We now provide a detailed description of the overall workings of spectrum generator and decay calculator programs. We begin with the first stage of such calculations, the supersymmet- ric mass spectrum generation. Mass spectrum generators solve a system of linked differential equations with boundary conditions at each end. These are the renormalisation group equations of the supersymmetric model with boundary conditions at the low electroweak scale provided by physical measurements, such as the top mass, Z mass, fine structure constant at mZ and
others; and theoretical boundary conditions on the soft supersymmetry breaking parameters at the high GUT scale. In addition, there are requirements on the solution of successful radiative electroweak symmetry breaking. In order to solve this system, to determine the supersymmetric masses and couplings at the SUSY scale, spectrum generators must run particle masses, cou- plings and mixing parameters between two disparate scales. To complete this process in full generality and with complete rigour, one would have to integrate out each particle below its mass and match the theory above each particle mass to a reduced effective theory below each particle mass; however given the number of additional particles present in the MSSM (with even more in its extensions), this is intractable. Moreover, with the particles able to order themselves in mass in all possible ways, N additional particles would therefore result in ∼ N! effective theories to be run, each needing its own renormalisation group equations. Given this situation there are two approaches used in the literature, each relevant in a different regime.
The first approach, and the method adopted in SoftSusy as well as SUSPECT (and also in versions of sPHENO prior to version 4.0), is to match the Standard Model parameters used as inputs at the low scale immediately onto the full MSSM at mZ 2. This matching involves the
conversion of the Standard Model parameters extracted from experiment into MSSM parameters. For example, considering αs, this is determined via jet cross-sections with vertices such as
g → q¯q, gg. The measurements for such cross-sections are then used to determine the vertex factors (proportional toαsat the scale it is measured) including Standard Model loop corrections
up to the desired order in perturbation theory. However, if the theory is taken to be the MSSM rather than the Standard Model, there are additional loop corrections which we must subtract off the calculated αs value to obtain our boundary condition on αs(mt) for the MSSM. At 1-
loop, such corrections come from gluino loops via processes such as that in Figure 3.3. These “finite term” corrections are proportional to m2
Z/(16π2m2SUSY) and are included by matching
straight onto the MSSM at the low scale3. However, by matching at an electroweak scale, logarithmic pieces in the β function are not resummed which arise in the RGEs between mt
and mSUSY due to mass splittings between the various supersymmetric particles (i.e. as not all
supersymmetric particles appear atmSUSY); these are proportional to (1/16π2) log[(∆m)2/m2Z]
and alter the gradient of the running. In order to account for such missing pieces, SoftSusy and other programs that use this method add “threshold corrections” to a given order, these account
2
Traditionally the low matching scale is mZ, however as of SoftSusy version 4.1.1 the matching is done at mt,
this may have effects on the numerical values of the parameters obtained, such as mh[83]. 3
for the difference in gradient over themZtomSUSYrunning with additional intercept corrections
on top of the finite pieces4. This approach deals with the effects of sparticle thresholds by using the MSSM (or its extensions as appropriate) as an effective theory between mZ and mSUSY.
Even within this prescription, there are choices which represent different higher order terms - for the gluino correction of Figure 3.3 for example, the question of which gluino mass value should one use in the loop ensues. Using the pole mass or the DR running mass will lead to distinctαs values essentially corresponding to 2-loop effects. In order to minimise such effects,
higher orders must be included; in this respect SoftSusy is state of the art, containing 3-loop RGEs and many 2-loop threshold corrections to the third generation Yukawas and the strong gauge couplingαs as these have particularly large effects on the Higgs mass [117].
Figure 3.3: Supersymmetric correction to the vertex used to provide the value ofαs(mt), the contribution
of this diagram must be factored in to obtain our value ofαsat the low scale in the MSSM. There is also
a similar contribution from squark loop corrections.
