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Upper limit on Bd0 − B d0 mixing as evidence for the existence of the top quark
Volume 243, number 3
PHYSICS LETTERS B
28 June 1990
Upper limit on B ° -l)°o mixing as evidence for the existence of the top quark D . P . R o y a n d S. U m a S a n k a r Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005 Bombay. India
Received 26 February 1990
We consider the mixing between the neutral B° mesons in an SU (2)L × U ( 1) model without a top quark, in which bL is an SU (2)u singlet. The mixing parameter xd = AmBS/Fb, arising from the resulting flavour changing neutral currents is found to be one to two orders of magnitude larger than the value measured experimentally. This is a very strong evidence for the existence of the top quark, comparable to that coming from the b--*~t+la X branching ratio.
The standard model ( S M ) of electroweak interactions [ 1 ], based on the gauge group SU ( 2 ) e × U ( 1 ), has been very successful in explaining all the current experimental data [ 2 ]. In this model the left-handed fermions transform as doublets under SU (2)L. O f the five observed quarks, the four lighter ones are paired into two doublets (UL, de) and (Ce, Se). It is postulated that the fifth quark, b o t t o m ( b ) , must have an SU (2) e partner top ( t ) , to make the model complete and consistent. So far the top quark has not been observed and the limits on its mass are m r > 2 8 GeV from e+e - annihilations [ 3], and mt > 44 GeV from pf) collisions [4]. Moreover, the p r e l i m i n a r y results from the recent plb collision experiments at T E V A T R O N and A C O L seem to push the latter limit upto m, > 70 GeV [ 5 ]. In the light o f these negative results from top search experiments, it is imperative to examine all indirect evidences for its existence carefully. The theoretical impetus for the top quark is the cancellation o f the chiral anomalies [6], which spoil the renormalizability of the SM. W i t h o u t the top quark one is forced to cancel the anomalies through some other mechanism which usually spoils the simplicity of the SM. F r o m a purely experimental point o f view, there are two pieces of data so far which indicate the existence o f the top quark. In the SM, the b quark couples to the Z boson both via vector and 296
axial vector currents and the axial vector coupling is predicted to be ab = T 3 = - 0.5, where T 3 is the third c o m p o n e n t of the weak isospin group SU (2)L. If the top quark does not exist, the be would be an SU (2) L singlet. In this case the axial vector coupling a b = 0 , i.e. the b quark couples to Z only via vector current. This would imply that the differential cross section for e+e--~bl3 would be symmetric under 0 ~ z - 0 . The observed f o r w a r d - b a c k w a r d a s y m m e t r y [7] in this process at PEP and P E T R A has led to a determ i n a t i o n o f a b = --0.54_+0.15. This value is in good agreement with the SM prediction and differs from zero by more than 3.5 standard deviations. The other experimental evidence is the observed suppression of flavour changing neutral currents in b--. ~t+~t-X. If bL is an S U ( 2 ) L singlet, its Cabibbo mixing [8] via mass terms with doublet members de and SL, induces flavour changing neutral currents. These neutral currents give rise to large rates for flavour changing processes. The BR(b--.~t+~t-X) is predicted to be [9] >~ 1% if the bL is a singlet and the experimental upper limit is [ 10 ] ~<0.1%, a discrepancy o f a factor o f 10. This suppression of flavour changing neutral currents, along with the measurement ofab, indicates that bL is a m e m b e r of an S U ( 2 ) L doublet, i.e. the top quark exists. In this letter we calculate the Bo-Bd o -o mixing in a model without the top quark. This mixing can now arise (i) from the box diagram as in the SM and (ii)
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 243, number 3
PHYSICS LETTERS B
due to the flavour changing neutral current interactions mentioned above. In a similar situation of K ° - I 4 ° mixing without the charm quark (i.e. without the G I M cancellation [ 11 ], both the box diagram and the flavour changing neutral currents give very large contributions to AmK. In the model without the top quark, the value o f AmK remains small due to the cancellation between up and charm couplings to d - s (i.e. the G I M cancellation still holds). The box diagram contribution to the Bd-Bd o -o mixing is small enough to be compatible with the mixing measured experimentally [ 12]. But the mixing arising due to the flavour changing neutral currents is found to be one to two orders o f magnitude larger than the measured value. Thus, the smallness of the observed value of B °-I] ° mixing provides a very strong evidence for the existence o f the top quark, which is at least as strong as that coming from B (b --. ~t+ ~t- X ) [ 9,10 ]. The gauge couplings in an S U ( 2 ) L × U ( 1 ) model without the top quark are g 5Pg~ug~= eJeumA u + ~
g
+ c~s O~
( J + W ;'- + J ~ W u+ )
( j 3 -sin2OwJ~,m)z ;~
~
Qq(Ft'cTuq'c +~]hTuqk),
(5)
S'L = UcddL + Uc~SL + Ucbbc .
