Oblique corrections from less-Higgsless models in warped space
aa r X i v : . [ h e p - ph ] D ec KIAS-P15064
Oblique corrections from less-Higgsless modelsin warped space
Hisaki Hatanaka
Quantum Universe Center,Korea Institute for Advanced Study,Seoul 130-722, Republic of Korea
Abstract
The Higgsless model in warped extra dimension is reexamined. Dirichlet boundary conditions onthe TeV brane are replaced with Robin boundary conditions which are parameterized by a massparameter M . We calculate the Peskin-Takeuchi precision parameters S , T and U at tree level. Wefind that to satisfy the constraints on the precision parameters at 99% [95%] confidence level (CL)the first Kaluza-Klein excited Z boson, Z ′ , should be heavier than 5 TeV [8 TeV]. The Magnitudeof M , which is infinitely large in the original model, should be smaller than 200 GeV (70 GeV) forthe curvature of the warped space R − = 10 GeV (10 GeV) at 95% CL. If the Robin boundaryconditions are induced by the mass terms localized on the TeV brane, from the 99% [95%] boundwe find that the brane mass interactions account for more than 97% [99%] of the masses of Z and W bosons. Such a brane mass term is naturally interpreted as a vacuum expectation value of theHiggs scalar field in the standard model localized on the TeV brane. If so, the model can be testedby precise measurements of HW W , HZZ couplings and search for 1st Kaluza-Klein excited states. . INTRODUCTION Even after the discovery of the Higgs scalar with 125GeV mass [1, 2], mechanisms tomaintain the hierarchy between the electroweak scale and Planck scale is still unknown.Warped extra dimension is one of the way to explain such a large hierarchy between theelectroweak scale and Planck scale [3]. In this scenario, such hierarchy is obtained from theexponentially large warp factor of the metric of the space. In this direction, the standardmodel in the warped space is considered in [4]. Such models, however, suffer from largedeviation of oblique S and T parameters [5, 6]. To suppress the T parameter models areextended so as to possess the custodial symmetry[7]. To suppress the S parameter, brane-localized kinetic terms [8] and the soft-wall warped extra dimension are also considered in[9]. Most cases, nevertheless, Kaluza-Klein (KK) scale is needed to be higher than 3 TeV tosuppress the S parameter.Although some excesses with invariant masses around or below 2 TeV in di-boson channelshave been reported [10–15], experimental results in the LHC Run-1 [16–19] and first-yearresults of LHC Run-2 [20–25] seem to exclude the Z ′ and W ′ bosons which are lighterthan 3 TeV in fermionic decay channels. Therefore in this paper we focus on warped extradimensional scenarios in which KK particles are heavier than 3 TeV.In this paper we reconsider the Higgsless model in warped extra dimension [26–28]. Inthe Higgsless model the electroweak symmetry breaking is caused by the boundary condi-tions on the TeV brane, and this model also yields large value of S parameter[28–30], andexperimentally excluded by the discovery of the Higgs boson. In order to suppress the S parameter, some Dirichlet boundary conditions on the brane are replaced with generalizedRobin boundary conditions. A mass parameter M is introduced to parameterize the Robinboundary conditions. In the M → ∞ limit the model reduces to the original model, whereas M = 0 reproduce the unbroken electroweak symmetry. As M decreases from + ∞ to zero, weobtain smaller magnitudes of S , T and U parameters while the Kaluza-Klein scale becomeslarger.In this paper we also study the mass structure of weak bosons in detail. The Robinboundary conditions can be induced by the mass terms localized on branes [31, 32]. In theoriginal model where M → ∞ , the mass of weak bosons are coming from their momentaalong the extra spacial dimension. As M