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CMU-HEP-98-01 New Physics E?ects in CP-Violating B Decays

L. WOLFENSTEIN

arXiv:hep-ph/9801386v1 21 Jan 1998

Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

? Contributions to B ? B mixing from physics beyond the standard model may be detected from CP-violating asymmetries in B decays. There exists the possibility of large new contributions that cannot be detected by ?rst generation experiments because of a discrete ambiguity. Some possible strategies for resolving this are discussed.

A major goal of the experiments on B mesons is to check the standard

model, or conversely, to discover new physics. In many models beyond the ? standard model, there exist new contributions to B ? B mixing [1]. In this

paper, we assume that this is the only new physics and discuss strategies to

detect it. An important conclusion is that even large new contributions due ? to Bd ? Bd mixing may be di?cult to detect. Of course in some models, the

existence of such large new contributions might imply other deviations from

the standard model such as the rates for rare decay processes [2]. 1

? The ?rst information on B ? B mixing comes from the measurement of

? m or xd = ? m/Γ. This is proportional to

2 A2 (1 ? ρ)2 + η 2 ) BB η2 fB

2 where BB η2 fB involves the hadronic matrix element. Given the hadronic

uncertainty and conservative limits on the CKM matrix parameters (ρ, η) the

standard model predicts xd only within a factor of about ten. The experimental

result xd = 0.7 ?ts very nicely but provides weak constraints on new physics.

The next information, a major goal of B factories, is the phase of M12 ? in the standard phase convention. This is given by 2β and determined from ? measuring the CP-violating asymmetry sin2β in the decay B → ψKs . In the ? standard model β = β, the phase of Vtd , and is constrained to lie between

8? and 32? corresponding to sin 2β between 0.3 and 0.9. Thus a magnitude ? clearly below 0.3 or a negative value of sin 2β would indicate new physics. 2

? To proceed we assume that measurements yield sin2β between 0.4 and 0.8

corresponding in the standard model to a value

? ? β = β1 = 12? to 27?

? There exists the possibility that the true value of β is

π ? ? β2 = ? β1 , = 78? to 63? 2

(1a)

This would mean a large new physics contribution that reverses the sign of

Re M12 . Within the standard phase convention this new physics contribution

could be approximately CP invariant. As we now proceed to show this large

new physics e?ect is not easy to detect. ? The next goal of B factories is the measurement of sin 2 ( β + γ ) from

the asymmetry in decays like B 0 → π + π ? . For the moment we neglect the

penguin problem and assume this is measured. In the standard model there 3

? is almost no constraint [3] on the possible value of sin 2 ( β + γ ) for a value ? of sin 2β in the range we have assumed. Within the standard model there ? will in general be only one set of angles ( β1 , γ1 ) consistent with these two

measurements, although in general there is an eight-fold ambiguity [4]. In ? ? particular, corresponding to the choice β = β2 there is a corresponding choice

γ2 = π ? γ1

(1b)

? Since the allowable values of γ, which are independent of B ? B mixing

and in our scenario are unchanged by the new physics, are approximately

symmetric with respect to 90

?

the choice γ2 is always allowable. A number

of experiments are directed at determining sinγ; this does not distinguish γ1 ,

from γ2 .

If γ1 is far from 90

?

corresponding to |ρ| ≥ 0.2 then γ2 is distinguished 4

from γ1 by the sign of ρ and thus by the magnitude of Vtd . The best prospect

for determining this is from the rate [5] of K + → π + ν ν which is approximately ?

proportional to

[ (1 ± .15) + (2 ± .25) (1 ? ρ ? i η) ]

2

where the ?rst conservative error is due to the charm contribution and the

second to uncertainty in mt and Vcb . For |ρ| = 0.2 the di?erence between the

two signs of ρ is almost a factor of 2 in the K + → π + ν ν rate. ?

Another possibility is to look for interfering amplitudes that can be used

to determine cosγ. An example is the penguin-tree interference in the decay

B 0 → π ? K + . In contrast one expects that the decay B + → π + K 0 is pure

penguin. One then ?nds [6]

R =

Γ ( B 0 → π? K + ) = 1 ? 2r cos γ + r2 Γ ( B + → π+ K 0 ) 5

(2)

where r is the ratio of tree to penguin. If we accept the sign of r as given by

1 3

factorization and note that we expect r ≤

then the sign of (1 ? R) gives

the sign of cos γ which can distinguish γ1 from γ2 .