(6)
In the next paragraph, we will see that the first two rows of the matrix U have the same physical significance as the first two rows of the C K M matrix V [ 13 ] occurring in the SM with three generations. Anticipating this, we have used the subscripts uq and cq to label the elements of U. All quarks of identical charge have identical terms in the electromagnetic current. Therefore, jhm can be rewritten in terms of the mass eigenstates by simply replacing a weak eigenstate (primed one) by the corresponding mass eigenstate (unprimed one). To obtain the charged current in terms of the mass eigenstates, we substitute the expressions for d~. and s~_ in (3). The charged current terms are
+ eLTu(UcddL+ UcsSL+ UcbbL) •
(2) (3)
and j 3 = ½(uLTuuL +g'Ly;,C'I. - d ; ~,;,dL - gLy~,sL) •
d'L = UuddL + U~sSL+ Uubbc,
(7)
(1)
q = u,c,d,s,b
J ; = a'LT~d'L +gLT~SL,
states of the charge - ] quarks can be written as dL = U *dL, ' where U is a unitary matrix. For later use we write out the explicit expressions for d [ and s~_,
J ; =ULTu( UuddL + UusSL + UubbL)
where j~m =
28 June 1990
(4)
The primes in ( 2 ) - ( 4 ) indicate that the quarks are not in the mass eigenstates but in the weak eigenstates. The currents J + , J~- and j 3 form an isotriplet of the SU (2) L, and couple to the three gauge bosons associated with it. All the quarks of the same charge can mix with each other via mass terms. One has to make unitary transformations on the column vectors d~.,R = ( d ' , s ' , b ' )L,R T in order to diagonalize the charge quark mass matrix and to convert weak eigenstates into mass eigenstates. Without loss of generality, we can choose the weak and the mass eigenstates to be identical for the charge -~ quarks. The mass eigen-
By measuring the relative strengths of the various charged current interactions, the six elements of U occurring in (7) can be obtained and these are identical to the elements of the C K M matrix V that describe the strenghts o f various charged currents in the SM [ 13 ]. Therefore Uuq-= Vuq and Uc, = Vcq, where q = d , s, b ~. In terms of the mass eigenstates, j 3 is the sum of a flavour diagonal piece and a flavour changing piece. The flavour diagonal part is J3(f.d. ) = ½[dL~',( I Uu~ 12+ I Ucd 12)alL
+ eLY;,( I Uu~ 12+ I Ucs I~)SL +SLY~( I Uub 12+ I Ucb [2)bLl ,
(8)
and the flavour changing part is "J Note that only the elements of the first two rows of U have physical significance because they describe the strengths of various charged currents. The elements of the third row can be expressed in terms of the elements of the first two rows using unitarity but they have no direct physical significance. In the case of the CKM matrix V, the elements of the third row also have physical significance as they describe the strengths of the transitions t~u, t-,s and t~b. 297
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PHYSICS LETTERS B
28 June 1990
J~,(f.c.) = ~ [d~ y.( ~ uus + ~,, ucs)sL + dL y, ( U% Uub + U*a Ucb)bL
(9)
+ SLy~( Uu*s Uub + Uc* U c b ) b L ] + h . c .
The terms in (9) lead to flavour changing processes at tree level. Comparing the coefficients of the dominant charged current b decay in (7) and the corresponding flavour changing neutral current decay in (9), i.e. b ~ c and b ~ s , one sees that the two processes should have comparable rates. Multiplying the latter with the branching ratio for the off-shell Z--, g+p. - leads to a sizeable overall branching ratio for [ 9 ] b ~ p.÷ M- X ~> 1%, which is an order of magnitude larger than the experimental upper bound [10] as mentioned earlier. We shall see below that the flavour changing neutral current o f (9) also leads to an untenably large B° - g ° mixing. We now calculate the Bd°-B° mixing both from the box diagrams and the tree diagrams due to the flavour changing neutral current terms in (9). The box diagrams are shown in fig. 1. In the SM, Bd-Ba° -o mixing arises from similar box diagrams, with the intermediate quark labels i a n d j running over u, c and t, plus the associated box diagrams where one or both of the W bosons are replaced by the charged Goldstone bosons, Bd-Bd o -o mixing is typically expressed in terms of the parameter xd = AmBs/Fb = rb Ama8. Experimentally xa is measured to be [ 12,14 ] 0.7_+ 0.2. The expression for Xd in the SM with three generations is [ 14] 2
