BubbleDeterminant

class BubbleDet.BubbleDeterminant(bubble, particles, gauge_groups=None, renormalisation_scale=None, thermal=False)

Bases: object

Class for computing functional determinants.

The main class of the BubbleDet package.

Parameters for initialisation below.

Parameters:

• bubble (BubbleConfig) – Object describing the background field.

• particles (list of ParticleConfig) – List of ParticleConfig objects, describing the fluctuating particles. Can also be a single ParticleConfig object.

• gauge_groups (list, optional) – List of length two describing the full and unbroken gauge groups, separated by spaces for product groups e.g. [“SU5”, “SU3 SU2 U1”]. Applies to the simple Lie groups SU(N), U(N), SO(N) and SP(N). Default is None.

• renormalisation_scale (float, optional) – The renormalisation scale in the \(\text{MS}\) bar scheme. If None, set to the mass of the nucleating field.

• thermal (boole, optional) – If True, includes the thermal dynamical prefactor. Default is False.

Methods Summary

findDerivativeExpansion(particle[, NLO])	Derivative expansion of determinant
findDeterminant([eig_tol, full, gy_tol, ...])	Full determinant
findGelfandYaglomFl(particle, l[, gy_tol, rmin])	\(F=\log(\psi/\psi_{\rm FV})\) for given \(l\)
findGelfandYaglomTl(particle, l[, gy_tol, rmin])	\(T=\psi/\psi_{\rm FV}\) for given \(l\)
findLogPhiInfinity([log_phi_inf_tol, tail])	\(\log\phi_\infty\) coefficent for the background field
findNegativeEigenvalue([eig_tol])	Negative eigenvalue of the Higgs operator \(\mathcal{O}_H(\phi_\text{b})\)
findRenormalisationTerm(particle)	Renormalization of divergent terms
findSingleDeterminant(particle[, eig_tol, ...])	Single particle determinant
findWKB(particle, l_max[, Delta_W_inf, ...])	WKB approximation for large l

Methods Documentation

findDerivativeExpansion(particle, NLO=False)

Derivative expansion of determinant

The derivative expansion is an expansion in a ratio of length scales, which in turn can be related to a ratio of masses: the mass of the background scalar divided by the mass of the fluctuating field.

The leading order (LO) and next-to-leading order (NLO) of the expansion are

\[\int \mathrm{d}^d x \underbrace{\left[ V_{(1)}(\phi_\text{b}) - V_{(1)}(\phi_\text{F}) \right]}_\text{LO} + \int \mathrm{d}^d x \underbrace{\left[ \frac{1}{2} Z_{(1)}(\phi_\text{b}) \nabla_\mu\phi_\text{b}\nabla_\mu\phi_\text{b} \right]}_\text{NLO},\]

where \(V_{(1)}\) and \(Z_{(1)}\) are the heavy particle’s contribution to the one-loop effective potential and field normalisation factor.

Parameters:

• particle (ParticleConfig) – The heavy particle for which to carry out the derivative expansion. Must not have zero modes.

• NLO (boole, optional) – If True, the derivative expansion is carried out to next-to-leading order, otherwise at leading order. Default is False.

Returns:

• S1 (float) – The result within the derivative expasion.

• err (float) – Estimated error in the result.

findDeterminant(eig_tol=1e-06, full=False, gy_tol=1e-06, l_max=None, log_phi_inf_tol=0.001, rmin=0.0001, tail=0.007)

Full determinant

This function computes the determinant for the full multi-particle case. Mathematically, the return value is the sum of single-particle determinants,

\[\texttt{findDeterminant()} = \sum_{\texttt{particle}} \texttt{findSingleDeterminant(particle)}.\]

The particles summed over are those in the initialisation of the BubbleDeterminant object. Additional parameters and metaparameters are as for the single-particle case.

Parameters:

• eig_tol (float, optional) – Relative tolerance for negative eigenvalue computation. Relevant to the thermal case.

• full (boole, optional) – If True, returns list of single particle determinants, split into orbital quantum number, else returns (extrapolated) sum. Default is False.

• gy_tol (float, optional) – Relative tolerance for solving Gelfand-Yaglom initial value problems.

• l_max (int, optional) – The maximum value of the orbital quantum number. If None, then this value is estimated based on the radius of the bubble and the Compton wavelength of the fluctuating field.

