Soft Collinear EffectiveTheory Scot

Bauer Fleming Pirjol Stewart Zoo o

Outline

Lecture b

Introduction

Soft EffectiveTheory

Factorization of soft j's in QED

lecture 2

Momentum regions in the

Sudekou form factor

Soft Collinear EffectiveTheory

Lecture 3

vector current in SCET

Factorization of Sudoku FF

PDF factorization in Dhs

Soft Collinear Effective Theory

On Monday we obtained the factorization

theorem4 Es

T H Ime Efts4

in meggine QED In QCD and messless

QED often also the collinear region is

relevant Factorization then takes the

schematic form Q2 pj 77 5et e

r HCQ Tfp Tfp 545

with A s P5pE p I see later

An important complication is that

we encounter aw interplay of two

low energy regionsAs in the QED case

we'll construct the EFT by expanding

full theory diagrams The appropriate

tool is the method of regions Beneke

Smirnov 97 for loop phasespaceintegrals

a Expand integrand in relevant

momentum regions

b Integrate over the full momentum

space fddle

c Add up the different contributions

Will illustrate this for Sudakou form

factor and then construct Leff

with field for the different regions

Method of regions for the Sudakou

form factor

A

Q2 Ip e5

P2 P2 L e e

v P2 Cs Q2

uh It o o 1 a PIpo

him 1 o O 1 Effo

ht O IZ e O Tr h 2

Mh h

gm h 9 Iz t n g g t of

check h g EX t h g1

TO

yds V

m a9 t g I t 9 s

g 29 9 t 95

h g ii gZhi 2

4t 94

LnL

Expansion parametera n Pyon Toi

O 1

p2 R2 Q ni p n p t PI

Seahingighn g

ti g 91

pn f X 1 d Q

e n l s d d Q

Loop momentum

herd 1h K f I I 1 Q

coli K k f it 1 d Qto pre

coli re kn f 1 A X Q

folk

soft r s Kh f it d d Q

alewter kn n f N N R Q

fddk n

gAd fork u

o

k Ak

I f da k1

15 Ktp Lte2

Expand in eed region

hard I k t p 5 k t p I toldtete f k te I t 01117

courcy Ik tp j I Ktp 5 weep

late 2K et t old

coli E Ktp 2 Ktp told

k e2

ho exp

soft is Ktp p t 2p letd p

T d

oras

k e 2 e t 2kt k

1 o dB

The expanded loop integrals are

Kt et T k Il l

mQ2

T coli Seele

ai Er EpQ2

I

sit

Effective Lagrangian

4 Ye t Ye t 4s

An A t AE t Ai

Saling of fieldse b

L ol T thx Ar o to 7

ik x

f DI e f g t34UT

in Aa Krh As I As A J n M A M

n Ac in Ae AE r f d I d

Lol ex Is 1 x o i

sa ei

i

TA4

at RG

4g A 3

Collinear quarks

k e k w I t k I f t kiar to d

Ye e t ye Pt http 4

Pt 4 p 4

I

Pf Pf I

P t P I

T 4 KYI23

1,1 2yL

Lo I T Six 370 I 07K I

Sg ei

4 X EfT

K ft t kiar a A

X 4 I d

B 3 a R

tf e 12

Insert into QCD action

Jaa Ss t Se t Se

t Se st

Ss fd4x Ig IDs 4g testT p T T

5 as a as

So fd4X I 3 a tile

in Do t i ri De4g ti J

3 e t ya

fd9x Ic in Dc 23 e t I ii De 4

JaiDls yet ifi 3

Shift

he yo HIaDk e

R Q

Le 3 Elin Dot IDs zi0D3

icE in

Tdet