Estimates of long-distance contributions to the Bs→γγ decay

Estimates of long-distance contributions to the Bs→γγ decay

6 August 1998 Physics Letters B 433 Ž1998. 102–108 Estimates of long-distance contributions to the Bs ™ gg decay Debajyoti Choudhury a a,1 , John ...

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6 August 1998

Physics Letters B 433 Ž1998. 102–108

Estimates of long-distance contributions to the Bs ™ gg decay Debajyoti Choudhury a

a,1

, John Ellis

b,2

Mehta Research Institute of Mathematics and Mathematical Physics, Chhatnag Road, Jhusi, Allahabad 211019, India b Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Received 21 April 1998 Editor: R. Gatto

Abstract We present first calculations of new long-distance contributions to Bs ™ gg decay due to intermediate Ds and Ds) meson states. The relevant g vertices are estimated using charge couplings and transition moment couplings. Within our uncertainties, we find that these long-distance contributions could be comparable to the known short-distance contributions. Since they have different Cabibbo-Kobayashi-Maskawa matrix-element factors, there may be an interesting possibility of observing CP violation in this decay. q 1998 Elsevier Science B.V. All rights reserved.

A new era of experiments on rare B decays is about to dawn. The large data sets already obtained at LEP and CESR will be dwarfed by those provided by the eqey B factories, and by the hadron experiments HERA-B, CDF, D0 and LHC-B. Among the rare B decays with particularly clean experimental signatures is Bs ™ gg , whose branching ratio presently has the experimental upper limit BŽ Bs ™ gg . - 1.48 = 10y4 w1x. Higher-sensitivity measurements of Bs decays are not among the highest priorities of the eqey B factories, whose physics programmes are focussed initially on searches for CP violation in Bd decays, but the hadron experiments cannot avoid being sensitive to BŽ Bs ™ gg . to levels several orders of magnitude below the present experimental upper limit w1x. The lowest-order short-distance contributions to the Bs ™ gg decay arise from two sets of graphs: Ž i . box diagrams, and Ž ii . triangle diagrams with an external photon leg. These have been calculated w2–4x, and yield, for m t f 175 GeV, 3 BŽ Bs ™ gg . ; 3.8 = 10y7 , the next generation of experiments. Interestingly, the branching fraction can be enhanced substantially in extensions of the Standard Model such as a generic 2-Higgs scenario w5x. Emboldened by the fact that the short-distance QCD corrections are not too big w6,5,7x, one may then even attempt to use such data as may be forthcoming as probes for new physics 4 .

1

E-mail: [email protected] E-mail: John. [email protected] 3 Over a wide range of the top-quark mass, the partial width grows linearly with m t w4x. 4 Please note, however, that Reina et al. w8x claim that the short-distance QCD corrections may be as large as 100%. 2

0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 7 9 - 0

D. Choudhury, J. Ellis r Physics Letters B 433 (1998) 102–108

103

However, it is not immediately obvious that this decay is dominated by such short-distance contributions. Estimates have been made of some long-distance contributions to the decay amplitudes, for example via intermediate charmonium states Jrc and c X w9x. These were found to be very small, indicating that the short-distance contribution might be correct in order of magnitude. But, before concluding this to be the case, one must examine other possible long-distance contributions. In this paper we study such contributions due to intermediate Ds and Ds) states via the diagrams shown in Figs. 1 and 2. These include loops of Ds mesons alone, loops of Ds) mesons alone, and diagrams involving radiative Ds ™ Ds) transitions. The vertices are estimated using data on B ™ DD, DD ) and D ) D ) decays, the electromagnetic charge couplings of the Ds and Ds) , and phenomenological estimates of the Ds) ™ Ds q g decay rate for the inelastic transition-matrix element. The imaginary part of the diagrams can be calculated easily while the determination of the real part requires the use of dispersion relations. We find that these long-distance contributions to Bs ™ gg decay may be comparable to the short-distance contributions calculated previously, within the considerable uncertainties inherent to the phenomenological inputs we use. We find no evidence that BŽ Bs ™ gg . decay should occur at a rate very close to the present experimental upper limit, but even this possibility cannot be ruled out. Since the short- and long-distance contributions have different Cabibbo-Kobayashi-Maskawa matrix-element factors, the fact that they may be of comparable magnitudes offers the interesting possibility of observing CP violation in this decay mode. However, a detailed exploration of this possibility lies beyond the scope of this paper. At the quark level, the operators responsible for the BŽ Bs ™ gg . decay are bg 5 sFmn F mn and bg 5 sFmn F˜ mn , where F is the electromagnetic fileld strength tensor, and F˜ is its dual. The matrix element for the decay Bs ™ g Ž q1 m .g Ž q2 n . can then be parametrized as M s a G Ž R1 Smn q i R2 Pmn . ,

