- Email: [email protected]
B0 − B̄0 mixing and related CP violation in supersymmetric models
Volume 194, number 4
PHYSICS LETTERS B
20 August 1987
B ° - B ° M I X I N G A N D R E L A T E D CP V I O L A T I O N I N S U P E R S Y M M E T R I C M O D E L S
S. B E R T O L I N I Physics Department, Carnegie-Mellon University, Pittsburgh, PA 15213, USA
F. B O R Z U M A T I ~ and A. M A S I E R O 2 Physics Department, New York University, New York, NY 10003, USA
Received 4 May 1987
We analyze the supersymmetric contributions to B°-l)d° mixing. For squark and gluino masses which respect the bounds coming from the CERN pp collider, the supersymmetric contribution turns out to be substantially smaller than the corresponding standard model prediction. However, we suggest that the presence of supersymmetry may sizeably contribute to the observability of large CP asymmetries related to the mixing.
One of the most surprising experimental results o f this last period concerns the large Bo-Bd o -o mixing that has been announced by the A R G U S collaboration [ 1]. The mixing that they find is well above the previous theoretical expectations which were based on the standard model (SM) with three families and m r ~ 4 5 GeV [ 2 - 4 ] . The first obvious question is then: can such a large mixing be a c c o m m o d a t e d in SM? The three relevant input parameters which enter the computation of Bd-Bd o - o mixing in SM are the mass of the top quark, the entry Ftd o f the Kobayashi - M a s k a w a ( K M ) matrix and, finally, the A B = 2 hadronic wavefunction which can be expressed in terms o f Bf~, where B = 1 corresponds to the "vacuum saturation" and fa is the B meson decay constant characteristic o f the pure leptonic decays. Two papers [ 5 ] have recently discussed the allowed ranges for P'~aand x/BfB in order to infer a lower bound on mr. Only for extreme values o f these parameters can m, be as low as 60 GeV, in order to accommodate a 20% mixing. For more "central" values of V~d and x/~fB a lower bound m, >~80-100 GeV is obtained. These drastic implications for SM motivate cont Address after September 1, 1987: International School for Advanced Studies, 1-34100 Trieste, Italy. 2 Address after September 1, 1987: INFN, Sezione di Padova, 135131 Padua, Italy.
sideration of possible extensions o f SM with three generations as candidates for an enhancement o f 0 -0 Bd-Bd mixing. In particular, as one might conceivably expect, the introduction o f a fourth generation can play such an enhancing role. Here we examine whether the presence of low energy supersymmetry (SUSY) can significantly modify the SM prediction for Bd-Bo o -o mixing. A second issue which now deserves a careful analysis is the possibility o f observing CP violation in the B system. Indeed, in the light o f the presently acertained large B-B mixing, the option o f sizeable onshell CP asymmetries involving the mixing gets a renewed interest [ 6 ]. In the second part of this paper we shall c o m m e n t on the appealing possibility that some CP violating decay channels which involve 0 -0 Bo-Bo mixing, may be enhanced in the presence o f low energy SUSY. Our results can be summarized as follows: (i) for gluino (~) and squark (~) masses which respect the bounds coming from the C E R N UA1 data (i.e., m~.~ > 6 0 - 7 0 GeV, if one assumes the photino to be the lightest superpartner) the SUSY contribution to 0 -0 Bd-Bd mixing is always sizeably smaller than the SM prediction (for instance for m , = 1 0 0 GeV and rhg ~rh,---65 GeV, the mass difference Am in the 0 -0 Bd-Bo system receives a SUSY contribution which is one order of magnitude smaller than the SM con-
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
545
Volume 194, number 4
PHYSICS LETTERS B
tribution); (ii) SUSY can provide, however, a substantial contribution to the CP violating asymmetry tr(B°l)°~KsX~-Y) - tr(B°B°--,KsX~+Y'), where KsX constitutes a CP eigenstate; indeed, this asymmetry can be as large as 47%. Combining this result with SUSY branching ratios possibly at the level of 10-3 provides sound hope for experimental detectability in future machines. The mixing r d = - F ( B ° ~ f ~ + X ) / F ( B ° - - . f ~ - X ) is given by [ 7 ] (Am)2 + (AF/2) 2
(1)
where x = A m / F and F - ~ = z a is measured to be (1.11 _+0.14) × l 0 - ~2 s. The final expression of rd in (l) is obtained by taking into account that (AF/2)2'~ (Am) 2 [3].
The mixing rd in ( l ) can be determined experimentally by measuring the proportion of like-sign dilepton events at the Y(4s): rd=
N(+ +)+N(--) N(+-)
(2)
The ARGUS collaboration has indeed observed rd to be 0.234+0.067+0.031 [1]. In SM, the off-diagonal element M~2 of the B°B ° mass matrix, which gives rise to Am, is computed from the usual A B = 2 box diagrams with two W boson and two Q = 2/3 quark propagators in the loop. From the detailed study of the analogous supersymmetric contribution in the K ° - I ( ° case [8-10], we know that the dominant diagrams are given by gluino and Q = - 1/3 scluark exchanges in the loop (fig. 1 ). The key element in the construction of these SUSY contributions is the fact that gluinos can couple to quarks and squarks belonging to different generations [ 11 ]. The origin of this peculiar source of flavour-changing neutral currents (FCNC) in SUSY can be traced back to the different renormalization of the quark and squark mass matrices from the scale of supergravity breaking down to the S U ( 2 ) L × U ( 1 ) breaking scale. In the class of models based on spontaneously broken N = 1 supergravity ~, where canonical kinetic terms for the scalars are present in the original N = 1 supergravity lagrangian (the so-called "minimal" models), the mixing matrix which "~ See ref. [12] for a review.
