We compute the supersymmetric contribution to
  $R_{b}\equiv \Gamma (Z\to b{\bar b})/\Gamma (Z\to {\rm hadrons})$ in a
variety of supergravity models. We find $R^{\rm susy}_b\lsim0.0004$, which does
not shift significantly the Standard Model prediction
  ($R^{\rm SM}_b=0.2162$ for $m_t=160\GeV$). An improvement in experimental
precision by a factor of four would be required to be sensitive to such an
effect.Comment: 5 pages, LaTeX, 1 figure (included)

Lopez, J.

Nanopoulos, D.

Wang, X.

English

arXiv

We compute the supersymmetric contribution to R_{b}\equiv \Gamma (Z\to b{\bar b})/\Gamma (Z\to {\rm hadrons}) in a variety of supergravity models. We find R^{\rm susy}_b\lsim0.0004, which does not shift significantly the Standard Model prediction (R^{\rm SM}_b=0.2162 for m_t=160\GeV). An improvement in experimental precision by a factor of four would be required to be sensitive to such an effect.We compute the supersymmetric contribution to $R_{b}\equiv \Gamma (Z\to b{\bar b})/\Gamma (Z\to {\rm hadrons})$ in a variety of supergravity models. We find $R~{\rm susy}_b\lsim0.0004$, which does not shift significantly the Standard Model prediction ($R~{\rm SM}_b=0.2162$ for $m_t=160\GeV$). An improvement in experimental precision by a factor of four would be required to be sensitive to such an effect.We compute the supersymmetric contribution to $R_{b}\equiv \Gamma (Z\to b{\bar b})/\Gamma (Z\to {\rm hadrons})$ in a variety of supergravity models. We find $R^{\rm susy}_b\lsim0.0004$, which does not shift significantly the Standard Model prediction ($R^{\rm SM}_b=0.2162$ for $m_t=160\GeV$). An improvement in experimental precision by a factor of four would be required to be sensitive to such an effect

Wang, Xu

Lopez, Jorge L.

Nanopoulos, Dimitri V.

