Vevacious: a tool for finding the global minima of oneloop effective potentials with many scalars
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Abstract
Several extensions of the Standard Model of particle physics contain additional scalars implying a more complex scalar potential compared to that of the Standard Model. In general these potentials allow for charge and/or colorbreaking minima besides the desired one with correctly broken SU(2)_{ L }×U(1)_{ Y }. Even if one assumes that a metastable local minimum is realized, one has to ensure that its lifetime exceeds that of our universe. We introduce a new program called Vevacious which takes a generic expression for a oneloop effective potential energy function and finds all the treelevel extrema, which are then used as the starting points for gradientbased minimization of the oneloop effective potential. The tunneling time from a given input vacuum to the deepest minimum, if different from the input vacuum, can be calculated. The parameter points are given as files in the SLHA format (though is not restricted to supersymmetric models), and new model files can be easily generated automatically by the Mathematica package SARAH. This code uses HOM4PS2 to find all the minima of the treelevel potential, PyMinuit to follow gradients to the minima of the oneloop potential, and CosmoTransitions to calculate tunneling times.
Keywords
Minimal Supersymmetric Standard Model Renormalization Scale Vacuum Expectation Value Field Configuration Model File1 Introduction
A major part of the phenomenology of the incredibly successful standard model of particle physics (SM) is the spontaneous breaking of some (but not all) of the gauge symmetries of the Lagrangian density by the vacuum expectation value (VEV) of a scalar field charged under a subgroup of the SM gauge group. The entire scalar sector of the SM consists of a doublet of SU(2)_{ L } which also has a hypercharge under U(1)_{ Y } equal in magnitude to that of the lepton SU(2)_{ L } doublet. The potential energy of the vacuum is minimized by the scalar field taking a constant nonzero value everywhere. The presence of this VEV radically changes the phenomenology of the theory, and allows for masses for particles that would be forced to be massless if the gauge symmetries of the Lagrangian density were also symmetries of the vacuum state.
Since this scalar field is the only field in the SM that can possibly have a nonzero VEV while preserving Lorentz invariance, finding the minima of the potential energy is straightforward, though of course evaluating it to the accuracy required is quite involved [1, 2, 3].
Also, with the current measurements for the masses of the top quark and Higgs boson, one finds that the SM potential at oneloop order is actually unbounded from below for a fixed value of the renormalization scale. The value of the Higgs field for which the potential is lower than the desired vacuum is so high that one may worry that large logarithms of the Higgs field over the electroweak scale would render the loop expansion nonconvergent. However, the effect of large logarithms can be resummed, and the conclusion that our vacuum is only metastable persists using the renormalizationgroupimproved effective potential [1, 2, 3, 4, 5].
The existence of multiple nonequivalent vacua both raises technical challenges and introduces interesting physics. The technical challenges are now that one has to find several minima and evaluate which is the deepest, as well as calculate the tunneling time from a false vacuum to the true vacuum. However, this is an important ingredient in theories where a firstorder phase transition explains the baryon asymmetry of the universe through the sphalerons occuring in the nucleation of bubbles of true vacuum (see [6] and references therein).
Many extensions of the SM introduce extra scalar fields. Sometimes these fields are introduced explicitly to spontaneously break an extended gauge symmetry down to the SM gauge group [7, 8], and they are assumed to have nonzero VEVs at the true vacuum of the theory. Other times they are introduced for other reasons, such as supersymmetry [9], and often nonzero VEVs for such fields would be disastrous, such as breaking SU(3)_{ c } and/or U(1)_{EM}, which excludes certain parts of the parameter space of the minimal supersymmetric standard model (MSSM) from being phenomenologically relevant.
The technical challenges are much tougher when multiple scalar fields are involved. Even a treelevel analysis involves solving a set of coupled cubic equations, the socalled minimization or tadpole equations. It has generally only been attempted for highly symmetric systems such as two Higgs doublet models (2HDM) [10, 11] or with only a minimal amount of extra degrees of freedom such as the (assumed) three nonzero VEVs of the nexttominimal supersymmetric standard model (NMSSM) [12, 13, 14].
Since a general solution is usually too difficult, the question of the stability of VEV configurations against tunneling to other minima of the potential is often ignored. Instead, potentials are often engineered to have a local minimum at a desired VEV configuration through ensuring that the tadpole equations are satisfied for this set of VEVs. This approach allows one to go beyond tree level straightforwardly, and oneloop tadpoles are the norm, and in supersymmetric models twoloop contributions are often included [15]. This local minimum is implicitly assumed to be stable or longlived enough to be physically relevant. Unfortunately, as some examples will show, local minima which are not the global minimum of their parameter point are often extremely shortlived, excluding some benchmark parameter points for some models.
The program Vevacious has been written to address this. Given a set of tadpole equations and the terms needed to construct the oneloop effective potential,^{1} first all the extrema of the treelevel potential are found using homotopy continuation (HOM4PS2), which are then used as starting points for gradientbased minimization (PyMinuit) of the (real part of the) oneloop potential, and finally, if requested, the tunneling time from an input minimum to the deepest minimum found is estimated at the oneloop level (CosmoTransitions). The program is intended to be suitable for parameter scans, taking parameter points in the SLHA format [16, 17] and giving a result within seconds, depending on the number of fields allowed to have nonzero VEVs and the accuracy of the tunneling time required. Vevacious is available to download from http://www.hepforge.org/downloads/vevacious.
2 The potential energy function at tree level and oneloop level
The terminology of minimizing the effective potential of a quantum field theory is rather loaded. Hence first we shall clarify some terms and conventions that will be used in the rest of this article. In the following we consider models where only scalars can get a VEV as required by Lorentz invariance.
In principle, the effective potential is a realvalued^{2} functional over all the quantum fields of the model. However, under the assumption that the vacuum is homogeneous and isotropic, for the purposes of determining the vacua of the model, the effective potential can be treated as a function of sets of (dimensionful) numbers, which we shall refer to as field configurations. Each field configuration is a set of variables which correspond to the classical expectation values for the spinzero fields which are constant with respect to the spatial coordinates.
