding the so called R ξ gauge term and integrating by parts. For v = v(z), adding this same (now z−dependent) gaugefixing term does not get rid of mixing entirely but localises it to the region of the wall

Particle content in the asymptotic regions
We briefly remind ourselves of the spectrum of the theory in the asymptotic regions v = 0 andṽ at z → −∞ and z → ∞ respectively, before discussing the full interpolating space.
Symmetric phase: The theory eq. (60) around the symmetric point minimum v = 0 describes two scalars h 1,2, with equal mass by symmetry The gauge-fixing affects only the Maxwell equations of motion for the massless vector A µ . Whatever the value of ξ, we identify two physical degrees of freedom with polarisation vectors given by for k µ = (k 0 , k ⊥ , 0, k z ). The only subtlety is one has to impose constraints on the physical states, the Gupta-Bleuler condition.
Broken phase(s): At a symmetry breaking minimum ∂ 2 2 V (ṽ) = 0 and h 2 describes the would-be NGB. A particularly convenient choice is the 'unitary gauge', corresponding to ξ → ∞ in which the the NGB decouples completely making manifest the spectrum. We are left with a single massive scalar (the Higgs) h with mass squared equal to ∂ 2 1 V (ṽ) and a massive vector boson A µ with mass m ≡ gṽ , and satisfying the Proca equation, which reduced to a Klein-Gordon equation for each component of A µ supplemented by the Lorentz condition.
Solving this is straightforward and one adds to the transverse polarisations of eq. (62) a third longitudinal one parallel to 3−momentum

Global degrees of freedom
Here will analyse the fields defined over the entire region and identify the appropriate global modes to quantise, where by global we simply mean they are good across the wall.
In principle, one could choose a convenient value of ξ in eq. (60) and push ahead with quantisation. However, we would have to deal with mixing when solving for the mode functions as well as take care to impose a Gupta-Bleuler like condition on the physical states. Luckily, we will argue that even when asymptotically approaching the symmetric point as z → −∞, it is possible to work with unitary gauge ξ → ∞ with impunity. This approach was already made at the classical level in ref. [70], and we will re-derive and tweak some of their results using a slightly different language, before quantising.
In unitary gauge the h 2 degree of freedom decouples and the theory becomes and the equations of motion for eq. (58) reduce to just two uncoupled equations The first one is the equation of motion for the physical Higgs boson h and we will not have anymore to say about it. The second will be the focus of our attention. While v(z) > 0, the theory is always in a broken phase and A µ describes a massive vector. Unitary gauge is then manifestly a valid choice. Note that the usual transversality condition for massive vector bosons in this case becomes which, in the presence of the bubble, generalises the standard Lorentz condition ∂ µ A µ = 0 for massive electrodynamics. The constraint above ensures this vector field has three polarisation degrees of freedom. Subbing this back into eq. (69) we get The general Fourier mode can be written as where l runs over three indices, a l are some constant Fourier coefficients and the functions χ µ l,k 0 ,k ⊥ (z) have to be found by solving eq. (72).
τ polarisations: It is useful to define what we call τ −polarisations by the condition A z = 0, since for these we recover the Lorentz condition ∂ µ A µ = 0, which in Fourier space reduces to and has solutions in terms of two constant vectors where ϵ µ τ 1 =(0, 0, 1, 0), and the equations of motion eq. (72) become the Klein-Gordon-like where we remind the reader of the definition E = k 2 0 − k 2 ⊥ . Note that ϵ µ τ 1 is one of the standard transverse polarisations, but ϵ τ 2 is not, since it has non-zero time component, and is not orthogonal to three momentum. The wave equation to solve across the wall for τ d.o.f. is thus identical to the scalar case studied in section 2.
λ polarisation: It remains to solve for the remaining degree of freedom with A z ̸ = 0. The wave equation to solve for A z is obtained by setting µ = z in eq. (72). We will analyse it more in the next section but we note that it is significantly more complicated than what we found for the τ d.o.f.. Whatever the solution, the other components of the vector are fixed. Requiring orthogonality with χ µ τ implies the form χ µ λ (z) = (−ik n α(z), χ z λ ) where n = 0, ⊥. The generalised Lorentz condition in eq. (71) immediately leads to the relation Plugging this back into the eq. (72) we obtain the equation in terms of the χ z only We can get rid of the linear in derivative term if we introduce a new function λ(z) and eq. (79) becomes Schrodinger-like with effective potential Note that the solutions in terms of λ function unlike χ z λ will satisfy the usual orthogonality relations It is easy to prove − and we do so explicitly in appendix D − that for an interpolating solution v(z) of a completely arbitrary Higgs potential V , we have (v ′ /v) 2 , v ′′ /v → m 2 h,s as z → −∞ so that U λ (z) is always finite even if v = 0 and more precisely We see that λ is the perfect cross-wall field. It interpolates between one of the massive (Higgs) degrees of freedom on the symmetric v = 0 side z → −∞ and a third component of the massive vector in the broken region z → ∞. If instead v ̸ = 0, λ simply interpolates between the different mass vectors. Let us look at the A µ vector formed by the λ field. In general we need to solve the equations of motion, however even without a complete solution z component of eq. (69) forces the relations on-shell: So that the total vector can be written as a complete derivative plus a term subleading in energy. This expansion turns out to be very useful in calculating the amplitudes for the physical processes. Far from the wall when v → const we can introduce the polarisation vectors ϵ µ λ such that We would like to emphasise that these λ and τ differ from the conventional transverse and longitudinal polarisations. In the case of no domain wall all three polarisations satisfy the same equations of motion and one can use any linear combination of them as a basis to decompose the vector field. Outside of the wall we can relate τ, λ to the longitudinal and transverse polarisations and two basis of polarisations are related by the rotation matrix In the case of a very large k z , E ≫ k ⊥ , m the mixing angle between τ, λ transverse and longitudinals scales as m/E. We can see that two basis of polarisations are exactly the same for the case k ⊥ = 0. This is expected since for this configuration of momenta T 1 , T 2 polarisations have zero component in z direction. Using the unbroken part of the Lorentz symmetry (boosts in x − y direction) we can obtain the polarisations for generic momenta, which indeed agrees with τ, λ basis derived before. Our goal in this paper is to calculate the total pressure acting on the domain wall and for this calculation we need to sum the contributions from of all polarisations. At this point we can perform all of the calculations in the τ, λ basis, without even reporting the results for T, L polarisations.

