TRI-PP-93-66 There has been much interest in the decay 6 —» sy because of new results from July 1993 the CLEO collaboration which bound the inclusive branching ratio, B(b —* sy), CA9800085 below 5.4 x 10"* at the 95% confidence level, and give a non-zero branching ratio for the exclusive decay B —» A'*7 of about 5 x 10~5[l]. One expects this exclusive channel to make up 5% -40% of the inclusive rate [2], so B(b —» 37) must be greater than about \Q~*. The Standard Model (SM) contribution depends slowly on the top quark mass and is of order 4 x 10~* for m( of 140 GeV. Given this, some recent + Supersymmetric b —* 57 with Large Chargino Contributions works [3,4] claim that the charged Higgs (H ) masses in supersymmetric theories [5] must be very large to avoid exceeding the upper bound on B(b —» 37). We show that this is not always the case—that chargino (x+) loop contributions can R. Garisto and J. N. Ng - ^ 3 SQ O S cancel the H+ contributions and give B(b —» sy) near or below the SM prediction. In particular, we show that such destructive interference effects are important for large tan /? (which is the ratio of Higgs vacuum expectation values), and when there TRIUMF, 4OO4 Wesbrook Mall, Vancouver, B.C., V6T £AS, Canada is a large stop mass splitting. We show that the latter effect is due to the breaking of a GIM cancellation. Abstract The sign of certain mixing angles is also important because one needs the chargino contribution to interfere destructively, rather than constructively. From this one can obtain an approximate condition on the soft SUSY breaking parame- ters A and B, which may have implications for supergravity theories (for a review see [9]). Calculations for B(b —> 37) in SUSY can be found in the literature [6, 7]. Bertolini ei al. [6] perform a thorough but very constrained analysis which im- Supersymmetric (SUSY) theories are often thought to give large branching ratios poses radiative breaking, in the minimal model, with Bo = Ao — \. They also do not consider large tan/9, where chargino effects can become much more important. for b —> s-j from charged Higgs loops. We show that in many caseB chargino loop Barbieri and Giudice [7] make the important point that B(b —» sy) vanishes in the exact supersymmetric limit. However, the scenarios they consider (which are contributions can cancel those of the Higgs, and SUSY can give 5(6 -> sy) at or indeed close to the SUSY limit) with gaugino mass (m») and Higgs mixing mass below the Standard Model prediction. We show this occurs because the large stop (/i) set to zero, are not phenomenologically viable because they give chargino and neutralino masses which are too small (one of the higgsinos is even massless in this maS6 splittings usually found in SUSY break a G1M mechanism suppression. These case). These approaches are understandable since there are many parameters in SUSY theories. Our approach is to concentrate on those parameters which tend to effects are strongly enhanced by large tan/3, so that B(b -> sy) is very sensitive make the chargino contribution large and destructively interfering, so as to make qualitative statements about what areas of parameter space are favored. We show to the value of tan/3, contrary to what has been claimed. We also note that the that one cannot neglect the chargino contributions and that there are large areas of parameter space where B(b —» sy) in SUSY is at or below the SM prediction. supergravity relation fl0 = Ao - 1 is somewhat disfavored over the general case. Contrary to what is claimed in [7], we find that B(b —* sy) is very sensitive to tan /3, and one can even find regions for large tan fi where the chargino destructive interference is too larje [8]. •The inclusive decay b —* sy comes from the operator sia^bnF^. When one runs the scale from Mz to mj, this operator mixes with the gluon operator a 5i,<7'"'T 6nG°1/, as well as four quark operators. We use the notation of [7] through- (submitted to Physics Letters D) out, up to an overall sign in the amplitude. They define the coefficients of the 3 1 2 photon (and gluon) operators as Gp(o/8ir ) / V,*Vi6mt A^ (and A, —» As). We will concentrate on the photon coefficient A^ because the gluon coefficient contri- bution is relatively suppressed by QCD factors [10), as can be seen from the ratio of inclusive branching ratios [7]: which follows from the unitarity condition V^tVoi, = 0 and the condition that the first two generations of up