SUSY breaking scales in the gauge-Higgs unification

In the $SO(5) \times U(1)$ gauge-Higgs unification in the Randall-Sundrum (RS) warped space the Higgs boson naturally becomes stable. The model is consistent with the current collider signatures only for a large warp factor $z_L>10^{15}$ of the RS space. In order for stable Higgs bosons to explain the dark matter of the Universe the Higgs boson must have a mass $m_h = 70 \sim 75$ GeV, which can be obtained in the non-SUSY model with $z_L \sim 10^5$. We show that this discrepancy is resolved in supersymmetric gauge-Higgs unification where a stop mass is about $300 \sim 320 $GeV and gauginos in the electroweak sector are light.

The Higgs boson, necessary for inducing spontaneous symmetry breaking in the standard model (SM) of electroweak interactions, is yet to be discovered. It is not clear at all if the Higgs boson appears as described in the SM. New physics may be hiding behind it, the Higgs boson having properties quite different from those in the SM.
In the gauge-Higgs unification scenario the 4D Higgs boson becomes a part of the extradimensional component of gauge fields. [1]- [4] Many models have been proposed with predictions to be tested at colliders. Among them the SO(5) × U (1) gauge-Higgs unification in the Randall-Sundrum (RS) warped space is most promising. [5]- [14] One of the most striking results in the model is that the 4D Higgs boson naturally becomes stable. [10] The vacuum expectation value of the Higgs boson corresponds to an Aharonov-Bohm (AB) phase θ H in the fifth dimension. With bulk fermions introduced in the vector representation of SO(5) the value of θ H is dynamically determined to be 1 2 π, at which the Higgs boson becomes stable while giving masses to quarks, leptons, and weak bosons. There emerges H parity (P H ) invariance at θ H = 1 2 π. All particles in the SM other than the Higgs boson are P H -even, while the only P H -odd particle at low energies is the Higgs boson, which in turn guarantees the stability of the Higgs boson. [12,14] As a consequence the Higgs boson cannot be seen in the current collider experiments, since all experiments so far are designed to find decay products of the Higgs boson.
The model has one parameter to be determined, namely the warp factor z L of the RS spacetime. With z L given, the mass of the Higgs boson m h is predicted. It is found that m h = 72, 108 and 135 GeV for z L = 10 5 , 10 10 and 10 15 , respectively. We note that the LEP2 bound, m h > 114 GeV, is evaded as the ZZH coupling exactly vanishes as a result of the P H invariance.
There appears slight deviation in the gauge couplings of quarks and leptons from those in the SM. It turns out that the gauge-Higgs unification model gives a better fit to the forward-backward asymmetries in e + e − collisions on the Z pole than the SM. However, the branching fractions of Z decay are fit well only for z L > ∼ 10 15 . The gauge-Higgs unification model gives predictions for Kaluza-Klein (KK) excitation modes of various particles. In particular, the first KK Z has a mass 1130 GeV and a width 422 GeV for z L = 10 15 . The current limit on the Z production at the Tevatron and LHC indicates z L > 10 15 . All of the collider data prefer a large warp factor in the gauge-Higgs unification model. [13] These analyses have been done at the tree level so far.
The fact that Higgs bosons become stable leads to another important consequence. They become the dark matter of the Universe. [10] It has been shown that in order for stable Higgs bosons to account for the entire dark matter of the Universe observed by WMAP, [15]  Of course nothing is wrong with m h ∼ 135 GeV. It simply implies that Higgs bosons account for a tiny fraction of the dark matter of the Universe. Yet it is curious and fruitful to ask if there is a natural way in the gauge-Higgs unification scenario to satisfy the two requirements; (i) to be consistent with the collider data, and (ii) to explain the entire dark matter of the Universe.
In this paper we would like to show that the two requirements are naturally fulfilled if the model has softly broken supersymmetry (SUSY) such that SUSY partners of observed particles acquiring large masses. It will be found that the mass of a stop (t), a SUSY partner of a top quark (t), needs to be 300 ∼ 320 GeV, when SUSY partners of W , Z and γ are light.
The key observation is that the nonvanishing Higgs boson mass m h in the gauge-Higgs unification arises at the quantum-level, whereas the dominant part of collider experiments is governed by the structure at the tree-level. If SUSY is exact, the contributions of bosons and fermions to the effective potential V eff (θ H ) cancel so that V eff (θ H ) = 0, the Higgs boson remaining massless. As SUSY is broken, the cancellation becomes incomplete. If the SUSY breaking scale is much larger than the KK mass scale, the model is reduced to the non-SUSY model. In particular, m h becomes ∼ 135 GeV for z L = 10 15 . Put differently, one can ask how large the SUSY breaking scale should be to have m h = 70 ∼ 75 GeV with z L = 10 15 ∼ 10 17 so that the relic abundance of Higgs bosons saturate the dark matter of the Universe.
The RS warped spacetime is given by ds 2 = e −2ky dx µ dx µ + dy 2 for 0 ≤ y ≤ L. [16] The AdS curvature in 0 < y < L is −6k 2 . The warp factor is z L = e kL . In the SO(5) × U (1) gauge-Higgs unification there appears an AB phase, or the Wilson line phase θ H , in the fifth dimension, as the RS spacetime has topology of R 4 × (S 1 /Z 2 ). The 4D Higgs field appears as a zero mode in the SO(5)/SO(4) part of the fifth-dimensional component of the vector potential A y (x, y). A y = A4 y T4 = 0 when the EW symmetry is spontaneously broken, The effective potential V eff (θ H ) at the one-loop level is determined by the mass spectrum {m n (θ H )} in the presence of the phase θ H = 0. It is given in d dimensions, after Wick rotation, by The upper (lower) sign corresponds to bosons (fermions). The second equality is understood by analytic continuation for large Re d in the complex d-plane.