v → 0 limit
All of the previous discussion applied most manifestly for the case when the vev of the symmetry breaking field is v ̸ = 0. What will happen in the case when the domain wall separates the vacua where in one of the the gauge symmetry is unbroken? We have seen in the previous section that the potential of the λ mode has a property that in v → 0, U λ (z) → m 2 h,s , which together with our expectations from the Higgs phenomena hints that λ mode on the unbroken side should correspond to the would be Goldstone boson To understand this matching better let us look at the χ µ λ vector in the limit v → 0 where we have used that λ becomes a plane wave far from the wall and v ′ /v → m h,s . Note that the factor (−im h,s + k z )/E is a pure phase if the λ dof is on shell. Let us see whether we can build exactly the same vector but from the Goldstone fields h 2 . Indeed if we consider the vector So we can see, comparing with eq. (90), that the two vectors χ µ λ and ∂ µ (h 2 /gv) are exactly the same apart from the constant phase factor, so indeed λ field in the z → −∞ limit corresponds to the Goldstone boson.
What about starting from a finite value and taking v → 0? For concreteness let us consider the potential First of all the potential has a cusp at |h| = 0 so the limit v → 0 becomes discontinuous. This can be seen also by the form of U λ (z). As v becomes smaller a longer finite plateau develops in the potential with value ≈ m 2 h,s , as is shown in fig. 6. No matter how small v eventually the potential turns down and asymptotes to g 2 v 2 as is to be expected. Thus for any finite though tiny v the asymptotic states at z → −∞ are those of a massive vector.

The step wall case
In order to proceed further we need to solve the equations of motion. In general it is a complicated problem depending on the shape of the potential. One needs to find the solitonic solution v(z) connecting false and true vacuum and later the wavemodes describing perturbations of each field on this background. In the particular case of the domain wall the solutions were found in [70] in terms of hypergeometric functions. In this paper we will consider an even simpler case, namely a step function ansatz for the wall. This approximation will of course be valid only if the momentum of the particle during the passage is (much) less than inverse width of the wall k,k ≪ L −1 w . Typically, this width is controlled by the mass of the Higgs L −1 w ≈ m h . The solution of the equations of motion can be written down on each side immediately and the only challenge becomes deriving and implementing matching conditions. In this section we report the matching conditions for τ and λ polarisations and write down the corresponding wave functions. We will do so first for the broken to broken case v > 0. As an explicit example we can imagine distorting eq. (93) to We then comment on the v → 0 limit, which is straightforward for τ degrees of freedom but more delicate for λ.

(τ ) polarisations
For the τ polarisations, we showed in section 3.2 that the equations of motion are exactly the same as for the scalar field (□A µ (τ ) = −g 2 v 2 A µ (τ ) ) and so for the step wall the matching conditions become: with i = 1, 2. The reflection and transmission coefficients are thus the same as for scalars and the wave functions become where Taking the v → 0 limit for these degrees of freedom is simple and we approach the v = 0 case smoothly.

(λ) polarisation
The λ modes require more work. Again we first focus on the broken to broken case of v > 0. Matching conditions are easy to derive by integrating the wave equation for λ once and twice respectively (most easily done at the level of eq. (79)). These are which allows us to write down the expressions for (in-state) 'left' and 'right' movers: Notice that in the relativistic limit so that λ maintains a finite reflection probability as long as the step function is a valid approximation, as was pointed out in [68]. Interestingly we can see that r λ k → 1 in the limit v → 0, i.e. the wall becomes completely non-transparent for the λ polarisations in this limit (step wall ∪ v → 0 ∪ k z ≫ m,m). This in-penetrability of the wall deserves some discussion. It becomes more clear if we consider the explicit form of U λ (z) for the case of the tanh profile of the wall (see eq. (94)). As was mentioned in the section 3.2.1 and is sketched in fig. 6, in the limit of v → 0, U λ (z) develops a growing plateau with a height ≈ L −2 w ∼ m 2 h,s and width L plateau ∼ L w logṽ/v. In the step function approximation both these scales are sent to infinity. For v = 0 exactly then there are no oscillating λ R,k modes at all in the step approximation, while λ L,k are completely reflected.
For v tiny but non-zero the potential eventually does instead relax to m 2 and both oscillating λ L,R solutions exist, but are constrained to live on opposite sides of the wall 13 . 13 One might question the validity of the step wall approximation when the potential function U λ (z) has a very long plateau. However, the solutions eqs. (101) and (102) capture exactly the qualitative behaviour described.
A comment on bound states: We have so far considered 'scattering state' solutions to the equations of motion, i.e. those which are plane waves far from the wall. What about bound states? In principle such states are possible for λ polarisation. Ref [70] found that there is one for the case when v = 0 and the potential satisfies some specific constraints. Examining the form of the potential for v > 0 in 6 it appears that a bound state might generically appear. The mass of these bound states is controlled by the scale L −1 w as is obvious by absence of the bound state in the step wall limit. One could in principle calculate in the WKB limit the amplitude for an incoming particle to excite this bound state. We leave this interesting exercise to future work.

