Probing Supergravity Grand Unification in the Brookhaven G-2 Experiment

PHYSICAL REVIEW D VOLUME 53, NUMBER 3 1 FEBRUARY 1996

Probing supergravity grand unification in the Brookhaven 9 - 2 experiment

Utpal Chattopadhyay Department of Physics, flovthe&tem University, Boston, Massachusetts 02115

Pran Nath Department of Physics, Northeastern University, Boston, Massachusetts 0211S and Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030 (Received 24 July 1995)

A quantitative analysis of a, 3 f(g - 2), within the framework of supergravity grand unification and radiative breaking of the electroweak symmetry is given. It is found that a;“sy is dominated by the chiial interference term from the light chargino exchange, and that this terni carries a signature which correlates strongly with the sign of p. Thus as a rule azusy > 0 for fl > 0 and I$“‘~ < 0 for ti < 0 with very few exceptions when tanp N 1. At the quantitative level it is shown that if the ES21 BNL experiment can reach the expected sensitivity of 4 x lo-” and there is a reduction in the hadronic error by a factor of 4 or more, then the experiment will explore a majority of the parameter space in the mo - mg plane in the region mo < 400 GeV, mg < 700 GeV for tanp > 10 assuming the experiment will not discard the standard model result within its 2~ uncertainty limit. For smaller tanp, the SUSY reach of ES21 will still be considerable. Further, if no effect within the 2~ limit of the standard model v&e is seen, then large tanp scenarios will he severely constrained within the current naturalness criterion, i.e , mo, mp < 1 TeV.

PACS number(s): 12.6O.Jv, 04.65.+e, 13.40.Em, 14.60.Ef

I. INTRODUCTION symmetry [13] and without the benefit of the recent high- precision data from the CERN &+e- collider LEP [14,15] The high level of experimental accuracy of the mea- to constrain the coupling constants [16]. The purpose sured value of the anomalous magnetic moment of the of the present work is to include these features in the muon [l] has provided verification to several orders in analysis. Additionally, we investigate the effects on the the perturbation expansion of quantum electrodynamics results due to b + sy constraint [17], and dark matter (QED) [2] as well as put constraints on physics beyond constraint. We shall find that the expected accuracy of the standard model (SM) [3]. Further the E821 experi- a, q g9 - %I, in ES21 BNL experiment would either ment which is to begin soon at Brookhaven is expected to allow for supergravity grand-unification effects to be vis- improve the accuracy over the previous measurement by ible in the BNL experiment, or if no effect beyond the 2~ a factor of 20 [4]. Simultaneously, it is expected that im- level is seen, then there will be a significant constraint on proved analyses of existing data on (e+e- +hadrons) as the model. well as new data from ongoing experiments in the Novosi- The most recent experimental value of a, is averaged birsk collider VEPP-2M [5] together with future exper- to be aTpt = 1165 923(8.4) x lOme. The quantity within iments in at DAaNE [6], the Beijing Electron-Positron .the parenthesis refers to the uncertainty in the last digit. Collider (BEPC) [7], etc., will reduce the uncertainty in The standard model results consist of several parts: the hadronic contributions to a significant level so as to allow for a test for the fast time of the electroweak corrections in the standard model. It was pointed out in [S,9] (see also [lo] for a discussion of previous work and Here azED [2] is computed to O(a5) QED corrections. [ll] for a discussion of recent work) that supersymmetric a?:“” consists of O(a’) [l&19] and O(a3) [3] hadronic corrections to (g - 2), are in general the same size as the vacuum polarizations, and also light-by-light hadronic electroweak corrections in supergravity grand unification contributions [21]. We exhibit two different evaluations [12]. Thus improved experiments designed to test the in Table I. Case A uses the analysis of Martinovic and standard model electroweak corrections can also provide Dubnicka [18] who use a detailed structure of pion and a probe of supersymmetric contributions. kaon form factors in their fits to get the O(cy’) correc- The analyses of [8,9], however, were done without using tions. Case B uses the O(a’) analysis of Eidelman and the constraints of radiative breaking of the electroweak Jegerlehner [19,20]. We note that there is almost a factor of 2 difference between the hadronic errors of case A and case B. For the electroweak corrections we have used the recent analyses of Czarnecki et al. (221, which include one- loop electroweak corrections of the standard model [23], *Permanent address. two-loop corrections with fermion loops [22,24], and par-

