Warped Electroweak Breaking Without Custodial Symmetry
aa r X i v : . [ h e p - ph ] M a r CPHT-RR096.1110UAB-FT-684
Warped Electroweak Breaking WithoutCustodial Symmetry
Joan A. Cabrer a , Gero von Gersdorff b and Mariano Quir´os a, ca Institut de F´ısica d’Altes Energies (IFAE), Universitat Aut`onoma de Barcelona08193 Bellaterra, Barcelona, Spain b Centre de Physique Th´eorique, ´Ecole Polytechnique and CNRSF-91128 Palaiseau, France c Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA)
Abstract
We propose an alternative to the introduction of an extra gauge (custodial)symmetry to suppress the contribution of KK modes to the T parameterin warped theories of electroweak breaking. The mechanism is based on ageneral class of warped 5D metrics and a Higgs propagating in the bulk.The metrics are nearly AdS in the UV region but depart from AdS inthe IR region, towards where KK fluctuations are mainly localized, andhave a singularity outside the slice between the UV and IR branes. Thisgravitational background is generated by a bulk stabilizing scalar field whichtriggers a natural solution to the hierarchy problem. Depending on themodel parameters, gauge-boson KK modes can be consistent with presentbounds on EWPT for m KK & . Introduction Warped models in five dimensions were proposed by Randall and Sundrum(RS) [1] as an elegant way of solving the Planck/weak hierarchy problem. In theRS setup, we live in a slice of AdS with two flat branes in the ultraviolet (UV)and infrared (IR) regions whose four-dimensional theory is Poincar´e invariant.If the Higgs field is localized on (or towards) the IR brane, in the 4D theoryits Planckian mass is warped down to the TeV scale, and this is how the largehierarchy is generated.When gauge bosons propagate in the 5D bulk their Kaluza-Klein (KK) exci-tations can contribute to the Standard Model electroweak precision observables–in particular the T parameter [2, 3]– and their masses and couplings have to becontrasted with the Standard Model electroweak precision tests (EWPT). SinceKK modes decouple when they are heavy, EWPT translate into lower bounds ontheir masses. If these bounds are much larger than LHC scales, they make thetheories phenomenologically unappealing and create a “little hierarchy” problem,which translates into some amount of fine-tuning to stabilize weak masses. Inparticular, in order to avoid large volume-enhanced contributions to the T pa-rameter it was proposed to enlarge the gauge symmetry in the bulk by addingthe SU (2) R × U (1) B − L gauge group [4]. In this case one has to worry only forlarge contributions to the S parameter, which yield bounds on KK masses of O (3)TeV. Moreover, the presence of extra matter, in particular the SU (2) R -symmetricpartner of the RH top quark, generate large anomalous (volume enhanced) con-tributions to the Z ¯ bb vertex which should be controled by introducing a discreteleft-right symmetry [5].The presence of large IR brane kinetic terms was proposed in Ref. [6] as away of reducing the T parameter. Generically they include bare contributions,which encode physics above the cutoff scale, as well as radiative contributions thatare calculable within the effective 5D theory. However on the IR brane radiativeeffects are small since the local cutoff is around the TeV scale, there is no room fora large logarithmic enhancement and a large IR brane kinetic term would have toarise from the unknown UV physics. Let us remark though that such unknown UVphysics could just as well directly modify the T parameter, by contributing to anIR brane localized operator of the type | H † D µ H | . In order to keep calculabilitywe will assume such degrees of freedom to be absent.In this letter we will propose a simple alternative to the introduction of anextra custodial symmetry (or large IR brane kinetic terms) based on generalizedmetrics and a bulk Higgs field, to keep the T parameter under control, whichshould allow us to construct a pure 5D Standard Model. Although there arenegative results in the literature based on general 5D metrics when the Higgs islocalized on the IR brane [7, 8], we will circumvent them by assuming a Higgspropagating in the bulk of the fifth dimension in the background of a singular2etric (asymptotically AdS near the UV brane) with a singularity outside the slicebetween the branes, but close enough to the IR brane. In fact, as we will see, the T parameter will be suppressed by the combined effect of the non-localized bulkHiggs as well as the vicinity of the singularity to the IR brane. This effect can betraced back to a large wave function renormalization for a light Higgs mode. Thisalternative to the custodial symmetry is purely based on a modification of the 5Dgravitational background, which requires a bulk propagating scalar field playingthe role of the Goldberger-Wise field [9] in RS theories, and does not interferewith the electroweak physics. This kind of metrics have been widely studied inthe literature in the past [10] and more recently they constitute the backgroundof the so-called soft-wall models [11–19]. Let us mention that a reduction in the Sparameter in a particular class of soft-wall models with custodial symmetry wasalready pointed out in Ref. [12].