CERN Document Server

arXiv:hep-ph/9501258v1  10 Jan 1995CERN-TH.7553/95CTP-TAMU-67/94ACT-24/94hep-ph/9501258Rb in supergravity modelsXU WANG1,2, JORGE L. LOPEZ1,2, and D. V. NANOPOULOS1,2,31Center for Theoretical Physics, Department of Physics, Texas A&M UniversityCollege Station, TX 77843–4242, USA2Astroparticle Physics Group, Houston Advanced Research Center (HARC)The Mitchell Campus, The Woodlands, TX 77381, USA3CERN Theory Division, 1211 Geneva 23, SwitzerlandAbstractWe compute the supersymmetric contribution to Rb ≡ Γ(Z → bb¯)/Γ(Z →hadrons) in a variety of supergravity models. We find Rsusyb <∼ 0.0004, whichdoes not shift significantly the Standard Model prediction (RSMb = 0.2162 formt = 160GeV). An improvement in experimental precision by a factor of fourwould be required to be sensitive to such an effect.CERN-TH.7553/95CTP-TAMU-67/94ACT-24/94January 1995Precision tests of the electroweak interactions at LEP have provided the mostsensitive checks of the Standard Model of particle physics. The pattern that hasemerged is that of consistent agreement with the Standard Model predictions. Thispattern seems to have so far only one apparently dissonant note, namely in the mea-surement of the ratio Rb ≡ Γ(Z → bb¯)/Γ(Z → hadrons), where the latest global fitto the LEP data (0.2192± 0.0018 [1]) lies about two standard deviations above theone-loop Standard Model prediction [2] for all preferred values of the top-quark mass(e.g., RSMb = 0.2162 for mt = 160GeV). Further experimental statistics will revealwhether this is indeed a breakdown of the Standard Model. In the meantime, it isimportant to explore what new contributions to Rb are expected in models of newphysics, such as supersymmetry.The study of supersymmetric contributions to Γ(Z → bb¯) has proceeded in twophases. Originally the quantity ǫb [3] was defined as an extension of the ǫ1,2,3 scheme[4] for model-independent fits to the electroweak data. More recently it has becomeapparent that the ratio Rb is more directly calculable [5, 6] and readily measurable. Ithas been made apparent [6] that supersymmetric contributions to Rb are not likely toincrease the total predicted value for Rb in any significant manner, as long as typicalassumptions about unified supergravity models are made. On the other hand, ifthese assumptions are relaxed, it is possible for supersymmetry to make a significantcontribution to Rb, if certain conditions on the low-energy supersymmetric spectrumare satisfied: the lightest chargino and lightest top-squark should be below 100GeV,with the chargino being mostly Higgsino and the top-squark mostly right-handed.One should keep in mind that novel supersymmetry breaking scenarios, such as thosearising from string models, may provide such otherwise ad-hoc conditions. In anyevent, for the purposes of this paper we assume that the experimental data willsettle at values for Rb which do not require extreme choices of the supersymmetryparameters for its explanation, and yet still allow for a discrimination among thevarious supergravity models on the basis of their prediction for Rb. If this assumptionturns out to be invalid, all models discussed in this paper (as well as the StandardModel) would be seriously disfavored.We consider unified supergravity models with universal soft supersymmetrybreaking at the unification scale, and radiative electroweak symmetry breaking (en-forced using the one-loop effective potential) at the weak scale [7]. These constraintsreduce the number of parameters needed to describe the models to four, which canbe taken to be mχ±1, ξ0 ≡ m0/m1/2, ξA ≡ A/m1/2, tanβ, with a specified value forthe top-quark mass (mt). In what follows we take mpolet = 160GeV which is thecentral value obtained in fits to all electroweak and Tevatron data in the context ofsupersymmetric models [8]. Among these four-parameter supersymmetric models weconsider generic models with continuous values of mχ±1and discrete choices for theother three parameters:tan β = 2, 10, 20 ; ξ0 = 0, 1, 2, 5 ; ξA = 0 . (1)The choices of tanβ are representative; higher values of tanβ are likely to yield1values of B(b → sγ) in conflict with present experimental limits [9]. The choices ofξ0 correspond to mq˜ ≈ (0.8, 0.9, 1.1, 1.9)mg˜. The choice of A has little impact onthe results. We also consider the case of no-scale SU(5) × U(1) supergravity [7]. Inthis class of models the supersymmetry breaking parameters are related in a string-inspired way. In the two-parameter moduli scenario ξ0 = ξA = 0 [10], whereas in thedilaton scenario ξ0 =1√3, ξA = −1 [11]. A series of experimental constraints andpredictions for these models have been given in Ref. [12]. In particular, the issue ofprecision electroweak tests in this class of models has been addressed in Refs. [13, 14].Besides the one-loop Standard Model contributions to Rb, in supersymmetricmodels there are four new diagrams, as follows:• The charged-Higgs–top-quark loop, depends on the charged Higgs mass andthe t− b−H± coupling. For a left-handed b quark the coupling is ∝ mt/ tanβwhereas for a right-handed b quark it is ∝ mb tanβ. Therefore, for small1 (large)tanβ left- (right)-handed b-quark production is dominant. (For tanβ ≫ 1, thevalue of mb impacts the contribution significantly.) It has been shown that theH± − t contribution is always negative [15], a fact which makes the predictionfor Rb in two Higgs-doublet models always in worse agreement with experiment.• The chargino–stop loop, is the supersymmetric counterpart of the H±–t loopdiscussed above. The chargino mass eigenstate is a mixture of (charged) Hig-gsino and wino, and the coupling strength is a complicated matter now becauseit involves the stop mixing matrix and the chargino mixing matrix. However,because only the Higgsino admixture in the chargino eigenstate has a Yukawacoupling to the t–b doublet, generally speaking a light chargino with a significantHiggsino component, and a light stop with a significant right-handed compo-nent are required for this diagram to make a non-negligible contribution to Rb,as pointed out in Ref. [6].