1648 01996 The American Physical Society 53 PROBING SUPBRGRAVITY GRAND UNIFICATION IN THE.. 1649

TAB&E I.. Contributions to a,, x 10”.

i N&tie of contribution Case A Case B QED to O(a/+ 11658470.8(0.5) 11658470.8(0.5) Kinoshita ,et al. [22,2]

Hadronic vat. polarization 705;2(7.56) 702.35(15.26) al O(a/*)2 M&o& and Dubnicl&[lS] Eidelman,and Jegerlehner [19,20] Hadronic MC.’ pol&ation -9.0(0.5) ” -9.0(0.5) to O(allrP Kino~hit~and’MaCiano [3] Light-by-light hadronic O.S(O.9) ” O.S(O.9) amplitude Bijnens et al. 1211 Total hadronic ~697.0(7.6) 694.2(15.3)

Electroweak one loop 19.5 19.5 Fujikawa,et al. [23] Electroweak two-fermion loops ‘-2.3(0.3) -2.3(0.3) Czankki et al. [22] Electroweak two-boson loops (-2.0 + 0.045~XRa) (-2.0 + 0.045 x Rb) Czarnecki et al. [22] Total electroweak (15.2(0.3)+0.045xRb) (15.2(0.3)+~.045xRe,) up to two-loops

Total (with Rb = 0) 11659 183(7.6) 11659 lSO(15.3)