XdISM=Zb GFmtfamB * Vtb I2F(yt) , 67r2 I Vtd
4 (1-yt) 2
the expression in (10) for the experimental measurement one has the limit on the product I Vtd I mt ~ 1 GeV or mt~> 50 GeV if one takes I Vtd I -%<0.02, which follows from the unitarity of the CKM matrix [ 15 ]. In the model without the top quark, the internal quark indices i and j in the box diagrams can be up or charm only. The graphs with the Goldstone boson exchange make a negligible contribution to the mixing because the masses of the internal as well as the external quarks are much smaller than Mw ( M 2 >> m 2 ,m 2, m,], m2). Therefore it is enough to consider the two W boson exchange graphs shown in fig. 1. The mixing parameter from the box diagrams is calculated to be Xd Ibox=ZbAmB~ Ibox
GFMwfBmB2 2 2 -
-
Zb
6~ 2
U,d U~bU]d Ujbl(yi, yj) *
*
,
(12)
where y~ = m~/MZw and the function I(yi, yj) is given by y2 In Yi 1(y,, yj) = (y, _ y j ) ( 1 - y , ) ~
y~lnyj + 1 (yj-y,) ( 1 _yj)2 ( 1 -y~) ( 1 - y j ) "
(13)
b
I(yi, yj)~-I for Yi, Yj<< 1. Since m~ <
d
i =u,c
b
w j= u ~c:
d 0 d o mixing. Fig. 1. Box diagrams for Bd-]~
298
(11)
(10)
where y~ = m t2/ M w2 and the function F(y~) is given by
W
1
F ( y t ) ~ 1 for yt<< 1, and ~ ] for yt>> l. Comparing
+
2 2
d
3 y2 lny~ "~ 2 (l-y,)3] "
3yt(l+yt)
F(yt)=
Volume 243, number 3 2 Xd I box = "(b
2
PHYSICS LETTERS B 2
G v M 6~2 wfamB
I U% Uub +
. Ucb 12 Ucd
(14) For almost all the allowed values o f U o, available from the charged current data [ 15 ], the expression in (14) yields a value Of Xd that is consistent with the experimental result ~2. Therefore the box d i a g r a m result for 0 -0 Bo-Bd mixing is roughly the same in the m o d e l with the top quark and that without, unlike in the case o f K°-I~ ° mixing, where the box result for ArnK is larger by a factor ( M w2 / m ¢ )2 if the c h a r m quark were not there. Turning to the tree diagrams generated by the flayour changing neutral current terms in (9), which are shown in fig. 2, we obtain Xd [tree = Tb
orders of magnitude larger than the experimental measurement. A m o r e precise estimate o f Xd Itree can be obtained by substituting the values o f the p a r a m e t e r s o f refs. [14,15] in ( 1 5 ) , i.e. rb--~ 1.4× 10 -12 S, GF = 1. 166× 10 -5 GeV -2, f B = 0 . 1 - 0 . 1 5 GeV, mB=5.27 GeV, Ucd= 0.23 and Ucb~--- 0.05. The resulting Xd, o b t a i n e d by ignoring the first term in the modulus in ( 15 ), is xo ] t r e e = 5 0 - 1 0 0 ,
(17)
about two orders of magnitude larger than the experimental value o f xo = 0 . 7 _+0.2. One can get a lower b o u n d on the predicted Xd[tr~e by assuming that the two terms in the modulus in ( 1 5 ) have exactly opposite phase and the ratio [Uub/Ueb[ t o be equal to its u p p e r limit o f 0.14 ~2. Even in this case one gets Xd ]tree>/7 ,
Amn8 ]tree
* . = rb'~( x/~ Gv.ffBmB)[UudUub+UcdU~b[ 2 (15) Comparing this expression with that Of Xd [box, we find the ratio to be Xd Itree
28 June 1990
(18)
at least an order o f magnitude larger than the experimental value. In terms of the directly measurable quantity 0 0 -0 -0 e + e -- --+BdBd+BoB o x2 rd= e+e___,BOi)O - 2+x~'
(19)
N//2 7"(2
Xd [box -- G F M ~ ~- 184 .
(16)
We saw before that Xd[box is consistent with the observed value. Therefore from (16) we find that the predicted value for Xd, in a m o d e l without the top quark [ where bL is an SU ( 2 ) L singlet ] is roughly two ~2 The only exception is if I Uub/UcbI is very close to the conservative upper limit [ 15 ] of 0.20 and the two terms U~*dUub and Uc*dU~b have opposite phase. In this case the two terms could effectively cancel and the resulting xd could be too small to account for the experimental value. However, a more reasonable upper limit on I U,b//-/coI is 0.14 [ 16 ]. This is consistent with a recent estimate of this ratio from the preliminary ARGUS data giving ] U~b/U~b [ = 0.09 _+0.02 [ 17 ].
