Goldstone Boson Effects on Vacuum Decay

We study the effects of Goldstone modes on the decay of the vacuum in a $U(1)$ theory for a complex scalar field. The dynamics of the field resemble those of Keplerian motion in the presence of time-dependent friction. In flat spacetime, the equations of motion imply a conserved quantity, $L$, reminiscent of conserved angular momentum. They also imply a persistent infinite barrier at $\rho=0$ and a divergent value of the field at the origin of coordinates, rendering any solution physically unattainable. However, in a spacetime punctured at the origin of coordinates, we find finite-action solutions to the equations of motion. These correspond to critical values of $L$, these values depend on the size of the punctured hole and the value to which the field arrives immediately after tunnelling. Goldstone modes accelerate the field away from $\rho=0$ in the radial direction, which results in an asymmetry in tunnelling rates for potentials that differ by the order in which the false and true vacua are placed. We find that tunnelling rates differ by large orders of magnitude for different orderings. Moreover, we show that, with the inclusion of Goldstone modes, it is possible to obtain solutions for any potential, including ones that provide no tunnelling solutions for real scalar fields. This occurs because contributions from Goldstone modes provide the necessary energy to overcome drag forces.


I. INTRODUCTION
Instantons in D-dimensions are O(D) symmetric classical solutions of the equations of motion corresponding to a quantum tunnelling process of a scalar field through a potential that has more than one vacuum state. Classically, the evolution from the false vacuum to the true vacuum of the theory is forbidden by energy conservation. However, the laws of Quantum Mechanics allow for such a process to occur through means of tunnelling. Energetics favour the tunnelling from false vacua to the true vacuum of the theory as a decay process, and the probability of such a decay depends on the value of the Euclidean action.
In Quantum Mechanics, while adopting the WKB approximation, consistent with the path integral formulation of quantum field theory, we find that the probability of tunnelling through a finite potential is proportional to e iS , where S is the action defined as the integral of the Lagrangian corresponding to a given potential [1]. By Wick rotating the time coordinate into Euclidean time τ ≡ −it, we deduce the Euclidean action to be S E ≡ −iS. Hence, the tunnelling rate is given by [1] where A is a prefactor. This means that only solutions that correspond to a finite Euclidean action lead to physically viable tunnelling solutions with a non-zero tunnelling rate.
Such solutions for real scalar fields with a range of different potentials have been studied extensively in the literature. The mathematical formulations of vacuum decay for a real scalar field in flat spacetime were first found by Coleman and Callan in two seperate papers [1] [2]. The first of which introduced thin wall solutions corresponding to potentials where the difference in potential energy between the false vacuum and the true vacuum is small compared to the height of the barrier, while the second aimed to calculate the prefactor A by considering quantum corrections. These papers were followed by another paper by Coleman and De Luccia who studied gravitational effects on such decay [3]. This paper will recap these formulations, and apply them on complex scalar fields with non-zero Goldstone boson modes.
Goldstone bosons appear in theories which exhibit spontaneous symmetry breaking of continuous symmetries. They are necessarily massless and play an important role in observed physical phenomena such as giving massive particles in the standard model (SM) their mass through the Higgs mechanism. Potentials which are independent of Goldstone fields give out the same vacuum expectation value (VEV) regardless of the evolution of the Goldstone field, a rotating vacuum would yield the same VEV as a vacuum with a non-evolving Goldstone field. However, the energy content of the complex field as a whole would differ corresponding to the kinetic energy of the rotating vacuum. Therefore, it is our interest to study the effects of non-zero Goldstone modes on the decay of the vacuum, particularly on the Euclidean action that appears in the tunnelling rate exponent.
The layout of this paper is as follows: Section II will present theoretical and mathematical preliminaries, starting with a subsection considering the flat spacetime case where we will highlight the impossibility of finding finite-energy solutions due to an unavoidable divergence at the origin of coordinates. The following subsection will explore solutions in punctured spacetime which depend on the parameter space of the size of the hole and the value to which the field arrives after tunnelling. These topological holes resemble primordial wormholes that are well motivated and studied [24] [29]. We will show that theses holes decay with the decay of the vacuum, and that spacetime assumes a flat-like behavior at the end of tunnelling.
Section III explores numerical solutions for different potentials. We consider Coleman potentials where the energy difference between the false and true vacua is small compared to the height of the potential. We will highlight the fact that the order in which the false and true vacua are placed within the potential makes a significant difference, as we find that tunnelling rates differ by large orders of magnitude. We then consider Fubini potentials with a mass term, it has been proven that no tunnelling solutions exist in flat spacetime. However, we find that Goldstone modes provide the necessary energy to overcome friction, and thus provide solutions to such problem. The concluding section will summarise the results of the paper and discuss possible applications and implications of such solutions in the cosmological picture.

