We present first calculations of new long-distance contributions to $B_s \to
\gamma \gamma$ decay due to intermediate $D_s$ and $D_s^*$ meson states. The
relevant $\gamma$ 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.Comment: 9 pages, 2 figures (embedded), all style files include

Bauer

Debajyoti Choudhury

Herrlich

John Ellis

Reina

Simma

Physics Letters B

English

arXiv

Crossref

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

We present first calculations of new long-distance contributions to Bs→γγ decay due to intermediate Ds and Ds∗ meson states. The relevant γ 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

Choudhury, Debajyoti

Ellis, John

Publications of the IAS Fellows

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MRI-PHY/P980443
CERN-TH/98-76
hep-ph/9804300
Estimates of Long-distance Contributions
to the Bs → γγ Decay
Debajyoti Choudhurya)∗ and John Ellisb)†
a) Mehta Research Institute of Mathematics and Mathematical Physics,
Chhatnag Road, Jhusi, Allahabad – 211019, India
b) Theory Division, CERN, CH 1211 Geneva 23, Switzerland
ABSTRACT
We present first calculations of new long-distance contributions to Bs → γγ decay
due to intermediate Ds and D
∗
s meson states. The relevant γ vertices are estimated
using charge couplings and transition moment couplings. Within our uncertain-
ties, 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.
CERN-TH/98-76
April 1998
∗debchou@mri.ernet.in
†John.Ellis@cern.ch
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 e+e− 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 → γγ, whose branching ratio presently has the
experimental upper limit B(Bs → γγ) < 1.48×10−4 [1]. Higher-sensitivity measurements of Bs
decays are not among the highest priorities of the e+e− 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 B(Bs → γγ) to levels several orders of magnitude below the
present experimental upper limit [1].
The lowest-order short-distance contributions to the Bs → γγ decay arise from two sets of
graphs: (i) box diagrams, and (ii) triangle diagrams with an external photon leg. These have
been calculated [2–4], and yield, for mt ≈ 175 GeV,1 B(Bs → γγ) ∼ 3.8 × 10−7 , 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 [5]. Emboldened by the
fact that the short-distance QCD corrections are not too big [5–7], one may then even attempt
to use such data as may be forthcoming as probes for new physics.
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 J/ψ and ψ′ [8]. 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 D
∗
s states via the
diagrams shown in Figures 1 and 2. These include loops ofDs mesons alone, loops ofD
∗
s mesons
alone, and diagrams involving radiative Ds → D∗s transitions. The vertices are estimated using
data on B → DD,DD∗ and D∗D∗ decays, the electromagnetic charge couplings of the Ds
and D∗s , and phenomenological estimates of the D
∗
s → Ds + γ 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 → γγ decay may be comparable to
the short-distance contributions calculated previously, within the considerable uncertainties in-
herent to the phenomenological inputs we use. We find no evidence that B(Bs → γγ) decay
should occur at a rate very close to the present experimental upper limit, but even this pos-
sibility 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 compara-
ble 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 B(Bs → γγ) decay are b¯γ5sFµνF µν
