# R-symmetry for Higgs alignment without decoupling

## Abstract

It has been observed that an automatic alignment without decoupling is predicted at tree-level in a Two-Higgs Doublet Model (2HDM) with extended supersymmetry in the gauge and Higgs sectors. Moreover, it was found that radiative corrections preserve this alignment to a very good precision. We show that it is the non-abelian global \(SU(2)_R\) R-symmetry that is at the origin of this alignment. This differs from previously considered Higgs family symmetries as it is present only in the quartic part of the Higgs potential. It can not be imposed to the quadratic part which has to be generated by \(N=1\) supersymmetry breaking sectors. This absence of symmetry does not spoil alignment at the minimum of the potential. We show how the (small) misalignment induced by higher order corrections can be described as the appearance of non-singlet representations of \(SU(2)_R\) in the quartic potential.

## 1 Introduction

In contrast to fermions and vectors, there is only one known fundamental spin zero particle in Nature: the Standard Model (SM) Higgs boson. Additional fundamental scalars are ubiquitous in Early Universe cosmological and supersymmetric models. In the latter, matter fermions have scalar partners. Also, additional Higgs scalars appear in their electroweak symmetry breaking sectors. Mixings of the observable Higgs with the new scalars are subject to strong constraints by present LHC experiments data. They imply that the observed Higgs, an eigenstate of the scalars mass matrix, is *aligned* with the direction acquiring a non-zero vacuum expectation value (v.e.v). Making all the additional scalars heavy enough, thus decoupling them, is a trivial way to achieve this. But a more interesting option is that alignment emerges as a consequence of specific patterns of the model. The benefits are that less constraints on masses allow to keep new scalars within the reach of future searches at the LHC. Such an *alignment without decoupling* was obtained in [1] and further discussed in [2, 3].

The effective low energy scalar potential of [1], studied in details in [4], corresponds to a peculiar case of a Two Higgs Doublet Model (2HDM)^{1}. Symmetries in the 2HDM ( e.g. [8, 9, 10]) can in particular cases imply alignment without decoupling [11, 12, 13]. Unfortunately, these symmetries have been quoted to lead to problematic phenomenological consequences, as massless quarks [14]. When not due to symmetries, alignment without decoupling remains viable. This situation was discussed for example in [15, 16, 17, 18, 19] for the MSSM and NMSSM. This remains not totally satisfactory as it is due to an ad-hoc specific choice of the model parameters. It is to be contrasted with [1] where the alignment at tree-level is a prediction. It is the purpose of this work to uncover the symmetry at the origin of the automatic Higgs alignment in [1], a success that was shown to survive with a impressive precision when radiative corrections, up to two-loop, are taken into account [3].

The crucial ingredient in [1] is the presence of \(N=2\) extended supersymmetry. Early models have required that \(N=2\) supersymmetry acts on the whole SM states and, as a consequence, suffered from the non-chiral nature of quarks and leptons [20, 21]. Overcoming this issue by allowing both \(N=2\) supersymmetry and chirality is possible in models inspired by superstring theory: orbifold fixed points and brane localizations enable to construct models where different parts preserve different amounts of supersymmetries. In [1] the (non-chiral) gauge and Higgs states appear in a \(N=2\) supersymmetry sector while the matter states, quarks and leptons, appear in an \(N=1\) sector. An important feature of such constructions considered here [1, 4, 22, 23, 24] is that gauginos have Dirac masses [25, 26, 27, 28, 29]. Also these \(N=2\) extended models have implication for Higgs boson physics as discussed in [4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].

It is not totally satisfactory to explain the predicted alignment only by the presence of \(N=2\) extended supersymmetry because this is realized only at a very high energy, the fundamental scale of the theory, while it should be possible to trace back the alignment to a symmetry manifest in the scalar potential of 2HDM. In this work, we will exhibit the relevant symmetry.

The paper is organized as follows. In Sect. 2, the main ingredients of the model are presented succinctly. This allows to define the notation used through this work. In Sect. 3, we show that the potential can be written as a sum of two \(SU(2)_R\) R-symmetry singlet representations. We also show how this implies alignment. In Sect. 4, we review how higher order corrections induce a small misalignment. Section 5 presents our conclusions.

## 2 A glimpse of the model

*SU*(2) triplet \({\mathbf {T}}\). We define

*R*-symmetry. Unfortunately, it leads to tachyonic directions for the adjoint scalars. To overcome this, we must restrict to particular \(N=1\) breaking and mediating sectors [29, 43, 44] (see also [31, 45, 46, 47]). As a consequence, the \(SU(2)_R\) structure is not preserved by the quadratic part of the scalar potential given by the soft breaking terms. Furthermore, we will consider that the remaining \(U(1)_R\) symmetry is broken by the presence of a non-vanishing \(B\mu \) term (keeping zero the coefficients of supersymmetric terms as \(M_S\), \(M_T\), and \(M_O\)). We also take \(\kappa =0\) for simplicity. This avoids then the introduction of extra-doublets as in the MRSSM [31] and allows us to consider an effective 2HDM. Note also that the trilinear terms in the last line of (2.4) will be neglected here, as they were shown to be generically small [29, 44].

