Soft-Collinear Effective Theory

Soft-Collinear Effective Theory
Part I: The strategy of regions
Thomas Becher
University of Bern
The Infrared structure of Gauge Theories, ETHZ, Feb. 1-12, 2010

Literature
Unfortunately, no SCET review article is available and to make
matters worse two different formalisms are commonly used. A few
selected original references are:
• Original papers (using the label formalism):
• C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An effective field
theory for collinear and soft gluons: Heavy to light decays, PRD 63,
114020 (2001) [hep-ph/0011336]
• C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, Soft-Collinear
Factorization in Effective Field Theory, PRD 65, 054022 (2002) [hep-ph/
0109045]
• SCET in position space:
• M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann,``Soft-collinear
effective theory and heavy-to-light currents beyond leading power,''
Nucl. Phys. B643, 431 (2002) [arXiv:hep-ph/0206152]
2

Literature
Collider physics applications:
• Factorization analysis for DIS, Drell-Yan and other processes
• C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard
scattering factorization from effective field theory, PRD 66, 014017 (2002)
[hep-ph/0202088].
• Threshold resummation in momentum space using RG evolution
• TB and M. Neubert, Threshold resummation in momentum space from
effective field theory, PRL 97, 082001 (2006), [hep-ph/0605050]
• EFT analysis of the IR structure of gauge theory amplitudes
• TB and M. Neubert, Infrared singularities of scattering amplitudes in
perturbative QCD, PRL 102, 162001 (2009) [0901.0722]; On the
Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 0906,
081 (2009) [0903.1126]; Infrared singularities of QCD amplitudes with
massive partons, PRD 79, 125004 (2009) [0904.1021]
3

Outline
Part I: The strategy of regions
Part II: Scalar SCET
Part III: Generalization to QCD
Part IV: Resummation by RG evolution
Part V: IR divergences of gauge theory amplitudes
4

Introduction
5

IR singularities
*
On-shell amplitudes in theories with massless
particles (such as gauge theories) have infrared (IR)
divergences. They arise from configurations of soft
and collinear loop momenta.
In physical observables, they cancel against real
radiation contributions involving soft and collinear
emissions.
To understand the structure of these singularities
in QCD, we thus need to understand the theory in
the soft and collinear limit.
* Here: IR refers to soft or collinear. Lorenzo Magnea reserves IR for soft divergences.
6

Soft-Collinear Effective Theory (SCET)
As the name suggests, this effective field theory
(EFT) is constructed to reproduce QCD in the
limit where particles become soft or collinear.
Each QCD field is represented by several fields in
SCET:
• Soft field
• Collinear fields describe partons propagating
with large energy along some reference
directions. (n reference directions for an n-jet
process)
7

Diagrammatic vs. EFT approach
Instead of using an effective field theory, we can
just expand the full theory diagrams around the
given limit. →Lorenzo Magnea’s lecture
There is a close relation between the expanded full
theory diagrams and the effective theory
• Soft and collinear regions are reproduced by the
EFT diagrams
• High energy region is abosorbed into Wilson
coefficients (i.e. coupling constants) of the EFT
8

Strategy of regions
• A technique to perform asymptotic expansions
of loop integrals in dimensional regularization
around various limit.
• The expansion is obtained by splitting the
momentum integration into different regions.
• There is a one-to-one correspondence
between the strategy of regions technique and
Feynman diagrams of EFTs in dimensional
regularization
Beneke and Smirnov NPB522, 321 (1998) [hep-ph/9711391]
9

Advantages of the EFT approach
Convenient language to derive all order results
• Factorization theorems
• Renormalization group: resummation of
logarithmically enhanced contributions
Gauge invariance is manifest on the Lagrangian level
and it is possible to go to higher twist by
systematically including subleading terms in the
effective Lagrangian.
10

Strategy of regions
11

A simple example
Consider the integral
I =
∫ ∞
0
dk
k
(k 2 + m 2 )(k 2 + M 2 ) = ln M m
M 2 − m 2
= ln M {
}
m
M 2 1+ m2
M 2 + m4
M 4 + ...
for m 2 ≪ M 2
How can we expand the integral before
performing the integration?
12

