Table Of ContentTTP02-29
hep-ph/0210248
February 1, 2008
Subleading Light-cone distributions functions
→ →
3 in the decays Λ X γ and Λ X ℓν¯
b s b u ℓ
0
0
2
Michael Kraetz Thomas Mannel
n and
a
J
8
2
2
v Institut fu¨r Theoretische Teilchenphysik,
8
Universit¨at Karlsruhe, D–76128 Karlsruhe, Germany
4
2
0 Abstract
1
2 In this paper we investigate the subleading twist corrections to the photon energy
0
spectrum in the decay Λ → X γ and to the lepton energy spectrum in Λ → X ℓν¯ .
/ b s b u ℓ
h
As a first step we rederive the matching coefficients of the subleading twist us-
p
- ing a much simpler method. Parametrizing the matrix elements for Λb and using
p
reparametrization invariance, we show that the energy spectra close to the endpoints
e
h are given in terms of only a single universal function.
:
v
i
X
r
a
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 baryon Λ is the simplest
b
heavy state, although it contains three quarks. The reason is that the spin of the heavy b
quark decouples in the limit m → ∞ and thus the light degrees of freedom have to be in
b
a spin-zero state [1]. In other words, the spin of a Λ is the spin of the heavy quark, which
b
in principle allows interesting polarization studies. Some of these studies require polarized
Λ ’s which may become available at GIGA-Z [2], a high-luminosity linear collider running
b
at the Z resonance.
0
In the present paper we investigate energy spectra of inclusive heavy-to-light decays
of the Λ , in particular the photon spectrum of the rare decay Λ → X γ and the lepton
b b s
energy spectrum of the semileptonic decay Λ → X ℓν¯ . The situation here is very similar
b u ℓ
to the one in the decays B → X γ and B → X ℓν¯ : The photon spectrum in B → X γ
s u ℓ s
is mainly concentrated in the endpoint of small hadronic mass of X , where the strict
s
1/m expansion breaks down. This forces us to switch to an expansion in twists similar
to deep inelastic scattering. The leading twist contribution is given in terms of a light-
cone-distribution function which can be formally written as a forward matrix element of a
non-local operator [3]. This function describes all inclusive heavy to light transitions and
is universal. Applying this to b → uℓν¯ and b → sγ one may obtain a model independent
ℓ
determination of |V /V | [4].
ub ts
Recently subleading terms have been included into the analysis of B meson decays [5],
and the purpose of the present paper is to include subleading terms for the inclusive decays
of heavy baryons. The main result of the paper 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
Λ → X γ and Λ → X ℓν¯ are given in terms of one universal function up to terms of
b s b u ℓ
order 1/m2. This has to be compared to the case of the corresponding B decays, where
b
three universal functions are needed [5].
In the next section 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 Λ states. In section 4 we discuss our result
b
and conclude.
2
1
2 Matching on non-local operators to order /m
The tree-level matching of full QCD to the non-local operators to order 1/m has been
b
performed for both b → sγ [5] and b → uℓν¯ [6, 7] and the results of these papers can be
ℓ
used also for the case of the Λ . However, we find it useful to rederive the results of [5] in
b
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
p = m v +k (1)
b b
where the components of the residual momentum k are small compared to m . At tree
b
level, the inclusive decay b → sγ requires to consider the propagator of the (massless) s
quark, which is
i i
S = = (2)
/p −/q m v/+k/ −/q
b b
where q is the photon momentum
In position space and in terms of the static heavy-quark field h (x) the residual mo-
v
mentum k corresponds to a covariant derivative, since the large part m v has been removed
b
by a field redefinition. This leads us to consider the following (non-local) operator
i
T = h¯ (0)Γ† Γh (x) (3)
v v
mbv/+iD/ −/q!
where D is the (QCD) covariant derivative and Γ is the Dirac matrix of the corresponding
heavy-to-light current s¯Γb
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
1
2 2
v = (n+n¯), n = 0 = n¯ , n·n¯ = 2, q = (v·q)n¯ (4)
2
In this way we can decompose any vector into its components according to
1 1
g = n n¯ + n¯ n +g⊥ (5)
µν 2 µ ν 2 µ ν µν
We want to expand (3) in the kinematical region where
(m v −q)·n = m−n·q ∼ Λ (6)
b QCD
which is the kinematical region for the shape function. In order to perform this expansion
we write
1 m v/+iD/ −/q
b
= (7)
m v/+iD/ −/q (m v/+iD/ −/q)2
b b
3
and consider the denominator, which can be written in the form
1
(m v/+iD/ −/q)2 = m Q + {iD ,Q }+(iD⊥)2 −iσµνiD iD (8)
b b + − + µ ν
2
where we have defined
Q = m −n·q +iD , iD = i(n·D), iD = i(n¯ ·D), D⊥ = g⊥ Dν (9)
+ b + + − µ µν
Since m − n · q ∼ Λ we have Q ∼ Λ and thus the term m Q is the leading
b QCD + QCD B +
term (of order m Λ ) in the denominator. All other terms are of order Λ2 and are
b QCD QCD
expanded as
1 1 1 1 1
= − {iD ,Q } (10)
(mv/−/q +iD/)2 mQ mQ 2 − + mQ
+ + +
1 1
− ((iD⊥)2 −iσµνiD iD ) +...