this corresponds to a predicted lower limit of rd >/0.96, c o m p a r e d to the experimental value o f 0.2 +_0.1. Thus we see from the B ° - ~ ° mixing that the flavour changing neutral currents are suppressed to at least as great an extent as the m e a s u r e m e n t o f B ( b ~ t + ~ t - X ) indicates. This is a very strong evidence for be being an SU (2) c doublet m e m b e r and hence for the existence o f the top quark. Let us briefly discuss Bs-B~ o -o mixing. The SM predicts 2
x__~sls~_M_ Vt~ ~ 2 0 . Xd [ SM
(2O)
Vt,~ d
b
Z
b
d
Fig. 2. Tree graphs for B°-l) ° mixing in a model without the top quark. 299
Volume 243, number 3
PHYSICS LETTERS B
References
In the m o d e l w i t h o u t the top quark, we h a v e *
Xs Ibox
Xd [bo~
Xs [ tree
*
2
[U*usUub+UcsUcb
-- Xdltre~ -- [UudU, T T bE+ - UcdUcb] - * '
(21)
T h e values o f xs I SM and xs I box are roughly e q u a l for the allowed values o f the U o [ 15 ]. T h e r e f o r e Xs [tree is nearly 200 t i m e s larger t h a n the SM p r e d i c t i o n . H o w ever, the e x p e r i m e n t a l i n f o r m a t i o n on this m i x i n g p a r a m e t e r xs is v e r y poor. Finally it should be e m p h a s i z e d that the e v i d e n c e for the presence o f the t o p q u a r k discussed h e r e and the c o n s t r a i n t on the top q u a r k mass discussed extensively in the literature b e f o r e [14] are two distinct results following f r o m the B do - B-od m i x i n g data. T h e r e l e v a n t c o n t r i b u t i o n for the first is f r o m the tree diagram, which is i n d e p e n d e n t o f rnt a n d is m u c h too large c o m p a r e d to the e x p e r i m e n t a l v a l u e o f Bd°-Bd-° mixing. T h e r e l e v a n t c o n t r i b u t i o n for the s e c o n d result is f r o m the box d i a g r a m , w h i c h is t o o small c o m p a r e d to the e x p e r i m e n t a l v a l u e o f B ° - l ] ° m i x i n g for mt ~< 5 0 G e V and is c o m p a t i b l e with it for mt >t 50 G e V , but is n e v e r too large. L o o k i n g purely f r o m the p o i n t o f v i e w o f the data, it is the e x p e r i m e n t a l u p p e r l i m i t on 0 -0 Bd-Bd m i x i n g that p r o v i d e s the e v i d e n c e for the presence o f the top q u a r k while the e x p e r i m e n t a l lower l i m i t on this q u a n t i t y p r o v i d e s the c o n s t r a i n t on mr. We t h a n k P r o f e s s o r G. R a j a s e k a r a n for a critical r e a d i n g o f the m a n u s c r i p t .
300
28 June 1990
[ 1 ] S.L. Glashow, Nucl. Phys. B 22 ( 1961 ) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particle theory, ed. N. Svartholm (Almquist and Wiksells, Stockholm, 1969) p. 367. [2] P. Langacker, Comm. Nucl. Part. Phys. 19 (1989) 1. [3] T. Kamae, in: Proc. XXIV Intern. Conf. on High energy physics, eds. R. Kotthaus and J.H. Kiihn (Munich, August 1988)p. 156. [4] UAI Collab., C. Albajar et al., Z. Phys. C 37 (1988) 504. [5 ] P. Sinervo, Report Intern. Symp. on Lepton and photon interactions at high energies (Stanford, CA, 1989). [ 6 ] See, for example, T.-P. Cheng and L.-F. Li, Gauge theory of elementary particle physics (Oxford U.P., Oxford, 1984) pp. 347, 348. [7]S.L. Wu, in: Intern. Syrup. on Lepton and photon interactions at high energies (Hamburg, 1987 ), Nucl. Phys. (Proc. Suppl.) B 3 (1988) 39; R. Marshall, Rutherford preprints RAL 87-037, 88-051. [8] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531. [9] G.L. Kane and M. Peskin, Nucl. Phys. B 195 (1982) 29; V. Barger and S. Pakvasa, Phys. Lett. B 81 (1979) 195. [ 10] CLEO Collab., A. Bean et al., Phys. Rev. D 35 (1987) 3533. [ 11 ] S.L. Glashow, J. llliopoulos and L. Maiani, Phys. Rev. D 2 (1970) 1285. [ 12 ] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 192 ( 1987 ) 245; CLEO Collab., M. Artuso et al., Phys. Rev. Lett. 62 (1989) 2233. [ 13] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [ 14] K. Hagiwara, KEK preprint KEK-TH-170, 87-47; P.J. Franzini, Phys. Rep. 173 (1989) 1, and references therein. [ 15] Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lett. B 204 (1988) 107. [ 16] G. Altarelli and P.J. Franzini, Z. Phys. C 37 (1988) 271. [ 17] M.V. Danilov, Report Intern. Symp. on Lepton and photon interactions at high energies (Stanford, CA, 1989).