546
bL
~L
"
..... t
dL
~iL
dL
Pt
bL
P4
P2
' PS
(b) .bL,
,
,,dL
I I
I I
dL
~L
.dL I
I
I
=
bL
(c)
x2
rd = 2 F 2 + ( A m ) 2 _ ( A F / 2 ) 2 " 2 + x 2 ,
20 August 1987
(d)
Fig. 1. Box diagrams contributing to B°-B ° mixing in minimal SUSY models. The crossed diagrams are allowed because of the Majorana nature of the gluino. The graphs on the second line, (c) and (d), appear with a minus sign relative to the graphs (a) and (b).
appears at the g--qL--qLvertices essentially coincides with the standard KM matrix. The G I M cancellation among the contributions of the u, c and t quarks in the internal lines in SM is here replaced by a superG I M cancellation of the dL, gL and bL SCluarks that circulate in the loop. In order to be more quantitative, let us take a closer look at the squark mass matrix. In particular, the 3 × 3 mass matrix of the scalar partners of the left-handed quarks is of the form rh 2t = m 2 + m ~ m o + cm~ m u ,
(3)
where mD (too) is the Q = - 1/3 (2/3) quark mass matrix and m is the scale of low energy global SUSY breaking. The parameter c can be computed by solving the set of renormalization group equations which determine the evolution of the SUSY parameters in the soft breaking potential from the scale of local SUSY breaking down to the Fermi scale. For m ~ row, i.e. the region of SUSY masses of interest in our present discussion, and m t between 50 and 100 GeV, making use of the results of ref. [ 9 ] we find that c varies between - 0.7 and - 0.4 (let us recall that Ic I increases with m and decreases with increasing m r ) . Notice that the 3 × 3 block of the Q = - 1/3 squark mass matrix which refers to the right-handed down squarks gets a contribution ~ c m ~ m D and therefore no FCNC effects occur in this sector. In addition, since the off-diagonal m"2DLDRblock is proportional t o mo, we can safely neglect the I~L--l~ R mixings (in the case of the B meson no enhancement for L - R
Volume 194, number 4
PHYSICS LETTERSB
currents originates in the evaluation of the box matrix elements). In conclusion, the leading SUSY contribution arises when all the external quarks in the box diagrams are left-handed, which is at all analogous to the SM case. Given the expression (3) of the squark mass matrix there are only two extra parameters that enter the computation of the SUSY contribution: m and r~g. Indeed ,,,~2b- ,,,d'z'2'~cm2-- and r~2b,~mZ+cm2; therefore once we fix r~b, for instance, rhd is automatically determined for a given value of m~. The SUSY contribution to Am arises from the four diagrams in fig. 1. In particular, the contributions of the first two diagrams (figs. la, lb) and the last two (figs. lc, ld) have a relative minus sign. Indeed, they derive from contractions which differ by a fermion field exchange: d or b (this is the analogue of the relative minus sign between the s and t channels, for instance, in positron-electron elastic scattering). Explicit calculation of the four diagrams of fig. 1 leads, after a convenient color and spinor Fierz rearrangement, to
20 August 1987
erators normalized to Tr( TaT b) = ½0ab, Vij are the elements of the KM matrix and .%, )7, - m~ 2, / m ~ 2g ( i = d, s, b). The functionsf(x, y) and g ( x , y) are given by X ( ~ - x ~ In x - ( y _Y1 )-----~l n y f ( x , y ) = y__~l
x_l +
g(x,y)=x-~y[(x-f---~)2lnx-(yY~_l)2
x- 1 +
lny
"
(9)
Expressions forf(x, y) and g ( x , y) for some limiting values of the arguments are given in ref. [ 13 ]. In particular we have f(1, 1) = 1/6 and g(1, 1) = 1/3. By subtracting (6) and (7) from (4) and (5) we obtain
Mlz = - ( ot21r~) ~b l~jdl:,~, V*d
M ~] ) = - ( a~ l rh ~) V:, V~jdV,b V~mg(~j, )7,) )< { [
)< ( ~ S +
hO),
~ g( xj, )vJ) -- ~ f ( xj, )vi) ] S
(4)
+ [ ~g(£j, jV,)+ -~f(£j, )7,)] O}. M i ~ ) = - (a,z/,"h~) V:,, V*d V,-,, V'd f ( % , )73
x(-~s+~8o),
(5)
MI~) = - ( as2 ~rag) - 2 Vj~, Vjd , V,~, V~mg(xj, ~ )7i)
x(-~S-~O),
M.•, .2
= - - ( o q l m2g )
. -2 ~bl].jdVibV.idf(.~i,)Ti )
X(~S-]O),
(7)
S ~ [Od(P2)vl, PLVb(--p3)]
O = [ f-ld(P2)}'uTael-Vb( --P3)]
)< [Vd( --P4)~" TaPLUb(P, )] .