2. General results
We will now consider the Standard Model (SM) propagating in a 5D spacewith an arbitrary metric A ( y ) such that ds = e − A ( y ) η µν dx µ dx ν + dy , (1)in proper coordinates and two branes localized at y = 0 and y = y , at theedges of a finite S / Z -interval. We define the 5D SU (2) L × U (1) Y gauge bosonsas W iM ( x, y ), B M ( x, y ) [or in the weak basis A γM ( x, y ), Z M ( x, y ) and W ± M ( x, y )],where i = 1 , , M = µ,
5, and the SM Higgs as H ( x, y ) = 1 √ e iχ ( x,y ) (cid:18) h ( y ) + ξ ( x, y ) (cid:19) , (2)where the matrix χ ( x, y ) contains the three 5D SM Goldstone bosons. The Higgsbackground h ( y ) as well as the metric A ( y ) will be for the moment arbitraryfunctions which will be specified later on.We will consider the 5D action (in units of the 5D Planck scale M ) for theHiggs field H and other possible scalar fields of the theory, generically denoted as φ : S = Z d xdy √− g (cid:18) − ~W MN − B MN − | D M H | −
12 ( D M φ ) − V ( H, φ ) (cid:19) − X α Z d xdy √− g ( − α λ α ( H, φ ) δ ( y − y α ) , (3)where V is the 5D potential and λ α ( α = 0 ,
1) the 4D brane potentials whichdepend on the scalar fields of the theory. For the sake of simplicity, we will assume3ll the fermions of the SM living on the UV brane. One can then construct the4D effective theory out of (3) by making the KK-mode expansion [20] A µ ( x, y ) = a µ ( x ) · f A ( y ) / √ y where A = A γ , Z, W ± and the dot product denotes an expansionin modes. The functions f A satisfy the equations of motion (EOM) m f A f A + ( e − A f ′ A ) ′ − M A f A = 0 , (4)where the functions f A ( y ) are normalized as R y f A ( y ) dy = y and satisfy theboundary conditions e − A f ′ A | y =0 ,y = 0 . We have defined the 5D y -dependentgauge boson masses M W ( y ) = g h ( y ) e − A ( y ) , M Z ( y ) = 1 c W M W ( y ) , M γ ( y ) ≡ . (5)where c W = g / p g + g ′ , and g and g ′ are the 5D SU (2) L and U (1) Y couplingsrespectively.Equations (4) will in general not have analytic solutions. However we canfind an approximation for the light mode of (4) with mass m f A ≡ m A in thelimit where the breaking is small ( m A ≪ m KK ) and thus there is a zero modewith almost constant profile. Expanding around this limit we can write f A ( y ) =1 − δ A + δf A ( y ) , where δf A ( y ) = Z y dy ′ e A ( y ′ ) Z y ′ dy ′′ (cid:2) M A ( y ′′ ) − m A, (cid:3) , (6) δ A = 1 y Z y dy δf A ( y ) , (7) m A, = 1 y Z y M A ( y ) dy , (8)and where m A, is a zeroth order approximation for the zero mode mass m A .Including the first order deviations from a constant profile we obtain for the lightmode mass the expression m A = m A, − y Z y dy M A ( y ) [ δ A − δf A ( y )] . (9)In the SM the three Lagrangian parameters ( g, g ′ , v ), or equivalently ( e, s w , v )are normally traded for the well measured parameters ( α, G F , m Z ) in such a waythat all other SM observables can be written as functions of them. In our 5D modelwe have to fix the physical Z mass using Eq. (9) m Z = 91 . ± . S, T, U ) variables in Ref. [21]. They canbe given the general expressions [20]: αT = s W m Z ρ k y Z y (1 − Ω( y )) e A ( y ) − A ( y ) , (10) αS = 8 c W s W m Z ρ k y Z y (cid:18) − yy (cid:19) (1 − Ω( y )) e A ( y ) − A ( y ) , (11) αU = O ( δ Z ) ≃ , (12)where ρ = ke − A ( y ) , Ω( y ) = U ( y ) U ( y ) , U ( y ) = Z y h e − A , (13)We have reexpressed the hierarchy exp[ − A ( y )] by the ratio of an IR scale ρ anda UV scale k which we will take to be of the order of the TeV and Planck scalesrespectively. The parameter ρ is related to the gauge boson KK-mode mass m KK by m KK = F ( A ) ρ where F is a function which depends on the metric A andit is entirely determined from the solution to the EOM (4). The function