arXiv:1103.0081v1 [hep-ph] 1 Mar 2011
The exclusive rare decay b → sγ of heavy b-Baryons
Yong-Lu Liu, Long-Fei Gan, and Ming-Qiu Huang
Department of Physics, College of Science,
National University of Defense Technology,
Changsha, Hunan 410073, People’s Republic of China
(Dated: March 2, 2011)
We present an analysis on the exclusive rare radiative decay modes Σb → Σγ
and Ξb→ Ξγ. The transition form factors which parameterize these processes are
calculated using QCD light-cone sum rules. The decay widths we predict are Γ(Σb→
Σγ) = (7.21±0.04)×10−18GeV and Γ(Ξb→ Ξγ) = (1.34±0.07)×10−16GeV. The
Branching ratio of Ξb→ Ξγ is predicted to be Br(Ξb→ Ξγ) = (3.03 ± 0.10) × 10−4.
PACS numbers: 14.20.-c, 11.25.Hf, 11.55.Hx, 13.40.-f
Heavy flavor physics plays an important role both in the precise test of the standard
model in the relatively high energy region and in the investigation of the hadronization
of quarks at the low energy. Hence, a lot of effort has been paid into it and a large
amount of experimental data have been accumulated [1–4]. Theoretically, much progress
has been made in the heavy flavor meson sector for its comparatively simple structure
while knowledge about baryons appears to be limited. Although much literature have
been provided to decipher these heavy flavor states (such as Refs. [5–8]), a deep under-
standing of them undoubtedly demands the information on the dynamical details which
are encoded in various decay modes [9, 10]. Among these modes, the rare radiative de-
cay processes of the b-baryons are important in that they are not only the ways to study
the Cabibbo-Kobayashi-Maskawa matrix elements Vtsand Vtbwhich are closely attached
with the dynamics inside the baryons, but also the ways to probe new physics beyond the
These types of processes (such as b → sγ), which are forbidden at the tree level in
the standard model of electroweak theory, are induced by the flavor-changing neutral cur-
rent (FCNC) of the b-quark. Their amplitudes are dominated by the one-loop diagrams
with a virtual top quark and a W boson, and thus are strongly suppressed by Glashow-
Iliopoulos-Maiani mechanism. The relative b-meson rare radiative decay modes have been
investigated experimentally since the early 1990’s [11–14], while not so many experimental
data are available for the corresponding b-baryon processes. Theoretical studies on the
exclusive processes are available for both b-mesons and b-baryons [15–19], despite the fact
that the dynamics of the b-baryons decays are far less clarified in comparison with that of
the b-mesons. However, most of the existing literature is about the process Λb→ Λγ, the
branching ratio of which has been predicted to be Br(Λb→ Λγ) ≤ 1.3 × 10−3experimen-
tally . Unfortunately, this decay mode is not expected to be measured easily in the
experiments due to the fact that the final state Λ baryon is of neutral charged, as argued in
Ref. . For this reason, we turn to study the possible decay modes of other octet heavy
baryons Σband Ξb, in which charged final states arise and may be easily tested in experi-
ments. It has been estimated early in the 1990′s that an amount number of b-baryons may
be produced at the c.m. energy level of the LHC . Thus we can expect that these rare
decay modes could be measured by the LHC experiments in the near future, the updated
energy of which is expected to be ∼ 14TeV.
The remainder of this paper is organized as follows. We give an introduction to the
exclusive rare decay mode b → sγ and derive the formula of the decay widths in Sec. II.
Then the light-cone QCD sum rules for the relative transition form factors are derived in
Sec. III. Finally, Sec. IV is devoted to the numerical analysis and a summary is given at
the end of this section.
II.PARAMETRIZATION OF THE TRANSITION FORM FACTORS
In the standard model, the process of the exclusive rare decay b → sγ can be described
by the following effective Hamiltonian :
Heff(b → sγ) = −4GF
16 π2¯ s σµν(mbR + msL) b Fµν,(2)
where L/R = (1 ∓ γ5)/2 and Fµνis the field strength tensor of the photon. GF is the
Fermi coupling constant and C7(µ) is the Wilson coefficient at the scale µ. Considering
the general form beyond the standard model, O7can be represented as
32 π2mb¯ s σµν(gV+ γ5gA) b Fµν.