Smn ' q1 n q2 m y q1 P q2 gmn ,

Pmn ' emn a b q1a q2b ,

Gs

GF

'2

VcbVc)s

Ž 1.

where R1 and R2 are the yet-to-be-determined hadronic matrix elements, and Ž a G . has been factored out in anticipation of the results to be derived later in the text. The partial width is then given by

G Ž Bs ™ gg . s

Ž aG.

2

64p

m 3B Ž < R1 < 2 q < R2 < 2 . .

Ž 2.

Since G Ž Bs . f 5 = 10y1 0 MeV, we have, for < Vcb < s 0.04, y7

Br Ž Bs ™ gg . f 10

ž

R1

2

q

100 MeV

R2 100 MeV

2

/

.

Ž 3.

It can be argued that the long-distance contributions should be dominated by two-meson intermediate states. Now, the quark-level transitions with the least Cabibbo suppression are b ™ ccs and b ™ cud. The second process obviously leads to very dissimilar mesons which may rescatter into two photons only through higher-order terms. Consequently, we concentrate on the b ™ ccs case, with the corresponding weak Hamiltonian being given by Hwk s G Jm j m ,

Jm s cgm Ž 1 y g 5 . b ,

j m s sgm Ž 1 y g 5 . c ,

Ž 4.

)q )y where V denotes the CKM matrix. The relevant intermediate states are then DsqDy Ds , Ds)q Dsy, etc. 5. s , Ds To calculate the Bs ™ gg decay amplitude exactly, we need to determine the corresponding matrix elements of

5

Since the contributions due to charmonia have already been investigated w9x, we shall not consider these here.

D. Choudhury, J. Ellis r Physics Letters B 433 (1998) 102–108

104

Ž4., a non-trivial task. However, as a first approximation, we may assume naive factorization 6 . The Hwk matrix elements are then expressible in terms of simpler quantities given by ² Dsy Ž q . < j m <0: s yifq m ,

² Ds)y Ž k , e . < j m <0: s m ) f ) e ) m ,

² Dsy Ž k . < Jm < Bs Ž p . : s 0 ,

² Ds)y Ž k , e . < Jm < Bs Ž p . : s 0 ,

² Dsq Ž k . < Jm < Bs Ž p . : s fq Ž q 2 . Ž p q k . m q fy Ž q 2 . qm , ² Ds)q Ž k , e . < Jm < Bs Ž p . : s

V m)

emn a b e ) n p a k b q

i m)

½

m2B A1 en) y e ) P p A2 Ž k q p . m q

m2) q2

A3 qm

5

,

Ž 5.

where q s p y k in the last two terms, and m ) ' m D ) . The quantities V and Ai are related to the usual form factors through Vs A1 s

2 m) mB q m)

VB s D s) Ž q 2 . ,

m) Ž mB q m) . m2B

A1B s D s) Ž q 2 . ,

A2 s

m) mB q m)

A2B s D s) Ž q 2 . ,

A3 s 2 A3B s D s) Ž q 2 . y A0B s D s) Ž q 2 . .

Ž 6. Finally, we have ² Dsq Ž k . Dsy Ž q . < Hwk < Bs Ž p . : s yi G f

Ž p 2 y k 2 . fq Ž q 2 . q q 2 fy Ž q 2 .

