- Email: [email protected]
Effects of 4th generation quarks in the B meson system
494
Nuclear Physics B (Proc. Suppl.) 13 (1900) 494--498 North-Holland
EI~T~
OT 4TR GEa~ERATZON Q~U~KS ZN THE B ~ S O N
SYSTEM*
C. ~ A O U I Conco:~a
University, Montreal, Quebec, Canada H3G-IM8
I. SANDA Rockefeller University, 1230 York Avenue, New York, N.Y.
A.
10021
and A. SONI
University of California, Los Angeles, CA
90024-1547
The B-B mixing is unusually large, 1 and it may be the first sign that the effect of
The 3X3 K M m a t r i x
shows a definite pattern.
Let us w r i t e down the 4×4 KM matrix as we know it
physics beyond the standard model is in sight. One obvious candidate for such new physics is the new generation of quarks.
today:
The contribution
of (b',t') quarks to B-B mixing and rare decays of B mesons have been studied extensively in 2 the literature. They take a viewpoint that experimental checks of unitarity of 3×3 KM
1
~
0(A~ 3)
Vub ,
-k
1
Ak 2
Vcb ,
Vtd
Vts
Vtb
Vtb,
Vt, d
Vt, s
Vt, b
Vt~ b,
VKM =
(1)
matrix allows considerable freedom in the coupling between (t',b') to 3 generations of quarks.
For ~pacial range of parameters, there
Here ~ is the Cabibbo angle and A = i.
Note
that the off-diagonal terms Vcb and Vub are
will be additional contributions from loop
noticeably smaller than the diagonal terms.
diagrams involving {t',b') which dominat~ over
trying to guess the pattern of the KM matrix,
the predictions based on 3 families.
it is unrealisitic to hope that all values of
In this
talk, we shall impose several restrictions on
the matrix element consistent with unitarity
the K M m a t r i x elements which we feel are quite
will be realized.
We shall instead assume that
reasonable based on the existing pattern of the
Vtd , Vts , Vtb are roughly the same order of
known matrix elements. 3
magnitude as implies by the Wolfenstein
Our investigation seems to indicate that,
In
representation:
if (t',b') are responsible for at least some part of B-B mixing, mr, ~ 500 GeV.
We also
Vtd = 0(k 3} ; Vts = 0(~ 2) ; Vtb ffi 0(I).
(2)
find that their contributions to rare B decays are minimal.
Finally, their effect on CP
violation of B meson decays is modified in a
As for Vt, b , we assume that the observed trend for the off diagonal elements hold:
well controlled manner. vt, b ~ 0(k)
0920-5632/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
(3)
C. Hamzaoui et al. / 4th generation quarks in the B meson system
495
Since Vt, d is further away from the r E (Xc)ll;
diagonal element than Vt, s ,
'Vt.sVt,d' <- t
1/2k3 t' ~
(7)
"
E (xt,)113 Vt, d < kVt, s
(4) The key point which leads to a very strong
Finally, note that one virtue of the KM
constraint on IVt,sVt,d i is the fact that
matrix is that the smallness of ~ (CP violating
Imk 2 = 0(k6), while Rek 2 = 0(k2), and our c c assumption about the phase stated in (5).
parameter for the K meson system) is readily
Now use assumptions (3) and (4) to convert
explained by the fact that £ ~ 0(k 5) and the phase parameter is the K M m a t r i x maximal, E/2.
can be nearly
this bound to
We assume that this virtue
continues to hold in the 4x4 K M m a t r i x .
(8)
To be Elxt,lq ~'
more specific which is relevant for the size of t' sin2~td
contribution to the B-B mixing.
K = Isin2(~t, d _ ~t,s ) I < 1O
(5)
E(x )/E(x. ,)
(m /m_,) so that the coupling c 6c is suppressed by ~k mc/mt, .
c
where ~ij = arg Vii-
t
Lowez: B o u n d for mr,.
We emphasize that assumptions stated in Eqs.
