ScannerS: constraining the phase diagram of a complex scalar singlet at the LHC
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Abstract
We present the first version of a new tool to scan the parameter space of generic scalar potentials, ScannerS (Coimbra et al., ScannerS project., 2013). The main goal of ScannerS is to help distinguish between different patterns of symmetry breaking for each scalar potential. In this work we use it to investigate the possibility of excluding regions of the phase diagram of several versions of a complex singlet extension of the Standard Model, with future LHC results. We find that if another scalar is found, one can exclude a phase with a dark matter candidate in definite regions of the parameter space, while predicting whether a third scalar to be found must be lighter or heavier. The first version of the code is publicly available and contains various generic core routines for tree level vacuum stability analysis, as well as implementations of collider bounds, dark matter constraints, electroweak precision constraints and tree level unitarity.
Keywords
Dark Matter Higgs Boson Large Hadron Collider Higgs Doublet Vacuum Expectation Value1 Introduction
The recent discovery of a scalar particle [2, 3] at CERN’s Large Hadron Collider (LHC) has boosted the activity in probing extensions of the scalar sector of the Standard Model (SM). So far, the experimental results indicate that this scalar is compatible with the SM Higgs boson. However, many of its extensions are also compatible with the present experimental results. In fact, we know that several of them will never be completely disproved even if the Higgs has all the properties one expects from a SM Higgs. The limit where this occurs is known as the decoupling limit and is characterised by a light Higgs with SM couplings to all other known particles while the remaining scalars are very heavy. It is possible though that other scalars are waiting to be found at the LHC or that a more precise measurement of production cross sections and branching ratios of the newly found 125 GeV scalar reveals meaningful deviations from the SM predictions. As one needs to be prepared to address that scenario we have developed a new code to deal with the vacuum structure of scalar extensions of the SM. The code presented in this work, ScannerS [1], is intended to contribute to the analysis of the different scenarios that can be suggested by the LHC experiments.
In the SM, electroweak symmetry breaking (EWSB) is achieved by one complex SU(2)_{ L } scalar doublet. Although the pattern of EWSB can be correctly reproduced by this one scalar doublet, the SM is not able to accommodate a number of experimental results such as the existence of dark matter or the measured baryon asymmetry of the universe. Adding a scalar singlet to the potential could provide a viable dark matter candidate [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] as well as means of achieving electroweak baryogenesis by allowing a strong first-order phase transition during the era of EWSB [15, 16, 17, 18, 19]. Although minimal, this extension provides a rich collider phenomenology leading to some distinctive signatures that can be tested at the LHC [20, 21, 22, 23, 24, 25, 26, 27].
In this work we focus on the conditions on the parameters that lead to the existence of a global minimum in the scalar potential. However, contrary to previous works, we impose these conditions not to single out a particular model (or phase), but rather use that information to distinguish between possible coexisting phases. This way, we expect to identify properties of the models by classifying the possible phases as function of the parameter space point. We have also included in the code the most relevant theoretical and experimental constraints both from dark matter and collider searches. Our goal is therefore to investigate the possibility of using measured experimental quantities (e.g. the mass and branching ratios of a new light scalar at the LHC), to automatically exclude one of the possible phases (e.g. a phase with a dark matter candidate) using the phase diagram of the model. To that purpose we scan over the entire parameter space subject to the most relevant constraints and plot the results in projections that include physical quantities whenever possible.
As the ultimate goal of ScannerS is to serve as a tool that can be used for general scalar sectors, the core routines of the code can already be used for some larger extensions of the scalar sector in terms of its field content. The core includes generic local minimum generation routines (with Goldstone/flat direction identification as well as a-priori curved directions/symmetries) and tree level unitarity check routines. However, since the first version of the code has been tested extensively with the complex singlet models we present here, routines for testing electroweak observables can be used only for extensions with n-singlets, and global minimum and boundedness from below routines are defined on a model by model basis by the user. The same naturally applies for some of the analysis of experimental bounds (since such analysis will depend on the model that is being analysed) though some data tables from several experiments are generic and included in the core. We expect in the near future to automatise global minimum and boundedness from below routines to be in the core of the program. Also note that some tasks, such as the choice of experimental bounds analysis and model specific theoretical constraints, are naturally user dependent. For this reason, some routines of the program are editable by the user, as well as an input Mathematica file where the model is specified. Details on the structure of the program and on how to use it can be found in [1] (see also Sect. 3).