[0)
)< ( O l v d ( - - P 4 ) Y u P L U b ( P l ) l B ° ) - - m~ lf 2 a m B 2 .
(8a)
(11 )
We are left now with the evaluation of the matrix element (~°lM~21B°). If we assume the vacuum dominance, there are no contributions from the "octet" component in (10) and the color singlet operators give
( BO [tla(p2)YuPLVb(--P3)
where
X [Vd(--P4)7"PLUb(P,)],
The B°-l] ° eigenstate mass difference is related to the mass matrix elements M o by the relation m 2 _ m ~ = - 2 Re(B°lM~z]B°), from which it follows that Aml2 - - (l/ma) R e ( B ° IMl2 IB ° ) •
(6)
(10)
(12)
By implementing (12) and (10) into (11 ) we finally obtain: ~2 1 Amsusv --- (as2 ~rag) gB f B rn8 ,
(8b)
PL= (1 --?5)/2, T ~ ( a = 1..... 8) are the S U ( 3 ) c gen-
)
(13) 547
Volume 194, number 4
PHYSICS LETTERS B
where the coefficient B' equals 1 for the vacuum saturation estimate and we have defined
E(x, y) - ~g(x, y) - ~f(x, y) .
(14)
20 August 1987 I0
,
,
,
E
Taking now into account that, in the case of the B meson, the contribution from the third generation (s)quarks is dominant, we can write Amsusv = ( ot~/r~ 2) ~B' fEamB × R e ( Eb Vt*d)2Esusv0~b, )~b) ,
(15)
,6'30
40
50 60 m'~ = m~'(GeV)
7b
80
where /~susv (~b,)~b) -- E()~b, )~b) - 2E(~b, ~d) + E ( $ d , ~d) •
(16)
It is worth tO notice that (16) is symmetric in the interchange ~b '--' ~d or, in other words, does not depend on the sign of the parameter c, as it is expected from the presence of the double GIM. In order to compare (15) with the SM prediction, we use the result of ref. [ 14]. We have checked explicitly that the expression given in ref. [ 14] (for the kaon case) is consistent with ours ~2. By taking B' ~-B, we find Amsusv AmSM
3
as
mw
/~S.=_U Sy ( )~b, -~b ) E s M ( X t , Xt) '
(17)
where x , = m~/m~v and
F.sM(X,X)-2\X_I] (191 +x
4x-1
lnx 3 1 ) 2 (x--l) 2 "
(18)
In fig. 2 we report the values of Amsusv/AmsM, taking r~g = rhb (the lightest "down" squark), for two representative values of mr, m t = 60 and 100 GeV. It is clear that for r ~ _ r~b > 60 GeV, the SUSY contribution to Am is negligible in comparison to AmsM. The above result may seem rather unexpected at ~2 It is worth to notice that, although our derivation is consistent with the SM result obtained by the method of the effective lagrangian, we are at variance, in the numerical coefficients, with previous estimates of the KL-Ks mass difference in SUSY models. Numerically however, the final results turn out to be comparable.
548
Fig. 2. The ratio Amsvsv/Amsu for the B° - ~ ° system is shown as a function of the gluino and the lightest squark masses (taken here to be equal for the purpose of illustration), for representative values of mt.
first sight. Even for values of squark and gluino masses comparable to row, one would expect the SUSY contribution, being the suppression due to the loop integration of the same order, to be enhanced by a factor O(a2/a2w ) - 10 with respect to SM. What is neglected in this naive estimate are two factors. First, the double G I M mechanism evidentiates here the stronger suppression present in SUSY. Indeed, a direct comparison of/~sus¥ versus/~SM in (17) shows that the double G I M suppression, which is typical of the box diagrams, is much stronger in the SUSY case than in SM. This can be related in part to the extra factor ~ c 2 present in SUSY and in part to the fact that for a fixed m i2- r n j ,2 when one considers the exact functional dependence, the G I M suppression becomes stronger as m/m:-, 1. Already for squarks in the rnw range this makes a difference with respect to SM, which is here enhanced by the presence of the double GIM. Secondly, one has to take into account the effect of destructive interference in (14) between the crossed and uncrossed diagrams (we recall that in SM crossed diagrams are forbidden by charge conservation). In particular, it is peculiar that, when rhg increases, the behaviour o f g ( x , y) a n d f ( x , y) is different, since the r~g dependence o f f ( x , y) is softened by the double mass insertion present in the crossed diagrams, figs. 2b, 2d. As a consequence the effect of destructive interference becomes stronger. It is indeed because of this effect, that the value of the ratio Amsusv/AmsM drops so rapidly, about one order of magnitude, when gluino and squark masses
Volume 194, number 4
PHYSICS LETTERSB
are brought from 40 to 60 GeV #3. It is worth to remark that this cannot be cured by increasing the top mass. In general one should not expect a peculiar enhancement of the SUSY contributions over the SM ones due to very heavy rot. Indeed, the flavour changing suppression in the minimal SUSY models here considered is proportional to cm2t and the parameter c decreases with increasing mt. Therefore the behaviour of flavour-changing SUSY amplitudes is considerably softened, for increasing mr, with respect to the corresponding behaviour of the SM contributions to processes with "hard" GIM suppression. We did not consider in (I 7) the effects of higher order QCD corrections [ 3 ] since they do not change our conclusions. Indeed, in the SUSY case, due to the aforementioned degeneracy of the squark masses in the loop, we expect QCD corrections not to play any substantial role. Finally, let us remark that the results here presented for Amsusv/AmsM hold as well for B°-l] ° mixing. Before concluding this paper however, let us mention the possibility that supersymmetry may help to evidentiate large CP asymmetries related to the mixing ~4. On this respect, we recall that in the class of minimal models here considered, no large CP violating phases appear beyond the standard KM phase (for a discussion on the CP properties of minimal supergravity models, see ref. [ 16 ]). However, supersymmetry may contribute indirectly to the observation of CP violation effects through the enhancement of decay rates involved in specific CP asymmetries. In previous papers we pointed out that potential large SUSY enhancements may be present in b~s), [ 17] and b-~s gluon [ 13] decays. In particular, for the latter process, branching ratios as high as 10-20% can be obtained even with squark and gluino masses of 70-80 GeV and m, = 80 GeV. This could help in evidentiating a CP asymmetry of the form a(B°B°-*KsX~-Y)- a(B°B°-~KsX~+Y'), with KsX a CP eigenstate. Indeed, if we call f a final state common to both ~3Crossed diagrams, for instance, were neglected in the analysis of ref. [ 15]. where only light gluinoswere considered. ~4Wethank A. Sanda for a stimulating discussion on this point.
