application/xmlρ parameter constraints on radion phenomenology and a lower bound on Higgs massPrasanta DasUma MahantaPhysics Letters B 528 (2002) 253-258. doi:10.1016/S0370-2693(02)01209-1journalPhysics Letters BCopyright © 2002 Elsevier Science B.V. All rights reserved.Elsevier B.V.0370-26935283-47 March 20022002-03-07253-25825325810.1016/S0370-2693(02)01209-1http://dx.doi.org/10.1016/S0370-2693(02)01209-1doi:10.1016/S0370-2693(02)01209-1http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB18385S0370-2693(02)01209-110.1016/S0370-2693(02)01209-1Elsevier Science B.V.PhenomenologyFig. 1Feynman rules for one and two radion couplings to EW gauge bosons.Fig. 2Feynman diagrams giving the radion contribution to T parameter in the unitary gauge.Fig. 3ρ parameter constraints on radion 〈φ〉 and radion mass mφ. The allowed region lies above both curves.Fig. 4Plots of λ(μ) at μ=100 GeV against 〈φ〉 for different radion masses under non-perturbative UVBC on λ.Fig. 5Lower bound on Higgs mass from ρ parameter constraint plotted against the radion mass. RG1: Region forbidden by T=0.18 but allowed by direct search of LEPII; RG2: Region allowed by T=0.18 but forbidden by direct search of LEPII; RG3: Region forbidden by T=0.0.4 but allowed by direct search of LEPII; RG4: Region allowed both by T=0.04 and direct search of LEPII.Fig. 6Plots showing the sensitivity of the lower bound on mh to the UVBC on λ for different values of mφ.ρ parameter constraints on radion phenomenology and a lower bound on Higgs massPrasantaDasapdas@iitk.ac.inUmaMahantabmahanta@mri.ernet.inaDepartment of Physics, Indian Institute of Technology, Kanpur 208 016, IndiabMehta Research Institute, Chhatnag Road, Jhusi Allahabad-211019, IndiaEditor: T. YanagidaAbstractIn this Letter we determine the contribution of a light stabilized radion to the weak isospin violating ρ parameter by using an ultraviolet momentum cut off as the regulator. The LEPI bound on ρnew is then used to derive constraints on the radion mass mφ and its vev 〈φ〉. Finally by using the beta function of the Higgs self coupling we have determined a lower bound on the Higgs mass from the rho parameter constraints on mφ and 〈φ〉. Our results show that for mφ< 600 GeV the rho parameter bound on mh is stronger than the present direct bound from LEPII.1IntroductionRecently several attractive proposals [1,2] based on theories in extra dimensions have been put forward to explain the hierarchy problem. Among them the Randall–Sundrum (RS) model is particularly interesting because it considers a five-dimensional world based on the following non-factorizable metric (1)ds2=e−2krc|θ|ημνdxμdxν−rc2dθ2. Here rc measures the size of the extra dimensions which is an S1/Z2 orbifold. xμ are the coordinates of the four-dimensional space–time. −π⩽θ⩽π is the coordinate of the extra dimension with θ and −θ identified. k is a mass parameter of the order of the fundamental five-dimensional Planck mass M. Two 3-branes are placed at the orbifold fixed points θ=0 (hidden brane) and θ=π (visible brane). Randall and Sundrum showed that any field on the visible brane with a fundamental mass parameter m0 gets an effective mass (2)m=m0e−krcπ due to the exponential warp factor. Therefore, for krc≈14 the electroweak (EW) scale is generated from the Planck scale by the warp factor.In the Randall–Sundrum model rc is the vacuum expectation value (vev) of a massless scalar field T(x). The modulus was, therefore, not stabilized by some dynamics. Goldberger and Wise [3] later showed how to generate a potential for the modulus