B decays in SCET
Gaber Faisel
National Central University
B decays in SCET – p. 1/24
Outline
Introduction to SCET
Momenta and filed scaling in SCET
SCET Lagrangian
SCET effective Hamiltonian for ∆B = 1 decays
B → M1 M2 decay amplitudes in SCET
B decays in SCET – p. 2/24
Introduction to SCET
The main idea of effective field theories is the presence of lower limit of the distances that
can be resolved through a process of given energy. In this limit, the heavy modes can be
integrated out and the non-local interactions mediated by these heavy modes are reduced to
local interactions.
SCET is an effective field theory describing the dynamics of highly energetic particles
moving close to the light-cone interacting with a background field of soft quanta
SCET provides a systematic and rigorous approach for calculating processes with
several relevant energy scales. For instance, in B decays to light mesons we have the B
energy scale, the jet scale and the low energy QCD scale.
SCET has the ability to sum up all large radiative corrections appears in high energy
scattering processes and thus preserving the perturbation theory at each order. As an
example, in processes with highly energetic hadron jets SCET can sum up the enhanced
corrections which are proportional to large logarithms of ratios of mass scales .
The systematic power counting in SCET reduce the complexity of calculations: we start
by defining a small parameter λ as the ratio of the lowest and largest energy scales in
the process under consideration and then we make a scaling for the momenta and fields
in terms of λ and finally, the Lagrangian, Effective Hamiltonian can be expanded into
terms with different orders in λ.
B decays in SCET – p. 3/24
Momenta and field scaling in SCET
Consider a process in which B meson decays into two light energetic quarks. In the rest
frame of the B meson and due to the conservation of momentum, the two quarks will be
emitted in opposite directions which can be chosen as Z direction for simplicity.
As the motion of the emitted quarks is in the Z direction, it is appropriate to define two
vectors nµ = (1, 0, 0, 1) and n̄µ = (1, 0, 0, −1). The two emitted quarks move in nµ and
n̄µ directions and they are called collinear quarks.
In terms of nµ and n̄µ we can write any vector P µ as
µ
µ
µ
µ
µ
µ
= P+
+ P−
+ P⊥
P µ = (n · P ) n̄2 + (n̄ · P ) n2 + P⊥
The notation P = (P + , P − , P⊥ ) is usually used and this decomposition of momentum
is referred as light cone decomposition.
For any given momentum, the scaling of any of its light cone components depends on a
small parameter λ.
lowest energy scale n
Usually, λ is defined as λ = ( largest!energy
) where n can be 1/2 or 1 depending
scale
on the process under consideration and so we will have two types of SCET as we will
show in the following.
B decays in SCET – p. 4/24
Consider the energetic collinear quark moves along the nµ direction: its different light
cone components are widely separated, with P− ∼ E being large, P⊥ ∼ λE being small
2 for fluctuations near
and P+ ∼ λ2 E being very small where we have used P+ P− ∼ P⊥
the mass shell. So the scaling of the momentum of the collinear quark moves in nµ
µ
µ
2
0
1
2
direction is : pµ = n · p n̄2 + n̄ · p n2 + pµ
⊥ = O(λ ) + O(λ ) + O(λ ) ≡ (λ , 1, λ)E.
The other partons in the B meson carry momenta that scale like
p
(ΛQCD , ΛQCD , ΛQCD ). If we choose λ = ΛQCD /E we can write the scaling of
these partons as (λ2 , λ2 , λ2 )E and the momentum is referred as ultrasoft momentum
mode.
Λ
we can write the scaling of these partons as (λ, λ, λ)E and the
If we choose λ = QCD
E
momentum is referred as soft momentum mode.
We can classify two different effective theories SCETI and SCETII according to the
momenta modes in the process under consideration:
SCETI : When we have only collinear and ultrsoft momentum modes as inclusive decay
of a heavy meson such as B → Xs∗ γ at the end point region and e− p → e− X at the
threshold region.
SCETII : When we have only collinear and soft momentum modes as in semi-inclusive
or exclusive decays of a heavy meson such as B → Dπ, B → Kπ,....etc.
B decays in SCET – p. 5/24
In order to write the scaling of the collinear quark field we rewrite the momentum of the
collinear quark as
p = p̃ + k
(1)
where
(2)
p̃ ≡
1
(n̄ · p)nµ + p⊥ .