decreases, contributions from the brane mass terms2ominates in the weak boson masses. Such a brane mass can also be identified with thevacuum expectation value (VEV) of a scalar field, namely the Higgs boson observed in theLHC. Based on such identification we also estimate the Higgs couplings to the weak bosonsin this model.This paper is organized as follows. In Section II, an extension of the Higgsless model inwarped space is introduced. In Section III, the model is numerically studied. Section IV isdevoted to a summary and discussion. In Appendix A, formulas for the wave function ofthe gauge field are collected. II. MODEL
The model [26, 27] is a SU (2) L ⊗ SU (2) R ⊗ U (1) B − L gauge theory in a slice of fivedimensional (5D) anti-de Sitter space AdS . The metric of AdS bulk is given by ds = R z [ η MN dx M dx N ] , R ≤ z ≤ R ′ , (II.1)where M, N = 0 , , , , η MN = diag(1 , − , − , − , −
1) and x ≡ z . R is the AdS curva-ture radius. A large hierarchy between R and R ′ appears as ln( R ′ /R ) = O (10). Boundariesat z = R and z = R ′ are referred as the Planck (UV) brane and the TeV (IR) brane, respec-tively. Gauge fields propagate in AdS bulk. Let A LaM , A
RaM , B M ( a = 1 , ,
3) be 5D gaugefields of SU (2) L , SU (2) R and U (1) B − L , respectively. The action of the gauge fidlds in thebulk is given by S bulk = S [ A L ] + S [ A R ] + S [ B ] ,S [ A ] ≡ Z d x Z R ′ R dz Rz (cid:26) − F aµν F aµν + 12 ( D µ A a ) † ( D µ A a ) − ξ A (cid:20) ∂ µ A aµ − ξ A z∂ (cid:18) z A a (cid:19)(cid:21) (cid:27) , (II.2)where µ, ν = 0 , , , µ, ν are done with η µν . F aµν ≡ ∂ µ A aν − ∂ ν A aµ + g A f abc A bµ A cν , and D µ A a = ∂ µ A a + g A f abc A bµ A c . f abc is the structure constant of thegauge group, and vanishes for U (1) B − L . g A = g L , g R , ˜ g denote the 5D gauge couplings of SU (2) L , SU (2) R and U (1) B − L . Hereafter we impose SO (4) ≃ SU (2) R ⊗ SU (2) L symmetryand set g L = g R ≡ g . ξ A ( A = A L , A R , B ) are the gauge fixing parameters. We take theunitary gauge, ξ A = ∞ , and concentrate ourselves only on the physical components, i.e., A Lµ , A Rµ and B µ . 3he boundary conditions of gauge fields at z = R are given by ∂ A Laµ = 0 , a = 1 , , ,A Raµ = 0 , a = 1 , ,∂ ( g B µ + ˜ g A R µ ) = 0 , ˜ g B µ − g A R µ = 0 . (II.3)The boundary conditions at z = R ′ are ∂ z ( A Laµ + A Raµ ) = 0 , a = 1 , , ,∂ B µ = 0 , (II.4)and for A Lµ − A Rµ we assign Robin boundary conditions( M + ∂ )( A Laµ − A Raµ ) = 0 , a = 1 , , , (II.5)where we have introduced a parameter M with mass dimension one. Boundary conditions(II.3) (II.4) are same as ones in [27]. When M → ∞ , (II.5) becomes the Dirichlet b.c. ∂ ( A Laµ − A Raµ ) = 0 as in the original model [27].When M = 0 the model has unbroken SU (2) L × U (1) Y ( Y = T R + B − L ) gaugesymmetry. Therefore M can be related with a dynamics of the electroweak symmetrybreaking, which lies on the z = R ′ brane. Actually some boundary conditions in (II.3)-(II.5)can be reproduced by introducing a mass term localized on each branes [31, 32]. Togetherwith the surface terms, the boundary action is partly given by S bdr ⊃ Z d x (cid:26) (cid:20)(cid:16) zR (cid:17) A aµ ∂ A aµ + (cid:16) zR (cid:17) M IR A aµ A aµ (cid:21) z = R ′ − (cid:20)(cid:16) zR (cid:17) ( A Raµ ∂ A Raµ + B µ ∂ B µ ) + (cid:16) zR (cid:17) M UV u † X µ X µ u (cid:21)(cid:27) , A aµ ≡ A Laµ − A Raµ √ , X µ ≡ g A Raµ T Ra + 12 ˜ g B µ , u = , (II.6)where M UV and M IR are the mass parameters. When we set M IR = ( R ′ /R ) M,M UV → ∞ , (II.7)the boundary action reproduces boundary conditions (II.3)-(II.5). We note that even M IR is as large as 1 /R , we have a small value of M = O (1 /R ′ ) ≪ M IR thanks to the suppression4actor R/R ′ . In the M → ∞ limit, the wave functions A Laµ − A Raµ vanish and decouplecompletely with the source of the electroweak symmetry breaking on z = R ′ brane, as thename “Higgsless” stands for.In the low-energy effective four dimensional (4D) theory there are the photon, Z and W ± bosons, and their Kaluza-Klein (KK) excitations. The expansions to Kaluza-Klein modesare given by A ( L ± ) µ ( x, z ) = ∞ X n =1 ψ ( W ) n ( z ) W ± ( n ) µ ( x ) ,A ( R ± ) µ ( x, z ) = ∞ X n =1 ψ ( W ) n ( z ) W ± ( n ) µ ( x ) ,A ( L µ ( x, z ) = ∞ X n =0 ψ ( L γ ) n ( z ) γ ( n ) µ ( x ) + ∞ X n =1 ψ ( L Z ) n ( z ) Z ( n ) µ ( x ) ,A ( R µ ( x, z ) = ∞ X n =0 ψ ( R γ ) n ( z ) γ ( n ) µ ( x ) + ∞ X n =1 ψ ( R Z ) n ( z ) Z ( n ) µ ( x ) ,B µ ( x, z ) = ∞ X n =0 ψ ( Bγ ) n ( z ) γ ( n ) µ ( x ) + ∞ X n =1 ψ ( BZ ) n ( z ) Z ( n ) µ ( x ) , (II.8)where γ ( n ) µ ( x ), Z ( n ) µ ( x ) and W ± ( n ) µ ( x ) are KK excited states with masses m ( γ ) n , m ( Z ) n and m ( W ) n , respectively. W ± µ ≡ ( W µ ∓ iW µ ) / √ γ (0) µ , Z (1) µ and W ± (1) µ correspond tothe photon, Z -boson and W ± bosons in the SM, respectively. Wave functions ψ ( A ) ( z ) satisfybulk equations of motion (EOM) (cid:18) ∂ − z ∂ + q (cid:19) ψ ( A ) ( z, q ) = 0 , (II.9)where we have assumed that solutions take the form of A ( A ) µ ( q ) e − iqx ψ ( A ) k ( z ). Solutions of theEOM are written in the form of ψ ( A ) = z [ C J ( qz ) + C Y ( qz )], where J and Y are Besselfunctions of the first kind and second kind, respectively. Boundary conditions (II.3), (II.4)and (II.5) determine the KK masses and eigenfunctions of KK excitations except for theirnormalizations. They are summarized in Appendix A.Just same as the original model [27], we assume that fermions are localized on the z = R brane. The couplings of the fermions to the gauge bosons are read from the covariantderivatives at z = R ,( gA µ ( x, z ) + g ′ Y B µ ( x, z )) | z = R (cid:18) g ψ ( L ± )1 ( z ) T L ± W ± µ ( x ) + g T L h Z µ ( x ) ψ ( L Z )1 ( z ) + γ µ ( x ) ψ ( L γ )0 ( z ) i +˜ g Y h Z µ ( x ) ψ ( BZ )1 ( z ) + γ µ ( x ) ψ ( Bγ )0 ( z ) i(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z = R = gT L ± W ± µ ( x ) + gT L [ Z µ ( x ) c w + γ µ ( x ) s w ] + g ′ Y [ − Z µ ( x ) s w + γ µ ( x ) c w ] , (II.10)where c w = cos θ W , s w = sin θ W and θ W is the Weinberg angle. g and g ′ are the 4D couplingsof SU (2) L and U (1) Y , respectively. In the last line of (II.10) the couplings to the photon isgiven by eQ where Q = T L + Y is the electric charge and e = g/ sin θ W is the electromagneticcoupling constant. Hence we obtain normalization conditions at z = R as follows g ψ ( L ± )1 ( R ) = g,g ψ ( L Z )1 ( R ) = g cos θ W , ˜ g ψ ( BZ )1 ( R ) = − g ′ sin θ W , (II.11)and photon wave functions (given in (A.11)) are fixed by g ψ ( L γ )0 ( z ) = g ψ ( R γ )0 ( z ) = ˜ g ψ ( Bγ )0 ( z ) = e. (II.12)Here let us relate 5D and 4D couplings. From the boundary conditions (II.11) and wavefunctions given in (A.12), we obtain g ′ g = − ˜ g g ψ ( BZ )1 ( R ) ψ ( L Z )1 ( R ) = ˜ g g + ˜ g . (II.13)The wave-function normalization of the photon is given by Z γ ≡ Z R ′ R dz Rz (cid:26)(cid:16) ψ ( Bγ )0 ( z ) (cid:17) + (cid:16) ψ ( L γ )0 ( z ) (cid:17) + (cid:16) ψ ( R γ )0 ( z ) (cid:17) (cid:27) = R ln( R ′ /R ) (cid:18) g + 2 g (cid:19) e = 1 . (II.14)From (II.13) and (II.14), we obtain relations between 4D and 5D gauge couplings as g = g R ln( R ′ /R ) , g ′ = g ˜ g ( g + ˜ g ) R ln( R ′ /R ) , (II.15)and sin θ W is given by sin θ W ≡ g ′ g + g ′ = ˜ g g + 2˜ g . (II.16)6ith these relations one also finds that the normalized wave functions satisfy (cid:12)(cid:12) Ψ ( W ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) z = R = (cid:12)(cid:12) Ψ ( Z ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) z = R = 1 R ln( R ′ /R ) , (II.17)where (cid:12)(cid:12) Ψ ( Z ) (cid:12)(cid:12) ≡ (cid:12)(cid:12)(cid:12) ψ ( L Z )1 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( R Z )1 