1Over a wide range of the top-quark mass, the partial width grows linearly with mt [4].
1
and b¯γ5sFµνF˜
µν , where F is the electromagnetic fileld strength tensor, and F˜ is its dual. The
matrix element for the decay Bs → γ(q1µ)γ(q2ν) can then be parametrized as
M = αG (R1Sµν + iR2Pµν)
Sµν ≡ q1νq2µ − q1 · q2gµν
Pµν ≡ ǫµναβqα1 qβ2
G = GF√
2
VcbV
∗
cs
(1)
where R1 and R2 are the yet-to-be-determined hadronic matrix elements, and (αG) has been
factored out in anticipation of the results to be derived later in the text. The partial width is
then given by
Γ(Bs → γγ) = (αG)
2
64π
m3B
(
|R1|2 + |R2|2
)
. (2)
Since Γ(Bs) ≈ 5× 10−10 MeV, we have, for |Vcb| = 0.04,
Br(Bs → γγ) ≈ 10−7
(∣∣∣∣ R1100 MeV
∣∣∣∣2 + ∣∣∣∣ R2100 MeV
∣∣∣∣2
)
. (3)
It can be argued that the long-distance contributions should be dominated by two-meson in-
termediate states. Now, the quark-level transitions with the least Cabibbo suppression are
b→ cc¯s and b→ cu¯d. 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→ cc¯s case, with the corresponding weak Hamiltonian being given by
Hwk = GJµjµ
Jµ = c¯γµ(1− γ5)b
jµ = s¯γµ(1− γ5)c ,
(4)
where V denotes the CKM matrix. The relevant intermediate states are then D+s D
−
s , D
∗+
s D
∗−
s ,
D∗+s D
−
s , etc.
2. To calculate the Bs → γγ decay amplitude exactly, we need to determine the
corresponding matrix elements of (4), a non-trivial task. However, as a first approximation,
we may assume naive factorization3. The Hwk matrix elements are then expressible in terms of
2Since the contributions due to charmonia have already been investigated [8], we shall not consider these
here.
3Based on experience elsewhere, we may hope that the associated errors are < O(10%).
2
simpler quantities given by
〈D−s (q)|jµ|0〉 = −ifqµ
〈D∗−s (k, ǫ)|jµ|0〉 = m∗f∗ǫ∗µ
〈D−s (k)|Jµ|Bs(p)〉 = 0
〈D∗−s (k, ǫ)|Jµ|Bs(p)〉 = 0
〈D+s (k)|Jµ|Bs(p)〉 = f+(q2)(p+ k)µ + f−(q2)qµ
〈D∗+s (k, ǫ)|Jµ|Bs(p)〉 =
V
m∗
ǫµναβǫ
∗νpαkβ
+
i
m∗
[
m2BA1ǫ∗ν − ǫ∗ · p
{
A2(k + p)µ + m
2
∗
q2
A3qµ
}]
,
(5)
where q = p− k in the last two terms, and m∗ ≡ mD∗ . The quantities V and Ai are related to
the usual form factors through
V = 2m∗
mB +m∗
VBsD∗s (q
2)
A1 = m∗(mB +m∗)
m2B
A1BsD∗s (q
2)
A2 = m∗
mB +m∗
A2BsD∗s (q
2)
A3 = 2
[
A3BsD∗s (q
2)− A0BsD∗s (q2)
]
.
(6)
Finally, we have
〈D+s (k) D−s (q)|Hwk|Bs(p)〉 = −iGf [(p2 − k2)f+(q2) + q2f−(q2)]
〈D∗+s (k) D∗−s (q)|Hwk|Bs(p)〉 = Gf∗ ǫ∗µ+ ǫ∗ν−
[
V(q2)ǫνµαβpαkβ
+i
{
A1(q2)p2gµν − 2A2(q2)pµpν
}]
〈D∗+s (k, ǫ) D−s (q)|Hwk|Bs(p)〉 =
Gf
m∗
ǫ∗ · p
[
p2A1(q2)− (p2 − k2)A2 −m2∗A3(q2)
]
〈D+s (k)D∗−s (q, ǫ)|Hwk|Bs(p)〉 = 2Gm∗f∗f+(q2) ǫ∗ · p .
(7)
Within the factorization approximation,the relations in (7) can be interpreted as a parametriza-
tion of the BD(∗)s D
(∗)
s vertices.
The form factors themselves are nonperturbative quantities. Apart from lattice calcula-
tions [9], the best arenas for determining these are phenomenological approaches such as the
BSW model [10] or heavy-quark effective theory [11]. Some of these parameters are also mea-
sured experimentally [12], although a somewhat large spread persists in the data. Typically,
though, we have f, f∗ ∼ 200 MeV, f±,V ∼ 0.6 and A1,2 ∼ 0.25. On 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
3
fixes the charge coupling uniquely for each angular-momentum state of the (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.
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
Bs(p)
D+s (k)
D−s
Ds
γµ(q1)
γν(q2)