II. THEORETICAL BACKGROUND
In this section we will develop the equations of motion for a tunnelling field corresponding to a general action and find their solutions, starting from the flat spacetime case followed by the punctured curved spacetime case.

A. Euclidean flat spacetime
We start by explicitly showing how performing a Wick rotation leads us to the correspondence between the action S and the Euclidean action S E : where the indices now run over the coordinates (τ, x). By defining the evolution parameter r ≡ τ 2 + | x| 2 and using the relation for the volume element [6] dV D = 2π dr , (II. 2) we find that the Euclidean action can be written as To properly describe the tunnelling process, we say that the field was trapped in the false vacuum at time τ i = −∞, tunnels through the barrier and arrives at time τ = 0. However, by time reversal symmetry, we find that τ and −τ correspond to the same evolution parameter r. Therefore, as r → ∞, the field "bounces" back to the false vacuum state. The mechanism for tunnelling can be visualised as the materialisation, or the nucleation, of a bubble near the true vacuum at r = 0, which grows at the speed of light. Far away from the origin, the vacuum is unperturbed [1].
We write the complex scalar field as Φ = ρe iχ , where ρ, χ are real fields, with the latter resembling the Goldstone field. This gives where the primes denote differentiation with respect to r. It is clear that we must have a finite value for the Euclidean action for the tunnelling process to take place since e iS = e −S E . For that to happen we must have a vanishing potential as goes r → ∞. Moreover, to conserve energy, the field must tunnel through the barrier with zero kinetic energy. Thus, we impose the condition dρ/dτ | τ =0 = 0 which translates to dρ/dr| r=0 = 0 since Denoting the point where the false vacuum occurs as ρ fv , we summarise the initial conditions needed to find solutions to the equations of motion as [1] The equations of motion give [5] ρ We can solve for χ and write where L is a positive constant of integration. Plugging χ into the first equation of motion, we get The Euclidean action now is given by Interpreting the parameter r as "time", we can make the classical analogy of a particle sliding through a time-dependent effective potential given by subject to a time-dependent drag force. This is analogous to Keplerian dynamics defined by a potential given by −V (ρ) with a time-dependent friction term. We would like to find the position from which the particle starts from rest, slides down the potential and slows down gradually to settle at the false vacuum at r → ∞.
Setting L = 0 is equivalent to eliminating the Goldstone bosons and the fields become real. The parameter L = χ ρ 2 r D−1 must be constant at all "times" r, which implies that if L is non-zero, then (for D > 1) at r = 0, at least one of χ and ρ must be singular. The divergence at the origin is problematic, since it requires infinite energy for the field to tunnel through. As a classical analogue, we say that you cannot shrink a rotating sphere to a point while conserving its angular momentum, because that would require infinite rotational energy.
Moreover, there is an infinite barrier at ρ = 0 due to the term in the effective potential that is proportional to ρ −2 . Therefore, fields undergoing potentials where the false vacuum occurs at ρ = 0 can never reach this value asymptotically without undergoing infinitely many oscillations which gives an infinite value for the action.
To avoid these problems, we can consider a minimal extension to the action that includes gravitational effects in curved spacetime, and introduce a hole at the origin of coordinates resembling a punctured spacetime. One must also consider potentials where the false vacuum occurs at some positive value ρ > 0. For a polynomial potential, this implies adding a linear term that displaces the minimum.