In supersymmetric theory with SUSY breaking each KK tower with a spectrum {m n } is accompanied by its SUSY partner with a spectrum {m n }. With a SUSY breaking scale Λ the latter is well mimicked by which has a property that m n ∼ m n for m n Λ. Suppose that the spectrum {m n } (m n > 0) is determined by the zeros of an analytic function ρ(z); ρ(m n ) = 0. Then the spectrum {m n } is determined by the zeros ofρ(z) = ρ( √ z 2 − Λ 2 ). When ρ(iy) = ρ(−iy) for real y and | ln ρ | < |z| q with some q for |z| → ∞, a convenient formula for V eff in (1) has been derived. [17] In the present case the functionρ(z) has a branch cut between Λ and −Λ in the z-plane so that elaboration of the argument there is necessary.
In the dimensional regularization, (1) with {m n } is transformed into Here the contour C encircles the zeros ofρ(z), {m n }, clockwise. The contour is transformed, for sufficiently small Re d, to C which runs from −i∞ to +i∞, avoiding the cut, as shown in fig. 1. With C the formula is analytically continued to a larger Re d. The contribution coming from an infinitesimally small circle (C ) around the branch point at Λ vanishes. The integral just below the cut (C cut − ) cancels the one just above the cut (C cut + ) as ρ(iy) = ρ(−iy). The contributions coming from the integrals along the imaginary axis combine to give the expression involving an integral ∞ 0 dy y d−1 ln ρ(i y 2 + Λ 2 ). Combining contributions from the KK tower with a spectrum {m n } and its SUSY partner with {m n }, one finds that Previously V eff (θ H ) in the gauge-Higgs unification in the RS spacetime has been evaluated by making use of the formula with only y d−1 ln ρ(iy) in the integrand. [8,12,17,18] Re z Im z The SO(5) × U (1) gauge-Higgs unification model has been specified in Refs. [8,9]. In the The relevant contributions to V eff (θ H ) come from the W and Z towers of the fourdimensional components A µ (x, y), the Nambu-Goldstone towers of the fifth-dimensional components A y (x, y), and the top quark tower. Contributions coming from other quark and lepton towers are negligible. [8,12] Brane fields give no contribution to V eff (θ H ), as they do not couple to A y . We recall the way the quarks and leptons acquire masses is different from that in SM. A y connects the left-handed and right-handed components of the up-type quarks and charged leptons directly, whereas those of the down-type quarks and neutrinos are intertwined through both gauge couplings and additional interactions with brane fermions and scalars. Thus the effective Higgs couplings of the down-type quarks and neutrinos appear after integrating heavy brane fermions. It is notable that only the ratio of two large mass scales of the brane fermion couplings appears in the effective Higgs couplings. [8,9] We comment that in the supersymmetric extension of the model two brane scalar fields, Φ u andΦ d , need to be introduced. Further in 5D SUSY there appear 4D scalar fields, associated to the zero mode of A y , to form a 5D N = 1 (4D N = 2) vector multiplet. In this paper we assume that such scalar fields acquire large SUSY breaking masses, giving little effect on the Wilson-line dynamics.
In the supersymmetric extension two SUSY breaking scales become important for It is most convenient to express the function ρ(z) in (4) in the form ρ(iy) = 1 + Q(q) where y = kz −1 L q. The effective potential is expressed in terms of whereΛ is related to the SUSY breaking scale by Λ = kz −1 LΛ . We define where I α (q) and K α (q) are the modified Bessel functions. Then V eff (θ H ) in the model is given by where r t = (m b /m t ) 2 , and θ W and c t are the Weinberg angle and the bulk mass parameter for the top multiplet, respectively. The θ H -dependence enters through Q 0 (q; c). The values of the parameters with a given z L are summarized in Table I. The effective potential has the global minima at θ H = ± 1 2 π for z L 1.  The mass of the Higgs boson m h is related to the effective potential Noting that the KK mass scale is given by m KK ∼ πkz −1 L , one finds not affect the above result. There arises no constraint for gluino masses from this analysis so that gluinos can be heavier than 1 TeV. mt = 300 ∼ 320 GeV with Λ gh < ∼ 100 GeV is in the range allowed by current experiments. [19] In the above analysis we have supposed that the stop masses are degenerate. Though unnecessary in the current scheme, it may be of interest to see the effect of large stop mixing, which plays an important role in MSSM to obtain a desired Higgs mass [23]. In the presence of the left-right squark mixing, the stop masses become non-degenerate. The spectra of their KK towers are approximated by m stop,i n = (m top n ) 2 + Λ 2 sotp,i , i = 1, 2, and accordingly the last terms in Eq. (7) and in the first equation of (8) are separated into two parts. As an extremal case we consider the case where one of the stops is very heavy and decouple. In such a case the curves in Fig. 2  In the present analysis we adopted a mass spectrum of a SUSY KK partner in the form (2) for convenience. Depending on how SUSY is broken, the spectrum may deviate from (2).
However, the detailed form of the spectrum is not relevant in the present analysis, provided that m n > m n and m n ∼ m n for m n Λ. Only low lying modes in the KK towers give relevant contributions to the θ H -dependent part of V eff (θ H ). Contributions coming from the modes with m n , m n m KK are irrelevant.
We need a further consideration of the consistency with the current electroweak precision measurements, especially with Peskin-Takeuchi S and T parameters. [20] There are many studies on these parameter in the models of extra dimensions. Besides studying the electroweak precision observables, it is necessary to complete the model by incorporating flavor mixings [24] and implementing light neutrinos by the seesaw mechanism in the bulk-brane system. We hope to report on these in the near future.
Acknowledgements: The authors would like to thank K. Oda