Quantisation
Summarising the previous sections, we can expand the field into a complete basis of eigenmodes of the free theory in the background vev v(z), where I = R, L denote right and left movers, ℓ = τ 1 , τ 2 , λ sums over different polarisations. The wave modes χ µ (z) are in general constructed via where∂ µ ≡ (−ik n , ∂ z ), with the explicit form of scalar fields χ τ i ,I,k (z) and λ I,k (z) obtained by solving the respective Schrodinger-like wave eqs. (77) and (81) with appropriate R, L-mover boundary conditions. In the step wall approximation for the vev v(z), these are given analytically in eqs. (97), (98), (101) and (102). In complete analogy to the case of fundamental scalars 14 , the modes χ µ ℓ,I,k should be thought of as describing incoming (early time) eigenstates of momentum (particles) in the plane wave limit with physical z−momentum k z > 0 and −k z for R and L respectively. Modes describing outgoing (late time) eigenstates of momenta ζ µ ℓ,I,k are instead obtained via where we have dropped k labels to not clutter the notation. Notice the switch in L, R -labels. Both sets of eigenmodes form a complete orthonormal basis and can 14 See section 2.3 and appendix A.
be used to expand the field operator in eq. (105). The associated Fourier operators carry in and out labels to emphasise that they create / annihilate in and out states in the S−matrix language Both satisfy the usual algebra eq. (19) upon quantisation.
Ward identity and current conservation: We now comment on current conservation in the case of spontaneously broken Lorentz symmetry. If the gauge symmetry is preserved, vector bosons can couple only to conserved currents. This is not the case when it is spontaneously broken, but we may still choose to consider coupling to a conserved current 15 . In the Lorentz invariant theory the statement of the current conservation can be expressed in terms of amplitudes. Given an arbitrary process with an external vector leg with momentum k µ , we have the following identity where the (4, J) label indicates full 4-momentum conservation and the process is mediated by the conserved current J µ , and ϵ µ k is the external particle's polarisation vector. This Ward identity implies that substituting the latter for the particle's momentum k µ makes the amplitude vanish.
In the presence of a domain wall in the z direction, the generalised matrix element M (3) as defined in eq. (26) includes an integral over z and the polarisation tensor is also a function thereof. The expression of conservation closest to eq. (112) is and f (z) is an arbitrary function.
To make this discussion more concrete we consider the coupling of the gauge field to the conserved current made out of ψ fields we introduced in eq. (56): Then the amplitude (defined in eq. (26)) corresponding to the emission of the (l, I) polarisation from the current will be equal to where as usual p, q are the initial and final momentum of the ψ particle. Note, modulo a numerical factor, the same expression will be valid for emission of the vector boson from an arbitrary conserved current (not necessary one made from scalars). The current conservation imposes that any interaction which can be written in the form has a vanishing matrix element. We see now the use of writing the polarisation vector for the λ d.o.f. as we did in eq. (109). The dangerous-looking first term is actually a total derivative and can be subtracted when computing amplitudes (see appendix B). We comment further on this in the next section as well as discuss the case of non-conserved current in appendix C.

Subtleties with WKB regime
Before we start computing the amplitudes of interest for our application we highlight some important subtleties related to the calculation in the WKB regime. As we have discussed in the section 2.6 our formulas are valid only if the contribution inside the wall can be ignored. Let us check whether this is a reasonable approximation for the vector emission. Let us look at the λ and τ cases separately For the τ polarisations we can estimate the contribution to the amplitude inside and outside of the wall and we find: We can see that similarly to the scalar case discussed in the section 2.6 the contribution inside the wall can be safely ignored.
• λ polarisation Now let us look at the λ polarisation and the interactions between the current and λ field. Using the expansion for χ µ λ field (see for example eq. (90)) we get: outside of the wall, when λ = e ikx this becomes see eq. (86). Let us consider the domain wall connecting the vacua with broken and restored gauge symmetry. In this case the (λ) polarisation will interact with the ψ particle only on the broken side. Then the amplitude originating from the integration outside of the wall will be equal to where we have kept only the leading term in energy in polarisation vector and simplified using the conservation of the current, p 2 = q 2 . We can see that this matrix element is growing with energy and is singular in the limitṽ → 0, which are very worrisome properties since the limitṽ → 0 corresponds to the no domain wall and therefore no transition radiation, i.e. M → 0! Let us look now the contribution coming from the integration inside the wall, for it we will use the interaction form of eq. (118) In the first integral there is a term ∝ ∂ z v(z), which upon integration will necessarily lead to the contribution to which is of the same size as the contribution outside of the wall. We see that the amplitude M wkb red. λ will definitely lead to the incorrect results, so how can we proceed? One possibility would be to take some ansatz for the domain wall and then perform full WKB calculation keeping the terms inside the wall, which will lead to correct results without bad properties of eqs. (121)−(120). However we can still make progress even without the knowledge of the shape of the wall using the following trick. By construction we have been focusing on the case where the current built out of ψ fields is conserved On the other hand λ mode can be written as a complete derivative plus term subleading in energy (eq. (85)) the part ∂ µ α cannot couple to conserved current this means that With this simplification we immediately see that all of the problems with λ polarisations are cured contribution inside the wall is suppressed and the matrix element is not growing with energy and is vanishing in the limitṽ → 0.
So far we have been focusing only on the case when the current made out of ψ fields is conserved on both sides of the wall. This is not the case generically and in particular for SM fermions where the Yukawa interactions will lead to the current non-conservation, so how one should proceed in that case? It turns out that with very minor modification very similar trick can be used we provide more details about it in the appendix C.