The example of the SM is relatively simple: the field configuration is simply a set of two complex numbers, which are the values of the neutral and charged scalar fields assuming that each is constant over all space. These four real degrees of freedom can be reduced to a single degree of freedom by employing global SU(2)_{ L } and phase rotations, leaving an effective potential that is effectively a function of a single variable.
Henceforth we shall assume that each complex field is treated as a pair of real degrees of freedom, and note that this may obscure continuous sets of physically equivalent degrees of freedom which are manifestly related by phase rotations when expressed with complex fields.
Also we shall refer to the local minima of the effective potential as its vacua, and label the global minimum as the true vacuum, while all the others are false vacua. A potential may have multiple true vacua, either as a continuous set of minima related by gauge transformations as in the SM for example, or a set of disjoint, physically inequivalent minima, each of which may of course be a continuous set of physically equivalent minima themselves. In cases where there is a continuous set of physically equivalent minima, we assume that a single exemplar is taken from the set for the purposes of comparison of physically inequivalent minima.
Furthermore, the term vacuum expectation value can be used in many confusing ways. In this work, VEVs will only refer to the sets of constant values which the scalar fields have at the field configurations which minimize the effective potential. Hence we do not consider the effective potential to be a “function of the VEVs”, rather a function of a set of numbers that we call a field configuration.
2.1 The treelevel potential and tadpole equations
Although we assume renormalized potentials here for simplicity, the methods used by Vevacious are equally applicable to nonrenormalizable potentials, as long as V ^{tree} is expressed as a finitedegree polynomial. The value of loop corrections to a nonrenormalizable potential may be debatable, but Vevacious can be restricted to using just the treelevel potential.
While closedform solutions for cubic polynomials in one variable exist, solving a coupled system in general requires very involved algorithms, such as using Gröbner bases to decompose the system [20, 21], or homotopy continuation to trace known solutions of simple systems as they are deformed to the complicated target system of tadpole equations.
2.1.1 The homotopy continuation method
The homotopy continuation method [22, 23] has found use in several areas of physics [24, 25, 26], in particular to find string theory vacua [27, 28] and extrema of extended Higgs sectors [29], where the authors investigated a system of two Higgs doublets with up to five singlet scalars in a general treelevel potential, and [30], where systems of up to ten fields were allowed to have nonzero VEVs. In contrast, the Gröbner basis method is deemed prohibitively computationally expensive for systems involving more than a few degrees of freedom [20].
The numerical polyhedral homotopy continuation method is a powerful way to find all the roots of a system of polynomial equations quickly [31]. Essentially it works by continuously deforming a simple system of polynomial equations with known roots, with as many roots as the classical Bézout bound of the system that is to be solved (i.e. the maximum number of roots it could have). The simple system with known roots is continuously deformed into the target system, with the position of the roots updated with each step. While the method is described in detail in [22, 23], a light introduction can be found for example in [29].
2.2 The oneloop potential
For scalar degrees of freedom, the \({\bar{M}}^{2}_{n}( \varPhi)\) are the eigenvalues of the second functional derivative of V ^{tree}, i.e. the eigenvalues of \(({\bar{M}}^{2}_{s=0})_{ij} = ( {\lambda}_{ijkl} + {\lambda}_{ikjl} + {\lambda}_{iklj} + \cdots) {\phi}_{k} {\phi}_{l} + ( A_{ijk} + A_{ikj} + A_{kij} + \cdots) {\phi}_{k} + {\mu}^{2}_{ij} + {\mu}^{2}_{ji}\). Thus these \({\bar{M}}^{2}_{n}( \varPhi)\) are the eigenvalues of the treelevel scalar “masssquared matrix” that would be read off the Lagrangian with the scalars written as fluctuations around the field configuration. Unless the field configuration corresponds to a minimum of the effective potential, these do not correspond to physical masses in any way, of course.
Likewise, the \({\bar{M}}^{2}_{n}( \varPhi)\) for fermionic and vector degrees of freedom are the eigenvalues of the respective “masssquared matrices” where the scalar fields are taken to have constant values given by the field configuration. (The fermion masssquared matrix is given by the mass matrix multiplied by its Hermitian conjugate.)
The terms c _{ n } depend on the regularization scheme. In the \(\overline{\mathrm{MS}}\) scheme, c _{ n } is 3/2 for scalars and Weyl fermions, but 5/6 for vectors, while in the \(\overline{\mathrm{DR}}'\) scheme [33, 34], more suitable for supersymmetric models, c _{ n } is 3/2 for all degrees of freedom. Since this is a finiteorder truncation of the expression, the renormalization scale Q also appears explicitly in the logarithm, as well as implicitly in the scale dependence of the renormalized Lagrangian parameters.
Much of the literature on oneloop potentials (including [33]) assumes a renormalization scheme where V ^{counter} is zero; however, such a scheme is often inconvenient for other purposes, such as ensuring tadpole equations have a given solution at the oneloop level, see e.g. the appendix of [35]. (SARAH automatically generalizes this approach to extended SUSY models as explained in [36, 37].) Finally, we also note that models can be constructed where spontaneous symmetry breaking does not happen at tree level, but does exist when one takes loop corrections into account [18, 38].
2.2.1 Scale dependence
As noted, the oneloop effective potential depends on the renormalization scale. Ideally one would use the “renormalizationgroup improved” expression for the potential [32] as this is invariant under changes of scale; however, this is often totally impractical except for potentials with only a handful of parameters and a single scalar field.
If one must use a scaledependent expression for the potential, as is often the case, the renormalization scale should be chosen carefully: if one chooses a scale too high or too low, one may find that with a finitelooporder (and thus scaledependent) effective potential, there is no spontaneous breaking of any symmetry, or even that the potential is not bounded from below [39]! This is often simply due to the fact that higher orders become more important in such a case, especially when corrections from the next order would introduce new, large couplings, such as often happens when going from tree level to one loop. It can also be that the scale is so large or small that the loop expansion is no longer a reliable expansion. We also note that rather undermines arguments that radiative effects do not change treelevel conclusions on the absolute stability of vacua such as in [40] (where the argument also fails to take into account that there may not even be a scale at which the renormalization condition used can be satisfied).