The decay amplitude is given by the expectation value of the effective Hamiltonian
between the initial and final states at the hadron level
M(Xb→ Xγ) = ?Xγ|Heff|Xb? , (4)
where X stands for the baryon involved in the process.
By considering the Lorentz structure, the contribution of the hadronic part to the pro-
cess, which is written as the hadronic matrix elements, is generally parameterized in terms
of the following form factors:
?Xb(P′)|jν|X(P)? =¯ Xb(P′)[f1γν−f2iσµνqµ+f3qν−(g1γν+g2iσµνqµ+g3qν)γ5]X(P), (5)
where Xband X are the spinors of the baryons and the weak current jνis defined as
jν(x) = i¯b(x)σµν(1 − γ5)qµs(x).(6)
In fact, form factors f3and g3do not contribute to the process due to the conservation of
the vector current. Therefore, the form factors we need to calculate are f1(g1) and f2(g2),
which can be determined from the QCD light-cone sum rules. It is noted that the processes
are only related to the form factors at the point q2= 0, thus we just consider this case in
the following analysis.
With the form factors defined above, the decay width is represented as
Γ(Xb→ Xγ) =G2
III.LIGHT-CONE SUM RULES FOR THE FORM FACTORS
Now we apply the light-cone QCD sum rule approach to calculate the transition form
factors f1(g1) and f2(g2). The interpolating currents to the heavy baryons are chosen as
jΣb(0) = ǫijk[qi(0)C/ zqj(0)]γ5/ zbk(0) for Σband jΞb(0) = ǫijk[si(0)C/ zbj(0)]γ5/ zqk(0) for Ξb,
respectively. Herein q stands for u or d quark, C is the charge conjugaion matrix, and z is
the vector defined on the light-cone z2= 0. The normalization of these currents is defined
by the parameters fΣband fΞb:
?0|jΣb|Σb(P′)? = fΣb(z · P′)/ zΣb(P′),
?0|jΞb|Ξb(P′)? = fΞb(z · P′)/ zΞb(P′).(8)
In the following part, we will take Σ+
calculating the form factors is the correlation function
b→ Σ+γ as an example. Our starting point for
at q2= 0 and with Euclidean m2
procedure of the light-cone sum rule method, we need to express the correlation function
both phenomenologically and theoretically. By inserting a complete set of intermediate
states and using the definitions (5) and (8), the phenomenological side is represented as
b− P′2of about several GeV2. Following the standard
zνTν(P,q) =2fΣb(z · P′)2
Σb− P′2[f1/ z − f2/ z/ q − g1/ zγ5+ g2/ z/ qγ5]Σ(P) + ...,(10)
where “...” stands for the continuum contributions. The correlation function (9) is con-
tracted by zνto remove contributions proportional to the light-cone vector zνwhich is
subdominant on the light-cone.
On the other hand, the theoretical side is obtained by contracting the heavy b quarks in
the correlation function and using the distribution amplitudes presented in Ref.[24–26]. To
make the paper self-contained, we present in the Appendix the definition and the explicit
expressions of the distribution amplitudes of Σ and Ξ used in this paper. After assuming
the quark-hadron duality and performing the Borel transformation, we arrive at the final
light-cone sum rule of the form factor f2(0):
where s = (1 − α3)M2+ m2
parameter which is introduced to suppress the contributions from the higher resonances
and the continuum states. Our calculation shows that f1= g1= 0 and f2= g2. In Eq.
(11), the following abbreviations are used for convenience:
B1(α3) = (2?V1−?V2−?V3−?V4−?V5)(α3),
b/α3, M is the mass of the final baryon, and M2
Bis the Borel
B0(α3) = dα1V1(α1,1 − α1− α3,α3),
B2(α3) = (−??V1+??V2+??V3+??V4+??V5−??V6)(α3).