+
crossed
diagram + Bs(p)
D+s (k)
D−s
γµ(q1)
γν(q2)
Figure 1: One-loop contribution to the Bs → γγ amplitude due to Ds mesons alone.
G(1)µν =
∫ d4k
(2π)4
(−iG)f
[
(p2 − k2)f+ + (p− k)2f−
] i
k2 −m2 (ie) [2k + q1]µ
i
(k − q)2 −m2 (ie) [2k − q1 − p]µ
i
(k − p)2 −m2 ,
whilst the third gives
G(3)µν =
∫
d4k
(2π)4
(−iG)f
[
(p2 − k2)f+ + (p− k)2f−
] i
k2 −m2
i
(k − p)2 −m2 (2ie
2gµν) .
In each of the above, the form-factors f± are to be evaluated at (p− 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
Im(G(Ds)µν ) = 2αGm˜1f [(1− m˜1)f+(m2) + m˜1f−(m2)] I111Sµν
Iijk = ln
1 + 2m˜k − m˜i − m˜j + λij
1 + 2m˜k − m˜i − m˜j − λij
λij =
[
(1− m˜i − m˜j)2 − 4m˜im˜j
]1/2
.
(8)
Here m˜i = m
2
i /m
2
B with m1 = m ≡ mDs and m2 = m∗ ≡ mD∗s . The form of (8) 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 (8), and
ImR1(Ds) = 0.39 f
[
f+(m2) + 0.16f−(m2)
]
ImR2(Ds) = 0
(9)
4
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 [14], 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 (3), we then have
Br(Bs
2Ds−→ γγ) ∼ 2.5× 10−8 . (10)
This, by itself, is small compared to the short-distance contribution [6], though the interference
term can be significant.
What about the other long-distance effects? Of the ones calculated in the literature, Bs →
φγ → γγ has an amplitude somewhat smaller than the one calculated here, whilst Bs →
J/ψφ→ γγ is seen to contribute only at the 1% level [8]. We are thus in a situation where the
2Ds 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 → γγ case with that for KS → γγ
decay. Whereas there it is the 2π intermediate state that dominates the decay amplitude [15],
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 D∗s . As long as we neglect any D
∗
sDsγ coupling, the additional diagrams involving
the D∗s field are exact analogues of those in Fig. 1, and lead to
ImR1(D∗s) =
f∗
8m˜22
[
A1(m2∗)
{
(1− 4m˜2)λ22 + m˜2
(
1− 12m˜2 + 48m˜22
)
I222
}
−A2(m2∗)
{
(m˜2 − 5)λ22 + m˜2
(
1− 10m˜2 + 32m˜22
)
I222
}]
= f∗
[
1.57A1(m2∗) + 15.3A2(m2∗)
]
ImR2(D∗s) =
f∗V(m2∗)
8m˜2
{
(12m˜2 − 1)λ22 +
(
4m˜2 − 32m˜22
)
I222
}
= 0.13 f∗ V(m2∗)
(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 ImR1(D∗s) ∼ 800 MeV, on the basis of
which (3) then gives us
Br(Bs
2D∗
s−→ γγ) ∼ 6.5× 10−6 . (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 (5), it would be preferable to use the data to parametrize the BsD
∗
sD¯
∗
s
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.
5
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
D∗sDsγ 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, the relevant part
of which can be expressed as
Leff = i eD
m∗
ǫρσαβq
αpβ . (13)
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 δR1,2 given by
4m˜2 Im δR1 = |D|2f
[
(1− m˜1)f+(m2) + m˜1f−(m2)
]
{λ11 − m˜2I112}
− |D|2f∗
[
A1(m2∗) {λ12 − m˜1I221}
+A2(m2∗)
{
(m˜2 − m˜1 − 0.5)λ12 +
(
(m˜2 − m˜1)2 + m˜1
)
I221
}]
8m˜2 ImδR2 = Df∗ f+(m2∗)
{
2(m˜31 − m˜21 − 4m˜21m˜2 + 2m˜1m˜2 + 5m˜1m˜22 − 2m˜32 + 3m˜22 − m˜2)I122
+8m˜1m˜2I121 + 3(m˜1 − m˜21 + m˜22 − 3m˜2)λ12
}
+ Df
[
A1(m2) + (m˜2 − 1)A2(m2)− m˜2A3(m2)
]{
4m˜1I211 + (1− 3m˜1)I112 +
(
2m˜2 − m˜1 − 3− m˜1 − m˜
2
1
m˜2
)
λ12
}
+ |D|2f∗V(m2∗)
{(
m˜22 − m˜21
)
I221 + (m˜2 + m˜1 − 0.5)λ12
}
.
Numerically, then,
Im δR1 = |D|2
{
0.077 f
[
f+(m2) + 0.16f−(m2)
]
+ f∗
[
−0.67A1(m2∗) + 0.15A2(m2∗)
]}
Im δR2 = 0.15 Df
[
A1(m2)− 0.845A2(m2)− 0.155A3(m2)
]
− 0.098 V(m2∗) f∗ |D|2 − 0.12Df∗ f+(m2∗)
(14)