PHYSICS LETTERS B
28 June 1990
Upper limit on B ° -l)°o mixing as evidence for the existence of the top quark D . P . R o y a n d S. U m a S a n k a r Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005 Bombay. India
Received 26 February 1990
We consider the mixing between the neutral B° mesons in an SU (2)L × U ( 1) model without a top quark, in which bL is an SU (2)u singlet. The mixing parameter xd = AmBS/Fb, arising from the resulting flavour changing neutral currents is found to be one to two orders of magnitude larger than the value measured experimentally. This is a very strong evidence for the existence of the top quark, comparable to that coming from the b--*~t+la X branching ratio.
The standard model ( S M ) of electroweak interactions [ 1 ], based on the gauge group SU ( 2 ) e × U ( 1 ), has been very successful in explaining all the current experimental data [ 2 ]. In this model the left-handed fermions transform as doublets under SU (2)L. O f the five observed quarks, the four lighter ones are paired into two doublets (UL, de) and (Ce, Se). It is postulated that the fifth quark, b o t t o m ( b ) , must have an SU (2) e partner top ( t ) , to make the model complete and consistent. So far the top quark has not been observed and the limits on its mass are m r > 2 8 GeV from e+e - annihilations [ 3], and mt > 44 GeV from pf) collisions [4]. Moreover, the p r e l i m i n a r y results from the recent plb collision experiments at T E V A T R O N and A C O L seem to push the latter limit upto m, > 70 GeV [ 5 ]. In the light o f these negative results from top search experiments, it is imperative to examine all indirect evidences for its existence carefully. The theoretical impetus for the top quark is the cancellation o f the chiral anomalies [6], which spoil the renormalizability of the SM. W i t h o u t the top quark one is forced to cancel the anomalies through some other mechanism which usually spoils the simplicity of the SM. F r o m a purely experimental point o f view, there are two pieces of data so far which indicate the existence o f the top quark. In the SM, the b quark couples to the Z boson both via vector and 296
axial vector currents and the axial vector coupling is predicted to be ab = T 3 = - 0.5, where T 3 is the third c o m p o n e n t of the weak isospin group SU (2)L. If the top quark does not exist, the be would be an SU (2) L singlet. In this case the axial vector coupling a b = 0 , i.e. the b quark couples to Z only via vector current. This would imply that the differential cross section for e+e--~bl3 would be symmetric under 0 ~ z - 0 . The observed f o r w a r d - b a c k w a r d a s y m m e t r y [7] in this process at PEP and P E T R A has led to a determ i n a t i o n o f a b = --0.54_+0.15. This value is in good agreement with the SM prediction and differs from zero by more than 3.5 standard deviations. The other experimental evidence is the observed suppression of flavour changing neutral currents in b--. ~t+~t-X. If bL is an S U ( 2 ) L singlet, its Cabibbo mixing [8] via mass terms with doublet members de and SL, induces flavour changing neutral currents. These neutral currents give rise to large rates for flavour changing processes. The BR(b--.~t+~t-X) is predicted to be [9] >~ 1% if the bL is a singlet and the experimental upper limit is [ 10 ] ~<0.1%, a discrepancy o f a factor o f 10. This suppression of flavour changing neutral currents, along with the measurement ofab, indicates that bL is a m e m b e r of an S U ( 2 ) L doublet, i.e. the top quark exists. In this letter we calculate the Bo-Bd o -o mixing in a model without the top quark. This mixing can now arise (i) from the box diagram as in the SM and (ii)
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 243, number 3
PHYSICS LETTERS B
due to the flavour changing neutral current interactions mentioned above. In a similar situation of K ° - I 4 ° mixing without the charm quark (i.e. without the G I M cancellation [ 11 ], both the box diagram and the flavour changing neutral currents give very large contributions to AmK. In the model without the top quark, the value o f AmK remains small due to the cancellation between up and charm couplings to d - s (i.e. the G I M cancellation still holds). The box diagram contribution to the Bd-Bd o -o mixing is small enough to be compatible with the mixing measured experimentally [ 12]. But the mixing arising due to the flavour changing neutral currents is found to be one to two orders o f magnitude larger than the measured value. Thus, the smallness of the observed value of B °-I] ° mixing provides a very strong evidence for the existence o f the top quark, which is at least as strong as that coming from B (b --. ~t+ ~t- X ) [ 9,10 ]. The gauge couplings in an S U ( 2 ) L × U ( 1 ) model without the top quark are g 5Pg~ug~= eJeumA u + ~
g
+ c~s O~
( J + W ;'- + J ~ W u+ )
( j 3 -sin2OwJ~,m)z ;~
~
Qq(Ft'cTuq'c +~]hTuqk),
(5)
S'L = UcddL + Uc~SL + Ucbbc .