Ω ismonotonically increasing from Ω(0) = 0 to Ω( y ) = 1. In the case of an IRbrane localized Higgs it is actually a step function and in particular it vanishesidentically in the bulk, Ω = 0. More generally, due to the presence of the warpfactor, the integral will be dominated near the IR and one could approximateΩ( y ) ≃ − k ( y − y ) Z , (14)where Z = k Ω ′ ( y ) = k Z y dy h ( y ) h ( y ) e − A ( y )+2 A ( y ) . (15)The integrals in Eqs. (10)-(12) will be approximated well whenever Z is largeenough. One finds αT = s W m Z ρ ky Z I, (16) αS = 8 c W s W m Z ρ Z I , (17)with the dimensionless integral I = k Z y [ k ( y − y )] e A ( y ) − A ( y ) dy . (18)Notice in particular the standard volume enhancement of the T parameter. Thisapproximation is valid whenever Z is a parametrically large number (but not5ecessarily as large as the volume ky ). One can see that the T parameter issuppressed with two powers of Z , while the S parameter is only suppressed withone power. As we will see below, in theories with a light Higgs mode the quantity √ Z can in fact be interpreted as a wave function renormalization in the effectiveLagrangian of that mode. Note that the operators contributing to T have fourpowers of the Higgs field (e.g. | H † D µ H | ), while the ones contributing to S haveonly two powers ( e.g. H † W µν HB µν ), thus nicely explaining the observed suppres-sion in models with large values of Z . In case of pure RS with a bulk Higgs profile h ( y ) ∼ e aky one can easily evaluate Z − = 2( a − a >
2, this is not a small number. Nevertheless,we will see below that there are theories which can have sizable Z factors despitethe fact that a >
2, and hence display suppression of the precision observables.In the remainder of this Letter we will always use for numerical calculations theexact expressions, Eq. (10) and (11).The SM fit on the (
S, T ) plane, assuming U = 0, for a reference Higgs mass m refH = 117 GeV, provides [22] T = 0 . ± . , S = 0 . ± . , (19)which should constrain any particular model.For the Higgs fluctuations in (2) one can write from the action (3) an EOMsimilar to that of gauge bosons (4). In fact making the KK-mode expansion ξ ( x, y ) = H ( x ) · ξ ( y ) / √ y the functions ξ ( y ) satisfy the bulk equation ξ ′′ ( y ) − A ′ ξ ′ ( y ) − ∂ V∂h ξ ( y ) + m ξ e A ξ ( y ) = 0 , (20)as well as the boundary conditions (BC) ξ ′ ( y α ) ξ ( y α ) = ∂ λ α ( h ) ∂h (cid:12)(cid:12)(cid:12)(cid:12) y = y α . (21)If we compare Eq. (20) with the bulk EOM for the Higgs background [20] for aquadratic bulk Higgs potential h ′′ ( y ) − A ′ h ′ ( y ) − ∂V∂h = 0 , (22)with BC h ′ ( y α ) h ( y α ) = ∂λ α ( h ) ∂h (cid:12)(cid:12)(cid:12)(cid:12) y = y α , (23)we see that the Higgs wave function for m H ≡ m ξ = 0 is proportional to h ( y ). Forsmall values of m H we can correct this to O ( m H ) which yields the corresponding6properly normalized) wave function ξ H ( y ) = r ky Z h ( y ) h ( y ) e A ( y ) (cid:20) − m H (cid:18)Z y e A ΩΩ ′ + Z y e A ΩΩ ′ (Ω − (cid:19)(cid:21) , (24)where the function Ω( y ) was defined in Eq. (13). The true value of m H is of coursedetermined by the boundary conditions. With the usual choice of the boundarypotentials λ = M | H | , − λ = − M | H | + γ | H | , (25)and after using the BC (21) and (23), as well as the definition of Z , Eq. (15),one obtains for the light Higgs mass m H = ( kZ ) − (cid:18) M − h ′ h (cid:19) ρ . (26)Using the definition of the W W ξ n coupling [20] h W W ξ n = gy Z y dy e − A ( y ) M A ( y ) f ( y ) ξ n ( y ) (27)and the wave function (24), one can deduce that h W W H = h SMW W H (cid:2) − O ( m H /m KK , m W /m KK ) (cid:3) , (28)so the light Higgs unitarizes the theory in a similar way to the SM Higgs.
