In this paper we investigate the subleading twist corrections to the photon
energy spectrum in the decay Lambda_b -> X_s \gamma and to the lepton energy
spectrum in Lambda_b -> X_u l \bar{\nu}. As a first step we rederive the
matching coefficients of the subleading twist using a much simpler method.
Parametrizing the matrix elements for Lambda_b and using reparametrization
invariance, we show that the energy spectra close to the endpoints are given in
terms of only a single universal function.Comment: LATEX,The complete paper is also available via the www at
  http://www-ttp.physik.uni-karlsruhe.de/Preprints/ . Discussion section
  improve

Kraetz, Michael

Mannel, Thomas

Physics Letters B

English

arXiv

AbstractIn this Letter we investigate the subleading twist corrections to the photon energy spectrum in the decay Λb→Xsγ and to the lepton energy spectrum in Λb→Xuℓν̄ℓ. As a first step we rederive the matching coefficients of the subleading twist using a much simpler method. Parametrizing the matrix elements for Λb and using reparametrization invariance, we show that the energy spectra close to the endpoints are given in terms of only a single universal function

Elsevier - Publisher Connector 

Physics Letters B 556 (2003) 163–168
www.elsevier.com/locate/npe
Subleading light-cone distribution functions in the decays
Λb→Xsγ and Λb→Xuν¯
Michael Kraetz, Thomas Mannel
Institut für Theoretische Teilchenphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Received 23 October 2002; received in revised form 13 January 2003; accepted 27 January 2003
Editor: G.F. Giudice
Abstract
In this Letter we investigate the subleading twist corrections to the photon energy spectrum in the decay Λb →Xsγ and to
the lepton energy spectrum in Λb →Xuν¯. As a first step we rederive the matching coefficients of the subleading twist using
a much simpler method. Parametrizing the matrix elements for Λb and using reparametrization invariance, we show that the
energy spectra close to the endpoints are given in terms of only a single universal function.
 2003 Elsevier Science B.V.
1. Introduction
The existence of baryons with bottom quarks is
well established by now, and at future facilities a
detailed study of the decay modes of such states will
become feasible. However, the B factories are running
below the threshold for the production of these modes,
which means that the measurements we are going to
discuss will be—at least in the foreseeable future—the
domain of hadron colliders.
From the theoretical point of view, the ground state
bottom baryonΛb is the simplest heavy state, although
it contains three quarks. The reason is that the spin
of the heavy b quark decouples in the limit mb →∞
and thus the light degrees of freedom have to be in a
spin-zero state [1]. In other words, the spin of a Λb is
E-mail address: thomas.mannel@phys.uni-karlsruhe.de
(T. Mannel).
the spin of the heavy quark, which in principle allows
interesting polarization studies. Some of these studies
require polarized Λb’s which may become available at
GIGA-Z [2], a high-luminosity linear collider running
at the Z0 resonance.
In the present Letter we investigate energy spectra
of inclusive heavy-to-light decays of the Λb , in partic-
ular the photon spectrum of the rare decay Λb →Xsγ
and the lepton energy spectrum of the semileptonic de-