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( BZ )1 (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12) Ψ ( W ) (cid:12)(cid:12) ≡ (cid:12)(cid:12)(cid:12) ψ ( L ± )1 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( R ± )1 (cid:12)(cid:12)(cid:12) . (II.18)For later use, we define wave function renormalizations of W and Z bosons by Z W ≡ Z R ′ R dz Rz (cid:12)(cid:12) Ψ ( W ) ( z ) (cid:12)(cid:12) ,Z Z ≡ Z R ′ R dz Rz (cid:12)(cid:12) Ψ ( Z ) ( z ) (cid:12)(cid:12) . (II.19)Masses of W and Z bosons, M W , M Z correspond to m ( W )1 , m ( Z )1 and those are determinedby KK mass conditions (A.7) and (A.14), respectively. For m ( V )1 ≪ /R ′ ( V = W, Z ), theKK mass conditions are approximately written as (cid:16) m ( V )1 (cid:17) ≃ x R ′ (1 + MR ′ ) ln( R ′ /R ) (cid:18) x (1 + MR ′ ) ln( R ′ /R ) (cid:19) , (II.20)where x = 1 [ ( g + 2˜ g ) / ( g + ˜ g ) = 1 / cos θ W ] for V = W [ Z ]. In the M R ′ → ∞ limitthey agree with results in [27], m ( W )1 ≃ / ( R ′ p ln( R ′ /R )) and m ( Z )1 ≃ / ( R ′ p R ′ /R cos θ W ).When M R ′ ≪
1, ( m ( W,Z )1 ) are suppressed by a factor (1 + MR ′ ) − ≃ M R ′ / m ( V )1 = M V essentially normalizes R and R ′ , i.e., the size and shape of theextra dimension, and also determine the shapes of wave functions ψ ( A ) n . Contrary, one canread masses of W and Z bosons from the bulk and boundary actions. In the boundaryaction the mass terms at z = R ′ serve masses for the W and Z bosons. Such brane massesfor W and Z bosons m ( V )brane ( V = W, Z ) can be read from the boundary interaction (II.6) as (cid:16) m ( W )brane (cid:17) W − µ W + µ ( x ) = R ′ R M ψ ( L ± )1 − ψ ( R ± )1 √ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = R ′ W − µ W + µ ( x ) , (cid:16) m ( Z )brane (cid:17) Z µ Z µ ( x ) = 12 R ′ R M ψ ( L Z )1 − ψ ( R Z )1 √ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = R ′ Z µ Z µ ( x ) . (II.21)Using wave functions in Appendix A, we obtain m ( V )brane m ( V )1 ≃ s M R ′ (1 + MR ′ ) , V = Z, W, (II.22)7here C ′ ( R, m ( V )1 ) ≃ ( m ( V )1 ) R ln( R ′ /R ) has been used. In the M R ′ → m ( V )brane = m ( V )1 ( V = Z, W ) is satisfied and hence brane masses account for masses of the W and Z bosons. When M R ′ → ∞ , on the other hand, m ( V = W,Z )brane vanish. Alternatively, one candefine P W ≡ Z R ′ R dz Rz (cid:20)(cid:16) ∂ ψ ( L ± )1 (cid:17) + (cid:16) ∂ ψ ( R ± )1 (cid:17) (cid:21) ,P Z ≡ Z R ′ R dz Rz (cid:20)(cid:16) ∂ ψ ( L Z )1 (cid:17) + (cid:16) ∂ ψ ( R Z )1 (cid:17) + (cid:16) ∂ ψ ( BZ )1 (cid:17) (cid:21) , (II.23)each of which measures the contribution of extra-dimensional component of the momentum, p , to the mass-squared of the vector boson. Contrary to the brane masses (II.21), in the M R ′ → ∞ limit we obtain [27]( P W , P Z ) MR ′ →∞ = ( M W , M Z ) . (II.24)Hence in this limit P W and P Z account for the W and Z boson masses M W and M Z ,respectively.Now we consider the precision observables. The S , T and U parameters are defined in[5, 6] by S ≡ π [Π ′ (0) − Π ′ Q (0)] ,T ≡ πc w s w M Z [Π (0) − Π (0)] ,U ≡ π [Π ′ (0) − Π ′ (0)] . (II.25)Since S and U parameters are related with wave-function renormalizations [33, 34], justfollowing [27], we write S = 16 π Π ′ (0) = 16 π − Z Z g + g ′ ,U = 16 π (cid:20) − Z W g − − Z Z g + g ′ (cid:21) , (II.26)where Z W and Z Z are defined in (II.19), and we have used Π ( ′ )3 Q = 0 at tree level.There are a few possible expressions of the T parameter. At first, following [27] one canidentify P W and P Z with vacuum polarizations at zero momentum P W ⇔ g Π (0) , Z ⇔ ( g + g ′ )Π (0) , (II.27)and define T = T (A) ≡ α EM M Z (cid:20) P W cos θ W − P Z (cid:21) . (II.28)We note that in the M R ′ → ∞ limit we have (II.24) and hence identifications (II.27) arenaturally allowed. For M R ′ ≪