One may, in principle, estimate D from the partial decay width of D∗s :
Γ(D∗s → Dsγ) =
α
24
|D|2 m∗
(
1− m
2
m2∗
)3
= (1.46× 10−3 MeV) |D|2 (15)
6
Bs(p)
D+s (k)
D∗−s
D∗s
γµ(q1)
γν(q2)
Bs(p)
D+s (k)
D∗−s
Ds
γµ(q1)
γν(q2)
Bs(p)
D+s (k)
D−s
D∗s
γµ(q1)
γν(q2)
Bs(p)
D∗+s (k)
D−s
Ds
γµ(q1)
γν(q2)
Bs(p)
D∗+s (k)
D−s
D∗s
γµ(q1)
γν(q2)
Bs(p)
D∗+s (k)
D∗−s
Ds
γµ(q1)
γν(q2)
Figure 2: One-loop contribution to the Bs → γγ amplitude involving the D∗sDsγ vertex. The
crossed diagrams are not shown.
Unfortunately, this decay mode is not yet well measured. All we know is that Γ(D∗s) < 1.9MeV
and that (Ds + γ) and (Ds + π) are the only decay modes seen. This implies that
|D|2 < 1300 Br(D∗s → Dsγ) .
A branching ratio of even 1% could then lead to an order of magnitude change in Γ(Bs → γγ)!
However, a stricter bound forD can be obtained if one relates it to the corresponding form factor
for the S = 0 charm meson. Noting that Γ(D∗) < 0.131MeV and Br(D∗+ → D+γ) ≈ 1.7% [13],
a relation analogous to (15) leads to
|DD∗Dγ |2 < 0.95 .
A value of D of this order obviously cannot negate the conclusions of (12). Examining (14)
closely, we see that the contribution due to a pair of on-shell Ds exchanging a D
∗
s is actually
much smaller than that of (9). This result is similar again to the case of KS → γγ decay [14],
where the analogous contributions due to the flavour-octet vector mesons are small. The other
diagrams, where at least one D∗s is on shell, can, however, compete with the 2Ds contribution.
To conclude, we have estimated the long-distance contributions to the Bs → γγ decay
arising from charmed-meson intermediate states. We find that the 2Ds contribution, by itself,
is larger than the other long-distance contributions calculated hitherto in the literature [8].
It is, however, still smaller than the short-distance amplitude. More interestingly, the 2D∗s
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 Cabibbo-Kobayashi-
Maskawa structures might offer interesting prospects for observing CP violation.
DC acknowledges useful discussions with Ahmed Ali, Tariq Aziz, Leo Stodolsky and York-
Peng Yao.
7
References
[1] M. Acciari et al. (L3 Collab.), Phys. Lett. B363 (1995) 127.
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D42 (1990) 2314; Mod. Phys. Lett. A6 (1991) 1333.
[3] H. Simma and D. Wyler, Nucl. Phys. B344 (1990) 283.
[4] S. Herrlich and J. Kalinowski, Nucl. Phys. B381 (1992) 502.
[5] T.M. Aliev and G. Turan, Phys. Rev. D48 (1993) 1176;
T.M. Aliev, G. Hiller and E.O. Iltan, hep-ph/9708382.
[6] C.-H.V. Chang, G.-L. Lin and Y.-P. Yao, hep-ph/9705345.
[7] S.R. Choudhury and Y.-P. Yao, hep-ph/9802403.
[8] G. Hiller and E.O. Iltan, Phys. Lett. B409 (1997) 425; hep-ph/9708477.
[9] C. Bernard, hep-ph/9709460;
A.X. El-Khadra et al. hep-ph/9711426.
[10] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C34 (1987) 134.
[11] For a review of HQET results, see e.g., M. Neubert, Phys. Rep. 245 (1994) 259.
[12] S. Aoki et al. (WA75 Collab.), Prog. Theor. Phys. 89 (1995) 131;
J.Z. Bai et al. (BES Collab.), Phys. Rev. Lett. 74 (1995) 4599;
K. Kodama et al. (E653 Collab.), Phys. Lett. B382 (1996) 299;
M. Acciari et al. (L3 Collab.), Phys. Lett. B396 (1997) 327;
F. Parodi et al. (DELPHI Collab.), DELPHI 97-105 CONF 87;
M. Chadha et al. (CLEO Collab.), hep-ex/9712014.
[13] J. Bartelt et al. (CLEO Collab.), hep-ex/9711011.
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R.L. Goble, Phys. Rev. D7 (1973) 931.
8