I =
∫
Naive
∞
k
dk
expansion
(k 2 + m 2 )(k 2 + M 2 ) = ln M m
M 2 − m 2
0
= ln M {
m
M 2
1+ m2
The naive expansion of the integrand leads to illdefined
expressions:
I =
≠
∫ ∞
0
∫ ∞
0
}
M 2 + m4
M 4 + ...
k
dk
(k 2 + m 2 )(k 2 + M 2 )
k
dk
{1
k 2 (k 2 + M 2 − m2
) k 2
for m 2 ≪ M 2
+
m4
k 4 + ... }
IR divergence! For k ~ m, the expansion was not justified.
To be expected, since the result depends nonanalytically
on m/M...
13

≠
Regions
k 0k
{1 − m2
Split integration I = into two dk + regions dk
In the low-energy region (I), k ∼ m ≪ M ,
expand:
I (I) =
I (II) =
∫ Λ
0
∫ ∞
I =
∫ ∞
k
(k 2 + m 2 )(k 2 + M 2 )
0 I = (kdk
2 + m 2 )(k 2 + M 2 )
0
≠ dk
∫ ∞
≠dk
{1
k 2 dk [
(k + M 2 − m2
∫ Λ ) ∫
0 k 2 (k 2 + M 2 ) ∞ k 2
∫ ∞
0
[ ∫ I = Λ
0
dk +
0
∫ ∞
Here,
Λ
Λ acts as a UV cut-off.
Λ
0
[ ∫ Λ 0
dk +
Λ
∫ ∞
(I) + (II)
Λ
dk
]
dk
]
m ≪ Λ ≪ M
k
k 2 (k 2 + M 2 )
Λ
]
+ m4
+ m4
{1 − m2
k 4 + ... }
k 4 + ... }
m ≪ Λ ≪ M
k
(k 2 + m 2 )(kk
2 + M 2 )
(k 2 + m 2 )(k 2 + M 2 )
m ≪ Λ ≪ M
k ∼ m ≪ M
m ≪ k ∼ M
m ≪ Λ ≪ M
k 2
k
+ m4
k 4 +
(k 2 + m 2 )(k 2 + M 2 )
∫
k
Λ
{
}
m ≪ k ∼ M
dk
(k 2 + m 2 )(k 2 + M 2 ) = k
dk
(k 2 + m 2 )M 2 1 − k2
M 2 + k4
M 4 + ...
∫
k
∞
dk
(k 2 + m 2 )(k 2 + M 2 ) = dk
0
k
k 2 (k 2 + M 2 )
{1 − m2
k 2
+ m4
k 4 + ... }
14

∫ Λ
0
∫ ∞
Λ
Regions
k ∼ m ≪ M
k
k ∼ m ≪ M
∫ Λ
∫
k
Λ
{
}
I (I) = dk
0 (k 2 + m 2 )(k 2 + M 2 ) = dk
m0
≪ k(k ∼ 2 + Mm 2 )M 2 1 − k2
M 2 + k4
M 4 + ...
In ∫ ∞ the high-energy region ∫ (II), m ≪ k ∼ M
∫ ,
Λ k
∫ ∞
}
k
Λ
{
I (II) = dk
I (I) expand: = dk
(k 2 + m 2 )(k 2 + M 2 ) = dk
(k 2 + m 2 )M 2 1 k2
M 2 k4
Λ (k 2 + m 2 )(k 2 + M 2 ) = k
dk
{1
Λ k 2 (k 2 + M 2 − m2
) k 2 + m4
k
M 4 + 4 ... ...
0
∫ ∞
0
∫ {1
kk
Λ
dk
(k 2 + m 2 )(k + M 2 ) = k
dk I ∫ ∞
dk
(k 2 + m 2 )M 2 − Λ (k2 + m2 )(k2 + M 2 ) = k
dk
Λ k 2 (k 2 + M 2 ) k 2 + m4
(I) = − Λ2
2M 4 − 1
M 2 ln m Λ
M
I (II) =
m ≪ k ∼ M
I (II) =+ Λ2
2M 4 − 1
Λ acts as a IR cut-off. M 2 ∫ ln Λ M
k
∞
dk Result: (k 2 + m 2 )(k 2 + M 2 ) = k
I = I dk
(I) + I (II) = − 1
M 2 ln m M k 2 (k 2 + M 2 )
0
Λ
{1 − m2
I (I) = − Λ2
2M 4 − 1
M 2 ln m ⎫⎪
Λ , ⎬
I = I (I) + I (II) = − 1
I (II) =+ Λ2
2M 4 − 1
M 2 ln Λ M , ⎪ M 2 ln m M
⎭
k 4 + ... }
✓
{
1 −
15

