Perturbative $\lambda$-Supersymmetry and Small $\kappa$-Phenomenology

For the minimal $\lambda$-supersymmetry, it stays perturbative to the GUT scale for $\lambda \leq 0.7$. This upper bound is relaxed when one either takes the criteria that all couplings close to $\sim 4\pi$ for non-perturbation or allows new fields at the intermediate scale between the weak and GUT scale. We show that a hidden $U(1)_X$ gauge sector with spontaneously broken scale $\sim10$ TeV improves this bound as $\lambda\leq1.23$ instead. This may induce significant effects on Higgs physics such as decreasing fine tuning involving the Higgs scalar mass, as well as on the small $\kappa$-phenomenology.


I. INTRODUCTION
The standard model (SM)-like Higgs boson with mass around 126 GeV [1] reported at the LHC strongly favors extension beyond the minimal supersymmetric model (MSSM). A simple idea is adding a SM singlet S to the MSSM, with superpotential Due to the Yukawa coupling λ in Eq.(1), the Higgs boson mass can be naturally uplifted to observed value for moderate value λ ≥ 0.5 at the electroweak (EW) scale. This specific extension is referred as the next-to-minimal supersymmetric model (NMSSM) [3] for λ ≤ 0.7, or λ-supersymmetry (λ-SUSY) [4] for 0.7 ≤ λ ≤ 2. Either the NMSSM or λ-SUSY is very attractive from the viewpoint of naturalness argument [5,6], as the stringent tension on stop masses required by the Higgs mass in the MSSM is dramatically reduced. The origin of upper bound on λ above can be seen from the beta function β λ for λ. Since β λ is dominated by the top Yukawa coupling y t , the sign of which is positive, it imposes an upper bound on the EW value of λ in order to be still in the perturbative region at high energy scale. Given this scale near the grand unification (GUT) scale ≃ 1.0 × 10 15 GeV, the critical value was found to be ∼ 0.7 [7] in terms of the one-loop beta function 1 , where t ≡ ln(µ/µ 0 ), µ being the running renormalization scale and µ 0 ≡ 1 TeV. On the other hand, the observed Higgs mass favors large value of λ [8].
The intuition for solving the tension between the observed Higgs mass and perturbativity is obvious in the context of SUSY. The main observation is that Yukawa couplings in superpotential receive only wave-function induced renormalizations due to the protection of SUSY, in contrast to non- 1 In this paper, we follow the convention in [2], and present our β functions in the DR scheme. supersymmetric case 2 . By following the fact that the beta functions of Yukawa couplings are related to the anomalous dimensions of chiral fields [2], and the anomalous dimensions are proportional to quadratic Yukawa couplings with positive coefficients, the sign of contribution to β function due to Yukawa interactions is thus always positive. A way to alleviate this is introducing hidden super-confining gauge dynamics [13,14], from which Yukawa couplings are asymptotically free, and S is a composite other than fundamental state, with large λ at low energy as a result of this asymptotical freedom. In this context the Higgs doublets can be either fundamental [13] or composite states [14].
In this paper, we will explore λ-SUSY that stays perturbative up to the GUT scale in alternative way. In this framework we will obtain a well defined λ-SUSY on the realm of perturbative analysis, and operators such as Higgs doublets and singlet S are all fundamental other than composite states. This is our main motivation for this study.
The study of well defined λ-SUSY was initially addressed in Ref. [7], in which it was found that for tan β smaller than ∼ 10, λ is the first Yukawa coupling running into the nonperturbative region at high energy scale. It was also understood that either introducing new matter fields at the intermediate scale between the weak and GUT scale or adopting smaller EW value κ(µ 0 ) can decrease the evolution rate for λ. Given an initial EW value κ(µ 0 ), one can derive the upper bound on λ(µ 0 ) or vice versa, once new matters which appear at the intermediate scale are identified explicitly. The minimal content of new fields only includes the messenger sector which communicates the SUSY breaking effect to the visible sector, namely the λ-SUSY.
