Spontaneous Twin Symmetry Breaking

We consider a twin Higgs scenario where the $Z_2$ twin symmetry is broken spontaneously, not explicitly. This scenario provides an interesting interpretation of the Higgs metastability in the standard model; the $SU(4)$ breaking scale $f$ is determined by the scale where the Higgs self quartic coupling flips its sign. However, for the misalignment of nonzero vacuum expectation values of the twin Higgs fields, parameter tuning of ${\cal O}(m_h^2/f^2)$ is required like explicit twin symmetry breaking scenarios. For the minimal model with the exact twin symmetry, $f$ is ${\cal O}$($10^{10}$ GeV), so the model is very unnatural. We point out that the tuning can be significantly reduced ($f\gtrsim 2.7$ TeV) if there are twin vector-like leptons with large Yukawa coupling to twin Higgs fields.


I. INTRODUCTION
The approach of the twin Higgs scenarios to the little hierarchy problem is a realization of the Higgs boson as a pseudo-Goldstone boson [1]. With introducing the twin/mirror sector of the standard model (SM), twin Higgs fields H A and H B of each sector form a fundamental representation of a global group SU (4) whose spontaneous symmetry breaking down to SU (3) generates seven Goldstone bosons. Six of them are eaten by SU (2) LA and SU (2) LB gauge bosons of each sector, and one is identified as the observed Higgs boson.
An important ingredient for the twin Higgs mechanism to work is the Z 2 twin symmetry under which each particle of one sector is interchanged with the corresponding particle of the other sector. The role of the twin symmetry is to prevent the explicit breaking terms of SU (4) symmetry from introducing a quadratic divergence of the Higgs boson.
However, the twin symmetry has to be broken either explicitly or spontaneously for various phenomenological reasons including the observed Higgs signal strength [2][3][4]. The SU (4) breaking scale f should be at least about three times larger than the SM Higgs vacuum expectation value (vev). Introducing soft Z 2 symmetry breaking term f 2 |H A | 2 can easily provide a misalignment of twin Higgs vevs, but it requires fine tuning of parameters with an order of v 2 SM /f 2 where v SM 246 GeV is the SM Higgs vev.
In this paper, we focus on the possibility that the twin symmetry is exact but broken spontaneously. Fig. 1 describes the desired situation that we consider in this paper. In the twin Higgs field space (h A , h B ), the twin symmetry (Z 2 ) corresponds to the mirror symmetry along the diagonal dashed line. There are two degenerate minima in the flat direction h 2 A + h 2 B = f 2 whose locations in the field space are symmetric under the Z 2 transforma- * thjung0720@ibs.re.kr tion. Once scalar fields fall down to one of the minima, the sector with smaller Higgs vev becomes what we call the SM. While there are several realizations in twin two Higgs doublet model [5,6] or singlet extended twin Higgs model [7], we focus on the realization in the minimal twin Higgs setup without additional scalar fields. Similar idea is discussed in Refs. [8,9] in the context of the strong CP problem. We provide a more systematic approach to the construction of the effective potential.
Cosmological history of spontaneous twin symmetry breaking is strongly restricted by dark radiation constraints from the cosmic microwave background [10], and by the domain wall problem [11]. To avoid these problems, we assume that Z 2 symmetry is spontaneously broken before the end of the inflation, and reheaton decays mostly to the sector with smaller Higgs vev. Possible realizations of such an asymmetric reheating can be seen in Ref. [12,13]. For preventing twin sector particles from being produced thermally, reheating temperature should be less than around the bottom quark mass when f ∼ 10 v SM . Otherwise, twin sector particles can be produced through the bottom quark annihilation to the twin muon production process, and twin photons and twin neutrinos will finally contribute to the dark radiation [2,[12][13][14][15].

II. MINIMAL MODEL WITH EXACT TWIN SYMMETRY
Explicit breaking of the global SU (4) is necessary for the nonzero Higgs mass. Without violating Z 2 symmetry, there are three sources of explicit SU (4) breaking: quartic interaction, gauge interaction and Yukawa interaction. To be more specific, let us consider an effective scalar potential, preserving Z 2 . A mixed quartic term h 2 A h 2 B is the leading term to break SU (4) 1 . Gauge and Yukawa interactions will also contribute to ∆V through loops.