cay Λb → Xuν¯. The situation here is very similar
to the one in the decays B → Xsγ and B → Xuν¯:
The photon spectrum in B → Xsγ is mainly concen-
trated in the endpoint of small hadronic mass of Xs ,
where the strict 1/m expansion breaks down. This
forces us to switch to an expansion in twists similar to
deep inelastic scattering. The leading twist contribu-
tion is given in terms of a light-cone-distribution func-
tion which can be formally written as a forward ma-
trix element of a non-local operator [3]. This function
describes all inclusive heavy to light transitions and
0370-2693/03  2003 Elsevier Science B.V.
doi:10.1016/S0370-2693(03)00129-1
Open access under CC BY license.
Open access under CC BY license.
164 M. Kraetz, T. Mannel / Physics Letters B 556 (2003) 163–168
is universal. Applying this to b→ uν¯ and b→ sγ
one may obtain a model independent determination of
|Vub/Vts| [4].
Recently subleading terms have been included into
the analysis of B meson decays [5], and the purpose
of the present Letter is to include subleading terms for
the inclusive decays of heavy baryons. The main result
of the Letter is that only a single distribution function
is needed to parametrize the differential decay rates
to subleading order; in other words, the endpoints of
the photon energy spectrum and the charged lepton
energy spectrum in Λb →Xsγ and Λb →Xuν¯ are
given in terms of one universal function up to terms
of order 1/m2b. This has to be compared to the case
of the corresponding B decays, where three universal
functions are needed [5].
In Section 2 we consider the matching including
the terms of subleading twist. We reproduce the
results of [5] using a much simpler method based
on a background field, in which the light quark
propagates. In Section 3 we parametrize the forward
matrix elements for the non-local operators between
Λb states. In Section 4 we discuss our result and
conclude.
2. Matching on non-local operators to order 1/m
The tree-level matching of full QCD to the non-
local operators to order 1/mb has been performed for
both b→ sγ [5] and b→ uν¯ [6,7] and the results
of these papers can be used also for the case of the
Λb . However, we find it useful to rederive the results
of [5] in a different way, which is very transparent and
is manifestly gauge invariant. The method resembles
the one used in [8] for the shape function for mesons.
The starting point is the usual decomposition of the
heavy quark momentum according to
(1)pb =mbv + k,
where the components of the residual momentum k
are small compared to mb. At tree level, the inclusive
decay b→ sγ requires to consider the propagator of
the (massless) s quark, which is
(2)S = i
/pb − /q =
i
mb/v + /k − /q ,
where q is the photon momentum.
In position space and in terms of the static heavy-
quark field hv(x) the residual momentum k corre-
sponds to a covariant derivative, since the large part
mbv has been removed by a field redefinition. This
leads us to consider the following (non-local) opera-
tor
(3)T = h¯v(0)Γ †
(
i
mb/v + i/D− /q
)
Γ hv(x),
where D is the (QCD) covariant derivative and Γ is
the Dirac matrix of the corresponding heavy-to-light
current s¯Γ b.
It has been shown in [8] that (3) yields the correct
tree level matching for the shape function and we shall
use the same arguments to rederive the subleading