Together with the messengers a spontaneously broken U (1) X gauge group will be considered as the new fields. We consider a class of models different from those studied in the literature [16], in which SM fermions and sfermions except the Higgs doublets and S are all charged under the hidden U (1) X . The abelian gauge coupling g X and the U (1) X -breaking scale M X enter into the parameter space as two new free parameters. In comparison with traditional NMSSM or λ-SUSY, in our model the beta function for top Yukawa coupling β t receives new and negative contribution due to the hidden U (1) X sector, the magnitude of which is determined by the hidden gauge coupling g X and scale M X . As long as g X is large enough but still valid for perturbative analysis, these new effects decrease the slope of λ as function of renormalization scale µ above scale M X , and therefore lead to larger critical value λ(µ 0 ).
The paper is organized as follows. In section 2, in terms of one-loop renormalization group equations (RGEs) for relevant coupling constants we estimate the critical values of κ(µ 0 ) and λ(µ 0 ) due to the hidden U (1) X effect. In section 3, we revise the phenomenology for large λ but small κ. Finally we conclude in section 4. In appendix A, we show the details of the hidden sector and briefly review collider constraints on the model parameters.

II. PERTURBATIVE λ-SUSY
As mentioned in the introduction, the beta function for Yukawa couplings of superpotential can be easily estimated by using the non-renormalization of superpotential. Firstly, in the case without hidden U (1) X gauge group the one-loop beta functions for Yukawa couplings in λ-SUSY are given by 3 , The one-loop beta functions for the SM gauge couplings are the same as the MSSM, where In the case with a hidden U (1) X gauge group, we study the model in which SM fermions and sfermions all carry a hidden U (1) X charge, with the spontaneously broken scale M X ≃ 10 TeV. The Higgs doublets, however, are singlets of this U (1) X symmetry. Anomaly free conditions require three hidden matters X 1,2,3 with the same U (1) X charge added to the model. For details about the matter representations, see together with where n g = 3 denotes the number of SM fermion generations. It is obvious that large EW value g X (µ 0 ) favors small | a |. The RG running for g X with different values at µ 0 is plotted in Fig.1. In this figure non-perturbative region lies below the red line. It shows the critical value g X (µ 0 ) ≤ {2.0, 1.18, 0.82} for a = { 1 6 , 1 3 , 1 2 }, respectively. Given the upper bound on g X (µ 0 ), the upper bound on Yukawa coupling λ can be estimated in terms of RGEs in Eq.(5) and Eq.(3). The input parameters related to the critical value λ(µ 0 ) are composed of where M denotes the messenger mass scale, n 55 represents the number of5 + 5 representation of SU (5) for messengers. We will use the updated pole mass of top quark m t = 174 GeV [15] instead of m t = 180 GeV in [7] for our analysis, and take the criteria that the theory enters into the nonperturbative region when any of coupling constants in the theory is bigger than ∼ 4π.
We want to emphasize that (i), below scale M X , the modifications to RGEs arising from U (1) X can be ignored. (ii), above scale M , coefficients b i in beta functions for SM gauge couplings should change as b i → b i + n 55 . Fig. 2 shows the modifications to the upper bound on κ(µ 0 ) due to the hidden U (1) X sector given fixed λ(µ 0 ). We choose the initial EW value λ(µ 0 ) = 1.0, g X (µ 0 ) = 0.82 for a = 1/2, and messenger paramaters M = 10 7 GeV and n 55 = {1, 4} for illustration. Given the critical value κ(µ 0 ), the solid line in gray (green) represents the RG running for λ −1 without (with) U (1) X effect. The RG runnings of κ −1 (dotted) and y −1 t (dotdashed) shown in the figure imply that the upper bound κ 0 ≤ 0.8 is improved to κ 0 ≤ 0.85 ∼ 0.86 when the hid-den U (1) X effect is taken into account. Alternatively, given the same κ(µ 0 ), the upper bound on λ(µ 0 ) can be improved by the U (1) X effect. Combination of the left and right panel shows that the deviation due to the change of n 55 is small, in comparison with the U (1) X effect. Note that plots in Fig.1 are actually critical lines, because the change from the perturbative to the non-perturbative region is rather abrupt. In table I, we show the improvement on the upper bound on λ(µ 0 ) due to the hidden U (1) X . Without U (1) X sector, the model is not well defined up to the GUT scale when λ > 0.80. Instead, λ ≤ 1.23 stays perturbative up to the GUT scale when the hidden U (1) X is added to the model. The critical value on λ may be modified due to different choices on parameters {n 55 , M, a}. As shown in Fig.2, the deviation to λ(µ 0 ) due to messenger parameters M and n 55 is rather small. Similar conclusion holds when one tunes a. Because a larger than 1/2 leads to larger contribution to δβ yt but unfortunately smaller g X (µ 0 ), and vice versa.
