Dynamical Higgs Field Alignment in the NMSSM

Experimental probes of the recently discovered Higgs boson show that its behavior is close to that of the Standard Model (SM) Higgs particle. Extensions of the SM which include extra Higgs bosons are constrained by these observations, implying either the decoupling of the heavy non-standard Higgs particles or the realization of alignment, associated with vanishing mixing of the SM-like Higgs boson with the non-standard ones. Quite generally, alignment is not enforced by symmetry considerations and hence it is interesting to look for dynamical ways in which this condition can be realized. We show that this is possible in the Next-to-Minimal Supersymmetric Standard Model (NMSSM), in which alignment is achieved for values of the coupling of the Higgs fields to the singlet field that become large close to the Grand Unification (GUT) scale. This, in turn, can be explained by the composite nature of the Higgs fields, with a compositeness scale close to the GUT scale. In this article we present this dynamical scenario and discuss its phenomenological properties.


I. INTRODUCTION
With the discovery of the Higgs boson in 2012, the Standard Model (SM) is complete and the theory of electroweak symmetry breaking confirmed [1,2]. The primary focus of the Large Hadron Collider (LHC) since this discovery has been measurements of the precise properties of the Higgs boson [3], as well as searches for new physics. However, no evidence of new physics beyond the SM has yet been found, and the LHC Higgs boson so far appears to be SM-like.
In light of these results, extensions of the SM have become further constrained, and an interesting area of study is the examination of how extended Higgs theories may include a SM-like Higgs boson [4]. This can be achieved in two ways: either by decoupling of the nonstandard physics, rendering the SM as the effective low energy theory, or by the condition of alignment, associated with the cancellation of the mixing of the non-standard Higgs bosons with the SM-like one. The condition of alignment has been studied in several extensions of the SM, including two Higgs doublet models, the Minimal Supersymmetric Standard Model extension (MSSM) and the next-to-minimal one (NMSSM) [4][5][6][7][8][9][10][11]. While the necessary parameter spaces have been identified and studied in the past, of further interest is the manner through which these parameter spaces may be obtained. Although in certain cases alignment may be associated with symmetry properties [12][13][14][15][16], this is not the case in most extensions of the SM. It is therefore of interest to study whether alignment could be achieved dynamically.
In this paper, we focus on the NMSSM and investigate how one may dynamically obtain Higgs alignment in this theory. We concentrate on the running of the NMSSM parameters up to the Grand Unification (GUT) scale, and examine general implementations of the high-energy theories suggested by such running. Particular focus is placed on Fat Higgs models, which we show may naturally include the elements necessary to satisfy the alignment limit for the doublet sector as well as limited mixing with the singlet. For this to happen, the compositeness scale must be close to the GUT scale. We therefore also examine the interesting coincidence of bottom and tau Yukawa unification at the GUT scale, which may be realized for the same parameter space as the one associated with Higgs alignment. We also consider the possibility of including a well behaved Dark Matter candidate within this scenario. This paper is structured as follows. In Section II, we review the alignment limit of the NMSSM and the relevant conditions on the parameters necessary for alignment. In Section III, we present results from the running of the NMSSM parameters and examine the range of GUT-scale parameter values for which alignment is obtained in the doublet and singlet sectors. We then present an implementation of a Fat Higgs theory which runs down to alignment at the weak scale in Section IV. In Section V, we examine the bottom-and tau-Yukawa unification for our set of low-energy parameters. Finally, in Section VI we present our conclusions.

II. THE ALIGNMENT LIMIT OF THE NMSSM
Within the NMSSM Higgs sector, which contains two doublets and a singlet, there are two methods through which one may obtain a SM-like Higgs of 125 GeV: decoupling and alignment. In the decoupling case, the heavier non-standard Higgs bosons are pushed to high masses, such that the mixing with the SM-like Higgs boson is suppressed. In the case of alignment, the parameters of the Higgs sector are such that the mixing terms of the squared-mass matrix between the SM-like Higgs boson and the neutral, non-SM-like one and singlet are small. More specifically, if we work in the Higgs basis [17,18] in which only one of the doublets acquires a vacuum expectation value and hence is aligned with the SM Higgs doublet, here denoted by the subscript 1, the symmetric CP-even Higgs mass-squared matrix is given generally by and the alignment condition is With minimal mixing, we also therefore have that The alignment limit of the NMSSM and its phenomenological properties have previously been thoroughly investigated in Ref. [9]. Here we give a brief review of the relevant properties.