B. Punctured curved spacetime
As we include the Einstein-Hilbert action and allow for a general metric, the action in the presence of gravity (we choose to work in D = 4) is given by [4] where κ = 8πG = 8π/M 2 p . We Wick rotate the time coordinate, and obtain the Euclidean action: (II.14) We choose to work in an O(4) symmetric configuration by constructing a general rotationally invariant metric defined by the line element [3] The variation with respect to the metric gives the Einstein equations: while the equations of motion with respect to ρ, χ give where the primes denote the derivative with respect to r, and L is some positive constant. Combining these results we get the following coupled differential equations: Again, we make the classical analogy of a particle undergoing an effective potential of the form subject to a time-dependent drag force. The Euclidean action S E can now be written as (II.23) Using equation (II.17), we write Unlike the case in which gravity is absent, provided that a(r = 0) = 0, there is nothing that stops us from obtaining finite energy solutions that satisfy the boundary condition ρ (0) = 0. For the other condition ρ(∞) = 0 to be satisfied, we need the function a to grow faster than the decay of ρ. Moreover, as the field ρ vanishes, the metric should become flat, which reads a(r) − r| r→∞ = constant. We choose following the boundary conditions: Setting L = 0 is equivalent to giving the Goldstone field angular momentum. Since Goldstone fields do not appear in the potential then they do not contribute to the vacuum expectation value. However, their kinetic energy allows us to tunnel through the barrier to a field value ρ 0 where V (ρ 0 ) > V (ρ fv ) = 0. Thus, we conclude that for every possible field value ρ 0 to which the field arrives after tunnelling, there is a critical value L c which provides us with a solution that asymptotes to the false vacuum with a finite action, which is also dependent on the size of the hole a 0 .

III. NUMERICAL ANALYSIS
In this section, we will numerically find finite-action solutions satisfying the boundary conditions for different potentials, starting with a Coleman potential considering both ways of which the false and true vacua are ordered, followed by a Fubini potential with a non-vanishing mass term.
The constants µ, λ and are all positive, with the latter being very small compared to the height of the potential V (ρ max ). We choose ρ max > µ 2 /λ such that all minima lie on positive values of the field ρ. We can easily see that V (ρ fv ) = 0 and V (ρ tv ) = − . We choose the values µ = 3 GeV, λ = 50, = 0.01 GeV 4 , and ρ max = µ 2 /λ + 1 GeV to parametrise the potential (see Fig 1).
For a function of this form, we obtain the thin wall solution corresponding to L = 0. The field starts at a value that is very close to the true vacuum ρ tv where it settles until a very large time r = R, when the time-dependent friction term becomes negligible. It then starts sliding rapidly down the potential and asymptotes towards the false vacuum ρ fv . Mathematically, this is expressed as The calculations for the Euclidean action corresponding to the thin wall solution were laid out in detail by Coleman and De Luccia [3]. We quote their results below The first three lines come from the interior of the wall, while the last line comes from the wall. The validity of the approximation relies on satisfying the condition µ 4 / λ 1. Minimising the action with respect to a(R) gives [3] a(R) = 3S where Λ = (κ /3) −1/2 . Hence, in the limit a 0 → 0, the Euclidean action takes the simple form [3] S E = 27π 2 S 4 We start our analysis by fixing the size of the hole a 0 , and then numerically find finite-action solutions that asymptote to the false vacuum corresponding to critical values L c , for different field values ρ 0 to which the field arrives after tunnelling.    The term 1 2 L 2 ρ −3 a −6 in (II.20) accelerates the field ρ towards positive values, which means that it drives the field away from the false vacuum before it dies out. The field climbs up the potential at first, then settles down near the true vacuum until a very large time r = R while the time-dependent friction term dies out. It then slides rapidly down the potential and asymptotes towards the false vacuum, reminiscent of the thin wall solution (see Fig 2). In this case, the lifetime of the vacuum is longer than the L = 0 case as shown in Table I . On the other hand, we may modify the potential such that ρ fv > ρ tv and write where V + (ρ) is defined by equation (III.3). The term 1 2 L 2 ρ −3 a −6 now accelerates the field towards ρ fv . This means that the field ρ asymptotes to the false vacuum rapidly compared to the thin wall solution as it does not need the time-dependent friction time to die out before rolling down the potential (see Fig 4).    As a result, the lifetime of the vacuum is comparatively very short-lived as table III reports.
Comparing these results to the ones obtained earlier highlights that the order of which the false and true vacua are placed within the potential makes a significant difference, since tunnelling rates differ by large orders of magnitude.
C. Fubini potential with a mass term We construct a shifted Fubini potential of the form where λ < 0 so that the potential has no true vacuum and is unbounded from below. We choose to parametrise the potential by setting λ = −0.01, m = 0.2 GeV, and ρ fv = 1 GeV (see Fig 5). For L = 0, using scale invariance arguments [10], it is possible to show that no solutions exist for a non-vanishing m in flat spacetime. The friction term will always dominate and prevent the field from climbing up the hill and reaching the false vacuum at infinity. Therefore, one needs extra operators to find tunnelling solutions. For example, it can be achieved by adding a suppressed ρ 6 term to the potential [10], or by including gravitational effects and introducing a cosmological constant [4].
In punctured curved spacetime with L = 0, the energy content coming from the rotational kinetic energy of the Goldstone field would provide the necessary energy to overcome drag forces. Thus, we are able to obtain finite-action tunnelling solutions for such a potential (see Fig 6). The numerics are shown in Table III.    This result is not exclusive to massive Fubini potentials. In fact, we can extend this to any potential because we can always compute the kinetic energy of the Goldstone field that is needed to overcome the time-dependent friction force which dies out at infinity.
Another possible implication of the existence of these Goldstone modes is the tunnelling from a true vacuum to a false vacuum within a potential, since the kinetic energy of these modes can make up for the energy difference between the two minima.
We conclude this section by noting that we necessarily have a (r → ∞) = 1, since these solutions asymptote to the false vacuum, and hence spacetime will be flat at the end of the tunnelling process, signaling the decay of these holes.