squarks are nearly degenerate. The functions g(x) are B{b 6a given by (6, 7): (1) B(b -> ceu) ~ IT I(mc/mb) (l - £ft. 2 (1), _ 12J- 18i )lni 36(z-l)< (8) where the inclusive semileptonic branching ratio is B(b —» cev) ~ 0.107; the QCD _ factor 77 = a,(Mz)/a,(mi,) ~ 0.546; a QCD correction factor f(mc/mt,) ~ 2.41; and (3) (9) /(x) = 1 — 812 + 816 - i8 - 24a:'1 In x is a phase space factor. The constant C comes 9 K ' from mixing of four quark operators as we run down to mi, and is about 0.175 [7]. + Both of these are positive, and fall off as z becomes large. One finds that top quark, an H+ and a top quark, and charginos xf (j = l,2) and up squarks. bigger than g (i) by at least a factor of 4, for all 1. There are also contributions from flavor changing neutral current (FCNC) vertices We have broken Ay(x) up into four pieces to see when it can significantly reduce due to squark flavor mixing, but these contributions tend to be very small in the B(b —» 37). The sum Ai(x) + Ai(x) is almost never large enough to cancel the //+ + + minimal model [6]. Squark flavor mixings can be large in certain non-minimal contribution. On the other hand, J43(X ) and A<(x- ) can be large because they models, but they are constrained to be small by other FCNC observables so that are enhanced by large tan/?. However, if the stop squarks are degenerate in mass their contribution to b —t $7 is generally small [11]. Thus we can write with the other up squarks, these large contributions exactly cancel, due to a GIM cancellation. To see how the sum A3(x) + A^(x) depends upon the stop mass splittings, let (2) (3) 3> us define fOj = ) = fo, + A/tj- (>,* = 1,2). Defining sin 0i = TIJ, we can write + + + where A~,(W ) and A^(H ) are always greater than zero, while A7(x ) can be of either sign. In the limit of degenerate up and charm squark masses, we can write Mx+) = Mx*) + Mx*) + Mx*) + Mx*), with [7] 1 (cos2 0,-A/ij + sin2 0,-A/jj) (3) ^ sin 0-t cos8; (A/, J - A/2>)]. (10)

+ (4) One sees immediately that if all the squark masses axe degenerate (m,, = m,3 = m), ^(x )^-E^fk + then A/ij = A/2> = 0 which means that A3(x ) + AA(x+) - 0. From (7) it is clear that this cancellation arises from a GIM mechanism; if IOJ = xt^ the unitarity of (5) + + the CKM matrix ensures that A3(x ) + A,(x ) = 0. One can get a sense for the behavior of (10) by considering only the light chargino piece (j = 1), which tends to contribute more than the heavier chargino (6) 3 m cos/3 since J' '(I) is larger for small x. A careful analysis of the chargino mass matrix X) diagonalization reveals that where Uij and Vij are the unitary matrices which diagonalize the chargino mass , signl/uVjj = +sign/i. matrix (see [5]), Tu diagonalizes the stop mass matrix, vu = v^mwsin^, and we define xOj = fhP/rfi^ and x^j = m^/m^. We have used

The only exception is for p < —mwino tan j9, where C/IJV12 is positive but very small. 1 (7) If we use the large tan/? approximation cos" /? ~ tan/?, we can estimate that

t{X+) 5^ tan/? 2 2 + Isign^i \Ui7 V,, | (cos 9-t A/, + sin 9-t A/2) of the amplitude—cancelling the /f , W* and C contributions—so that certain regions of parameter space are ruled out because the value of — A-,(\) is too large! >,cos0,-| —(A/, -. (12) Conversely, the regions of AB > 0 tend to give larger 27(6 —» 37) due to constructive interference from A^(\). This allows one to understand the gross behavior of the sum. For moderate to large We have shown what happens when one varies tan/3, A, n and m\. We took |U| stop splittings, |#,-| ~ 45 , m,, will be less than m, and rhl7 will be greater than such that mj;+ was of order the weak scale—if |Z?| is larger (smaller) than in Figures or of order rh [12]. One sees that the chargino contribution tends to have a large 1-4, wi^+ will be larger (smaller), and all the values for B(b —* 37) will be smaller + + destructive interference with the W and H pieces if it is light (i.e. there is a large (larger). This simply demonstrates the point stressed by [3, 4] that B(b -+ 37) can stop mass splitting), tan/3 is large, and if 0ti fi > 0, i.e. be suppressed by large mH+. If m, is heavier (lighter) than 140 GeV, all of the values for B(b —> sj) will be shifted up (down) slightly, but for the SUSY result < 0, 9-t < 0, or /i > 0,6-t > 0. (13) of m( ~ 134 ± 25 GeV [14], there is no qualitative change in our results. Lastly, 2 one can make a different choice for m0. Increasing mo makes both Am, and m /+ larger, so that B(b —» 37) generally decreases in the AB < 0 regions. One must be 2 The /i > 0 case gives a smaller B(b —• s-f) because both pieces in (12) help to reduce careful for large Am0 that m is greater than zero. the overall amplitude. One can show that signtf,- = —sign(Amo — /Jcot/3), so that The branching ratio for b —» 37 in SUSY theories is near or below the SM A\i < 0 implies that 9-t JL > 0 (though the converse is not necessarily true). Finally value if the charged Higgs mass is large, or the chargino contribution destructively we note that the sign of ft is just the sign of Z?