Note that
2
(2)-(5) are pure speculation at this time. cf we define
We feel, however, that these are very mild assumptions and almost certainly valid if 4th
AM B
generation of quarks exist.
XB-
Constzaints f z o m E
XB
(9)
F =
2
(10)
2
BbdfbdGrMw mB¢ BRB/6E2
He now impose constraint that the 4th family of quarks do not change the prediction
where ~B' and mB are the B meson lifetime and
of £.
Considering the uncertainty in
mass, respectively, fbd is the B meson decay ~"
evaluating the h a d r o n i c m a t r i x element, this
constant and Bbd is the B meson bag factor and
constraint will be satisfied if the
finally,
contributions from the box diagrams with c
R B = l(VtdVtb)2qttE(xt )
quark is l~rger than those with t' quark, (11) + 2VtdV~bVt,dVt,b~tt,E(xt , ,x t)
(Im~) (.~E(xc))~
( I n ~ , ) ( q ~ E(xt,))
(6)
.
2
+ (Vt,dVt,b) qt't ' E(xt')l" ,
q
where qi = VisVid' qi are QCD correction coeff i c i e n t s and finally E(x) is a functions which 2 2 arises from computing the box diagram, 4 x = m./m 1 W (for small x, E(x) - x). is for the imaginary part.
Note that the inequality With the assumption
(5), we change the constraint into that for JV~,sVt,dl-
using the inequality (9), and the fact that
IV~V~I
(12)
2 ~k6D(l_p) + iql 2 < 2.6k 6 t'
C .85, = .86, ,-1, ql = qtt qt't q3
> .5 -
496
C. Hamzaoui et ad./ 4th generation quarks in the B meson system
we have shown in Fig. 1, the value of mr,
contributions.
needed in order to explain X B ~ .52 as a
b ~ s£+~ -, b ~ s + gluon, b w sT decays can also
function of m t .
obtain appreciable contributions from the
If Vt,sVt, b is large enough,
virtual t' quark. |
q
g
|
In this connection note that
J
(3), (4), and (7) when combined yields
E (Xc)ql K] 114k2 IV~'sVt'bl < [ E(xt,)~3'
(14) < .12k 2 for m t, ~500 GeV. I;
10~
This coupling is nearly a factor of 10 smaller _k 2 than VtsVtb expected in the 3 family model.
E
We do not, therefore, expect the effect of t' to be easily observable in rare (loop) decays of b.
The Pattezn of C P V i o l a t i o n im B Decays
20
To investigate the pattern of CP violation
eo
40
120
ml ( ~ v ) F-;~.
in B decays, let us start with the unitarity relation
I
We notice that the masses required for t'
vtJtb + V:#ob + VuJub + *.tdtbv. = o.
are rather large unless m t is large enough so that there is no need for t' to make up the
Using an approximation Vud = 1, Vcd = ~,
deficiency in X B .
and Vtb = 1,
Such a heavy ~' would, of
course, not be consistent with perturbative unitarity (and indeed retention of graphs to one loop order becomes invalid).
Vtd* + ~Vcb +rub + V~,dVt ,b ~ 0 .
(16)
Our result
~hould therefore be taken to imply that, if t'
Using
(4) and (14), we obtain
is needed to explain the deficiency in X B , then mr, must be above or close to the value
IV* t,dVt,b i < .12k 3
(17)
given by the perturbative unitarity -500 GeV. We also note that the constraint from the p
Noting that ali terms in (16) are 0(A3), it is
parameter 5
likely that vectors in the complex plane representing Vtd,, ~Vcb , Vub will close to a fairly
Imt,-mb, I < ] 80 GeV(90%CL)
(13)
good approximation.
The concept of unitarity
triangle in the 3 family case should remain combined with our result implies a rather high mass for the b'.
useful.
This traingle is shown in Fig. 2.
To see how (t',b') influence the pattern of CP violation in B decays, consider a CP
Raz~BDecays 6 It is well known that the loop decays of b quark are sensitive to virtual t quark
eigenstate f such that B, B ~ f. f = rE, ~K s etc.