1, however, both P W and P Z can be much smaller than M W and M Z and the above identifications cannot be justified. As one of alternatives to T (A) , weexpress T parameter by a deviation of a tree-level ρ parameter from the unity. To make acontrast with T (A) , here we write the ρ parameter in terms of m ( W )1 and m ( Z )1 as ρ ≡ θ W m ( W )1 m ( Z )1 ! , (II.29)where m ( W )1 and m ( Z )1 are determined by the KK conditions (A.7) and (A.14) with couplingssatisfying (II.16), respectively. Then we define T = T (B) ≡ sin θ W − sin θ ′ W α EM cos θ W , sin θ ′ W ≡ − m ( W )1 m ( Z )1 ! . (II.30)Using (II.20), we estimate T (B) ≃ sin θ W α EM cos θ W · MR ′ ) ln( R ′ /R ) ∼ . · MR ′ ) ln( R ′ /R ) . (II.31)In the M R ′ → ∞ limit we obtain T (B) ∼ . · / ln( R ′ /R ). This is a considerably largevalue even for ln( R ′ /R ) = O (30), although m ( W,Z )1 are not directly related to masses of W and Z bosons. For M R ′ ≪
1, on the other hand, T (B) is suppressed by the factor M R ′ . Wealso note that in the M R ′ → ∞ limit we have S = 6 π/ ( g ln( R ′ /R )) [27] hence S = 4 cos θ W · T (B) (II.32)is satisfied. In the followings, we use both T (A) and T (B) as reference values of the T parameterin this model. 9 ABLE I: Boundary condition parameter M , masses of the first KK states M Z ′ ,W ′ ,γ ′ , KK momen-tum mass-squared P Z,W , and oblique parameters S , T and U . M Z ′ , M W ′ and M γ ′ are masses of1st KK Z , W and photon, respectively. P Z,W are KK momentum mass-squared of Z and W bosons(see text). For the T parameter, two different values T = T (A) and T (B) are shown (see text). Asinput parameters, R − = 10 GeV, 10 GeV and
M R ′ ≥ .
01 are chosen. R − = 10 GeV R − = 10 GeV
M R ′ ∞
10 1 0 . . ∞
10 1 0 . . M [GeV] ∞ . ∞ . M Z ′ ,W ′ [TeV] 1 .
07 1 .
17 1 .
85 4 .
81 14 . .
69 0 .
76 1 .
18 3 .
00 8 . M γ ′ [TeV] 1 .
09 1 .
19 1 .
87 4 .
90 14 . .
71 0 .
78 1 .
23 3 .
16 9 . P Z [GeV ] 8190 6824 2723 387 . . . . P W [GeV ] 6365 5306 2122 302 . . . . S .
36 1 .
21 0 .
57 0 .
09 0 .
01 3 .
15 2 .
80 1 .
33 0 .
22 0 . T (A) − .
002 0 .
05 0 .
11 0 .
03 0 . − .
014 0 .
11 0 .
27 0 .
07 0 . T (B) .
45 0 .
40 0 .
19 0 .
03 0 .
003 1 .
10 0 .
97 0 .
45 0 .
07 0 . U × − − − − . − . − − − − . − . III. NUMERICAL STUDY
In the numerical study, to see the tree level effects we use α EM = e / π = 1 / θ W = M W /M Z , M W = 80 . M Z = 91 . R and M R ′ as inputparameters. R ′ is normalized so that m ( Z )1 = M Z is satisfied.In Table I, we have tabulated M , and masses of the first KK Z ′ , W ′ , γ ′ . Here Z ′ , W ′ and γ ′ are first KK Z , W , and photon and correspond to Z (2) µ , W (2) µ and γ (1) µ in (II.8),respectively. We also note that masses of W ′ and Z ′ are almost degenerate. We have alsoshown P Z and P W , which are defined in (II.23). Finally, in Table I, we have tabulated S , T = T (A) , T (B) and U parameters. We also plotted the ( M, M Z ′ ) and ( M, S ) with respect to
M R ′ in Figures 1 and 2, respectively.From Table I one finds that M Z ′ , M W ′ ≃ . /R ′ , (III.1)10 æ æ æ æ æ æà à à à à à à
50 100 500 1000 5000 1 ´ ´ ´ M @ GeV D M Z ' @ G e V D R - = GeVMR' = R - = GeV MR' = FIG. 1: (
M, M Z ′ ) as functions of M R ′ . Blue circles and red squares correspond to R − = 10 GeVand 10 GeV, respectively. and that γ ′ is slightly heavier than Z ′ and W ′ . From plots in Figure 1, we see that M Z ′ (or R ′ ) is in inverse proportion to M when M . M R ′ ≪ R ′− ≃ M W M ln[ R − /µ ′ ]= 3 . · (cid:18) M (cid:19) (cid:20) (cid:26) ln (cid:18) R − GeV (cid:19) − ln (cid:18) µ ′ GeV (cid:19)(cid:27)(cid:21) , (III.2)where µ ′ = O (1TeV), or one can solve µ ′ = R ′− by an iteration.Experimental lower bound for masses of heavy charged vector bosons at LHC Run-1 are, M W ′ ≥ . M W ′ ≥ . M Z ′ ≥ . Z ′ with SM-like coupling to fermions, and M Z ′ ≥ . √ s = 13TeV), similar orslightly stringent bounds are obtained [20–25]. Hence we safely put the experimental boundsas M Z ′ , M W ′ & M R ′ . . .