-one rotates the Higgs fields so as to interferes with the charged Higgs and IV loops. We found that the latter occurs in make the Higgs potential coefficient /i,2 positive, and then signal equals the signB regions of parameter space where AB < 0 (or equivalently when An < 0), and is before that rotation [13]. Thus AB < 0 implies 6;fi > 0, which is the favorable accentuated by large tan/3 and large |A|. One can have B(b —> 37) at or below the region for destructive interference from the chargino loops. If \Amo\ tan /3 > \(i\, the SM prediction in supersymnietric models without requiring a large charged Higgs converse is also true. mass. Finally, we have noted that if m//+ and tan /3 were found to be small, it might In the simplest SUGRA theories, one has the relation at the Planck scale Bo = be possible to rule out minimal SUSY models which satisfy the SUGRA relation Ao — 1 [9]. One can show using general properties of the renormalization group Bo = Ao - 1. equations that this relation implies one cannot have A < 0 and B > 0 at the weak scale, which is the most favored region for small B(b —• 37). If mj/+ and tan/3 were Acknowledgements found experimentally to be small, it might be possible to rule out minimal SUSY We thank G. Kane and G. Poulisfor useful discussions. This work was supported models which satisfy Bo = Ao — 1. in part by a grant from the National Sciences and Engineering Research Council of To illustrate these results, we consider some supersymmetric scenarios. In Fig- Canada. ure 1, we consider the heuristic parameters Am, and d\. We see that for the given choice of parameters with Q\\i < 0, B(b —» 37) is always greater than the CLEO bound (in the region allowed by LEP, above the unlabeled curves). The case &-t /i > 0 has lower B(b —• 37), especially for the n > 0 case, and there are regions where the CLEO bound is satisfied. Increasing Am, lowers B(b —» 37) in the 9; (i > 0 regions because Aa(x+) + A«(x+) becomes more important. Radiative corrections lower both mtl and m(, relative to rh [12], so we take (m(l — m)/Am, to be —2/3 and 1/3, respectively. If one raises (lowers) rn,, while holding mtl constant, the difference between the /J > 0 and fi < 0 regions tends to become less (more) pronounced. Figures 2-4 show more realistic scenarios where one inputs A instead of Am< 2 and signtf;. Increasing tan/3 will increase m /+, so that large tan/3 gives smaller B(b —» 37) just by suppressing the H+ loop contribution. To examine the different values for tan/3 on equal footing, we have taken \B\ = 1.5/ tan/3 so that Ti;y+ is about the same in each graph (mj/+ — 260 GeV at \fi\ = 400). Even so, B(b —• 37) gets much smaller in the AB < 0 (i.e. An < 0) regions as tan/3 increases, again because Az(x*) + -^(v"1") becomes more important. Larger |.4| also reduces B(b —» 37) in those regions. For A < 0, B > 0 (which is not allowed if Bo = Ao — 1) and tan/3 > 10, there are even regions where the chargino contribution flips the sign REFERENCES Fig. 2. Contour plots of B(b —» sy) (labeled lines) in units of lO"1 for tan£ = 3, m, = 140, B = 0.5, and m0 = 100. Graphs (a), (b), (c), (d) have A = +1, —1, +2, -2, respectively. Unlabeled solid lines are mx» = 25, and m + = 45. All masses in GeV. [1] E. Thorndike, CLEO Collab., talk given at the 1993 Meeting of the American Fig. 3. Same as Figure 2 for tan /3 = 10. The contours in the /1 > 0 region of Physical Society, Washington, D.C., April 1993. 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Fig. 1. Contour plots of B(b —» 37) (labeled lines) in units of 10"* for tan/3 = 5, m, = 140, B = 0.3, and m0 = \f\- The current CLEO bound in these units is 5.4. Graphs (a) and (c) ((b) and (d)) use 0; < 0 (9-t > 0). Graphs (a) and (b) ((c) and Fig. 1 (d)) use a fixed stop mass splitting of 100 (200). Unlabeled solid lines are mxa = 25, and m + = 45. All masses in GeV. 8 S3 8 S g

g i