The distribution amplitudes with tildes which come from the integration by parts in α3
are defined as
dα1Vi(α1,1 − α1− α′
dα2Vi(α1,1 − α1− α′′
The same procedure is also carried out to calculate the transition form factors of the
process Ξb→ Ξγ. We obtain the final sum rule as follows:
where s′= (1 − α2)M2+ m2
b/α2, M is the mass of Ξ, and the following abbreviations are
dα1T1(α1,α2,1 − α1− α2),
C1(α2) = (2?T1−?T2−?T5− 2?T7− 2?T8)(α2),
C3(α2) = (−??T1+??T2+??T5−??T6+ 2??T7+ 2??T8)(α2).
The functions with tildes are defined as
C2(α2) = (??T2−??T3−??T4+??T5+??T7+??T8)(α2),
2,1 − α1− α′
2,1 − α1− α′′
IV. NUMERICAL ANALYSIS AND THE SUMMARY
Before the numerical evaluation of the sum rules (11) and (14), we need to determine
the input parameters. Two important parameters are the decay constants fΣband fΞb,
which can be calculated with the QCD sum rule approach. Using the same expressions
in Refs.  and  with the replacements ms→ mbfor fΣband mc→ mbfor fΞb, we
get the estimations fΣb= (6.18 ± 0.03) × 10−3GeV2and fΞb= (3.32 ± 0.46) × 10−3GeV2.
Other input parameters needed in our calculation can be read from Ref. :
mb = 4.8GeV, ms= 0.15GeV, MΣ= 1.189GeV,
MΞ = 1.314GeV, MΣb= 5.729GeV, MΞb= 5.81GeV,
Vts = 0.0403,Vtb= 0.9992,(17)
αem= 1/137,GF= 1.166364 × 10−5GeV−2,C7(mb) = −0.31.(18)
An important step in the numerical analysis of the QCD sum rules is to determine the
Borel mass parameter M2
be chosen by demanding that the continuum contribution is subdominant in comparison
with that of the ground state which we are concerned about. Simultaneously, the resulting
form factors should not vary drastically along with the threshold. Thus s0 is generally
connected with the first resonance which has the same quantum numbers as the particle we
care about. Here we fix the threshold s0in the region 39GeV2≤ s0≤ 41GeV2. As for the
Borel parameter M2
efficiently, we also demand that the higher twists contributions are less significant and the
form factors should vary mildly along with it. Our calculation shows that the working
windows can be chosen properly in the region 8GeV2≤ M2
Using the distribution amplitudes given in Refs.  and , we obtain the form factors
at the zero momentum transfer f2(g2)(0) as functions of the Borel parameter M2
Band the continuum threshold s0. The continuum threshold s0can
B, which is introduced to suppress the higher resonance contributions
B≤ 11GeV2for Σb→ Σγ and
B≤ 12GeV2for Ξb→ Ξγ.
9 9.510 10.5 11 11.512
39, 40, 41GeV2from the top down.
The dependence of the form factors f2(0)’s on the Borel parameter with s0 =
8 8.59 9.5 1010.5 11
99.5 1010.5 11 11.512
eter with s0= 40GeV2.
The contributions to the form factors f2(0)’s from different twists on the Borel param-
are displayed in Fig. 1 . We have also analyzed the contributions from the distributions
of different twists, which are shown in Fig. 2. The results show that the contributions
of the leading-order and next-to-leading-order twists are dominant while the contributions
from higher twists are suppressed efficiently. This implies that the light-cone expansion is
reasonable in the cases we considered in this paper.
By using of the form factors we have estimated above, the decay widths of the processes
can be easily evaluated with the formula (7), which turn out to be Γ(Σb→ Σγ) = (7.21 ±
0.04) × 10−18GeV and Γ(Ξb→ Ξγ) = (1.34 ± 0.07) × 10−16GeV. Although the mean life
in Ref.  to estimate the branching ratio of the process Ξ−
of the threshold, the sum rule windows, and the uncertainties in the decay constants fΣb
and fΞb. It is worth noting that errors from other sources are not considered here because
the sum rule method itself brings in an amount of uncertainties (about 20%), which makes
it less significant to take into account the errors of the input parameters.
bhas been estimated experimentally [3, 4, 28, 29], here we use the average value given
b→ Ξ−γ which turns out to
b→ Ξ−γ) = (3.03±0.10)×10−4. The errors in the widths come from the choices
TABLE I: Decay widths and Branching ratios at different points of mb.