The structure of the paper is the following. In Sect. 2 we describe the scalar potential of the models we will study. We derive the conditions to be fulfilled for the minimum to be global at tree level for those specific models, based on symmetries, and classify the various phases for each model. In Sect. 3 we address the problem of efficiently performing a scan of the parameters space in a more general perspective. We propose a method to generate a local minimum by generating vacuum expectation values (VEVs), mixings and masses uniformly, and obtaining some of the (dependent) couplings from the linear systems of equations which characterise the minimum. In Sect. 3.2 we discuss the generic implementation of tree level unitarity bounds. In Sect. 4 we implement constraints from various experimental sources, such as electroweak precision observables, LEP bounds, dark matter bounds, and constraints from Higgs searches at the LHC. Finally, we conclude with a discussion of our main results in Sect. 5.
2 The models
We consider an extension of the scalar sector of the SM that consists on the addition of a complex singlet field to the SM field content. Our starting point is the most general renormalisable model, invariant under a global U(1) symmetry. We then consider two models with explicit breaking of this symmetry, but where \(\mathbb{Z}_{2}\)-type symmetries are preserved for one or two of the components of the complex scalar singlet. Each model is classified according to phases associated with the spontaneous symmetry breaking pattern of the vacuum.
Considering the allowed parameter space, each model can exhibit more than one global minimum with the correct pattern of EWSB for fixed couplings. In these models, the mixing matrix is usually lower dimensional, allowing for one or two unstable scalar bosons that could be detected at the LHC and at future colliders, together with two or one dark matter candidates, respectively. However, if all \(\mathbb {Z}_{2}\) symmetries are broken we can end up with three unstable scalars (and no dark matter candidate) complicating the signatures and making a clear identification of the Higgs boson a much more difficult task. Our aim is to investigate the parameter space of these extensions as to identify the properties of the models which are not yet excluded by theoretical and experimental constraints. Furthermore, we want to investigate the possibility of, given a set of experimental measurements related to the scalar sector at the LHC, that we can automatically use the phase diagram of the model to exclude one of the phases, e.g. a phase with dark matter or a phase where all the scalars are mixed.
As previously stated, simpler versions of this model were already discussed in the literature. In most cases a specific model is singled out by imposing additional symmetries on the model. Because one of the main motivations for adding a scalar singlet to the SM is to provide a dark matter candidate, models with one or two dark matter candidates are then analysed and confronted with experimental results. Here, we are interested in applying our new tool ScannerS to various versions of such models, while applying the latest experimental bounds, together with theoretical constraints. The points that pass all constraints during the scan are then classified according to the phase they are in (see classification below) and plotted, whenever possible, in a physical projection of the parameter space, such as a mass of a new particle or a measurable rotation angle between group and mass eigenstates. A physical measurement allows us in some cases to discriminate a phase with dark matter candidates from one with no dark matter candidates. Consequently, the phase which is realised for a given model, can in such cases be decided by experiment.
- Model 0, U(1) symmetry with up to two dark matter candidates. This is obtained by imposing a U(1) symmetry on the complex singlet field, which eliminates the soft breaking terms, thus a _{1}=b _{1}=0. There are two possible phases:The purpose of this article is two fold: (i) describe a way to use the program to compare phases of a model, (ii) apply the program to a variety of simples cases. Since in Model 0 the broken phase is not allowed, this does not address purpose (i). Furthermore, the unbroken phase has a similar couter-part in Model 1 (unbroken phase—see below). Thus we will not study Model 0 in the remainder of our discussion.
- 1.
\(\langle\mathbb{S}\rangle=0\) at the global minimum (symmetric phase). Then we have two degenerate dark matter particles. In this scenario the model is equivalent to two independent real singlets of the same mass and quantum numbers.
- 2.
\(\langle\mathbb{S}\rangle\neq0\) at the global minimum (broken phase). The U(1) symmetry is spontaneously broken, then there is an extra scalar state mixing with the Higgs and a (massless) Goldstone boson associated with the breaking of the symmetry. Since the phase of the complex singlet is unobservable, without loss of generality we can take 〈S〉≠0 and 〈A〉=0 for such phase and A is the dark Goldstone particle. This phase is however strongly disfavoured by observations of the Bullet Cluster [7, 25, 28, 29]. These observations can be used to constrain the mass of the dark matter particle as a function of the value of δ _{2}. Hence, unless δ _{2} is vanishingly small, a zero mass dark matter particle is ruled out.