• log_phi_inf_tol (float, optional) – An estimate of the relative uncertainty on the tail of the profile, for computing \(\log\phi_\infty\).

• rmin (float, optional) – Size of first step out from the orgin, relative to Compton wavelength.

• tail (float, optional) – A parameter determining the fraction of the bubble to consider when fitting for the asymptotic behaviour, \(\log\phi_\infty\).

Returns:

• res (float) – The result of computing the full determinant.

• err (float) – Estimated error of the result.

findGelfandYaglomFl(particle, l, gy_tol=1e-06, rmin=0.0001)

\(F=\log(\psi/\psi_{\rm FV})\) for given \(l\)

This function solves the ode:

\[F'' + (F')^2+U F' - \Delta W = 0,\]

as part of Gelfand-Yaglom method to compute functional determinants. Here dash denotes a derivative with respect to the radial coordinate \(r\).

Parameters:

• particle (ParticleConfig) – The particle for which to compute the determinant.

• l (int) – Orbital quantum number.

• gy_tol (float, optional) – Relative tolerance for solving Gelfand-Yaglom initial value problem.

• rmin (float, optional) – Size of first step out from the orgin, relative to Compton wavelength.

Returns:

• res (float) – The result \(F(r_\text{max})\).

• err (float) – Estimated error in the result.

findGelfandYaglomTl(particle, l, gy_tol=1e-06, rmin=0.0001)

\(T=\psi/\psi_{\rm FV}\) for given \(l\)

This function solves the ode:

\[T'' + U T' - \Delta W T = 0,\]

as part of Gelfand-Yaglom method to compute functional determinants. Here dash denotes a derivative with respect to the radial coordinate \(r\).

Parameters:

• particle (ParticleConfig) – The particle for which to compute the determinant.

• l (int) – Orbital quantum number.

• gy_tol (float, optional) – Relative tolerance for solving Gelfand-Yaglom initial value problem.

• rmin (float, optional) – Size of first step out from the orgin, relative to Compton wavelength.

Returns:

• res (float) – The result \(T(r_\text{max})\).

• err (float) – Estimated error in the result.

findLogPhiInfinity(log_phi_inf_tol=0.001, tail=0.007)

\(\log\phi_\infty\) coefficent for the background field

We fit for the unknown constant \(\phi_\infty\) in the asymptotic behavior of the numerical bubble profile,

\[\phi(r) \sim \phi_{\mathrm{F}} + \phi_\infty K(d/2 - 1, m_\text{F} r)\left(\frac{m_\text{F}}{r}\right)^{d/2 - 1},\]

assuming the potential has a positive mass term in the false vacuum, \(\phi = \phi_{\mathrm{F}}\).

First, we find four approximate values for the asymptotic behavior of the numerical bubble profile. Then, we extrapolate linearly from these values to a more precise and robust value than obtainable directly from the numerical bubble solution.

If the bubble profile is precise near the false vacuum, i.e. at large radii, the argument tail can be decreased from the default of 0.015, which can then be used to increase the precision of the result. This corresponds to performing the linear extrapolation with points closer to the false vacuum, and correspondingly closer to the end of the profile. However, note that the default setting is already very precise and works well with the default settings of the CosmoTransitions package set in this package. A more detailed description for the tail parameter can be found from the correspoding article.

Parameters:

• log_phi_inf_tol (float, optional) – A parameter determining an accuracy goal for the error caused by choosing points that are close to the end of the numerical bubble profile, < result * log_phi_inf_tol. If the goal cannot be met, it is weakened with an internal algorithm.

• tail (float, optional) – A parameter determining the chosen bubble-tail points for fitting the asymptotic behaviour. Shrinking tail \(\to 0\) corresponds to \(r\to \infty\) for the chosen points.

Returns:

• res (float) – The value of \(\log\phi_\infty\).

• err (float) – Estimated error in the result.

findNegativeEigenvalue(eig_tol=1e-06)

Negative eigenvalue of the Higgs operator \(\mathcal{O}_H(\phi_\text{b})\)

In continuum notation, the eigenvalue equation takes the form

\[\left(-\partial^2-\frac{d-1}{r}\partial +V''(\phi_\text{b})\right)f(r)=\lambda_- f(r),\]

and for bubble nucleation, or vacuum decay, this operator has a single negative eigenvalue, some finite number of zero eigenvalues and an infinite number of positive eigenvalues.