Indeed, it is crucial that the scale is chosen so that the loop expansion is valid. Explicitly, large logarithms should not spoil the perturbativity of the expansion in couplings. Loops with a particle n typically come with a factor of \(\ln( {\bar{M}}^{2}_{n}( \varPhi)/ Q^{2} )\) along with the factor of \({\alpha}_{n} = [\mathrm{relevant\ coupling}]^{2} / ( 4 \pi)\), and thus \({\alpha}_{n} \ln( {\bar{M}}^{2}_{n}( \varPhi)/ Q^{2} )\) should remain sufficiently smaller than one such that the expansion can be trusted [32]. A rough first estimate then of the region of validity, assuming that the dimensionful Lagrangian parameters are all of the order of the renormalization scale to some power, is where ln(v ^{2}/Q ^{2})/(4π)≤1/2, say, for a field configuration with vector length v, so where the VEVs are within a factor of e ^{ π }≃20 of the renormalization scale.
Furthermore, it is in general not valid to compare the potential for different field configurations using a different scale for each configuration, if one is using a scaledependent effective potential. The reason is that there is an important contribution to the potential that is fieldindependent yet still depends on the scale^{3} [41, 42]. Of course, if one knows the full scale dependence of all the terms of the Lagrangian regardless of whether they lead to fielddependent contributions to the effective potential, then one can correctly evaluate different field configurations at different scales.
2.3 Gauge dependence
The oneloop potential is explicitly gaugedependent [43, 44]. However, as shown in [44, 45], the values it takes at its extrema are independent of the gauge chosen, except for spurious extrema of poorlychosen gauges. The popular R _{ ξ } gauges are wellbehaved and do not have fictitious gaugedependent extrema for reasonable choices of ξ [45].
It is also possible to formulate the effective potential in terms of gaugeinvariant composite fields [46], though this may not always be practical. One can also verify the gaugeindependence of extrema using more complicated gauges and applying BRST invariance [47].
2.4 Convexity
As shown in [18], the effective potential, which can be thought of as the quantum analogue of the classical potential energy for constant fields, is real for all values of the fields. However, the loop expansion leads to complex values in regions where the classical potential is nonconvex. While one can take the convex hull of the truncated expansion of the potential when evaluating the potential for configurations of fields in the convex region, it is not particularly helpful for the purpose of computing tunneling transition times. Fortunately, the oneloop truncation of the effective potential as a function of constant values for the fields can consistently be interpreted as a complex number with real part giving the expectation value of the potential energy density for the given field configuration and imaginary part proportional to the decay rate per unit volume of this configuration [19].
2.5 Comparing two vacua
If there are two or more physically inequivalent minima of a potential, then it is vitally important to know if the phenomenologically desired minimum is the global minimum, or, if not, how long the expected tunneling time to the true vacuum is.
Given the issues raised above in Sects. 2.2.1 and 2.3, it is safe to use a scale and gaugedependent oneloop effective potential to compare two inequivalent minima provided that the scale is held fixed and that the two minima are within the region of validity determined by the renormalization scale. Of course, an explicitly gauge and scaleindependent expression for the effective potential would obviously be unburdened by such concerns, but unfortunately it is rare to be able to formulate such an expression.
3 Tunneling from false vacua to true vacua
Given a path through the field configuration space from one vacuum to another with a lower value, which for convenience we shall label as the false vacuum and true vacuum respectively, one can solve the equations of motion for a bubble of true vacuum of critical size in an infinite volume of false vacuum [48, 50]. This allows one to calculate the bounce action and thus the major part of the tunneling time.
Unfortunately, this means that to calculate the tunneling time from a false vacuum to a true vacuum, one needs to evaluate the potential along a continuous path through the field configuration space, and even though the extrema of the potential are gaugeinvariant as noted above, the paths between them are not. However, it has been proved that at zero temperature, the gauge dependence at oneloop order cancels out [51].
At finite temperature, the situation is not so clear, though the Landau gauge may be most appropriate [52]. While some studies have shown that for “reasonable” choices of gauge, the differences in finitetemperature tunneling times are small [53], it is still possible to choose poor gauges that can even obscure the possibility of tunneling [54].
4 Vevacious: objectives, outline, features and limitations
4.1 Objectives
The Vevacious program is intended as a tool to quickly evaluate whether a parameter point with a given set of VEVs, referred to henceforth as the input vacuum, has, to oneloop order,^{4} any vacua with lower potential energy than the input vacuum, and, optionally, to estimate the tunneling time from the input vacuum to the true vacuum if so.
A typical use envisaged is a parameter scan for a single model. Some effort needs to be put into creating the model file in the first place, though this is straightforward if using SARAH as described in Sect. 6.1; once the model file has been created, parameter points given in the form of SLHA files should be evaluated within a matter of seconds, depending on how complicated the model is, what simplifications have been made, and how accurately the tunneling time should be calculated if necessary.
Given a model (through a model file) and a parameter point (through an SLHA file), Vevacious determines the global minimum of the oneloop effective potential, and a verdict on whether the input minimum is absolutely stable, by it being the global minimum, or metastable. The user provides a threshold for which the metastability is rated as longlived or shortlived. Whether the tunneling time or just an upper bound is calculated depends on whether the upper bound is above or below the threshold, or may be forced by certain options (see Sect. 6.3).
4.2 Outline
 (1)
An input file in the SLHA format [16] is read in to obtain the Lagrangian parameters defining the parameter point, required to evaluate the potential. We emphasize that even though the SUSY Les Houches Accord is used as the format, the model itself does not need to be supersymmetric, as long as the SLHA file contains appropriate BLOCKs.
