Natural supersymmetric Twin Higgs
Marcin Badziak Keisuke Harigaya





^1 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL–02–093 Warsaw, Poland

^2 Institute for Advanced Study, Natural Sciences, 1 Einstein Drive, Princeton, New Jersey 08540, United States

mbadziak@fuw.edu.pl


A new class of supersymmetric Twin Higgs (TH) models where new gauge symmetry is responsible for the TH mechanism is reviewed. In this class of models the Higgs mass is naturally in agreement with the LHC measurement while the electroweak symmetry breaking is realised without excessive tuning despite of strong lower bounds on masses of supersymmetric particles set by the LHC. Assuming particular non-abelian structure of the new gauge symmetry the model remains perturbative up to the energy scale of gravity, in contrast to all previously proposed UV completions of the TH model.  




§ INTRODUCTION


The lack of finding any new particles beyond the Standard Model (SM) at the LHC has substantially increased the fine-tuning of the electroweak symmetry breaking (EWSB) in models addressing the hierarchy problem of the SM such as supersymmetry (SUSY) or Composite Higgs.  This is indication of the little hierarchy problem.

A possible solution to the little hierarchy problem is provided by models of Neutral Naturalness in which top partners responsible for the cancellation of quadratic divergences to the Higgs mass parameter do not carry the SM color charge. The experimental constraints on such top partners are strongly relaxed making the fine-tuning of the EW scale much smaller. 

Arguably the most widely studied model of Neutral Naturalness is the Twin Higgs (TH) model <cit.>. In this scenario, the Higgs is a pseudo-Nambu-Goldstone boson of a global SU(4) symmetry emerging from ℤ_2 symmetry exchanging the SM with its mirror (or twin) copy. We denote mirror objects with supersctripts '. The Higgs potential in this class of models can be quite generally written in the following way:

    V = λ (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + Δλ(|H'|^4 + |H|^4) + Δ m^2 |H^2|  ,

where H is the SM Higgs doublet while H' its mirror counterpart.  The first two terms are both ℤ_2 and SU(4) symmetric, Δλ preserves ℤ_2 but breaks SU(4) which is necessary to make the Higgs massive. Since the LHC Higgs measurements show that the discovered Higgs has properties similar to the SM Higgs ℤ_2 must be broken which is parameterised by Δ m^2 in the above potential. This ℤ_2 breaking introduces some tuning in the potential:

    Δ_v/f = 1/2( f^2/v^2 -2)  ,

where H≡ v, H'≡ v', and f ≡√(v^2 + v^'2) is the decay constant of the spontaneous SU(4) breaking.
However, currently this results only in a minor tuning of 𝒪(20%) because the LHC Higgs data set a constraint f≳3v <cit.>. 

It should be emphasised that the TH model solves only the little hierarchy problem. A fully satisfactory TH model should be UV completed in a way that solve also the big hierarchy problem of the SM. Existing UV completions  of the TH model involve either Composite Higgs <cit.> or supersymmetry <cit.>. The UV completed TH models have additional source of tuning on top of that from Z_2 breaking mentioned above.  This is because in UV completed models ordinary top partners that carry QCD charge must be introduced to solve the big hierarchy problem. Due to the TH mechanism the tuning resulting from introducing such states  is suppressed by a factor λ_ SM/(2λ) (where λ_ SM≈0.13 is the SM Higgs quartic coupling) as compared to corresponding models without the TH mechanism. In any UV completion the SU(4) invariant coupling λ cannot be arbitrarily large so color charged top partners may introduce non-negligible tuning if they are too heavy. 

In the context of SUSY UV completions of the TH model the upper bound on λ, hence the lower bound on the fine-tuning, originates from perturbativity constraints.  Another constraint on SUSY TH models that may impact the fine-tuning comes from the Higgs mass measurement.  In initially proposed SUSY UV completions of the TH model  in which λ is generated from an F-term potential of some new singlet <cit.> these constraints make the TH mechanism totally inefficient and the fine-tuning is not improved with respect to SUSY models without TH mechanism <cit.>.