However, if cos γ is close to zero, corresponding to ρ close to zero, which

is in the center of the allowed (ρ, η) region, then neither of the above methods

can distinguish the solutions in Eq. (2) from the standard model. ? Instead of relying on γ one can try to ?nd a method of distinguishing β1 ? from β2 . Grossman and Quinn [4] suggest comparing the asymmetry in the

decay B → D+ D? to that of B → ψ Ks . Including a penguin contribution to

the D+ D? decay they ?nd

? ? ? a (D+ D? ) = sin 2β ? 2r cos 2β sin β cos δ

(3)

where r is the penguin to tree amplitude ratio and δ is the strong phase dif-

ference between penguin and tree. If one assumes r < 0 from factorization 6

? ? and cos δ > 0 then if β = β1 the asymmetry is reduced due to the penguin ? ? whereas if β = β2 the asymmetry is increased. ? ? Actually if β = β2 Eq. (3) is not correct since it assumes that the phase of ? the penguin amplitude, given by the phase β of Vtd , equals β. However in the ? scenario we consider while β is given by Eq. (1a), the phase β is constrained

to lie between 12? and 27? . In this case Eq. (3) becomes to ?rst order in r

? ? ? a (D+ D? ) = sin 2β2 ? 2r cos 2β2 sin (2 β2 ? β) cos δ

(4)

The previous conclusion that if r < 0 the asymmetry is increased by the ? ? penguin if β = β2 still holds. ? ? Another way to directly distinguish β2 from β1 in this scenario involves

decays dominated by the b → d penguin graph. Assuming t dominance the ? asymmetry of a decay like Bd → K 0 K 0 is given by sin 2 (β ? β). If we assume ? ? β is around 70? , corresponding to typical β2 value then any allowable value 7

of β gives an asymmetry greater than 0.9. In contrast in the standard model ? β = β and the asymmetry vanishes. Fleischer [7] has pointed out that there

may be signi?cant contributions from u and c quarks such that the standard

model value may not be zero. Nevertheless a very large asymmetry of 80% or

greater would be strong evidence for new physics. While the branching ratio is

small not so many events are needed just to show that the asymmetry is very

large.

We turn now to the Bs system. The ?rst quantity of interest that can be

measured is xs . The ratio xd /xs is given in the SM by

xd = λ2 xs

( 1 ? ρ )2 + η 2

K

(5)

2 where K is the ratio of BB η2 fB for the Bd as compared to Bs . In the SU(3)

limit K = 1 and estimates from lattice and other calculations give K between

0.7 and 0.9. Thus the measurement of xs can be used to put a constraint 8

on (ρ, η), primarily on ρ. In fact the present limit on xs disfavors values

ρ < ?0.2. A small value of xs leads to a signi?cant negative value of ρ and

a large value of xs to a positive value ρ. If this is inconsistent with the value

of (ρ, η) determined from the asymmetry measurements it could be a sign of

new physics in Bd ? Bd mixing. Note that this new physics in general would ? cause β to be di?erent from β and change the value of xd invalidating Eq. (5).

However, the larger new contribution to Bd ? Bd mixing implied by Eq. (1a)

could not be demonstrated in this way.

It would also be possible to compare the values of (ρ, η), mainly (1 ? ρ),

that ?ts xs /xd with that from K + → π + ν ν . If these are inconsistent it would ?

be probably a sign of a new physics contribution to xd .

If ?ms is not too large one can study the CP violating asymmetries from

the sin (?ms t) term in tagged Bs decays. For decays such as Bs → ψ η the

asymmetry is given by sin θs where θs = 2 λ2 η which is between .02 and .05. 9

If the asymmetry is signi?cantly larger that would be a sign of new physics

in Bs ? Bs mixing. For decays governed by b → u?d, such as Bs → ρ0 KS , u

the asymmetry in the tree approximation is sin (θs + 2 γ). If θs is consistent

with zero this gives sin 2 γ, the sign of which distinguishes γ2 from γ1 . There

is likely a sizable penguin contribution to Bs → ρ0 Ks , but the fact that one

wants only the sign of sin γ may make this useful in spite of the penguin.

In analyzing prospective B asymmetry experiments its is natural and ap-

propriate to assume the standard model and see how well these can constrain

the parameters (ρ, η). The purpose of the present note is to emphasize that it

is also important to look at new physics e?ects and see whether or not a given

set of experiments can detect them.

In particular we have looked at one particular ambiguity given by Eqs.

(1), which implies large new physics e?ects which may prove very di?cult to

10

detect. Proposed experiments should be analyzed from the point of view of

resolving such ambiguities.

Acknowledgments

This research was supported in part by the U.S. Dept. of Energy under DE-

FG02-91-ER-40682.

11

References

1. M. Gronau and D. London, Phys. Rev. D 55, 2845 (1997); and references

therein..

2. Y. Grossman and M. P. Worah, Phys. Lett. B395, 241 (1997).

3. We use as a guide the results given by D. Silverman, Int. J. Mod. Phys.

A 11, 2253 (1996).

4. Y. Grossman, Y. Nir, and M. P. Worah, Phys. Lett. B407, 307 (1997); Y.

Grossman and H. Quinn, Phys. Rev. D56, 7259 (1997); L. Wolfenstein

“Searching for New Physics From CP Violation in B Decays”, B Physics

Conference, Honolulu 1997 (in press).

5. G. Buchalla and A. J. Buras, Phys. Rev. D54, 6782 (1996).

6. R. Fleischer and T. Mannel, Korlsrule preprint, hep-ph/9704423.

7. R. Fleischer, Phys. Lett. B341, 205 (1994).

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