## 3 *R*-symmetric Higgs alignment

*SU*(2) gauge couplings, respectively.

*l*the spin and

*m*its projection along the

*z*axis. They were classified in [9]. Here

^{2}:

*SU*(2) Higgs family symmetry [8, 9]. The potential contains only terms that are invariant (singlet) under \(SU(2)_R\).

*S*and

*T*[29]. Avoiding these instability requires peculiar structure of these sectors that is not compatible with the

*R*-symmetry.

These results also allow us to understand why the simple identification of couplings in the Higgs couplings with their \(N=2\) expected values, as it was attempted for the MRSSM in [3], fails to achieve alignment. There, the integration out of the additional doublets breaks the \(SU(2)_R\) R-symmetry explicitly at tree level.

## 4 *R*-symmetry breaking and misalignment

*U*(1) subgroup of \(SU(2)_R\) is not sufficient for alignment as we have contribution from \(|i,0\rangle \) combinations. Here, we have \(\lambda _5 =0\) thus there is no contribution from \(|2,\pm 2>\). The breaking of \(SU(2)_R\) symmetry leads then to a contribution to the \(Z_6\) parameter of order:

*U*(1) and weak interaction

*SU*(2) are small but not zero:

Next, we consider the misalignment from quantum corrections. Radiative corrections to different couplings are generated when supersymmetry is broken inducing mass splitting between scalars and fermionic partners. This happens for instance through loops of the adjoint scalar fields *S* and \(T^a\). However, these scalars are singlets under the \(SU(2)_R\) symmetry and at leading order, when their couplings \(\lambda _S\) and \(\lambda _T\) are given by their \(N=2\) values, their interactions with the two Higgs doublets preserve \(SU(2)_R\). Therefore we do not expect them to lead to any contribution to \(Z_6\). In fact, explicit calculations of these loop diagrams was performed in Eq. (3.5) of [3] and it was found that when summed up their contribution to \(Z_6\) cancels out. This unexpected result is now easily understood as the consequence of the \(SU(2)_R\) symmetry.

*SU*(2). Then the set:

*Q*is the renormalisation scale. This leads to a \(\delta Z_6^{(1\rightarrow 0)}\) induced by \( \delta \lambda _{|1,0>}^{(stops)} \sim \sqrt{3} \delta \lambda _{|2,0>}^{(stops)}\) in (4.6). Thus, chiral matter through their Yukawa couplings contribute to \(Z_6\) with both of \(|1,0\rangle \) and \(|2,0\rangle \) combinations of doublets. The contributions \(\delta Z_6^{(1\rightarrow 0)}\) and \(\delta Z_6^{(2\rightarrow 1)}\) have similar strength but opposite sign so that they lead to a small misalignment compatible with LHC bounds as was shown in [3] by explicit computation.

Note that the the origin of misalignment is not the quadratic terms breaking \(SU(2)_R\). In Sect. 3, we explained how this breaking of \(SU(2)_R\) does not imply loss of alignment but fixes \(\tan \beta \). The misalignment comes the presence of chiral matter and the most important single contribution arises from a large top Yukawa coupling.

## 5 Conclusions

In the 2HDM, the LHC experiments data require one the Higgs squared-mass matrix eigenstates to be aligned with the SM-like direction. If this alignment comes without decoupling, then the additional scalars in the electroweak sector are subject to milder constraints. They might have masses in an energy range that can be reached in future LHC searches, leaving open the possibility of new discovery of fundamental scalars. This alignment was not only achieved but also predicted at tree-level in [1]. This success calls for understanding the main mechanism behind it. We have shown in this work that it is an \(SU(2)_R\) R-symmetry that acts as a Higgs family symmetry in the quartic scalar potential and enforces an automatic alignment. Another new result of this work is that we have written the CP-even Higgs squared-mass matrix off-diagonal element as a linear combination of the coefficients of non-singlet of \(SU(2)_R\) representations. These are generated in the quartic potential by tree level threshold and loop corrections. Their numerical values have been computed in [3] where it was proven that the alignment is preserved to an unexpected precision level. A new way of expressing the observables is presented here. We have found that using \(SU(2)_R\) symmetry allows to shed light on some of the existing results. For instance, we understand the origin of the cancellation between loop contributions of the adjoint scalars. When interested by alignment, only some contributions need to be evaluated or computed explicitly in the future: those contributing to coefficients of particular combinations of terms in the potential that break the \(SU(2)_R\) symmetry. This sheds light not only on how the alignment is realized here, but also on why \(N=2\) realizations of other models as the MRSSM attempted in [3] have not been successful. The extension of the quartic potential to an \(N=2\) sector with a particular attention to preserving the \(SU(2)_R\) symmetry at tree-level can be a way to pursue in order to implement alignment in extended Higgs sector models.

## Footnotes

## Notes

### Acknowledgements

We are grateful to P. Slavich for discussions. We acknowledge the support of the Agence Nationale de Recherche under grant ANR-15-CE31-0002 “HiggsAutomator”. This work is also supported by the Labex “Institut Lagrange de Paris” (ANR-11-IDEX-0004-02, ANR-10-LABX-63).

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