,

² Ds)q Ž k . Ds)y Ž q . < Hwk < Bs Ž p . : s G f ) eq) mey) n V Ž q 2 . e nm a b pa kb q i  A1 Ž q 2 . p 2 g mn y 2 A2 Ž q 2 . p m p n 4 , ² Ds)q Ž k , e . Dsy Ž q . < Hwk < Bs Ž p . : s

Gf m)

e ) P p p 2 A1 Ž q 2 . y Ž p 2 y k 2 . A2 y m2) A3 Ž q 2 . ,

² Dsq Ž k . Ds)y Ž q, e . < Hwk < Bs Ž p . : s 2 G m ) f ) fq Ž q 2 . e ) P p .

Ž 7.

Within the factorization approximation,the relations in Ž7. can be interpreted as a parametrization of the BDsŽ ) . DsŽ ) . vertices. The form factors themselves are nonperturbative quantities. Apart from lattice calculations w10x, the best arenas for determining these are phenomenological approaches such as the BSW model w11x or heavy-quark effective theory w12x. Some of these parameters are also measured experimentally w13x, although a somewhat V ; 0.6 and A1,2 ; 0.25. On large spread persists in the data. Typically, though, we have f, f ) ; 200 MeV, f ",V the other hand, A3 is very small. As a next step, we need to determine the meson-photon couplings. Since we are dealing here with charged mesons, the simplest course is to assume minimal substitution. This then fixes the charge coupling uniquely for each angular-momentum state of the Ž cs . system. Of course, the fact that these mesons are not fundamental particles means that we should ideally discuss a series of form factors, including charge radii as well as higher moments. As a first approximation, however, we neglect such aspects of the meson-photon coupling.

6

Based on experience elsewhere, we may hope that the associated errors are - O Ž10%..

D. Choudhury, J. Ellis r Physics Letters B 433 (1998) 102–108

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The one-loop contribution due to the Ds alone can then be expressed in terms of the diagrams in Fig. 1. Computing these diagrams, the first one gives Ž1. Gmn s

d4 k

H Ž 2p .

4

2 Ž yi G . f Ž p 2 y k 2 . fqq Ž p y k . fy

i

=

2

Ž kyq. ym

2

Ž ie . w 2 k y q1 y p x n

i 2

k y m2

i

Ž ie . w 2 k q q1 x m

,

2 Ž k y p . y m2

whilst the third gives Ž3. Gmn s

d4 k

H Ž 2p .

4

2 Ž yi G . f Ž p 2 y k 2 . fqq Ž p y k . fy

i 2

i 2

k y m Ž k y p . 2 y m2

Ž 2 ie 2 gmn . .

In each of the above, the form-factors f " are to be evaluated at Ž p y k . 2 . We note that an exact calculation of the above integrals needs knowledge of the momentum dependence of the form factors. Furthermore, the integrals are formally divergent, and to calculate the real parts we would need to consider a cutoff scale that sets the limit of validity of a theory with ‘fundamental’ meson fields. Rather than attempt this, we take recourse to the optical theorem and calculate only the absorptive parts of the diagrams above. On using the Cutkosky rules, the sum of the three diagrams reduces to Ž Ds. Im Ž Gmn . s 2 a G m˜ 1 f

Ii jk s ln

Ž 1 y m˜ 1 . fq Ž m2 . q m˜ 1 fy Ž m2 .

1q2m ˜ k y m˜ i y m˜ j q l i j 1q2m ˜ k y m˜ i y m˜ j y l i j

,

I111 Smn , 2

li j s Ž1 y m ˜ i y m˜ j . y 4 m˜ i m˜ j

1r2

.

Ž 8.

Here m ˜ i s m2i rm2B with m1 s m ' m D s and m 2 s m ) ' m D s) . The form of Ž8. is a testimonial to the fact that the three diagrams of Fig. 1 together form a gauge-invariant set. Thus, if Ds were the only meson that could contribute, R1,2 would be determined by Ž8., and Im R1 Ž Ds . s 0.39 f

fq Ž m 2 . q 0.16 fy Ž m 2 . ,

Im R2 Ž Ds . s 0 .

Ž 9.

The CP-violating form factor is thus identically zero. What about the real parts of the amplitudes R1? These may be computed using dispersion relations w15x, and, for the case in hand, are seen to be smaller than the imaginary parts. Substituting for the form factors in the above and using Ž3., we then have 2 Ds

Br Bs ™ gg ; 2.5 = 10y8 .