- 1.
- Model 1, \(\mathbb{Z}_{2}\times\mathbb{Z}_{2}'\) symmetry with up to two dark matter candidates. This model is obtained by imposing a separate \(\mathbb{Z}_{2}\) symmetry for each of the real components of the complex singlet. The \(\mathbb{Z}_{2}\) symmetries imply that the soft breaking couplings are a _{1}=0 and \(b_{1}\in\mathbb{R}\) (6 real couplings in the scalar potential & no other couplings are generated through renormalisation). Specialising the minimum conditions (3) we obtain the following qualitatively different possibilities for minima with v ^{2}≠0:
- 1.
\(\langle\mathbb{S}\rangle =0\), no mixing and two dark matter candidates (symmetric phase).
- 2.
〈S〉=0 or 〈A〉=0, a singlet component mixed with the Higgs doublet and a dark matter candidate (spontaneously broken phase). One can show, by noting that swapping S↔A only changes the sign of b _{1}, that without loss of generality we can take 〈A〉=0, while still covering the full parameter space (this is so because \(b_{1}\in \mathbb{R}\), and the potential only depends on squares of the VEVs). This is true in our scans only because we will adopt the strategy of first generating a locally viable minimum and, only after, to check all possibilities for minima below the one generated.
- 1.
- Model 2, One \(\mathbb{Z}_{2}'\) symmetry with up to one dark matter candidates. This is obtained by imposing a \(\mathbb{Z}_{2}'\) symmetry on the imaginary component A. Then the soft breaking couplings must be both real, i.e. \(a_{1}\in\mathbb{R}\) and \(b_{1}\in\mathbb{R}\). Looking at the minimum conditions we find the following cases:
- 1.
〈A〉=0, i.e. mixing between h (SM Higgs doublet fluctuation) and S only (symmetric phase). In this case we can take \(\langle S\rangle\in\mathbb{R}^{+}\) as long as a _{1} runs through positive and negative values.
- 2.
\(\langle \mathbb{S}\rangle \neq0\), i.e. both VEVs non-zero and mixing among all fields (broken phase).
- 1.
Phase classification for the three possible models
Model | Phase | VEVs at global minimum |
---|---|---|
\(\mathbb{U}(1)\) | Higgs + 2 degenerate dark | \(\langle \mathbb{S}\rangle =0\) |
2 mixed + 1 Goldstone | 〈A〉=0 ( Open image in new window ) | |
\(\mathbb{Z}_{2}\times\mathbb{Z}_{2}'\) | Higgs + 2 dark | \(\langle \mathbb{S}\rangle =0\) |
2 mixed + 1 dark | 〈A〉=0 ( Open image in new window ) | |
\(\mathbb{Z}_{2}'\) | 2 mixed + 1 dark | 〈A〉=0 |
3 mixed | \(\langle \mathbb{S}\rangle \neq0\) ( Open image in new window ) |
We have chosen to start the studies with our new tool with these models, because they already have a great diversity of physically different phases (which provide some interesting results—see Sect. 5), while allowing us to test the routines of the code we propose to develop. We have reproduced several results presented in the literature with a very good agreement. In the next sections, we describe the ScannerS scanning strategy and the main tree level theoretical constraints to generate a stable vacuum with the correct symmetry breaking pattern.
3 The scanning method
To implement the scan of the parameter space of the models, we have developed a dedicated tool ScannerS which can be used for more generic potentials. Our method is based on a strategy of reducing as many steps as possible to linear algebra, since these are computationally less expensive. In its present form, the program requires the user to decide the symmetry breaking pattern to scan over, as well as a choice of ordering of the couplings in the theory (details below). This is done in a Mathematica notebook which generates an input file for the C++ program. Before describing the method in detail, let us discuss the general idea.