The dependence on the regulator (or better
separator) Λ has cancelled among the different
regions, and we obtain the correct expansion of the
integral.
While the method works, putting hard cut-offs
between the momentum regions is impractical for
complicated loop integrals.
Fortunately, one can get the same result using
dimensional regularization!
16

Dimensional dk + dk regularization
I =
[ ∫ Λ
0
∫ ∞
Λ
]
Consider our integral in “dimensional regularization”
k
(k 2 + m 2 )(k 2 + M 2 )
I =
∫ ∞
0
dk k −ε
k
(k 2 + m 2 )(k 2 + M 2 )
and calculate the contributions obtained by
expanding the integral in regions (I) and (II), but
without putting an explicit cutoff Λ.
Dimensional regulator ε regulates the IR and UV
divergences which appear in the expanded integrals.
17

m ≪ Λ ≪ M
Λ
dk
∞
dk
Result:
k ∼ m ≪ M
∫ ∞
{
}
∫ ∞ m
I (I) = dk k −ε ≪ k ∼ kM
{
0 (k 2 + m 2 )M 2 1 − k2
M 2 + k4
M 4 + ... }
I (I) = dk k −ε k
0 (k 2 + m 2 )M 2 1 − k2
M 2 + k4
{
∫ M 4 + ...
k
Λ
{
}
}
= m−ɛ { 1 }
2 ɛ + O(ɛ) ···= 1 { }
1
= m−ɛ 1
M 2 ɛ − ln m + O(ɛ) + ...
M 2 ɛ + O(ɛ) + ···= 1 { }
(k 2 + m 2 )(k 2 + M 2 ) = k
dk
1
0 (k 2 + m 2 )M 2 1
M 2 ɛ − − k2
ln M m 2 + + k4
O(ɛ) M 4 + + ... ...
∫
k
∞
}
UV div.
(k 2 + m 2 )(k 2 ∫+ ∫ ∞
}
∞
M 2 ) = k
dk
{1
I (II) = dk k −ε Λ k 2 k
(k 2 + M }
I
{1
0 k 2 (k 2 + M 2 − m2
) k 2 + m4
k 4 + ... (II) = dk k −ε k
{1 2 − m2
) k 2 + m4
k 4 + ...
0 k 2 (k 2 + M 2 − m2
) k 2 + m4
k 4 + ... I (I) = − Λ2
= M −ɛ {
M 2 − 1 }
ɛ + O(ɛ) + ···= 1 {
M 2 − 1 }
= M −ɛ { 2M
ɛ +lnM + ...
M 2 − 1 4 − 1
M}
2 ln m Λ
ɛ + O(ɛ) + ···= 1 {
M 2 − 1 }
ɛ +lnM + ...
I (II) =+ Λ2
2M 4 − 1
M 2 ln Λ M
I (I) = − Λ2 4 − 1 2 ln m ,
I = I (I) + I (II) = − 1
M 2 ln m M + ...
⎫
⎪⎬
18
✓
IR div.