(6)
In the next paragraph, we will see that the first two rows of the matrix U have the same physical significance as the first two rows of the C K M matrix V [ 13 ] occurring in the SM with three generations. Anticipating this, we have used the subscripts uq and cq to label the elements of U. All quarks of identical charge have identical terms in the electromagnetic current. Therefore, jhm can be rewritten in terms of the mass eigenstates by simply replacing a weak eigenstate (primed one) by the corresponding mass eigenstate (unprimed one). To obtain the charged current in terms of the mass eigenstates, we substitute the expressions for d~. and s~_ in (3). The charged current terms are
+ eLTu(UcddL+ UcsSL+ UcbbL) •
(2) (3)
and j 3 = ½(uLTuuL +g'Ly;,C'I. - d ; ~,;,dL - gLy~,sL) •
d'L = UuddL + U~sSL+ Uubbc,
(7)
(1)
q = u,c,d,s,b
J ; = a'LT~d'L +gLT~SL,
states of the charge - ] quarks can be written as dL = U *dL, ' where U is a unitary matrix. For later use we write out the explicit expressions for d [ and s~_,
J ; =ULTu( UuddL + UusSL + UubbL)
where j~m =
28 June 1990
(4)
The primes in ( 2 ) - ( 4 ) indicate that the quarks are not in the mass eigenstates but in the weak eigenstates. The currents J + , J~- and j 3 form an isotriplet of the SU (2) L, and couple to the three gauge bosons associated with it. All the quarks of the same charge can mix with each other via mass terms. One has to make unitary transformations on the column vectors d~.,R = ( d ' , s ' , b ' )L,R T in order to diagonalize the charge quark mass matrix and to convert weak eigenstates into mass eigenstates. Without loss of generality, we can choose the weak and the mass eigenstates to be identical for the charge -~ quarks. The mass eigen-
By measuring the relative strengths of the various charged current interactions, the six elements of U occurring in (7) can be obtained and these are identical to the elements of the C K M matrix V that describe the strenghts o f various charged currents in the SM [ 13 ]. Therefore Uuq-= Vuq and Uc, = Vcq, where q = d , s, b ~. In terms of the mass eigenstates, j 3 is the sum of a flavour diagonal piece and a flavour changing piece. The flavour diagonal part is J3(f.d. ) = ½[dL~',( I Uu~ 12+ I Ucd 12)alL
+ eLY;,( I Uu~ 12+ I Ucs I~)SL +SLY~( I Uub 12+ I Ucb [2)bLl ,
(8)
and the flavour changing part is "J Note that only the elements of the first two rows of U have physical significance because they describe the strengths of various charged currents. The elements of the third row can be expressed in terms of the elements of the first two rows using unitarity but they have no direct physical significance. In the case of the CKM matrix V, the elements of the third row also have physical significance as they describe the strengths of the transitions t~u, t-,s and t~b. 297
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J~,(f.c.) = ~ [d~ y.( ~ uus + ~,, ucs)sL + dL y, ( U% Uub + U*a Ucb)bL
(9)
+ SLy~( Uu*s Uub + Uc* U c b ) b L ] + h . c .
The terms in (9) lead to flavour changing processes at tree level. Comparing the coefficients of the dominant charged current b decay in (7) and the corresponding flavour changing neutral current decay in (9), i.e. b ~ c and b ~ s , one sees that the two processes should have comparable rates. Multiplying the latter with the branching ratio for the off-shell Z--, g+p. - leads to a sizeable overall branching ratio for [ 9 ] b ~ p.÷ M- X ~> 1%, which is an order of magnitude larger than the experimental upper bound [10] as mentioned earlier. We shall see below that the flavour changing neutral current o f (9) also leads to an untenably large B° - g ° mixing. We now calculate the Bd°-B° mixing both from the box diagrams and the tree diagrams due to the flavour changing neutral current terms in (9). The box diagrams are shown in fig. 1. In the SM, Bd-Ba° -o mixing arises from similar box diagrams, with the intermediate quark labels i a n d j running over u, c and t, plus the associated box diagrams where one or both of the W bosons are replaced by the charged Goldstone bosons, Bd-Bd o -o mixing is typically expressed in terms of the parameter xd = AmBs/Fb = rb Ama8. Experimentally xa is measured to be [ 12,14 ] 0.7_+ 0.2. The expression for Xd in the SM with three generations is [ 14] 2