In these proceedings we review a new class of SUSY TH models in which the SU(4) invariant quartic term is generated by a D-term potential of a new gauge symmetry <cit.>. In this class of models the Higgs mass constraint is easily satisfied even for light stops while EWSB does not require excessive fine-tuning. 

   







§ SUSY U(1)_X D-TERM TWIN HIGGS


We start with a model in which a large SU(4) invariant quartic term originates from a
non-decoupling D-term of a new U(1)_X gauge symmetry <cit.>. 
Such a non-decoupling D-term may be present if the mass of a scalar field responsible for
the breaking of the U(1)_X gauge symmetry is dominated by a SUSY breaking soft mass. 
We introduce chiral multiplets
Ξ, S and S̅ whose U(1) charges are 0, +q and -q, respectively, and the superpotential,


    W = κΞ (SS̅- M^2),


where κ, M are constants,
and soft masses,


    V_ soft = m_S^2 (|S|^2 + |S̅|^2  ).


Here we assume that the soft masses of S and S̅ are the same. Otherwise, the asymmetric VEVs of S and S̅ give a large soft mass to
the Higgs doublet through the D-term potential of U(1)_X.


Assuming that all Higgs bosons are charged under the new U(1)_X gauge symmetry and  integrating out S fields 
we obtain the non-decoupling D-term potential:

    V_U(1)_X=g_X^2/8( |H_u|^2-|H_d|^2 + |H'_u|^2-|H'_d|^2 )^2 (1-ϵ^2)  ,   ϵ^2 ≡m_X^2/2m_S^2 + m_X^2,

where m_X^2= 4 g_X^2 q^2 v_S^2 is the U(1)_X gauge boson mass with g_X the U(1)_X  gauge coupling and v_S the VEV of S and S̅.
This term gives the following SU(4) invariant coupling:

    λ=g_X^2cos^2( 2β)/8(1-ϵ^2)  .

Note that λ is maximized in the limit of large tanβ which makes it easier to satisfy
the Higgs mass constraint. Large λ prefers also g_X as large as possible and ϵ≪1. 

The magnitude of g_X is constrained from above by perturbativity.  The beta function of the 
U(1)_X gauge coupling constant depends on the charge assignment of particles in the visible and mirror sectors. It turns out that the beta function is minimized when the U(1)_X charges of the MSSM
particles and the mirror particles are the following linear combination of U(1)_Y and U(1)_B-L charges:


    q_X = q_Y -1/2 q_ B-L .


Then the beta function of the U(1)_X gauge coupling constant is given by


    d/ dlnμ8π^2/g_X^2 = b_X,   b_X =
    
    - 14    mirror
    
    -10    fraternal


where in fraternal Twin Higgs models <cit.> the first and the second generation fermions do not have their twins so the RG running is slightly slower.

 
The scale of the Landau pole M_c is given by


    M_c = m_X × exp [  - 8π^2/b_Xg_X(m_X)^2].


We require that M_c is larger than the mediation scale of the SUSY breaking,
which is typically at the smallest around 100 m_ stop. 
The electroweak precision as well as the production of di-muon at the LEP put a lower bound m_X/ g_X ≳ 4 TeV.
This requires M_c ≳ 10m_X which sets an upper bound on g_X(m_X) of
about 1.6 (1.9) for the mirror (fraternal) Twin Higgs model.






In order to quantify the fine-tuning of EWSB we use the following measure:

    Δ_v ≡Δ_f ×Δ_v/f,

where the tuning in percent is 100%/Δ_v and

    Δ_v/f = 1/2( f^2/v^2 -2),
    Δ_f =   max_i ( |∂ ln f^2/∂ ln x_i(Λ)|, 1 ) .


Δ_v/f measures the tuning to obtain v < f
via explicit soft ℤ_2 symmetry breaking which is required by the Higgs coupling measurements <cit.>. Δ_f measures the tuning to obtain the scale f
from the soft SUSY breaking. x_i(Λ) are the parameters of the theory evaluated at the mediation scale of the SUSY breaking Λ. More details about our procedure to calculate Δ_v can be found in <cit.>. Apart from the usual tuning from stops, the tuning may also
arise from a threshold correction to the soft Higgs mass which is proportional to
a new gauge boson mass squared:

    (δ m_H_u^2)_X= g_X^2/64 π^2 m_X^2 ln(ϵ^-2)  .