We define the relevant couplings defining the interaction of the Higgs fields through the superpotential where the Higgsino mass parameter is proportional to the vacuum expectation value of the singlet field µ = λv s . We shall follow the conventions of Refs. [9], [19].
In the Higgs basis {H SM , H N SM , H S }, where H SM denotes the SM-like Higgs, H N SM the non-standard Higgs doublet contribution and H S the singlet contribution, the CP-even Higgs tree-level squared-mass matrix can be explicitly written as Values of the µ parameter close to the weak scale and therefore much lower than the stop masses are preferred in order to obtain a mostly Bino or singlino Dark Matter (DM) candidate and to reduce the fine tuning associated with electroweak symmetry breaking [28,29].
As shown in Eq. (10), the stop loop corrections to M 2 12 not included in M 2 11 are suppressed by µ/M S 1, and one may therefore neglect the stop corrections to find an approximate relation between the values of λ and tan β which satisfy exact alignment. Taking M 2 11 = m 2 h , Eq. (10) gives [9] ( (12)  we have included an uncertainty of 3 GeV characterizing the theoretical uncertainties in the determination of the Higgs mass. Points within this region will be close to fulfilling exact alignment, while points close to this region should have small mixing between the two doublets. We will better define "small" mixing quantitatively in our later analyses.
In order to analyze a possible dynamical origin of these parameters, we are interested in identifying the high energy-scale values of NMSSM parameters which naturally run down to this alignment limit at low energies.
Although the above conditions of alignment have been derived by performing an analysis by including only one loop corrections, models that lead to an appropriate phenomenology at low energies tend to be consistent with those conditions, as shown by the similarity of the phenomenological properties of the benchmark scenarios derived in Ref. [9] compared with more complete numerical analysis as those performed in Refs. [30]- [44].  the suppression of the singlet mixing will be examined in the next section. To quantify how well the points fall within the alignment limit, we vary along M S and X t curves to examine the quantity which reflects the mixing between the two doublets and reduces to − and therefore decrease the mixing. In the effective 2HDM, the deviations of the SM-like coupling may be parametrized by [7,8] From Eqs. (14)- (16) we see that for tan β > 1, the tree-level bottom coupling is the one mostly affected by mixing with the non-standard states and, due to the relevant decay branching ratio of the SM-like Higgs to bottom quarks, it has a relevant effect on all Higgs branching ratios. We plot the quantity |η|, which parametrizes the variation of the bottom coupling, for our weak-scale points in Fig. 4.
Inspection of Fig. 4 shows that the deviation of the parameter |η| is below 0.1 for the majority of points, restricting the deviations of all couplings to values below ten percent, in agreement with current experimental observations [52][53][54] (in this work, we shall not consider the region in which the bottom Yukawa coupling acquires a wrong sign, η 2, which can also be achieved within the NMSSM for heavy singlets [55]). The points on the extreme ends of the tan β region reach larger values of |η|, but do not exceed a deviation of 0.16. Following the same analysis with a value of M A = 400 GeV, we find a maximum value of |η| = 0.08, which follows the expected scaling of approximately 1/M 2 A . We therefore find that a composite Higgs model with a compositeness scale near the GUT scale may naturally lead to the alignment limit for the doublet sector at low energies. In Section IV, we will describe a general implementation of an NMSSM Fat Higgs model with a scale Λ of the order of M GUT .

C. Alignment Condition
As discussed above, the alignment condition in the NMSSM does not arise from a symmetry condition. To further investigate the origin of the alignment in the doublet sector, one can write the effective two Higgs doublet potential For small values of the Higgsino mass parameter compared to the stop mass scale µ/M S 1 -the dependence of the quartic couplings on the stop mass parameters is given, for instance, in Refs. [56,57] -one may take λ 6 λ 7 ∼ 0 as a good approximation. The condition of alignment can then be rewritten as [7] m 2 h = λ 1 cos 4 β + 2λ 3 sin 2 β cos 2 β + λ 2 sin 4 β In the literature, symmetry considerations have been invoked to relate the quartic couplings [12][13][14][15][16]. In particular, the condition λ 1 = λ 2 =λ 3 ensures alignment whenever In the NMSSM, however, the couplings λ 1 and λ 2 differ by the sizable stop loop corrections and these conditions cannot be fulfilled. For moderate or large values of tan β > ∼ 2.5, however, the alignment conditions reduce approximately to λ 2 λ 3 , with Taking into account that one recovers the previously-obtained relation, Eq. (12), which in this regime of tan β reads Moreover, as said above, λ 2 v 2 differs from its tree-level value M 2 Z λ 1 v 2 due to the sizable stop radiative corrections.