XdISM=Zb GFmtfamB * Vtb I2F(yt) , 67r2 I Vtd
4 (1-yt) 2
the expression in (10) for the experimental measurement one has the limit on the product I Vtd I mt ~ 1 GeV or mt~> 50 GeV if one takes I Vtd I -%<0.02, which follows from the unitarity of the CKM matrix [ 15 ]. In the model without the top quark, the internal quark indices i and j in the box diagrams can be up or charm only. The graphs with the Goldstone boson exchange make a negligible contribution to the mixing because the masses of the internal as well as the external quarks are much smaller than Mw ( M 2 >> m 2 ,m 2, m,], m2). Therefore it is enough to consider the two W boson exchange graphs shown in fig. 1. The mixing parameter from the box diagrams is calculated to be Xd Ibox=ZbAmB~ Ibox
GFMwfBmB2 2 2 -
-
Zb
6~ 2
U,d U~bU]d Ujbl(yi, yj) *
*
,
(12)
where y~ = m~/MZw and the function I(yi, yj) is given by y2 In Yi 1(y,, yj) = (y, _ y j ) ( 1 - y , ) ~
y~lnyj + 1 (yj-y,) ( 1 _yj)2 ( 1 -y~) ( 1 - y j ) "
(13)
b
I(yi, yj)~-I for Yi, Yj<< 1. Since m~ <
d
i =u,c
b
w j= u ~c:
d 0 d o mixing. Fig. 1. Box diagrams for Bd-]~
298
(11)
(10)
where y~ = m t2/ M w2 and the function F(y~) is given by
W
1
F ( y t ) ~ 1 for yt<< 1, and ~ ] for yt>> l. Comparing
+
2 2
d
3 y2 lny~ "~ 2 (l-y,)3] "
3yt(l+yt)
F(yt)=
Volume 243, number 3 2 Xd I box = "(b
2
PHYSICS LETTERS B 2
G v M 6~2 wfamB
I U% Uub +
. Ucb 12 Ucd
(14) For almost all the allowed values o f U o, available from the charged current data [ 15 ], the expression in (14) yields a value Of Xd that is consistent with the experimental result ~2. Therefore the box d i a g r a m result for 0 -0 Bo-Bd mixing is roughly the same in the m o d e l with the top quark and that without, unlike in the case o f K°-I~ ° mixing, where the box result for ArnK is larger by a factor ( M w2 / m ¢ )2 if the c h a r m quark were not there. Turning to the tree diagrams generated by the flayour changing neutral current terms in (9), which are shown in fig. 2, we obtain Xd [tree = Tb
orders of magnitude larger than the experimental measurement. A m o r e precise estimate o f Xd Itree can be obtained by substituting the values o f the p a r a m e t e r s o f refs. [14,15] in ( 1 5 ) , i.e. rb--~ 1.4× 10 -12 S, GF = 1. 166× 10 -5 GeV -2, f B = 0 . 1 - 0 . 1 5 GeV, mB=5.27 GeV, Ucd= 0.23 and Ucb~--- 0.05. The resulting Xd, o b t a i n e d by ignoring the first term in the modulus in ( 15 ), is xo ] t r e e = 5 0 - 1 0 0 ,
(17)
about two orders of magnitude larger than the experimental value o f xo = 0 . 7 _+0.2. One can get a lower b o u n d on the predicted Xd[tr~e by assuming that the two terms in the modulus in ( 1 5 ) have exactly opposite phase and the ratio [Uub/Ueb[ t o be equal to its u p p e r limit o f 0.14 ~2. Even in this case one gets Xd ]tree>/7 ,
Amn8 ]tree
* . = rb'~( x/~ Gv.ffBmB)[UudUub+UcdU~b[ 2 (15) Comparing this expression with that Of Xd [box, we find the ratio to be Xd Itree
28 June 1990
(18)
at least an order o f magnitude larger than the experimental value. In terms of the directly measurable quantity 0 0 -0 -0 e + e -- --+BdBd+BoB o x2 rd= e+e___,BOi)O - 2+x~'
(19)
N//2 7"(2
Xd [box -- G F M ~ ~- 184 .
(16)
We saw before that Xd[box is consistent with the observed value. Therefore from (16) we find that the predicted value for Xd, in a m o d e l without the top quark [ where bL is an SU ( 2 ) L singlet ] is roughly two ~2 The only exception is if I Uub/UcbI is very close to the conservative upper limit [ 15 ] of 0.20 and the two terms U~*dUub and Uc*dU~b have opposite phase. In this case the two terms could effectively cancel and the resulting xd could be too small to account for the experimental value. However, a more reasonable upper limit on I U,b//-/coI is 0.14 [ 16 ]. This is consistent with a recent estimate of this ratio from the preliminary ARGUS data giving ] U~b/U~b [ = 0.09 _+0.02 [ 17 ].
