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JORGE L. LOPEZ, D. V. NANOPOULOS, and GYE T. PARK,

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ABSTRACT

We present a calculation of the branching ratio $\brbsg$ in two well motivated supersymmetric models: the minimal $SU(5)$ and the no-scale flipped $SU(5)$ supergravity models. We find that the improved CLEO upper bound ($\brbsg<5.4\times10^{-4}$ at 95% CL) does not yet constrain the minimal $SU(5)$ supergravity model, where $\brbsg_{minimal}=(2.3-3.6)\times10^{-4}$. In the flipped $SU(5)$ model the CLEO bound is constraining, although still not very significantly, even for light charged Higgs masses. An improvement in sensitivity by a factor of two will probe all (more than half) of the parameter space of the minimal (flipped) $SU(5)$ supergravity model. This “resolution" of the $\bsg$ microscope far surpasses that of present collider experiments. In the flipped model there exists a significant region of parameter space where $\brbsg$ is highly suppressed due to a new phenomenon involving a complicated cancellation against the QCD correction parameter.

There has recently been a renewed surge of interest on the flavor-changing-neutral-current (FCNC) $\bsg$ decay, prompted by the CLEO bound $\brbsg<8.4\times10^{-4}$ at $90\%$ CL . Since the Standard Model (SM) prediction looms around $(2-5)\times10^{-4}$ depending on the top-quark mass ($m_t$), a reappraisal of beyond the SM contributions has become topical. Most recently it has been pointed out that the charged Higgs ($H^\pm$) contributions in two-Higgs-doublet models (like the minimal supersymmetric standard model (MSSM)) would violate the CLEO bound if $m_{H^\pm}$ is too light. The precisely excluded region is $m_t$- and $\tan\beta$-dependent and appears to depend non-negligibly on the bottom-quark mass ($m_b$) which partially determines the QCD enhancements to the $\bsg$ amplitude. The recently announced improved CLEO bound of $\brbsg<5.4\times10^{-4}$ at 95% CL , leads to yet larger excluded domains in the charged Higgs parameter space . In these papers it remained an open question whether inclusion of the full set of supersymmetric contributions to the $\bsg$ amplitude would change significantly the quoted bounds, and thus the outlook for supersymmetric Higgs searches at LEPII. The full supersymmetric contributions to $\bsg$ have been known for some time  and have been recently given in a justifiably simplified form in Ref. . The latter paper argues that the bound on $m_{H^\pm}$ in Refs. could well be evaded in the full supersymmetric calculation since in the unbroken supersymmetric limit the $\bsg$ amplitude vanishes.

The purpose of this paper is to study $\brbsg$ in two supergravity models: (i) the minimal $SU(5)$ supergravity model , and (ii) the no-scale flipped $SU(5)$ supergravity model . We show that the [*new*]{} CLEO bound does not yet constrain the minimal $SU(5)$ model, while some as-yet-mild constraints are imposed on the flipped $SU(5)$ model, where a new phenomenon can drastically suppress the $\bsg$ amplitude. We present the results for $\brbsg$ in these two models and show that improved sensitivity could probe them in ways not possible at present collider experiments.

We use the following expression for the branching ratio $\bsg$ where $\eta=\alpha_s(M_Z)/\alpha_s(m_b)$, $I$ is the phase-space factor $I(x)=1-8x^2+8x^6-x^8-24x^4\ln x$, and $f(m_c/m_b)=2.41$ the QCD correction factor for the semileptonic decay. The $A_\gamma,A_g$ are the coefficients of the effective $bs\gamma$ and $bsg$ penguin operators evaluated at the scale $M_Z$. Their simplified expressions are given in the Appendix of Ref. , where the gluino and neutralino contributions have been justifiably neglected  and the squarks are considered degenerate in mass, except for the $\tilde t_{1,2}$ which are significantly split by $m_t$. This is a fairly good approximation to the actual result obtained in the two supergravity models we consider below, since $m_{\tilde
q}>200\GeV$ in these models. To include at least partially the QED corrections to the $\bsg$ operator, in Eq.  we take $\alpha=\alpha(m_b)=1/131.2$. We use the 3-loop expressions for $\alpha_s$ and choose $\Lambda_{QCD}$ to obtain $\alpha_s(M_Z)$ consistent with the recent measurements at LEP. For the numerical evaluation of Eq. (1) we take $\alpha_s(M_Z)=0.118$, ${\rm BR}(b\to ce\bar\nu)=10.7\%$, $m_b=4.8\GeV$ and $m_c/m_b=0.3$.