In summary, in terms of introducing a hidden U (1) X sector with gauge symmetry broken scale ≃ 10 TeV well defined λ-SUSY up to the GUT scale is allowed for λ ≤ 1.23. In this class of models all fields including singlet S and the Higgs doublets are fundamental. The implication for λ ∼ 1 − 2 has been addressed, e.g., in [8]. In the next section, we revise the phenomenological implication for such large λ together with small κ.

III. PQ SYMMETRY AND SMALL κ-PHENOMENOLOGY
This section is devoted to study the phenomenology in λ-SUSY with small κ. Although it is a consequence of taking large λ for well defined λ-SUSY, the study of small κphenomenology can be considered as an independent subject from the viewpoint of phenomenology. The smallness of κ can be understood due to a broken U (1) global symmetry, i.e., Peccei-Quinn (PQ) symmetry. Without κ, the model is invariant under the following U (1) symmetry transformation, Terms like δV = m 2 S 2 + B µ H u H d explicitly breaks this symmetry. For these breaking small, we have a pseudo-Goldstone boson with small mass.
The soft terms in the scalar potential for small κphenomenology is given by, where κ relevant terms are ignored. The EW symmetry breaking vacuum can be determined from Eq. (9). For details on analysis of EW breaking vacuum, see, e.g., [3,21]. There are five free parameters which define the small κ-phenomenology. The first two can be traded for υ = 174 GeV and tan β. In what follows we explore the constraints on these parameters, the Higgs scalar spectrum, and their couplings to SM particles. The 3 × 3 squared mass matrix of CP-even neutral scalars reads 4 , in the basis (H, h, s), where x ≡ m 2 S /(λυ) 2 , H = cos βh 2 − sin βh 1 and h = cos βh 1 + sin βh 2 under the compositions The precision measurement of h including its couplings to SM gauge bosons and fermions powerfully constrains the magnitude of mixing effect. After mixing, we define the mass eigenstates as h 1 and h 2 , where h 1 = cos θh − sin θs, h 2 = cos θs + sin θh. Normalized to SM Higgs boson couplings, h 1 and h 2 couple to SM particles as, where V = {W, Z, t, b, · · ·}. At present status, LHC data suggests that 0.96 ≤ cos 2 θ ≤ 1 [22]. Alternatively we have sin 2 θ ≤ 0.04. In Fig.3 we show the parameter space in the two-parameter plane of m S and A λ , for λ = 1.2 and tan β = {4.5, 5, 5.5}, respectively. For each tan β region below the color contour 4 Here we follow the conventions and notation in Ref. [21]. is excluded by the condition of stability of potential and mass bound on chargino mass m χ > 103 GeV [23]. Region in the right up corner is excluded by the precision measurement of Higgs coupling presently. In particular, m S heavier than ∼90, 105, and 110 GeV is excluded for tan β = 4.5, 5 and 5.5, respectively. In comparison with the choice λ = 0.7 discussed in [21], the main difference is that for λ = 1.2 it allows larger m S .
As for the Higgs mass constraint, the discrepancy between M 2 22 and 126 GeV is compensated by the stop induced loop correction. The stop mass for the zero mixing effect (i.e., A t ∼ 0) is ∼ 340 GeV for tan β = 4.5, ∼ 550 GeV for tan β = 5 and ∼ 800 GeV for tan β = 5.5. Stop mass beneath 1 TeV is favored by naturalness. Nevertheless, there has tension for such light stop with present LHC data. This problem can be resolved in some situation, which we will not discuss here.