 (2)
All the extrema of the treelevel potential are found using the homotopy continuation method [22] to solve the treelevel tadpole equations. The publicly available program HOM4PS2 [31] is used for this.
 (3)
The treelevel extrema are used as starting points for gradientbased minimization of the oneloop effective potential. The MINUIT algorithms [55] are used here through the Python wrapper PyMinuit [56]. The points where PyMinuit stops are checked to see if they are really minima.^{5} Any saddle point is then split into two further points, displaced from the original in the directions of steepest descent by amounts given by the <saddle_nudges> arguments (see Sect. 6.3), which are then also used as starting points for PyMinuit. By default, PyMinuit is restricted to a hypercube of field configurations where each field is only allowed to have a magnitude less than or equal to one hundred times the renormalization scale of the SLHA file. This is rather excessive by the reasoning of Sect. 2.2.1, which would lead one to take at most maybe ten or twenty times the scale as an upper limit; however, it was considered better to allow the user to decide whether the results of Vevacious are within a trustworthy region.
 (4)
The minima are sorted, and, if necessary, the tunneling time from the input vacuum to the true vacuum is calculated. The A factor of Eq. (5) is taken to be equal to the fourth power of the renormalization scale, as this is expected to be the typical scale of the potential and thus the expected scale of the solitonic solutions, and the bounce action is calculated with the code CosmoTransitions [50]. To save time, first CosmoTransitions is called to calculate the bounce action with a bubble profile given by a straight line in field configuration space from the false vacuum to the true vacuum to get an upper bound on the tunneling time. If this upper bound is below the usergiven threshold <direct_time> (see Sect. 6.3), then no refinement is pursued. If, however, the upper bound is above the threshold, the bounce action is calculated again allowing CosmoTransitions to deform the path in field configuration space to find the minimal surface tension for the bubble. If one wishes to calculate the tunneling time with a different A factor, one can edit a line of Python code as described in Sect. 4.3.
 (5)
The results are printed in a results file and also appended to the SLHA input file.
4.3 Features
Finds all treelevel extrema
The homotopy continuation method is guaranteed to find all the solutions of the system of tadpole equations (to the limitations of the finite precision of the machine following the algorithm) [22]. One does not have to worry that there may be solutions just beyond the range of a scan looking for the solutions.
Rolls to oneloop minima
Vevacious rolls from the treelevel extrema to the minima of the oneloop effective potential before comparing them, because in general the VEVs get shifted. In addition, extrema that change their nature with radiative corrections, such as the field configuration with zero values for all the fields in the Coleman–Weinberg model of radiative spontaneous symmetry breaking [38], which is a minimum of the treelevel potential but a local maximum of the oneloop effective potential, are found.
Calculates tunneling times or upper bounds on them
A parameter point in a model is not necessarily ruled out on the basis that the desired minimum of the potential is not the global minimum, since a metastable configuration with a lifetime of roughly the observed age of the Universe or longer is compatible with the single data point that we have. Vevacious creates CosmoTransitions objects with its effective potential function to evaluate the bounce action and thus the tunneling time from a false vacuum to the true vacuum of a parameter point.
Fast
An important aspect of Vevacious is that it is fast enough to be used as a check in a parameter scan of a model. For example, on a laptop with a 2.4 GHz processor, a typical parameter point for the MSSM allowing six real nonzero VEVs (two Higgs, two stau, two stop) can report within 3.2 seconds that no deeper vacuum than the input vacuum was found, or, for a different parameter point, can report an upper bound on the tunneling time within 18 seconds. However, borderline cases which require a full calculation of the minimal bounce action can take up to 500 seconds. Reducing the number of degrees of freedom to four (fixing the stop values at zero) reduces the calculation times to 0.6, 2.3 and 27 seconds respectively.
Flexible
4.4 Limitations
Garbage in, garbage out
Vevacious performs very few sanity checks, so rarely protects the user from their own mistakes. For instance, Vevacious does not check if the potential is bounded from below. However, there are some checks, such as those which result in the warning that the given input minimum was actually rather far from the nearest minimum found by Vevacious (though Vevacious carries on regardless after issuing the warning). Another important sanity check that is not performed is to check that the SLHA BLOCKs required by the model file are actually present in the given SLHA input file. The user is fully responsible for providing a valid SLHA file to match the model. As model files are expected to be produced automatically by software such as SARAH, it is expected that the SLHA files for the model will also be prepared consistently with the expected BLOCKs. Unfortunately it is quite easy to miss this point when using the example model files provided by default with Vevacious: if one does use these files, one must use the correct model file for the input SLHA files, as described in Sect. 7.
May be excessively optimistic about the region of validity
Not guaranteed to find minima induced purely by radiative effects
While Vevacious does find all the extrema of the treelevel effective potential, there is no guarantee that these correspond to all the minima of the effective potential at the oneloop level. The strategy adopted by Vevacious will find all the minima of the oneloop effective potential that are in some sense “downhill” from treelevel extrema, but any minima that develop which would require “going uphill” from every treelevel extremum will not be found. Such potentials are not impossible: if the quadratic coefficient in the Coleman–Weinberg potential [38] is small enough while still positive, the single treelevel minimum can remain a minimum at the oneloop level while deeper minima induced by radiative corrections still appear. However, if the treelevel minimum is sufficiently shallow then the finite numerical derivatives used by Vevacious may be enough to push it over the small “hills” into the oneloop minima.
Extreme slowdown with too many degrees of freedom
Like many codes, the amount of time Vevacious needs to produce results increases worse than linearly with the number of degrees of freedom. A proper quantification of exactly how Vevacious scales with degrees of freedom remains on the todo list, but as a guide, some typical running times (again on a 2.4 GHz core) for the HOM4PS2 part are: 3 degrees of freedom: 0.03 seconds; 5: 0.28 seconds; 7: 5.1 seconds; 10: 20 minutes; 15: 10 days. The PyMinuit part depends on the number of solutions found by HOM4PS2, but in general takes several seconds. The CosmoTransitions part is strongly dependent on the details of a particular potential, and how rapidly the path deformations converge; models with the same degrees of freedom can vary wildly from seconds to hours to be computed. For this reason, Vevacious only calls the full calculation of CosmoTransitions if the quick estimate of the upper bound on the tunneling time is over the usergiven threshold, and also gives the user the option to never use the full calculation, rather only the quick upperbound calculation, which in general takes only a few seconds at most.
