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Next-To-Leading Order Corrections to Gauge-Mediated Gaugino Masses

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Next-To-Leading Order Corrections to Gauge-Mediated Gaugino Masses View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Feb 1998 IFUP{TH 63/97 hep-ph/9802446 Next-to-leading order corrections to gauge-mediated gaugino masses Marco Picariello and Alessandro Strumia Dipartimento di Fisica, Universita di Pisa and INFN, sezione di Pisa, I-56126 Pisa, Italia Abstract We compute the next-to-leading order corrections to the gaugino masses M in gauge- i mediated mo dels for generic values of the messenger masses M and discuss the predic- tions of uni ed messenger mo dels. If M < 100 TeV there can b e up to 10% corrections to the leading order relations M / . If the messengers are heavier there are only i i few % corrections. We also study the messenger corrections to gauge coupling uni - cation: as a result of cancellations dictated by sup ersymmetry, the predicted value of the strong coupling constant is typically only negligibly increased. 1 Intro duction The \gauge mediation" scenario for sup ersymmetric particle masses [1] can b e realized in reasonable mo dels [2, 3]. Furthermore, with a uni ed sp ectrum of messenger elds, it gives rise to some stable and acceptable prediction for the sp ectrum of sup ersymmetric particles. One of these predictions is the `uni cation prediction' for the gaugino masses M (i =1;2;3): at one-lo op order the RGE-invariant ratio M = is the same for all the i i i i N three factors of the SM gauge group G = G = U(1) SU(2) SU(3) . This prediction is suciently SM i Y L C i stable that it is interesting to compute it with more accuracy. A detailed computation allows a comparison with the gaugino sp ectrum predicted by the alternative scenario known as \uni ed supergravity" [4]. Uni cation relations for the (E ) are infact the more stable prediction i of this second scenario: gauge couplings and gaugino masses could receive sizeable GUT threshold corrections; but these corrections largely cancel out in the ratios (M ), since and M have the same one-lo op RGE i GUT i i dr scheme [5] are (including evolution. The testable predictions for the low-energy running ratios in the i NLO RGE corrections [6], but neglecting p ossible unknown O (%) GUT-scale threshold e ects [7]) (M ) (M ) 2 Z 1 Z 1:02; 0:97: (1) (M ) (M ) 2 Z 3 Z This should be compared with the corresp onding prediction in gauge-mediation mo dels for the ratios = , i j plotted in g.s 2 and 3. The computation of gaugino masses with NLO precision is done in sections 2 and 3 for generic values of the messenger mass M . It requires the following main steps: 1. compute the renormalized running gaugino masses, M (E )atE M, in the e ective theory without i H H messengers. 1 2. compute the running gaugino masses, M (E )atE M , evolving M (E )down to the Fermi scale i L L Z i H with 2 lo op RGE equations. p ole 3. use M (E ) to compute directely measurable quantities, like the gaugino p ole masses, M , including i L i all 1 lo op e ects at the electroweak scale. The computation necessary for step 1 is done in section 2. We will employ sup ersymmetric dimensional regu- dr scale, . The RGE necessary for step 2 (recalled larization, so that the renormalization scale E will b e the in app endix B) can b e read from the literature [6]. The one-lo op expressions for p ole gaugino masses in terms of running parameters are also well known [8]. Since various unmeasured and unpredicted parameters (like the so-called -term) would enter the nal step 3, we prefer to show our nal predictions for the running MSSM gaugino masses renormalized at = M , without including the gauge corrections at the electroweak scale. L Z We will compute these predictions in uni ed messenger models. One more step is necessary to imp ose the uni cation constraints on the messenger sp ectrum, namely 0. compute the messenger sp ectrum, M (E ), evolving the uni ed M (M )down to the messenger scale n H n GUT E M with 2 lo op RGE