• The neutralino–sbottom loop, is the supersymmetric counterpart of the neutral-Higgs–bottom-quark loop. The coupling strength of χ01−b˜−b is also rather com-plicated, since it involves the sbottom mixing, the neutralino mixing, and theirmasses. However, it can be non-negligible since it is proportional tomb tan β forthe left-handed b-quark. Therefore in the high tanβ region we have to includethis contribution.2• The neutral-Higgs–bottom-quark loop, involves the three neutral scalars, twoCP-even (h,H) and one CP-odd (A). For the h (H) neutral Higgs boson thecoupling to the bottom quark is ∝ mb sinα(cosα) which in the absence of atanβ enhancement makes its contribution negligible. For the A Higgs boson,the coupling to bb¯ is ∝ tanβ and the A-dependent contribution can be largeand positive if mA <∼ 90GeV and tan β >∼ 30 [15]. Since we restrict ourselves to1The radiative electroweak symmetry breaking mechanism requires tanβ > 1.2This term was not included in our previous study in terms of the parameter ǫb [14], although itwas pointed out that its effects were non-negligible for tanβ ≫ 1.2tanβ <∼ 20, and mA >∼ 100GeV in these models, this contribution is neglectedin what follows.Our computations of Rsusyb have been performed using the expressions givenin Ref. [6]. Even though these formulas are given explicitly, the details are quitecomplicated by the presence of various Passarino-Veltman loop functions. As a checkof our calculations, we have verified numerically that the results are independent ofthe unphysical renormalization scale that appears in the formulas. The predictionsfor Rsusyb in the generic supergravity models are shown in Fig. 1. Only curves fortan β = 2, 10 are shown, since the corresponding sets of curves for other values oftan β fall between these two sets of curves. Moreover, in the interest of brevity we donot show explictly the results in SU(5)×U(1) supergravity since for the same valuesof the parameters the predictions differ little from those in the generic models.In almost all cases the largest positive contribution to Rsusyb comes from thechargino–stop loop. As expected, the largest contribution from this diagram happensfor points with the lightest chargino masses (which correspond to the lightest t˜1masses) since supersymmetry is a decoupling theory. However, even the largest value(≈ 10−4) is still very small compared with the largest possible result in a genericlow-energy supersymmetric model [6]. The reason for this is that while the smallestpossible chargino and a stop masses are required for an enhanced contribution, itis also necessary that the chargino has a significant Higgsino component and thatthe stop be mostly right-handed. The latter requirement is in fact attainable in thesemodels (i.e., the stop mixing angle is not small), but the former is not. Indeed, Rsusyb ∝|V12|2, where V12 is the (12) element of the chargino mixing matrix V , which does notexceed ≈ 0.3, since for light charginos |µ| ≫ M2 which makes the lightest charginomainly a wino instead of a Higgsino. The charged-Higgs–top-quark loop is alwaysnegative, and is enhanced for either small or large values of tanβ. The neutralino–sbottom contribution is almost always smaller than the chargino–stop contributionand not of definite sign.We conclude that the presently known supersymmetry breaking scenarios inthe context of supergravity shift only slightly the Standard Model prediction for Rb,and would require an improvement in experimental sensitivity by a factor of four tobe observable. Also, if the experimental value of Rb remains essentially unchangedover time, the models explored in this paper and the Standard Model as well, wouldfall into disfavor. If the experimental value changes in the direction of the StandardModel prediction, our calculations should help in discriminating among the varioussupergravity models on the basis of their prediction for Rb. It is also possible that newscenarios for supersymmetry breaking may entail low-energy spectra which satisfy thenecessary conditions for an enhanced supersymmetric contribution to Rb.AcknowledgementsThis work has been supported in part by DOE grant DE-FG05-91-ER-40633. Thework of X. W. has been supported by the World Laboratory. We would like to thank3Chris Kolda for helpful communications. X. W. also would like to thank J. T. Liufor helpful discussions.References[1] The LEP Collaborations, CERN/PPE/94-187 (November 1994).[2] J. Bernabeu, A. Pich, and A. Santamaria, Phys. Lett. B 200 (1988) 569; W.Beenaker and W. Hollik, Z. Phys. C40, 141(1988); A. Akhundov, D. Bardin, andT. Riemann, Nucl. Phys. B 276 (1986) 1; F. Boudjema, A. Djouadi, and C.Verzegnassi, Phys. Lett. B 238 (1990) 423; A. Blondel and C. Verzegnassi, Phys.Lett. B 311 (1993) 346.[3] G. Altarelli, R. Barbieri, and F. Caravaglios, Nucl. Phys. B 405 (1993) 3.[4] G. Altarelli and R. Barbieri, Phys. Lett. B 253 (1990) 161; G. Altarelli, R.Barbieri, and S. Jadach, Nucl. Phys. B 369 (1992) 3; G. Altarelli, R. Barbieri,and F. Caravaglios, Phys. Lett. B 314 (1993) 357.[5] M. Boulware, D. Finnel, Phys. Rev. D 44 (1991) 2054; A. Djouadi, G. Girardi,C. Vergzegnassi, W. Hollik and F. Renard, Nucl. Phys. B 349 (1991) 48.[6] J. D. Wells, C. Kolda, and G. L. Kane, Phys. Lett. B 338 (1994) 219.[7] For a recent review see e.g., J. L. Lopez, D. V. Nanopoulos, and A. Zichichi,Prog. Part. Nucl. Phys. 33 (1994) 303.[8] J. Ellis, G.L. Fogli and E. Lisi, Phys. Lett. B 333 (1994) 118; J. Erler and P.Langacker, UPR–0632T (hep-ph/9411203).[9] See e.g., J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, Phys. Rev.D 51 (1995) 147.[10] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. D 49 (1994) 343.[11] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 319 (1993) 451.[12] J. L. Lopez, D. V. Nanopoulos, G. Park, X. Wang, and A. Zichichi, Phys. Rev.D 50 (1994) 2164.[13] J. L. Lopez, D. V. Nanopoulos, G. T. Park, H. Pois, and K. Yuan, Phys. Rev.D 48 (1993) 3297; J. L. Lopez, D. V. Nanopoulos, G. T. Park, and A. Zichichi,Phys. Rev. D 49 (1994) 355.[14] J. L. Lopez, D. V. Nanopoulos, G. T. Park, and A. Zichichi, Phys. Rev. D 49(1994) 4835.[15] W. Hollik, Int. J. Mod. Phys. A 5 (1990) 1909; A. Denner, R. Guth, W. Hollik,and J. Ku¨hn, Z. Phys. C51 (1991) 695.4Figure 1: The supersymmetric contribution to Rb as a function of the chargino massin generic supergravity models with tan β = 2, 10, ξ0 = 0− 5, and A = 0. Curves forother values of tan β fall between the two sets shown.5

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