Homotopy continuation method requires discrete extrema
The homotopy continuation method relies on tracking the paths of a discrete number of simple solutions to a discrete number of target solutions. There is no guarantee that a system with a continuous set of degenerate solutions will be solved by HOM4PS2, and unfortunately Vevacious can not check that the system has redundant degrees of freedom such as those corresponding to a gauge transformation. Hence the user must choose the degrees of freedom of the model appropriately.
Homotopy continuation path tracking resolution
The homotopy continuation method guarantees that there is a path from each solution of the simple system to its target solution, however, there is a danger that a finiteprecision pathtracking algorithm will accidentally “jump” from the path it should be following onto a very close other path to a different solution, possibly leading to one or more solutions remaining unfound (though not necessarily, since several simple solutions may map to the same (degenerate) target solution).
Tunneling path resolution
Calculating the tunneling time requires finding a continuous path in field configuration space from the false to the true vacuum. However, this must be discretized to a finite number of points on a finite machine, and may even lead to a very small barrier between the vacua disappearing entirely. However, in such cases the tunneling time should be very short indeed, so Vevacious notes this and takes a fixed very small tunneling time as the result.
5 Subtleties: renormalization schemes and allowed degrees of freedom
Currently, Vevacious performs very few sanity checks. In particular, it remains blissfully ignorant of any physical meanings the user intends for the values of the Lagrangian parameters which are given. Thus is it entirely up to the user to ensure that these values correspond correctly to the intended renormalization conditions. However, Vevacious does assume some form of dimensional regularization (e.g. switching between \(\overline{\mathrm{DR}}'\) and \(\overline{\mathrm{MS}}\) depends on the value for c _{ n } given for the vector masssquared matrix in the model file, see Appendix B, and providing explicit terms for the “ϵ scalars”). So far, Lagrangian parameters as appearing in the model files and as printed by the SPheno produced by SARAH 4 are consistent with the renormalization conditions as specified in the appendix of [35]. One should note though that the VEVs from Vevacious are given for the Landau gauge by default, and have slightly different values to those they have in the Feynman–’t Hooft gauge that is used within SPheno, for example.
In principle, every single degree of freedom should be checked for a VEV, but this is often totally impractical, given that about ten degrees of freedom is at the limit of what might be considered tolerable with current processors. Thus the user will often want to consider only a subset of scalars as being allowed nonzero VEVs. For example, when considering a supersymmetric model, one might restrict oneself to possible VEVs only for the third generation, or when considering a model specifically engineered for light staus but all other sfermions being very heavy, one might only worry about stau VEVs as a first check. Hence Vevacious should be used bearing in mind the caveat that it will not find vacua with nonzero VEVs for degrees of freedom which are not allowed nonzero VEVs in the model file. It is up to the user to decide on the best compromise between speed and comprehensiveness by choosing which degrees of freedom to use.
Importantly, the user is responsible for ensuring that the model file has a treelevel potential which has a discrete number of minima. Mostly this means that the user has to identify unphysical phases and hence remove the associated degrees of freedom from the imaginary parts of such complex fields. Another way that problems can arise is when there are flat directions such as the tanβ=1 direction of the MSSM with unbroken supersymmetry, even though the degeneracy of these directions may be lifted by loop corrections.
6 Using Vevacious
Vevacious needs at least two input files: the model file and the parameter file. The model file contains the information about the physical setup. This file is most easily generated by SARAH, as explained in Sect. 6.1. Should the user intend to modify this file by hand, we give more information about the format in Appendix B. The parameter file should be in the SLHA format, extended within the spirit of the format, as required in general for extended models, as described in Sect. 6.2. Finally, there is the option of supplying an initialization file in XML to save giving several commandline arguments, and this file is described along with these arguments in Sect. 6.3.
6.1 Preparing the input file for SARAH

ComplexParameters, Value: list of parameters, Default: {}: By default, all parameters are assumed to be real when writing the Vevacious input files. However, the user can define those parameters which should be treated as complex.

IgnoreParameters, Value: list of parameters, Default: {}: The user can define a list of parameters which should be set to zero when writing the Vevacious input.

OutputFile, Value: String, Default MyModel.vin, where MyModel here is the same name as is given in Start["MyModel"]; above: The name used for the output file.
Example: MSSM with stau VEVs
 1.Defining the particles which can get a VEV where it is important that the VEVs have names that are at least two characters long. The first two lines are the standard decomposition of the complex Higgs scalar \(H_{i}^{0} \to\frac{1}{\sqrt{2}}(v_{i} + i \sigma_{i} + \phi_{i})\) and are the same as in the chargeconserving MSSM. The third and fourth line define the decomposition of the three generations of left and righthanded charged sleptons. The last three lines define the decomposition of the charged Higgs fields and the sneutrinos into CPeven and odd eigenstates. This is necessary for the adjacent mixing, see below. There are two new features in SARAH 4 which are shown here: (i) it is possible to give VEVs just to specific generations of a field, such as in this example we only use the third one (vLR[3])—in the same way, one can allow for smuon and stau VEVs using vLR[2,3]; (ii) VEVs can have real and imaginary parts. Until now, complex VEVs in SARAH had been defined by an absolute value and a phase (ve ^{ iϕ }); however, it is easier to handle within Vevacious in the form v _{ R }+iv _{ I }.
 2.Changing the rotation of the vector bosons: With nonzero stau VEVs the photon won’t be massless any more but will mix with the massive gauge bosons. In general, there can be a mixing between the B gauge boson (VB) and the three W gauge bosons (VWB[i]) to four mass eigenstates (VB1 …VB4). The mixing matrix is called ZZ.