Here, we use the finite difference matrix representation of the differential operator accurate to \(1/N^4\), with two different boundary conditions, "Neumann" and "Dirichlet", at the maximal numerical radius.

The leading, \(1/N^4\) numerical error is extrapolated away using a fit to direct numerical estimates of the eigenvalue, and the residual error is estimated. The different boundary conditions provide additional information for the error estimation, appended to the residual numerical error.

Parameters:

eig_tol (float, optional) – Relative tolerance for the direct numerical eigenvalues used for the extrapolation.

Returns:

• res (float) – The value of the negative eigenvalue.

• err (float) – Estimated error in the result.

findRenormalisationTerm(particle)

Renormalization of divergent terms

This routines renormalizes divergent terms and adds the appropriate counterterm to render the result finite. These terms are only nonzero in even dimensions. The present implementation gives the nonzero results for \(d = 2, 4, 6\), and dimensional regularization is used with \(d=2n-2\epsilon\).

The divergent term can be found from the WKB approximation, for example for a scalar field in \(d = 4\)

\[-\frac{1}{2} \sum_l \deg(d;l) \log \frac{\psi^l_{b}(\infty)}{\psi^l_{F}(\infty)} \sim \frac{1}{16}\sum_l \deg(d;l)\frac{1}{ \overline{l}^3} \int \mathrm{d}r r^3\left[W(r)^2-W(\infty)^2\right]\]

The sum over \(l\) has an \(\epsilon\) pole and gives a contribution

\[\left(\frac{\exp (\gamma ) \mu ^2}{4 \pi }\right)^{\epsilon} \sum_{l=2}^{\infty}\deg(d;l) \overline{l}^{-3} =\frac{1}{2\epsilon} + \log\mu - \frac{1}{2}\left(\log 4\pi-\gamma\right)+\mathcal{O}(\epsilon)\]

The counterterm contribution is

\[S_\text{ct}[\phi] = \frac{1}{32 \epsilon} \int \mathrm{d}r r^{3-2 \epsilon} \frac{\pi ^{-\epsilon}}{\Gamma (2-\epsilon)} W(\phi)^2.\]

After adding adding the counterterm contribution to the divergent WKB term all \(\epsilon\) poles cancel and one finds the finite result:

\[-\frac{1}{16}\int \mathrm{d}r rr^3 \left[W(\phi_{b}(r))^2-W(\phi_{F})^2\right] \left[\log \left(\frac{\mu r}{2}\right)-a-\frac{1}{2}+\gamma\right]\]

The contributions are analogous in \(d=2\) and \(d=6\).

Parameters:

particle (ParticleConfig) – The particle for which to compute the renormalisation term.

Returns:

• res (float) – The renormalisation scale dependent term.

• res_eps (float) – The additional finite renormalisation scale dependent term which arises due to factors of \(d/\epsilon\) for vector fields.

• err (float) – Estimated error in the result.

findSingleDeterminant(particle, eig_tol=1e-06, full=False, gy_tol=1e-06, l_max=None, log_phi_inf_tol=0.001, rmin=0.0001, tail=0.007)

Single particle determinant

The functional determinant is the one-loop correction to the action induced by fluctuations of a field. It is also the statistical part of the bubble nucleation rate.

The computation factorises based on orbital quantum number, \(l\). For each value of \(l\), the functional determinant is computed using the Gelfand-Yaglom method. This is carried out for \(l\in [0, l_\text{max}]\), and then extrapolated to \(l_\text{max} \to \infty\).

Mathematically, the return value is

\[\begin{split}\texttt{findSingleDeterminant(particle)} &= \\ \frac{\text{dof}(d,s,n)}{2}& \log\frac{\det {'} (-\nabla^2 + W(r))}{\det (-\nabla^2 + W(\infty))} - \log \mathcal{J} \mathcal{V},\end{split}\]

where the dash denotes that zero eigenvalues have been dropped from the first term (if present). Their effect is captured by the Jacobian \(\mathcal{J}\) and volume \(\mathcal{V}\) factors. The volume factor is only included for finite internal groups. The factor \(\text{dof}(d,s,n)\) is the total number of internal and spin degrees of freedom of the field. Ultraviolet divergences are regulated in the \(\text{MS}\) bar scheme.