(I) =
0
= m−ɛ
dk k −ε
k
(k 2 + m 2 )M 2 1 − k M 2 + k M 4 + ...
{ } 1
MAt 2 first ɛ sight, + O(ɛ) + ···= 1 { }
1
it is surprising that M 2 the ɛprocedure
− ln m + O(ɛ)
works. It looks like we are double counting, since we
+ ...
integrate both contributions over the full phase space!
∫ ∞
{1
}
I However, note that the two parts scale differently.
(II) = dk k −ε k
The 0 low energy k 2 (k integrals 2 + M 2 − m2
behave ) as km 2 + m4
-ε , the k 4 + ... highenergy
integrals as M -ε .
= M −ɛ {
Full Mintegral 2 − 1 }
ɛ for + O(ɛ) + ···= 1 {
arbitrary ε M 2 − 1 }
ɛ +lnM + ...
I = Γ
(
1 − ε )
2
Γ
(I) + (II)
( ε
2) m −ε − M −ε
M 2 − m 2
19

∫ ∞
Λ
= M I = Γ 1 − Γ 2 2 2
M 2 − 1 1
+ O(ɛ) + ···=
ɛ M 2 − 1 m+lnM ɛ 2 + ...
∫ (
Λ
{
}
I
Keeping
(I) = dk k −ε I = Γk1 − ε ) ( ε m
Γ
2
an explicit cutoff would generate additional
0 (k 2 + m 2 )M 2 1 2) −ε − M −ε
− k2
M 2
M 2 + − k4
m 2
M 4 + ...
Λ -ε [∫
pieces,
∫ ∞Λ
∫
which
∞
] { { }
}
have to cancel among the two parts
=
− dk k −ε k
I
since the full integral
Λ
is Λ
(k
independent. 2 + m 2 )M 2 1 − k2
M
E.g.
2 + k4
M 4 + ... (I) = dk k −ε k
0 (k 2 + m 2 )M 2 1 − k2
M 2 + k4
M 4 + ...
[∫ ∞ ∫ ∞
]
{
}
=
∫
dk − dk k −ε k
Λ
{
}
0
I (I) = dk k −ε Λ (k
k
2 + m 2 )M 2 1 − k2
0 (k 2 + m 2 )M 2 1 − k2 M 2 + k4
M 4 + ...
M 2 + ... ∫
[∫ Λ
{
}
∞ ∫ ∞
]
{
}
I (I) = dk k −ε k
= dk − dk k −ε k
0
Λ (k 2 + m 2 )M 2 1 − k2
0 (k 2 + m 2 )M 2 1 − k2
M 2 + ... M 2 + ... [∫ ∞ ∫ ∞
]
{
}
= dk − dk k −ε k
Cut-off part:
{ 0
Λ } (k ∫ 2 + m 2 )M 2 1 − k2
dk k −ε k
∞
{ M 2 + ... }
(k 2 + m 2 )M 2 1 − k2
M 2 + ... = dk k −ε k
Λ k 2 M 2 1 − m2
k 2 − k2
M 2 + ... ∼ Λ −ε
{
} ∫
dk k −ε k
∞
{
}
(k 2 + m 2 )M 2 1 − k2
M 2 + ... = dk k −ε k
k 2 M 2 1 − m2
k 2 − k2
M 2 + ... ∼ Λ −ε
∫ ∞
Λ
Since the Λ -ε pieces cancel in the end, we might as
well leave them out from the beginning...
Λ
20

Strategy of regions
For a review review: V.A. Smirnov Springer, Tracts Mod.Phys.177:1-262,2002
In general, the expansion is obtained as follows:
• Identify all regions of the integration which lead
to singularities in the limit under consideration.
• Expand the integrand in each region and
integrate over the full phase space.
• Summing the contribution from the different
regions gives the expansion of the original
integral.
21