tial two-loop c,o&i&s Li& b&on &ops 1251. The rei GUT scale MG is given by maining unknown t&lo& &re&ns with boson loops are denoted by Rb following the notation of 1221. Pending I& = n# + (A,@) + BoW@) + H.c.) , (2) the full two-loop bosonic contributiond in the EW co&c- tions, the total azM for case A is 11659 lSO(7.6) x lo-lo where I+‘(‘), W@) are the quadratic and the cubic parts while for case Bone has 116592OO(15.3)x1O-‘o. We see of the effective superpotential. Here WC21 s &I,H,, that the sverall error, which is dominated by the hadronic with HI, Hz being two Higgs doublets, and WC31 con- corrections, is about one-half for case A relative to that tains terms cubic in fields and involves the interactions for case B. of quarks, leptons, and Higgs bosom with strength deter- The new high-precision E821 Brookhaven exl&mmt mined by Yukawa couplings. In addition the low-energy [4] has an anticipated design sensitivity of 4 x10-“. This theory has a universal gaugino mass term -~z~I~X~:X”. is about 20 times more accurate than that of the CERN The minimal supergravity model below the GUT scale measurement conducted earlier. However, as mentioned depends on the following set of parameters: already one needs in addition an improvement in the computation of the hadronic contribution ap”, where uncer- mo,ml/z,Ao,Bo;/lo,ac,M~, (3) tainties primarily arise due to’hadronic vacuum polarization effects. This problem is expected to be over- where MG and o1~ are the GUT mass and the cou- come soon through further accurate measurements of pling constant, respectively. Among these parameters CYG and MG are determined with the high-precision LEP re- gtt,t(e+e- + hadrons) for the low-energy domain. sults by using two-loop renormalization-group equations a;(Mz), i = 1,2,3, up to MC [26]. Renormalization- group analysis is used to break the electroweak symmetry II. THE MINIMAL SUPERGRAVITY MODEL 1131and radiative breaking allows one to determine p,, by fking MZ and to find BO in terms of tanP = (H2)/(Hl). The framework of OUI‘ analysis is N = ,l .supergravity Thus the model is completely parametrized by just four grand-unified theory (121 where the supergravity interac- quantities (271 tions spontaneously break supersymmetry at the Planck scale (Mp, 5 2.4 x lOIs GeV) via a hidden sector [12]. mo,m;>At,t=nO. (4) Further we assume that the grand-unified theory (GUT) group G breaks at scale & = I& to the standard model Here the universal gaugino mass mllz is replaced by the gauge group: G + SU(3)c x Sum x U(l)y. In the gluino mass mg and Ao by At, which is the t-quark A analysis the nature of the group G is left arbitrary and parameter at the electroweak scale A&w. In addition to we do not impose proton stability constraint. At low’en-~ these four parameters and the top quark maas rn*, one ergy the symmetry-breaking effective potential below the has to specify the sign of p, since the radiative breaking 1650 UTPAL CHATTOPADHYAY AND PRAN NATH equations determine only $. There are 32 new parti- masses rniv which are charged spin-i Dirac fields and cles in this model (12 squarks, 9 sleptons, 2 charginos, 4 a sneutrino state. In Fig. l(b) one exchanges four neu- neutralinos, 1 gluino, 2 CP-even neutral Higgs bosom, tralinos .?&, k = 1,2,3,4 with masses rn++., which are 1 CP-odd neutral Higgs boson, and 1 charged Higgs bo- spin-i Majorana fields (our labeling of particles satisfy son). The masses of these 32 new particles and all their fii < riLj for i < j), and two scalar amuon mass eigen- interactions can be determined by the four parameters states. The mass spectra of the charginos, neutralinos, mentioned above. Thus the theory makes many new pre- smuons, and of the sneutrino are given in the Appendix dictions 1271,and has led to considerable activity [28,29] for convenience. The one-loop supersymmetric contribu- to explore the implications of supergravity grand unification to (g - 2), is then given by tion and its signals [30]. The allowed parameter space of the model is further constrained by (i) charge and color .yy = gg - qJSY = ggfi + gz) conservation [31], (ii) the absence of tachyonic particles, (5) (iii) a lower bound on SUSY particle masses as indicated by the Collider Detector at Fermilab (CDF), DO, and For the chargino-sneutrino part referring to [S] we find LEP data, and (iv) an upper limit on SUSY masses from (with summation over repeated indices implied) the naturalness criterion which is taken as mo,mg < 1 TeV. Our analysis automatically takes into account important Landau pole effects that arise due the top being heavy and thus in proximity to the Landau pole that lies in the top Yukawa coupling (321. Additionally, we con- sider the constraint on b + sy decay rate from the recent CLEO data, and the neutralino relic density constraint.

III. ANALYSIS OF (g - Z);“= AND RESULTS The mass of the sneutrino required in Eq. (7) can be determined from Eq. (33). The first term in Eq. (6) contains We use [8] to compute SUSY contributions to (g-Z),. terms diagonal in chirality whereas the second term has These contributions arise from Figs. l(a) and l(b). In R-L interference terms arising from Yukawa interactions. Fig. l(a) one exchanges two charginos fi*, a = 1,2, with The functions FL(z) and Fz(z) are defined by

FI(z) = (1 - 5x - 2r2)(1 - z)-~ -6&1- ~)-~ln(r) , (8) F&) = (1 - 3z)(l -zr)-” - 2271 -z)-%(z) .