Parameters:

• particle (ParticleConfig) – The particle for which to compute the determinant.

• eig_tol (float, optional) – Relative tolerance for negative eigenvalue computation. Relevant to the thermal case.

• full (boole, optional) – If True, returns results split into orbital quantum number, else returns (extrapolated) sum. Default is False.

• gy_tol (float, optional) – Relative tolerance for solving Gelfand-Yaglom initial value problems.

• l_max (int, optional) – The maximum value of the orbital quantum number. If None, then this value is estimated based on the radius of the bubble and the Compton wavelength of the fluctuating field.

• log_phi_inf_tol (float, optional) – An estimate of the relative uncertainty on the tail of the profile, for computing \(\log\phi_\infty\).

• rmin (float, optional) – Size of first step out from the orgin, relative to Compton wavelength.

• tail (float, optional) – A parameter determining the fraction of the bubble to consider when fitting for the asymptotic behaviour, \(\log\phi_\infty\).

Returns:

• res (float) – The result of computing the single particle determinant.

• err (float) – Estimated error of the result.

findWKB(particle, l_max, Delta_W_inf=None, a_inf=None, separate_orders=False)

WKB approximation for large l

Our implementation is an higher-orders generalization of [GO].

The routine solves the differential equation

\[\Psi''(x)=(x^2 W(e^x)+\overline{l}^2)\Psi(x)\]

in powers of \(\overline{l}=l+(d-2)/2\), where \(r=e^x\). The false vacuum equation has \(W(\infty)\) instead. The WKB approximation of \(\log \frac{\Psi(\infty)}{\Psi_{FV}(\infty)}\) is then computed up to \(O\left(\overline{l}^{-10}\right)\).

Sums of the form

\[\sum_{l=2}^{\infty}{\rm deg}(d,l)\overline{l}^{-a}\]

are also returned, where

\[{\rm deg}(d,l) = \frac{(d+2 l-2) \Gamma (d+l-2)}{\Gamma (d-1) \Gamma (l+1)}.\]

If \(d=2n-2\epsilon\), then these sums contain \(\epsilon\) poles if \(2n-2-a=-1\). In cases when this happens findWKB() replaces the sum with

\[{\rm deg}(d,1) \left(d/2\right)^{-a} +{\rm deg}(d,2)\left(d/2+1\right)^{-a}\]

The divergent sum

\[\sum_{l=0}^{\infty}{\rm deg}(d,l)\overline{l}^{-a}\]

is instead returned by findRenormalisationTerm.

If the sum is finite in dimensional regularization the code returns

\[\lim_{\epsilon \rightarrow 0} \sum_{l=0}^{\infty}{\rm deg}(d,l)\overline{l}^{-a}.\]

In cases where the bounce approaches the false minima as \(\phi \sim r^{-a}\), the routine improves the WKB approximation by approximating \(W(r) \sim W_\infty r^{-a_\infty}\) for large \(r\)

Parameters:

• particle (ParticleConfig) – The particle for which to compute the determinant.

• l_max (int) – Maximum orbital quantum number.

• Delta_W_inf (float, optional) – The prefactor \(W_\inf\) in the fit at large radii \(\Delta W \approx W_\inf r^{-a_\inf}\). Note that these values are only needed if the particle is massless.

• a_inf (float, optional) – The exponent \(a_\inf\) in the fit at large radii \(\Delta W \approx W_\inf r^{-a_\inf}\). Note that these values are only needed if the particle is massless.

• separate_orders (boole, optional) – If True returns the terms in the WKB expansion at each power of \(1/l\) separately. Default is False.

Returns:

• WKB (float) – The ratio of determinants \(\log \frac{\Psi(\infty)}{\Psi_{FV}(\infty)}\) up to \(l^{-9}\).

• err_WKB (float) – Estimated error on WKB.

• WKBSum (float) – The sum \(\frac{1}{2}\sum_{l=2}^{\infty}{\rm deg}(d,l)\log\frac{\Psi(\infty)}{\Psi_{FV}(\infty)}\) within the WKB approximation.

References

[GO]

Gerald V. Dunne, Jin Hur, Choonkyu Lee, Hyunsoo Min. Instanton determinant with arbitrary quark mass: WKB phase-shift method and derivative expansion, Phys.Lett.B 600 (2004) 302-313