126 GeV Higgs boson and universality relations in the SO(5)×U(1) gauge-Higgs unification
aa r X i v : . [ h e p - ph ] A p r Toyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013
126 GeV Higgs boson and universality relationsin the SO (5) × U (1) gauge-Higgs unification Yutaka Hosotani
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
The Higgs boson mass m H = 126 GeV in the SO (5) × U (1) gauge-Higgs unification in the Randall-Sundrum space leads to important consequences. An universal relation is found between the Kaluza-Klein (KK) mass scale m KK and the Aharonov-Bohm phase θ H in the fifth dimension; m KK ∼ / (sin θ H ) . . The cubic and quartic self-couplings of the Higgs boson become smallerthan those in the SM, having universal dependence on θ H . The decay rates H → γγ, gg are evaluatedby summing contributions from KK towers. Corrections coming from KK excited states turn outvery small. With θ H = 0 . ∼ .
35, the mass of the first KK Z is predicted to be 2 . ∼ I. INTRODUCTION
The discovery of a Higgs-like boson with m H = 126 GeV at LHC may give a hint for extra dimensions. Weshow [1] that the observed Higgs boson mass in the gauge-Higgs unification scenario leads to universal relationsamong the AB phase θ H , the KK mass m KK , the Higgs self couplings, and the KK Z boson mass m Z (1) ,independent of the details of the model.The gauge-Higgs unification scenario is predictive. As a result of the Hosotani mechanism [2–6] the Higgsboson mass emerges at the quantum level without being afflicted with divergence. The Higgs couplings tothe KK towers of quarks and W/Z bosons have a distinctive feature that their signs alternate in the KK level,significant departure from other extra dimensional models such as UED models. As a consequence contributionsof KK modes to the decay rate Γ( H → γγ ) turn out very small. Surprisingly the gauge-Higgs unification givesnearly the same phenomenology at low energies as the standard model (SM).The gauge-Higgs unification can be confirmed by finding the KK Z boson in the range 2 . ∼ II. SO (5) × U (1) GAUGE-HIGGS UNIFICATION IN RS
The model is given by SO (5) × U (1) gauge theory in the Randall-Sundrum (RS) warped space ds = e − σ ( y ) η µν dx µ dx ν + dy (1)where η µν = diag( − , , , σ ( y ) = σ ( y + 2 L ) = σ ( − y ), and σ ( y ) = k | y | for | y | ≤ L . The RS space is viewedas bulk AdS space (0 < y < L ) with AdS curvature − k sandwiched by the Planck brane at y = 0 and theTeV brane at y = L . The SO (5) × U (1) model was proposed by Agashe et al [7, 8]. It has been elaborated inrefs. [9, 10], and a concrete realistic model has been formulated in ref. [1]. The schematic view of the gauge-Higgsunification is given below.5D A M four-dim. components A µ ∈
4D gauge fields γ, W, Z extra-dim. component A y ∈
4D Higgs field H ∼ AB phase θ H in extra dim. Hosotani mechanism ⇓ Dynamical EW symmetry breaking oyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 L = L gaugebulk ( A, B ) + L fermionbulk (Ψ a , Ψ F , A, B )+ L fermionbrane ( ˆ χ α , A, B ) + L scalarbrane ( ˆΦ , A, B ) + L intbrane (Ψ a , ˆ χ α , ˆΦ) . (2) SO (5) and U (1) X gauge fields are denoted by A M and B M , respectively. The two associated gauge couplingconstants are g A and g B . Two quark multiplets and two lepton multiplets Ψ a are introduced in the vectorrepresentation of SO (5) in each generation, whereas n F extra fermion multiplets Ψ F are introduced in thespinor representation. These bulk fields obey the orbifold boundary conditions at y = 0 and y = L given by (cid:18) A µ A y (cid:19) ( x, y j − y ) = P j (cid:18) A µ − A y (cid:19) ( x, y j + y ) P − j , (cid:18) B µ B y (cid:19) ( x, y j − y ) = (cid:18) B µ − B y (cid:19) ( x, y j + y ) , Ψ a ( x, y j − y ) = P j Γ Ψ a ( x, y j + y ) , Ψ F ( x, y j − y ) = ( − j P sp j Γ Ψ F ( x, y j + y ) ,P j = diag ( − , − , − , − , , P sp j = diag (1 , , − , − . (3)The orbifold boundary conditions break SO (5) × U (1) X to SO (4) × U (1) X ≃ SU (2) L × SU (2) R × U (1) X .The brane interactions are invariant under SO (4) × U (1) X . The brane scalar ˆΦ is in the ( , ) − / representa-tion of [ SU (2) L , SU (2) R ] U (1) X . It spontaneously breaks SU (2) R × U (1) X to U (1) Y