Aa”’ and At’ of Eq. (6) are given as

A(‘) = - R &ow cOs”fl ’ (9) A(‘) = (-I)@ ’ np cos 72 L fi sin 6~ &MW cos 0 ’

Ac21 = _ e R sin-71 ,

FIG. 1. (a) Chargino-sneutrino one-loop exchange diagram which contributes to (L~‘~~. (b) Neutral&-smuons one-loop where -ir < 2/3,,2 5 ?T. exchange diagram which contributes to a~usy. The neutralino-smuon loop correction results in PROBING SUPERGRAVITY GRAND UNIFICATION IN THE. 1651

Here s = sins, c = cos6, and [see Eq. (26)]. For fi > 0, one finds X1 < 0 invariably and for /I < 0 one has X1 > 0 for almost all the regions of 2 parameter space of interest. This can be seen by writing , r=l,2, k=l,2,3,4. (13) Xl,z in the form

Xl,2 = $([2M& + (p - 6~2)~ - 2M&sin2#/’ The functions G,(r) and Gz(z) are given by ?[2M$ + (p + 7h~)~ + 2M& sin2fl]“2) (17) GI(I) = $(2 + 52 -x2)(1 -z)-” + 3z(l- ~)-~ln(z) , and noting that the terms containing sin 20 are only ap- (14) preciable when tano is small (- 1). As a result, because G&z) - (1 +x)(1 -z)-” + 241 - ~)-~ln(z) of this unique dominance of the chiral interference term involving the lighter chargino mass, one finds as a rule The Yukawa coefficients C’h are found from a~““” > 0 for /I > 0 and azusy < 0 for p < 0 with some very few exceptions for the latter case when tanP - 1 resulting in a very small la~“syI. The observation on the relation between the sign of p and the sign of aE:‘I” and Bf and Bt are found by using was previously noted in [ll] on the basis of numerical analysis. The analysis here is in agreement with this ob- Bk” = E[--Olk + cot28wOzk], sew&ion and provides an analytic understanding of the (16) phenomenon. B; = e[-Olk - tan0wOzk](-1)81 . The generic dependence of ~cz~“‘~I on the remaining parameters mo, rni, and At .iL as follows; Regarding The &ngle 6, BE and the quantities 0, are defined in the mo, the dependence results primarily from the chargino- Appendix. sneutrino sector because the mass spectra depend on mo. Among the two sources of one-loop contributions to This results in a decreasing ja~“syI with increase in mo. aps” the chargino-sneutrino loop contributions more Among the other two basic parameters a large gluino than the neutralino-smuon loop. This occurs mainly due mass mg in general again leads to a smaller la?/, due to the smallness of the mixing angle 6 [see Eq. (12)] which to the resulting larger sparticle masses entermg in the itself arises from the smallness of the muon to the spar- loop. The At dependence enters implicitly via the SUSY tick mass ratio [see Eq. (31)]. Partial cancellations on mass spectra, and also explicitly in the neutralino-smuon the right-hand side of Eq. (12) are also responsible for exchange diagrams. Figures 2(a) and 2(b) show the up- a reduction of the neutralino-smuon contribution. For per limit of Icz~“~~] in the minimal supergravity model the chargino-sneutrino contribution [see Eq. (S)] one has by mapping the entire parameter space for mo, ms 5 1 comparable magnitudes for chirality diagonal and non- TeV, tanP < 30 and both signs of p. One finds that diagonal terms for small tanp(- 1). The nondiagonal m=+, s”syl increases with increasing tanP for fixed mi. terms become more important as tan@ starts to deviate As discussed above this happens due to the essentially significantly from unity. Indeed, a large value of tan0 re- linear dependence of the chirality nondiagonal term on sults in a large contribution of the chirality nondiagonal tan& Furthermore, for large tan/? one finds similar mag- term in the chargino-sneutrino part and hence to (a~“sy( nitudes of max(a~“sY( for p > 0 and p < 0. This can be due to the enhancement arising from I/ cos p - tan0 in understood by noting that for a large tanP the lighter the Yukawa coupling. While this result is implicit in the chargino maSses for p > 0 and ~1< 0 cases have almost analyses of (81, [9] it was first emphasized in [ll]. similar magnitudes [see Eq. (17)]. This, along with the There exists a very strong correlation between the sign explanation of the dependence of the sign of CZ~“‘~ on of a~“” and the sign of /I which we now explain. While the sign of /I accounts for similar IIZ~“~~Ivalues for both the chiral interference chargino part of a:? dominates p < 0 and p > 0 when tanP is large. This similarity over the other terms, it is the lighter chargmo mass which between the p > 0 and fi < 0 Casey is less apparent for contributes most dominantly. In fact this part depends small tanp (i.e., tanP 5 2) and small mg values. on At’ Ag’ which from Eq. (9) can be seen to have a front Next we include in the analysis 6 + sy constraint and factor of (-I)~,, where .G = O(1) for X1 > 0(< 0) where the dark matter constraint which have been shown in re- X1 is the smaller eigenvalue of the chargino mass matrix cent work [33,34] to generate strong constraints on the 1652 UTPAL CHA’ITOPADHYAY AND PRAN NATH 53