3. RS model
The previous formalism can be applied to any particular 5D model. Thesimplest and best known case is the RS model where the space is a slice of AdSspace, with metric A ( y ) = ky . Assuming an exponential background for the Higgsfield as h ( y ) = h ( y ) e ak ( y − y ) , (29)the T and S parameters can be readily computed from Eqs. (10) and (11) yielding α ( m Z ) T RS = s W m Z ρ ( ky ) ( a − a (2 a −
1) + . . . , (30) α ( m Z ) S RS = 2 s W c W m Z ρ a − a + . . . , (31) As already mentioned above, the quantity √ Z can be viewed as a wave function renor-malization in the effective theory, obtained by integrating over the true zero mode ξ ( x, y ) = H ( x ) h ( y ) /h to obtain −L eff = e − A k − Z | D µ H| + e − A [( h ′ /h − M ) |H| + γ |H| ]. ky and ρ = k exp( − ky ). In the holographic dual, the quantity a corresponds to thedimension of the Higgs condensate and we demand a ≥ F relating m KK to ρ is givenby F ( A RS ) ≃ .
4. These expressions agree precisely with the recent result inRef. [23]. One can see from (30) that the contribution to the T parameter isvolume enhanced while that to the S parameter is not. This translates into a verystrong bound on ρ when we compare these expressions with the experimental data(19). In particular for a Higgs localized on the IR brane (which corresponds tothe a → ∞ limit) the expression (30) and the experimental fit (19) leads to thebound ρ > . m KK > . a = 2 they are loweredby a factor √ ρ > . m KK > . SU (2) R × U (1) B − L symmetry in the bulk such thatthere is a residual custodial symmetry which protects the T parameter. In thatcase the experimental bound on the S parameter (19) translates into the bound ρ > . m KK > . ρ > . m KK > . a = 2.
4. The model
In the rest of this Letter we will explore an alternative solution to the problemof the T parameter based on singular metrics. We will see that the combinedeffect of the Higgs delocalization and the strength and vicinity of the IR brane tothe singularity makes the T parameter comparable to the S parameter. We willthen consider the metric singular at y = y s [17] A ( y ) = ky − ν log (cid:18) − yy s (cid:19) , (32)where ν > ν → ∞ limit it coincides with the RSmetric. Moreover, it is AdS near the UV brane and has a singularity at y s = y +∆,outside the slice between the UV and IR branes and at a distance ∆ from the IRbrane. We will also assume that the Higgs background is exponential and givenby Eq. (29). The gravitational setup which can provide such background willbe considered later on. For the moment we will just analyze the impact of thedeparture from AdS in the IR region in the EWPT parameters T and S . Thisdeparture is controlled by the parameters ν and ∆. The smaller ν and/or ∆ themore the IR brane feels the nearby singularity.As we can see in Fig. 1, the singularity affects both the T and the S parame-ter . In the left [right] panel we plot the ratio T /T RS [ S/S RS ] where T [ S ] is the In this letter we will consider the independent bounds on S and T . Notice that for T /S ∼ +1, a=2a=3 a= ∞ ν . . . . . . . T / T R S . . . . . . . . . PSfrag replacements a=2a=3a= ∞ ν . . . . . . . S / S R S . . . . . . . . Figure 1:
Plots of the T (left panel) and S (right panel) parameters as a functionof ν for different values of a , k ∆ = 1 and keeping the masses of the first heavyKK-mode of gauge bosons constant. The parameters are normalized to when ν = ∞ (i.e. the RS case). parameter obtained from Eq. (10) [Eq. (11)], as a function of ν and for differentvalues of a , and T RS [ S RS ] is the corresponding parameter in the RS-model asgiven by Eq. (30) [Eq. (31)]. In the plots we consider the same value of m KK inboth theories. For the metric (32) the relation between m KK and ρ is given bythe function F ( ν ) which is a monotonically decreasing function and reproducesthe RS-result in the limit ν → ∞ . An approximate fit for the function F ( ν ) andvalues ν & . k ∆ = 1 is provided by F ( ν ) ≃ .