Transition radiation and pressure from vectors
We are now ready to calculate transition radiation and the resultant pressure from vector boson emission. We are working in the Abelian Higgs theory of eq. (56) and calculate the average momentum transfer during the radiation of a gauge boson from an incoming ψ particle of energy p 0 . We evaluate the amplitudes of interest in the next section, comment on the final state phase space and masses employed in section 4.2, and finally present our results in section 4.3.

Amplitudes
All the relevant amplitudes for the particle process ψ → ψA obtained from eq. (115) are reported here. These are τ and λ polarisation emission for left and right movers in the step wall and WKB regimes, M wkb red.
where the scattering coefficients r τ,λ k , t τ,λ k relevant for the amplitudes in the step wall regime are defined in sections 3.3.1 and 3.3.2 and the ∆p factors in denominator are defined in eq. (4). We presented amplitudes for v ̸ = 0. However, the symmetrybreaking transition case can be obtained smoothly at this level by sending v → 0. Note that in this limit. The discontinuity in asymptotic d.o.f. (and therefore masses) is hidden here inside the kinematic factors and are addressed in the following section.
For λ emission we used current conservation to simplify the computation of these amplitudes by subtracting the total derivative piece in the wavemode eq. (109), as explained in section 2.2. For M step this simplification does not change the final expression since it is exact (in the limit of a step wall). However, we emphasise again that it does for M wkb red. λ , which is an approximation as described in section 2.6, and the subtraction is necessary to be consistent with the approximations and avoid unphysical divergences, as explained in section 3.5.

Phase space integration for vector emission
In going from the amplitudes above to the averaged exchanged momentum ⟨∆p ℓ ⟩, where ℓ = τ, λ, we integrate over final state phase space following the prescriptions and kinematics summarised in section 2.7, using M step and M wkb red. in their respective regimes of validity. However, there are some important subtleties to discuss compared to the simple theory of scalars of section 2, particularly for a symmetric to broken transition. In this case, the mass of the vector (and therefore τ d.o.f.) in the old phase (z → −∞) is zero by gauge invariance since v = 0. As shown explicitly in appendix E.4, in principle we can get finite results working with m = 0 and integrating over the full phase space as long as the mass of the emitter is kept finite m ψ ̸ = 0. However, thermal corrections ought to be important. Strictly speaking, we should expect our calculation to break down for momenta that are too soft -to be defined precisely -where thermal field theory becomes important. For example, one might cut phase space integration below the thermal soft scale [66] | ⃗ k| 2 ∼ g 2 T 2 , which is roughly equivalent to using the following thermally-corrected asymptotic masses in all kinematics (symmetric to broken) Note however that ∼ gT is not the only scale possible 16 . The self energy for transverse vectors receives ('magnetic mass') thermal corrections only at two loops of parametric order ∼ g 2 T from charged matter [73]. Using this instead would slightly change the dependence of our results on the coupling g. However, since we work in the frame of the wall, the background plasma is boosted and it is likely that the usual thermal field theory results for propagating particles are distorted anyway. We use eq. (134) here, in line with literature, and leave the rigorous inclusion of finite temperature to a separate study. In eq. (134), m h,s is the mass of the Higgs d.o.f. in the symmetric phase (see eq. (84)) which will also be temperature dependent 17 . So now, for example, the integration limits for the k z integral for ⟨∆p λ,step R ⟩ in eq. (46) become explicitly It is worth stressing that for τ polarisation what appears in the amplitude and in the kinematics boundaries of the PS are always the masses m andm as defined in eq. (134). However, for the λ polarisation the coupling appearing in the amplitude is really the bare gv(→ 0) and does not receive thermal corrections, while in the kinematics and PS integration what appears is m andm as defined as in eq. (134). So eq. (133) still holds even at finite T . For broken to broken transitions the vector masses are, for both λ and τ fields In summary, ⟨∆p⟩ is computed as in section 2.7 with phase space integration limits in Table 1 and asymptotic masses defined here. In general there is a total of six contributions:

Pressure on the bubble wall
We now report and comment on our results for the average NLO momentum transfer ⟨∆p⟩ due to transition radiation from a ψ particle travelling across the wall and thereby compute the total pressure on the bubble wall. We show a break down of figs. 2 and 3 into all their contributions, as well as provide some analytical formulae. A comprehensive comparison (via numerical integration -see appendix E for some analytical evaluation of phase space integrals) of all the different parts in eq. (136) is shown in figs. 7 and 8, as a function of the energy of the incoming emitter particle, ψ. In the asymptotic p 0 → ∞ limit all contributions are constant. Their relative importance is displayed in fig. 9. We now discuss the two cases of interest v = 0 and v ̸ = 0 separately. 17 We do not discuss the explicit form of it since our results are largely insensitive to it. 18 When m h,s (T ) <m(T ) we are in line with our set up which assumed m <m, but if it is the other way around the lower limit of k z becomes imaginary. This is signalling the fact that modes with 0 <k < √ m 2 −m 2 are now exponentially decaying in the old phase (right) side of the wall. Then it is more appropriate to define the step wall regime according to k <k ≲ L −1 w and parameterise the PS integrals in terms of (k z , k 2 ⊥ ). The integration limits fork z for R−mover emission would bek z ∈ [0,k z max ], wherek z max = (p 0 − m ψ ) 2 −m 2 .  Symmetric to broken case (v = 0). We observe in fig. 7 that when the emitting particle is m ψ ≲ m the contribution from τ saturates quickly to the constant p 0 → ∞ value (top left in fig. 7), but if there is a hierarchy m ≪ m ψ , we observe an interrelativistic regime of logarithmic dependence on p 0 up until p 0 ∼mm ψ /m. This can be traced to a collinear log divergence of phase space integration in the limit m = 0 & p 0 → ∞ as explained in appendix E. This behaviour is not present in contributions from λ polarisation emission, which are insensitive to m ψ for relativistic p 0 , even when the symmetric side mass is set to zero m h,s = 0.
In the asymptotic p 0 → ∞ limit τ contributions depend significantly only on the ratio 19 m/m, which can be translated to T /ṽ. While individual λ contributions depend also on L −1 w their sum is constant. The total momentum transfer (summing over all contributions) can be fitted by the following expression We recall that to obtain this expression we cut the phase space integrals in the IR at energies ∼ gT . The expression in eq. (137) becomes valid for the energies of initial particle p 0 ≳ṽ T × m ψ , which in the case of massless emitter m ψ → 0 becomes p 0 ≳ṽ. The contribution of the longitudinal modes is sub-leading except perhaps for mild super-coolingṽ/T ∼few, and is equal approximately to An analytical form for the function c λ is given in eq. (212). 19 See for example the exact evaluation of the dominant contribution eq. (202).  Figure 9: Here is presented the same plot as in fig. 2, but this time disentangling all the contributions. Solid lines refer to τ polarisation, while dashed to λ. Red lines are the total sum, for each polarisation. Top: Symmetric → Broken. It is shown two cases with different wall width and it is worth noticing that while the total contribution for τ and λ does not change appreciably, the single contributions do, like R step wall versus WKB. Bottom: Broken → Broken. Here it happens the same as in the top panels, but, since in this case L step contribution from λ polarisation has to be taken into account, we see that, depending on the value of L w and the hierarchy between the different vevs, can be the most relevant contribution.
So far we were calculating the momentum transfer from individual collisions. In order to find the pressure acting on the bubble wall we need to perform the integration over the flux of incoming particles. This can be easily done in the thermal case since we know the distributions If the average momentum transfer is a constant the integration is simple and we find, since in the ultra-relativistic case p z /p 0 → 1, where n ψ is the density of emitters ψ defined in the plasma frame. Then, for the symmetric to broken transition we obtain the following expression for the pressure: where the th. stands for thermal.
Broken to broken case (v ̸ = 0). In fig. 8 we show the evolution of ⟨∆p⟩(p 0 ) for a broken to broken transition. We focus only on λ contributions since the ones from τ are essentially the same as in fig. 7 with suitable re-interpretation of what m,m mean (see section 4.2). Again the curves eventually saturate to a constant value but we highlight the strongly L w −dependent novel contribution from L−mover emission, which easily comes to dominate in the thin wall regime. As in fig. 3, we highlight that in the left panel we can clearly distinguish the inter-relativistic region where this last contribution develops a linear growth in p 0 . In fig. 9 we see the dependence of the saturation value of the averaged exchanged momentum (in the limit p 0 → ∞) on the ratio between vevs (lower panels). Broken to broken transitions were recently studied at leading order by [68] and it was found that reflection of the longitudinal vectors is efficient for the energies below L −1 w (inverse width of the wall). This in its turn lead to the pressure scaling as γ 2 w , as long as p 0 < L −1 w . We find that a reminiscent effect happens at NLO level, the main difference is that the momentum of the vector is not fixed by the speed of the bubble expansion and is always integrated over all possible values. We find that the momentum transfer is dominated by left-mover modes and for the values of the energies of incoming particle In the case of large hierarchy 0 < m ≪ m ψ , not only is the saturation point delayed, but a further slight distortion occurs at low p 0 , as shown in fig. 8. Once the energy of the initial particle becomes larger than ∼ L −1 w Max[1, m ψ /m], we have Momentum transfer stops growing and reaches the saturation value. Note that the maximal value of this pressure is controlled by L −1 w and not by the mass of the vector gṽ. This is related to the fact that at high energies Goldstone Boson equivalence theorem relates the longitudinal vectors to the Goldstone bosons, and the strength of their interaction with the bubble wall (Higgs field) is controlled by the mass of the Higgs (wall width). Consequently, in the case of broken to broken transition, there is an additional contribution to the pressure which scales as with an intermediate regime scaling as P λ ∝ γ 2 w for the values of boost factor γ w < (L w T ) −1 .