µ ν
mQ mQ
+ +
Next we reinsert the numerator, which we do in a symmetrized fashion. After some simple
algebra we get
mv/−/q +iD/ n/ n/¯ 1 1
= + + iD/⊥, (11)
(mv/−/q +iD/)2 2Q+ 2m 2m ( Q+)
1 1 1 1
− n/(iD⊥)2 − {n/,iσµν}iD iD
µ ν
2mQ 2 Q
+ (cid:18) (cid:19) +
The last term can be rewritten in terms of the perpendicular components of the covariant
derivative. We have
i
{n/,σµν}n n = (n/(n/n/¯ −n/¯n/)+(n/n/¯ −n/¯n/)n/) = 0 (12)
µ ν
2
i
{n/,σµν}n g⊥ = (n/(γ⊥n/ −n/γ⊥) = 0 (13)
µ να 2 α α
from which we derive
{n/,iσµν}iD iD = iσαβn n¯ γµ[iD⊥,Q ]+2n/iσµνiD⊥iD⊥ (14)
µ ν α β ⊥ µ + ⊥ µ ν
Combining this with the other terms we get
mv/−/q +iD/ n/ n/¯ γµ 1 iσαβn n¯ γµ 1
= + + ⊥ iD , + α β ⊥ iD ,
(mv/−/q +iD/)2 2Q+ 2m 2m ( µ Q+) 4m " µ Q+#
1 n/ 1 1 1
− gµν{iD⊥,iD⊥}− iσµν[iD⊥,iD⊥] (15)
2mQ 2 ⊥ µ ν 2 ⊥ µ ν Q
+ (cid:18) (cid:19) +
4
This result can now be used to perform the matching on the non-local operators as
they have been defined in [5]. These operators are
O (ω) = h¯ δ(ω +iDˆ )h (16)
0 v + v
Oµ(ω) = h¯ iD ,δ(ω +iDˆ ) h (17)
1 v µ + v
Oµ(ω) = h¯ niD ,δ(ω +iDˆ )oh (18)
2 v µ + v
Oµν(ω ,ω ) = h¯ δh(ω +iDˆ ){iDµ,iiDν}δ(ω +iDˆ )h (19)
3 1 2 v 2 + ⊥ ⊥ 1 + v
Oµν(ω ,ω ) = h¯ δ(ω +iDˆ )[iDµ,iDν]δ(ω +iDˆ )h (20)
4 1 2 v 2 + ⊥ ⊥ 1 + v
and
P (ω) = h¯ δ(w+iDˆ )γ γ h (21)
0,α v + α 5 v
Pµ (ω) = h¯ iD ,δ(ω +iDˆ ) γ γ h (22)
1,α v µ + α 5 v
Pµ (ω) = h¯ niD ,δ(ω +iDˆ )oγ γ h (23)
2,α v µ + α 5 v
Pµν(ω ,ω ) = h¯ δh(ω +iDˆ ){iDµ,iiDν}δ(ω +iDˆ )γ γ h (24)
3,α 1 2 v 2 + ⊥ ⊥ 1 + α 5 v
Pµν(ω ,ω ) = h¯ δ(ω +iDˆ )[iDµ,iDν]δ(ω +iDˆ )γ γ h (25)
4,α 1 2 v 2 + ⊥ ⊥ 1 + α 5 v
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 Λ .