(4 Ce)

mgmo (GeV) Cb) (4 Cf)

FIG. 2. (a) Thd upper limit of IayyI vs me for the case ,J < 0 in the minimal supergravity model for different values of tanp in the region tanp < 30; when one allows the remaining parameters At and mo to vary over the parameter space subject to the naturalness criterion of mo 5 1 TeV. The dashed horizontal line is the current 20 limit as given by Eq. (23). (b) Same as (a) except that p > 0. (c) Same as (a) except that the constraint from b + sy and dark matter as discussed in the text are included. (d) Same as (c) except that p > 0. (e) Excluded regions (regions enclosed by the curves towards the origin) in the (rn, - mp) plane under the current theoretical and experimental limits of au for various values of tanp, for the case @ < 0. (f) Same as (e) except that fi > 0.

parameter space. For the b + sy decay the CLEO Col- to the presence of different SUSY thresholds [38]. In the laboration [17] gives a value of present analysis we limit ourselves to the leading-order evaluation. In a similar fashion we can define B(b --t ~7) = (2.32 f 0.57 f 0.35) x 1O-4 (16) ~xpt = B(b + w)exptlB(b + -vh (20) Combining the errors in quadrature one has B(b + sr) = (2.32,+ 0.66) ‘x lo@. The standard model pre- Using the experimental result of Eq. (18) and the stan- diction for this branching ratio has an -30% uncertainty dard model values given above one finds that ~..~t lies in [35] mainly from the currently unknown next-to-leading the range [36] (NLO) order QCD corrections. Recent standard model evaluations give [35] B(b + ~7) = (2.9 f 0.8) x 10m4 at ~.x,,t = p.46-2.2 (21) mt z 170 GeV. The SUSY effects in B(b + ~7) can be Now normally in SUSY theory one can get rather large conveniently parametrized by introducing the parameter deviations from the SM results so that ‘~susy ,can lie TSUSY which we define by the ratio [36] in a rather large range, i.e., = (0,lO). Thus the constraint TSUSY = 7.xpt is an important constraint on the TSUSY = B(b -+ sy)susu/B(b + ~7)s~ (19) theory. This constraint then excludes a part bf the pa- Several uncertainties that are present in the individual rameter space [34,36] and reduces the magnitude of the branching ratios cancel out in the ratio ~susy. However, max Io.S;,,~] for a given mg. we point out that the NLO corrections would in general The cosmological constraint on the neutralino relic be different for the SUSY case than for the SM case due density [39] also plays a very significant role in limit- 53 PROBING SUPERGRAVITY GRAND UNIFICATION IN THE. 1653 ing the parameter space of the model. Theoretically the value of tanP and constraints become more severe the quantity of interest is S2i,h2 where na = ps,/p,. as tanP increases. For tanP 5 10 the constrai@s on Here pi, is the neutralino mass density, pc is the criti- (mo,mg) are very modest. A similar analysis holds for cal mass density to close the imiverse and h = H/(100 fi > 0 as shown in Fig. 2(f), except that here the con- km/sMpc) where H is the Hubble constant. Astronom- straints on (mo, ma) are generally less severe. ical observations indicate h =0.4-0.8 which results in a We analyze now the effect of the implications of the spread of value for S$,h’. For our analysis we use a predicted experimental accuracy of a, (- 4 x lo-lo) to mixture of cold and hot dark matter (CHDM) in the ra- be attained in the Brookhaven experiment for supergrav- tio &DM : &DM = 2 : 1 consistent with the data from ity grand unification. Of course, the present theoretical the Cosmic Background Explorer (COBE). Assuming to- uncertainty in the hadronic contributions to a,,, mostly tal 0 = 1 as is implied by the inflationary scenario and arising from the hadronic vacuum polarization effects as using for the baryonic matter 02~ = 0.1 we get 136,391 discussed earlier, limits the usefulness of such a precise measurement of a,. The hadronic uncertainty arises from 0.1 5 n&v 5 0.4 (22) the uncertainty in the computation of [4]