this corresponds to a predicted lower limit of rd >/0.96, c o m p a r e d to the experimental value o f 0.2 +_0.1. Thus we see from the B ° - ~ ° mixing that the flavour changing neutral currents are suppressed to at least as great an extent as the m e a s u r e m e n t o f B ( b ~ t + ~ t - X ) indicates. This is a very strong evidence for be being an SU (2) c doublet m e m b e r and hence for the existence o f the top quark. Let us briefly discuss Bs-B~ o -o mixing. The SM predicts 2
x__~sls~_M_ Vt~ ~ 2 0 . Xd [ SM
(2O)
Vt,~ d
b
Z
b
d
Fig. 2. Tree graphs for B°-l) ° mixing in a model without the top quark. 299
Volume 243, number 3
PHYSICS LETTERS B
References
In the m o d e l w i t h o u t the top quark, we h a v e *
Xs Ibox
Xd [bo~
Xs [ tree
*
2
[U*usUub+UcsUcb
-- Xdltre~ -- [UudU, T T bE+ - UcdUcb] - * '
(21)
T h e values o f xs I SM and xs I box are roughly e q u a l for the allowed values o f the U o [ 15 ]. T h e r e f o r e Xs [tree is nearly 200 t i m e s larger t h a n the SM p r e d i c t i o n . H o w ever, the e x p e r i m e n t a l i n f o r m a t i o n on this m i x i n g p a r a m e t e r xs is v e r y poor. Finally it should be e m p h a s i z e d that the e v i d e n c e for the presence o f the t o p q u a r k discussed h e r e and the c o n s t r a i n t on the top q u a r k mass discussed extensively in the literature b e f o r e [14] are two distinct results following f r o m the B do - B-od m i x i n g data. T h e r e l e v a n t c o n t r i b u t i o n for the first is f r o m the tree diagram, which is i n d e p e n d e n t o f rnt a n d is m u c h too large c o m p a r e d to the e x p e r i m e n t a l v a l u e o f Bd°-Bd-° mixing. T h e r e l e v a n t c o n t r i b u t i o n for the s e c o n d result is f r o m the box d i a g r a m , w h i c h is t o o small c o m p a r e d to the e x p e r i m e n t a l v a l u e o f B ° - l ] ° m i x i n g for mt ~< 5 0 G e V and is c o m p a t i b l e with it for mt >t 50 G e V , but is n e v e r too large. L o o k i n g purely f r o m the p o i n t o f v i e w o f the data, it is the e x p e r i m e n t a l u p p e r l i m i t on 0 -0 Bd-Bd m i x i n g that p r o v i d e s the e v i d e n c e for the presence o f the top q u a r k while the e x p e r i m e n t a l lower l i m i t on this q u a n t i t y p r o v i d e s the c o n s t r a i n t on mr. We t h a n k P r o f e s s o r G. R a j a s e k a r a n for a critical r e a d i n g o f the m a n u s c r i p t .
300
28 June 1990
[ 1 ] S.L. Glashow, Nucl. Phys. B 22 ( 1961 ) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particle theory, ed. N. Svartholm (Almquist and Wiksells, Stockholm, 1969) p. 367. [2] P. Langacker, Comm. Nucl. Part. Phys. 19 (1989) 1. [3] T. Kamae, in: Proc. XXIV Intern. Conf. on High energy physics, eds. R. Kotthaus and J.H. Kiihn (Munich, August 1988)p. 156. [4] UAI Collab., C. Albajar et al., Z. Phys. C 37 (1988) 504. [5 ] P. Sinervo, Report Intern. Symp. on Lepton and photon interactions at high energies (Stanford, CA, 1989). [ 6 ] See, for example, T.-P. Cheng and L.-F. Li, Gauge theory of elementary particle physics (Oxford U.P., Oxford, 1984) pp. 347, 348. [7]S.L. Wu, in: Intern. Syrup. on Lepton and photon interactions at high energies (Hamburg, 1987 ), Nucl. Phys. (Proc. Suppl.) B 3 (1988) 39; R. Marshall, Rutherford preprints RAL 87-037, 88-051. [8] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531. [9] G.L. Kane and M. Peskin, Nucl. Phys. B 195 (1982) 29; V. Barger and S. Pakvasa, Phys. Lett. B 81 (1979) 195. [ 10] CLEO Collab., A. Bean et al., Phys. Rev. D 35 (1987) 3533. [ 11 ] S.L. Glashow, J. llliopoulos and L. Maiani, Phys. Rev. D 2 (1970) 1285. [ 12 ] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 192 ( 1987 ) 245; CLEO Collab., M. Artuso et al., Phys. Rev. Lett. 62 (1989) 2233. [ 13] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [ 14] K. Hagiwara, KEK preprint KEK-TH-170, 87-47; P.J. Franzini, Phys. Rep. 173 (1989) 1, and references therein. [ 15] Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lett. B 204 (1988) 107. [ 16] G. Altarelli and P.J. Franzini, Z. Phys. C 37 (1988) 271. [ 17] M.V. Danilov, Report Intern. Symp. on Lepton and photon interactions at high energies (Stanford, CA, 1989).