Summary
We conclude by summarising the main results of our work. We analyzed in details the phenomena of transition radiation in the presence of domain walls. We quantised from first principles scalar and vector theories on a translation-violating background and identified the correct asymptotic states. We split emission into soft and UV regimes and used a step wall and WKB approximation respectively to compute the desired matrix elements for transition radiation. Quantisation of vector field theories was naturally performed via the introduction of new degrees of freedom which do not coincide with the traditional transverse and longitudinal polarisation but are a convenient admixture. In this way we have resolved some puzzles regarding the inclusion of longitudinal polarisations in the calculation of transition radiation.
We applied these results to calculate the pressure experienced by the bubble wall during the ultra-relativistic expansion. For the phase transitions with spontaneous breaking of the gauge symmetries in the regime of strong supercooling we find the pressure which scales as and is dominated by the emission back out of the wall of transverse-like polarisations with momenta k z ∈ (0, ∆m). This result qualitatively agrees with the previous literature on the subject. For moderate ratioṽ/T ∼ few we find that the contribution from the longitudinal-like polarisations can lead to significant corrections. We provide an updated fitted formula for the total pressure in eq. (142). We also analyzed the pressure in the case of transition between two vacua with broken gauge symmetry. Interestingly we find in this case the contribution from the longitudinal-like polarisation can esily become dominant for thin walls, with the asymptotic value controlled by the inverse wall width P max ∝ γ w g 2 T 3 L −1 w . Moreover, we find a transient intermediate regime of P ∝ γ 2 w scaling for p 0 ≲ L −1 w . Our results make an advance in understanding the balance between bubble acceleration and friction which plays a crucial role in determining most phenomenological consequences of FOPTs as well as their detection prospect at upcoming gravitational wave detectors.
Future outlook: The work can be improved and generalised in several ways. An important remaining question is the inclusion of finite temperature effects in a robust first principles fashion (see for example the discussion in section 4.2). It is relatively easy, though cumbersome, to allow also for the emitting particle to feel the wall m ψ → m ψ (z). Though in the γ w → ∞ limit, any dependence on m ψ (z) should drop out, we saw how m ψ can distort the intermediate shape of P NLO (γ w ) and ultimately finding the equilibrium velocity v w will require a full knowledge of this curve. Similarly one could rigorously quantise fermionic fields that change mass across the wall.
A possibly important direction is the analysis of multiple vector emissions, particularly in regimes with large logs, possible IR enhancements and back-reaction effects coming from the overdensity of soft vector bosons around the wall (see discussion in [66]). Furhtermore it would be interesting to compare our wall-shapeindependent results with a full numerical calculation using a specific smooth wall ansatz (for example tanh). Finally, with some tweaks our expressions can be used to analyse pressure in qualitatively different types of FOPTs, such as spontaneous breaking of global symmetries, or even symmetry restoring transitions.
While our interest here was in friction, we emphasise that our set up is useful for rigorously computing any process / Feynman diagram 20 in the presence of background walls which are not treated as a perturbation. Just as an example, one can easily re-purpose our expressions to compute the number of particles of a given species produced from collisions between an expanding bubble and surrounding particles 21 . The spontaneous breaking of Lorentz/translational symmetry in the early universe results in a rich phenomenology which is only starting to be explored systematically.

A Wavepackets and asymptotic states
No wall: Let us recall a few things in the usual manifestly translation-invariant case (no wall background). For clarity we will first focus on 1 + 1 dimensions. For a scalar field theory, the field operator ψ(t, z) is interpreted as creating a particle at position z at time t = 0. The position space wavefunction of a state is thus given by its inner product with ⟨0|ψ. The wavefunction of a single particle eigenstate of momentum |k⟩ = √ 2k 0 a † k |0⟩ is, as expected, To derive formulae for observables such as scattering cross sections or emission probabilities involving realistic particles we have to crucially go through appropriately defined wavepackets that describe those asymptotic states, taking a limit of sharp momentum only at the end. A wavepacket state describing a particle with momentum peaked around p and localised in space is given by where f p (k z ) is like a sharp Gaussian in k z peaked at p z . Note there is no spacetime dependence in this expression (we are always in the Heisenberg picture). The limit to recover a momentum eigenmode is . However, the more appropriate limit used in the derivation of physical rates makes use of the normalisation condition above To see the localisation we can look at the wavefunction of the wavepacket state: At t = 0 this is a an oscillating function of z (with wavelength controlled by p) with a Gaussian envelope so that it is indeed localised at z = 0. As a function of time, the wavepacket moves in the direction of sign(p). Because each mode has a slightly different dispersion relation, the spatial width of the wavepacket tends to widen in time (dispersion) but this can be counteracted by making f p (k) a sharper Gaussian, taking an appropriate order of limits.