The relation λ 2 λ 3 m 2 h /v 2 is therefore an emergent condition arising dynamically in the infrared limit, and it is not coming from any fundamental symmetry. Alignment for smaller values of tan β emerges in a similar way in the infrared limit.

D. Alignment in the singlet sector
We must additionally examine how the mixing with the singlet Higgs might be naturally For the region of λ and tan β obtained by running down from the GUT scale, the value of sin(2β) is approximately 1. We may thusly reduce the singlet-sector alignment condition to the approximate relation where we have assumed that κ/2λ is significantly lower than one, as necessary to obtain a singlino state lighter than the Higgsino one, 2κ/λ < 1, for which a natural Dark Matter candidate may be obtained [29]. Alignment for the singlet therefore additionally depends on the relationship between the parameters M A and µ, which is not determined by the running down from M GU T performed above. We therefore conclude that this alignment condition cannot obviously be imposed through choices in the high-energy theory.
We thusly examine whether one may effectively decouple the singlet due to aspects of the high-energy theory. We note that the addition of a tadpole term can effectively decouple the singlet from the doublet sector by increasing the singlet mass. In particular, the general form for M 2 33 is given by [19] where ξ S is the constant in a tadpole term in the Higgs potential of the form ξ S S ⊂ V H .
A large value of ξ S can lead to large M 2 33 , thereby decoupling the singlet and limiting the mixing with the doublet sector. If the high-energy theory produces a singlet tadpole term in the Higgs potential, as we will examine in the next section, then the singlet mixing may be efficiently suppressed. and (T 5 , T 6 ) transform as singlets under SU (2) L . The tree-level superpotential is given by where S and S are new singlet superfields included to ensure dynamic electroweak symmetry breaking. Making the identifications one obtains a dynamically-generated superpotential of Using Naive Dimensional Analysis [61][62][63][64], one expects that Of particular interest in our case is the very small value of m required to obtain v 0 ≈ O(100) GeV for a compositeness scale of Λ H ≈ 10 16 GeV; in particular, m must be on the order of 10 −1 eV.
We note that a term of the form mT 5 T 6 may arise from the vev of a scalar superfield, in which case one would have a term of the form gΦT 5 T 6 , where g is a dimensionless coupling.
As a scalar superfield, Φ may have the form Φ = ϕ + θθF , where ϕ and F have some vacuum expectation values. When integrating to obtain the potential, one therefore finds an additional term linear in the Higgs singlet S arising from the F −term. Thus, the presence of a tadpole term of the form ξ FŜ in the superpotential may naturally give rise to a tadpole term in the potential of the form ξ S S.
The necessary scales can be estimated based on the values of m we require due to the compositeness scale, as well as the scale of ξ S required to decouple the singlet from the doublet sector. We write the Higgs singlet terms with the vev of Φ = ϕ + θθ F φ by where the first term generates the supersymmetric mass term mT 5 T 6 while the second term generates the tadpole term in the potential. We estimate that ϕ and | F | should both be on the order of a TeV. In order to obtain m ∼ O(10 −1 ) eV, we therefore require g ∼ O(10 −13 ). The scalar part ofŜ then acquires a tadpole term in the potential with we require ξ S on the order of 10 9 GeV 3 for decoupling, which indicates that Λ H is around 10 15 GeV. We thus obtain a similar compositeness scale to the one that matches the NMSSM running, as described in Section III.
The problem now reduces to the generation of the small coupling g. Such a small coupling may be explained by using a seesaw mechanism, similar to the one associated with the Majorana neutrino mass models. In order to propose such a model, let's follow Ref. [58] and introduce two additional SU (2) H doublets T 7 and T 8 . We shall assume the presence of certain flavor symmetries which forbid an explicit T 5 T 6 mass term, but allow mixing between these states and the T 7 and T 8 term via the analogue of a Giudice Masiero mechanism [65] and a T 7 T 8 mass term via the interaction with an additional superfield, Ψ. Under these assumptions, the superpotential reads where the m SUSY term comes from the Giudice Masiero relation between the effective bilinear superfield term and the supersymmetry breaking scale. We shall assume that where F is proportional to the square of the supersymmetry breaking scale, such that the superpartner masses m SUSY F/M GUT , and M is of the order of the GUT scale. Integrating out the heavy superfields T 7 and T 8 , one can identify the supersymmetry conserving and breaking terms that appear at low energies. This can be done diagrammatically. For instance, the supersymmetry breaking tadpole term may be obtained by considering the presence of the scalar mixing terms in the scalar potential, where the first four terms arise from F terms in the superpotential, of the form |∂W/∂T 7 | 2 and |∂W/∂T 8 | 2 , and we replace Ψ by its vacuum expectation value. After integrating out the heavy fields, the above terms lead to a supersymmetry breaking term This induces a tadpole of the right size for the scalar component of S.
