1 Introduction Radiative corrections can ave a significant impact on te predicted values of Higgs masses and couplings. Te radiative corrections invol

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1 RADCOR November 15, 2000 Radiative Corrections to Pysics Beyond te Standard Model Clint Eastwood 1 Department of Radiative Pysics California State University Monterey Bay, Seaside, CA USA Loop corrections for pysics beyond te Standard Model are computed. Dazzling tecniques are employed to calculate extremely difficult loop diagrams. Implications for precision experiments at future colliders tat will never be built are examined. PRESENTED AT 5t International Symposium on Radiative Corrections (RADCOR-2000) Carmel CA, USA, September, Work supported by te Carmel Mission and te Banana Slug Preservation Society.

2 1 Introduction Radiative corrections can ave a significant impact on te predicted values of Higgs masses and couplings. Te radiative corrections involve bot loops of Standard Model particles and loops of supersymmetric partners. Te dominant effects arise from loops involving te tird generation quarks and squarks and are proportional to te corresponding Yukawa couplings. Tus, we first review te parameters tat control te masses and mixing of te tird-generation squarks. 2 Higgs mass radiative corrections Te radiative corrections to te Higgs squared-masses ave been computed by a number oftecniques, and using a variety of approximations suc as te effective potential at one-loop and two-loops [only te O(m 2 t 2 t ff s ) and O(m 2 t 4 t )two-loop results are known], and diagrammatic metods. Complete one-loop diagrammatic computations of te MSSM Higgs masses ave been presented by a number of groups; te resulting expressions are quite complex, and depend on all te parameters of te MSSM. Partial two-loop diagrammatic results are also known. Tese include te O(m 2 t 2 t ff s ) contributions to te neutral CP-even Higgs boson squared-masses in te on-sell sceme. One of te most striking effects of te radiative corrections to te MSSM Higgs sector is te modification of te upper bound of te ligt CP-even Higgs mass. Consider te region of parameter space were tan fi is large and m A fl m Z. In tis limit, te tree-level prediction for m corresponds to its teoretical upper bound, m = m Z. Including radiative corrections, te teoretical upper bound is increased. Te dominant effect arises from an incomplete cancellation of te top-quark and top-squark loops (tese effects actually cancel in te exact supersymmetric limit). Te qualitative beavior of te radiative corrections can be most easily seen in te large top squark mass limit, were in addition, te splitting of te two diagonal entries and te off-diagonal entry of te top-squark squared-mass matrix are bot small in comparison to te average of te two stop squared-masses, denoted by MSUSY. 2 In tis case, te upper bound on te ligtest CP-even Higgs mass is approximately given by m 2 < ο m2 Z + 3g2 m 4 t 8ß 2 m 2 W " ln ψ! ψ!# M 2 SUSY + x 2 m 2 t 1 x2 t ; (1) t 12 were x t X t =M SUSY. Te more complete treatments of te radiative corrections cited above sow tat eq. (1) somewat overestimates te true upper bound of m. Neverteless, eq.(1) correctly reflects some noteworty features of te more precise result. First, te increase of te ligt CP-even Higgs mass bound beyond m Z can be significant. Tis 1

3 is a consequence of te m 4 t enancement of te one-loop radiative correction. Second, te dependence of te ligt Higgs mass on te stop mixing parameter X t implies tat (for a given value of M S ) te upper bound of te p ligt Higgs mass initially increases wit X t and reaces its maximal value X t = 6M S. Tis point is referred to as te maximal mixing case (wereas X t = 0 corresponds to te minimal mixing case). In a more complete computation tat includes bot two-loop logaritmic and non-logaritmic corrections, te X t values corresponding to maximal and minimal mixing are sifted and exibit an asymmetry under X t! X t. Tird, note te logaritmic sensitivity totestop masses. Naturalness arguments tat underlie lowenergy supersymmetry imply tat te supersymmetric particles masses sould not be larger tan a few TeV. Still, te precise upper bound on te ligt Higgs mass depends on te specific coice for te upper limit of te stop masses. As noted above, te largest contribution to te one-loop radiative corrections is enanced by a factor of m 4 t and grows logaritmically wit te top squark mass. Tus, iger order radiative corrections can be non-negligible for large top squark masses, in wic case te large logaritms must be resummed. Compact analytic expressions ave been obtained for te dominant one and two-loop contributions to Figure 1: Te radiatively corrected ligt CP-even Higgs mass is plotted as a function of tan fi, for M SUSY = m A =1TeV, for te maximal mixing [upper band] and minimal mixing [lower band] bencmark cases [see table 1]. Te impact of te top quark mass is exibited by te saded bands; te central value corresponds to M t = 175 GeV, wile te upper [lower] edge of te bands correspond to increasing [decreasing] m t by 5 GeV. 2