The combining effects of the b + sy constraint and otot(e+e- --f hadrons) R(s) = relic constraint put severe limits on the parameter space o(e+e- --f @/A-) (24) [36,40,41]. Their effect on I$“~~ is shown in Figs. 2(c) and Z(d) which are similar to Figs. 2(a) and 2(b) except for the low-energy tail of (e + e- + hadrons) cross sec- for the inclusion of the combined effects of b + sy con- tion. Ongoing measurements in VEPP-2M together with straint and relic constraint. Comparison of Figs. 2(a) future experiments in DA@NE, BEPC, etc., are expected and 2(c) and similarly of Figs. 2(b) and 2(d) show that to reduce this hadronic uncertainty to a significant ex- typically max j+~“syI for large tanp (i.e., tanp 2 2) is tent, perhaps by a factor of 4 or more, enhancing corre- reduced by about a factor of $ for gluino masses below spondingly the usefulness of the precision measurement the dip under the combined effects of b + ST and dark of a,. matter constraints. However, as is obvious, the most We present here two analyses. For the first analysis striking effect arises due to the appearance of the dip it- we used the more optimistic estimate of [18] where the self. The existence of such a dip was first noted in [36] authors made an improved evaluation of R(s) and ok:“” and is due to the relic density constraint. It is caused through the use of global analytical models of pion and by the rapid annihilation of neutralinos nex ,the Z pole kaon form factors in addition to the use of a better experi- which reduces the relic density below the lower limit in mental input of the three-pion e+e- annihilation data in Eq. (22) and hence part of the parameter space gets elim- comparison to previous determinations [3]. This corre- inated due to this constraint. In order to get the correct sponds to case A of Table I and gives (setting Rb = 0) position and depth for this dip one must use the accurate the result method 139,421 for calculation of the relic density. The analysis of Figs. 2(c) and 2(d) show that it would be dif- azM = 11659’183(7.6) x 10-l’ (25) ficult to discern SUSY effects in the (g - 2), experiment for gluino masses around the dip or correspondingly for For the second analysis we shall make a comparative neutralino masses of rni - Mz/2. (There is a similar study over an assumed range of errors which includes dip at rni - rn,,/2 which does not appear ,in the graphs analyses of both cases A and B of Table I as well as the because the remaining parameters have been allowed to possibility of even more constrained errors. range over the full space, but would be manifest once the In order to analyze the effect of the predicted accu- Higgs boson mass is fixed.) racy level of the Brookhaven experiment on the models Interestingly, as already noted in [ll] even the cur- of our discussion we have assumed that the experiment rent limits’ on (g - 2), including the present, experi- will not discard the standard model result within its 2a mental and theoretical errors place some constraint on uncertainty limit. As in the analysis of the current ex- the parameter space of supergravity grand unification. periment above we ascribe any new physics to supersym- Ascribing any new physics to supersymmetry by using metry by using o~“sy + oEM = oFpt and constrain the .p,Y + g” = ,y,t one may constrain the parameter parameter space of our model in the (mo - mg) plane space of the model in the (mo - ma) plane for different for different tan@ with a consideration of all possible At. tan/3 with a consideration of all possible At. We have Following the same procedure as before, we combine the combined the uncertainty of theoretical estimate and the uncertainty of theoretical estimate and the expected ex- current experimental uncertainties in quadrature to find perimental accuracy level of 4 X IO-lo in quadrature. In that our first analysis we use Eq. (25) as the theoretical input and a.s~ume that the predicted accuracy in the experi- -11.7 x lOKg < .;“sy < 22.5 x lO@ (23) mental determination of a@ will be achieved, we then determine the constraints on the SUSY particle spectrum Figure 2(e) exhibits for the case /I < ~0 the excluded if o~“sy lies within the 20 of the combined theoretical regions in the (mo - rni) plane for different tan@ val- and experimental error. In Figs. 3(a)-3(f) we exhibit the ues where the excluded domains lie below the curves. regions of the (rnti1 -ms*) plane which will be excluded We note that the excluded domains depend strongly on (dark shaded region) if (L~“~Y lies within the 2~ limit. 1654 UTPAL CHA’ITOPADHYAY AND PRAN NATH