“Problems in the strategy of regions”
• Need to identify all regions of the integration
which lead to singularities.
• There are examples where additional regions
appear at higher loop order.
• Make sure the expanded integrals are regularized.
• Sometimes dimensional regularization is not
enough. Can introduce additional analytic
regulators or perform subtractions.
• Avoid double counting...
V.A. Smirnov, Phys. Lett. B465:226-234,1999
22

parton carrying longitudinal momentum fraction x inside a fast light hadron
E(λ 2 , 1, λ) toE(λ 2 , λ,...). Similarly, in the region l + ≪ Λ the scaling of the m
asoftpartoninsidetheB meson changes from E(λ, λ, λ) toE(λ 2 , λ,...). In bo
natural to describe these endpoint configurations in terms of soft-collinearfields.
of factorization for the exclusive decay B → K ∗ γ this will be illustrated in [15].
Application to the Sudakov problem
Let us now perform the expansion in a situation,
where 2 Relevance particles have oflarge the soft-collinear energies, but small mode
invariant masses. Simplest example is the integral
It will be instructive to demonstrate the relevance of soft-collinear modes wit
example. Consider the scalar triangle graph shown in Figure 2inthekinematic
the external momenta are l µ ∼ (λ, λ, λ) soft,p µ ∼ (λ 2 , 1, λ) collinear,andq µ
(λ, 1, λ) hard-collinear. Wedefinetheloopintegral
k + l ∫
k I = l k + l
k +
iπ −d/2 µ 4−d dk d 1
k+ pk p
+ p
(k 2 + i0) [(k + l) 2 + i0] [(k + p) 2 + i0]
in d =4− 2ɛ space-time dimensions and analyze it for arbitrary external mom
the above scaling l relations. l Itk
willk
be convenient p p p
to define the invariants
l
L 2 ≡−l 2 − i0 , P 2 ≡−p 2 − i0 , Q 2 ≡−(l − p) 2 − i0 =2l + · p − − i0
which scale like L 2 ,P 2 ∼ λ 2 and Q 2 ∼ λ. (Inphysicalunits,L 2 ,P 2 ∼ Λ 2 and Q
E ≫ Λ.) As long as these momenta
L 2 ∼are P 2 off-shell
Qthe 2
integral is ultra-violet and in
and can be evaluated setting ɛ =0,withtheresult
23
[ ]
We consider the limit .
k
reFigure 2: Scalar 2: Scalar triangle triangle diagram diagram with with an external an external soft momentum soft momentum l andl and a a
collinear momentum momentum p. Theloopmomentumisdenotedbyk.
p. Theloopmomentumisdenotedbyk.
2: Scalar triangle diagram with an external soft momentum l and a
ar momentum p. Theloopmomentumisdenotedbyk.
rom he region the region of hard-collinear of hard-collinear loop momenta loop momenta k µ ∼k E(λ, µ ∼ E(λ, 1, λ 1/2 1, ), λ 1/2 and ), power and power counti co
e region of hard-collinear loop momenta k µ ∼ E(λ, 1, λ 1/2 ), and power co