A.1 Asymptotic states in the wall background
In the presence of the wall we should still define asymptotic particle states as appropriate wavepackets to compute formulae for physical rates. A slight complication comes from the fact that the one particle states |k L,R ⟩ we quantise are not eigenstates of momentum. Moreover, in general the same state describes different types of particles in different regions of space 22 . Defining the asymptotic state carefully will get rid of any ambiguity.
We now consider an operator that feels the wall ϕ(t, ⃗ x) expanded in left and right mover modes as in eq. (17). As before, the action of this on the vacuum should be thought of as creating a particle localised at x µ = 0. Contracting a state with ⟨0|ϕ still gives the wavefunction understood in the usual sense. In fact with I = R, L. To gain physical intuition of the one particle states |k I ⟩, consider constructing wavepackets by superimposing exclusively right (left) movers: where again we focus on 1 + 1 dimensions and where the lower limit is 0 (∆m) for R (L) movers respectively. The 'in' labels will become clear shortly. Their wavefunctions are respectively Ignoring the slight dispersion mentioned at the end of the previous subsection, the time evolution of eq. (152) for I = R(L) describes an isolated localised wavepacket travelling in towards the wall from z < 0 (z > 0) when t < 0. This wavemode scatters off the wall and for t > 0 splits into reflected and transmitted wavemodes travelling in opposite directions. Thus, |ϕ in R,p ⟩ (|ϕ in R,p ⟩) is a good asymptotic state for an incoming particle with positive (negative) z−momentum. It cannot however be used as an asymptotic state for any single outgoing asymptotic particle since at late times it describes a superposition.
So what about single localised outgoing particles? Clearly, when the wave equation is reduced to Schrodinger-like form with a real potential, these can be obtained by complex conjugation of the spacial part of the wavefunction. In other words we require states |Φ out I,p ⟩ such that In time, these corresponds to two waves coming from opposite sides of the wall, hitting the wall around t ≈ 0 and interfering in just the right way so that at late time there is only one wavemode travelling towards z → −∞ or z → ∞ respectively. Notice the swap in labels. We adopt the convention that the right-mover R (leftmover L) label always denotes a particle of positive (negative) z−momentum. It is an easy exercise to write the ζ R,L in terms of a linear composition of the complete basis χ R,L as explicitly done in eqs. (22) and (23). The appropriate wavepacket for an outgoing state is thus where |k out I ⟩ are the one-particle states with wavefunctions ζ I,k (z) and are related to the |k I ⟩ by where the reflection and transmission coefficients were defined in general for scalar d.o.f. in section 2.1, and given explicit form in the case of a step wall for fundamental scalars also in section 2.1 and for different vector polarisations in section 3.3.
Reflection and transmission probabilites: As a consistency check we can compute reflection and transmission probabilities in this language. Focusing on the identity part of the S matrix S = 1 + iT , the amplitude for a particle incoming from z < 0 to reflect and transmit should be respectively where the temporary label (3) on M emphasises only three of 4−momentum are conserved. in the the last step we take the peaked momentum limit of eq. (148). For the particular case of a 1 → 2 process as discussed in this work, we have derivative must give zero matrix element,

C WKB regime in the case of current non-conservation
In the main text we focus only on the transition radiation from the conserved current. How we can perform a similar calculation in the case when the current is not conserved? Let us consider the following example with scalar fields The charges under the gauged U (1) symmetry are as follows: Q U (1) (H) = 1, Q U (1) (ϕ) = −1/2. In the section 3.5, in order to perform the WKB calculation and get rid off interactions which can potentially lead to the divergences we have used the current conservation equations to modify the expression for the matrix elements. In the case of the system in eq. (173) the divergence of the current becomes equal to: The interaction between the λ polarisation and J µ ϕ and can be written as follows: We can see that on top of the term λJ z present in the conserved current case, there is an additional interaction. However this interaction is not growing with energy, and in the limit v(z) → 0, it is finite (see discussion in the appendix D), thus the calculation of the vector emission becomes straightforward.

D Properties of the potential for λ field
In the main text we have shown that λ field satisfies the following equation of motion eq. (81)−(82) Let us investigate the properties of the function U λz .
Broken→ Broken In this case and the potential has the limits : where we have used that v ′ , v ′′ → 0 outside of the wall. Physics wise this result is expected since λ mode must have the mass of the vector gv outside of the wall.
Symmetric→Broken Here we will assume On the broken side as expected the potential becomes equal to the mass square of the vector boson. To find its limit on the symmetric side we need to look at the equation defining v(z): where the prime stands for a derivative with respect to v(z). Integrating this equation we get , which need not to be zero. Now we look at the limits as z → −∞. At this point, using eqs. (181) and (182) we can write the various terms of the U λ function in terms of the potential V and its derivatives V ′ : In the limit z → −∞, v → 0, however we know that the first derivative of the potential V must be equal to zero at this point since there is a local minimum at v = 0. Thus in the region v → 0 we can write down: Combining these results we can see that where m h,s is the mass of the scalar on the symmetric side. We can see that as was discussed in the section 3.2 U λ has the correct properties for a potential of a d.o.f interpolating between the scalar field and λ polarisation on different sides of the wall.

E Evaluation of phase space Integrals
In this appendix we provide some details regarding the evaluation of the phase space integrals, and how to derive analytical expressions. We will always be interested in the limit of large incoming energy p 0 ≫ all masses. The type of integrals we deal with are all of the form where the dependence on couplings and masses m i is implicit. When I(p 0 ) admits an expansion around infinity, this can be obtained in principle straightforwardly as and one can sometimes take this 'exact' approach. However, such an expansion does not always exist 23 , or if it does, its coefficients might in practice be difficult to evaluate. Moreover, in the presence of large mass hierarchies, the regime where eq. (188) is actually a good approximation can begin at arbitrary high energies 24 . One can often successfully use a different expansion instead, which we call 'collinear expansion', of p 0 , k z ≫ k ⊥ , m i where m i stands for all masses. Keeping the leading term, this reduces integrands to the form where The function N depends on the contribution in question but we wish to highlight the factor in denominator. This form makes most manifest the properties of the phase space structure in particular for τ polarisation (and scalars), most importantly when m (and possibly m ψ ) is very small. For a given k z , the integrand is peaked for k 2 ⊥ ≲ (k IR ⊥,m ) 2 . Notice the role of k IR ⊥,m -its presence regulates an otherwise logarithmically IR divergent integral. Indeed the momentum transfer diverges in the limit of m, m ψ → 0. It is easy to prove that k IR ⊥,m < k z for small m, justifying the expansion. When m/m ψ is very small, clearly there are two relativistic regimes, the first given by where k z should be taken in the dominant region. Typically k z ∼m for step function contributions and ∼ L −1 w for WKB. Above this scale instead one reaches the true asymptotic behaviour where eq. (188) is valid.