by non-vanishing h ˆΦ i whosemagnitude is supposed to be much larger than the KK scale m KK . At this stage the residual gauge symmetryis SU (2) L × U (1) Y . Brane fermions ˆ χ α are introduced in the ( , ) representation. The quark-lepton vectormultiplets Ψ a are decomposed into ( , ) + ( , ). The ( , ) part of Ψ a , ˆ χ α in ( , ) and ˆΦ in ( , ) form SO (4) × U (1) X invariant brane interactions. With h ˆΦ i 6 = 0 they yield mass terms. The resultant spectrum ofmassless fermions is the same as in the SM. All exotic fermions become heavy, acquiring masses of O ( m KK ).Further with brane fermions all anomalies associated with gauge fields of SO (4) × U (1) X are cancelled.[10]With the orbifold boundary conditions (3) there appear four zero modes of A y in the components ( A y ) a = − ( A y ) a ( a = 1 , · · · , SO (4) vector, or an SU (2) L doublet, corresponding to the Higgs doubletin the SM. The AB phase is defined with these zero modes by e i Θ H / ∼ P exp (cid:26) ig A Z L dy A y (cid:27) . (4)At the tree level the value of the AB phase Θ H is not determined, as it gives vanishing field strengths. At thequantum level its effective potential V eff becomes non-trivial. The value of Θ H is determined by the locationof the minimum of V eff . This is the Hosotani mechanism and induces dynamical gauge symmetry breaking. Itleads to gauge-Higgs unification, resolving the gauge-hierarchy problem.[6] Without loss of generality one canassume that ( A y ) component develops a non-vanishing expectation value. Let us denote the correspondingcomponent of Θ H by θ H . If θ H takes a non-vanishing value, the electroweak symmetry breaking takes place. III. V eff ( θ H ) AND m H Given the matter content one can evaluate V eff ( θ H ) at the one loop level unambiguously. The θ H dependentpart of V eff ( θ H ) is finite, being free from divergence. V eff ( θ H ) depends on several parameters of the theory; V eff = V eff ( θ H ; ξ, c t , c F , n F , k, z L ) where ξ is the gauge parameter in the generalized R ξ gauge, c t and c F are thebulk mass parameters of the top and extra fermion multiplets, n F is the number of the extra fermion multiplets,and k, z L are parameters specifying the RS metric (1). Given these parameters, V eff is fixed, and the locationof the global minimum of V eff ( θ H ), θ min H , is determined.With θ min H determined, m Z , g w , sin θ W are determined from g A , g B , k, z L and θ min H . The top mass m t isdetermined from c t , k, z L , θ min H , whereas the Higgs boson mass m H is given by m H = 1 f H d V eff dθ H (cid:12)(cid:12)(cid:12)(cid:12) min , f H = 2 g w s kL ( z L − . (5) oyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 ξ = 1. Then the theory has seven parameters { g A , g B , k, z L , c t , c F , n F } . Adjusting theses parameters,we reproduce the values of five observed quantities { m Z , g w , sin θ W , m t , m H } . This leaves two parameters, say z L and n F , free. Put differently, the value of θ min H is determined as a function of z L and n F ; θ min H = θ H ( z L , n F ).We comment that contributions from other light quark/lepton multiplets to V eff are negligible. V eff ( θ H ) in the absence of the extra fermions ( n F = 0) was evaluated in refs. [9, 11]. It was found there thatthe global minima naturally appear at θ H = ± π at which the Higgs boson becomes absolutely stable. It isdue to the emergence of the H parity invariance.[11, 12] In particular the Higgs trilinear couplings to W , Z ,quarks and leptons are all proportional to cos θ H and vanish at θ H = ± π .[13–18]This, however, conflicts with the observation of an unstable Higgs boson at LHC. To have an unstable Higgsboson the H parity invariance must be broken, which is most easily achieved by introducing extra fermionmultiplets Ψ F in the spinor representation of SO (5) in the bulk.[1]Let us take n F = 3 , z L = e kL = 10 as an example. { g w , sin θ W } are related to { g A , g B } by g w = g A √ L , tan θ W = g B p g A + g B , (6)where z L = e kL . The observed values of { m Z , g w , sin θ W , m t , m H } are reproduced with k = 1 . × GeV, c t = 0 . c F = 0 .
353 for which the minima of V eff are found at θ H = ± . m KK = πkz − L = 3 .