Γ(Σb→ Σγ)(×10−18GeV) 6.92 ± 0.03 7.21 ± 0.04 7.26 ± 0.07
Γ(Ξb→ Ξγ)(×10−16GeV) 0.98 ± 0.04 1.34 ± 0.04 1.75 ± 0.05
b→ Ξ−γ)(×10−4) 2.21 ± 0.08 3.03 ± 0.10 3.96 ± 0.11
We also investigate the sensitivity of the form factors to the variation of mbat different
points mb= 4.7, 4.8, and 4.9GeV. The corresponding predictions for the decay widths
and branching ratios are given in Table I.
In summary, we have investigated the exclusive rare decay processes Σb → Σγ and
Ξb→ Ξγ. The corresponding transition form factors are estimated through the light-cone
QCD sum rule approach and the decay widths of these processes are predicted to be Γ(Σb→
Σγ) = (7.21 ± 0.04) × 10−18GeV and Γ(Ξb→ Ξγ) = (1.34 ± 0.04) × 10−16GeV. We also
estimate the branching ratio of Ξ−
As we can see, our prediction is larger than the theoretical estimations for the branching
ratio of the Λb→ Λγ. Therefore it is reasonable to assume that this mode may be tested
easily, provided that a good source of Σbor Ξbis available in the future experiments, such
as the LHC experiments.
b→ Ξ−γ, which is Br(Ξ−
b→ Ξ−γ) = (3.03±0.10)×10−4.
This work was supported in part by the National Natural Science Foundation of China
under Contract Nos.10975184 and 11047117.
In the following we give the distribution amplitudes of Σ and Ξ used in the paper. In
general, the distribution amplitudes are defined by the matrix element of the three-quark
where M is the mass of the baryon X and C is the charge conjugation matrix. Note that
the other Lorentz structures which do not contribute to the calculations are omitted. For
each distribution amplitudes Fi= Vi,Tidefined above, it can be presented as
with the relationship 0 < xi< 1,?
baryon momentum on the quarks. The integration measure is defined as
F(aip · z) =Dxe
xi= 1, and xicorresponds to the distribution of the
dx1dx2dx3δ(x1+ x2+ x3− 1). (21)
The distribution amplitudes can be expanded with a conformal spin. The detailed
process is referred to Refs. [24–26]. The explicit expressions of the distribution amplitudes
are collected below:
V1(xi) = 120x1x2x3φ0
V3(xi) = 12x3(1 − x3)ψ0
V5(xi) = 6x3φ0
3,V2(xi) = 24x1x2φ0
V4(xi) = 3(1 − x3)ψ0
V6(xi) = 2φ0
T1(xi) = 120x1x2x3φ
3,T2(xi) = 24x1x2φ
T4(xi) = −3
T6(xi) = 2φ
T3(xi) = 6x3(1 − x3)(ξ0
T5(xi) = 6x3φ
T7(xi) = 6x3(1 − x3)(ξ
The parameters in the expressions are as follows:
The nonperturbative parameters fX, λ1and λ2are determined with QCD sum rules to be
fΣ = (9.4 ± 0.4) × 10−3GeV2,
λ2 = (4.4 ± 0.1) × 10−2GeV2,
λ1= −(2.5 ± 0.1) × 10−2GeV2,
λ3= (2.0 ± 0.1) × 10−2GeV2
for Σ and
fΞ = (9.9 ± 0.4) × 10−3GeV2,
λ2 = (5.2 ± 0.2) × 10−2GeV2,
λ1= −(2.8 ± 0.1) × 10−2GeV2,
λ3= (1.7 ± 0.1) × 10−2GeV2
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