E.1 Scalars
Scalar R-mover emission (WKB) We begin with the most dominant contribution to scalar emission at the highest of energies p 0 → ∞. The amplitude is the simplest that one can encounter, and For m ̸ = 0, properly evaluating the expansion eq. (188) for this case is tricky 25 . However the collinear expansion eq. (189) works extremely well, giving where k IR ⊥ is given by eq. (190). We see indeed that for large p 0 the integrand is relatively flat in k z while k ⊥ is strongly IR dominated as anticipated in the preamble above, justifying the expansion. The integrals can now be computed exactly and we obtain eq. (51). We can see clearly however that there is also an intermediate regime for small m/m ψ characterised by a plateau: where we remind ourselves that there is no WKB contribution at all unless we can emit modes greater than the inverse wall length. The value of the plateau is easily obtained by setting m = 0 at the start and computing the leading order in eq. (188). However, in this regime, we find the step function contributions are more important and we move on there now.
Scalar R and L-mover emission (step wall) The step wall case is an interesting theory exercise in its own right, since everything can be done exactly. For scalars the contributions are 25 The coefficient of the 1/p 0 term in eq. (188) is the first non-zero contribution as expected from fig. 5. Out of the three terms to evaluate only one is non-zero in the x → 0 limit. Said limit cannot be taken before the integral is evaluated however and such a task proves unnecessarily difficult.
where the dimensionless function defined is Symmetric to Broken (v = 0): The symmetric side mass of the λ degree of freedom is m h,s -the mass of the Higgs in that phase. We find however that taking m h,s = 0 incurs an error which is at most ∼ 10% when m h,s ∼ few ×m. One can rigorously prove that in the limit p 0 → ∞ we have where G ≈ 0.916 is Catalan's constant. F step λ,R has a closed form in terms of (hyperbolic) trig functions and the dilogarithm, but we have deemed it more useful to explicitly report only its smallmL w limit -an excellent approximation, as can be seen in fig. 10. There we also see that the sum of the two contributions is roughly constant, giving ⟨∆p λ,total ⟩ ≈ ⟨∆p λ,step R ⟩ + ⟨∆p λ,wkb to leading orders inmL w . A more accurate expression whenmL w ∼ 1 is eq. (211).

F Pressure in the EW phase transition
In this work we computed the pressure in the context of an Abelian toy model, where the U (1) gauge boson was emitted by a complex scalar. Emission from fermions will not significantly change our result per degree of freedom. Moreover, the emission can be straightforwardly expanded to the non-Abelian case. For the SM case, the pressure at 1 → 2 level originates from the vertices inducing ψ → Zψ (where the first particle in the final state is soft and ψ is some fermion of the SM), ψ → W ± ψ for the gauge bosons emitted from fermions H → W ± H, H → ZH for the gauge bosons emitted from the Higgs and A → W W, W → W A, Z → W W, W → W Z, W → ZW for gauge bosons emitting gauge bosons. The careful counting of all the processes involved in the pressure was presented in Appendix C of [45]. This leads to final pressure of the form where every contribution has is normalised to the fine structure constant α em and the Z boson mass M Z in broken phase. We caution the reader that this expression is still an estimate and may incur future revision, for example, from a better understanding of finite temperature corrections -see section 4.2.

G Sensitivity to wall width
In this work we separated the phase space of particles according to when the step wall and WKB approximations are justified where L w is the width of the wall. However, this is a somewhat arbitrary quantity, significant up to some order 1 factor. Although we have discussed the sensitivity of our results to L w in several places, we summarise it in this dedicated appendix. We study numerically how pressure changes with wall width. For scalars, the dominant contribution comes from the WKB regime and dependence on L w vanishes, as can be seen in explicitly by the analytical formulae eqs. (51) and (198). The results for vector emission are highlighted in fig. 11. The left and right columns show left and right mover emission respectively, while top and lower panels correspond to τ and λ polarisations. The first thing to notice is that ⟨∆p τ, step L ⟩, which is the dominant contribution for super-cooled symmetric to broken transitions is largely insensitive to L w . Secondly, ⟨∆p λ, step R ⟩ and ⟨∆p λ, wkb R ⟩ show almost linear dependence around L w ∼ fewm but their sum is largely constant. This point was made also in fig. 10 using analytical asymptotic formulae. Instead, ⟨∆p λ, step L ⟩, which exists only in broken to broken transitions is linearly dependent on the cut-off (see eq. (143)).  Figure 11: Dependence of various contributions to average momentum transfer, due to the emission of a vector boson, on the inverse wall width L −1 w in the limit p 0 → ∞. Top (Bottom) panels refers to τ (λ) emission, while left (right) column refers to L (R) emission. In this limit, any dependence on the mass of the emitter (m ψ ) vanishes.