95 TeV. V eff ( θ H ) is depicted in Fig. 1 with red curves. For comparison V eff in the case of n F = 0 is also plotted with a blue curve. When n F = 0 and z L = 10 , the minima are located at θ H = ± π .The observed values of { m Z , g w , sin θ W , m t } are reproduced with k = 3 . × GeV and c t = 0 . m H = 87 . m KK = 993 GeV. One can see howthe position of the minima is shifted from θ H = ± π to θ H = ± . π = ± .
258 by the introduction of theextra fermions. - - Θ H Π- - - H a L Θ H Π- - - H b L FIG. 1: The effective potential V eff ( θ H ) for z L = 10 . U = 16 π m − V eff is plotted. The red curves are for n F = 3 with m H = 126 GeV. V eff has minima at θ H = ± .
258 and m KK = 3 .
95 TeV. The blue curve is for n F = 0 in which case m H = 87 . m KK = 993 GeV. IV. UNIVERSALITY
As explained above, the AB phase θ H (= θ min H ) is determined as a function of z L and n F ; θ H ( z L , n F ). The KKmass scale m KK = πkz − L is also determined as a function of z L and n F ; m KK ( z L , n F ). The relation betweenthem is plotted for n F = 1 , , n F .Similarly one can evaluate the cubic ( λ ) and quartic ( λ ) self-couplings of the Higgs boson H by expanding V eff [ θ H + ( H/f H )] around the minimum in a power series in H . They are depicted in the bottom figure in Fig. 2.Although the shape of V eff ( θ H ) heavily depends on n F , the relations λ ( θ H ) and λ ( θ H ) turn out universal,independent of n F .It is rather surprising that there hold universal relations among θ H , m KK , λ and λ . Once θ H is determinedfrom one source of observation, then many other physical quantities are fixed and predicted. The gauge-Higgs unification gives many definitive predictions to be tested by experiments. We tabulate values of variousquantities determined from m H = 126 GeV with given z L for n F = 3 in Table I. The relation between θ H and oyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 Θ H m KK @ T e V D nf = ò ò ò òàà à à à à à à àæ æ æ æ æ æ æ æ æ Θ H m KK @ T e V D æ n F = à n F = ò n F = çç ò ò ò ò àà à à à à à à àææ æ æ æ æ æ æ æ Θ H Λ @ G e V D æ n F = à n F = ò n F = ç SM çç ò ò ò ò àà à à à à à à àææ æ æ æ æ æ æ æ - - - Θ H Λ æ n F = à n F = ò n F = ç SM FIG. 2: Universality relations. [Top] KK scale m KK ( θ H ). [Bottom left] Higgs cubic self-coupling λ ( θ H ). [Bottom right]Higgs quartic self-coupling λ ( θ H ). The universality, independent of n F , is seen in all relations. m KK is well summarized with m KK ∼ θ H ) . . (7) TABLE I: Values of the various quantities with given z L for n F = 3. m Z (1) and m F (1) are masses of the first KK Z bosonand the lowest mode of the extra fermion multiplets. Relations among θ H , m KK and m Z (1) are universal, independentof n F . z L θ H m KK m Z (1) m F (1) .
360 3 .
05 TeV 2 .
41 TeV 0 .
668 TeV10 .
258 3 .
95 3 .
15 0 . .
177 5 .
30 4 .
25 1 . .
117 7 .
29 5 .
91 2 . V. H → γγ, gg In the gauge-Higgs unification all of the 3-point couplings of W , Z , quarks and leptons to the Higgs boson H at the tree level are suppressed by a common factor cos θ H compared with those in the SM.[13–18] The decay ofthe Higgs boson to two photons goes through loop diagrams in which W boson, quarks, leptons, extra fermionsand their KK excited states run.The decay rate Γ[ H → γγ ] is given byΓ( H → γγ ) = α g w π m H m W (cid:12)(cid:12) F total (cid:12)(cid:12) , oyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 F total = F W + 43 F top + (cid:16) Q ( F ) X ) + (cid:17) n F F F , F W = cos θ H ∞ X n =0 I W ( n ) m W m W ( n ) F ( τ W ( n ) ) , I W ( n ) = g HW ( n ) W ( n ) g w m W ( n ) cos θ H , F top = cos θ H ∞ X n =0 I t ( n ) m t m t ( n ) F / ( τ t ( n ) ) , I t ( n ) = y t ( n ) y SM t cos θ H , F F = sin θ H ∞ X n =1 I F ( n ) m t m F ( n ) F / ( τ F ( n ) ) , I F ( n ) = y F ( n ) y SM t sin θ H , (8)where W (0) = W , t (0) = t , τ a = 4 m a /m H . The functions F ( τ ) and F / ( τ ) are defined in Ref. [19], and F ( τ ) ∼ F / ( τ ) ∼ − for τ ≫ Q ( F ) X is the U (1) X charge of the extra fermions. I W (0) and I t (0) are ∼ I W ( n ) , I t ( n ) , and I F ( n ) are plotted. One sees that the values of these I ’s alternate in sign as n increases, which gives sharp contrast to the UED models. I W ( n ) ∼ ( − n I ∞ W , I t ( n ) ∼ ( − n I ∞ t , I F ( n ) ∼ ( − n I ∞ F for n ≫ n ) p corrections. This is special to the gauge-Higgs unification models. It has been known in the modelsin flat space as well.[20, 21] As a consequence of the destructive interference due to the alternating sign, theinfinite sums in the rate (8) converges rapidly. There appears no divergence. ìììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììì ààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ - - n I I t I W I F FIG. 3: I W ( n ) , I t ( n ) , and I F ( n ) for n F = 3, Q ( F ) X = 0 and θ H = 0 .
