# Investigating multiple solutions to boundary value problems in constrained minimal and non-minimal SUSY models

## Abstract

We investigate the physical origins of multiple solutions to boundary value problems in the fully constrained MSSM and NMSSM. We derive mathematical criteria that formulate circumstances under which multiple solutions can appear. Finally, we study the validity of the exclusion of the CMSSM in the presence of multiple solutions.

## 1 Introduction

After the discovery of the Higgs boson with a mass of \(M_h = (125.10 \pm 0.14)\,\text {GeV}\) [1, 2, 3] and the non-discovery of supersymmetric (SUSY) particles at the LHC, it becomes clearer that pure weak-scale supersymmetry may not be realized in nature. Although the general Minimal Supersymmetric Standard Model (MSSM) may be difficult to fully exclude, the constrained MSSM (CMSSM), which is inspired by minimal supergravity, is in tension with precision observables at more than \(90\%\) confidence level [4, 5]. One reason for the tension is that in the CMSSM, by construction, all sfermion masses are of the same order. In this case, however, observables such as the Higgs boson mass, Dark Matter relic density and the anomalous magnetic moment of the muon cannot be explained simultaneously by the MSSM, because \(M_h \approx 125\,\text {GeV}\) requires multi-TeV stops, while other observables like \((g-2)_\mu \) prefer sub-TeV sleptons.

However, in Refs. [6, 7, 8] it was discovered that there may be multiple MSSM parameter sets which fulfill the same CMSSM boundary conditions. The mathematical reason for this phenomenon is that the CMSSM is formulated as a boundary value problem (BVP), where the running MSSM \(\overline{\text {DR}}'\) parameters are fixed by input values at different renormalization scales. The parameters at the different scales are connected via a set of differential equations, the so-called renormalization group equations (RGEs). Formally, such a BVP may have no, one, or multiple solutions for the MSSM parameters.

In order to make a statement about the validity of the CMSSM, all possible solutions to the BVP must be studied. However, the BVP solving algorithm used in the global fitting analyses of Refs. [4, 5] can at most find one solution and may miss further ones. This raises the question to which extent the CMSSM remains challenged in the presence of multiple solutions of the BVP.

In the present paper we systematically study the physical origin of the multiple solutions in the CMSSM. In doing this, we go beyond the scope of Refs. [6, 7] and derive mathematical criteria that formulate circumstances under which multiple solutions can appear. In addition we study the influence of the chosen low-energy observables that fix the electroweak gauge couplings, which appear to play in important role for the occurrence and the number of multiple solutions. Next, we apply our newly gained insights to the results presented in Ref. [4] and investigate the influence of multiple solutions on the global fit performed therein. Finally, we extend our analysis to the fully constrained Next-to-Minimal Supersymmetric Standard Model (CNMSSM) and demonstrate that multiple solutions can also occur in constrained non-minimal SUSY models.

## 2 Boundary value problems

### 2.1 CMSSM boundary conditions

*i*th \(\overline{\text {DR}}'\) stop mass, the parameters \(|\mu |\) and \(B\mu \) are fixed by the two electroweak symmetry breaking (EWSB) equations. This leaves the following five free parameters of the CMSSM:

### 2.2 CNMSSM boundary conditions

### 2.3 Matching to low-energy observables

- 1.The \(Z^0\) pole mass \(M_Z\) and the Fermi constant \(G_F\) can be chosen as input, in which case the weak mixing angle is calculated aswhere \(\varDelta \hat{r}(Q)\) is a function of the one-loop \(Z^0\) and the \(W^\pm \) self-energies [17].$$\begin{aligned} \theta _W(Q) = \frac{1}{2}\arcsin {\sqrt{\frac{2\sqrt{2}\pi \alpha (Q)}{G_FM_Z^2 [1 - \varDelta \hat{r}(Q)]}}}, \end{aligned}$$(6)
- 2.The \(Z^0\) pole mass \(M_Z\) and the \(W^\pm \) pole mass \(M_W\) can be chosen as input, in which case the weak mixing angle is calculated asIn Eq. (7) \(m_W(Q)\) and \(m_Z(Q)\) denote the \(\overline{\text {DR}}'\) \(W^\pm \) and \(Z^0\) masses, respectively, calculated as$$\begin{aligned} \theta _W(Q) = \arccos \left( \frac{m_W(Q)}{m_Z(Q)}\right) . \end{aligned}$$(7)with \({V \in \{ W, Z \}}\) and \(\varPi _V\) being the corresponding one-loop vector boson self-energy.$$\begin{aligned} m_V^2(Q^2) = M_V^2 + \frac{1}{(4 \pi )^2}\mathfrak {R}\varPi _V(Q^2, p^2 = M_V^2) \end{aligned}$$(8)