and stabilize it at the right value (krc) that is needed for solving the hierarchy problem without any excessive fine tuning of the parameters.In their original model Randall and Sundrum assumed that SM fields are localized on the visible brane located at θ=π. However, small fluctuations of the modulus field from its vev gives rise to non-trivial couplings of the modulus field with the SM fields. Using such couplings the effect of a light radion on low energy phenomenology has been studied [4].In this Letter we shall determine the radion contribution to the weak isospin breaking ρ parameter. The LEPI data imposes stringent constraints on new physics contribution to the ρ parameter. We have, therefore, used the LEPI bound on ρnew to put bounds on the two unknown parameters mφ and 〈φ〉. Throughout our analysis the cut off Λ will be assumed to be related to the expansion parameter 1〈φ〉 of the non-renormalizable radion interaction to SM particles by the naive dimensional analysis (NDA) estimate Λ=4π〈φ〉 [5]. In the RS model the cut-off Λ corresponds to the mass of the lightest KK graviton mode. The beta function of the Higgs self-coupling is also modified in the presence of a light stablized radion. In this Letter we have used this beta function to derive a lower bound on mh from the LEP1 bounds on mφ and 〈φ〉.1.1Radion couplings to electroweak gauge bosons in unitary gaugeIn order to determine the radion contribution to the T parameter we need to determine the radion couplings to the EW gauge bosons localized on the visible brane. The relevant radion couplings to the EW gauge bosons can be determined from the following action (3)S=∫d4x−gv(DμH)†(DνH)gμνv−14WaμνWaρσgμρvgνσv−14BμνBρσgμρvgνσv+Lgf, where −gv=φf4,gμνv=φ(x)f−2ημν and (4)Lgf=−12ξ∂μWaνgμνv+ig2ξH′†τa2〈H〉−H†τa2H′2−12ξ∂μBνgμνv+ig1ξ2H′†〈H〉−H†H′2. The SM Higgs field in unitary gauge is given by H=H′+〈H〉=0v+h(x)2.Using the above expression of the Higgs field it can be shown that the gauge fixing Lagrangian Lgf vanishes in the unitary gauge (ξ→∞). Consider first the radion coupling to the KE of the gauge bosons. We have (5)−gvVμνVρσgvμρgvνσ=VμνVρσημρηνσ, where Vμ=(Waμ,Bμ). At the classical level the radion, therefore, does not couple to the gauge boson KE in four dimensions. Note that the Christoffel symbol Γλμν in the expression for the general covariant derivative DμVν does not contribute to the field strength tensor of Waν or Bν because Γλμν is symmetric in (μ,ν).Consider next the radion coupling to the KE of the Higgs boson. We have −gv(DμH)+(DνH)gvμν=φ〈φ〉2DμH+DμH, where H=H〈φ〉f and 〈H〉=〈H〉〈φ〉f. In the following we shall assume that the Higgs field and its vev has been properly rescaled as above and drop the tilde sign. We then get (6)−gv(DμH)+(DνH)gvμν=ig2WaμH+τa2∂μH′−∂μH′+τa2〈H〉φ〈φ〉2+ig12BμH+∂μH′−∂μH′+〈H〉φ〈φ〉2+m2wW+μW−μ+12m2zZμZμ1+2φ̂〈φ〉+φ̂2〈φ〉2+⋯.In the unitary gauge the first two terms on the r.h.s. of the above expression vanishes, leaving only the gauge boson mass terms to couple to radion fluctuations. The couplings of one and two radions to the EW gauge bosons that are relevant for computing the radion contribution to Πμνvv(q) in unitary gauge can, therefore, be expressed by the Feynman rules shown in Fig. 1(a) and Fig. 1(b).The contribution of new physics to the vectorial isospin violating parameter ρ[6] is given by (7)ρnew=αTnew=Πww(0)m2w−Πzz(0)m2z.The function Πvv(q) is defined through the gauge boson self energy tensor (8)iΠμνvv(q)=iημνΠvv(q)−iqμqνΠvv(q).The Feynman diagrams that give rise to radion contribution to ρ parameter in unitary