B FEYNMAN RULES WITH A MASS
differential cross section for this process can be written as
Z
X
1
dσ
2
=
H
(q
,
µ)
dka+ dkb+ Q2 Bi [ωa (Ba+ − ka+ ), xa , µ]Bj [ωb (Bb+ − kb+ ), xb , µ]
ij
σ0 dq 2 dY dBa+ dBb+
ij
!#
"
p
B
ω
Λ
a,b a,b
QC D
(12.40)
× Si hemi (ka+ , kb+ , µ) 1 + O
,
Q
Q
where ωa,b = xa,b Ecm andBi is defined as our ”Beam Function.”
Z
θ(ω)
n
dy − ib+ y/2 n
/̄
Bq (ωb+ , ω/p̂− , µ) =
e
pn (p̂− ) |χ̄n (y − )δ(ω − P̄) χn (0)| pn (p̂− )
ω
4π
2
2
(12.41)
We recll the definitions of jet function
n n
/¯
(0| |χ̄n, ω (y − ) χn (0)| |0)
2 2
(12.42)
and pdf
n
/¯
(12.43)
(p| |χ̄n, ω (0) χn (0)| |p)
2
We see that the Jet Function is a mix of both. The proton is a collinear field in SCETII and the jet is
collinear in SCETI . Matching SCETI to SCETII gives us
"
!#
X Z 1 dξ
Λ2QCD
x
Bi (t, x, µ) =
Iij (t, , µ)fj (ξ, µ) 1 + O
(12.44)
ξ
t
x ξ
i
bµa = (ξ − x)Ecm
n̄a
na
+ b+
+ ba ⊥
a
2
s
(12.45)
At tree level the Beam Function is simply
Bi (t, x, µ) = δ(t)fi (x, µ)
as in the pdf case we can write the RGE for the beam function
Z
d
µ Bi (t, x, µ) = dt0 γi (t − t0 , µ)Bi (t0 , x, µ)
dµ
(12.46)
(12.47)
Like the jet function Bi is independent of mass evolution. The RGE sums ln2 (t/µ), is independent of x
and has no mixing.
A
More on the Zero-Bin
A.1
0-bin subtractions with a 0-bin field Redefinition
A.2
0-bin subtractions for phase space integrations
B
Feynman Rules with a mass
If we add a mass the collinear Lagrangian becomes
(0)
¯
Lξξ = ξn (x) in·D + (iD
/c⊥ − m)
1
n̄/
c
(iD
/⊥ + m)
ξn (x) ,
c
in̄·D
2
and the modified Feynman rules are shown in Fig. 12.
82
(B.1)
D
(p̃, pr )
n
/
=
FEYNMAN RULES FOR SUBLEADING LAGRANGIANS
n̄·p
i 2 n·p n̄·p + p2 −m2 +i
r
⊥
μ,A
= ig T A nµ n̄2/
μ,A
= ig T A nµ +
γµ⊥ (p/⊥ +m)
n̄·p
+
0
(p/⊥
−m)γµ⊥
n̄·p 0
−
0
(p/⊥
−m)(p/⊥ +m)
n̄·p n̄·p 0
n̄µ
n̄/
2
pɂ
p
ν,B
μ,A
=
ig 2 T A T B
n̄·(p−q)
γµ⊥ γν⊥
−
⊥ (p
γµ
/⊥+m)
n̄ν
n̄·p
−
0 −m)γ ⊥
(p/⊥
ν
n̄µ
n̄·p 0
+
0 −m)(p
(p/⊥
/⊥+m)
n̄µ n̄ν
n̄·p n̄·p 0
n̄
/
2
q
+
pɂ
p
ig 2 T B T A
n̄·(q+p0 )
γν⊥ γµ⊥ −
γν⊥ (p
/⊥+m)
n̄µ
n̄·p
−
0 −m)γ ⊥
(p/⊥
µ
n̄ν
n̄·p 0
+
0 −m)(p
(p/⊥
/⊥+m)
n̄µ n̄ν
n̄·p n̄·p 0
n̄
/
2
Figure 12: Order λ0 Feynman rules as in Fig. 6, but with a collinear quark mass.