2
We can remove the large momenta p̃ by defining a new field ψn,p for the collinear quark
as follows
X
(3)
ψ(x) =
e−ip̃·x ψn,p
p̃
ψn,p contains only the component k that will be treated as a dynamical degree of
freedom while p̃ becomes a label on the field.
B decays in SCET – p. 6/24
The four component field ψn,p can be expressed in terms of two two-components
spinors ξn,p and ξn̄,p defined as follows
(4)
ξn,p
=
ξn̄,p
=
n
6 n̄
6
ψn,p
4
n̄
6 n
6
ψn,p
4
The scaling ξn,p and ξn̄,p can be obtained using
(5)
h0|T ψi (x), ψ̄j (y) |0i =
Z
ip
6 ij −ip·(x−y)
d4 p
e
.
(2π)4 p2 + iǫ
Assuming collinear momentum scaling ∼ (λ2 , 1, λ), one finds that p2 ∼ O(λ2 ) and d4 p
scales like O(λ4 ).
Thus, the two-component spinor fields scale as ξn,p ∼ O(λ) and ξn̄,p ∼ O(λ2 ).
The scaling momenta modes and their corresponding fermion fields they can be
obtained in a similar way and it is given in Table (1).
B decays in SCET – p. 7/24
Momenta mode
Momentum scaling
Fermion field scaling
Hard (h)
(1,1,1)E
Collinear (c)
(λ2 , 1, λ)E
λ
Hard-collinear (hc)
(λ, 1, λ1/2 )E
λ1/2
Soft (s)
(λ, λ, λ)E
λ3/2
Ultrasoft (us)
(λ2 , λ2 , λ2 )E
λ3
Soft-collinear (sc)
(λ2 , λ, λ3/2 )E
λ2
Table 1: Scaling of different momenta modes and their corresponding fermion fields
B decays in SCET – p. 8/24
SCET Lagrangian
Few Remarks about construction of the SCET Lagrangian:
Consideration of the kinematics:
Kinematics allows only collinear collinear and collinear ultrasof interactions between
quarks or gluons or a quark and gluon. So we have Lagrangian for collinear collinear
interaction and Lagrangian for collinear ultrasof interaction.
Matching with the QCD Lagrangian and Doing expansion in orders of λ and keeping
order 0,1,2 terms.
integrating out the off shell fluctuation through introducing Wilson lines.
Recalling that the QCD Lagrangian for massless quarks and gluons is given by
(6)
LQCD
=
ψ̄ i D
6 ψ−
1
Gµν Gµν ,
4
where the covariant derivative Dµ is defined as Dµ = ∂µ − igT a Aa
µ , and Gµν is the gluon
field strength. Using
(7)
µ
γ µ = nµ n̄
6 /2 + n̄µ n
6 /2 + γ⊥
the quark part in the Lagrangian (6) can be expressed in terms of ξn,p and ξn̄,p as follows
B decays in SCET – p. 9/24
L
=
X
e−i(p̃−p̃
′
)·x
p̃,p̃′
+ξ̄n,p′
(8)
"
n̄
6
n
6
in · D ξn,p + ξ̄n̄,p′
n̄ · p + in̄ · D ξn̄,p
ξ̄n,p′
2
2
p
6 ⊥+iD
6 ⊥ ξn̄,p + ξ̄n̄,p′
#
p
6 ⊥+iD
6 ⊥ ξn,p .
Using the equation of motion, one can eliminate the small component ξn̄,p in favor of ξn,p
(9)
L=
X
p̃,p̃′
e−i(p̃−p̃
′
)·x
ξ̄n,p′
1
n̄
6
n · iD + (6p⊥ + i D
6 ⊥)
(6p⊥ + i D
6 ⊥)
ξn,p .
n̄ · p + n̄ · iD
2
µ
The covariant derivative Dµ includes only collinear and ultrasoft gluons, Aµ = Aµ
c + Aus as
the interaction of soft gluons and collinear quarks is forbidden kinematically.