44 + 1 . ν . (33)Since the observables are quadratically dependent on m KK this means that onecan read the corresponding reduction on the value of m KK , with respect to theRS-bound, by taking the square root of the vertical axis in the respective plot ofFig. 1.We can see from the left panel of Fig. 1 that there is no reduction on the T parameter for a localized Higgs in agreement with the general results of Refs. [7,8].On the other hand for a bulk Higgs the corresponding reduction in the T and S parameters can be qualitatively understood in the limit of large wave-functionrenormalization Z from the approximate expressions in Eqs. (16) and (17) in thelimit of large volume suppression. To see this better let us integrate out the KKmodes of the gauge bosons. This produces dimension-six operators quadratic inthe Higgs and fermion electroweak currents. The operator quadratic in the Higgs by reducing m KK from infinity to finite values, we move in the T − S plane from the originalong the major (long) axis of the ellipse, taking advantage of that particular correlation in thefit. T , while the operator proportional to the product of oneHiggs and one fermionic current contributes to S after going to the ”obliquebasis” [3]. Notice that the coupling of the KK modes to the Higgs current isproportional to Z − . The question is whether there are other effects that influencethese couplings. They can be calculated as integrals over KK wave functions,which can be obtained only numerically. However, we can easily evaluate thequantity I appearing in our expressions for S and T . Numerically it turns out that I m KK /ρ is fairly constant (in fact, it shows some additional mild suppression)over the interesting range of ν and ∆, and hence it is reasonable to expect thatthe couplings are mainly affected by the parameters ν , ∆ and a through thedependence on Z . It is noteworthy that the IR dimension of the Higgs condensatein the holographic 4D dual is reduced with respect to its UV conformal value[dim( O UVH ) = a ] leading to the enhanced Z factors responsible for the suppressionof the S and T parameters. Indeed, identifying the logarithm of the RG scale withthe metric as log Q = − A ( y ), the IR dimension of the Higgs can be expressed asdim( O IRH ) = a k ∆ ν . (34)Moreover, the number of colors N c of the effective theory is also reduced in the IRregion since the curvature increases along the extra dimension due to the presenceof the singularity : N IRc N UVc = (cid:18) R IR R UV (cid:19) − / = "(cid:18) k ∆ ν (cid:19) −
25 1( k ∆ ν ) − / . (35)It is interesting that both effects, small N c and dim( O IRH ) <
2, also play a majorrole in reducing S and T in the conformal technicolor model proposed in Ref. [26],the main difference being that our proposal implies a strong deformation of theconformal theory in the IR. More details will be presented elsewhere [20].A 5D setup leading to the background (32) and (29) can be easily obtainedby using the formalism of Ref. [27], where first-order gravitational EOM and thebulk potential can be obtained from a superpotential. We will introduce on top ofthe SM Higgs field H a scalar field φ which will generate a singularity at y = y s ,with the superpotential W ( φ, H ) related to the scalar potential by V ( φ, h ) ≡ "(cid:18) ∂W∂φ (cid:19) + (cid:18) ∂W∂h (cid:19) − W ( φ, h ) . (36) This phenomenon has been extensively studied in warped throat geometries with a singular-ity which appear e.g. in the Klebanov-Tseytlin solution of type IIB string constructions [24, 25]. A ′ ( y ) = 16 W ( φ ( y ) , h ( y )) , φ ′ ( y ) = ∂ φ W ( φ, h ) , h ′ ( y ) = ∂ h W ( φ, h ) . (37)In terms of the boundary potentials λ α ( φ, h ), the boundary conditions are A ′ ( y α ) = 23 λ α ( φ, h ) (cid:12)(cid:12)(cid:12)(cid:12) y = y α , φ ′ ( y α ) = ∂λ α ∂φ (cid:12)(cid:12)(cid:12)(cid:12) y = y α , h ′ ( y α ) = ∂λ α ∂h (cid:12)(cid:12)(cid:12)(cid:12) y = y α . (38)We postulate the superpotential W ( φ, H ) = W φ ( φ ) + W H ( h ) where W φ ( φ ) = 6 k (1 + be νφ/ √ ) , W H ( h ) = 12 akh , (39)and a and b are real arbitrary parameters. This leads to the background configu-ration (29) and [10] φ ( y ) = − √ ν log[ ν bk ( y s − y )] , (40) A ( y ) = ky − ν log (cid:18) − yy s (cid:19) + 124 ( h ( y ) − h (0)) , (41)where we are using the normalization A (0) = 0. Notice that the Higgs contri-butions to the metric near the UV brane and near the singularity at y = y s areoverwhelmed by that of the φ background. Thus, the metric (41) agrees to allpractical purposes with (32).Using the superpotential formalism amounts to some fine-tuning among thedifferent coefficients of the bulk potential, unless they are protected by someunderlying 5D supergravity [27]. The quadratic Higgs term which is generated by(36) can be written as k h a ( a − − abe νφ/ √ i | H | and the coefficients of thetwo operators | H | and e νφ/ √ | H | can be considered as independent parameters .However, since the parameter b can be traded by a global shift in the value of the φ field, or in particular by a shift in its value at the UV brane φ , for simplicitywe will fix its value to b = 1 hereafter.We assume that the brane dynamics λ αφ fixes the values of the field φ at φ = φ , φ on the UV and IR branes respectively. The inter-brane distance y , aswell as the location of the singularity at y s and the warp factor A ( y ), are relatedto the values of the field φ α at the branes by the following expressions: ky = 1 ν h e − νφ / √ − e − νφ / √ i , k ∆ = 1 ν e − νφ / √ ,A ( y ) ≃ ky + 1 ν ( φ − φ ) / √ , (42) Of course the coefficients of the operators not involving the Higgs field remain fine-tuned. m KK ( T e V ) ν . . . . . . . ST PSfrag replacements k∆0 . . . . . ST Figure 2:
Plots of the 95% CL lower bounds on the first KK-mode mass of elec-troweak gauge bosons from experimental bounds on the T and S parameters for A ( y ) = 35 and a = 2 as a function of ν (left panel, with k ∆ = 1 ) and as afunction of k ∆ (right panel, with ν = 1 ). which shows that the required large hierarchy can be naturally fixed with valuesof the fields φ & φ , φ < φ ≫ , y → y s . As for the branepotentials λ αH , they are given by Eq. (25). The BC (38) at the UV brane imply M − ak = 0, while the BC at y fixes the value of the Higgs background at theIR brane h as γh = M − ak and thus triggers electroweak symmetry breaking.Also note that due to its exponential dependence on φ , ∆ can be small or, inother words, the IR brane naturally occurs very close to the singularity.We will now explore numerically the predictions of the model defined by thebackground (29), (40) and (41). In particular, the KK-modes of the gauge bosonssatisfy Eq. (4). This equation can be solved numerically and the free parametersare ( y , ∆ , a, ν ). We will fix the Planck/weak hierarchy by imposing A ( y ) ≃ y = y (∆ , ν ) by which y increases with∆ and ν . The rest of parameters are free and they have a clear physical meaning.The parameter a indicates the departure from a localized Higgs, which is the limit a → ∞ . The parameter ν indicates the departure from the RS case, which is thelimit ν → ∞ . The parameter ∆ indicates the distance between the IR brane andthe singularity. All the effects that we consider are enhanced when a , ν and ∆decrease. We plot in Fig. 2 the lower bounds obtained on the mass m KK of the firstKK-mode for the electroweak gauge bosons W, Z and γ from the 95% CL bounds This apparent fine tuning is an artifact of the first order formulation and has no observablephysical consequences for a >
2. Indeed for a > M has no physical impactneither on the background nor in the spectrum. The fact that y is an increasing function of ∆ and ν helps lowering the bounds on ρ fromthe T parameter for small ν and ∆, since the T parameter is volume enhanced [see Eq. (16)].However this effect is small in the parameter region shown in the figures. m KK ( T e V ) gaugeHiggs ν . . . . . . .