Estimates of long distance contributions to the B<SUB>s</SUB>→γγ decay

We present first calculations of new long-distance contributions to $B_s \to relevant $\gamma$ 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.We present first calculations of new long-distance contributions to $B_s \to \gamma \gamma$ decay due to intermediate $D_s$ and $D_s^*$ meson states. The relevant $\gamma$ 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.We present first calculations of new long-distance contributions to B s → γγ decay due to intermediate D s and D s ∗ meson states. The relevant γ 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

Ellis, John R.

CERN Document Server

Estimates of Long-distance Contributions to the $B_{s}\to\gamma\gamma$ Decay

Indian Academy of Sciences

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MRI-PHY/P980443
CERN-TH/98-76
hep-ph/9804300
Estimates of Long-distance Contributions
to the Bs → γγ Decay
Debajyoti Choudhurya)∗ and John Ellisb)†
a) Mehta Research Institute of Mathematics and Mathematical Physics,
Chhatnag Road, Jhusi, Allahabad – 211019, India
b) Theory Division, CERN, CH 1211 Geneva 23, Switzerland
ABSTRACT
We present first calculations of new long-distance contributions to Bs → γγ decay
due to intermediate Ds and D
∗
s meson states. The relevant γ vertices are estimated
using charge couplings and transition moment couplings. Within our uncertain-
ties, 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.
CERN-TH/98-76
April 1998
∗debchou@mri.ernet.in
†John.Ellis@cern.ch
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 e+e− 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 → γγ, whose branching ratio presently has the
experimental upper limit B(Bs → γγ) < 1.48×10−4 [1]. Higher-sensitivity measurements of Bs
decays are not among the highest priorities of the e+e− 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 B(Bs → γγ) to levels several orders of magnitude below the
present experimental upper limit [1].
The lowest-order short-distance contributions to the Bs → γγ decay arise from two sets of
graphs: (i) box diagrams, and (ii) triangle diagrams with an external photon leg. These have
been calculated [2–4], and yield, for mt ≈ 175 GeV,1 B(Bs → γγ) ∼ 3.8 × 10−7 , 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 [5]. Emboldened by the
fact that the short-distance QCD corrections are not too big [5–7], one may then even attempt
to use such data as may be forthcoming as probes for new physics.
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 J/ψ and ψ′ [8]. 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 D
∗
s states via the
diagrams shown in Figures 1 and 2. These include loops ofDs mesons alone, loops of D
∗
s mesons
alone, and diagrams involving radiative Ds → D∗s transitions. The vertices are estimated using
data on B → DD,DD∗ and D∗D∗ decays, the electromagnetic charge couplings of the Ds
and D∗s , and phenomenological estimates of the D
∗
s → Ds + γ 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 → γγ decay may be comparable to
the short-distance contributions calculated previously, within the considerable uncertainties in-
herent to the phenomenological inputs we use. We find no evidence that B(Bs → γγ) decay