it is convenient to separate the collinear and ultrasoft gluon field such that the covariant
derivative Dµ contains only the ultrasoft collinear gluons. We define the collinear gluon
−iq̃·x Aµ (x) in a similar way to the collinear quark field and so Eq. (9)can
as Aµ
n,q
c (x) = e
be written as
B decays in SCET – p. 10/24
L
=
X
e−i(p̃−p̃
′
)·x
p̃,p̃′ ,q̃

ξ̄n,p′  n · iD + ge−iq̃·x n · An,q +
1
n̄ · p + n̄ · iD + ge−iq̃·x n̄ · An,q
(10)
p
6 ⊥+iD
6 ⊥ + ge−iq̃·x A
6 ⊥
n,q

n̄
6

p
6 ⊥+iD
6 ⊥ + ge−iq̃·x A
6 ⊥
ξn,p
n,q
2
Expanding Eq. (10) in powers of gAc , we obtain at order λ0
Lcus
(11)
=
−
p2⊥ n̄
6
n̄
6
ξ̄n,p
ξn,p + ξ̄n,p n · iD ξn,p
n̄ · p 2
2
"
p
6 ⊥
p
6 ⊥ + q6 ⊥
6 ⊥
+ξ̄n,p+q gn · An,q + g A
+
gA
6 ⊥
n,q
n,q
n̄ · p
n̄ · (p + q)
#
p
6
n̄
6
p
6 ⊥ + q6 ⊥
gn̄ · An,q ⊥
ξn,p + . . . + O(λ)
n̄ · (p + q)
n̄ · p 2
The first term in Eq. (11) gives the propagator for the collinear quarks, the second term
gives its interaction with an ultrasoft gluon and the third term gives its interaction with a
collinear gluons.
B decays in SCET – p. 11/24
The order λ lagrangian of the quark collinear field is given by
(12)
(1)
Lξξ = ξ̄n i D
6 us
⊥
n̄
6
1
iD
6 c⊥ ξn + h.c.
in̄ · Dc
2
We need also to carry the matching of the full QCD lagrangian to the mixed usoft
quark-collinear quark Lagrangian.
We split the quark field into the ultrasoft quark field qus and the collinear quark field
(13)
ψ = ξn + ξn̄ + qus
Decompose the Dirac matrix γ µ in light-cone coordinates as before and using
n
6 ξn = n
6 ¯ ξn̄,p = 0, the kinetic term in the Lagrangian L = ψ̄ i D
6 ψ + . . . can be matched to
1
n̄
6
6 n ξn
6 ⊥
gA
6 n qus + q̄us g A
Lξq =
ξ̄n g A
6 n qus + ξ̄n i D
2
in̄ · D
1
n̄
6
+ q̄us g A
6 n
(14)
iD
6 ⊥ ξn
in̄ · D
2
Expanding Eq. (14) to second order in λ gives
B decays in SCET – p. 12/24
(1)
Lξq
=
(2)
=
Lξq
(15)
1
gn̄ · Ac qus + h.c.
6 c⊥
6 c⊥ − i D
ξ̄n g A
in̄ · Dc
n̄
6
1
1
c
c
us
ξ̄n
gA
6 ⊥
gn̄ · Ac qus + h.c.
gn · Ac + i D
6 ⊥
qus − ξ̄n i D
6 ⊥
2
in̄ · Dc
in̄ · Dc
µ
Introducing the label operators P̄ ∼ λ0 and P⊥
∼ λ such that, P̄ ξn,p = (n̄·p) ξn,p allows us
to redefine the collinear covariant derivatives as follows
(16)
in̄ · Dc = P̄ + gn̄ · An ,
µ
iDc⊥µ = P̄⊥
+ gA⊥µ
n ,
The ultrasoft covariant derivatives can be written as
(17)
µ
⊥µ
= i∂⊥
in̄ · Dus = in̄ · ∂ + gn̄ · Aus , iDus
+ gA⊥µ
us .
In SCET , the off-shell fluctuations resulting from attaching gluons to quarks can be
integrated out with the help of the so-called Wilson lines.
