# Semi-analytic techniques for calculating bubble wall profiles

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## Abstract

We present semi-analytic techniques for finding bubble wall profiles during first order phase transitions with multiple scalar fields. Our method involves reducing the problem to an equation with a single field, finding an approximate analytic solution and perturbing around it. The perturbations can be written in a semi-analytic form. We assert that our technique lacks convergence problems and demonstrate the speed of convergence on an example potential.

## Keywords

Mass Matrix Perturbative Correction Bubble Wall Euclidean Action False Vacuum## 1 Introduction

The decay of a false vacuum is a complex problem with numerous applications in cosmology [1, 2, 3, 4, 5, 6] and is particularly important in the study of baryogenesis [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] (although there are mechanisms for producing the baryon asymmetry that do not require calculating the decay of the false vacuum [27, 28, 29, 30, 31, 32, 33]). Calculating tunneling rates is also an important problem in the study of vacuum stability [34, 35, 36, 37, 38, 39, 40].

While the physics of the tunneling process is qualitatively well understood [41], quantitatively it is a complicated problem that involves solving a set of highly nonlinear coupled differential equations usually requiring a numerical solution. The two techniques that are most commonly used to solving the tunneling problem are path deformation [42, 43] and minimizing the integral of the squared equations of motion for a set of parametrized functions [44, 45], although other methods also exist [46]. Exact solutions only exist for specialized cases [47, 48].

In this paper we offer a new approach by solving the tunneling problem semi-analytically. First, we give an analytic solution to an ansatz for an arbitrary potential. Since the ansatz is only an approximate solution, in the next step we derive a perturbative expansion that converges to the exact solution. For each term in the perturbative series we provide a semi-analytic solution. Since our starting potential is arbitrary, and because the convergence of the perturbative expansion is independent of the potential, our method is completely general.

To derive the ansatz we take advantage of the fact that the multi-field problem can be approximated by finding the solution to a single-field potential. The approximate tunneling solution can be found along the field direction that connects the true and false vacua. This single-field potential can be solved in terms of a single parameter. To improve the initial ansatz we compute correction functions to the ansatz in a manner analogous to Newton’s method of finding roots. The result is a perturbative series of corrections that are expected to converge quadratically.

The differential equations that define these corrections can be solved analytically in terms of eigenvalues of the mass matrix and a function of the initial ansatz. In doing so we use techniques that were recently employed to analytically solve number densities across a bubble wall [49]. Although the technique has elements in common with Newton’s method it does not share its trouble with null derivatives giving divergent corrections or oscillations around the solution. We also argue that the other problems with Newton’s method are generically not relevant to our method.

The layout of this paper is as follows. In Sect. 2 we give a brief overview of the false vacuum problem. In Sect. 3 we develop an ansatz form that approximately solves a general variety of multi-field potentials with a false vacuum, where the potential is specified by a single parameter. In Sect. 4 we derive the perturbative corrections to the ansatz forms and discuss the convergence. In Sect. 5 we use this method to solve a problem which can be directly compared with the literature. Concluding remarks are given in Sect. 6.

## 2 Fate of the false vacuum

Consider a potential of multiple scalar fields \(V(\phi _i )\) with at least two minima. The trivial solution to the classical equations of motion is stationary extremizing the potential. This solution typically gives the field a non-zero vacuum expectation value and is responsible for giving standard model particles their mass via electroweak symmetry breaking. The other, less obvious solution is one where the fields continuously vary from one minimum to another. In this case, the false vacuum decays into the true vacuum via tunneling processes, and is termed the ‘bounce solution’ [41]. If this is achieved within a first order phase transition, regions of the new vacuum appear and expand as bubbles in space. In this paper we are interested in calculating the profile of the bubble, that is, the spacetime dependence of the bubble nucleation.

*A*(

*T*) is a temperature dependent prefactor proportional to the fluctuation determinant,

*T*is the temperature and \(S_E\) is the euclidean action for the bounce solution which satisfies the classical equations of motion. In the case of a spherically symmetric bubble the classical equations of motion are

^{1}Here \(\rho \) is the ordinary 3D spherical coordinate, as we are considering finite temperature, and \(v_i^\mathrm {true}\) and \(v_i^\mathrm {false}\) are the vacuum expectation values of the field \(\varphi _i\) in the true and false vacua, respectively. The equations of motion resemble the classical solution of a ball rolling in a landscape of shape-

*V*with \(\rho \) playing the role of time, but including a \(\rho \)-dependent friction term.

## 3 Approximate solution to the multi-field potential

### 3.1 Reducing to a single-field potential

^{2}

^{3}To illustrate this point, we present in Fig. 1 the potential in \(\phi \) given in Eq. (6) for both the edge choices of \(\alpha \) and the mean allowed choice, using several choices of

*E*. One can see that, for \(\alpha =\frac{1}{2}\), we have exactly the Mexican hat potential (albeit shifted to \(\phi =0.5\)) with degenerate minima, and for \(\alpha = \frac{3}{4}\), there is no potential barrier between false and true minima.

### 3.2 Developing ansatz solutions

In deriving an approximate solution to the potential in Eq. (6), we first note that the effective potential is proportional to *E*. Thus, one can factor |*E*| out of the equations of motion by further rescaling \(\rho \mapsto \rho / \sqrt{|E|}\). Then the equations of motion only depend on \(\alpha \).