A possible way of performing the scan for a generic scalar potential (strategy 1), and determining the spectrum of scalars is: (i) scan the couplings λ _{ a } uniformly in chosen ranges, (ii) determine all stationary points by solving the (non-linear) stationarity conditions, (iii) check if any are minima and choose the global one, (iv) if yes, accept the point, compute the mass matrix, diagonalise and check if the masses of the scalars (\(m_{i}^{2}\)) and mixing matrix (M _{ ij }) are consistent with the symmetry breaking imposed. This method contains two steps which are quite expensive, computationally, which are executed before we know if the minimum has the desired properties. The first is the determination of the stationary points. For a generic multi-scalar model this involves finding the solutions of a polynomial system of equations in the VEVs.^{1} The second, not as expensive, is the diagonalisation of the mass matrix at the global minimum.
Finally, this set of parameters (VEVs, masses and mixings) is more directly related to physical properties of the scalar states, so it is a more natural choice of parametrisation.
3.1 Generation of a local minimum
With this procedure, we end up generating a point in parameter space, and all the properties of the physical states are determined (such as masses and mixing matrices), having avoided the problem of solving a system of non-linear equations. The price to pay is that we have delayed checking if the minimum is global. However, the advantage of this procedure is that we do not spend any time in points of parameter space where there is no stationary point with the correct properties. Furthermore, we can add to this procedure more constraints on the local minimum (if they are computationally quick to check), before checking if it is global. If we use strategy 1, the first steps involve a computationally intensive non-linear problem, which will be wasted each time a point in parameter space is rejected, whereas with strategy 2 we generate a local minimum with the desired properties quickly and only then do we have to perform the computationally intensive steps.
3.2 Tree level unitarity
4 Constraints and phenomenological potential
4.1 Electroweak precision observables
4.2 LEP and LHC bounds
With the recent discovery of a Higgs like boson by the ATLAS and CMS collaborations, we now have a very strong constraint on BSM models. In any extension of the SM one of the scalars is bound to have a mass of approximately 125 GeV. At the same time both LHC experiments have also constrained any new scalar couplings to the SM particles, that is, they have provided us an exclusion region for μ _{ i } as function of the mass of the scalar H _{ i }. These bounds are also included in the program and the details are as follows. We assume a Higgs boson with mass m _{ h }=125 GeV, and allow for a signal strength μ _{ i } in the interval 1.1±0.4 [36]. In the mass regions where a scalar particle is excluded, we apply the 95 % CL combined ATLAS upper limits on μ _{ i } as a function of m _{ i } [36] for all other non 125 GeV scalars, as long as their production is allowed at the LHC.
4.3 Dark matter experimental bounds
Another important constraint on these models is the elastic cross section from DM scattering off nuclear targets. The most recent direct detection DM experiments have placed limits in the spin-independent (SI) scattering cross section (σ _{SI}) of weakly interacting massive particles (WIMPS) on nucleons. The most restrictive upper bound on the SI elastic scattering of a DM particle is the one from XENON100 [40]. In ScannerS we have included the limits of σ _{SI} as a function of the dark matter mass as presented in [40].
5 Discussion
In this section we analyse the results of full scans over all parameter space, for the various phases of each model. We focus on models 1 and 2, which we denote by cSM1 and cSM2 respectively. We have performed two main scans for each model. One with a smaller hyper-cubic box in parameter space to allow for a better resolution of the region being scanned over, and another wider scan to check which boundaries did not change significantly. Unless stated otherwise, we always use the wider scan.^{6} We will also present a scan with some of the constraints removed to clarify the appearance of certain boundaries in the allowed regions of parameter space.
We start by presenting some results for model 1 (cSM1) where we label the phase with two dark matter candidates as “2DM” and the phase with only one dark matter candidate by “1DM”. The key of all figures contains an extra label for each colour of the points, indicating whether all the bounds/constraints discussed in previous sections have been included, and when not, the corresponding constraints that have been removed, are indicated.
There are regions exclusive to the phase with two dark matter candidates, and also very small regions where just one dark matter state is possible. A measurement of the mass of a dark matter candidate could give us a hint on the phase the model is in. However, even if it is possible in some cases to distinguish between phases with actual physical measurements, it is rather hard to discriminate between phases in model 1, even more so because in some regions it requires independent measurements of the properties of a dark matter candidate and an unstable scalar mixing with the Higgs.