For example
The time dependent CP violating
C. Hamzaoui et a l . / 4th generation quarks in the B meson system
497
~VK s = 2 (~+G) ~EE = 2 ( - ~ + G )
we emphasize here that ~ K •
-~EE = 2(~+~) and S.
is dependent only on the unltar~ty triangle for the 3 family case.
$mz~ We have investigated the role of 4th generation quarks
(t',b') on the B-B mixing, B
meson rare decay and B .meson CP violation. Deviating from previous analysis of many Fig. 2
authors,
we impose a set of assumptions in the
KM matrix elements which we feel are very
asymmetry is given by
reasonable.
These mild assumptions together
with the constraint that the description of K F(B(t)wf)-F(B(t)~f)
= sin~mt sin~f
(18)
meson decays is not grossly modified lead us to the conclusion that mr, > 500 GeV, if it were
F(B (t)wf) +F(B(t)~f}
to give nontrivial contribution to the B-B mixing.
where
Our analysis also shows that B meson
rare decays will have negligible effect from #f = arg[M22A(B~f)/A(Bgf) ] •
(19)
loop diagrams containing t' if our assumptions are valid.
For the case of three families it is known that
Also, we conclude that CP violation
in B meson decays will be modified in a controlled manner.
sin#~ K •
sln~ E
of asymmetries
= sin2~ S
=
(20)
-sin2~.
In particular,
measurement
in both B ~ EE and B ~ ~K s will
reveal the existence of new physics contribution to M12.
Note that (21) argMl2 = 2argV td
Figuze Captions 2'ig. 1. in the 3 family case.
For the case of 4 families,
we assumed that the box diagrams containing t' contribute substantially longer valid.
to M12 and
(21) is no
horizontal line shows the upper bound (500 GeV) for m t, taken from perturbative unitarity. Fig. 2.
Let us write
Curves show lower
bounds on m t, as in function of m t. The
The unitarity triangle.
P~fezenoes
argMl 2 = 2(argVt d + G)
(22)
where G is the additional phase of MI2 due to the t' contribution. ~o
Eqs.
(20) will be modified
*
Presented by A. Soni.
1.
H. Albretcht et al., Phys. Lett. $91B, 245 (1987), A. Jawahery, p. 545, Proceedings of the XXIV Int. Conf. on High Energy Physics, Eds. R. Kotthaus and J. H. Kuhn (1988}.
498
C. Hamzaoui et al. / 4th generation quarks in the B meson system
2.
See e . g . A . Datta, E. A. Paschos and U. Turke Phys. Lett. 1 9 ~ 376 (1987)~ G. Altarelli and P. Franzinl, Z. Phys. 37, 271 (1988); W. S. Hou and A. Sonl, Phys. Lett. 196B, 92 (1987). J. L. Hewett, Phys. Lett. 193B, 327 (1987); W.-S. Hou, R. S. willey and A. Soni, Phys. Ref. Lett 5a, 1608 (1987); G. Eilam, J. L. Hewett, T. G. Rizzo, Phys. Rev. D34, 2774 (1986); R. M. Godbole, U. Turke, M. Wirbel, Phys. Left. 194B, 302 (1987); X. G. He, S. Pakvasa, and H. Sugawara, Int. Symp. on the Fourth Family of Quarks and Leptons, Santa Monica, CA (1987).
3.
This talk is based on our paper: C. Hamzaoui et al. Phys. Rev. Lett. 62, 128 (1989).
4.
See T. Inami and C. S. Lim, Prog. Theor. Phys. ~5, 297 (1981), for functions associated with box diagrams. See F. J. Gilman and M. B. Wise, Phys. Rev. D27, 1128 (1983) for QCD corrections.
5.
W. J. Marciano, Proceedings of the First Int. Symp. on the Fourth Family of Quarks and Leptons. Eds. D. B. Cline and A. Soni, Annals of the New York Academy of Sciences, Vol. 518 (1987).
6.