external momenta are l µ ∼ (λ, λ, λ) soft,p µ ∼ (λ 2 , 1, λ) collinear,andq µ =(l − p)
, λ) hard-collinear. Wedefinetheloopintegral
∫
discuss QCD factorization
I = iπ −d/2 µ 4−d theorems
d d for these processes 1 [11] and power correc
k
(k 2 + i0) [(k + l) 2 + i0] [(k + p) 2 + i0]
urrently much effort devoted to applications of soft-collinear effective theory (SC
, 5] to exclusive B decays [6, 7, 8, 9, 10]. SCET provides a systematic framewo
language of effective field theory. The hadronic states relevant to exclusive de
→ K ∗ γ or B → ππ contain highly energetic, collinear partons inside the final-
=4− 2ɛ space-time dimensions and analyze it for arbitrary external momentaobe
ns, andWe soft partons consider inside the the
above scaling relations. It will be scalar initial
convenient integral B meson.
to define I, Understanding but the the invariants same the intricate inte
oft and collinear degrees of freedom is a challenge that one hopes to address u
momentum regions appear in tensor integrals.
ive L 2 ≡−l theory. 2 − i0 This , interplay P 2 ≡−pis 2 −relevant i0 , Qeven 2 ≡−(l in simpler − p) 2 −processes i0 =2l + · psuch − − i0+... as semilep ,
near q 2 ≈ 0, which are described in terms of heavy-to-light form factors at
To obtain the expansion introduce light-like
h scale like L 2 ,P 2 ∼ λ 2 and Q 2 ∼ λ. (Inphysicalunits,L 2 ,P 2 ∼ Λ 2 and Q 2 ∼ EΛ w
counting Λ.) Asreference long in SCET as theseis vectors momenta based onare in
anthe off-shell expansion
directions the integral parameter
of isp ultra-violet λand ∼ Λ/E,
l and whereE infra-red ≫ fiΛ
scale
can be(typically evaluatedEsetting ∼ m
ɛ =0,withtheresult
n µ =(1,
b in B decays), and Λ ∼ Λ
0, 0,
I = 1 1) [ ¯n µ =(1,
QCD is of order the QCD scal
0, 0, ] −1)
ln Q2 Q2
with the large scale E. To ln
Q 2 makeL this 2
Pscaling +
π2
2 3
explicit + O(λ) one . introduces two ligh
ion in SCET is that different components of particle momentaandfieldsmay
n 2 =¯n 2 =0 n · ¯n =2
µ and ¯n µ satisfying n 2 =¯n 2 =0andn · ¯n =2. Typically,n µ is the direction
fast et us hadron now try
Any
(or to
vector
areproduce jet of this result by evaluating the contributions from different
can
hadrons).
be decomposed
Any 4-vector
as
can then be decomposed as
tum modes. The method of regions [16, 17] can be used to find the momentum config
s giving rise to leading-order contributions to the integral I. Thereisnohardcontribut
p µ =(n · p) ¯nµ nµ
+(¯n · p)
e for k µ ∼ E(1, 1, 1) the integrand 2 can be 2 Taylor-expanded + pµ ⊥ ≡ pµ + + p µ − + p µ
in the
⊥
external , momenta,
scaleless integrals that vanish in dimensional regularization. A short-distance contribu
· n = p ⊥ · ¯n = 0. The light-like vectors 24 p µ ± are defined by this relation.

Introduce expansion parameter
λ 2 ∼ P 2 /Q 2 ∼ L 2 /Q 2
The different components of p μ scale differently.
Since
p 2 = n · p ¯n · p + p 2 ⊥ ∼ λ 2 Q 2
and
, we must have
p µ ≈ 1 2 Qnµ (n · p, ¯n · p, p ⊥ )
p µ ∼ λ 2 , 1 , λ Q
l µ ∼ 1 , λ 2 , λ Q
25

I s = iπ µ d k (k2 + i0) (2k − · l + + l 2 + i0) (2k + · p − +
( 2µ 2 ) ɛ
l + · p −
= − Γ(ɛ) Γ(−ɛ)
2l + · p − L 2 P 2
(
Γ(1 + ɛ) 1
=
Q 2 ɛ 2 + 1 ɛ ln µ2 Q 2
L 2 P 2 + 1 µ 2 Q 2 )
2 ln2 L 2 P 2 + π2
6
Regions
Γ(1
in
+ ɛ)
the Sudakov problem
The following momentum regions contribute to
the expansion of the integral
+ O(ɛ) .
• Hard (h)
(n · k, ¯n · k, k ⊥ )
k µ ∼ ( 1 , 1 , 1 ) Q
• Collinear to p (c1)
k µ ∼ ( λ 2 , 1
, λ ) Q
• Collinear to l (c2)
k µ ∼ ( 1
, λ 2 , λ ) Q
• Soft (s)
k µ ∼ ( λ 2 , λ 2 , λ 2) Q
All other possible scalings (λ a , λ b , λ c ) lead to
scaleless integrals upon expanding.
26

Soft region
Note that the soft region has
Interestingly, loop diagrams involve a lower scale
than what is present on the external lines.
Sometimes scaling
p 2 s ∼ λ 4 Q 2 ∼ L2 P 2
p 2 s ∼ λ 4 Q 2
Q 2
is called ultra-soft.
Implies e.g. that jet-production processes can
involve non-perturbative physics, even when the
masses of the jets are perturbative.
27