B decays in SCET – p. 13/24
For example, attaching collinear gluons to a heavy quark results in intermediate states (
propagators ) which are off-shell by an amount of order E. These intermediate states
must be integrated out as SCET is an effective theory below the energy scale E. This
can be done by a field redefinition q(x) → W (x)q 0 (x) where q(x) is the quark field and
W is the collinear Wilson line defined as
X
g
(18)
W =
n̄ · An,q (x)
,
exp −
P̄
perms
where the label operators only act on fields inside the square brackets. Performing this
field redefinition will lead to a lagrangian that contains the field q 0 (x) which no longer
couples to the collinear gluons.
we can write eq.(15) as:
(19)
(1)
Lξq = ig ξ̄n
(2a)
Lξq
(20)
=
1
c
B̄
6 ⊥ W qus + h.c.
in̄ · Dc
ig ξ̄n
1
M̄
6 W qus + h.c. ,
in̄ · Dc
B decays in SCET – p. 14/24
(2b)
Lξq
(21)
n̄
6
1
c
c
= ig ξ̄n i D
B̄
6
6 ⊥
⊥ W qus + h.c.
2
(in̄ · Dc )2
where the following operators are introduced
c
6 c⊥ ].
ig B̄
6 ⊥ = [in̄ · Dc , i D
(22)
and
ig M̄
6 = [in̄ · Dc , i D
6 us +
(23)
n̄
6
gn · Ac ].
2
Finally, we give here the Lagrangian for the collinear gluons
Lcg =
(24)
1
µ
ν 2
tr
[iD
,
iD
]
2g 2
The order λ collinear gluons Lagrangian which can be obtained by Expanding eq.(24) in
powers of λ
o
2 n
(1)
µ
⊥ν
⊥
(25)
Lcg = 2 tr [iD , iDc ][iDµ , iDusν ]
g
where
(26)
Dµ = Dcµ + n̄µ n ·
Dus
2
B decays in SCET – p. 15/24
SCET Hamiltonian for ∆B = 1 decays
We start by writing the Weak Effective Hamiltonian in the SM describing the weak
interactions of the B meson namely ∆B = 1 decays.
GF
Heff = √
2
X
(D)
λp
C1 Qp1 + C2 Qp2 +
p=u,c
X
Ci Qi + C7γ Q7γ + C8g Q8g
i=3,...,10
!
+h.c.
(27)
(D)
where λp
Qp1
=
Q3
=
Q5
=
Q7
=
Q9
=
(28)
∗ with D = d, s and
≡ Vpb VpD
(p̄b)V −A (D̄p)V −A ,
X
(D̄b)V −A
q (q̄q)V −A ,
X
(D̄b)V −A
q (q̄q)V +A ,
X
3
(D̄b)V −A
q 2 eq (q̄q)V +A ,
X
3
(D̄b)V −A
q 2 eq (q̄q)V −A ,
Qp2 = (p̄α bβ )V −A (D̄β pα )V −A ,
X
Q4 = (D̄α bβ )V −A
q (q̄β qα )V −A ,
X
Q6 = (D̄α bβ )V −A
q (q̄β qα )V +A ,
X
3
Q8 = (D̄α bβ )V −A
q 2 eq (q̄β qα )V +A ,
X
3
Q10 = (D̄α bβ )V −A
q 2 eq (q̄β qα )V −A
B decays in SCET – p. 16/24
Q7γ
=
e
µν
m
D̄σ
(1
+
γ
)F
b,
µν
5
b
8π 2
Q8g =
gs
A
A
G
m
D̄
σ
(1
+
γ
)t
α
µν
5
b
µν bβ
αβ
8π 2
(29)
where Ci (MW ) are the Wilson coefficients α and β stand for color indices, tA
αβ are the
SU (3)c color matrices and σ µν = 21 i[γ µ , γ ν ]. eq are quark electric charges in units of e,
(q̄q)V ±A ≡ q̄γµ (1 ± γ5 )q, and q runs over u, d, s, c, and b quark labels.
Matching of the weak effective Hamiltonian Heff to the corresponding SCET gauge invariant
operators requires two step matching:
First the full QCD effective weak Hamiltonian is matched to the corresponding weak
Hamiltonian in SCETI by integrating out the hard scale mb . Hence, after integrating out
the b̄b pairs in the electroweak Penguin operators, we can write
(30)
Q9
=
Q10
=
3 u
Q +
2 2
3 u
Q +
2 1
3 c
Q −
2 2
3 c
Q −
2 1
2
Q3
2
2
Q4 .
2
B decays in SCET – p. 17/24
which leads to remove the operators Q9 and Q10 from the effective Hamiltonian.