In the right panel of Fig. 2 we display the effects of applying the different bounds to the “1DM” phase obtained for a wide scan. In this case the horizontal axis has the variable |M _{2h }| which is the mixing matrix element with the SM Higgs doublet component of the non-dark matter state. The points have been overlaid following the order in the key—first in the key list is the bottom layer in the plot. It can be seen that EWPO constraints cut out a considerable portion of the parameters space and are responsible for the upper right boundary between the allowed region for the one dark matter phase as compared to the mixed phase in the left panel. Nevertheless, the theoretical constraints alone are responsible for the bottom boundary as seen from comparing the left and the right plots. Note that the apparent top left boundary is not meaningful, and should shrink to the vertical axis if we perform increasingly wider scans in parameter space, as discussed in the previous paragraph.
6 Conclusions
We have presented a new tool, ScannerS, devoted to the search for global minima in multi-Higgs models. In this work we have applied it to some versions of a simple extension of the SM—the addition of a complex singlet to the SM doublet, with some symmetries. The code includes the most relevant theoretical and experimental bounds. Our main focus is in distinguishing the possible phases of each model by using the present experimental data both from the LHC and from dark matter experiments.
Once we have identified our working models based on symmetries, we have excluded all phases that did not display the correct electroweak symmetry breaking pattern. We ended up with two possible phases for each model. In the first model, which we called model 0, one of the phases leads to a massless dark matter candidate which is already excluded by the Bullet Cluster results. The other phase has two dark matter candidates. Because in the allowed phase there is no mixing between scalars, the only way to tell them apart is by actually detecting a dark matter particle. Hence, a study with ScannerS would add no advantageous information to what we know experimentally.
The cases of model 1 and (especially) of model 2 are the most interesting. We have shown that by measuring physical quantities like the particle masses, mixing angles, or quartic couplings we are able, in some particular cases, to pinpoint the phase that is realised in Nature if one of these models applies. Most importantly, in model 2 a simultaneous measurement of the mass of a non-dark matter scalar together with its mixing angle could be enough to exclude a dark matter phase, and simultaneously indicate whether we are observing the lightest or the heaviest of the new scalar states expected in the model. Nevertheless, as we move closer and closer to the SM limit the phases become more and more indistinguishable.
In summary, although the differences that we found between phases of these singlet extensions are restricted only to some measurable quantities, we found it possible to fall into regions where measurements will definitely exclude one of the phases and predict properties of the scalar spectrum. An interesting question is whether such differences between phases can be identified for more complicated scalar extensions or if they can even become more impressive and predictive.
The ultimate goal of ScannerS is to provide the community with a tool that can be used to search for global minima in general scalar sectors. Although the core routines can already be used for an arbitrary scalar sector, the present release [1] requires the user to define the boundedness from below and global minimum routines. In the next release, we expect to provide core routines for such tasks, as well as further analysis examples such as the important case of the two-Higgs doublet model. The present publicly available code includes examples with all the analysis used in this article, which we hope will be a useful starting point for users who intend to explore this tool.
Footnotes
- 1.
The models we study in the next sections offer the advantage of providing closed form solutions for all stationary points, which can be used to test the core routines of the program for consistency.
- 2.
Note: One can always favour keeping couplings of lower dimensional operators as independent, since they are usually easier to interpret in terms of observable quantities. Or alternatively they can be ordered according to any other criterion.
- 3.
This procedure can be continued until all “a priori” eigen-directions are found.
- 4.
A rotation matrix can be generated with uniform probability with respect to the Haar measure, by generating its entries with a Gaussian distribution and then performing a QR decomposition to extract it [30].
- 5.
Note: The latest fit central values differ only very slightly to these numbers so we do not expect a noticeable difference.
- 6.
The exact ranges of the scans that we have performed are presented in Appendix C.
Notes
Acknowledgements
We would like to thank Augusto Barroso, Pedro Ferreira and João P. Silva for comments and suggestions. The work of R.C., M.S. and R.S. is supported in part by the Portuguese Fundação para a Ciência e a Tecnologia (FCT) under contract PTDC/FIS/117951/2010 and by the FP7 Reintegration Grant, number PERG08-GA-2010-277025. R.S. is also partially supported by PEst-OE/FIS/UI0618/2011. R.C. is funded by FCT through the grant SFRH/BPD/45198/2008. M.S. is funded by FCT through the grant SFRH/BPD/ 69971/2010.
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