should occur at a rate very close to the present experimental upper limit, but even this pos-
sibility 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 compara-
ble 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 B(Bs → γγ) decay are b̄γ5sFµνF µν
1Over a wide range of the top-quark mass, the partial width grows linearly with mt [4].
1
and b̄γ5sFµνF̃
µν , where F is the electromagnetic fileld strength tensor, and F̃ is its dual. The
matrix element for the decay Bs → γ(q1µ)γ(q2ν) can then be parametrized as
M = αG (R1Sµν + iR2Pµν)
Sµν ≡ q1νq2µ − q1 · q2gµν
Pµν ≡ ǫµναβqα1 qβ2
G = GF√
2
VcbV
∗
cs
(1)
where R1 and R2 are the yet-to-be-determined hadronic matrix elements, and (αG) has been
factored out in anticipation of the results to be derived later in the text. The partial width is
then given by
Γ(Bs → γγ) =
(αG)2
64π
m3B
(
|R1|2 + |R2|2
)
. (2)
Since Γ(Bs) ≈ 5 × 10−10 MeV, we have, for |Vcb| = 0.04,
Br(Bs → γγ) ≈ 10−7
(∣∣∣∣
R1
100 MeV
∣∣∣∣
2
+
∣∣∣∣
R2
100 MeV
∣∣∣∣
2
)
. (3)
It can be argued that the long-distance contributions should be dominated by two-meson in-
termediate states. Now, the quark-level transitions with the least Cabibbo suppression are
b→ cc̄s and b→ cūd. 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→ cc̄s case, with the corresponding weak Hamiltonian being given by
Hwk = GJµjµ
Jµ = c̄γµ(1 − γ5)b
jµ = s̄γµ(1 − γ5)c ,
(4)
where V denotes the CKM matrix. The relevant intermediate states are then D+s D
−
s , D
∗+
s D
∗−
s ,
D∗+s D
−
s , etc.
2. To calculate the Bs → γγ decay amplitude exactly, we need to determine the
corresponding matrix elements of (4), a non-trivial task. However, as a first approximation,
we may assume naive factorization3. The Hwk matrix elements are then expressible in terms of
2Since the contributions due to charmonia have already been investigated [8], we shall not consider these
here.
3Based on experience elsewhere, we may hope that the associated errors are < O(10%).
2
simpler quantities given by
〈D−s (q)|jµ|0〉 = −ifqµ
〈D∗−s (k, ǫ)|jµ|0〉 = m∗f∗ǫ∗µ
〈D−s (k)|Jµ|Bs(p)〉 = 0
〈D∗−s (k, ǫ)|Jµ|Bs(p)〉 = 0
〈D+s (k)|Jµ|Bs(p)〉 = f+(q2)(p+ k)µ + f−(q2)qµ
〈D∗+s (k, ǫ)|Jµ|Bs(p)〉 =
V
m∗
ǫµναβǫ
∗νpαkβ
+
i
m∗
[
m2BA1ǫ∗ν − ǫ∗ · p
{
A2(k + p)µ +
m2∗
q2
A3qµ
}]
,
(5)
where q = p− k in the last two terms, and m∗ ≡ mD∗ . The quantities V and Ai are related to
the usual form factors through
V = 2m∗
mB +m∗
VBsD∗s (q
2)
A1 =
m∗(mB +m∗)
m2B
A1BsD∗s (q
2)
A2 =
m∗
mB +m∗
A2BsD∗s (q
2)
A3 = 2
[
A3BsD∗s (q
2) − A0BsD∗s (q
2)
]
.
(6)
Finally, we have
〈D+s (k) D−s (q)|Hwk|Bs(p)〉 = −iGf [(p2 − k2)f+(q2) + q2f−(q2)]
〈D∗+s (k) D∗−s (q)|Hwk|Bs(p)〉 = Gf∗ ǫ∗µ+ ǫ∗ν−
[
V(q2)ǫνµαβpαkβ
+i
{
A1(q2)p2gµν − 2A2(q2)pµpν
}]
〈D∗+s (k, ǫ) D−s (q)|Hwk|Bs(p)〉 =
Gf
m∗
ǫ∗ · p
[
p2A1(q2) − (p2 − k2)A2 −m2∗A3(q2)
]
〈D+s (k)D∗−s (q, ǫ)|Hwk|Bs(p)〉 = 2Gm∗f∗f+(q2) ǫ∗ · p .
(7)
Within the factorization approximation,the relations in (7) can be interpreted as a parametriza-
tion of the BD(∗)s D
(∗)
s vertices.
The form factors themselves are nonperturbative quantities. Apart from lattice calcula-
tions [9], the best arenas for determining these are phenomenological approaches such as the
BSW model [10] or heavy-quark effective theory [11]. Some of these parameters are also mea-