equations. H n The necessary RGE equations are given in app endix B. In sec. 3 we study the predictions of gauge mediation mo dels with an uni ed messenger sp ectrum. If the messenger sp ectrum is only negligibly splitted by sup ersym- metry breaking e ects, the NLO corrections to the LO uni cation-like relations M / are around few % and i i numerically not much di erent from the ones present in uni ed sup ergravity mo dels. Larger e ects (up to 10%) < can b e present if the messengers are very light, M 50 TeV. We also study the corrections to gauge coupling uni cation due to the presence of messenger elds b elow the uni cation scale. Messenger threshold e ects largely cancel messenger corrections to twoloopRGE running, as dictated by sup ersymmetry [9, 10]. 2 Computation In this section we do the computations necessary for step 1. We assume that the messenger elds sit in real L representations R = R of the SM gauge group and have a sup ersymmetric mass term M together with n n n a non-sup ersymmetric mass term F . For the moment we assume that the messengers lie in self-conjugate n complex representations, R = X X . See app endix A for a more detailed discussions of the notations, of n n n the mo del, of its sp ectrum, and of the relevant Lagrangian. < In order to compute the running gaugino masses M ()at M in the e ective theory we rst compute i n the gauge-indep endent pole gaugino masses in the twoversions of the theory. The computation in the full theory (with messengers) is done in section 2.1 | the computation in the e ective theory (with messengers integrated out) is done in section 2.2. Requiring that the two theories describ e the same gaugino masses up to second order in we get, in section 2.3 the gaugino mass terms in the e ective theory. i dr regularization [5] in b oth versions We employ the Feynman-Wess-Zumino gauge and the sup ersymmetric dr values of the gauge couplings and of the gaugino masses M can b e converted of the theory. The nal i i into the ms ones (i.e. the ones obtained with nave dimensional regularization) using C (G) ); M ) = (1 = M (1 + C (G) ms dr ms dr 3 4 4 N where the group factors are de ned as follows. For each representation R of a gauge group G = G we de ne i i a the \Dinkin index" T (R ) and the \quadratic Casimir" C (R ) in terms of the generators T as i i Ri X ab b a a a = T (R ) : (2) T = C (R )1I; Tr T T T i ij i Rj Ri Ri Ri a With generators canonically normalized so that T (n)=T(n)=1=2 for the fundamental n representation of a SU(n) group, the quadratic Casimir of the adjoint representation of a SU (n) group is C (G)=T(G)=n, while 2 graph and its value V graph and its value V 10 1 ~ XX C (X)( ) X X [C (X ) C (G)](2=" +4) j j j j ij i 3 2 g 4 1 g ~ XX C (X) (1=" +1) XX [C (X) C (G)](4) j j j j ij i 3 2 ~ ~ X X C (X )(3=" 2=3) X C (G)(4=" 1) j j i i g 8 ~ ~ ~ X X C (X ) (2=" 1) X C (G)(1=" 1) j j i i 3 g 5 1 4 0 ~ ~ X X C (X )( + ) C (G)(4=" +4) j j i i ir 3 " 3 (1) 2 Table 1: Contributions M = M V =4 of the single graphs in the limit F M . i i C (G)=0for G a U(1) factor. The values of the co ecients for the SM gauge group and for representations contained in 5 5 and 10 10 representation of SU(5) are given in table 2. 2.1 Computation in the full theory The NLO correction to the p ole gaugino masses are given by the ten two-lo op diagrams shown in gure 1 and some one-lo op renormalization factor. It is convenient to separate the renormalization factors due to the light MSSM lo ops from the ones due to messenger lo ops. We write the various contributions to the p ole gaugino masses as X X X F n i (1) (1) (1) (1) full ^ ^ ^ T (R )^g (x )=M + O (") (3) M j = M + M M = ; M i n 1 n i p ole i i i in i 4 M n n n P 2 where x = F =M and the sum extends over all the messengers. The one-lo op function g^ (x) g (x)+ n n 1 1 n n 2 "g (x)+O(" )is[11] 1 2 2 ln(1 + x) x 1+x 2 4 ln(1 + x) 1+" 1 1+"ln +(x !x)+O(" )=1+ +O(x ;") (4) g^ (x)= 1 2 2 x 2 M 6 All parameters are unrenormalized (`bare'). Here we list all the contributions to the p ole gaugino masses.
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