Note that it depends on a parameter ϵ which also enters the effective SU(4)-preserving quartic coupling, cf. eq.(<ref>). Therefore, the tuning is typically minimized for some intermediate value of ϵ^2≈𝒪(0.1). We show the contours of fine-tuning in the plane -g_X in fig. <ref>.
We see that for the mirror TH model with U(1)_X gauge symmetry the tuning may be at the level of 𝒪(10%) for stop masses above 1 TeV. Another interesting feature of fig. <ref> is that stops could be very light but in agreement with the Higgs mass measurement (even though stop mixing has been set to zero). For tanβ=10, the 125 GeV Higgs mass implies stop masses around 500 GeV.  This is in tension with the LHC direct searches for stops but heavier stops are possible, as seen from the right panel, e.g. 2 TeV stops imply tanβ≈3.  This is in sharp contrast to the Minimal Supersymmetric Standard Model (MSSM) where, even for large tanβ, 𝒪(10) TeV stops are required to get the 125 GeV Higgs mass without stop mixing which results in severe fine-tuning. This follows from the fact that the tree-level Higgs mass in supersymmetric TH models is enhanced with respect to the tree-level Higgs mass in the MSSM:

    (m_h^2)_ tree≈ 2 M_Z^2 cos^2 ( 2β) (1-v^2/f^2)   .

Since the LHC Higgs data enforces v^2≪ f^2 the tree-level Higgs mass is almost a factor of √(2) bigger than in the MSSM.







§ SU(2)_X D-TERM TWIN HIGGS




We may avoid the low Landau pole scale by generating the D-term potential by a non-Abelian gauge symmetry <cit.>.
The matter content of the minimal model is shown in Table <ref>.
In addition to the SU(3)_c× SU(2)_L× U(1)_Y gauge symmetry and its mirror counterpart, we introduce an SU(2)_X gauge symmetry which is neutral under the ℤ_2 symmetry.
We embed an up-type Higgs H_u into a bi-fundamental of SU(2)_L× SU(2)_X, H, and its mirror partner H_u' into that of SU(2)'_L× SU(2)_X, H'. 
The D-term potential of SU(2)_X is responsible for the SU(4) invariant quartic coupling of H_u and H_u'.
The SU(2)_X symmetry is broken by the vacuum expectation value (VEV) of a pair of SU(2)_X fundamental S and S̅. The relevant superpotential and soft terms is analogous to eqs.(<ref>)-(<ref>) generalized to the case of SU(2)_X. The resulting SU(4)-invariant coupling is given by:

    λ=g_X^2/8sin^4β(1-ϵ^2)  .
 
Except for S and S̅ all matter fields have their mirror partner.
The right-handed top quark is embedded into Q̅_R and allow for a large enough top yukawa coupling through the superpotential term HQ̅_R Q_3, where Q_3 is the third generation quark doublet. E̅ is necessary in order to cancel the U(1)_Y`-SU(2)_X^2 anomaly.
The VEV of ϕ_u is responsible for the masses of the up and charm quarks.
Q_1,2,3, u̅_1,2, e̅_1,2,3, d̅_1,2,3 and L_1,2,3 are usual MSSM fields.
To cancel the gauge anomaly of 
SU(3)_c^2 `-U(1)_Y and U(1)_Y^3
originating from the extra up-type right handed quark in Q̅_R and two extra right-handed leptons in E̅, we introduce U and E_1,2.
There are three up-type Higgses in H and ϕ_u, so we need to introduce three down-type Higgsses ϕ_d1,2,3.
Their VEVs are responsible for the masses of down-type quarks and charged leptons. SU(2)_L charged Higgses mix with each other by the couplings W ∼λ Hϕ_d  S + λ Hϕ_d  S̅ + m ϕ_u ϕ_d. More details about how the masses of the SM particles are generated can be found in <cit.>.