20 August 1987
B° and ~o (in addition we will assume f to be a CP eigenstate), it can be shown [ 18,19 ] that there may exist a time-integrated asymmetry fi
cP) +F(~O ~fce)
F( B ° ~ f ) - F ( B ° ~ f
f=
~
l+e 2 x Im {~]--~_Epf] - - 1 + [pf[ 2 I + X 2
(19)
where pf=A(f3°~f)/A(B°-~f), x=Am/F and ~ is defined in analogy with the kaon system. The decay we want to consider is b--,s gluon, leading, at the exclusive level, to final states KsX which are, neglecting the CP impurity in Ks, CP eigenstates containing a net strangeness. In this case we have t~ X f= l ~ x 2
im(( VtbV'd)2 ( Fib Vl*s)2 "~ \ ] ~ - ] ~ T ] -
(20)
For a mixing ra = 20% we find that c~fcan be as large as 47% (it is convenient to use here the Wolfenstein parametrization [ 20] of the KM matrix). Even taking into account that exclusive few-body channels, like Ksn°l] or Ks00, may be at the level of 1% of the inclusive b~sg, these branching ratios could nevertheless be, in SUSY models, as high as few parts times 10 -3, allowing for the possibility of detecting this asymmetry with ~ O ( 1 0 7) B°]3° pairs. It is worth to remark that, from the general considerations on the possible mechanisms of SUSY enhancement of weak meson decays that we have presented in refs. [ 13,17 ], CP asymmetries, in particular involving semileptonic or leptonic decays. Given the results on the B°-B ° mixing and related CP violation here presented, it seems to us that the decay b--,sg together with the analogous radiative decay b~sT, provide at present two of the most sensitive tests for a SUSY search in low energy physics. Observation ofb--,sg at the 10% level or b~s), at the 10 3 level would indeed provide a strong indication in favour of SUSY. On the other hand, in the event that no signals are observed, we think that the physics of the B mesons will still provide, before Tevatron reaches its full potentiality, the most stringent constraints on squark and gluino masses. We are grateful to P. Rakow, A. Sanda and U Wolfenstein for useful discussions. The work of S.B. 549
Volume 194, number 4
PHYSICS LETTERS B
is partially s u p p o r t e d by the U S D e p a r t m e n t o f Energy. T h e w o r k o f F.B. a n d A . M . is s u p p o r t e d in part by the N S F u n d e r G r a n t no. P H Y 8 1 1 6 1 0 2 .