b
Inserting (15) back into (3) and taking the imaginary part we obtain for the leading term
Γ¯n/Γ 1 π
Imh¯ h = − dω Tr[P Γn/Γ]δ(1−x−ω)h¯ δ(w+iDˆ )h
v v + v + v
2Q 2m 2
+ Z (cid:18)
π
− Tr[sαΓn/Γ]δ(1−x−ω)h¯ δ(w+iDˆ )γ γ h
v + α 5 v
2
(cid:19)
1
= − dω[C (ω)O (ω)+Cα (ω)P (ω)] (26)
2m 0 0 5,0 0,α
Z
from which we can read off the matching coefficients C and Cα as they have been derived
0 5,0
already in [5]1
In the same way we obtain for the subleading contributions
Γ¯γµΓ 1
Imh¯ ⊥ iD , h (27)
v µ v
2m ( Q+)
Γ¯γµΓ 1
= Im dωh¯ ⊥ {iD ,δ(ω +iDˆ )} h
v 2m2 µ + 1−x−ω v
Z
1Here we made us of the decomposition formula
1 1 1
P+ΓP+ = P+Tr[P+Γ]− sµTr[sµΓ] sµ =P+γµγ5P+ P+ = (1+v/).
2 2 2
5
1 π
= − dω Tr P Γ¯γµΓ δ(1−x−ω)h¯ {iD ,δ(ω +iDˆ )}h
2m2 2 + ⊥ v µ + v
Zπ (cid:18) h i
− Tr sαΓ¯γµΓ δ(1−x−ω)h¯ {iD ,δ(ω +iDˆ )}γ γ h
2 ⊥ v µ + α 5 v
h i (cid:19)
from which we obtain the matching coefficients Cµ and Cα,µ
1 5,1
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 O and Oµν in such a way
0 3
that only one function is needed to parametrize the effect of the two operators O and Oµν.
0 3
We shall use the relations from [10] in the next section when we will count the number of
of unknown functions needed to describe the spectrum of Λ → X γ including subleading
b s
contributions.
In addition to these operators we have also to take into account the subleading terms
of the Lagrangian. They are given as
dt
O (ω) = i d4x e−iωtT[h¯ (0)h (t)L (x)] (28)
T v v 1/m
2π
Z Z
dt
P (ω) = i d4x e−iωtT[h¯ (0)γ γ h (t)L (x)] (29)
T,α v α 5 v 1/m
2π
Z Z
where L is the subleading contribution to the Lagrangian
1/m
1
¯ 2
L = h (iD/) h (30)
1/m v v
2m
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.
Λ
3 Matrix Elements for Decays
b
After having computed the matching we have to consider the forward matrix elements
with a Λ . These matrix elements have to be evaluated in the static limit in which the Λ
b b
becomes a very simple object. We obtain for the matrix elements:
hΛ (v,s)|O (ω)|Λ (v,s)i = u¯(v,s)u(v,s)f (ω) (31)
b 0 b Λ
hΛ (v,s)|P (ω)|Λ (v,s)i = u¯(v,s)γ γ u(v,s)f (ω) (32)
b 0,α b α 5 Λ
where u(v,s) describes the static Λ with velocity v and spin s.
b
For the subleading operators the non-vanishing matrix elements are parametrized as
hΛ (v,s)|Oµ(ω)|Λ (v,s)i = −2u¯(v,s)u(v,s)(vµ−nµ)ωf (ω) (33)
b 1 b Λ
hΛ (v,s)|Pµ (ω)|Λ (v,s)i = −2u¯(v,s)γ γ u(v,s)(vµ−nµ)ωf (ω) (34)
b 1,α b α 5 Λ
hΛ (v,s)|Oµν(ω ,ω )|Λ (v,s)i = u¯(v,s)u(v,s)gµνg (ω ,ω ) (35)
b 3 1 2 b ⊥ Λ 1 2
6
hΛ (v,s)|Pµν(ω ,ω )|Λ (v,s)i = u¯(v,s)γ γ u(v,s)gµνg (ω ,ω ) (36)
b 3,α 1 2 b α 5 ⊥ Λ 1 2
hΛ (v,s)|O (ω)|Λ (v,s)i = u¯(v,s)u(v,s)t (ω) (37)
b T b Λ
hΛ (v,s)|P (ω)|Λ (v,s)i = u¯(v,s)γ γ u(v,s)t (ω) (38)
b T,α b α 5 Λ
Consequently, only two additional functions appear at order 1/m . Furthermore, these
b
subleading function are tied to the leading order function f by reparametrization invari-
Λ
ance, which means that it appears always the combination
1 δ(ω −ω )−δ(ω −ω )
1 2
F (ω) = f (ω)+t (ω)− dω dω g (ω ω ) (39)
Λ Λ Λ m2b Z 1 1 Λ 1 2 ω1 −ω2 !
with the leading order function f . In other words, due to heavy quark spin symmetry we
Λ
can express Λ → X γ and Λ → X ℓν¯ in terms of a single universal function, up to terms
b s b u ℓ
of order 1/m2.