(4 (4

rn ,,,=170 GeV

100 100 *w 800 m-charginol (GeV) rn-chargino 1 (GeV)

CbI (4

FIG. 3. (a) Display in the (rn+> - mu) plane of the allowed (shown in dots), disallowed (shown in squares), and partially allowed (shown in cross) regions corresponding to the 2u limit of case A of the theoretical evaluation of a,, and the predicted level of accuracy of the Brookhaven experiment, for the case tanp = 5 and w < 0. The white area between the excluded region and rnwl axis remains inaccessible due to the existence of a lower limit of sneutrino mass rnc* [36]. (b) Same as (a) except that p > 0. (c) Same as (a) except that tan/3 = 10. (d) Same as (c) except that p > 0. (e) Same as (c) except that tanP = 30. (f) Same as (e) except that p > 0.

We also exhibit the regions which will be partially ex- (rn,, ,- rn;) parameter space if the expected sensitivity of cluded (light shaded area) because a significant part of the measurement of a, is reached for large tanp +&es the parameter space is eliminated by the constraint, and [see Figs. 4(e) and 4(f)]. Of course a significant reduction the allowed region (dotted area). In addition, there is of the uncertainty in uzM which one expects to be pos- a region (white space) which is inaccessible due to the sible in the near future will more stringently constrain existence of a lower limit of sneutrino mass (ms.) [36]. the parameter space when combined with the expected Next we give a comparative analysis of the constraints sensitivity of the Brookhaven experiment. We have also for ca~es A and B of Table I and also for a case C where carried out a similar analysis for the aSUsY for the 7 lep- the error is reduced by factor of 4 over case B [see in ton. This gives max Ion”“’ 1 - 1.0 x 10-b for tano = 20. this context the analysis of [37] which gives a new eval- Even for this large value of tan@ the predicted value of u&ion of hadronic contributions of 699(4.5)x1O-1o]. In aSUsY is too small to observe with present experimental Figs. 4(a) and 4(b), we give a composite display of the ox- accuracy [43]. eluded regions in the (mo -rni) plane for the three cases. We observe that the forbidden region increases in propor- tion to the decrease in the combined error of theory and experiment. Further, the excluded region increases with IV. CONCLUSION increasing value of tano. Thus if the combined error de- creases by a factor of 4 (case C) and no effect beyond In this paper we have presented an analysis of (g - 2n is seen, then the 9 - 2 experiment will exclude most 2)Ps”sy within the framework of supergravity grand uni- of the region in the (ti0 - ms) plane for large tano as fication under the constraint of radiative breaking of the can be seen from Figs. 4(e) and 4(f). In fact even with electroweak symmetry, and the constraint of b -+ s’y the presently large uncertainty in the theoretical values and dark matter. One finds that over most of the pa- of os” one will be able to exclude a significant part of the rameter space the chiral interference light chargino part 53 PROBING SUPERGRAVlTY GRAND UNIFICATION IN THE. 1655