References [1 ] ARGUS Collab., H. Albrecht et al., DESY preprint, in preparation, reported by H. Albrecht at Intern. Symp. on the Fourth family of quarks and leptons (Santa Monica, CA, February 1987). [ 2 ] J. Ellis, M.K. Gaillard, D.V. Nanopoulos and S. Rudaz, Nucl. Phys. B 131 (1977) 285. [3] J.S. Hagelin, Nucl. Phys. B 193 (1981) 123. [4] I.I. Bigi and A.I. Sanda, Phys. Rev. D 29 (1984) 1393; V. Barger and J.N. Phillips, Phys. Lett. B 143 (1984) 259; A. Ali and C. Jadskog, Phys. Lett. B 144 (1984) 266; T. Brown and S. Pakvasa, Phys. Rev. D 31 (1985) 1661; A. Ali, preprints DESY-85-107, DESY-86-108. [5]J. Ellis, J.S. Hagelin and S. Rudaz, preprint CERNTH4679/87 (April 1987); I.I. Bigi and A.I. Sanda, SLAC preprint, in preparation. [6] I. Dunietz and J. Rosner, Phys. Rev. D 34 (1986) 1404; I.I. Bigi and A.I. Sanda, Nucl. Phys. B 281 (1987) 41. [7] A. Pais and S.B. Treimann, Phys. Rev. D 12 (1975) 2744. [8] M.J. Duncan and J. Trampetic, Phys. Lett. B 134 (1984) 439; J.M. Gerard, W. Grimus, A. Raychaudhuri and G. Zoupanos, Phys. Lett. B 140 (1984) 349;
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J.M. Gerard, W. Grimus, A. Masiero, D.V. Nanopoulos and A. Raychaudhuri, Phys. Lett. B 141 (1984) 79; Nucl. Phys. B253 (1985) 93; R. Langacker and R. Sathiapalan, Phys. Lett. B 144 (1984) 401. [9] A. Bouquet, J. Kaplan and C. Savoy, Phys. Lett. B 148 (1984) 69. [ 10] M. Dugan, B. Grinstein and L. Hall, Nucl. Phys. B 255 (1985) 413. [ ! 1] M.J. Duncan, Nucl. Phys. B 221 (1983) 285; J.F. Donoghue, H.P. Nilles and D. Wyler, Phys. Lett. B 128 (1983) 55. [ 12] H.P. Nilles, Phys. Rep. 110 (1984) 1. [ 13] S. Bertolini, F. Borzumati and A. Masiero, preprint CMUHEP03-87; preprint NYU/TH03/87 (March 1987). [ 14] T. Inami and C.S. Lira, Prog. Theor. Phys. 65 (1981) 297. [ 15 ] M.B. Gavela, A. Le Yaouanc, L. Oliver, O. Pene, J.C. Raynal, M. Jar[] and O. Lazrak, Phys. Lett. B 154 (1985) 147. [16] M. Dugan, B. Grinstein and L. Hall, Nucl. Phys. B 255 (1985) 413. [ 17] S. Bertolini, F. Borzumati and A. Masiero, Phys. Lett. B 192 (1987) 437. [ 18] A.B. Carter and A.I. Sanda, Phys. Rev. Lett. 45 (1980) 952; Phys. Rev. D 23 (1981) 1567; I.I. Bigi and A.I. Sanda, Nucl. Phys. B 193 (1981) 85. [ 19] L. Wolfenstein, Nucl. Phys. B 246 (1984) 45. [20] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945.
PHYSICS LETTERS B
20 August 1987
B ° - B ° M I X I N G A N D R E L A T E D CP V I O L A T I O N I N S U P E R S Y M M E T R I C M O D E L S
S. B E R T O L I N I Physics Department, Carnegie-Mellon University, Pittsburgh, PA 15213, USA
F. B O R Z U M A T I ~ and A. M A S I E R O 2 Physics Department, New York University, New York, NY 10003, USA
Received 4 May 1987
We analyze the supersymmetric contributions to B°-l)d° mixing. For squark and gluino masses which respect the bounds coming from the CERN pp collider, the supersymmetric contribution turns out to be substantially smaller than the corresponding standard model prediction. However, we suggest that the presence of supersymmetry may sizeably contribute to the observability of large CP asymmetries related to the mixing.
One of the most surprising experimental results o f this last period concerns the large Bo-Bd o -o mixing that has been announced by the A R G U S collaboration [ 1]. The mixing that they find is well above the previous theoretical expectations which were based on the standard model (SM) with three families and m r ~ 4 5 GeV [ 2 - 4 ] . The first obvious question is then: can such a large mixing be a c c o m m o d a t e d in SM? The three relevant input parameters which enter the computation of Bd-Bd o - o mixing in SM are the mass of the top quark, the entry Ftd o f the Kobayashi - M a s k a w a ( K M ) matrix and, finally, the A B = 2 hadronic wavefunction which can be expressed in terms o f Bf~, where B = 1 corresponds to the "vacuum saturation" and fa is the B meson decay constant characteristic o f the pure leptonic decays. Two papers [ 5 ] have recently discussed the allowed ranges for P'~aand x/BfB in order to infer a lower bound on mr. Only for extreme values o f these parameters can m, be as low as 60 GeV, in order to accommodate a 20% mixing. For more "central" values of V~d and x/~fB a lower bound m, >~80-100 GeV is obtained. These drastic implications for SM motivate cont Address after September 1, 1987: International School for Advanced Studies, 1-34100 Trieste, Italy. 2 Address after September 1, 1987: INFN, Sezione di Padova, 135131 Padua, Italy.
sideration of possible extensions o f SM with three generations as candidates for an enhancement o f 0 -0 Bd-Bd mixing. In particular, as one might conceivably expect, the introduction o f a fourth generation can play such an enhancing role. Here we examine whether the presence of low energy supersymmetry (SUSY) can significantly modify the SM prediction for Bd-Bo o -o mixing. A second issue which now deserves a careful analysis is the possibility o f observing CP violation in the B system. Indeed, in the light o f the presently acertained large B-B mixing, the option o f sizeable onshell CP asymmetries involving the mixing gets a renewed interest [ 6 ]. In the second part of this paper we shall c o m m e n t on the appealing possibility that some CP violating decay channels which involve 0 -0 Bo-Bo mixing, may be enhanced in the presence o f low energy SUSY. Our results can be summarized as follows: (i) for gluino (~) and squark (~) masses which respect the bounds coming from the C E R N UA1 data (i.e., m~.~ > 6 0 - 7 0 GeV, if one assumes the photino to be the lightest superpartner) the SUSY contribution to 0 -0 Bd-Bd mixing is always sizeably smaller than the SM prediction (for instance for m , = 1 0 0 GeV and rhg ~rh,---65 GeV, the mass difference Am in the 0 -0 Bd-Bo system receives a SUSY contribution which is one order of magnitude smaller than the SM con-
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
545
Volume 194, number 4
PHYSICS LETTERS B
tribution); (ii) SUSY can provide, however, a substantial contribution to the CP violating asymmetry tr(B°l)°~KsX~-Y) - tr(B°B°--,KsX~+Y'), where KsX constitutes a CP eigenstate; indeed, this asymmetry can be as large as 47%. Combining this result with SUSY branching ratios possibly at the level of 10-3 provides sound hope for experimental detectability in future machines. The mixing r d = - F ( B ° ~ f ~ + X ) / F ( B ° - - . f ~ - X ) is given by [ 7 ] (Am)2 + (AF/2) 2
(1)
where x = A m / F and F - ~ = z a is measured to be (1.11 _+0.14) × l 0 - ~2 s. The final expression of rd in (l) is obtained by taking into account that (AF/2)2'~ (Am) 2 [3].