 3.Changing the rotation of matter fields: The stau VEVs generate new bilinear terms in the scalar potential which trigger a mixing between the neutral and charged Higgs fields, the charged sleptons and the sneutrinos. Note that even if we had used above complex stau VEVs to present the new syntax, we don’t take the potential mixing between CPeven and odd eigenstates into account here. To incorporate this, the new basis would read Also, stau VEVs lead to a mixing of all the uncolored fermions, which are now also Majorana in nature. In this model file, they are now all labeled as L0 with mixing matrix ZN. The spinor sector also has to be adjusted:
6.2 Preparing the SLHA data
To check for the global minimum in a given model for a specific parameter point all necessary numerical values of parameters have to be provided via an SLHA spectrum file. The conventions of this file have to be, of course, identical to the ones used for preparing the Vevacious model file.
The SLHA (1 [16] and 2 [17]) conventions are only concerned with the MSSM and the NMSSM, but do specify that extra BLOCKs within the same format should be acceptable within SLHA files, and should be ignored by programs that do not recognize them. Vevacious is intended to be used for many other models, so accepts any BLOCKs that are mentioned in its model file and looks for them in the given SLHA file. In this sense, the user is free to define BLOCKs as long as the names are unbroken strings of alphanumeric characters (e.g. BLOCK THISISAVALIDNAME or BLOCK MY_BLOCK_0123 ^{6}). However, we strongly advise against redefining those BLOCKs specified by [16] and [17], or any of their elements, to have meanings other than those given in [16] and [17].
With this in mind, two sets of pregenerated model files for the MSSM (each file within a set allowing for different scalars to have nonzero VEVs) are provided: one set being that produced by SARAH 4, which assumes a certain extension of the SLHA BLOCK HMIX, the other restricted to quantities completely specified by the SLHA 1 and 2 conventions. The extensions of HMIX are three additional elements: 101, providing the value of \(m_{3}^{2}\) (often written as B _{ μ }) directly, and 102 and 103 providing the values of v _{ d } and v _{ u } respectively. All three can be derived from elements 2, 3, and 4 of HMIX, but are much more suited to conversion between renormalization schemes and different gauges (as v _{ d } and v _{ u }, and thus tanβ, are gaugedependent quantities, with values which also depend on the renormalization conditions).
6.2.1 Scale dependence in the SLHA file
Vevacious is a tool for finding minima for a oneloop effective potential evaluated at a single scale, and thus it is important that the Lagrangian parameters are provided consistently at this scale. The scale is also required to be given explicitly. Since the SLHA convention specifies that running parameters are given in BLOCKs each with their own scale, at first glance this may seem problematic. Even worse, the format allows for multiple instances of the same BLOCK, each with its own scale. However, the default behaviour of SPheno, SoftSUSY, SuSpect, and ISAJET when writing SLHA output is to give all running parameters consistently at a single scale. This is the behaviour that Vevacious assumes.
The explicit value of the scale Q used in V ^{mass} in Eq. (4) is that given by the BLOCK GAUGE. All other BLOCKs are assumed to be at this same scale Although it is not the default behaviour of any of the popular spectrum generators to give the same BLOCKs at different scales, if Vevacious finds multiple instances of the same BLOCK, the BLOCK with the lowest scale is used and the others ignored. In addition, Vevacious performs a consistency check that all the BLOCKs used have the same scale, aborting the calculation if not.
6.2.2 SLHA expression of parameters at different loop orders
The output BLOCKs enumerated in the SLHA papers are specified to be in the \(\overline{\mathrm{DR}}'\) renormalization scheme,^{7} but some users may prefer a different renormalization scheme. The SLHA does not insist on private BLOCKs adhering to the same standards of those explicitly part of the accord, so Vevacious allows for a certain pattern of private BLOCKs to give values for a different renormalization scheme. (Again, we strongly advise against using the BLOCKs explicitly mentioned in [16, 17] to convey values that do not adhere to the definitions in [16, 17].) The additional renormalization schemes that Vevacious allows are those where the finite parts of Lagrangian parameters are themselves apportioned into loop expansions, e.g. μ+δμ, where δμ is considered to be a parameter already of at least one order higher than μ.
To allow for different renormalization conditions of this type, Vevacious first looks for extra (“private”) SLHA BLOCKs that specify particular loop orders. Since Vevacious deals with oneloop effective potentials, it has two categories of parameters: “treelevel” and “oneloop”. When writing the minimization conditions for the treelevel potential, it uses exclusively the “treelevel” values. When writing the full oneloop effective potential, it uses both sets appropriately to avoid including spurious twoloop terms. With reference to Eqs. (2), (3), and (4), Vevacious writes combines the sum V ^{tree}+V ^{counter} by inserting the “oneloop” parameter values into V ^{tree} as part of V ^{1loop}. (The treelevel potential function that is also written automatically for convenience, as mentioned in Sect. 4.3, uses the “treelevel” values, of course.) The term V ^{mass} is already a loop correction, so “treelevel” values are used in the \({\bar{M}}^{2}_{n}( \varPhi)\) functions. If only a single value for any parameter is given, it is assumed to be in a scheme where it has a single value which is to be used in all parts of the effective potential.
As an example, within the renormalization used by SPheno3.1.12, at the point SPS1a′, μ has the value 374.9 GeV at tree level, and 394.4 GeV at one loop. Vevacious inserts the value 374.9 for μ in the minimization conditions (as the units are assumed to be in GeV, as per the SLHA standard), and also into the “masssquared” matrices that are part of the evaluation of the V ^{mass} contributions to the oneloop effective potential. Vevacious inserts the value 394.4 for μ in the polynomial part of the potential, accounting for the contributions of both V ^{tree} and V ^{counter} together.
6.3 Setting up and running Vevacious
There are many options that can be passed to Vevacious. If values other than the defaults are required, they can either be passed by commandline arguments or with an initialization file in XML format. If an option is given by both commandline argument and in an initialization file, the commandline argument takes precedence.