Second, the SCETI weak Hamiltonian is matched to the weak Hamiltonian SCETII by
integrating out the hard collinear modes with p2 ∼ Λmb as we will show in details below.
At leading power in (1/mb ) expansion, the effective Hamiltonian in eq.(27) is matched into
SCETI Hamiltonian as follows
Z
2GF X n X
(f )
(0)
√
[dωj ]3j=1 ci (ωj )Qif (ωj )
HW =
2 n,n̄
i
Z
o
X
(1)
(f )
4
+
[dωj ]j=1 bi (ωj )Qif (ωj ) + Qcc̄ + ....
(31)
i
(f )
where f = d or s, ci
(0)
(f )
and bi
are Wilson coefficients corresponding to O(λ0 ) operators
(1)
(Qif ) and O(λ) operator (Qif ). Qcc̄ denotes the operator corresponding to the long
(0)
(1)
distance charm Penguin. For B → ππ decays as an example, the operators Qif and Qif
in Eq.(31) are given by
(32)
(0)
Q1f
=
[ūn,ω1 6n̄PL bν ][f¯n̄,ω2nP
6 L un̄,ω3 ]
B decays in SCET – p. 18/24
(0)
Q2f,3f
=
[f¯n,ω1 6n̄PL bν ][ūn̄,ω2nP
6 L,R un̄,ω3 ],
(0)
=
6 L qn̄,ω3 ],
[q̄n,ω1 6n̄PL bν ][f¯n̄,ω2nP
Q4f
(33)
(1)
Q1f
=
Q2f,3f
(1)
=
(1)
=
Q4f
−2
⊥
P b ][f¯n̄,ω2nP
6 L un̄,ω3 ],
[ūn,ω1 ig6Bn,ω
4 L ν
mb
−2 ¯
⊥
P b ][ūn̄,ω2nP
6 L,R un̄,ω3 ],
[fn,ω1 ig6Bn,ω
4 L ν
mb
−2
⊥
P b ][f¯n̄,ω2nP
6 L qn̄,ω3 ],
[q̄n,ω1 ig6Bn,ω
4 L ν
mb
(34)
where a sum over q = u, d, s is understood and
(q)
qn,ω = [δ(ω − n̄.P )Wn† ξn ],
(35)
(36)
⊥µ
=
igBn,ω
1
µ
[Wn† [in̄.Dc,n , iDn,⊥
]Wn δ(ω − n̄.P † )].
(−ω)
B decays in SCET – p. 19/24
with n̄.P is the operator that project out the large momentum component of the collinear
quark field and P operates only inside the square brackets. The bν is the standard heavy
(0)
quark effective field (HQET). The Wilson coefficients corresponding to the operators Qif
are given by
1
3
1
(f )
(f )
(f )
(f )
c1,2 = λu C1,2 + C2,1 − λt
C9,10 + C10,9 + ∆c1,2
N
2 N
3 (f )
1
(f )
(f )
= − λt
c3
C7 + C8 + ∆c3
2
N
1
1
(f )
(f ) 1
C3 + C4 −
C9 − C1 0 + ∆c4
c4 (f ) = −λt
(37)
N
2N
2
(1)
and for Qif we have
b1,2
(f )
=
(f )
=
(f )
=
b3
b4
(38)
m
1
m
3
1
(f )
(f )
(f )
b
b
1−
C2,1 − λt
C10,9 +
1−
C9,10 + ∆b1,2
λu C1,2 +
N
ω3
2
N
ω3
mb 1
(f ) 3
(f )
−λt
C7 + 1 −
C8 + ∆b3
2
ω2 N
1
1
m
1
m
(f )
(f )
(f )
b
b
C4 +
−λt
1−
C3 + λt
C10 +
1−
C9 + ∆b4
N
ω3
2
N
ω3
B decays in SCET – p. 20/24
where ω2 = mb u and ω3 = −mb ū. u and ū = 1 − u are momentum fractions for the quark
(f )
(f )
and antiquark n̄ collinear fields. The ∆ci and ∆bi denote terms depending on αs
generated by matching from HW .