terms. In order to define the power counting properly
we introduce light-cone vectors as
v = 1
2
(n+ n¯), n2 = 0= n¯2,
(4)n · n¯= 2, q = (v · q)n¯.
In this way we can decompose any vector into its
components according to
(5)gµν = 12nµn¯ν +
1
2
n¯µnν + g⊥µν.
We want to expand (3) in the kinematical region where
(6)(mbv − q) · n=m− n · q ∼ΛQCD
which is the kinematical region for the shape function.
In order to perform this expansion we write
(7)1
mb/v + i/D− /q =
mb/v + i/D− /q
(mb/v + i/D− /q)2
and consider the denominator, which can be written in
the form
(mb/v + i/D− /q)2
=mbQ+ + 12 {iD−,Q+} +
(
iD⊥
)2
(8)− iσµνiDµiDν,
where we have defined
Q+ =mb − n · q + iD+, iD+ = i(n ·D),
(9)iD− = i(n¯ ·D), D⊥µ = g⊥µνDν.
Since mb − n · q ∼ ΛQCD we have Q+ ∼ ΛQCD and
thus the term mBQ+ is the leading term (of order
M. Kraetz, T. Mannel / Physics Letters B 556 (2003) 163–168 165
mbΛQCD) in the denominator. All other terms are of
order Λ2QCD and are expanded as
1
(m/v − /q + i/D)2
= 1
mQ+
− 1
mQ+
1
2
{iD−,Q+} 1
mQ+
(10)
− 1
mQ+
((
iD⊥
)2 − iσµνiDµiDν) 1
mQ+
+ · · · .
Next we reinsert the numerator, which we do in a sym-
metrized fashion. After some simple algebra we get
m/v − /q + i/D
(m/v − /q + i/D)2
= /n
2Q+
+ /¯n
2m
+ 1
2m
{
i/D⊥, 1
Q+
}
(11)
− 1
2m
1
Q+
(
/n
(
iD⊥
)2
− 1
2
{
/n, iσµν
}
iDµiDν
)
1
Q+
.
The last term can be rewritten in terms of the per-
pendicular components of the covariant derivative. We
have
(12)
{
/n,σµν
}
nµnν = i2
(
/n(/n/¯n− /¯n/n)+ (/n/¯n− /¯n/n)/n)= 0,
(13){/n,σµν}nµg⊥να = i2/n(γ⊥α /n− /nγ⊥α )= 0
from which we derive{
/n, iσµν
}
iDµiDν
(14)
= iσ αβnαn¯βγ µ⊥
[
iD⊥µ ,Q+
]+ 2/niσµν⊥ iD⊥µ iD⊥ν .
Combining this with the other terms we get
m/v − /q + i/D
(m/v − /q + i/D)2
= /n
2Q+
+ /¯n
2m
+ γ
µ
⊥
2m
{
iDµ,
1
Q+
}
+ iσ
αβnαn¯βγ
µ
⊥
4m
[
iDµ,
1
Q+
]
(15)
− 1
2m
/n
Q+
(
1
2
g
µν
⊥
{
iD⊥µ , iD⊥ν
}
− 1
2
iσ
µν
⊥
[
iD⊥µ , iD⊥ν
]) 1
Q+
.
This result can now be used to perform the match-
ing on the non-local operators as they have been de-
fined in [5]. These operators are
(16)O0(ω)= h¯vδ(ω+ iD̂+)hv,
(17)Oµ1 (ω)= h¯v
{
iDµ, δ(ω+ iD̂+)
}
hv,
(18)Oµ2 (ω)= h¯v
[
iDµ, δ(ω+ iD̂+)
]
hv,
O
µν
3 (ω1,ω2)
(19)
= h¯vδ(ω2 + iD̂+)
{
iD
µ
⊥, iD
ν⊥
}
δ(ω1 + iD̂+)hv,
O
µν
4 (ω1,ω2)
(20)= h¯vδ(ω2 + iD̂+)
[
iD
µ
⊥, iD
ν⊥
]
δ(ω1 + iD̂+)hv
and
(21)P0,α(ω)= h¯vδ(w+ iD̂+)γαγ5hv,
(22)Pµ1,α(ω)= h¯v
{
iDµ, δ(ω+ iD̂+)
}
γαγ5hv,
(23)Pµ2,α(ω)= h¯v
[
iDµ, δ(ω+ iD̂+)
]
γαγ5hv,
P
µν
3,α(ω1,ω2)
= h¯vδ(ω2 + iD̂+)
{
iD
µ
⊥, iD
ν⊥
}
(24)× δ(ω1 + iD̂+)γαγ5hv,
P
µν
4,α(ω1,ω2)
= h¯vδ(ω2 + iD̂+)
[
iD
µ
⊥, iD
ν⊥
]
(25)× δ(ω1 + iD̂+)γαγ5hv.
The matching onto these operators is performed for
the decay rate. In order to obtain the rate we have to
take the imaginary part of a forward matrix element
with the Λb . Inserting (15) back into (3) and taking
the imaginary part we obtain for the leading term
Im h¯v
Γ /nΓ
2Q+
hv
=− 1
2m
∫
dω
(
π
2
Tr[P+Γ /nΓ ]δ(1− x −ω)h¯v
× δ(w+ iD̂+)hv
− π
2
Tr
[
sαΓ /nΓ
]
δ(1− x −ω)h¯v
× δ(w+ iD̂+)γαγ5hv
)
(26)
=− 1
2m
∫
dω
[
C0(ω)O0(ω)+Cα5,0(ω)P0,α(ω)
]
166 M. Kraetz, T. Mannel / Physics Letters B 556 (2003) 163–168
from which we can read off the matching coefficients
C0 and Cα5,0 as they have been derived already in [5].1
In the same way we obtain for the subleading
contributions
Im h¯v
Γ γµ⊥Γ
2m
{
iDµ,
1
Q+
}
hv
= Im
∫
dω h¯v
Γ γµ⊥Γ
2m2
{
iDµ, δ(ω+ iD̂+)
}
× 1
1− x −ωhv
(27)
=− 1
2m2
∫
dω
(
π
2
Tr
[
P+ Γ γµ⊥Γ