The mixing rd in ( l ) can be determined experimentally by measuring the proportion of like-sign dilepton events at the Y(4s): rd=
N(+ +)+N(--) N(+-)
(2)
The ARGUS collaboration has indeed observed rd to be 0.234+0.067+0.031 [1]. In SM, the off-diagonal element M~2 of the B°B ° mass matrix, which gives rise to Am, is computed from the usual A B = 2 box diagrams with two W boson and two Q = 2/3 quark propagators in the loop. From the detailed study of the analogous supersymmetric contribution in the K ° - I ( ° case [8-10], we know that the dominant diagrams are given by gluino and Q = - 1/3 scluark exchanges in the loop (fig. 1 ). The key element in the construction of these SUSY contributions is the fact that gluinos can couple to quarks and squarks belonging to different generations [ 11 ]. The origin of this peculiar source of flavour-changing neutral currents (FCNC) in SUSY can be traced back to the different renormalization of the quark and squark mass matrices from the scale of supergravity breaking down to the S U ( 2 ) L × U ( 1 ) breaking scale. In the class of models based on spontaneously broken N = 1 supergravity ~, where canonical kinetic terms for the scalars are present in the original N = 1 supergravity lagrangian (the so-called "minimal" models), the mixing matrix which "~ See ref. [12] for a review.
546
bL
~L
"
..... t
dL
~iL
dL
Pt
bL
P4
P2
' PS
(b) .bL,
,
,,dL
I I
I I
dL
~L
.dL I
I
I
=
bL
(c)
x2
rd = 2 F 2 + ( A m ) 2 _ ( A F / 2 ) 2 " 2 + x 2 ,
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(d)
Fig. 1. Box diagrams contributing to B°-B ° mixing in minimal SUSY models. The crossed diagrams are allowed because of the Majorana nature of the gluino. The graphs on the second line, (c) and (d), appear with a minus sign relative to the graphs (a) and (b).
appears at the g--qL--qLvertices essentially coincides with the standard KM matrix. The G I M cancellation among the contributions of the u, c and t quarks in the internal lines in SM is here replaced by a superG I M cancellation of the dL, gL and bL SCluarks that circulate in the loop. In order to be more quantitative, let us take a closer look at the squark mass matrix. In particular, the 3 × 3 mass matrix of the scalar partners of the left-handed quarks is of the form rh 2t = m 2 + m ~ m o + cm~ m u ,
(3)
where mD (too) is the Q = - 1/3 (2/3) quark mass matrix and m is the scale of low energy global SUSY breaking. The parameter c can be computed by solving the set of renormalization group equations which determine the evolution of the SUSY parameters in the soft breaking potential from the scale of local SUSY breaking down to the Fermi scale. For m ~ row, i.e. the region of SUSY masses of interest in our present discussion, and m t between 50 and 100 GeV, making use of the results of ref. [ 9 ] we find that c varies between - 0.7 and - 0.4 (let us recall that Ic I increases with m and decreases with increasing m r ) . Notice that the 3 × 3 block of the Q = - 1/3 squark mass matrix which refers to the right-handed down squarks gets a contribution ~ c m ~ m D and therefore no FCNC effects occur in this sector. In addition, since the off-diagonal m"2DLDRblock is proportional t o mo, we can safely neglect the I~L--l~ R mixings (in the case of the B meson no enhancement for L - R
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currents originates in the evaluation of the box matrix elements). In conclusion, the leading SUSY contribution arises when all the external quarks in the box diagrams are left-handed, which is at all analogous to the SM case. Given the expression (3) of the squark mass matrix there are only two extra parameters that enter the computation of the SUSY contribution: m and r~g. Indeed ,,,~2b- ,,,d'z'2'~cm2-- and r~2b,~mZ+cm2; therefore once we fix r~b, for instance, rhd is automatically determined for a given value of m~. The SUSY contribution to Am arises from the four diagrams in fig. 1. In particular, the contributions of the first two diagrams (figs. la, lb) and the last two (figs. lc, ld) have a relative minus sign. Indeed, they derive from contractions which differ by a fermion field exchange: d or b (this is the analogue of the relative minus sign between the s and t channels, for instance, in positron-electron elastic scattering). Explicit calculation of the four diagrams of fig. 1 leads, after a convenient color and spinor Fierz rearrangement, to