C
Feynman Rules for the Wilson line W
Results for the Feynman rules for the expansion of the W Wilson line are also useful
gT A n̄·εA
n (q)
+ ... ,
n̄·q
gT A n̄·εA
n (q)
W† = 1 +
+ ... ,
n̄·q
W =1−
(C.1)
where here the momentum q is incoming and εA
n is the gluon-polarization vector.
D
Feynman Rules for Subleading Lagrangians
In this subsection Feynman rules are given for the subleading quark Lagrangians involving two collinear
quarks
(1)
=
(2)
=
Lξξ
Lξξ
⊥ 1
1
n̄
/ / † n̄
ξ¯n W iD
/us
W † iD
/⊥
ξn + ξ¯n iD
/⊥
iD
/⊥
ξn
c W
c
us W
n̄·P
2
n̄·P
2
⊥ 1
1
n̄/
n̄/
ξ¯n W iD
/us
iD
/⊥
W † ξn + ξ¯n iD
/⊥
in̄·Dus W † iD
/⊥
us
c W
c ξn ,
2
n̄·P
2
n̄·P
2
83
(D.1)
D
FEYNMAN RULES FOR SUBLEADING LAGRANGIANS
and for the mixed usoft-collinear Lagrangians from Eq. (??),
(1)
Lξq
(2a)
Lξq
(2b)
Lξq
1
igB
/⊥
c W qus + h.c. ,
in̄·Dc
1
/ W qus + h.c. ,
= ξ¯n
igM
in̄·Dc
n̄
/ c
1
= ξ¯n iD
/
igB
/⊥c W qus + h.c. .
2 ⊥ (in̄·Dc )2
= ξ¯n
(i)
(D.2)
(1)
All Feynman rules for Lξq involve at least one collinear gluon. From Lξq we obtain Feynman rules with
zero or one A⊥
n gluons and any number of n̄·An gluons. The one and two-gluon results are shown in Fig. 15.
(2a)
For Lξq we have Feynman rules with zero or one {n·An , A⊥
us } gluon and any number of n̄·An gluons. The
(2b)
one and two-gluon results are shown in Fig. 16. Finally, for Lξq one finds Feynman rules with zero, one,
or two A⊥
n gluons and any number of n̄·An gluons. In this case the one and two gluon Feynman rules are
shown in Fig. 17.
Finally, for the subleading terms in the mixed usoft-collinear gluon action we find
o
2 n µ
⊥ν
⊥
†
L(1)
=
tr
iD
,
iD
iD
,
W
iD
W
,
0µ
cg
c
us
ν
0
g2
o
1 n µ
⊥ν
†
⊥
†
iD
,
W
iD
W
L(2)
=
tr
iD
,
W
iD
W
0µ
cg
us
us
ν
0
g2
n
⊥µ
† ⊥
o
o
1 n
1
⊥ν
⊥
+ 2 tr iD0µ , in·D iD0µ , W in̄·Dus W †
, iDus
W iDcµ , iDcν
+ 2 tr W iDus
g
g
o
1 n
⊥µ
⊥
⊥
+ 2 tr W iDus
W † , iDc⊥ν iDcµ
, W iDusν
W† ,
g
where iD0µ = iDµ + gAnµ .
84
(D.3)
D
(p̃, pr ) (1)
FEYNMAN RULES FOR SUBLEADING LAGRANGIANS
n̄
/ 2p⊥ ·p⊥
r
n̄·p
=
i2
μ,A
¯ 2pµ
⊥
n̄·p
= ig T A n2/
p
μ,A
= ig T A
n̄
/
2
/⊥
γµ⊥ p
r
n̄·p
+
p/r0⊥ γµ⊥
n̄·p 0
+
n̄µ p/⊥
/⊥
r p
n̄·q n̄·p
−
n̄µ p/ 0⊥ p/r0⊥
n̄·q n̄·p0
−
n̄µ p/r0⊥ p/⊥
n̄·q n̄·p0
pɂ
p
μ,A
ν,B
=
/
ig 2 T A T B n̄
2
⊥ ⊥
γµ γν · · ·
q
+
p
pɂ
/
ig 2 T B T A n̄
2
⊥ ⊥
γν γµ · · ·
(1)
Figure 13: Order λ1 Feynman rules with two collinear quarks from Lξξ .