]
δ(1− x −ω)h¯v
× {iDµ, δ(ω+ iD̂+)}hv
− π
2
Tr
[
sα Γ γµ⊥Γ
]
× δ(1− x −ω)h¯v
× {iDµ, δ(ω+ iD̂+)}γαγ5hv),
from which we obtain the matching coefficients Cµ1
and Cα,µ5,1 .
A further restriction of the structure of the matching
coefficients, which should hold even beyond tree level,
is due to reparametrization invariance [9]. It turns out
that reparametrization invariance [10,11] relates the
coefficients of O0 and Oµν3 in such a way that only
one function is needed to parametrize the effect of the
two operators O0 and Oµν3 . We shall use the relations
from [10] in the next section when we will count
the number of unknown functions needed to describe
the spectrum of Λb → Xsγ including subleading
contributions.
In addition to these operators we have also to take
into account the subleading terms of the Lagrangian.
They are given as
(28)
OT (ω)= i
∫
d4x
∫
dt
2π
e−iωt
× T [h¯v(0)hv(t)L1/m(x)],
1 Here we made us of the decomposition formula
P+Γ P+ = 12P+ Tr[P+Γ ] −
1
2
sµ Tr[sµΓ ],
sµ = P+γµγ5P+, P+ = 12 (1+ /v).
(29)
PT,α(ω)= i
∫
d4x
∫
dt
2π
e−iωt
× T [h¯v(0)γαγ5hv(t)L1/m(x)],
where L1/m is the subleading contribution to the
Lagrangian
(30)L1/m = 12mh¯v(i/D)
2hv.
Once matrix elements are taken, these pieces can
be interpreted as the corrections to the states, which
means that they will appear always in a specific
combination with the leading terms, i.e., these will not
introduce any new unknown functions.
3. Matrix elements for Λb decays
After having computed the matching we have to
consider the forward matrix elements with aΛb . These
matrix elements have to be evaluated in the static limit
in which the Λb becomes a very simple object. We
obtain for the matrix elements:〈
Λb(v, s)
∣∣O0(ω)∣∣Λb(v, s)〉
(31)= u¯(v, s)u(v, s)fΛ(ω),〈
Λb(v, s)
∣∣P0,α(ω)∣∣Λb(v, s)〉
(32)= u¯(v, s)γαγ5u(v, s)fΛ(ω),
where u(v, s) describes the static Λb with velocity v
and spin s.
For the subleading operators the non-vanishing
matrix elements are parametrized as〈
Λb(v, s)
∣∣Oµ1 (ω)∣∣Λb(v, s)〉
(33)=−2u¯(v, s)u(v, s)(vµ − nµ)ωfΛ(ω),〈
Λb(v, s)
∣∣Pµ1,α(ω)∣∣Λb(v, s)〉
(34)=−2u¯(v, s)γαγ5u(v, s)
(
vµ − nµ)ωfΛ(ω),〈
Λb(v, s)
∣∣Oµν3 (ω1,ω2)∣∣Λb(v, s)〉
(35)= u¯(v, s)u(v, s)gµν⊥ gΛ(ω1,ω2),〈
Λb(v, s)
∣∣Pµν3,α(ω1,ω2)∣∣Λb(v, s)〉
(36)= u¯(v, s)γαγ5u(v, s)gµν⊥ gΛ(ω1,ω2),〈
Λb(v, s)
∣∣OT (ω)∣∣Λb(v, s)〉
(37)= u¯(v, s)u(v, s)tΛ(ω),〈
Λb(v, s)
∣∣PT,α(ω)∣∣Λb(v, s)〉
(38)= u¯(v, s)γαγ5u(v, s)tΛ(ω).
M. Kraetz, T. Mannel / Physics Letters B 556 (2003) 163–168 167
Consequently, only two additional functions appear
at order 1/mb. Furthermore, these subleading function
are tied to the leading order function fΛ by reparame-
trization invariance, which means that it appears al-
ways the combination
FΛ(ω)= fΛ(ω)+ tΛ(ω)
− 1
m2b
∫
dω1 dω1 gΛ(ω1ω2)
(39)×
(
δ(ω−ω1)− δ(ω−ω2)
ω1 −ω2
)
with the leading order function fΛ. In other words,
due to heavy quark spin symmetry we can express
Λb → Xsγ and Λb → Xuν¯ in terms of a single
universal function, up to terms of order 1/m2b.
Finally we discuss the moment expansion of the
resulting functions. For the leading term we have
(40)fΛ(ω)= δ(ω)− λΛ6m2b
δ′′(ω)+ · · · ,
where
(41)
λΛu¯(v, s)u(v, s)=
〈
Λb(v, s)
∣∣h¯v(iD)2hv∣∣Λb(v, s)〉
is the kinetic energy of the b quark inside the Λb
baryon. For the subleading contributions we obtain
(42)gΛ(ω1,ω2)= 2λΛ3 δ(ω1)δ(ω2)+ · · · ,
(43)tΛ(ω)=− λΛ2m2b
δ′(ω)+ · · · ,