sured experimentally [12], although a somewhat large spread persists in the data. Typically,
though, we have f, f∗ ∼ 200 MeV, f±,V ∼ 0.6 and A1,2 ∼ 0.25. On 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
3
fixes the charge coupling uniquely for each angular-momentum state of the (cs̄) 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.
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
Bs(p)
D+s (k)
D−s
Ds
γµ(q1)
γν(q2)
+
crossed
diagram + Bs(p)
D+s (k)
D−s
γµ(q1)
γν(q2)
Figure 1: One-loop contribution to the Bs → γγ amplitude due to Ds mesons alone.
G(1)µν =
∫ d4k
(2π)4
(−iG)f
[
(p2 − k2)f+ + (p− k)2f−
] i
k2 −m2 (ie) [2k + q1]µ
i
(k − q)2 −m2 (ie) [2k − q1 − p]µ
i
(k − p)2 −m2 ,
whilst the third gives
G(3)µν =
∫
d4k
(2π)4
(−iG)f
[
(p2 − k2)f+ + (p− k)2f−
] i
k2 −m2
i
(k − p)2 −m2 (2ie
2gµν) .
In each of the above, the form-factors f± are to be evaluated at (p− 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
Im(G(Ds)µν ) = 2αGm̃1f [(1 − m̃1)f+(m2) + m̃1f−(m2)] I111Sµν
Iijk = ln
1 + 2m̃k − m̃i − m̃j + λij
1 + 2m̃k − m̃i − m̃j − λij
λij =
[
(1 − m̃i − m̃j)2 − 4m̃im̃j
]1/2
.
(8)
Here m̃i = m
2
i /m
2
B with m1 = m ≡ mDs and m2 = m∗ ≡ mD∗s . The form of (8) 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 (8), and
ImR1(Ds) = 0.39 f
[
f+(m2) + 0.16f−(m2)
]
ImR2(Ds) = 0
(9)
4
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 [14], 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 (3), we then have
Br(Bs
2Ds−→ γγ) ∼ 2.5 × 10−8 . (10)
This, by itself, is small compared to the short-distance contribution [6], though the interference
term can be significant.
What about the other long-distance effects? Of the ones calculated in the literature, Bs →
φγ → γγ has an amplitude somewhat smaller than the one calculated here, whilst Bs →
J/ψφ→ γγ is seen to contribute only at the 1% level [8]. We are thus in a situation where the
2Ds 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 → γγ case with that for KS → γγ
decay. Whereas there it is the 2π intermediate state that dominates the decay amplitude [15],
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 D∗s . As long as we neglect any D
∗
sDsγ coupling, the additional diagrams involving
the D∗s field are exact analogues of those in Fig. 1, and lead to
ImR1(D∗s) =
f∗
8m̃22
[
A1(m2∗)
{
(1 − 4m̃2)λ22 + m̃2
(
1 − 12m̃2 + 48m̃22
)
I222
}
−A2(m2∗)
{
(m̃2 − 5)λ22 + m̃2
(
1 − 10m̃2 + 32m̃22
)
I222
}]
= f∗
[
1.57A1(m2∗) + 15.3A2(m2∗)
]
ImR2(D∗s) =
f∗V(m2∗)
8m̃2
{
(12m̃2 − 1)λ22 +
(
4m̃2 − 32m̃22
)
I222
}
= 0.13 f∗ V(m2∗)
(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 ImR1(D∗s) ∼ 800 MeV, on the basis of
which (3) then gives us
Br(Bs
2D∗
s−→ γγ) ∼ 6.5 × 10−6 . (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 (5), it would be preferable to use the data to parametrize the BsD
∗
sD̄
∗
s
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.
5
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
D∗sDsγ 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, the relevant part
of which can be expressed as
Leff = i
eD
m∗
ǫρσαβq
αpβ . (13)
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 δR1,2 given by
4m̃2 Im δR1 = |D|2f
[
(1 − m̃1)f+(m2) + m̃1f−(m2)
]