Low energy regions
In contrast to expansion problems in Euclidean
space, we encounter several low-energy regions.
Each one is represented by a field in SCET.
Q
QCD
hard
H
M
SCET
collinear
J
M 2 /Q
SET
soft
S
28

k
I h = iπ −d/2 µ 4−d ∫
=
=
Γ(1 + ɛ)
2l + · p −
Γ(1 + ɛ)
Q 2 ( 1
ɛ 2 + 1 ɛ
k
k 2 M 2 1 − m k 2 − k + ... ∼ Λ−ε
M
2
dk k −ε
Λ (k 2 + m 2 )M 2 1 − 2l k + ·
M 2 + p ... − = dk k −ε L 2 P 2
(
Λ
Hard
Γ(1 + ɛ)
contribution
1
sion of the Sudakov = integral Q 2 ɛ 2 + 1 ɛ ln µ2 Q 2
L 2 P 2 + 1 µ 2 Q 2 )
2 ln2 L 2 P 2 + π2
+ O(ɛ) .
6
d d 1
k
(k 2 + i0) (k 2 +2k − · l + + i0) (k 2 +2k + · p − + i0)
k µ ∼ ( 1 , 1 , 1 ) Q
Γ 2 ( )
(−ɛ) µ
2 ɛ
Γ(1 − 2ɛ) 2l + · p −
I c1 = iπ −d/2 µ 4−d ∫
Q 2 + 1 µ 2 )
2 ln2 Q 2 k −µ ∼ π2 ( 1 + O(ɛ) ,
6
ln
µ2
d d k
• Have expanded away (ksmall 2 + i0) [(k momentum + l) 2 + i0] (2k + components
· p − + i0)
Γ(1 + ɛ) Γ 2 (−ɛ)
= −
2l + · p − Γ(1 − 2ɛ)
(
Q 2
IR divergences!
( ) µ
2 ɛ
L 2
k µ ∼ ( λ 2 , 1
• The hard Γ(1 region + ɛ) is given by the on-shell form factor
= − 1 ɛ
integral.
2 − 1 µ2
ln
ɛ L 2 − 1 µ 2
2 ln2 L 2 + π2
+ O(ɛ) .
6
1
)
, λ ) Q
, λ 2 , λ ) Q
k µ ∼ ( λ 2 , λ 2 , λ 2) Q
p µ → (¯n · p) nµ
2 ≡ pµ − , l µ → (n · l) ¯nµ
2 ≡ lµ +
29

Collinear contribution
I c1 = iπ −d/2 µ 4−d ∫
d d k
1
(k 2 + i0) (2k − · l + + i0) [(k + p) 2 + i0]
= −
=
Γ(1 + ɛ)
2l + · p −
Γ 2 (−ɛ)
Γ(1 − 2ɛ)
(
Γ(1 + ɛ)
Q 2 − 1 ɛ 2 − 1 ɛ
( µ
2
P 2 ) ɛ
µ2
ln
P 2 − 1 µ 2
2 ln2 P 2 + π2
6
)
+ O(ɛ) .
• The other collinear contribution Ic2 is obtained
d d 1
k
(k
from exchanging 2 + i0) (2k
l ↔
− · l
p .
+ + l 2 + i0) (2k + · p − + p 2 + i0)
I s = iπ −d/2 µ 4−d ∫
(
Γ(1 + ɛ)
2µ 2 ) ɛ
l + · p −
= − Γ(ɛ) Γ(−ɛ)
2l + · p − (k L+ 2 P 2
l) 2 =2k − · l + + O(λ 2 )
( 1
Q 2 ɛ 2 + 1 ɛ ln µ2 Q 2
L 2 P 2 + 1 µ 2 Q 2 )
2 ln2 L 2 P 2 + π2
+ O(ɛ) .
6
• Have expanded
Γ(1 + ɛ)
•
=
Scales as (P 2 ) -ε
k µ ∼ ( 1 , 1 , 1 ) 30 Q