This subject is discussed extensively in the First Int. Symp. on the Fourth Family of Quark and Leptons, Eds. D. B. Cline and A. Soni, Annals of the New York Academy of
Nuclear Physics B (Proc. Suppl.) 13 (1900) 494--498 North-Holland
EI~T~
OT 4TR GEa~ERATZON Q~U~KS ZN THE B ~ S O N
SYSTEM*
C. ~ A O U I Conco:~a
University, Montreal, Quebec, Canada H3G-IM8
I. SANDA Rockefeller University, 1230 York Avenue, New York, N.Y.
A.
10021
and A. SONI
University of California, Los Angeles, CA
90024-1547
The B-B mixing is unusually large, 1 and it may be the first sign that the effect of
The 3X3 K M m a t r i x
shows a definite pattern.
Let us w r i t e down the 4×4 KM matrix as we know it
physics beyond the standard model is in sight. One obvious candidate for such new physics is the new generation of quarks.
today:
The contribution
of (b',t') quarks to B-B mixing and rare decays of B mesons have been studied extensively in 2 the literature. They take a viewpoint that experimental checks of unitarity of 3×3 KM
1
~
0(A~ 3)
Vub ,
-k
1
Ak 2
Vcb ,
Vtd
Vts
Vtb
Vtb,
Vt, d
Vt, s
Vt, b
Vt~ b,
VKM =
(1)
matrix allows considerable freedom in the coupling between (t',b') to 3 generations of quarks.
For ~pacial range of parameters, there
Here ~ is the Cabibbo angle and A = i.
Note
that the off-diagonal terms Vcb and Vub are
will be additional contributions from loop
noticeably smaller than the diagonal terms.
diagrams involving {t',b') which dominat~ over
trying to guess the pattern of the KM matrix,
the predictions based on 3 families.
it is unrealisitic to hope that all values of
In this
talk, we shall impose several restrictions on
the matrix element consistent with unitarity
the K M m a t r i x elements which we feel are quite
will be realized.
We shall instead assume that
reasonable based on the existing pattern of the
Vtd , Vts , Vtb are roughly the same order of
known matrix elements. 3
magnitude as implies by the Wolfenstein
Our investigation seems to indicate that,
In
representation:
if (t',b') are responsible for at least some part of B-B mixing, mr, ~ 500 GeV.
We also
Vtd = 0(k 3} ; Vts = 0(~ 2) ; Vtb ffi 0(I).
(2)
find that their contributions to rare B decays are minimal.
Finally, their effect on CP
violation of B meson decays is modified in a
As for Vt, b , we assume that the observed trend for the off diagonal elements hold:
well controlled manner. vt, b ~ 0(k)
0920-5632/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
(3)
C. Hamzaoui et al. / 4th generation quarks in the B meson system
495
Since Vt, d is further away from the r E (Xc)ll;
diagonal element than Vt, s ,
'Vt.sVt,d' <- t
1/2k3 t' ~
(7)
"
E (xt,)113 Vt, d < kVt, s
(4) The key point which leads to a very strong
Finally, note that one virtue of the KM
constraint on IVt,sVt,d i is the fact that
matrix is that the smallness of ~ (CP violating
Imk 2 = 0(k6), while Rek 2 = 0(k2), and our c c assumption about the phase stated in (5).
parameter for the K meson system) is readily
Now use assumptions (3) and (4) to convert
explained by the fact that £ ~ 0(k 5) and the phase parameter is the K M m a t r i x maximal, E/2.
can be nearly
this bound to
We assume that this virtue
continues to hold in the 4x4 K M m a t r i x .
(8)
To be Elxt,lq ~'
more specific which is relevant for the size of t' sin2~td
contribution to the B-B mixing.
K = Isin2(~t, d _ ~t,s ) I < 1O
(5)
E(x )/E(x. ,)
(m /m_,) so that the coupling c 6c is suppressed by ~k mc/mt, .
c
where ~ij = arg Vii-
t
Lowez: B o u n d for mr,.
We emphasize that assumptions stated in Eqs.
Note that
2
(2)-(5) are pure speculation at this time. cf we define
We feel, however, that these are very mild assumptions and almost certainly valid if 4th
AM B
generation of quarks exist.