which reproduces the known result from [12] with the
replacements λ1 → λΛ and λ2 → 0:
dΓ
dx
(Λb →Xsγ )
(44)
= Γ0
[
δ(1− x)− λΛ
2m2b
δ′(1− x)
− λΛ
6m2b
δ′′(1− x)+ · · ·
]
with
(45)Γ0 = G
2
F α|V ∗tbVts ||C7|2
32π4
m5b.
The moment expansion of the new universal function
reads
(46)FΛ(ω)= δ(ω)+ λΛ6m2b
δ′(ω)+ · · · .
4. Discussion and conclusions
The main result of this Letter is that the semilep-
tonic and radiative decays of a Λb baryon into light
hadrons are described by a universal function, where
the corrections to this statement are of order 1/m2b. In
this way we find the two differential decay rates:
dΓ
dx
(Λb →Xsγ )
(47)= G
2
F α|V ∗tbVts ||C7|2
32π4
m5b(2x − 1)FΛ(1− x),
dΓ
dy
(Λb →Xuν¯)
= G
2
F |Vub|2
96π3
m5b
(48)×
∫
dωΘ(1− y −ω)(1−ω)FΛ(ω),
where x = 2Eγ /mb is the rescaled photon energy and
y = 2E/mb is the rescaled lepton energy. Note that
the first moment of F still vanishes up to terms of
order 1/m2b, so we still get no contribution to the
total rates. This serves a check, since the contributions
of order 1/m may not contribute to the rate. As for
Λb → Xsγ , the spectrum of the semileptonic decay
can be expanded in terms of moments. Inserting (46)
into the spectrum, one arrives at
dΓ
dy
(Λb →Xuν¯)
(49)
= Γ0
[
θ(1− y)− λΛ
6m2b
δ(1− y)
− λΛ
6m2b
δ′(1− y)+ · · ·
]
,
where
(50)Γ0 = G
2
F |Vub|2
96π3
m5b.
With the replacements already mentioned above, this
agrees with the result obtained in [12].
Although this may be difficult from the experi-
mental point of view, we may investigate the sub-
leading corrections to a determination of Vub from
a comparison of Λb → Xsγ with Λb → Xuν¯.
Like in the case of B mesons, an energy cut Ec
on the charged lepton will be unavoidable to dis-
criminate the large charm background, i.e., Ec >
168 M. Kraetz, T. Mannel / Physics Letters B 556 (2003) 163–168
(M2[Λb] −M2[Λc])/(2M[Λb]). One may define ob-
servables (similar as for the decays of B mesons,
see [13]) involving partially integrated rates with suit-
able weight functions. Since the 1/mb terms appear
only a kinematic factors (the pre-factors (2x − 1) in
(47) and (1 − ω) in (48)) we have more complicated
weight functions as in [13]. We define
(51)
Γu(Ec)≡
mΛ/2∫
Ec
dE
(
4E2 −Ecmb
2E2
)
dΓ
Λb→Xuν¯
u
dE
,
(52)Γs(Ec)≡ 2
mb
mΛ/2∫
Ec
dEγ (Eγ −Ec)dΓ
Λb→Xsγ
s
dEγ
from which Vub can be determined as
(53)
∣∣∣∣ VubVtbV ∗t s
∣∣∣∣= 3απ ∣∣Ceff7 ∣∣2Γu(Ec)Γs(Ec) +O
(
1/m2b
)
.
Comparing our result (53) for the Vub determi-
nation with the one obtained in [6,7] one expects
substantially smaller corrections in the case of Λb
baryons, since the corrections are only of order 1/m2b.
However, this requires to measure the inclusive semi-
leptonic and radiative rare Λb decays, which is more
difficult experimentally.
Acknowledgements
This work was supported by the DFG Graduierten-
kolleg “Hochenergiephysik and Teilchenastrophysik”,
by the DFG Forschergruppe “Quantenfeldtheorie,
Computeralgebra und Monte Carlo Simulationen” and
by the Ministerium für Bildung und Forschung bmb+f.
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Subleading light-cone distribution functions in the decays Λb→Xsγ and Λb→Xuℓν̄ℓ 

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Subleading Light-cone distributions functions in the decays Lambda(b) --> X(s) Gamma and Lambda(b) --> X(u) Lepton Anti-Nu

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