gauge are shown in Fig 2.Let Π(1)vv(q) and Π(2)vv(q) denote the contributions arising from single and two radion vertices to Πvv(q). We then have (9)Π(1)vv(0)=−mv216π2〈φ〉2Λ2−mφ2lnΛ2mφ2−mv416π23lnΛ2mφ2−3m2vmφ2−m2vlnmφ2mv2 and (10)Π(2)vv(0)=mv216π2〈φ〉2Λ2−mφ2lnΛ2mφ2.It is clear from the above that Π(2)vv(q) will not contribute to the ρ parameter since Π(2)vv(0)/mv2 is independent of mv. Radion contribution to ρnew, therefore, arises only from Π(1)vv(q). We would like to note at this point that since the φ̂VV coupling is proportional to m2v, the radion tadpole diagrams do not contribute to the ρ parameter. We also find that although Π(1)vv(0) and Π(2)vv(0) are individually quadratically divergent the sum Πvv(0) is only log divergent. This cancellation of quadratic divergence is a consequence of gauge symmetry which protects gauge boson masses from receiving large power corrections.1.2Radion contribution to the ρ parameterThe radion contribution to the ρ parameter is, therefore, given by (11)ρnew=mw216π2〈φ〉2−3lnΛ2mφ2+3mw2mφ2−mw2lnmφ2mw2−mz216π2〈φ〉2−3lnΛ2mφ2+3mz2mφ2−mz2lnmφ2mz2.Note that the above expression for ρnew diverges logarithmically with the cut off. The cut-off dependence of the radion contribution to the rho parameter is easily understood. It arises from using the non-renormalizable dimension-five operator (DμH)+(DμH)φ̂〈φ〉 in the calculation of Π1vv(0). Secondly, the radion contribution to ρnew depends on three unknown parameters: the cut-off Λ, mφ and 〈φ〉. Naive dimensional analysis can, however, be used to relate Λ to the expansion parameter 1〈φ〉 through Λ=4π〈φ〉. Physically it means that the radion effective field theory would become non-perturbative and radion induced radiative corrections would become very large above the ultraviolet scale 4π〈φ〉 thereby implying a breakdown of the low energy effective theory. The NDA estimate of the cut off is known to work quite well for estimating chiral loops that arises in dealing with chiral Lagrangian. Further Luty et al. has shown that it gives reliable estimates for extra-dimensional gravity also. We shall assume it to hold good for the radion effective field theory also. This reduces the dependence of ρnew to two unknowns only: mφ and 〈φ〉. The LEPI bounds on ρnew can, therefore, be used to impose stringent constraints on mφ and 〈φ〉 [7]. Finally, the radion contribution to the T parameter must be a gauge invariant quantity, since the radion is a gauge singlet. Therefore, although the calculations presented in this Letter were done in unitary gauge, the final answer given by Eq. (11) must be independent of this gauge choice.2ρ parameter bounds on mφ and 〈φ〉The present value of the T parameter is given by [8]T=−0.10±0.14(0.09). In Fig. 3 we have shown the contour plots in mφ vs. 〈φ〉 plane for T=0.04 and T=0.18. The first value corresponds to +1σ deviation and the second value to 2σ deviation from the central value. We have chosen positive values of T only since the radion contribution to T is positive for Λ⪢mφ. The region allowed by the ρ parameter bound lies above both curves. We find that for T=0.18 and mφ= 10 GeV, 〈φ〉 must be greater than about 440 GeV. On the other hand, for T=0.04, 〈φ〉 must be greater than about 1000 GeV for the same mφ. The bound on 〈φ〉, however, decreases monotonically with increasing mφ and becomes about 320 GeV (for T=0.18) and 810 GeV (for T=0.04) when mφ increases to 500 GeV. The region allowed by the ρ parameter constraint lies above the relevant curve.3Lower bound on Higgs massThe beta function for the Higgs self coupling in the presence of a light radion and the ρ parameter constraints on mφ and 〈φ〉 can be used together to derive a lower bound on mh. In the following we briefly describe how this lower bound on mh can be determined.The beta function for the Higgs self coupling λ in the presence of a light radion is given by [9](12)β(λ)=μdλdμ=18π29λ2+402λ2v2〈φ〉2+144λ2v4〈φ〉4+5λm2φ〈φ〉2+λ6g2Y−92g22−32g12−6g4Y+18π2316g24+12g22+g122.