1] for R − = 10 GeV [10 GeV].For P Z and P W , one finds numerically that P V ≃ M V (cid:18) M R ′ (cid:19) − , V = W, Z, (III.3)and the correspondence (II.27) holds only when
M R ′ ≫ æ æ æ æ æ æà à à à à
50 100 500 1000 5000 1 ´ - @ GeV D S R - = GeVMR' = R - = GeVMR' = FIG. 2: (
M, S ) as functions of
M R ′ . Blue circles and red squares correspond to R − = 10 GeVand 10 GeV, respectively. The light-blue horizontal band shows allowed range of the S parameter. For the S parameter, as pointed out in [27] in the M → ∞ limit large value of S = O (1)is obtained. We also see that S shrinks as M R ′ decreases. In Figure 2, an allowed region of S parameter is also shown. Here current experimental bounds for S, T, U are given in [35]as S = 0 . ± . , T = 0 . ± . , U = 0 . ± . , (III.4)and S − T , S − U and T − U correlations are 0 . − .
59 and 0 .
83, respectively. Fromthe allowed range in Figure. 2 we obtain the bound
M R ′ . . .
05] for R =1 = 10 GeV[10 GeV].For the T parameter, T = T (A) is tiny for M R ′ = ∞ and this is consistent with the resultsin [27]. When M R ′ decreases, M R ′ monotonically increases [decreases] for ∞ > M R ′ & & M R > T (B) is monotonically decreasing for decreasing M R ′ , and one finds that T (B) is almostproportional to S , 3 T (B) ∼ S, (III.5)from which we find that (II.32) is well satisfied for finite M R ′ since 4 cos θ W = 3 .
1. Numer-12cally we also find that T (A) ≃ T (B) , (III.6)for M Z ′ & U parameter is very small and this also agrees with results in the originalmodel [27].As we have seen from Table I, Figures 1 and 2, both M Z ′ and oblique parameters dependlargely on both R and M R ′ . However, once we choose the free parameters as M Z ′ and R , wefind that the oblique parameters mainly depend on M Z ′ but weakly on R . We numericallyfind that S and M Z ′ are related by S ≃ (cid:18) M Z ′ (cid:19) . , (III.7)irrespective to the magnitude of R . This behavior is reasonably reflects the fact that S is adimension-six operator and should be inversely proportional to the square of a new physicsscale.From (III.5), (III.6) and (III.7), for M Z ′ & S and T as functionsof M Z ′ irrespective to the magnitude of R . In Fig. 3, we plot ( S, T ) with respect to M Z ′ .From the constraints for ( S, T ) with U ≃ M Z ′ ≥ R . Hereafter we refer (III.8) as 95% and 99% CL boundsof this model.From the Fig. 1 (or (III.2)) and the bounds (III.8), we obtain M . R − = 10 GeV70GeV [40GeV] for R − = 10 GeV (III.9)and
M R ′ . . .
03] for R − = 10 GeV0 .
03 [0 .
01] for R − = 10 GeV (III.10)for 99% [95%] CL bounds. 13 æææ àààà - - - T M Z ' @ TeV D = FIG. 3: (
S, T ) as functions of the first KK Z boson mass M Z ′ , M Z ′ = [2TeV , S, T ) = (0 ,
0) corresponds to the SM value. Blue circles and red squares correspond to(
S, T = T (A) ) for R − = 10 GeV and 10 GeV, respectively. Black dashed and dotted lines indicate(
S, T = T (B) ) for R − = 10 GeV and 10 GeV, respectively. Contours are 68%, 95% and 99% CLin (
S, T ) plane with U = 0. All points and curves meet at ( S, T ) = (0 ,
0) for M Z ′ = ∞ . Plots for M Z ′ > If we assume that the boundary condition (II.5) comes from the boundary mass termsgiven in (II.6) with (II.7), then from Eq. (II.22), we obtain m ( V )brane M V ≃ m ( V )brane m ( V )1 & .
976 [0 . R − = 10 GeV0 .