01 2345678
PSfrag replacements gaugeHiggs k∆0 . . . . .
00 1234567
Figure 3:
Plots of the 95% CL lower bounds on the first KK-mode mass of theHiggs boson from experimental bounds on the T parameter for A ( y ) = 35 and a = 2 as a function of ν (left panel, with k ∆ = 1 ) and as a function of k ∆ (rightpanel, with ν = 1 ). For comparison the corresponding bounds on the KK-gaugeboson masses are shown. on the T and S parameters of Eq. (19). On the left plot we see the dependenceon the ν parameter for a = 2 and k ∆ = 1. We see that for ν ≫ m KK > T and S parameters respectively. On the right plot we see the influenceon the bounds of the vicinity to the singularity. In summary, for ν . k ∆ . m KK = O (1 −
3) TeV.Higgs fluctuations satisfy the EOM (20) and the BC (21). Plots based onnumerical solutions are shown in Fig. 3, where the 95% CL bounds on the firstKK-mode Higgs masses are shown. From Fig. 3 we see that Higgs KK-modes areheavier than gauge boson ones, disfavoring its experimental detection at LHC.Moreover, as we have seen earlier in this Letter, there is a light Higgs eigenstatewith wave function given by (24) and mass eigenvalue given by (26). After usingthe background metric A ( y ) in Eq. (41) and the background Higgs h ( y ) in Eq. (29),the Higgs mass eigenvalue is given by m H ρ = 2 µZ , (43) Z = e a − ∆ − η [2( a − − η Γ( η, a − , (44)where µ ≡ M /k − a , η = 1 + ν and Γ stands for the incomplete gamma function.Equation (44) provides a measure of the required fine-tuning to get a light Higgs.In fact for µ = 0 (which amounts to an infinite fine-tuning and no EWSB) theHiggs is massless. In general the smallness of µ is a measure of the degree of fine-tuning required to achieve a light Higgs. In fact, the fine-tuning is smaller than inthe RS-model because the prefactor of µ in (44) is smaller than one. For instance,13or a = 2, ν = 0 . k ∆ = 0 . ρ from the T parameter is ρ ≃ .
49 TeV and m H ≃ √ µ GeV, providing a light Higgs for µ = O (10 − ).
5. Conclusions
In this Letter we have presented an alternative mechanism to the introduc-tion of an extra gauge symmetry to suppress the T parameter in warped extradimensional models. The mechanism is based on the introduction of a metric thatis nearly AdS in the UV region but departs from AdS in the IR region, towardswhere KK-fluctuations are mainly localized, and that has a singularity outside theslice between the UV and IR branes. The two main parameters which control thiseffect are then the departure of the metric with respect to the AdS metric and thevicinity of the IR brane to the singularity. Depending on these parameters, thelower bounds on the gauge boson KK-mode masses can be as low as around theTeV scale. This low bound should alleviate the little hierarchy problem that arisesin warped electroweak breaking models and facilitate experimental detection ofthese heavy gauge bosons at LHC. The model requires a scalar field φ propagat-ing in the bulk, which plays the role of a radion stabilizing field, similar to theGoldberger-Wise scalar in the RS theory. We have considered the back-reactionsof the φ and Higgs fields on the metric by means of the superpotential formalism,which requires some fine-tuning in the gravitational part of the bulk potential.The model also contains a naturally light Higgs that unitarizes the 4D theory inmuch a similar way as the SM Higgs. Acknowledgments