### 2.4 Boundary value problem solvers

The CNMSSM BVP, however, cannot be solved in this way with the TSS. There, if one chooses \(m_0^2\), \(M_{1/2}\), and \(A_0\) as input at the GUT scale, the parameters \(\lambda \), \(\kappa \) and \(v_s\) would need to be fixed by the EWSB equations at the scale \(M_S\). Since \(\lambda \) and \(\kappa \) are dimensionless parameters which enter most NMSSM \(\beta \) functions, the iteration between the scales becomes unstable and the TSS does not converge.

If the relation between input and output parameters is not injective (which occurs in both the CMSSM and CNMSSM), scanning over one parameter while obtaining the other one as output, and vice versa, allows the search for multiple solutions of the BVP. This procedure is shown by the red solid line in Fig. 3, where \(\mu (M_S)\) is used as input and \(m_0^2\) is output. One immediately sees that with the SAS one can find up to four solutions for \(\mu \) around \(m_0 \approx 3.3\,\text {TeV} \), which have not been found by the TSS. In Sect. 3 we use this procedure to study in depth the physical origin of the multiple solutions found in Refs. [6, 7, 8].

In models where the TSS would require dimensionless parameters to be output, as for example in the CNMSSM or CE\(_6\)SSM [19, 23], the SAS enables one to take them as input, which yields a stable iteration between the high and low scales. In Sect. 4 we use this feature in the CNMSSM to take the parameters \(\lambda (M_S)\), \(\kappa (M_S)\), and \(v_s(M_S)\) as input and obtain \(m_0^2\), \(M_{1/2}\), and \(A_0\) as output.

If the derivative \(\mu '(m_0)\) approaches zero, as happens for example in Fig. 3 around \(\mu \approx 800 \,\text {GeV}\), scanning over \(\mu \) is no longer suitable. Invoking the TSS to vary \(m_0\) instead and receiving \(\mu \) as an output allows to study a region of parameter space which cannot be accessed via the SAS as easily.

In cases where both solvers find the same unique solution it is a priori not clear whether the TSS or the SAS is the better choice. A combination of both allows a more complete study and a comparison validates the equivalence of the two solvers. In those cases, the TSS in general converges much faster than the SAS does.

## 3 Multiple solutions in the CMSSM

### 3.1 Effects from light SUSY particles

^{1}

First we consider the case where \(\{ M_Z, G_F\}\) are chosen as input, see Fig. 4. In the enlarged subplot of Fig. 4a one finds one spike at \(\mu \approx 48.2 \,\text {GeV}\). Zooming in further reveals one additional kink in Fig. 4b at \(\mu \approx 49.4 \,\text {GeV}\). If \(\{ M_Z, M_W\}\) are chosen as input (see Fig. 5), the shape of the curve \(m_0^2(\mu )\) is different: In the zoomed subplot in Fig. 5a one finds two spikes at \(\mu \approx 37.8 \,\text {GeV}\) and \(\mu \approx 48.16 \,\text {GeV}\), respectively. Zooming in further reveals two additional kinks in Fig. 5b for \(\mu \approx 48.15 \,\text {GeV}\) and \(\mu \approx 49.4 \,\text {GeV}\), respectively. In total one finds two kinks for positive \(\mu \) for \(\{ M_Z, G_F\}\) and four kinks for \(\{ M_Z, M_W\}\).

^{2}\(\lambda (p^2, m_1^2, m_2^2)\), has two roots

Since the entries of the chargino mixing matrices \(U^-\) and \(U^+\) are independent, there is no relation which would guarantee the cancellation \(C_L + C_R = 0\), as long as at least one chargino appears in the loop. Hence, the diagrams with \(\chi _1^0 \chi _1^\pm \), \(\chi _2^0 \chi _1^\pm \), and \(\chi _1^\pm \chi _1^\pm \) in the loop contribute non-vanishing singularities to \(\varPi _V\).