The subject of the QCD corrections to $\brbsg$, , the origin of the $\eta$ factors and the $C$-coefficient in Eq. , has received a great deal of attention over the years in the SM. The results for supersymmetric models are not known, but are not expected to differ significantly from the SM result in the models we consider, since all strongly interacting sparticle masses are above $M_Z$. In our analysis we use the leading-order QCD corrections to the $\bsg$ amplitude when evaluated at the $\mu=m_b$ scale , , $C=\sum_{i=1}^8 b_i\eta^{d_i}=-0.1766$ for $\eta=0.548$, with the $b_i,d_i$ coefficients given in Ref. . The result follows from the renormalization-group scaling from the scale $M_Z$ down to $\mu=m_b$ of the effective $\bsg$ operators at $M_Z$, which include the usual electromagnetic penguin operator plus some four-quark operators. Scaling introduces operator mixing effects (as exemplified by the appearance of the gluonic penguin operator in Eq. ) and the scaled coefficients at $\mu=m_b$ are given by linear combinations of the coefficients at scale $M_Z$. It has been pointed out  that the low-energy mass scale $\mu$ affects the leading-order results significantly. The combined effect on $\brbsg$ of the uncertainties in $\mu$ and $\Lambda_{QCD}$ has been estimated to be $\lsim25\%$ . Recently a partial next-to-leading order calculation of the QCD effects has appeared , which gives somewhat smaller enhancement factors than the complete leading-order result. A full next-to-leading order QCD calculation should decrease the above mentioned uncertainties significantly.

The two models we consider are built within the framework of supergravity with universal soft-supersymmetry breaking. The renormalization-group scaling from the unification scale down to low energies plus the requirement of radiative electroweak symmetry breaking using the one-loop effective potential, reduces the number of parameters needed to describe these models down to just five: $m_t$, $\tan\beta$, and three soft-supersymmetry breaking parameters ($m_{1/2},m_0,A$) (for a detailed account of this procedure see , Ref. ). The sign of the superpotential Higgs mixing term $\mu$ remains as a discrete variable. The models we consider belong to this general class of models but are further constrained making them quite predictive.

The minimal $SU(5)$ supergravity model  is strongly constrained by the proton lifetime  and the cosmological constraint of a not too young universe . A thorough exploration of the parameter space, including two-loop gauge coupling unification  yields a restricted region of parameter space to be subjected to further phenomenological tests.

The no-scale flipped $SU(5)$ supergravity model  assumes that the parameters $m_0$ and $A$ vanish, as is typically the case in no-scale supergravity models . This constraint reduces the dimension of the parameter space down to three, and we take $m_t=100,130,160\GeV$. The choice of gauge group (flipped $SU(5)$) is made to make contact with string-inspired models which unify at the scale $M_U\sim10^{18}\GeV$. We also consider a “strict no-scale" scenario (see Sec. 4 of Ref. ) where in addition the universal bilinear soft-supersymmetry breaking scalar mass parameter $B$ vanishes. This constraint reduces the dimension of the parameter space down to two, since $\tan\beta$ can now be computed as a function of $m_{1/2}$ and $m_t$. An interesting consequence of this scenario is that for $\mu>0$ only $m_t\lsim135\GeV$ is allowed, whereas for $\mu<0$ only $m_t\gsim140\GeV$ is allowed. Moreover, for $\mu>0$ the calculated value of $\tan\beta$ can be double-valued. In what follows we take $m_t=100,130\,(140,150,160)\GeV$ for $\mu>0\,(\mu<0)$.

In both models there are semi-quantitative relations among the lighter neutralino and chargino masses ($m_{\chi^0_1}\sim{1\over2}m_{\chi^0_2}
\sim{1\over2}m_{\chi^\pm_1}$) and it is found that these can be as light as presently experimentally allowed. The gluino mass is bounded below by $m_{\tilde g}\gsim200\GeV$ in both models, with $m_{\tilde q}\approx
m_{\tilde g}$ in the flipped model whereas $m_{\tilde q}>2m_{\tilde g}$ in the minimal $SU(5)$ model. Also, light (heavy) sleptons are characteristic of the flipped (minimal) $SU(5)$ model. The Higgs spectrum of the minimal model is to a good approximation SM-like , in that the lightest Higgs boson has close to SM couplings to gauge bosons and fermions, whereas the other Higgs bosons ($A,H,H^\pm$) are heavy. In the flipped model something similar occurs for a good portion of the allowed parameter space. However, significant deviations can occur  which allow, , charged Higgs masses as low as $\approx100\GeV$. In what follows we evaluate $\brbsg$ using as inputs the couplings and masses obtained by running over all points in the previously determined allowed parameters spaces for these models.

In Fig. 1 (top row) we present $\brbsg$ for the minimal $SU(5)$ supergravity model, plotted against one of the light sparticle masses in the model, namely the lightest chargino mass. We find for $\mu>0\,(\mu<0)$, which are all within the new CLEO bound. Since these numbers are within the SM range, in Fig. 1 (bottom row) we show their ratios to the corresponding SM contribution, which depends solely on $m_t$. We obtain for $\mu>0\,(\mu<0)$. This implies that $\brbsg$ would need to be measured with better than $20\%$ accuracy to start disentangling the minimal $SU(5)$ supergravity model from the SM. Moreover, for $\mu<0$ there is a band of points which will be difficult to tell apart (requires $<1\%$ accuracy). It is interesting to remark that an analogous set of plots versus $m_t$ instead, does not reveal any particular structure in $m_t$, thus an independent determination of $m_t$ is not likely to constrain the possible values of $\brbsg$. This is in sharp contrast with the SM, where such a measurement would determine $\brbsg$, up to QCD uncertainties. In other words, in the minimal $SU(5)$ supergravity model, a measurement of $\brbsg$ within the range of Eq. (3), is unlikely to shed much light on the value of $m_t$. Of course, if the measurement falls outside this range, the model would be excluded.