 1.
hom4ps2_dir=./HOM4PS2/ This is a string giving the path to the directory where the HOM4PS2 executable is.
 2.
homotopy_type=1 This is an integer used to decide which mode HOM4PS2 should run in: 1 is used for polyhedral homotopy and 2 for linear homotopy.
 3.
imaginary_tolerance=0.0000001 This is a floatingpoint number giving the tolerance for imaginary parts of VEVs found as solutions to the treelevel minimization conditions, since it is possible that a numerical precision error could lead to what should be an exact cancellation leaving behind a small imaginary part. It is in units of GeV, as the other dimensionful values are assumed to be so since that is how they are in the SLHA standard.
 4.
model_file=./MyModel.vin This is a string giving the name of the model file discussed in Sect. 6.1, including the (relative or absolute) directory path.
 5.
slha_file=./MyParameters.slha.out This is a string giving the name of the SLHA file discussed in Sect. 6.2, including the (relative or absolute) directory path.
 6.
result_file=./MyResult.vout This is a string giving the name of the XML output file, discussed in Sect. 6.2, to write, including the (relative or absolute) directory path.
 7.
<saddle_nudges>1.0,5.0,20.0</saddle_ nudges> (Unfortunately this option does not work very well as a commandline argument, so instead here we display the XML element as it should appear in the XML initialization file. No initialization file has to be used, of course, if the default 1.0, 5.0, 20.0 is fine.) As discussed in Sect. 4.2, PyMinuit may get stuck at saddle points, and Vevacious creates pairs of nearby points as new starting points for PyMinuit in an attempt to get it to roll away to minima. Using the default list shown, if Vevacious finds that PyMinuit stopped at a saddle point, it will create two new starting points displaced by 1.0 GeV either side of this saddle point. If PyMinuit rolls from either or both of these displaced points to new saddle points (or just does not roll as it is still in a region that is too flat), Vevacious will repeat the process for each new saddle point, but this time displacing the new starting points by 5.0 GeV. Vevacious can repeat this a third time, using 20.0 GeV, but after this gives up. Giving a longer commaseparated list of floatingpoint numbers will lead to Vevacious performing this “nudging” as many times as there are elements of the list.
 8.
max_saddle_nudges=3 This is an integer giving the length of the list of floatingpoint numbers of the saddle_nudges option: if it is larger than the length of the list Vevacious already has, the list is extended with copies of the last element; if it is shorter, the list is truncated after the given number of elements.
 9.
ct_path=./CosmoTransitions This is a string giving the path to the directory where the CosmoTransitions files pathDeformation.py and tunneling1D.py are.
 10.
roll_tolerance=0.1 This is a floatingpoint number giving a tolerance for extrema are identified with each other, since PyMinuit may roll to the same minimum from two different starting points, but not stop at exactly the same point numerically. If the length of the vector that is the difference of the two field configurations is less than the tolerance multiplied by the length of the longer of the two vectors that are the displacements of the two field configurations from the origin, then the two field configurations are taken to be the same minimum within errors; e.g. if A is v _{ d }=24.42, v _{ u }=245.0 and B is v _{ d }=24.39, v _{ u }=242.7, the length of A is 246.2140, the length of B is 243.9225, so the longer length is 246.2140; the length of their difference is 2.300196 which is less than 0.1∗246.2140, so A and B are considered to be the same extremum. (This is important for avoiding attempting to calculate a tunneling time from a point back to itself.)
 11.
direct_time=0.1 This is a floatingpoint number giving a threshold tunneling time as a fraction of the age of the Universe for whether the metastability is decided by the upper bound from the fast CosmoTransitions calculation taking a straight line from the false vacuum to the true vacuum as described in Sect. 4.2. If the upper bound resulting from this calculation is below this number, the input vacuum is considered to be shortlived and no refinement in calculating the tunneling time is pursued; e.g. if it is 0.1, and the upper bound on the tunneling time is found to be 10^{−20} times the age of the Universe, the input vacuum is judged to be shortlived since the tunneling time is definitely below 0.1 times the age of the Universe. If the value given for direct_time is negative, this fast calculation is skipped.
 12.
deformed_time=0.1 This is a floatingpoint number giving a threshold tunneling time as a fraction of the age of the Universe for whether the input vacuum is consider shortlived or longlived when the full CosmoTransitions calculation is performed. If the tunneling time was calculated to have an upper bound above the threshold given by the direct_time option (or if the calculation of the upper bound was skipped because direct_time was given a negative value), then CosmoTransitions is called to calculate the bounce action allowing it to deform the path in VEV space to find the minimal bounce action. If the tunneling time calculated from this bounce action is less than the deformed_time value, the input vacuum is considered shortlived, otherwise it is reported to be longlived. (In order to prevent overflow errors when exponentiating a potentially very large number, the bounce action is capped at 1000.) If deformed_time is set to a negative value, this calculation is skipped.
As mentioned above, an XML initialization file can be provided with values for these options. By default, Vevacious looks for ./VevaciousInitialization.xml for these options, but a different file can be specified with the commandline option input=/example/MyVevaciousInit.xml to use /example/MyVevaciousInit.xml as the initialization file. Any other commandline arguments take priority over options set in the initialization file.
An example XML initialization file called VevaciousInitialization.xml is provided with the download, which shows how to set each option.
6.4 The results of Vevacious
The element <stability> can have the values stable, if the input minimum is the global minimum, longlived if the lifetime of the input minima is longer than the specified limit, or shortlived if the tunneling time of the input minimum to the global minimum is shorter than the specified threshold.
The elements <global_minimum> and <input_minimum> contain the numerical values of all VEVs at the global and input minima respectively as well as the depth of the potential at this point, relative to the (real part of the) value of the oneloop effective potential for the field configuration where all fields are zero. The VEVs are given in units of GeV, while the potential depths are in units of (GeV)^{4}.
 No treelevel extrema were found. This might happen for example if HOM4PS2 did not find any real solutions (recalling that complex fields have already been written as pairs of real scalars) because it was given a system with continuous degenerate solutions—however, it is not necessarily the case that this is indicative of this problem, and also it is not necessarily guaranteed to result from such problematic systems.