The contours of fine-tuning in the Λ`-g_X plane is shown in fig. <ref>. The fine-tuning is not monotonously improved for larger g_X. This is because of the one-loop threshold correction analogous to eq. (<ref>) (which is three times larger) as well as the two-loop RGE correction from the soft masses of S and S̅. The tuning is at the level of  𝒪(10%) for low mediation scales. Because of the slower running of the SU(2)_X gauge coupling, the model remains perturbative up to the Planck scale while the tuning is at the level of few %.











§ ASYMPTOTICALLY FREE SUSY TWIN HIGGS








In the SU(2)_X model, the new gauge symmetry SU(2)_X is assumed to be ℤ_2 neutral, and mirror particles are charged under SU(2)_X. We may instead consider the case where SU(2)_X also has a mirror partner SU(2)_X', under which mirror particles are charged. As a result the number of SU(2)_X charged fields is reduced, so that the SU(2)_X× SU(2)_X' gauge interaction is asymptotically free, which is explicitly shown in fig. <ref>. See <cit.> for the matter content and details of the mass spectrum of the model. The SU(2)_X× SU(2)_X' symmetry is broken down into the diagonal SU(2)_D subgroup by the vacuum expectation value v_Σ of a bi-fundamental field Σ in  a supersymmetric way, e.g. by a
superpotential W ∼ Y (Σ^2 - v_Σ^2) where Y is a chiral multiplet, and that v_Σ is much larger than the TeV scale, say few tens of TeV. Then below the scale
v_Σ the theory is well-described by a SUSY theory with an SU(2)_D gauge symmetry. The symmetry breaking of SU(2)_D involves
SUSY breaking effect similar to model discussed in previous sections. We introduce chiral multiplets
Ξ, Ξ' and the superpotential


    W = κΞ (SS̅- M^2) + κΞ' (S'S̅'- M^2),


and soft masses,


    V_ soft = m_S^2 (|S|^2 + |S̅|^2 +  |S'|^2 + |S̅'|^2 ).


The diagonal subgroup is responsible for the SU(4) invariant quartic coupling which is given by eq. (<ref>).

The contours of fine-tuning in the Λ`- g_X plane is shown in fig. <ref>. The constraint from the Landau pole of the SU(2)_X interaction is absent. For 2 TeV stops and gluino the tuning is better than 5% even if the mediation scale is as large as the Planck scale. The tuning is relaxed by two to three orders of magnitude as compared to the MSSM. 

It should be emphasized that the huge improvement in tuning with respect to the MSSM originates not only from the TH mechanism, guaranteed by a large SU(4) invariant quartic term, and enhanced tree-level Higgs mass which allows for much lighter stops required by the 125 GeV Higgs mass. The tuning is additionally suppressed by a strong suppression of the top Yukawa coupling via RG effect of the large new gauge coupling. This suppression is demonstrated in  fig. <ref>. Similar effect is present also in a SUSY model without the TH mechanism and allows for tuning of 𝒪(1) % for gravity mediated SUSY breaking <cit.>. 

An intriguing feature of the model is that the up quark must be embedded in the SU(2)_X doublet together with the top quark to avoid the Landau pole for the hypercharge below the Planck scale. This has interesting phenomenological implications including flavor-violating top decays to the Higgs and the up quark which is potentially observable at the LHC <cit.>.








§ SUMMARY


We presented a new class of supersymmetric UV completions of the TH model in which the SU(4) invariant quartic term is generated by a D-term potential of a new gauge symmetry. We showed that the Higgs mass of 125 GeV is easily obtained both for light or heavy stops. The tuning can be at the level of 𝒪(10) % for stops and gluino masses that comfortably satisfy the LHC constraints. Even if there is no sign of supersymmetry in future LHC data the tuning will remain moderate in these models. If the new gauge group is abelian a low Landau pole scale for the new interaction is required for the TH mechanism to work. However, the model can be made perturbative up to the Planck scale assuming SU(2)_X× SU(2)_X' gauge symmetry.  





§ ACKNOWLEDGMENTS

MB has been partially supported by National Science Centre, Poland, under research grant no. 2017/26/D/ST2/00225. KH has been partially supported by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contracts DE-SC0009988.




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