Alternatively, one can also obtain the same result by doing a simple expansion considering the supersymmetry breaking terms like a perturbation of the values obtained in the supersymmetric limit. Let's start with the supersymmetric case, with superpotential Integrating out the heavy superfields, we get the effective superpotential This term, together with the supersymmetry breaking term, Eq.(34), leads to the supersymmetric and non-supersymmetric tadpole contributions of the singlet S. We can then formally identify the spectator field Φ introduced in Eq. (30) with where the above expression acquires meaning after decoupling the heavy superfields T 7 , T 8 and performing the above mentioned expansion [66], from which we obtain Hence, we reproduce the diagrammatic result for the supersymmetry breaking tadpole and obtain the required values of the coupling and the effective superfield Φ vacuum expectation values in a natural way.
While the interactions of the singlet field S with the Higgs field have the required structure to obtain alignment, the self interactions of S are not determined in a clear way from our discussion above. We shall assume that the flavor symmetries forbid a superpotential mass This model does not predict the exact value of the non-standard Higgs boson masses, but the moderate values of tan β imply that the production cross section is governed by top-Yukawa induced processes. Due to the alignment condition, which suppresses the decay into pairs of weak gauge bosons or SM-like Higgs bosons [9], and the absence of light singlets, the non-standard Higgs bosons decay mostly into fermion states. Therefore, the decay branching ratio depends on whether the decay into pairs of top-quarks and electroweakinos is allowed.
If top-quark decay is dominant, searches for the heavy Higgs doublets become difficult due to interference effects with the large top-quark production background [67]- [71]. Therefore, the only region that is currently constrained is for low values of tan β < 2 and values of the heavy Higgs mass below about 350 GeV, where the top-quark decay process is absent. The main constraint comes from the decay of the heavy Higgs bosons into τ pairs [72,73] which, however, can be efficiently suppressed if the electroweakinos are light [74].
Regarding the chargino and neutralino sectors, the model provides an acceptable Dark Matter candidate in terms of the lightest neutralino [19]. Assuming this particle to be either predominantly Bino or singlino, spin independent direct detection bounds may be efficiently suppressed provided [29] m χ ∼ ±µ sin 2β, A further phenomenological consideration is the charged Higgs contribution to the b → sγ rate. Within a basic Type II 2HDM model, a light charged Higgs on the order of a few hundred GeV enhances b → sγ rates and therefore becomes constrained by experimental measurements [76][77][78][79]. However, within supersymmetric theories these flavor rates also depend strongly on the contributions from other supersymmetric particles; these include charginos and stops, which can exactly cancel the SM contributions to the b → sγ transition in the limit of exact supersymmetry [80][81][82][83]. Furthermore, there are contributions arising from possible flavor violation in the scalar fermion sector; these can be large corrections arising from gluino-squark loops. This can occur when there is a misalignment between the bases in which the quark and squark mass matrices are diagonalized [84]. In light of this, we do not further consider flavor constraints; however, we have confirmed using NMSSMTools that the models described above can be in agreement with flavor constraints up to the SUSY contributions included in NMSSMTools.

V. UNIFICATION OF h b AND h τ
Although it is not directly related to the alignment in the Higgs sector, another intriguing aspect of the running of the RG evolution from the alignment limit is the unification of h b and h τ at the GUT scale. In addition to obtaining a Higgs sector which is consistent with current experimental constraints, the model also includes a Dark Matter candidate, which is mostly Bino-like and obtains the correct relic density through coannihilation with light singlinos. Moreover, for values of the Dark Matter mass close to −µ sin 2β, direct detection constraints can be avoided in the Bino case. All these conditions may be simultaneously satisfied within these models.
Finally, we stress that the relatively strong values of the top Yukawa coupling lead to the unification of the bottom and tau Yukawa couplings at the GUT scale. This suggests the possible embedding of this theory within a GUT scenario, like SU (5), in which the bottomquark and tau-lepton share the same multiplets. We reserve for future work the construction of such a theory. The large values of κ(M GU T ) do not significantly affect the weak-scale parameter values, which remain near the alignment region. The primary effect of increased κ is a lower value of λ(M Z ), which tends to be reduced by up to about 0.05 relative to the κ = 0 case. Based on the low variation in the results with large κ, we conclude that setting κ = 0 provides a representative analysis.