20 August 1987
erators normalized to Tr( TaT b) = ½0ab, Vij are the elements of the KM matrix and .%, )7, - m~ 2, / m ~ 2g ( i = d, s, b). The functionsf(x, y) and g ( x , y) are given by X ( ~ - x ~ In x - ( y _Y1 )-----~l n y f ( x , y ) = y__~l
x_l +
g(x,y)=x-~y[(x-f---~)2lnx-(yY~_l)2
x- 1 +
lny
"
(9)
Expressions forf(x, y) and g ( x , y) for some limiting values of the arguments are given in ref. [ 13 ]. In particular we have f(1, 1) = 1/6 and g(1, 1) = 1/3. By subtracting (6) and (7) from (4) and (5) we obtain
Mlz = - ( ot21r~) ~b l~jdl:,~, V*d
M ~] ) = - ( a~ l rh ~) V:, V~jdV,b V~mg(~j, )7,) )< { [
)< ( ~ S +
hO),
~ g( xj, )vJ) -- ~ f ( xj, )vi) ] S
(4)
+ [ ~g(£j, jV,)+ -~f(£j, )7,)] O}. M i ~ ) = - (a,z/,"h~) V:,, V*d V,-,, V'd f ( % , )73
x(-~s+~8o),
(5)
MI~) = - ( as2 ~rag) - 2 Vj~, Vjd , V,~, V~mg(xj, ~ )7i)
x(-~S-~O),
M.•, .2
= - - ( o q l m2g )
. -2 ~bl].jdVibV.idf(.~i,)Ti )
X(~S-]O),
(7)
S ~ [Od(P2)vl, PLVb(--p3)]
O = [ f-ld(P2)}'uTael-Vb( --P3)]
)< [Vd( --P4)~" TaPLUb(P, )] .
[0)
)< ( O l v d ( - - P 4 ) Y u P L U b ( P l ) l B ° ) - - m~ lf 2 a m B 2 .
(8a)
(11 )
We are left now with the evaluation of the matrix element (~°lM~21B°). If we assume the vacuum dominance, there are no contributions from the "octet" component in (10) and the color singlet operators give
( BO [tla(p2)YuPLVb(--P3)
where
X [Vd(--P4)7"PLUb(P,)],
The B°-l] ° eigenstate mass difference is related to the mass matrix elements M o by the relation m 2 _ m ~ = - 2 Re(B°lM~z]B°), from which it follows that Aml2 - - (l/ma) R e ( B ° IMl2 IB ° ) •
(6)
(10)
(12)
By implementing (12) and (10) into (11 ) we finally obtain: ~2 1 Amsusv --- (as2 ~rag) gB f B rn8 ,
(8b)
PL= (1 --?5)/2, T ~ ( a = 1..... 8) are the S U ( 3 ) c gen-
)
(13) 547
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where the coefficient B' equals 1 for the vacuum saturation estimate and we have defined
E(x, y) - ~g(x, y) - ~f(x, y) .
(14)
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,
,
,
E
Taking now into account that, in the case of the B meson, the contribution from the third generation (s)quarks is dominant, we can write Amsusv = ( ot~/r~ 2) ~B' fEamB × R e ( Eb Vt*d)2Esusv0~b, )~b) ,
(15)
,6'30
40
50 60 m'~ = m~'(GeV)
7b
80
where /~susv (~b,)~b) -- E()~b, )~b) - 2E(~b, ~d) + E ( $ d , ~d) •
(16)
It is worth tO notice that (16) is symmetric in the interchange ~b '--' ~d or, in other words, does not depend on the sign of the parameter c, as it is expected from the presence of the double GIM. In order to compare (15) with the SM prediction, we use the result of ref. [ 14]. We have checked explicitly that the expression given in ref. [ 14] (for the kaon case) is consistent with ours ~2. By taking B' ~-B, we find Amsusv AmSM
3
as
mw
/~S.=_U Sy ( )~b, -~b ) E s M ( X t , Xt) '
(17)
where x , = m~/m~v and
F.sM(X,X)-2\X_I] (191 +x
4x-1
lnx 3 1 ) 2 (x--l) 2 "
(18)
In fig. 2 we report the values of Amsusv/AmsM, taking r~g = rhb (the lightest "down" squark), for two representative values of mr, m t = 60 and 100 GeV. It is clear that for r ~ _ r~b > 60 GeV, the SUSY contribution to Am is negligible in comparison to AmsM. The above result may seem rather unexpected at ~2 It is worth to notice that, although our derivation is consistent with the SM result obtained by the method of the effective lagrangian, we are at variance, in the numerical coefficients, with previous estimates of the KL-Ks mass difference in SUSY models. Numerically however, the final results turn out to be comparable.