rn,,=170 GeV

400 Mx) 800 mw mgltio (GeV) Cb) (4

FIG. 4. (a) Excluded regions (regions enclosed by the curves towards the origin) for tano = 5 and p < II in the 20 limit, after combining in quadrature the predicted Brookhaven experimental uncertainty and different levels of uncertainty of theoretical estimates corresponding to cases A, B, and C as discussed in the text. (b) Same as (a) except that p > 0. (c) Same as (a) except that tanp = 10. (d) Same as (c) except that fi > 0. (e) Same as (a) except that tanp = 30. (f) Same as (e) except that !.d> 0.

of azusy dominates and imparts a signature to ayy. ACKNOWLEDGMENTS Thus as a rule one finds ap,” > 0 for /I > 0 and a~“~’ < 0 for fi < 0 with very few exceptions when We wish to acknowledge useful discussions with R. tanP - 1. At the qualitative level it is shown that the Amowitt, W. Marciano, and W. Worstell. This re- expected experimental sensitivity of (g - 2), measure- search was supported in part by NSF Grant Nos. PHY- ments combined with the expected reduction of error in 19306906 and PHY94-07194. (g - 2):” [by a 0(1/4)] will exclude the mo, rn~ parameters in the domain mo 5 700 GeV, mg 5 1 TeV for large tan/3(2 20) and stringently constrain the param- APPENDIX eter space for lower tan@ With the same assumptions one will be able to probe the domain mo 5 400 GeV, The chargino masses rn@<, = IX& i = 1,2, where XI,2 rn6 5 700 GeV for tanp > 10. The constraint becomes are the eigenvalues of the chargino mass matrix and are less severe for smaller values of tano. However, even for given by tan/3 = 5, the SUSY reach of the new experiment will be very substantial [see Figs. 4(a) and 4(b)]. Thus one X&Z = $([4vi + (jL - 7&)2]‘/2 finds that the Brookhaven experiment coupled with the T[4yt + (!J + 7j12)y2) , corroborating experiments and analyses designed to re- (Al) duce the hadronic error will complement SUSY searches where at colliders and provide an important probe of the parameter space of supergravity grand unification especially for W) large tanp. 1656 UTPAL CHATTOPADHYAY AND PRAN NATH 53 and T& is obtained from the relation &, = [aa(Mz)/cr~(M~)]m~ where a&!&), a = 1,2,3, are the SU(3), SU(2), and U(1) gauge coupling constants at the Z boson mass. (A5) The neutralino masses mi(k) = l&l where Xk are the eigenvalues tf the neutralino mass matrix which in the (W3, 2, I?*, HZ) basis reads 1121

/TL 0 a b\ where ml,2 = [ac/as(Mz)]q, L%C = LYC/~?(, f&) = t(2 + P&)/(1 + Pkt)’ with @,+ = &(33/5,1, -3) and t = 2 ln(Mc/Q) at Q = Mz. The mixing angle which describes the left-right mixing for the smuons is determined by the relations where a = M~cosB~cos~, b = -MzcosBwsin@, c = -MzsinBwcosp, and d = Mzsin6’wsinp while 2m,(A, - P W? sin26 = W) the quantities Bk that appear in Eq. (16) are defined by (TC? -+xX)” + 4m;(A, - ~cotfl)~ & = 0 (1) for Xk > 0 (< 0). The quantities O;j are the elements of the orthogonal matrix which diagonalizes the and neutralino mass matrix. Smuon masses are given by

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