By replacing h A = f cos θ and h B = f sin θ, we can obtain the potential along the flat direction for the case when V ∆V . We denote it by f 4V (θ) in this paper. SinceV (θ) is a periodic function with periodicity π/2 and symmetric under Z 2 : θ → π/2 − θ, we can apply the Fourier expansion with coefficients c n . The leading order contribution h 2 A h 2 B corresponds to (1 − cos 4θ)/8 whose extrema are 0 (h B = 0), π/4 (h A = h B ) and π/2 (h B = 0). If its minima are at 0 or π/2, h B or h A become zero and the electroweak symmetry breaking does not take place. In order to obtain a proper misalignment, there should be nonzero c n contributions with n ≥ 2 2 .
The simplest term to generate c n≥2 is the Coleman-Weinberg potential [16] which is proportional to 1 2 . It leads to c 1 ∝ (25 − 24 log 2)/96, c 2 ∝ −1/240, · · · when we take µ = f . Since we want c 2 to have a sizable effect, a suppression of c 1 is required. For this reason, we need a cancelation between contribution to c 1 from the h 2 A h 2 B term and the one from the Coleman-Weinberg potential. This cancelation causes an unavoidable tuning of parameters in this scenario. It will be shown that this cancelation is actually equivalent to the fine tuning of quadratic Higgs term in the infra-red (IR) theory.
In addition, the sign of c 2 should be positive for the spontaneous twin symmetry breaking. If c 2 were negative, the minima could be only at θ = 0, π/4 or π/2. The positive sign of c 2 can be obtained if the beta function of Higgs self quartic coupling is negative. It is noteworthy that the SM gauge and Yukawa interactions already provide the proper sign assignment. Therefore, just an exact copy of the SM with a renormalizable Higgs portal potential (1) is enough to realize the spontaneous twin symmetry breaking.
The SU (4) breaking effective potential ∆V can be parametrized by where we ommit contributions coming from λ mix because it will turn out to be negligible at µ = f . Note that λ does not contribute toV (θ) since λ repects SU (4) symmetry. In Eq. (3), we redefined λ mix (µ = f ) so that quartic operators coming from radiative contributions are absorbed, e.g. log y t /2 − 3/2. By taking h A = f cos θ, h B = f sin θ and µ = f , we obtain where we neglect the constant term for simplicity. In terms of Fourier expansion,V can be written aŝ (6) For the misalignment of vevs (i.e. h A = h B = 0), cos 4θ term should be suppressed. This suppression comes from the cancelation between 12λ mix and β(25 − 24 log 2). In terms of κ = λ mix /β, the condition for the misalignment becomes where the first condition β < 0 is satisfied by the large top Yukawa interaction. Here, we usedV (π/4) < 0 and V (0) < 0 withV in Eq. (5).
In Fig. 2,V (θ) is described for different κ values. If κ is too large, twin Higgs vevs become identical, i.e. v A = v B , If κ < 1/2, potential minima are located at θ = 0 and π/2 which correspond to vSM = 0. For κ > (3 − 2 log 2)/2 0.81, potential minimum is at θ = π/4 where Z2 symmetry is not boken spontaneously. and the twin symmetry is not broken spontaneously. On the other hand, too small κ leads one of twin Higgs vevs to zero, i.e. electroweak symmetry breaking does not occur.
In the minimal setup, β = β SM and we have only two free parameters (f and λ mix ), and they are fixed by two observational constraints (Higgs vev v SM and mass m h ). If we denote θ 0 as the minimum position ofV , In Fig. 3, m h /v SM is described as a function of κ with fixed beta functions β = β SM (red), β = 2β SM (blue) and β = 5β SM (blue). For the minimal case (β = β SM ), it is very difficult to obtain the observed value, m h /v SM 0.5 unless λ mix stands at the edge of the allowed region. In this plot, we estimated β SM at Z boson mass scale (µ = M Z ), so more precise estimation will make the situation worse.
To investigate further, we expand the potential around θ = 0 with assuming κ 1/2, and we obtain where the first two terms are negative and the last logarithmic term is positive near θ 0. By multiplying f 4 and replacing f θ = h, we can match Eq. (10) to the SM potential where For the observed Higgs mass and vev, we need m m h / √ 2 and λ h (µ = M Z ) m 2 h /2v 2 SM . Thus, we obtain and f is determined by the scale where λ h (µ = f ) = β/16 ∼ < 0 with boundary condition λ h (µ = M Z ) m 2 h /2v 2 SM . The prediction of the minimal model is f 10 10 GeV [17]. The metastability of the Higgs boson at the IR theory (SM) can be interpreted as a consequence of the spontaneous twin symmetry breaking.