IV. CONCLUSIONS
Evolving Goldstone modes resembling a rotating vacuum contribute to the energy content of the field. We have studied the effects of these modes on the decay of the vacuum in flat spacetime and concluded the impossibility of finding any physically viable solutions due to a divergence at the origin. However, in punctured spacetime it is possible to find finite solutions corresponding to critical values of the conserved quantity L, the size of the hole a 0 and the position to which the field arrives after tunnelling ρ 0 .
Implications of the existence of such modes include the possibility of tunnelling to a field value that corresponds to higher potential energy as the rotational kinetic energy of the Goldstone fields would make up for the energy difference, something that is not possible with real scalar fields. Moreover, we have found that the order in which the false and true vacua are placed within the potential is detrimental to the lifetime of the vacuum, since Goldstone modes accelerate the field towards positive values.
We were able to obtain finite-action solutions corresponding to potentials which are deemed unsolvable in flat spacetime. The necessary topological holes for such a decay process will decay away at the end of the tunnelling process and spacetime will assume a flat description. This means that we have provided a mechanism for such holes to decay through the decay of the vacuum.
For such solutions to occur in nature, we must assume the existence of primordial wormholes, which are yet to be observed. Our results are not conclusive, since we have not considered contributions from Goldstone modes to the value of the prefactor A in the decay rate. However, these decay mechanisms can explain different cosmological pheonomena. For instance, one can describe inflationary scenarios using scalar fields which settled in short-lived false vacua [20].
After tunnelling, the Universe will settle in a new vacuum changing the value of the VEV and leading to phase transitions. In the future, we might be able to detect signatures for such process in the form of resultant gravitational waves. The theoretical framework for predicting the shape of the power spectra is a work on progress [21]. The ESA is planning to build a laser interferometer and is scheduled to launch into space in the early 2030s under the LISA project, enabling us to probe low frequency ranges which are typical of gravitational waves generated from these cosmological phase transitions.