We show in Fig.4 the ratio ξ h2V V defined in Eq. (12), which determines the production rate for scalar h 2 . Its magnitude increases slowly as m S becomes larger. ξ h2V V reaches ∼ 0.03 at most when m S closes to its upper bound (suggested by Fig.  3). For such strength of coupling and mass ∼ 200 GeV, h 2 is easily out of reach of Run-I at the LHC and earlier attempts at the LEP2. Small ratio of strength of coupling similarly holds for heaviest CP-even neutral state H. In this sense, it is probably more efficient to probe charged Higgs scalar H ± , CP-odd scalar A, or light pseudo-boson G. Studies along this line can be found in, e.g, [24].
As for other scalar masses we show them in table II for three sets of A λ and m S , which are chosen in the parameter space shown in Fig.3. It is shown that for each case the mass of charged Higgs boson exceeds its experimental bound ∼ 300 GeV, and A scalar is always the heaviest with mass around 600 GeV. In comparison with scalar mass spectrum in fat Higgs model [14], the spectrum in table II is similar to it, although their high energy completions are rather different. As a final note we want to mention that the smallness of m S in compared with A λ can be achieved in model building, e.g., in gauge mediation. Because singlet S only couples to messengers indirectly through Higgs doublets.

IV. CONCLUSIONS
In this paper, we have studied λ-SUSY which stays perturbative up to the GUT scale. We find that the bound λ ≤ 0.7 ∼ 0.8 in the minimal model is relaxed to λ ≤ 1.23 if a hidden U (1) X gauge theory is introduced above scale ∼ 10 TeV and small κ(µ 0 ) is assumed at the same time. This improvement gives rise to several interesting consequences in phenomenology. For example, the fine tuning related to 126 Higgs mass can be reduced, and light stop beneath 1 TeV can be allowed. In the light of such single U (1) X , one may introduce multiple U (1)s or other gauge sectors at the intermediate scale, and further uplift the bound on λ.
In the second part of the paper, we have revised small κ-phenomenology for large λ. In comparison with fat Higgs model [14], the spectrum in the small κ-phenomenology is similar, although their high energy completions are different. The null result for signals of the other two CP-even neutral scalars h 2 and H is due to the perfect match between the scalar discovered at the LHC (Here its is referred as h 1 ) and the SM Higgs. Because the perfect fit dramatically reduces the mixing effect between h 1 and the others, which results in tiny strength of coupling for h 2 and H relative to the SM expectation. The studies on signals of charged scalar H ± , CP-odd scalar A and pseudo-boson G will shed light on this type of model. hidden matters X 1,2,3 with the same U (1) X charge anomaly free conditions such as U (1) X − Graviton − Graviton and U (1) X − U (1) X − U (1) X can not be satisfied 5 . Symmetry Breaking. There is no signal of Z ′ from broken U (1) X gauge group yet, so it should be spontaneously broken above the weak scale for g X of order SM gauge coupling. This can be achieved in various ways. Here we simply take the gauge mediation for example. If the hidden U (1) X gauge group communicates the D-type of SUSY-breaking effects into Xs, the sign of soft mass squared m 2 X would be negative [18]. For earlier application of such property in gauge mediation, see, e.g, [19]. Below SUSY-breaking scale the potential for X i we have which spontaneously breaks U (1) X , with U (1) X -breaking scale M X ∼ m X . Note that the magnitude of m X can be either larger or smaller than the soft breaking masses in the visible sector, which depends on the ratio of D term relative to F term and also the ratio of g X relative to SM gauge couplings. With a D term which gives rise to an order of magnitude larger than soft breaking masses and g X of the same order as SM gauge coupling, one can obtain M X ≃ O(10) TeV and mf i ∼ O(1) TeV in the visible sector. Limit on g X (µ 0 ). The experimental constraint on M X and gauge coupling g X is obtained from direct production of Z ′ at colliders through leptonic decays; and also indirect searches from flavor violation and electroweak precision tests. For a review on the status of hidden U (1) X , see e.g., [25]. The 5 We thank the referee for reminding us that the U (1) X charges in this model are identical to B − L quantum numbers of the SM fields, and the charges of the "hidden" matter, X 1,2,3 are same as those of right-handed neutrino fields.
parameter M X is constrained to be above 1 ∼ 2 TeV for coupling g X ∼ 1 − 2. For M X ≃ 10 TeV adopted in this note, we have the experimental limit g X ≤ M X /(| a | ·a few TeV) ∼| a | −1 [26].