In Fig. 2 we present the analogous results for the no-scale flipped $SU(5)$ supergravity model (general case; strict no-scale scenario discussed below). (In this figure $m_t=100,130,160\GeV$.) The results are strikingly different than in the prior case. One observes that part of the parameter space is actually excluded by the new CLEO bound, for a range of sparticle masses. However, light charged Higgs masses are not excluded altogether. Perhaps the most surprising feature of the results is the strong suppression of $\brbsg$ which occurs for a good portion of the parameter space for $\mu>0$. Note also that the results are generally quite different from the corresponding SM ones (see Fig. 2, bottom row) and not much accuracy would be needed to tell the models apart. Moreover, at least half of the parameter space should be accessible with an increase in sensitivity by a factor of two. The dependence on $m_t$ is also quite significant in this model, and for $m_t=130\GeV$ and $\mu<0$, the calculated $\brbsg$ values are quite large and are starting to be constrained: note the set or nearly parallel lines intersecting the CLEO bound for $\mu<0$ (top row), as well as the corresponding isolated branch (bottom row) for large values of the ratio.

In Fig. 3 we present the corresponding results for the strict no-scale scenario in the flipped $SU(5)$ model. The results fall within those in Fig. 2 as expected, but are a lot more predictive. For $\mu>0$ the two $\tan\beta$ solutions corresponding to $m_t=130\GeV$ give quite distinguishable results. For $\mu<0$ the results for $m_t=140\GeV$ overlap with those for $m_t=150\GeV$. Clearly, independent determinations of $m_t$ and $\brbsg$ would lead to very precise constraints, as follows: (i) if $m_t\gsim140\GeV$ then $\mu<0$ and $\brbsg\gsim3.5\times10^{-4}$; and (ii) if $m_t\lsim135\GeV$ then $\mu>0$ and a wide range of $\brbsg$ values are possible. In both cases, with sufficiently accurate measurements, one should be able to pin down the value of the chargino mass (with a possible two-fold ambiguity) and therefore the whole spectrum of the model.

The large suppression for $\mu>0$ in the flipped model (see Figs. 2,3) requires further explanation. What happens is that in Eq. , the $A_\gamma$ term nearly cancels against the QCD correction factor $C$; the $A_g$ contribution is small. The $A_\gamma$ amplitude receives three contributions: from the $W$-$t$ loop (, the SM contribution), from the $H^\pm$-$t$ loop, and from the $\chi^\pm_{1,2}$-$\tilde q$ loop. The first two contributions are always negative , whereas the last one can have either sign. When the chargino-squark contribution is positive and in the range $0.4-0.6$, the cancellation occurs. No cancellation occurs for $\mu<0$ since there the chargino-squark contribution is always below $\approx0.15$. Moreover, when this contribution is positive and large ($\gsim1.2$ for $\mu>0$) or sufficiently negative ($\lsim-0.05$ for $\mu<0$), the CLEO bound is exceeded. It has proven quite difficult to determine a simple reason for the behavior of the chargino-squark contribution and its sign-of-$\mu$ dependence. All we can do is to remark that the lightest stop eigenstate $\tilde t_1$ is lighter for $\mu>0$ and it can approach the $\chi^\pm_2$ mass, resulting in an enhancement in $A_\gamma$. A more detailed study of the circumstances under which the observed phenomenon occurs would be better motivated should the experimentally interesting range of $\brbsg$ fall below $\sim10^{-4}$. In fact, an actual observation of the exclusive $B\to K^*\gamma$ mode has also been recently reported (${\rm BR}(B\to K^*\gamma)=(4.5\pm1.5\pm0.9)\times10^{-5}$ ), which indicates that $\brbsg$ is probably not below $\sim10^{-5}$ and thus the points in Figs. 2,3 with very suppressed $\brbsg$ values are probably disfavored.

We conclude that the $\bsg$ microscope can probe the parameter spaces of the two models we have considered in ways not possible with present high-energy collider experiments. A sharpening of previous phenomenological predictions for processes at Fermilab , LEPII , and HERA  in the flipped $SU(5)$ model would be possible in the light of the constraints imposed on the parameter space of this model by the new CLEO bound . With further improvements in sensitivity to the inclusive $\bsg$ mode, it should not be long before these models begin to show their true colors.

[*Acknowledgments*]{}: This work has been supported in part by DOE grant DE-FG05-91-ER-40633. The work of J.L. has been supported by an SSC Fellowship. The work of D.V.N. has been supported in part by a grant from Conoco Inc. The work of G.P. has been supported by a World-Laboratory Fellowship. We would like to thank Kajia Yuan for spurring our interest on this subject and his participation at the early stages of this project.