b
Finally we discuss the moment expansion of the resulting functions. For the leading
term we have
λ
f (ω) = δ(ω)− Λ δ′′(ω)+··· (40)
Λ 6m2
b
where
¯ 2
λ u¯(v,s)u(v,s) = hΛ (v,s)|h (iD) h |Λ (v,s)i (41)
Λ b v v b
is the kinetic energy of the b quark inside the Λ baryon. For the subleading contributions
b
we obtain
2λ
Λ
g (ω ,ω ) = δ(ω )δ(ω )+··· (42)
Λ 1 2 1 2
3
λ
t (ω) = − Λ δ′(ω)+··· (43)
Λ 2m2
b
which reproduces the known result from [12] with the replacements λ → λ and λ → 0:
1 Λ 2
dΓ λ λ
(Λ → X γ) = Γ δ(1−x)− Λ δ′(1−x)− Λ δ′′(1−x)+··· (44)
dx b s 0" 2m2 6m2 #
b b
with
G2α|V∗V ||C |2
Γ = F tb ts 7 m5 (45)
0 32π4 b
The moment expansion of the new universal function reads
λ
F (ω) = δ(ω)+ Λ δ′(ω)+··· (46)
Λ 6m2
b
7
4 Discussion and Conclusions
The main result of this paper is that the semileptonic and radiative decays of a Λ baryon
b
into light hadrons are described by a universal function, where the corrections to this
statement are of order 1/m 2. In this way we find the two differential decay rates:
b
dΓ G2α|V∗V ||C |2
(Λ → X γ) = F tb ts 7 m 5(2x−1)F (1−x) (47)
dx b s 32π4 b Λ
dΓ G2|V |2
(Λ → X ℓν¯ ) = F ub m 5 dωΘ(1−y −ω)(1−ω)F (ω) (48)
dy b u ℓ 96π3 b Λ
Z
where x = 2E /m is the rescaled photon energy and y = 2E /m is the rescaled lepton
γ b ℓ b
energy. Notethat thefirst moment ofF stillvanishes uptotermsoforder 1/m 2, so westill
b
get nocontribution to thetotalrates. This serves a acheck, since thecontributions oforder
1/m may not contribute to the rate. As for Λ → X γ, the spectrum of the semileptonic
b s
decay can be expanded in terms of moments. Inserting (46) into the spectrum, one arrives
at
dΓ λ λ
(Λ → X ℓν¯ ) = Γ θ(1−y)− Λ δ(1−y)− Λ δ′(1−y)+... (49)
dy b u ℓ 0" 6mb2 6mb2 #
where
G2|V |2
Γ = F ub m 5. (50)
0 96π3 b
With the replacements already mentioned above, this agrees with the result obtained in
[12].
Although this may be difficult from the experimental point of view, we may investigate
the subleading corrections to a determination of V from a comparison of Λ → X γ
ub b s
with Λ → X ℓν¯ . Like in the case of B mesons, an energy cut E on the charged lepton
b u ℓ c
will be unavoidable to discriminate the large charm background, i.e. E > (M2[Λ ] −
c b
M2[Λ ])/(2M[Λ ]). One may define observables (similar as for the decays of B mesons, see
c b
[13]) involving partially integrated rates with suitable weight functions. Since the 1/m
b
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
mΛ/2 4E 2 −E m dΓΛb→Xuℓν¯ℓ
Γ (E ) ≡ dE ℓ c b u
u c ZEc ℓ 2Eℓ2 ! dEℓ
2 mΛ/2 dΓΛb→Xsγ
Γ (E ) ≡ dE (E −E ) s . (51)
s c γ γ c
m dE
b ZEc γ
from which V can be determined as
ub
V 3α Γ (E )
ub eff 2 u c 2
= |C | +O(1/m ) (52)
(cid:12)(cid:12)VtbVt∗s(cid:12)(cid:12) π 7 Γs(Ec) b
(cid:12) (cid:12)
Comparing our result ((cid:12)52) for(cid:12) the V determination with the one obtained in [6, 7] one
(cid:12) (cid:12) ub
expects substantially smaller corrections in the case of Λ baryons, since the corrections
b
are only of order 1/m 2. However, this requires to measure the inclusive semileptonic and
b
radiative rare Λ decays, which is more difficult experimentally.
b
8
Acknowledgements
This work was supported by the DFGGraduiertenkolleg “Hochenergiephysik and Teilchen-
astrophysik”, by the DFG Forschergruppe “Quantenfeldtheorie, Computeralgebra und
Monte Carlo Simulationen” and by the Ministerium fu¨r Bildung und Forschung bmb+f.
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