993 [0 . R − = 10 GeV( V = W, Z ) , (III.11)for 99% [95%] CL bounds. Since R − cannot exceed the reduced Planck mass M Pl , byextrapolating the above results up to R − = M Pl = 2 . × GeV we find that the branemass terms account for more than 97% [99%] of W and Z boson masses for 99% [95%] CLbounds with R − ≤ M Pl .The boundary mass term at z = R ′ may be interpreted as a VEV of a scalar field Φ,which is a scalar transforming as ( , ¯ ) of SU (2) L ⊗ SU (2) R and is localized on the z = R ′ brane. It is natural to identify this scalar with the SM-like Higgs field with 125GeV mass.If so, the ratios (II.22) are viewed as a ratios of the HW W and
ZW W couplings to their14M values, i.e., κ V ≡ g HV V /g SMHV V ≃ ( m ( V )brane /m ( V )1 ) ( V = W, Z ) where g SMHV V = gM V are HV V couplings in the SM. Then one obtain κ W = κ Z and1 − κ W,Z ≃ M R ′ M R ′ ≃ M W R ′ ln( R ′ /R )= (cid:18) . M W M Z ′ (cid:19) ln (cid:18) . M Z ′ R (cid:19) , (III.12)where (II.20) with M W ≃ m ( W )1 and (III.1) are used. For bounds 1 − κ V ≤ . M R ′ ≤ .
22, 0 . .
020 and 0 . M Z ′ ≥ R = 10 GeV [10 GeV] correspond to 1 − κ W,Z ≤ .
5% [5%, 1 . .
4% and 0 . κ V = W,Z will be precisely measuredat current and future collider experiments. Hence both the mass of first KK bosons and thecouplings between the Higgs and weak bosons will constrain the parameters of the model.
IV. SUMMARY AND DISCUSSION
In this paper we reconsidered the Higgsless model in the warped extra dimension. SomeDirichlet boundary conditions on the TeV brane are replaced with Robin boundary con-ditions which are parameterized by a mass parameter M . The Peskin-Takeuchi obliqueparameters in this model at tree level are evaluated. From the experimental bounds ofoblique parameters the lower bounds of the mass of the first Kaluza-Klein excited Z and W bosons M Z ′ ,W ′ are obtained. At 95 % [99 %] confidence level (CL), M Z ′ , M W ′ are greaterthan 8TeV [5TeV]. The magnitude of M , which is infinity in the original model, is smallerthan 120 [40] GeV for the curvature of the warped space R − = 10 GeV [10 GeV] at 95%CL. If we assume that the Robin boundary conditions come from the brane mass terms, itturns out that the brane mass accounts for more than 97% of the W and Z boson massesfor 99% CL bounds. If the brane mass is induced by the vacuum expectations value of theHiggs field Φ localized on the TeV brane, the model will also be tested by the precisionmeasurement of the Higgs-weak boson couplings.In this model fermions corresponding to the SM right-handed fermions have not beenintroduced. To obtain the Yukawa coupling, at least either left-handed fermions or Higgsfield Φ, or both must propagate in the bulk. When Φ propagates in the
AdS bulk andits kinetic term is given by R d x R dz ( R/z ) η MN tr( D M Φ) † ( D N Φ), then a steeply growing15EV, h Φ( z ) i ∝ z α , α >
1, eaasily mimics the boundary mass term at z = R ′ in (II.6).We also note that the hierarchy between M and R ′− which is expressed as M R ′ can beameliorated to O (( M R ′ ) /α ). In the α = 3 case, h Φ( z ) i can be viewed as a condensationwhich breaks SU (2) R × SU (2) R “chiral” symmetry in AdS/QCD [36–38] in the contextof AdS /CFT correspondence [39, 40]. In the α = 2 case, h Φ( z ) i may be interpreted asa VEV of 5th component of the SO (5) /SO (4) gauge fields in the context of the gauge-Higgs unification (GHU) in warped space [41–45], or as a pseudo Nambu-Goldstone bosonof SO (5) → SO (4) symmetry breaking [46–49]. In the GHU case, the electroweak symmetrywill be broken by the Hosotani mechanism [50], and the mass of the Higgs is stabilized bythe higher-dimensional gauge symmetry [51].In this paper contributions to oblique parameters at loop levels are not evaluated. In thismodel the mass and the mechanism to develop a VEV of the “Higgs” are also unexplained.These issues are model-dependent and will be discussed separately. Acknowledgements
This work was supported in part by National Research Fund of Korea(Grant No. 2012R1A2A1A01006053).