548
Fig. 2. The ratio Amsvsv/Amsu for the B° - ~ ° system is shown as a function of the gluino and the lightest squark masses (taken here to be equal for the purpose of illustration), for representative values of mt.
first sight. Even for values of squark and gluino masses comparable to row, one would expect the SUSY contribution, being the suppression due to the loop integration of the same order, to be enhanced by a factor O(a2/a2w ) - 10 with respect to SM. What is neglected in this naive estimate are two factors. First, the double G I M mechanism evidentiates here the stronger suppression present in SUSY. Indeed, a direct comparison of/~sus¥ versus/~SM in (17) shows that the double G I M suppression, which is typical of the box diagrams, is much stronger in the SUSY case than in SM. This can be related in part to the extra factor ~ c 2 present in SUSY and in part to the fact that for a fixed m i2- r n j ,2 when one considers the exact functional dependence, the G I M suppression becomes stronger as m/m:-, 1. Already for squarks in the rnw range this makes a difference with respect to SM, which is here enhanced by the presence of the double GIM. Secondly, one has to take into account the effect of destructive interference in (14) between the crossed and uncrossed diagrams (we recall that in SM crossed diagrams are forbidden by charge conservation). In particular, it is peculiar that, when rhg increases, the behaviour o f g ( x , y) a n d f ( x , y) is different, since the r~g dependence o f f ( x , y) is softened by the double mass insertion present in the crossed diagrams, figs. 2b, 2d. As a consequence the effect of destructive interference becomes stronger. It is indeed because of this effect, that the value of the ratio Amsusv/AmsM drops so rapidly, about one order of magnitude, when gluino and squark masses
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are brought from 40 to 60 GeV #3. It is worth to remark that this cannot be cured by increasing the top mass. In general one should not expect a peculiar enhancement of the SUSY contributions over the SM ones due to very heavy rot. Indeed, the flavour changing suppression in the minimal SUSY models here considered is proportional to cm2t and the parameter c decreases with increasing mt. Therefore the behaviour of flavour-changing SUSY amplitudes is considerably softened, for increasing mr, with respect to the corresponding behaviour of the SM contributions to processes with "hard" GIM suppression. We did not consider in (I 7) the effects of higher order QCD corrections [ 3 ] since they do not change our conclusions. Indeed, in the SUSY case, due to the aforementioned degeneracy of the squark masses in the loop, we expect QCD corrections not to play any substantial role. Finally, let us remark that the results here presented for Amsusv/AmsM hold as well for B°-l] ° mixing. Before concluding this paper however, let us mention the possibility that supersymmetry may help to evidentiate large CP asymmetries related to the mixing ~4. On this respect, we recall that in the class of minimal models here considered, no large CP violating phases appear beyond the standard KM phase (for a discussion on the CP properties of minimal supergravity models, see ref. [ 16 ]). However, supersymmetry may contribute indirectly to the observation of CP violation effects through the enhancement of decay rates involved in specific CP asymmetries. In previous papers we pointed out that potential large SUSY enhancements may be present in b~s), [ 17] and b-~s gluon [ 13] decays. In particular, for the latter process, branching ratios as high as 10-20% can be obtained even with squark and gluino masses of 70-80 GeV and m, = 80 GeV. This could help in evidentiating a CP asymmetry of the form a(B°B°-*KsX~-Y)- a(B°B°-~KsX~+Y'), with KsX a CP eigenstate. Indeed, if we call f a final state common to both ~3Crossed diagrams, for instance, were neglected in the analysis of ref. [ 15]. where only light gluinoswere considered. ~4Wethank A. Sanda for a stimulating discussion on this point.
20 August 1987
B° and ~o (in addition we will assume f to be a CP eigenstate), it can be shown [ 18,19 ] that there may exist a time-integrated asymmetry fi
cP) +F(~O ~fce)
F( B ° ~ f ) - F ( B ° ~ f
f=
~
l+e 2 x Im {~]--~_Epf] - - 1 + [pf[ 2 I + X 2
(19)
where pf=A(f3°~f)/A(B°-~f), x=Am/F and ~ is defined in analogy with the kaon system. The decay we want to consider is b--,s gluon, leading, at the exclusive level, to final states KsX which are, neglecting the CP impurity in Ks, CP eigenstates containing a net strangeness. In this case we have t~ X f= l ~ x 2
im(( VtbV'd)2 ( Fib Vl*s)2 "~ \ ] ~ - ] ~ T ] -
(20)
For a mixing ra = 20% we find that c~fcan be as large as 47% (it is convenient to use here the Wolfenstein parametrization [ 20] of the KM matrix). Even taking into account that exclusive few-body channels, like Ksn°l] or Ks00, may be at the level of 1% of the inclusive b~sg, these branching ratios could nevertheless be, in SUSY models, as high as few parts times 10 -3, allowing for the possibility of detecting this asymmetry with ~ O ( 1 0 7) B°]3° pairs. It is worth to remark that, from the general considerations on the possible mechanisms of SUSY enhancement of weak meson decays that we have presented in refs. [ 13,17 ], CP asymmetries, in particular involving semileptonic or leptonic decays. Given the results on the B°-B ° mixing and related CP violation here presented, it seems to us that the decay b--,sg together with the analogous radiative decay b~sT, provide at present two of the most sensitive tests for a SUSY search in low energy physics. Observation ofb--,sg at the 10% level or b~s), at the 10 3 level would indeed provide a strong indication in favour of SUSY. On the other hand, in the event that no signals are observed, we think that the physics of the B mesons will still provide, before Tevatron reaches its full potentiality, the most stringent constraints on squark and gluino masses. We are grateful to P. Rakow, A. Sanda and U Wolfenstein for useful discussions. The work of S.B. 549
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is partially s u p p o r t e d by the U S D e p a r t m e n t o f Energy. T h e w o r k o f F.B. a n d A . M . is s u p p o r t e d in part by the N S F u n d e r G r a n t no. P H Y 8 1 1 6 1 0 2 .
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