The other three relevant diagrams are pure \(\chi _i^0\chi _j^0\) contributions to the \(Z^0\) self-energy. For a \(Z^0\chi _i^0\chi _j^0\) vertex the relation \(C_L^* = -C_R\) holds [17]. Since the \(C_{L/R}\) are in general complex quantities, this property is not sufficient for \(C_L + C_R\) to vanish. If the neutralinos are identical, however, \(C_L\) and \(C_R\) become real and cancel when added.

We conclude that the \(Z^0\) self-energy diagrams with identical neutralinos in the loop do never give a spike, since the singular terms in \(\varPi _Z\) have a vanishing coefficient in the limit \(\lambda \rightarrow 0\). The singularities (kinks or spikes) originate only from diagrams with light \(\chi _1^0 \chi _1^\pm \), \(\chi _2^0 \chi _1^\pm \), \(\chi _1^\pm \chi _1^\pm \), or \(\chi _1^0 \chi _2^0\) in the loop.

Note, that at the singular points spikes or kinks can appear in the overall vector boson self-energies. However, only spikes lead to multiple solutions, because there the sign of the derivative around the singular point changes, which results in a turning point. Whether a singularity causes a spike or a kink depends on the relative sign between the \(B_0\) function which causes the singularity and other loop corrections to the self-energy. The relative signs generally depend on the regarded model as well as on the input parameters.

### 3.2 Effects from non-linear parameter inter-dependencies

The non-monotonic behavior of \(m_0(\mu )\) in the region \(\mu \approx -500 \,\text {GeV}\) is depicted in Fig. 8. Multiple solutions appear in this region, because the function \(m_0'(\mu )\) changes its sign such that \(m_0(\mu )\) has a minimum at \(\mu = -540.5 \,\text {GeV}\). For too small values of \(\mu \lesssim -545 \,\text {GeV}\) there is no physical solution because the running masses of the neutral Higgs bosons become tachyonic in this region of parameter space. In the following we study the parameter interplay which is responsible for the existence of the minimum.

### 3.3 Multiple solutions for \(M_{1/2}\) and \(A_0\)

CMSSM input/output parameters

Solver | Input | Output |
---|---|---|

TSS | \(m_0^2\), \(M_{1/2}\), \(A_0\), \(t_\beta \) | \(\mu \), \(B\mu \) |

SAS1 | \(\mu \), \(M_{1/2}\), \(A_0\), \(t_\beta \) | \(m_0^2\), \(B\mu \) |

SAS2 | \(\mu \), \(m_0^2\), \(A_0\), \(t_\beta \) | \(M_{1/2}\), \(B\mu \) |

SAS3 | \(\mu \), \(m_0^2\), \(M_{1/2}\), \(t_\beta \) | \(A_0\), \(B\mu \) |

Similarly to SAS2, \(A_0\) can be treated as output parameter by solving the EWSB equations for \(A_0\) (SAS3) instead. Also in this case the ansatz is quadratic in \(A_0\), so up to two distinct solutions are possible. In Fig. 10a, b we show \(A_0(\mu )\) and \(\mu (A_0)\) for \(M_{1/2} = 660 \,\text {GeV}\), \(t_\beta = 40\), and \(m_0 \in \{ 3000, 3500 \} \,\text {GeV}\). We find that for \(m_0 = 3500 \,\text {GeV}\) (Fig. 10b) the SAS can scan over both solution branches, while for \(m_0 = 3000 \,\text {GeV}\) (Fig. 10a) the two branches have merged. The multiple solutions around \(\mu \approx 0\) have vanished and a region without solutions (\(|\mu | \lesssim 300 \,\text {GeV}\)) has emerged. The curve \(\mu (A_0)\) becomes flat around \(A_0 \approx 1000 \,\text {GeV}\) and SAS3 is not suitable to find solutions in the region where the two solution branches merge; the two-scale solver, however, is able to find more solutions in this case. Besides this, the multiple solutions that we obtain for SAS3 are again the same as found for SAS1 in Sect. 3.1.

In conclusion, scanning over \(\mu \) while receiving either \(M_{1/2}\) or \(A_0\) as output does not give us any new solutions compared to the \(m_0^2\) output search strategy. We nevertheless have been able to reproduce the previous results in a consistent manner and also our expectation of finding two branches of solutions for parameters of mass dimension one has been fulfilled. Furthermore, the same singular structures for small values of \(\mu \) have appeared for SAS2 and SAS3. Note also, that both the two-scale and the semi-analytic solver had to be used to find all solutions in the considered parameter regions.