However, the Eq. (13) tells that λ mix needs to be very close to β/2 (remind that κ = λ mix /β). Since there is no reason for this relation, it should be regarded as a tuning. 3 The order of tuning is alleviated by the factor of β compared to other twin Higgs scenarios, but is basically For the theory to be natural, the scale f should not be very far away from the weak scale. In the next section, we will discuss one example to make f low.

III. VECTORLIKE LEPTONS
A possible way to alleviate tuning is introducing new Yukawa interactions. Additional Yukawa interactions can give negative contributions to β, and make f smaller. It can also be seen in Fig. 3 that if β is larger than the SM value (purple and blue curves), the slope at m h /v SM could be small, so the tuning of λ mix /β can be milder.
As an example, we consider a family of vectorlike leptons (VLL) in each sector: lepton doublets L Li ,L Li , charged lepton singlets E Ri ,Ē Ri , neutral lepton singlets N Ri andN Ri for i = A, B. Their interaction Lagrangian can be written as where the summation of i = A, B is ommitted in the expression. Although there are several implications of VLL for the case when they couple to the SM leptons [18][19][20][21][22][23][24][25][26][27], we do not discuss them in this paper because their coupling to the SM leptons should be small anyway, so their contributions to the effective potential are negligible.
Mass matrix of charged and neutral VLL are given by For simplicity, we assume that M L = M E = M N and for i = A, B. The effective potential from VLL in each sector is given by which can provide large enough Collider signatures of VLL are highly sensitive on the mixing with SM leptons [28][29][30][31][32][33][34]. The mass limit for charged leptons from the Large Electron-Positron collider (LEP) is 100.8 GeV when the charged lepton mostly decays to W ν. For neutral leptons, the mass limit from LEP is 101.3 GeV when they decay to W e [35,39]. At Large Hadron Collider (LHC), the most relevant search is Refs. [37] which provides constraints on the CKM matrix elemant |V eN | and |V µN | in the mass range from GeV to TeV. At this moment, LHC constraints are comparable to LEP constraints [37], but there will be much improvement in the future. Fig. 4 shows the SU (4) breaking scale f by colors in the parameter space of M L /f and y L . Black solid lines correspond to lightest lepton mass 100 GeV, 500 GeV and 1 TeV. We restrict Yukawa coupling to be smaller than 1.7 because the Landau pole arises below 100 TeV if it becomes larger. The smallest f within m L− > 100 GeV, 500 GeV and 1 TeV are f ∼ > 1.3 TeV, 2.7 TeV and 3.9 TeV, respectively.
Another advantage of VLL comes from changing RG running of top Yukawa coupling. In the minimal model, top Yukawa coupling rapidly drops because of large SU (3) c gauge coupling. If there are additional Yukawa interactions, they compensate negative contributions of gauge coupling and make the pseudo-IR fixed point smaller. Consequently, top Yukawa coupling can maintain its strength until µ ∼ f . We neglect this effect in Fig. 4, so f can be slightly smaller value in more precise calculations.
VLLs with large Yukawa couplings can modify electroweak precision parameters and Higgs to diphoton signal strength through the loop processes. We estimate constraints coming from the electroweak precision by using Peskin-Takeuchi parameters ∆S and ∆T [38,39]. Detailed calculations are summarized in the Appendix. A. Numerical results with various parameter choices are described in Fig. 5   i |y i | 4 1/4 = 1.26, which should correspond to y L in Fig. 4.
Since ∆T represents the strength of custodial symmetry breaking in the new physics, y E = y N andȳ E =ȳ N leads to ∆T = 0. Without tuning of Yukawa couplings, we conclude that M L− ∼ > 400 GeV is safe for large Yukawa couplings y eff = 1.26. For even larger Yukawa couplings, naive estimation would be M L− ∼ > y 2 eff ×350 GeV since ∆T ∝ y 4 in terms of Yukawa coupling differences among charged leptons and neutral leptons. If the charged lepton Yukawa coupling is large, Higgs to diphoton signal can be modified significantly. Here, we estimate Higgs to diphoton signal strength which can be obtained by where the denominator (numerator) is the decay rate of Higgs to diphoton in the SM (the model with VLLs). Formulas used in the numerical calculation can be found in the Appendix. B. In this paper, we estimate both of Γ SM h→γγ and Γ h→γγ in one loop order. The numerical results of µ γγ are depicted as functions of the lightest VLL mass M L− in Fig. 6. In this plot, we simply take y E =ȳ E . Horizontal lines correspond to the current 2σ bound obtained in Ref. [40] which yields µ γγ = 1.1 ± 0.10 in 1σ error. From the figure, we obtain a bound on the lightest VLL mass M L− ∼ > 150 -250 GeV for the Yukawa coupling y E =ȳ E = 1 -1.5 with the same sign. If y EȳE < 0, the corresponding bound becomes M L− ∼ > 450 -650 GeV.