 PyMinuit threw exceptions:
 PyMinuit got stuck at saddle points with very shallow descending or possibly flat directions:
 No oneloop extrema were found. (This shouldn’t ever happen, even for a potential that is unbounded from below, as by default, PyMinuit is restricted to a hypercube of field configurations where no field is allowed a value greater than a hundred times the scale Q.)
 The nearest extremum to the input field configuration is actually a saddle point:
 The nearest extremum to the input field configuration is further away than the permitted tolerance:
 The energy barrier between the false and true vacua is thinner than the resolution of the tunneling path:
7 Comparison with existing tools and examples with supersymmetric models
The major components of Vevacious have been tested and used already in the literature: HOM4PS2, MINUIT, and CosmoTransitions. The innovations of Vevacious are the automatic preparation and parsing of input and output of the various components in a consistent way, optimization of the running time with respect to how shortlived a metastable vacuum might be, and the feature that SARAH 4 can automatically generate Vevacious model files for any new model that can be implemented in SARAH.
Very few tools are currently available to do the same job as Vevacious. One can of course implement the minimization conditions of a treelevel potential in HOM4PS2 or other implementations of the homotopy continuation method, or any implementation of the Gröbner basis method, on a casebycase basis.
To our knowledge, there is only one publiclyavailable program that purports to find the global minimum of a potential of a quantum field theory: ScannerS [66]. However, at the time of writing, the routines to actually find the global minimum are still under development and are not available.

SARAHSPhenoMSSM_JustNormalHiggsVevs. vin with only the normal real neutral Higgs VEVs allowed;

SARAHSPhenoMSSM_RealHiggsAndStauVevs.vin with real VEVs allowed for the neutral Higgs components and for the staus;

SARAHSPhenoMSSM_RealHiggsAndStauAnd StopVevs.vin with real VEVs allowed for the neutral Higgs components, for the staus, and for the stops.

SARAHSPhenoNMSSM_JustNormalHiggsAnd SingletVevs.vin with only the normal real neutral Higgs and singlet VEVs allowed;

SARAHSPhenoNMSSM_RealHiggsAndSingletAndStauVevs.vin with real VEVs allowed for the neutral Higgs components, for the singlet, and for the staus;

SARAHSPhenoNMSSM_RealHiggsAndSingletAndStauAndStopVevs.vin with real VEVs allowed for the neutral Higgs components, for the singlet, for the staus, and for the stops.

SPS1a, the CMSSM point SPS1a from [65] (which is stable);

CMSSM_CCB, which corresponds to the CMSSM bestfit point including LHC and m _{ h }=126 GeV constraints from [67] (which has a global charge and colorbreaking minimum);

NUHM1_CCB, which corresponds to the NUHM1 bestfit point (“low”) from [68] (which also has a global charge and colorbreaking minimum);

CNMSSM_wrong_neutral, which corresponds to benchmark point P1 from [69] (which also has a neutral global minimum which however is not the input minimum).
We note that the CCB vacua found for CMSSM_CCB, for example, have VEVs of the order of five times the renormalization scale (shown in Sect. 6.4). This should not cause much concern, as ln(5^{2})/(4π)≃0.256 so the oneloop effective potential should still be reasonable around this minimum. However, even if one restricts the VEVs to be less than twice the scale, as described in Sect. 4.4, Vevacious still finds a tunneling time (upper bound) of 10^{−6} times the age of the known Universe to a CCB configuration at the edge of the bounding hypercube.
The MSSM model and parameter files are in the MSSM subdirectory and those of the NMSSM are in the NMSSM subdirectory.
8 Conclusion
Several extensions of the Standard Model contain additional scalar states. Usually one engineers the model such that one obtains a phenomenologically acceptable vacuum with the desired breaking of SU(2)_{ L }×U(1)_{ Y } to U(1)_{ EM }. However, in general it is not checked if the minimum obtained is the global minimum, as this in general quite an involved task already at tree level, where hardly any analytical conditions can be given. Often loop corrections become important too.
To tackle this problem, we have presented the program Vevacious as a tool to quickly evaluate oneloop effective potentials of a given model. It finds all extrema at tree level, which allows for a first check for undesired minima. Starting from these extrema, it calculates the oneloop effective potential to obtain a more reliable result. This is important as loop contributions can potentially change the nature of an extremum. In the case that the original minimum turns out to be merely a local minimum rather than the global minimum, the possibility is given to evaluate the tunneling time. As test cases we have considered supersymmetric models, but the program can be used for a general model provided that the treelevel potential is polynomial in the scalar fields.
Footnotes
 1.
The potential may include nonrenormalizable terms, as long as the oneloop effective potential is of the form of a polynomial plus V ^{mass} as given in Eq. (4).
 2.
 3.
For example, if one is comparing two field configurations of the MSSM potential where the squark fields happen to have zero values, the mass of the gluino is independent of the nonzero fields and yet provides a scaledependent contribution to the effective potential.
 4.
As mentioned in Sect. 2.2.1, a renormalizationgroupimproved effective potential would be better than the oneloop effective potential, but at the moment it seems infeasible to implement.
 5.
PyMinuit stops when it has found a sufficiently flat region without checking whether it is at a minimum.
 6.
The SLHA papers [16, 17] do not specify whether BLOCK names should be casesensitive (so that HMIX and Hmix would be considered equivalent for example) and many spectrum generators have already adopted different case conventions for their output, so Vevacious reads in BLOCK names as caseinsensitive.
 7.
 8.
Technically this is already an abuse of the BLOCK HMIX, since the value entering here is not exactly the value it should have in the \(\overline{\mathrm{DR}}'\) scheme, but the difference is a twoloop order effect.
Notes
Acknowledgements
The authors would like to thank Max Wainwright for advice on CosmoTransitions. This work has been supported by the DFG project No. PO1337/21 and the research training group GRK 1147.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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