(i)For a given value of mφ we use the above differential equation to determine the value of the renormalized coupling λ(μ) at μ=100 GeV. In this Letter we shall assume that the Kaluza–Klein modes of the graviton, which are much heavier than the radion, decouples at or above the cut-off scale Λ=4π〈φ〉. The value of λ at the cut-off Λ can be chosen to be either strong and non-perturbative (λ(Λ)>4π) or weak and perturbative (λ(Λ)<4π). In Fig. 4 we have plotted λ(μ) at μ=100 GeV against 〈φ〉 for seven different values of mφ starting from 5 GeV and going up to 600 GeV under the UVBC λ(Λ)=∞.(ii)For a given mφ we find the ρ parameter bound on 〈φ〉 from Fig. 3. The value of the renormalized coupling λ(μ) at that value of 〈φ〉 is then determined from the curve corresponding to the chosen mφ shown in Fig. 4. For each chosen mφ we, therefore, obtain a value for the renormalized coupling λ(μ) at μ= 100 GeV.(iii)The renormalized Higgs mass at μ= 100 GeV can be determined from the λ(μ) obtained in step (ii) via the relation mh(μ)=2λ(μ)v. Fig. 5 shows the lower bound on the Higgs mass as a function of mφ for T=0.04 and T=0.18. We find that for T=0.18 the bound on mh is greater than the lower bound on mh from direct search at LEPII [10] provided mφ< 600 GeV. This gives rise to two distinct regions in the mφ vs. mh plane: RG1 (which is allowed by LEPII but forbidden by T=0.18) and RG2 (which is allowed by T=0.18 but forbidden by LEPII). We also find that for T=0.04 the lower bound on mh is greater than the LEPII bound for the entire range of values of mφ relevant for a light radion. This gives rise to two distinct regions: RG3 (which is allowed by LEPII but forbidden by T=0.04) and RG4 (which is allowed both by T=0.04 and LEPII). The bound on mh decreases monotonically with increasing mφ due to the following reasons: (i) the bound on 〈φ〉 decreases with increasing mφ (Fig. 3) and (ii) the value of λ(μ) decreases with decreasing 〈φ〉 (Fig. 4).The results shown in Fig. 5 were obtained by using the UVBC λ(Λ)=∞. It is worthwhile, however, to investigate the sensitivity of the lower bound on mh to the UVBC on λ. In Fig. 6 we have plotted the bound on mh for two different UBVC λ(Λ)=∞ and λ(Λ)=e. We find that for T=0.18 the bound on mh is insensitive to the UVBC over the entire range of mφ. However, for T=0.04 the bound on mh is somewhat sensitive to the UVBC. To understand this feature we would like to refer the reader to [9] where it was shown that the value of λ(100) does not depend on the UVBC provided 〈φ〉 is less than 350 GeV. Although the last condition is more or less satisfied for T=0.18 it does not hold at all for T=0.04 over the entire range of values of mφ.AcknowledgementsWe would like to thank Prof. M. Einhorn, Prof. S. Raychaudhuri and Prof. T. Takeuchi for several useful discussions on this Letter. Uma Mahanta would also like to thank the Physics Department of IIT Kanpur for its generous support and hospitality while this work was being done.References[1]N.Arkani-HamedS.DimopoulosG.DvaliPhys. Lett. B4291998263I.AntoniadisN.Arkani-HamedS.DimopoulosG.DvaliPhys. Lett. 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