Work supported in part by the Spanish Consolider-Ingenio 2010 ProgrammeCPAN (CSD2007-00042) and by CICYT, Spain, under contract FPA 2008-01430.The work of JAC is supported by the Spanish Ministry of Education through aFPU grant. The work of GG is supported by the ERC Advanced Grant 226371, theITN programme PITN- GA-2009-237920 and the IFCPAR CEFIPRA programme4104-2. JAC and MQ wish to thank CPTH ( ´Ecole Polytechnique, Paris) wherepart of this work has been done for hospitality. We wish to thank J. Santiago andE. Pont´on for discussions and correspondence on the S , T and U parameters inwarped models. GG would like to thank Kaustubh Agashe for helpful discussions. References [1] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999)[arXiv:hep-ph/9905221].[2] S. J. Huber and Q. Shafi, Phys. Rev. D , 045010 (2001)[arXiv:hep-ph/0005286]. 143] H. Davoudiasl, S. Gopalakrishna, E. Ponton and J. Santiago, New J. Phys. (2010) 075011 [arXiv:0908.1968 [hep-ph]], and references therein.[4] K. Agashe, A. Delgado, M. J. May and R. Sundrum, JHEP (2003) 050[arXiv:hep-ph/0308036].[5] K. Agashe, R. Contino, L. Da Rold and A. Pomarol, Phys. Lett. B (2006)62 [arXiv:hep-ph/0605341].[6] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Rev. D (2003)045002 [arXiv:hep-ph/0212279]; M. S. Carena, E. Ponton, T. M. P. Tait andC. E. M. Wagner, Phys. Rev. D (2003) 096006 [arXiv:hep-ph/0212307].[7] A. Delgado and A. Falkowski, JHEP (2007) 097[arXiv:hep-ph/0702234].[8] P. R. Archer and S. J. Huber, JHEP (2010) 032 [arXiv:1004.1159 [hep-ph]].[9] W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. , 4922 (1999)[arXiv:hep-ph/9907447]; W. D. Goldberger and M. B. Wise, Phys. Lett. B , 275 (2000) [arXiv:hep-ph/9911457].[10] S. S. Gubser, Adv. Theor. Math. Phys. , 679 (2000) [arXiv:hep-th/0002160].[11] A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D , 015005(2006) [arXiv:hep-ph/0602229]; B. Batell and T. Gherghetta, Phys. Rev. D , 026002 (2008) [arXiv:0801.4383 [hep-ph]].[12] A. Falkowski and M. Perez-Victoria, JHEP , 107 (2008)[arXiv:0806.1737 [hep-ph]].[13] A. Falkowski and M. Perez-Victoria, Phys. Rev. D (2009) 035005[arXiv:0810.4940 [hep-ph]].[14] B. Batell, T. Gherghetta and D. Sword, Phys. Rev. D , 116011 (2008)[arXiv:0808.3977 [hep-ph]]; T. Gherghetta and D. Sword, Phys. Rev. D (2009) 065015 [arXiv:0907.3523 [hep-ph]]; T. Gherghetta and N. Setzer, Phys.Rev. D (2010) 075009 [arXiv:1008.1632 [hep-ph]].[15] A. Delgado and D. Diego, Phys. Rev. D (2009) 024030 [arXiv:0905.1095[hep-ph]].[16] S. Mert Aybat and J. Santiago, Phys. Rev. D (2009) 035005[arXiv:0905.3032 [hep-ph]]. 1517] J. A. Cabrer, G. von Gersdorff and M. Quiros, New J. Phys. (2010) 075012[arXiv:0907.5361 [hep-ph]].[18] M. Atkins and S. J. Huber, Phys. Rev. D (2010) 056007 [arXiv:1002.5044[hep-ph]].[19] G. von Gersdorff, Phys. Rev. D (2010) 086010 [arXiv:1005.5134 [hep-ph]].[20] J. A. Cabrer, G. von Gersdorff and M. Quiros, arXiv:1103.1388 [hep-ph].[21] M. E. Peskin and T. Takeuchi, Phys. Rev. D (1992) 381.[22] K. Nakamura [Particle Data Group], J. Phys. G (2010) 075021.[23] M. Round, Phys. Rev. D (2010) 053002 [arXiv:1003.2933 [hep-ph]].[24] I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B (2000) 123[arXiv:hep-th/0002159].[25] F. Brummer, A. Hebecker and E. Trincherini, Nucl. Phys. B (2006) 283[arXiv:hep-th/0510113]; B. Hassanain, J. March-Russell and M. Schvellinger,JHEP (2007) 089 [arXiv:0708.2060 [hep-th]].[26] M. A. Luty and T. Okui, JHEP (2006) 070 [arXiv:hep-ph/0409274].[27] O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch, Phys. Rev. D62