### 3.4 Is the CMSSM still challenged?

In this section we investigate the relevance of multiple solutions for globally fitting the CMSSM to experimental and observational data. As a reference point we use Ref. [4] (“Killing the CMSSM softly”), in which the program Fittino was used. The idea was to scan over a reasonable region of the CMSSM input parameter space and to determine the fit point which is the most compatible with the considered observables. Furthermore, this paper was the first to derive a consistent *p*-value for the CMSSM from toy experiment.

#### 3.4.1 The results of “Killing the CMSSM Softly”

We briefly list the observables which have been used in the analysis of Ref. [4]. As for the precision observables there are the anomalous magnetic moment of the muon \(a_\mu \), the effective weak mixing-angle \(\sin {\theta _\text {eff}}\), the top quark and *W* boson masses as well as the b quark/B meson branching ratios. Additionally, different combinations of Higgs observables, like e.g. the SM Higgs boson mass and its decay channels, as observed by the ATLAS/CMS experiments at the LHC, and the dark matter relic density \(\varOmega h^2\), as measured by the Planck collaboration, are incorporated.

^{3}FlexibleSUSY finds

The Standard Model prediction for the anomalous magnetic moment of the muon \(a_\mu \) deviates from the observed value at a \(3.5 \sigma \) level. To account for this, a successful SUSY model is expected to give an additional contribution to \(a_\mu \) of the order \({30 \times 10^{-10}}\) [38]. The best-fit point predicts a correction \({a_\mu ^\text {SUSY} \sim 4 \times 10^{-10}}\), which is far too small. This discrepancy is the main reason for the smallness of the *p* value; when taking \((g-2)_\mu \) into account, one finds \(p = (4.9 \pm 0.7) \%\), whereas leaving it out results in the much larger \(p = (51 \pm 3) \%\).

When performing a global fit of a model to known observables, all possible mathematical solutions to the formulated boundary value problem need to be taken into account. The TSS of SPheno, however, which was used in Ref. [4], does not necessarily find all solutions. For this reason we study in the following section whether further solutions to the CMSSM BVP can be found around the best-fit point with the semi-analytic approach.

#### 3.4.2 Multiple solutions around the best-fit point

In Fig. 11b, c the semi-analytic solver does not find any additional solutions compared to the two-scale solver. In the case of Fig. 11c the curve \(\mu (A_0)\) becomes too flat around \(A_0 \approx 1500 \,\text {GeV}\) to allow an efficient scan over \(\mu \) with the SAS3. For some values of the scan parameters, for instance the region where \(0< M_{1/2} < 400 \,\text {GeV}\) in Fig. 11b, we do not find a physical solution due to either tachyonic down-type sleptons, up-type squarks, or Higgs bosons.

Concluding, we find that the best-fit point is far off from the regions in which multiple solutions can occur. One reason is that around the best-fit point all SUSY particles are heavy enough to escape the experimental constraints, while our previous analysis has shown that additional solutions tend to occur in regions of parameter space where at least some superpartners become light.

## 4 Multiple solutions in the CNMSSM

In this section we study the occurrence of multiple solutions for a given set of parameters \(\{m_0^2\), \(\lambda \), \(\kappa \}\) in the CNMSSM. Exchanging \(m_0^2\) with \(\mu _{\text {eff}}\) allows us to search for multiple solutions in a similar fashion as we did for the CMSSM. It has to be noted, however, that we now also take the GUT parameters \(M_{1/2}\) and \(A_0\) to be output of our algorithm and not input as we did in the CMSSM.

### 4.1 Study of multiple solutions with the semi-analytic approach

In Fig. 12 we show a scan over \(\mu _{\text {eff}}(M_S)\) for fixed values of \(t_\beta \), \(\lambda \), and \(\kappa \). The output parameters \(\{m_0^2\), \(M_{1/2}\), \(A_0\}\) are shown on the abscissae. The curves are discontinuous due to the existence of two distinct solutions branches, which differ from each other in their sign of \(M_{1/2}\). In order to distinguish the branches, solutions with \({{\,\mathrm{sign}\,}}(M_{1/2}) = \pm 1\) are marked as red and cyan dots, respectively. If we were able to pre-select one of the branches, a scan should yield a continuous relation between the output parameters and \(\mu _{\text {eff}}\).