ž

/

Ž 10 .

This, by itself, is small compared to the short-distance contribution w6x, though the interference term can be significant.

Fig. 1. One-loop contribution to the Bs ™ gg amplitude due to Ds mesons alone.

D. Choudhury, J. Ellis r Physics Letters B 433 (1998) 102–108

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What about the other long-distance effects? Of the ones calculated in the literature, Bs ™ fg ™ gg has an amplitude somewhat smaller than the one calculated here, whilst Bs ™ Jrcf ™ gg is seen to contribute only at the 1% level w9x. We are thus in a situation where the 2 Ds intermediate state may, in fact, give the largest long-distance contribution to the decay amplitude. It is interesting, at this stage, to compare the Bs ™ gg case with that for K S ™ gg decay. Whereas there it is the 2p intermediate state that dominates the decay amplitude w16x, the analogous contribution to the Bs decay falls well short of the short-distance amplitude. We now turn our attention to the next higher state that can contribute to this process, namely the Ds) . As long as we neglect any Ds) Dsg coupling, the additional diagrams involving the Ds) field are exact analogues of those in Fig. 1, and lead to

Im R1 Ž Ds) . s

f) 8m ˜ 22

A1 Ž m 2) .  Ž 1 y 4 m ˜ 2 . l22 q m˜ 2 Ž 1 y 12 m˜ 2 q 48 m˜ 22 . I222 4

A2 Ž m 2) .  Ž m yA ˜ 2 y 5 . l22 q m˜ 2 Ž 1 y 10 m˜ 2 q 32 m˜ 22 . I222 4 A1 Ž m2) . q 15.3 A2 Ž m2) . , s f ) 1.57A Im R2 Ž Ds) . s

f ) V Ž m 2) . 8m ˜2

 Ž12 m˜ 2 y 1. l22 q Ž 4 m˜ 2 y 32 m˜ 22 . I222 4 s 0.13 f) V Ž m2) . .

Ž 11 .

Two new features confront us. One is that the CP-violating amplitude is non-zero, though small. More relevantly for our present purpose, we find Im R1Ž Ds) . ; 800 MeV, on the basis of which Ž3. then gives us 2 Ds)

Br Bs ™ gg ; 6.5 = 10y6 .

ž

/

Ž 12 .

This, of course, is much larger than the short-distance contribution, and suggests that this particular decay should be observable at the very first run of the hadronic machines! A few objections could possibly be raised against this conclusion. For one thing, the validity of the factorization approximation as applied to decays into two vector mesons is not apparent. Thus, instead of using Ž5., it would be preferable to use the data to parametrize the Bs Ds) Ds) vertex. Unfortunately, though, this exclusive mode is undetected so far. Once it is measured, it will be a straightforward task to reexpress our result as a prediction of the ratio of the two decay modes. Secondly, we have, until now, neglected quite a few possible contributions to the process. For example, the contributions due to the higher excited states could in principle be comparable in magnitude, and might interfere destructively, thus reducing the total long-distance contribution. However, in the absence of extensive data on these states, it is almost impossible to estimate such contributions to any degree of reliability. We can only hope that this lack of information does not invalidate the results presented here. Finally, there is another class of contribution that we have neglected so far, involving the Ds) Dsg vertex. Whilst this transition moment can, in principle, be calculated within a given model for the mesons, it is easier to work in terms of an effective Lagrangian of the form

Leff s

eD m)

e absh Dsy Ea Ab Es Ds)q h .

Ž 13 .

D. Choudhury, J. Ellis r Physics Letters B 433 (1998) 102–108

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Fig. 2. One-loop contribution to the Bs ™ gg amplitude involving the Ds) Dsg vertex. The crossed diagrams are not shown.