85
+
n̄µ p/ 0⊥ p/⊥
r
n̄·q n̄·p
Feynman rules for Jhl
D.1
(p̃, pr ) (2)
D
FEYNMAN RULES FOR SUBLEADING LAGRANGIANS
n̄
/ p2
=
r⊥
i 2 n̄·p
μ,A
= ig T A
¯/
n
2
= ig T A
¯/
n
2
μ,A
2p⊥µ
r
n̄·p
2
n̄µ pr⊥
n̄·p
−
n̄µ p2⊥
(n̄·p)2
2
n̄µ p0r⊥
n̄·p0
−
−
γµ⊥ p/⊥ n̄·pr
(n̄·p)2
−
0 ⊥
γµ n̄·pr
p/⊥
(n̄·p 0 )2
−
0
n̄µ p/⊥
p/⊥ n̄·pr
n̄·q(n̄·p)2
+
0
n̄µ p/⊥
p/⊥ n̄·pr
n̄·q(n̄·p 0 )2
pɂ
p
μ,A
ν,B
=
/
ig 2 T A T B n̄
2
γµ⊥ γν⊥
···
q
+
p
pɂ
/
ig 2 T B T A n̄
2
γν⊥ γµ⊥
···
(2)
Figure 14: Order λ2 Feynman rules with two collinear quarks from Lξξ .
µ,a
h
/q⊥ i
ig T a γµ⊥ − n̄µ
n̄·q
=
(q, t)
(p, k)
µ,a
( q1 , t 1 )
ν, b
( q2 , t 2 )
(p, k)
=
T b T a n̄µ n̄ν /p⊥
⊥
ig
− γν n̄µ
n̄·q1
n̄·p
a b
n̄µ n̄ν /p⊥
⊥
2 T T
+ ig
− γµ n̄ν
n̄·q2
n̄·p
2
(1)
Figure 15: Feynman rules for the subleading usoft-collinear Lagrangian Lξq with one and two collinear
gluons (springs with lines through them). The solid lines are usoft quarks while dashed lines are collinear
quarks. For the collinear particles we show their (label,residual) momenta. (The fermion spinors are
suppressed.)
D.1
Feynman rules for Jhl
Here we give Feynman rules for the O(λ) heavy-to-light currents J (1a) and J (1b) in Eq. (??) which are valid
in a frame where v⊥ = 0 and v·n = 1.
For the subleading currents the zero and one gluon Feynman rules for J (1a) and J (1b) are shown in
Figs. 18 and 19 respectively. (From the results in the previous sections the Feynman rules for the currents
86
D.1
Feynman rules for Jhl
D
FEYNMAN RULES FOR SUBLEADING LAGRANGIANS
µ,a
(q, t)
n̄/ n̄µ n·t nµ −
2
n̄·q
=
ig T a
=
−g 2 f abc T c n̄/
n̄µ nν
n̄·q
2
(p, k)
µ,a
ν, b
(q, t)
(p, k)
µ,a
( q1 , t 1 )
T aT b
n ·(t1 + t2 ) n̄
/
− nµ n̄ν + n̄µ n̄ν
ig
n̄·q2
n̄·p
2
ν, b
2
( q2 , t 2 )
=
T bT a
n ·(t1 + t2 ) n̄
/
+ ig
− nν n̄µ + n̄µ n̄ν
n̄·q1
n̄·p
2
2
(p, k)
(2a)
Figure 16: Feynman rules for the O(λ2 ) usoft-collinear Lagrangian Lξq with one and two gluons. The
spring without a line through it is an usoft gluon. For the collinear particles we show their (label,residual)
momenta, where label momenta are p, q, qi ∼ λ0,1 and residual momenta are k, t, ti ∼ λ2 . Note that the
result is after the field redefinition made in Ref. [?].