4 Mass parameters [TeV] (GeV) Bencmark μ X t A t μ cot fi A b M SUSY M ~g m max p 6 At maximal mixing 0:2 minimal mixing 0:2 0 A t Table 1: Description of te values of te MSSM parameters for te bencmark scenarios considered in tis report [6]. In all cases, te f W -gaugino ( wino") mass parameter as been taken to be M 2 = 200 GeV. te matrix elements of te radiatively-corrected CP-even Higgs squared-mass matrix. Diagonalizing tis matrix yields radiatively-corrected values for m 2, m2 H and te CP-even Higgs mixing angle ff. Explicit expressions for te M 2 ij, given in refs. [1, 2], include te dominant leading and sub-leading logaritms at two-loop order (te latter are generated by an iterative solution to te RG-equations). Also included are te leading effects at one loop of te supersymmetric tresolds and te corresponding two-loop logaritmically enanced terms, wic can again be determined by iteration of te RG-equations. Te most important effects of tis type are squark mixing effects in te tird generation. Te procedures described above produce a prediction for te Higgs mass in terms of running parameters in te MS sceme. It is a simple matter to relate tese parameters to te corresponding on-sell parameters used in te diagrammatic calculations [3, 4]. Te numerical results displayed in fig. 1 are based on te results of refs. [1, 2], wit improvements as described in refs. [5, 4]. Te supersymmetric parameters in te maximal and minimal mixing cases ave been cosen according to te first two bencmark scenarios of ref. [6], displayed in table 1. Of particular interest is te upper bound for te ligtest CP-even Higgs mass (m ). At fixed tan fi, te maximal value of m is reaced for m A fl m Z. Taking m A large, fig. 1 illustrates tat te maximal value of te ligtest CP-even Higgs mass bound is realized at large tan fi in te case of maximal mixing. For eac value of tan fi, we denote te maximum value of m by m max (tan fi) [tis value also depends on te tird-generation squark mixing parameters]. Allowing for te uncertainty in te measured value of m t and te uncertainty inerent in te teoretical analysis, one finds for M < SUSY ο 2 TeV tat m < ο mmax = m max (tan fi fl 1), were m max ' 122 GeV; if top-squark mixing is minimal, m max ' 135 GeV; if top-squark mixing is maximal. (2) Roug estimates of iger order corrections not yet computed suggest tat te results for Higgs masses sould be accurate to witin about 2 to 3 GeV over te parameter ranges displayed in fig. 1. 3

5 3 Conclusions Recent data from te LEP collider suggest te possibility of a Higgs boson wit mass of around 115 GeV. Altoug it is too soon to speculate, we note tat tis mass lies comfortably in te range predicted for te ligtest CP-even Higgs boson of te MSSM. Weter tis signal represents te first evidence for low-energy supersymmetry remains to be seen. Te Tevatron may ultimately be successful in its Higgs searc [7], altoug tis will require an intensive effort lasting at least six years, along wit some macine and detector improvements and a little bit of luck. If te Tevatron searc fails, te LHC will be poised to settle te issue of te existence of te Higgs boson. In eiter case, we must wait patiently for at least tree more RADCOR meetings before te trut is revealed. ACKNOWLEDGEMENTS I am grateful to Georgia Hamel and Jacqueline Pizzuti at SCIPP and Rut McDunn and Terry Anderson at SLAC for teir tireless efforts in elping to make RADCOR a success. References [1] M. Carena, J.R. Espinosa, M. Quiros and C.E.M. Wagner, Pys. Lett. B355 (1995) 209; M. Carena, M. Quiros and C.E.M. Wagner, Nucl. Pys. B461 (1996) 407. [2] H.E. Haber, R. Hempfling and A.H. Hoang, Z. Pys. C75 (1997) 539. [3] J.R. Espinosa and R.-J. Zang, JHEP 0003 (2000) 026. [4] M. Carena, H.E. Haber, S. Heinemeyer, W. Hollik, C.E.M. Wagner and G. Weiglein, Nucl. Pys. B580 (2000) 29. [5] S. Heinemeyer, W. Hollik and G. Weiglein, Pys. Rev. D58 (1998) ; Pys. Lett. B440 (1998) 296; Eur. Pys. J. C9 (1999) 343. [6] M. Carena, S. Heinemeyer, C.E.M. Wagner and G. Weiglein, CERN-TH [ep-p/ ]. [7] M. Carena, J.S. Conway, H.E. Haber, J.D. Hobbs et al., Report of te Tevatron Higgs Working Group, FERMILAB-CONF T and SCIPP [epp/ ]. 4