= − 2l+ · p − Γ(1 − 2ɛ) P 2
=
Soft
(
Γ(1 + ɛ)
contribution
Q 2 − 1 ɛ 2 − 1 ɛ P 2 − 1 µ 2 )
2 ln2 P 2 + π2
6
ln
µ2
+ O(ɛ) .
I s = iπ −d/2 µ 4−d ∫
d d k
1
(k 2 + i0) (2k − · l + + l 2 + i0) (2k + · p − + p 2 + i0)
= −
=
Γ(1 + ɛ)
2l + · p −
Γ(ɛ) Γ(−ɛ)
( 2µ 2 l + · p −
L 2 P 2
) ɛ
(
Γ(1 + ɛ) 1
Q 2 ɛ 2 + 1 ɛ ln µ2 Q 2
L 2 P 2 + 1 µ 2 Q 2
2 ln2 L 2 P 2 + π2
6
)
+ O(ɛ) .
UV divergences!
k µ ∼ ( 1 , 1 , 1 ) Q
• Scales as (Λsoft 2 ) -ε ~ k (P 2 L 2 /Q 2 ) -ε µ ∼ ( λ 2 , 1 , λ ) Q .
• Expand
k µ ∼ ( 1
, λ 2 , λ ) Q
(k + p) 2 =2k + · p − + p 2 + O(λ 3 )
k µ ∼ ( λ 2 , λ 2 , λ 2) Q
µ n µ µ µ ¯n µ µ
31

2 2
Grand total
(k − l) 2 =2k − · l + + O(λ 2 )
I h =
I c1 =
I c2 =
I s =
Γ(1 + ɛ)
Q 2 ( 1
ɛ 2 + 1 ɛ
(
Γ(1 + ɛ)
Q 2 − 1 ɛ 2 − 1 ɛ
(
Γ(1 + ɛ)
Q 2 − 1 ɛ 2 − 1 ɛ
µ2
ln
Q 2 + 1 µ 2 )
2 ln2 Q 2 − π2
6
µ2
ln
P 2 − 1 µ 2 )
2 ln2 P 2 + π2
6
µ2
ln
L 2 − 1 µ 2
2 ln2 L 2 + π2
6
(
Γ(1 + ɛ) 1
Q 2 ɛ 2 + 1 ɛ ln µ2 Q 2
L 2 P 2 + 1 µ 2 Q 2
2 ln2 L 2 P 2 + π2
6
I = I h + I c1 + I c2 + I s = 1 Q 2 (
ln Q2 Q2
ln
2
L
)
)
)
P 2 + π2
3 + O(λ)
Finite (and correct!)
32

I c1 =
I c2 =
(
Γ(1 + ɛ)
Q 2 − 1 ɛ 2 − 1 ɛ
(
Γ(1 + ɛ)
Q 2 − 1 ɛ 2 − 1 ɛ
IR divergences of the hard part are in one-to-one
correspondance to UV divergences of the lowenergy
regions
I s =
µ2
ln
P 2 − 1 µ 2 )
2 ln2 P 2 + π2
6
µ2
ln
L 2 − 1 µ 2 )
2 ln2 L 2 + π2
6
• True in general: IR divergences of on-shell )
amplitudes are equal to UV divergences of
(
Γ(1 + ɛ) 1
Q 2 ɛ 2 + 1 ɛ ln µ2 Q 2
L 2 P 2 + 1 µ 2 Q 2
2 ln2 L 2 P 2 + π2
6
soft+collinear contributions
The cancellation of divergences involves a
I = I h + I c1 + I c2 + I s = 1 Q 2 (
ln Q2 Q2
ln
2
nontrivial interplay of soft and collinear log’s
L
)
P 2 + π2
3 + O(λ)
− 1 ɛ
ln
µ2
P 2 − 1 ɛ
ln
µ2
L 2 + 1 ɛ ln µ2 Q 2
L 2 P 2 = − 1 ɛ
ln
µ2
Q 2
• Leads to constraints on IR structure of onshell
amplitudes. → last lecture
33