Appendix A: Wave functions1. Bulk functions
It would be useful to introduce bulk functions C ( z, q ) , S ( z, q ) which satisfy the equationof motion (II.9) and satisfy C ( R ′ , q ) = 1 , S ( R ′ , q ) = 0 ,C ′ ( R ′ , q ) = 0 , S ′ ( R ′ , q ) = q, (A.1)where C ′ ( z, q ) ≡ ∂ C ( z, q ) and S ′ ( z, q ) ≡ ∂ S ( z, q ). They can be written by C ( z, q ) = π qz [ Y ( qR ′ ) J ( qz ) − J ( qR ′ ) Y ( qz )] ,S ( z, q ) = π qz [ − Y ( qR ′ ) J ( qz ) + J ( qR ′ ) Y ( qz )] , ′ ( z, q ) = π q z [ Y ( qR ′ ) J ( qz ) − J ( qR ′ ) Y ( qz )] ,S ′ ( z, q ) = π q z [ − Y ( qR ′ ) J ( qz ) + J ( qR ′ ) Y ( qz )] , (A.2)where J ν and Y ν are Bessel functions of 1st and 2nd kind, respectively. C, S, C ′ and S ′ satifty C ( z, q ) S ′ ( z, q ) − C ′ ( z, q ) S ( z, q ) = qRR ′ . (A.3)From the boundary conditions at z = R ′ , Eqs. (II.4)(II.5), one can write the wave functionsin (II.8) as ψ ( LaU ) + ψ ( RaU ) = a ( U ) V C ( z, q ) ,ψ ( LaU ) − ψ ( RaU ) = a ( U ) A [ S ( z, q ) − ( q/M ) C ( z, q )] , a = 1 , , ,ψ ( BU ) = a ( U ) B C ( z, q ) , (A.4)for U = Z, γ, W . Subscripts for the KK number are omitted. Boundary conditions (II.3)determine q = m ( U ) n and a ( U ) V,A,B except for overall normalizations.
2. Charged bosons a. W -boson tower Wave functions for the W ± bosons and their KK excitation modesare ψ ( R ± ) n = N W n " S ′ ( R ) C ( z ) + C ′ ( R ) S ( z ) − m ( W ) n M C ′ ( R ) C ( z ) ,ψ ( L ± ) n = N W n [ S ′ ( R ) C ( z ) − C ′ ( R ) S ( z )] , (A.5)where C ( z ) = C ( z, m ( W ) n ) and so on. N W n is a normalization factor. The KK mass m ( W ) n isdetermined by − m ( W ) n M CC ′ + CS ′ + SC ′ = 0 , (A.6)where C ( ′ ) = C ( ′ ) ( R, m ( W ) n ) and so on. Using (A.3) we rewrite (A.6) as − m ( W ) n M CC ′ + 2 SC ′ + m ( W ) n RR ′ = 0 . (A.7)For the W bosons ( n = 1), with the normalization condition (II.11), the normalizeionfactor N W is determined to be N W = R ′ m ( W )1 R p R ln( R ′ /R ) . (A.8)17 . Neutral bosons b. Photon tower For n ≥ ψ ( L γ ) n ( z ) = ψ ( R γ ) n ( z ) = N γ n ˜ g C ( z, m ( γ ) n ) ,ψ ( Bγ ) n ( z ) = N γ n g C ( z, m ( γ ) n ) , (A.9)where N γ n is a normalization factor. The KK mass m ( γ ) n is determined by C ′ ( R, m ( γ ) n ) = 0 . (A.10)The photon correspond to the n = 0 mode and its wave functions are given by (cid:16) ψ ( R γ )0 ( z ) , ψ ( L γ )0 ( z ) , ψ ( Bγ )0 ( z ) (cid:17) = N γ (cid:18) g , g , g (cid:19) , (A.11)where N γ = e is fixed by (II.12). c. Z -boson tower wave functions of the Z boson and its KK excitations are given by ψ ( BZ ) n ( z ) = − g g N Z n " S ′ ( R ) − m ( Z ) n M C ′ ( R ) C ( z ) ,ψ ( L Z ) n ( z ) = N Z n [ S ′ ( R ) C ( z ) − C ′ ( R ) S ( z )] ,ψ ( R Z ) n ( z ) = N Z n " S ′ ( R ) C ( z ) + C ′ ( R ) S ( z ) − m ( Z ) n M C ′ ( R ) C ( z ) , (A.12)where C ( z ) = C ( z, m ( Z ) n ), and so on, and N Z n is a normalization factor. The KK mass m ( Z ) n is determined by − m ( Z ) n M ( g + ˜ g ) CC ′ + ( g + 2˜ g ) CS ′ + g SC ′ = 0 , (A.13)where C ( ′ ) ≡ C ( ′ ) ( R, m ( Z ) n ) and so on. Using (A.3) we rewrite (A.13) as − m ( Z ) n M CC ′ + 2 SC ′ + g + 2˜ g g + ˜ g m ( Z ) n RR ′ = 0 . (A.14)For the Z boson ( n = 1), the normalization factor is determined by (II.11) to be N Z = R ′ cos θ W m ( Z )1 R p R ln( R ′ /R ) . (A.15) [1] G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC” , Phys. Lett. B , 1(2012) [arXiv:1207.7214 [hep-ex]].
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