XB-
Constzaints f z o m E
XB
(9)
F =
2
(10)
2
BbdfbdGrMw mB¢ BRB/6E2
He now impose constraint that the 4th family of quarks do not change the prediction
where ~B' and mB are the B meson lifetime and
of £.
Considering the uncertainty in
mass, respectively, fbd is the B meson decay ~"
evaluating the h a d r o n i c m a t r i x element, this
constant and Bbd is the B meson bag factor and
constraint will be satisfied if the
finally,
contributions from the box diagrams with c
R B = l(VtdVtb)2qttE(xt )
quark is l~rger than those with t' quark, (11) + 2VtdV~bVt,dVt,b~tt,E(xt , ,x t)
(Im~) (.~E(xc))~
( I n ~ , ) ( q ~ E(xt,))
(6)
.
2
+ (Vt,dVt,b) qt't ' E(xt')l" ,
q
where qi = VisVid' qi are QCD correction coeff i c i e n t s and finally E(x) is a functions which 2 2 arises from computing the box diagram, 4 x = m./m 1 W (for small x, E(x) - x). is for the imaginary part.
Note that the inequality With the assumption
(5), we change the constraint into that for JV~,sVt,dl-
using the inequality (9), and the fact that
IV~V~I
(12)
2 ~k6D(l_p) + iql 2 < 2.6k 6 t'
C .85, = .86, ,-1, ql = qtt qt't q3
> .5 -
496
C. Hamzaoui et ad./ 4th generation quarks in the B meson system
we have shown in Fig. 1, the value of mr,
contributions.
needed in order to explain X B ~ .52 as a
b ~ s£+~ -, b ~ s + gluon, b w sT decays can also
function of m t .
obtain appreciable contributions from the
If Vt,sVt, b is large enough,
virtual t' quark. |
q
g
|
In this connection note that
J
(3), (4), and (7) when combined yields
E (Xc)ql K] 114k2 IV~'sVt'bl < [ E(xt,)~3'
(14) < .12k 2 for m t, ~500 GeV. I;
10~
This coupling is nearly a factor of 10 smaller _k 2 than VtsVtb expected in the 3 family model.
E
We do not, therefore, expect the effect of t' to be easily observable in rare (loop) decays of b.
The Pattezn of C P V i o l a t i o n im B Decays
20
To investigate the pattern of CP violation
eo
40
120
ml ( ~ v ) F-;~.
in B decays, let us start with the unitarity relation
I
We notice that the masses required for t'
vtJtb + V:#ob + VuJub + *.tdtbv. = o.
are rather large unless m t is large enough so that there is no need for t' to make up the
Using an approximation Vud = 1, Vcd = ~,
deficiency in X B .
and Vtb = 1,
Such a heavy ~' would, of
course, not be consistent with perturbative unitarity (and indeed retention of graphs to one loop order becomes invalid).
Vtd* + ~Vcb +rub + V~,dVt ,b ~ 0 .
(16)
Our result
~hould therefore be taken to imply that, if t'
Using
(4) and (14), we obtain
is needed to explain the deficiency in X B , then mr, must be above or close to the value
IV* t,dVt,b i < .12k 3
(17)
given by the perturbative unitarity -500 GeV. We also note that the constraint from the p
Noting that ali terms in (16) are 0(A3), it is
parameter 5
likely that vectors in the complex plane representing Vtd,, ~Vcb , Vub will close to a fairly
Imt,-mb, I < ] 80 GeV(90%CL)
(13)
good approximation.
The concept of unitarity
triangle in the 3 family case should remain combined with our result implies a rather high mass for the b'.
useful.
This traingle is shown in Fig. 2.
To see how (t',b') influence the pattern of CP violation in B decays, consider a CP
Raz~BDecays 6 It is well known that the loop decays of b quark are sensitive to virtual t quark
eigenstate f such that B, B ~ f. f = rE, ~K s etc.