If there are mixings between SM leptons and VLLs, there can be richer phenomenological implications such as lepton flavor violations or anomalous magnetic momentum of muon. However, the mixing should be small for the lepton flavor violations and thus contributions to the effective potential are negligible [21]. The smallness of the mixing could be understood in terms of symmetry argument, i.e. technical naturalness. For example, one can assign a charge of global symmetry in the the VLL sector.

IV. CONCLUSION
We have discussed the possibility of spontaneous twin symmetry breaking scenario. For the misalignment of nonzero twin Higgs vevs, there should be cos 8θ term in V (θ) which can come from the Coleman-Weinberg potential. In addition, we need a cancellation of O(v 2 /βf 2 ) between λ mix and β/2.
The SU (4) breaking scale f is determined by the scale where the Higgs self quartic coupling flips its sign in the IR theory. Since, in the SM, this flipping occurs around O(10 10 GeV), the minimal setup is not natural. For obtaining smaller f , we introduced twin VLLs with large Yukawa couplings and obtained f as small as 2.7 TeV when they are safe from the current bounds on VLLs.
Constraints on VLLs are estimated from the electroweak precision parameters ∆S and ∆T and Higgs to diphoton signal strength µ γγ . With strong enough Yukawa couplings, the lightest VLL mass should be larger than around 400 GeV. Potentially, VLLs are testable at the LHC or future lepton colliders through Higgs measurement and direct production depending on the mixing with SM leptons. Their signatures below the TeV scale would lend credence to this scenario.
Although we do not specify the inflation sector, we have assumed that reheaton decays mostly to the sector with smaller Higgs vev in order to avoid cosmological problems. The reheating temperature of the SM sector should be less than around bottom quark mass for preventing thermal production of twin sector particles. An interesting possibility is that cogenesis of baryon asymmetry and asymmetric dark matter could occur during the reheating process. We leave detailed studies about cosmological history and the inflation sector as future work. Acknowledge THJ is grateful to Chang Sub Shin, Kyu Jung Bae and Dongjin Chway for useful discussions. This work was supported by IBS under the project code, IBS-R018-D1.

Appendix A: Electroweak Precision Parameters
Yukawa couplings of VLLs to the Higgs field induce a mixing between left handed leptons and right handed leptons. Let us denote U EL and U ER by the mixing matrix such that U EL M E U † ER = diag(M E− , M E+ ), and U N L M N U † N R = diag(M N − , M N + ). VLL interactions with the Z boson are given by where sw and cw are sin and cos of the Weinberg angle, and g is the SU (2) W gauge coupling constant. For the final result to be consistent with the renormalization condition, we take cw = M W /M Z . With the W boson, we have Note that the photon interaction remains Q EM g sw, and diagonal, i.e. vectorlike, in the basis of mass eigenstates. The electroweak precision parameter S and T are defined as where Π AB (q 2 ) is the self energy diagram of external A boson and B boson, α is the fine structure constant, i.e. α = g 2 sw 2 /4π. The finite piece of the self energy diagram in the MS scheme is given by Π LL (q 2 ) = − 4 (4π) 2 − q 2 b 2 (m 2 1 , m 2 2 ) (A7) + 1 2 m 2 2 b 1 (m 2 1 , m 2 2 ) + m 2 1 b 1 (m 2 1 , m 2 2 ) , where the loop functions can be found in Ref. [41], with ∆(m 2 1 , m 2 2 , q 2 ) = xm 2 2 +(1−x)m 2 1 −x(1−x)q 2 . Here, LL or LR denotes how the projection operator P L and P R are inserted in the left and right vertices of the diagram. Since we have Π LL = Π RR and Π LR = Π RL , one should combine up Π LL and Π LR with proper coefficients given by Eq. (A1 -A3). If the couplings are correctly assigned, the RG scale µ dependence should be canceled out in S and T parameters. , where and M i is the VLL mass eigenvalue with i = ±, and the Yukawa matrix Y is given by