360 ( z L = 10 ) in the range 1 ≤ n ≤ I W (0) = 1 . I t (0) = 1 . Let F W only and F t only be the contributions of W = W (0) and t = t (0) to F total . The numerical values of theamplitudes F ’s are tabulated in Table II for n F = 3. It is seen that contributions of KK states to the amplitudeare small. The dominant effect for the decay amplitude is the suppression factor cos θ H .All Higgs couplings HW W, HZZ, Hc ¯ c, Hb ¯ b, Hτ ¯ τ are suppressed by a factor cos θ H at the tree level. Thecorrections to Γ[ H → γγ ] and Γ[ H → gg ] due to KK states amount only to 0.2% (2%) for θ H = 0 . . B ( H → j ) ∼ B SM ( H → j ) j = W W, ZZ, γγ, gg, b ¯ b, c ¯ c, τ ¯ τ , · · · γγ production rate: σ prod ( H ) · B ( H → γγ ) ∼ (SM) × cos θ H . (10)The signal strength in the γγ production relative to the SM is about cos θ H . It is about 0.99 (0.91) for θ H = 0 . . oyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 TABLE II: Values of the amplitudes F ’s in (8) for n F = 3 and Q ( F ) X = 0. θ H z L F W only F W / F W only F t only -1.372 -1.305 F t / F t only F F / F t only -0.0034 -0.033 F total F total / ( F W only + F t only ) 1.001 1.011 VI. SIGNALS OF GAUGE-HIGGS UNIFICATION
There are several constraints to be imposed on the gauge-Higgs unification.(i) For the consistency with the S parameter, we need sin θ H < . θ H < . Z ′ search at Tevatron and LHC. The first KK Z corresponds to Z ′ . No signal has been found so far,which implies that m Z (1) > θ H < . Z boson decay and the forward-backward asymmetry on the Z resonance has been investigated when n F = 0. Reasonable agreement was foundfor m KK > . n F ≥ θ H < .
4. When θ H is very small, the KK mass scale m KK becomes verylarge and it becomes very difficult to distinguish the gauge-Higgs unification from the SM. The range of interestis 0 . < θ H < .
35, which can be explored at LHC with an increased energy 13 or 14 TeV. The gauge-Higgsunification predicts the following signals.(1) The first KK Z should be found at m KK = 2 . ∼ θ H = 0 . ∼ . λ ( λ ) should be 10 ∼
20% (30 ∼ θ H = 0 . ∼ .
35, according to the universality relations. This should be explored at ILC.(3) The lowest mode ( F (1) ) of the KK tower of the extra fermion Ψ F should be discovered at LHC. Its massdepends on both θ H and n F . For n F = 3, the mass is predicted to be m F (1) = 0 . ∼ . θ H = 0 . ∼ . VII. FOR THE FUTURE
The SO (5) × U (1) gauge-Higgs unification model of ref. [1] has been successful so far. Yet further elaborationmay be necessary.(1) Flavor mixing has to be incorporated to explore flavor physics.[25](2) It is curious to generalize the model to incorporate SUSY. The Higgs boson mass becomes smaller thanin non-SUSY model. m H = 126 GeV should give information about SUSY breaking scales.[26](3) The orbifold boundary conditions ( P , P ) in (3) have been given by hand so far. It is desirable to havedynamics which determine the boundary conditions.[27, 28](4) Not only electroweak interactions but also strong interactions should be integrated in the form of grandgauge-Higgs unification.[29] Acknowledgments
This work was supported in part by scientific grants from the Ministry of Education and Science, Grants No.20244028, No. 23104009 and No. 21244036. oyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 References [1] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and T. Shimotani, arXiv:1301.1744 [hep-ph]. To appearin
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