For example
The time dependent CP violating
C. Hamzaoui et a l . / 4th generation quarks in the B meson system
497
~VK s = 2 (~+G) ~EE = 2 ( - ~ + G )
we emphasize here that ~ K •
-~EE = 2(~+~) and S.
is dependent only on the unltar~ty triangle for the 3 family case.
$mz~ We have investigated the role of 4th generation quarks
(t',b') on the B-B mixing, B
meson rare decay and B .meson CP violation. Deviating from previous analysis of many Fig. 2
authors,
we impose a set of assumptions in the
KM matrix elements which we feel are very
asymmetry is given by
reasonable.
These mild assumptions together
with the constraint that the description of K F(B(t)wf)-F(B(t)~f)
= sin~mt sin~f
(18)
meson decays is not grossly modified lead us to the conclusion that mr, > 500 GeV, if it were
F(B (t)wf) +F(B(t)~f}
to give nontrivial contribution to the B-B mixing.
where
Our analysis also shows that B meson
rare decays will have negligible effect from #f = arg[M22A(B~f)/A(Bgf) ] •
(19)
loop diagrams containing t' if our assumptions are valid.
For the case of three families it is known that
Also, we conclude that CP violation
in B meson decays will be modified in a controlled manner.
sin#~ K •
sln~ E
of asymmetries
= sin2~ S
=
(20)
-sin2~.
In particular,
measurement
in both B ~ EE and B ~ ~K s will
reveal the existence of new physics contribution to M12.
Note that (21) argMl2 = 2argV td
Figuze Captions 2'ig. 1. in the 3 family case.
For the case of 4 families,
we assumed that the box diagrams containing t' contribute substantially longer valid.
to M12 and
(21) is no
horizontal line shows the upper bound (500 GeV) for m t, taken from perturbative unitarity. Fig. 2.
Let us write
Curves show lower
bounds on m t, as in function of m t. The
The unitarity triangle.
P~fezenoes
argMl 2 = 2(argVt d + G)
(22)
where G is the additional phase of MI2 due to the t' contribution. ~o
Eqs.
(20) will be modified
*
Presented by A. Soni.
1.
H. Albretcht et al., Phys. Lett. $91B, 245 (1987), A. Jawahery, p. 545, Proceedings of the XXIV Int. Conf. on High Energy Physics, Eds. R. Kotthaus and J. H. Kuhn (1988}.
498
C. Hamzaoui et al. / 4th generation quarks in the B meson system
2.
See e . g . A . Datta, E. A. Paschos and U. Turke Phys. Lett. 1 9 ~ 376 (1987)~ G. Altarelli and P. Franzinl, Z. Phys. 37, 271 (1988); W. S. Hou and A. Sonl, Phys. Lett. 196B, 92 (1987). J. L. Hewett, Phys. Lett. 193B, 327 (1987); W.-S. Hou, R. S. willey and A. Soni, Phys. Ref. Lett 5a, 1608 (1987); G. Eilam, J. L. Hewett, T. G. Rizzo, Phys. Rev. D34, 2774 (1986); R. M. Godbole, U. Turke, M. Wirbel, Phys. Left. 194B, 302 (1987); X. G. He, S. Pakvasa, and H. Sugawara, Int. Symp. on the Fourth Family of Quarks and Leptons, Santa Monica, CA (1987).
3.
This talk is based on our paper: C. Hamzaoui et al. Phys. Rev. Lett. 62, 128 (1989).
4.
See T. Inami and C. S. Lim, Prog. Theor. Phys. ~5, 297 (1981), for functions associated with box diagrams. See F. J. Gilman and M. B. Wise, Phys. Rev. D27, 1128 (1983) for QCD corrections.
5.
W. J. Marciano, Proceedings of the First Int. Symp. on the Fourth Family of Quarks and Leptons. Eds. D. B. Cline and A. Soni, Annals of the New York Academy of Sciences, Vol. 518 (1987).
6.
This subject is discussed extensively in the First Int. Symp. on the Fourth Family of Quark and Leptons, Eds. D. B. Cline and A. Soni, Annals of the New York Academy of