We shall assume that D is real, i.e. there is no absorptive part associated with this vertex. The new contributions are given by the diagrams of Fig. 2, along with the crossed ones, and the resultant shifts d R1,2 given by 4m ˜ 2 Im d R1 s < D < 2 f Ž 1 y m˜ 1 . fq Ž m2 . q m˜ 1 fy Ž m2 .  l11 y m˜ 2 I112 4 y < D < 2 f ) A1 Ž m2) .  l12 y m ˜ 1 I221 4 A2 Ž m2) . qA

2

½ Ž m˜ y m˜ y 0.5. l q ž Ž m˜ y m˜ . q m˜ / I 5 2

1

12

2

1

1

221

,

½ž

8m ˜ 2 Im d R2 s D f ) fq Ž m2) . 2 m˜ 13 y m˜ 12 y 4 m˜ 12 m˜ 2 q 2 m˜ 1 m˜ 2 q 5m˜ 1 m˜ 22 y 2 m˜ 32 q 3m˜ 22 y m˜ 2 I122

/

q8m ˜ 1 m˜ 2 I121 q 3 Ž m˜ 1 y m˜ 12 q m˜ 22 y 3m˜ 2 . l12

5

q D f A1 Ž m2 . q Ž m ˜ 2 y 1 . A2 Ž m2 . y m˜ 2 A3 Ž m2 .

½

ž

= 4m ˜ 1 I211 q Ž 1 y 3m˜ 1 . I112 q 2 m˜ 2 y m˜ 1 y 3 y q < D < 2 f ) V Ž m 2) . Numerically, then, Im d R1 s < D < 2  0.077 f

2 2

2 1

½ Ž m˜ y m˜ . I

221 q

m ˜ 1 y m˜ 12 m ˜2

/ 5 l12

Ž m˜ 2 q m˜ 1 y 0.5 . l12 5 .

A1 Ž m2) . q 0.15 A2 Ž m2) . fq Ž m 2 . q 0.16 fy Ž m 2 . q f ) y0.67A

4,

Im d R2 s 0.15 D f A1 Ž m2 . y 0.845 A2 Ž m2 . y 0.155 A3 Ž m2 . y 0.098 V Ž m 2) . f ) < D < 2 y 0.12 D f ) fq Ž m 2) . . One may, in principle, estimate D from the partial decay width of Ds) :

a GŽ

Ds) ™ Dsg

.s

24

2

ž

< D < m) 1 y

m2 m2)

Ž 14 .

3

/

s Ž 1.46 = 10y3 MeV . < D < 2 .

Ž 15 .

Unfortunately, this decay mode is not yet well measured. All we know is that G Ž Ds) . - 1.9 MeV and that Ž Ds q g . and Ž Ds q p . are the only decay modes seen. This implies that < D < 2 - 1300 Br Ž Ds) ™ Dsg . .

D. Choudhury, J. Ellis r Physics Letters B 433 (1998) 102–108

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A branching ratio of even 1% could then lead to an order of magnitude change in G Ž Bs ™ gg .! However, a stricter bound for D can be obtained if one relates it to the corresponding form factor for the S s 0 charm meson. Noting that G Ž D ) . - 0.131 MeV and Br Ž D )q™ Dq g . f 1.7% w14x, a relation analogous to Ž15. leads to < DD ) Dg < 2 - 0.95 . A value of D of this order obviously cannot negate the conclusions of Ž12.. Examining Ž14. closely, we see that the contribution due to a pair of on-shell Ds exchanging a Ds) is actually much smaller than that of Ž9.. This result is similar again to the case of K S ™ gg decay w15x, where the analogous contributions due to the flavour-octet vector mesons are small. The other diagrams, where at least one Ds) is on shell, can, however, compete with the 2 Ds contribution. To conclude, we have estimated the long-distance contributions to the Bs ™ gg decay arising from charmed-meson intermediate states. We find that the 2 Ds contribution, by itself, is larger than the other long-distance contributions calculated hitherto in the literature w9x. It is, however, still smaller than the short-distance amplitude. More interestingly, the 2 Ds) contribution is much larger, and could enhance the branching fraction by more than an order of magnitude. This would lead to a very striking signal at the hadronic B factories. Moreover, if it is indeed comparable to the short-distance amplitude, the different CabibboKobayashi-Maskawa structures might offer interesting prospects for observing CP violation.

Acknowledgements D.C. acknowledges useful discussions with Ahmed Ali, Tariq Aziz, Leo Stodolsky and York-Peng Yao.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x

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