The alternative approach, as used by ISAJET, NMSSMTools, sPHENO (since version 4.0 released in March 2017) [158] and FlexbileSUSY, is to integrate out the sparticles from the RGEs at a higher scale (mSUSY) and then run in an effective theory betweenmSUSYandmZ. This naturally
resums the logarithmic terms due to mass splittings in the RGEs, but misses finite terms due to loop corrections via sparticles in loops. Generically, these two approaches have different regimes of validity, with the SoftSusy approach missing terms of order O(log[(∆m)2/m2
Z]), whilst the
NMSSMTools and sPHENO approach misses some terms of order O(m2Z/m2SUSY). Therefore the former approach will be most accurate for lower values of the mSUSY scale, whilst the latter is
more accurate for higher values ofmSUSY where the mass splittings increase but the finite terms
reduce in size. Where the exact boundary of the two approaches occurs is a model-dependent question, and one of increasing interest given the LHC constraints on low-scale supersymmetry. It has been addressed by the paper [83] in the context of the accuracy of the Standard Model-like Higgs mass, which offers a key constraint on supersymmetric models. The different approaches therefore offer another source of potential mass, mixing and coupling differences between spec- trum generators; these parameters are then used as inputs to the decay calculators and so may cause significant differences in partial widths obtained, depending upon the nature of the mass spectrum and model considered. Nonetheless, as previously mentioned, any such differences between codes can be used as an estimator of associated theoretical errors for these difficult calculations and offer an order of magnitude estimate for the size of higher order effects.
Aside from these differences in approach at the low end of the renormalisation group running, 4
Note we distinguish between these threshold corrections, and those obtained between mSUSY and the high
scale whose logarithms are resummed up to the order of the renormalisation group equations included: 3-loop with Next-to-Next-to-Leading-Logarithms.
the basic methodology of the fixed point iteration solution of the two boundary RGE problem to determine the supersymmetric and Higgs masses and couplings is the same and is illustrated in a simplified form in Figure 3.4. It is described here:
1. Match the low energy boundary conditions on fermion masses, gauge couplings and other electroweak parameters onto either the MSSM or the Standard Model, depending on which of the two approaches are used. Threshold corrections are included at this stage in SoftSusy to account for leading missing logarithmic pieces arising due to the sparticle mass splitting. Guesses are required for the parameters on which there are no boundary conditions, such as the supersymmetric masses, these are made approximately and are irrelevant, being overwritten the next time the iteration reaches the low scale.
2. If the latter approach is used match, onto the full MSSM atmSUSY; if the former is used,
there is no need for this step as the spectrum generator already runs in the full MSSM. 3. The particle masses, couplings and other parameters are then run in the full MSSM up to
the high scale (however it is defined) - often this is the GUT scale, defined as the point where theSU (2)L and U (1)Y coupling unify: α1(MGUT) =α2(MGUT).
4. At the high scale, the supersymmetric parameters are compared with the theoretical boundary conditions (such as unification of scalar masses, fermion masses and trilinear couplings in the case of minimal supergravity models); the parameters for which there are theoretical boundary conditions are replaced by the boundary condition values, leaving the remaining parameters unaltered.
5. The new set of parameters are all run down to the low scale in the full MSSM (perhaps via matching at mSUSY and running in the Standard Model as an EFT below this if the
second approach is used). These parameters at the low scale are compared with the low scale boundary conditions and replaced as appropriate, the whole new set of parameters is then run back to the high scale.
6. Steps 3-5 are then repeated in fixed point iteration until the parameters reach convergence within the level of the tolerance defined, by default this numerical precision is 10−4 but this may be changed in the input file5. Usually a self-consistent solution satisfying both low scale and high scale boundary conditions is found within 3-5 iterations. The number of iterations required is dependent on the model and the precise setup as well as the low and high energy scales set.
7. Finally, once the solution is found, the parameters are run to the supersymmetry scale mSUSY and the supersymmetric and Higgs masses, mixings and couplings are output in
the mass spectrum at this scale. This information is then used as an input to the decay calculator program.
There are potential issues which may arise from such an iterative approach, in particular can we be certain there is just one solution and if not does the fixed point iteration method necessarily produce the “best” solution, however that may be defined. For example, it may 5Specifically it is set in item 1 of the SOFTSUSY block, information on the SoftSusy input file is given in
Figure 3.4: Schematic overview of how a spectrum generator program solves the two boundary differential equation problem posed to determine the masses of the supersymmetric and Higgs particles. It does so by repeatedly running between the low and high scales in fixed point iteration, taking the boundary conditions at each end as inputs each time, until the masses are determined and consistent within a given tolerance.
appear that it would be prone to finding local minima in the solution “fit”, rather than the global minimum or even that the fixed point algorithm may be unstable in the region of some solutions. This has been studied in the literature, in particular in the context of SoftSusy itself an alternative “shooting” approach was investigated [172] and demonstrated that the fixed point iterative method may in some instances only provide one of several solutions, although cases where the phenomenology of these new solutions is markedly different are comparatively rare.