The matching of SCETI onto SCETII is performed by integrating out the hard collinear
modes with p2 ∼ Λmb . Therefore, the collinear n and collinear n̄ sectors are decoupled
(0,1)
and the operators Qif factor into
(0,1)
Qif
(39)
(0,1)
where, after dropping f , Q̃if
(40)
(0)
Q̃i
=
h
i
q̄n,ω
1
(0,1)
= Q̃if
Q̃n̄
if .
and Q̃n̄
if are given by
n̄
6 PL bv
i
(1)
Q̃i
i
−2 h i
⊥
=
q̄
ig B
6 n,ω4 PL bv
mb n,ω1
and
(41)
Q̃n̄
i
=

i
′i

6 PL qn̄,ω

3
 q̄n̄,ω2 n



′i
i
n
6 PR qn̄,ω
q̄n̄,ω
2
3
i = 1, 2, 4, 5
i = 3, 6
B decays in SCET – p. 21/24
B → M1M2 decay amplitudes in SCET
The leading order amplitude can be generated through the time ordered products of the
(0)
(1)
operators Qi and Qi and the subleading Lagrangians
Z
(1)
(1)
(0)
(0)
(1)
T1 [Q̃i ] =
d4 y d4 y ′ T [Q̃i (0), iLξn ξn (y ′ ) + iLcg (y ′ ), iLξn q (y)]
Z
(1,2)
(0)
+
d4 yT [Qif (0), iLξn q (y)],
Z
(1)
(1)
(1)
(42)
T2 [Q̃i ] =
d4 yT [Q̃i (0), iLξn q (y)].
The matrix elements of the time ordered products (T1,2 )and Q̃n̄
i can be expressed in terms
of the following hadronic parameters
h
i E
L
6 bv B
Mn T1 q̄nω1 n̄
h
i E
D
⊥
L
6 nω4 bv B
Mn T2 q̄nω1 ig B
E
D
′L
L Mn̄ q̄n̄ω2 n
6 qn̄ω3 0
E
D
′R
R Mn̄ q̄n̄ω2 n
6 qn̄ω3 0
D
(43)
=
BM
m
ζ
,
δ̄
CqBM
ω
B
1
L
=
δ̄ω1 ω4
−CqBM
L
=
=
mB BM
ζJ (z),
2
i M
C ′ δ̄ω ω , fM φM (u),
2 qL q 2 3
i M
C ′ δ̄ω ω fM φM (u).
2 qR q 2 3
B decays in SCET – p. 22/24
The parameters ζ BM , ζJBM are treated as hadronic parameters that can be determined
through the fit to the non leptonic decay data. Finally, the amplitude at leading order in 1/mb
expansion and to all orders in αS (mb ) is given by
E
D
LO
A (B̄ → M1 M2 ) = −i M1 M2 HW B̄
"Z
2
1
GF mB
√
=
fM1
dudzT1J (u, z)ζJBM2 (z)φM1 (u)
2
0

Z 1
(f ) M1 M2
+ ζ BM2
+ (1 ↔ 2).
(44)
duT1ζ (u)φM1 (u) + λc Acc
0
The hard kernels T1ζ and T1J are functions of linear combinations of the matching
(f )
(f )
coefficients ci (u) and bi (u, z) and depend on the final state mesons. T1ζ and T1J for
some specific decay channels and T1ζ and T1J are given in Table(2).
B decays in SCET – p. 23/24
Decay Mode (M1 M2 )
π0 K −
π+ K −
π 0 K̄ 0
T1ξ (u)
(s)
√1 (c
2 2
(s)
− c3 )
0
(s)
√1 (c
2 2
π − K̄ 0
0
K̄ 0 φ
0
K̄ 0 ηs
c4
K̄ 0 ηq
T2ξ (u)
(s)
− c3 )
(s)
(s)
√1 c
4
2
(s)
(s)
√1 (c
+
c
4 )
2 1
(s)
(s)
c1 + c4
(s)
− √1 c4
2
(s)
−c4
(s)
(s)
(s)
c4 + c5 + c6
(s)
(s)
(s)
c4 + c 5 − c 6
(s)
(s)
(s)
√1 (c
− c3 + 2c5 −
2
2
(s)
2c6 )
Table 2: Hard kernels for ∆S = 1 decays of B − , B̄ 0 and B̄s0 into some final states. The co(f )
efficients T1J,2J (u, z) for all these states are identical to T1ζ,2ζ (u) with each ci (u) replaced
(f )
by bi (u, z).
B decays in SCET – p. 24/24