µ,a
=
(q, t)
ig
T a n̄/ h
q2 i
/q⊥ γµ⊥ − n̄µ ⊥
n̄·q 2
n̄·q
(p, k)
µ,a
ν, b
γµ⊥ n̄ν /q2⊥
T a T b n̄/ ⊥ ⊥
p
/⊥ ⊥
(γµ n̄ν +γν⊥ n̄µ ) −
ig
γµ γν −
n̄·q2 2
n̄·p
n̄·q2
2
( q1 , t 1 )
( q2 , t 2 )
(p, k)
=
+n̄µ n̄ν
p2⊥
p/⊥ /q 2⊥
+
(n̄·p)2 n̄ ·p n̄·q2
+ (a, µ, q1 , t1 ) ↔ (b, ν, q2 , t2 )
(2b)
Figure 17: Feynman rules for the O(λ2 ) usoft-collinear Lagrangian Lξq with one and two gluons. For
the collinear particles we show their (label,residual) momenta, where label momenta are p, q, qi ∼ λ0,1 and
residual momenta are k, t, ti ∼ λ2 .
6 0 and v ·n 6= 1 can also be easily derived.) For J (1a) the Wilson coefficients depend only on
with v⊥ =
the total λ0 collinear momentum, while for J (1a) the coefficients depend on how the momentum is divided
between the quark and gluons. The J (1a) current has non-vanishing Feynman rules with zero or one A⊥
n
gluon and any number of n̄·An gluons. The possible gluons that appear in the J (1b) currents are similar,
but the current vanishes unless it has one or more collinear gluons present.
87
D.1
Feynman rules for Jhl
D
FEYNMAN RULES FOR SUBLEADING LAGRANGIANS
J (1a)
(d)α
=
(d)
−i Bi (n̄·p̂)
=
(d)
−i Bi
(p, k)
J (1a)
(p, k)
p⊥
α Υi
n̄·p
(d)α g Ta
n̄µ p⊥
(d)µ
α Υi
n̄·(p̂+q̂)
Υi +
n̄·(p+q)
n̄·q
(q, t)
µ, a
Figure 18: Feynman rules for the O(λ) currents J (1a) in Eq. (??) with zero and one gluon (the fermion
spinors are suppressed). For the collinear particles we show their (label,residual) momenta, where label
momenta are p, q ∼ λ0,1 and residual momenta are k, t ∼ λ2 . Momenta with a hat are normalized to mb ,
p̂ = p/mb etc.
J (1b)
=
0
=
(d)
( )
i Bi
(p, k)
J (1b)
(p, k)
�
�
((d)α
) g Ta
n̄
¯ µ qα⊥ Θi
(d)µ
( )
n̄·p,
¯ · ˆ n̄·q̂
¯·ˆ
Θi −
mb
n̄·q
¯·
(q, t)
µ, a
Figure 19: Feynman rules for the O(λ) currents J (1b) in Eq. (??) with zero and one gluon. For the
collinear particles we show their (label,residual) momenta, where label momenta are p, q, qi ∼ λ0,1 and
residual momenta are k, t ∼ λ2 . Momenta with a hat are normalized to mb , p̂ = p/mb etc.
88
E
E
INTEGRAL TRICKS
Integral Tricks
Feynman parameter tricks:
a
−1 −1
b
1
Z
−2
dx a + (b − a)x
=
(E.1)
0
Z
Γ(n + m) 1
xn−1 (1 − x)m−1
dx
Γ(n)Γ(m) 0
[a + (b − a)x]n+m
Z 1 Z 1−x
−3
= 2 dx
dy c + (a − c)x + (b − c)y
0
0
Z 1 Z 1
−3
= 2 dx dy x a + (c − a)x + (b − c)xy
a−n b−m =
a−1 b−1 c−1
0
0
−1
a−1
1 · · · an = (n − 1)!
(a1m1
n −1
· · · am
n )
1
Z
dx1 · · · dxn δ
X
xi − 1
X
xi ai
−n
0
P
Γ( mi )
=
Γ(m1 ) · · · Γ(mn )
Z
1
dx1 · · · dxn δ
X
xi − 1
X
xi ai
−n Y
0
i −1
xm
i
To get the fourth line from the third we let x0 = 1 − x and y 0 = y/x.