^{4}\(\widetilde{V} \equiv V/|E|\), and we have integrated over the angular variables assuming isotropy. The integral in Eq. (8) must only depend on \(\alpha \), as in

We first evenly sample values of \(\alpha \) within (0.5, 0.75), then numerically solve the full bubble profile using conventional techniques. Next, for each value of \(\alpha \), we fit the kink solution given in Eq. (10) to the full solution, extracting \(L_w\) and \(\delta \). Lastly, we numerically integrate to find *f* in the Euclidean action. This results in a tabulation of values for \(L_w\), \(\delta \), and *f*, for each value of \(\alpha \). Using the apparent \(\alpha \) dependence and intuition from our parametrization of the potential, we find ansatz functional forms in terms of \(\alpha \) for each of these parameters.

*r*and

*s*, and the coefficient

*c*are fit using the tabulated values from the full numerical calculation. Interestingly, we find almost exactly that \(r=18\). (The full fitted parameters are given in Table 1.)

The fitted values for the parameters that define the approximate ansatz functions for *f* which is used in the Euclidean action, the bubble wall width \(L_w\), and the offset \(\delta \) from the kink solution

\(f(\alpha )\) | \(L_w(\alpha )\) | \(\delta (\alpha )\) | |||
---|---|---|---|---|---|

Parameter | Value | Parameter | Value | Parameter | Value |

\(f_0\) | 0.0871 | \(\ell _0 \) | 1.4833 | \(\delta _0 \) | 2.2807 |

| 1.8335 | | 0.4653 | | \(-4.6187\) |

| 3.1416 | | 18.0000 | \(a_1\) | 0.5211 |

| 0.7035 | \(a_2\times 10^5\) | 7.8756 |

We note that |*E*| scales as \(|b \phi ^3|\) so \(S_E/T\) scales as \(\frac{\phi _m}{T} \sqrt{\frac{\phi _m}{b}}\), where *b* is the cubic coupling of the unscaled field, as in Eq. (5). Also *b* controls the height of the barrier separating the two minima.

## 4 Perturbative solution

In the previous section, we developed fitted curves to estimate the parameters of the well known kink solution. In this section, we will take advantage of rescaling to compute convergent perturbative corrections. The process is largely analogous to Newton’s method for finding roots of functions. Here, we iteratively determine functional corrections to the ansatz form.

### 4.1 Perturbative corrections to the ansatz

*i*refers to field \(\phi _i\), and the index

*k*will run over the

*n*eigenvalues of the mass matrix, and

*k*is not summed over. Substituting \(\varepsilon ^h_{ik}\) for \(\varepsilon _i\) in Eq. (16) with \(B_i=0\) yields

*M*is the mass matrix

*i*th element of the eigenvector of

*M*that has eigenvalue \(\lambda _k^2\). It is necessary, however, to use both positive and negative roots, \(\pm \lambda _k\). Thus we must further introduce \(\tilde{\lambda }_j\) and \(\tilde{z}_{ij}\) with

### 4.2 Observations on convergence

- 1.
If the initial ansatz function is too far from the true function, convergence will be slow.

- 2.
Oscillating solutions where \(\varepsilon ^{(n)}(\rho ) \approx - \varepsilon ^{(n+1)}(\rho )\).

- 3.
Divergent corrections that arise in Newton’s method if the function’s derivative becomes undefined or zero. The equivalent issue will be discussed in detail below.

- 4.
Being in the wrong basin of attraction and converging to the wrong function.

*n*, so that \(\varepsilon _i^{(n)} = -\varepsilon _i^{(n+1)}\). But this means that the equations of motion for \(A_i^{(n+2)} = A_i^{(n)} + \varepsilon _i^{(n)} + \varepsilon _i^{(n+1)}\) can be written before Taylor expanding as

## 5 Comparison with a solved example

*u*traces a straight line path from the origin to the global minimum and

*v*is of course orthogonal to

*u*. Our one dimensional potential is then given writing the potential in the rotated basis and setting

*v*to zero. We then rescale such that the minimum is at \(u=1\) and then we divide by |

*E*| to get

*x*,

*y*) basis the ansatz is

*x*and

*y*fields starting with the base ansatz forms \(x^{(0)}\) and \(y^{0}\), and then including the first three perturbative corrections, denoted by

*x*field, \(B_x(\rho )\), which arises from the inhomogeneous part of Eq. (16). The error function is given for the bare ansatz solution of \(x(\rho )\) and the first three perturbative corrections. We point out that the magnitude of the error is reduced by roughly a factor of 5–10 from each perturbative correction, and that the error function for \(x^{(3)}(\rho )\) has reduced in magnitude by a factor of 300 compared to that of \(x^{(0)}(\rho )\).

## 6 Conclusion

In this work we presented a new method to calculate the bubble profile in a bounce solution for a multi-field potential with a false vacuum. The method uses fitted functions to estimate the parameters of the single-field kink solution which is used as an ansatz form. It then applies this form to the full multi-field potential, which receive perturbative correction functions that are reduced to elementary numerical integrals. We have argued that the perturbative series of corrections should converge quadratically, and is immune to the issues of the analogous Newton’s method. This method is shown to be effective in solving a toy model with two scalar fields.

## Footnotes

- 1.
The first is not a boundary condition unlike the other two. It is instead the condition that differentiates the bounce from a trivial solution.

- 2.
This definition of \(\alpha \) differs from that of [50] but the physical principles are the same.

- 3.
This is assuming the three turning points are in the positive \(\phi \) direction with the local minimum at the origin. The rest of potentials with three turning points are covered by this analysis simply by making combinations of the transformations \(\phi \mapsto \phi + a\) and \(\phi \mapsto -\phi \).

- 4.
\(\widetilde{V}\) does not have any dependence on |

*E*|.

## Notes

### Acknowledgements

Funding was provided by Australian Research Council (Grant No. CE110001004).

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