The relation between \(M_{1/2}\) and \(\mu _{\text {eff}}\) is nearly linear, as one would expect for two parameters of mass dimension one. The two branches with different \({{\,\mathrm{sign}\,}}(M_{1/2})\) are related to one another by an approximate central symmetry, see Fig. 12b. The reason for the resulting discontinuous cross-like shape is that, as explained above, for a given value of \(\mu _{\text {eff}}\) the semi-analytic solver usually can find a solution for one value of \({{\,\mathrm{sign}\,}}(M_{1/2})\), but not simultaneously for the opposite one. A similar behavior can be found for \(A_0(\mu _{\text {eff}})\) in Fig. 12c, where the semi-analytic solver can find one solution for fixed \(\mu _{\text {eff}}\) at most. Here, however, the sign of \(A_0\) is fully determined by the sign of \(\mu _{\text {eff}}\) and the output parameter differs only slightly between the two solution branches.

### 4.2 Mass spectra for two different solutions of a single CNMSSM parameter point

CNMSSM parameter points

Open image in new window point 1 | \(\bigstar \) point 2 | |
---|---|---|

\(\mu _{\text {eff}}\)/GeV | \(-2013\) | 553 |

\(M_{1/2}\)/GeV | 1556 | 469 |

\(A_0\)/GeV | \(-1495\) | 401 |

\(M_h\)/GeV | 119.5 | 110.3 |

\(M_{\chi ^0_1}\)/GeV | 673.3 | 189.7 |

\(M_{\tilde{\tau }_1}\)/GeV | 558.1 | 169.8 |

In any case, our analysis shows that the different solutions can have a significantly different phenomenology and the SAS is a useful tool to find and study multiple solutions in this model. However, since \(M_{1/2}\) and \(A_0\) are output parameters in our SAS formulation of the CNMSSM BVP, viability conditions such as the ones given in Ref. [14] cannot be enforced from the start and have to be tested on the output parameters.

## 5 Conclusions

Light neutralinos and charginos can lead to singular points in the one-loop \(W^\pm \) and \(Z^0\) self-energies, which translate to singular points in the function \(m_0^2(\mu )\). At these points the derivative of \(m_0^2(\mu )\) can change its sign, which leads to multiple branches in the inverse functions \(\mu (m_0^2)\). The position of the singular points is given by the light neutralino/chargino masses. The number of singular points depends on their couplings to the \(W^\pm \) and \(Z^0\) bosons and on the formulation used to determine the \(\overline{\text {DR}}'\) weak mixing angle from physical observables.

A non-linear inter-dependence between the parameters \(m_0^2\) and \(\mu \) can lead to a minimum of the function \(m_0^2(\mu )\), resulting in the appearance of multiple branches in the inverse function \(\mu (m_0^2)\) around that minimum.

*p*value for the CMSSM remains unaltered. It has to be noted, however, that the ongoing “Muon \(g - 2\) Experiment” at Fermilab [39] might measure \(a_\mu \) to agree with the SM predictions and render the CMSSM fit perfectly good again by lifting the

*p*value to \((51 \pm 3)\%\).

Multiple solutions around \(\mu _{\text {eff}}\lesssim M_{W,Z}\) tend to not occur, because in the limit \(\mu _{\text {eff}}\rightarrow 0\) also \(m_0^2\) and other supersymmetry-breaking parameters vanish or become of the order of the electroweak scale, which leads to light or tachyonic scalar particles, i.e. unphysical solutions of the BVP.

For small \(m_0^2\) up to three solutions for \(\mu _{\text {eff}}\) can occur due to a non-linear inter-dependence between these two parameters, imposed by the EWSB equations and the \(\beta \) functions. The different solutions may have significantly different physical spectra because of different values for \(M_{1/2}\), \(A_0\) and \(\mu _{\text {eff}}\).

## Footnotes

- 1.
The singularity at \(\mu = 0\) originates from a massless chargino entering the 1-loop threshold correction for \(\alpha (M_Z)\). The region with \(\mu = 0\) is therefore strongly constrained by experimental data and is thus not discussed in the following.

- 2.
\(\lambda (a,b,c) = a^2+b^2+c^2-2ab-2ac-2bc\).

- 3.
The solution with negative \(\mu \) is located in an unphysical region where the

*CP*-odd Higgs boson becomes tachyonic.

## Notes

### Acknowledgements

We kindly thank Ben Allanach, Peter Athron, Dylan Harries and Werner Porod for helpful discussions.

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