Georgi parameter tricks (when one or more propagators are linear in loop momenta):
Z ∞
−2
−1 −1
a b =
dλ a + bλ
0
−q −1
a
b
∞
Z
−(q+1)
dλ a + bλ
= 2q
=q
0
∞
−(q+1)
dλ a + 2bλ
0
∞
−(p+q)
+ q)
dλ λp−1 a + 2bλ
Γ(p)Γ(q) 0
Z ∞
Z ∞
−3
−3
=2
dλ dλ0 c + aλ0 + bλ
=8
dλ dλ0 c + 2aλ0 + 2bλ
a−q b−p =
a−1 b−1 c−1
2p Γ(p
Z
Z
0
0
89
(E.2)
F
F
QCD SUMMARY
QCD Summary
The SU (Nc ) QCD Lagrangian without gauge fixing
¯ / − m)ψ − 1 GA GµνA ,
L = ψ(iD
4 µν
A A
Dµ = ∂µ + igAµ T ,
A
A
ABC B C
GA
A µ Aν
µν = ∂µ Aν − ∂ν Aµ − gf
(F.1)
A
[Dµ , Dν ] = igGA
µν T .
The equations of motion and Bianchi
(iD
/ − m)ψ = 0 ,
ABC Bµ C
¯ ν T Aψ ,
∂ µ GA
A Gµν + gψγ
µν = gf
µνλσ (Dν Gλσ )A = 0.
(F.2)
Color identites
[T A , T B ] = if ABC T C ,
Tr T A T B = TF δ AB ,
T¯A = −T A∗ = −(T A )T ,
i
f ABC T B T C = CA T A ,
2
C
40
5
0
0
A
T A T B T A = CF −
TB ,
dABC dABC =
,
dABC dA BC = δ AA ,
(F.3)
2
3
3
where CF = (Nc2 − 1)/(2Nc ), CA = Nc , TF = 1/2, and CF − CA /2 = −1/(2Nc ). The color reduction
formula and Fierz formula are
δ AB
i
1
1
1
T AT B =
(T A )ij (T A )k` = δi` δkj −
δij δk` .
(F.4)
1 + dABC T C + f ABC T C ,
2Nc
2
2
2
2Nc
Feynman gauge rules, fermion, gluon, ghost propagators, and Fermion-gluon vertex
T A T A = CF 1 ,
f AC D f BCD = CA δ AB ,
−ig µν δ AB
,
k 2 + i0
i(/p + m)
,
2
p − m2 + i0
k2
i
,
+ i0
−igγ µ T A .
(F.5)
B
C
Triple gluon and Ghost Feynman rules in covariant gauge for {AA
µ (k), Aν (p), Aρ (q)} all with incoming
C
momenta, and c̄A (p)AB
µ c with outgoing momenta p:
− gf ABC g µν (k − p)ρ + g νρ (p − q)µ + g ρµ (q − k)ν ,
gf ABC pµ .
(F.6)
2
A
B
C
Triple gluon Feynman rule in bkgnd Field covariant gauge Lgf = −(DµA QA
µ ) /(2ξ) for {Aµ (k), Qν (p), Qρ (q)}
with AA
µ a bkgnd field:
h q ρ
p ν − g f ABC g µν k − p −
+ g νρ (p − q)µ + g ρµ q − k +
.
(F.7)
ξ
ξ
Lorentz gauge:
(∂µ Aµ )2
−i µν
kµ kν ,
Dµν (k) = 2
g − (1 − ξ) 2 ,
L=−
(F.8)
2ξ
k + i0
k
where Landau gauge is ξ → 0. Coulomb gauge:
ν0 k 0 k µ + g µ0 k 0 k ν − k µ k ν ]
−i
[g
µν
µν
~ ·A
~ = 0,
g −
,
∇
D (k) = 2
~k 2
k + i0
i
i ij k i k j D00 (k) =
,
Dij (k) = 2
δ −
.
(F.9)
~k 2 − i0
~k 2
k + i0
Running coupling with β0 = 11CA /3 − 4TF nf /3 = 11 − 2nf /3:
αs (µ) =
1+
αs (µ0 )
β0
µ
2π αs (µ0 ) ln µ0
=
1
1
β0
µ
=
+
ln
.
αs (µ)
αs (µ0 ) 2π µ0
2π
,
µ
β0 ln ΛQCD
Bibliography References
90
(F.10)
REFERENCES
REFERENCES
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8.851 Effective Field Theory
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