{λ11 − m̃2I112}
− |D|2f∗
[
A1(m2∗) {λ12 − m̃1I221}
+A2(m2∗)
{
(m̃2 − m̃1 − 0.5)λ12 +
(
(m̃2 − m̃1)2 + m̃1
)
I221
}]
8m̃2 ImδR2 = Df∗ f+(m2∗)
{
2(m̃31 − m̃21 − 4m̃21m̃2 + 2m̃1m̃2 + 5m̃1m̃22 − 2m̃32 + 3m̃22 − m̃2)I122
+8m̃1m̃2I121 + 3(m̃1 − m̃21 + m̃22 − 3m̃2)λ12
}
+ Df
[
A1(m2) + (m̃2 − 1)A2(m2) − m̃2A3(m2)
]
{
4m̃1I211 + (1 − 3m̃1)I112 +
(
2m̃2 − m̃1 − 3 −
m̃1 − m̃21
m̃2
)
λ12
}
+ |D|2f∗V(m2∗)
{(
m̃22 − m̃21
)
I221 + (m̃2 + m̃1 − 0.5)λ12
}
.
Numerically, then,
Im δR1 = |D|2
{
0.077 f
[
f+(m2) + 0.16f−(m2)
]
+ f∗
[
−0.67A1(m2∗) + 0.15A2(m2∗)
]}
Im δR2 = 0.15 Df
[
A1(m2) − 0.845A2(m2) − 0.155A3(m2)
]
− 0.098 V(m2∗) f∗ |D|2 − 0.12 Df∗ f+(m2∗)
(14)
One may, in principle, estimate D from the partial decay width of D∗s :
Γ(D∗s → Dsγ) =
α
24
|D|2 m∗
(
1 − m
2
m2∗
)3
= (1.46 × 10−3 MeV) |D|2 (15)
6
Bs(p)
D+s (k)
D∗−s
D∗s
γµ(q1)
γν(q2)
Bs(p)
D+s (k)
D∗−s
Ds
γµ(q1)
γν(q2)
Bs(p)
D+s (k)
D−s
D∗s
γµ(q1)
γν(q2)
Bs(p)
D∗+s (k)
D−s
Ds
γµ(q1)
γν(q2)
Bs(p)
D∗+s (k)
D−s
D∗s
γµ(q1)
γν(q2)
Bs(p)
D∗+s (k)
D∗−s
Ds
γµ(q1)
γν(q2)
Figure 2: One-loop contribution to the Bs → γγ amplitude involving the D∗sDsγ vertex. The
crossed diagrams are not shown.
Unfortunately, this decay mode is not yet well measured. All we know is that Γ(D∗s) < 1.9MeV
and that (Ds + γ) and (Ds + π) are the only decay modes seen. This implies that
|D|2 < 1300 Br(D∗s → Dsγ) .
A branching ratio of even 1% could then lead to an order of magnitude change in Γ(Bs → γγ)!
However, a stricter bound for D can be obtained if one relates it to the corresponding form factor
for the S = 0 charm meson. Noting that Γ(D∗) < 0.131MeV and Br(D∗+ → D+γ) ≈ 1.7% [13],
a relation analogous to (15) leads to
|DD∗Dγ |2 < 0.95 .
A value of D of this order obviously cannot negate the conclusions of (12). Examining (14)
closely, we see that the contribution due to a pair of on-shell Ds exchanging a D
∗
s is actually
much smaller than that of (9). This result is similar again to the case of KS → γγ decay [14],
where the analogous contributions due to the flavour-octet vector mesons are small. The other
diagrams, where at least one D∗s is on shell, can, however, compete with the 2Ds contribution.
To conclude, we have estimated the long-distance contributions to the Bs → γγ decay
arising from charmed-meson intermediate states. We find that the 2Ds contribution, by itself,
is larger than the other long-distance contributions calculated hitherto in the literature [8].
It is, however, still smaller than the short-distance amplitude. More interestingly, the 2D∗s
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 Cabibbo-Kobayashi-
Maskawa structures might offer interesting prospects for observing CP violation.
DC acknowledges useful discussions with Ahmed Ali, Tariq Aziz, Leo Stodolsky and York-
Peng Yao.
7
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8


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Estimates of Long-distance Contributions to the $B_s \to \gamma \gamma$ Decay

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Estimates of Long-distance Contributions to the Bs→γγB_s \to \gamma \gammaBs​→γγ Decay

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Estimates of Long-distance Contributions to the $B_s \to \gamma \gamma$ Decay