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Beyond Feynman Diagrams Beyond Feynman Diagrams

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Beyond Feynman Diagrams - PPT Presentation

Lecture 2 Lance Dixon Academic Training Lectures CERN April 2426 2013 Beyond Feynman Diagrams Lecture 2 April 25 2013 2 Modern methods for trees Color organization briefly Spinor ID: 341181

diagrams feynman 2013 lecture feynman diagrams lecture 2013 april color helicity amplitudes mhv gluon complex massless spinor momenta formula

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Slide1

Beyond Feynman Diagrams

Lecture 2

Lance Dixon Academic Training LecturesCERNApril 24-26, 2013Slide2

Beyond Feynman Diagrams

Lecture 2 April 25, 20132

Modern methods for treesColor organization (briefly)

Spinor variablesSimple examples

Factorization properties

BCFW

(

on-shell)

recursion relationsSlide3

Beyond Feynman Diagrams

Lecture 2 April 25, 20133

How to organizegauge theory amplitudesAvoid tangled algebra of color and Lorentz indices generated by Feynman rules

Take advantage of physical properties of amplitudes

Basic tools:

dual (trace-based) color decompositions

spinor helicity formalism

structure constantsSlide4

Beyond Feynman Diagrams

Lecture 2 April 25, 20134

Standard color factor for a QCD graph has lots of structure constants contracted in various orders; for example:

Write every

n

-gluon

tree graph color factor as a sum

of traces of matrices

T

a

in

the

fundamental (defining)

representation of

SU(

N

c

):

+ all non-cyclic permutationsColor

Use definition:

+ normalization:

Slide5

Beyond Feynman Diagrams

Lecture 2 April 25, 20135

Double-line picture (’t Hooft)

In limit of large number of colors

N

c

, a gluon is

always

a combination of a color and a

different

anti-color.

G

luon tree amplitudes dressed by lines carrying color indices,

1,2,3,…,N

c

.

Leads to

color ordering

of the external gluons.

Cross section, summed over colors of all external gluons=

S |color-ordered amplitudes|2

Can still use this picture at Nc=3. Color-ordered amplitudes are

still the building blocks. Corrections to the color-summed cross section, can be handled exactly, but are suppressed by 1

/ Nc2Slide6

Beyond Feynman Diagrams

Lecture 2 April 25, 2013

6Trace-based (dual) color decomposition

Similar decompositions for amplitudes with external quarks.

Because comes from

planar diagrams

with cyclic ordering of external legs fixed to

1,2,…,

n

,

it only has singularities in

cyclicly

-adjacent

channels

s

i,i+

1

, …

For

n

-gluon

tree amplitudes

, the

color decomposition

is

momenta

color

helicities

color-ordered

subamplitude

only depends on

momenta

.

Compute separately for each

cyclicly

inequivalent

helicity

configurationSlide7

Far fewer factorization

channels with color orderingBeyond Feynman Diagrams

Lecture 2 April 25, 20137

1

1

2

3

k

k

+

1

n

o

nly

n

3

k

+

1

k+

2

n

-1

16

n

k+

2

2

4

(

n

k

)

partitions

…Slide8

Beyond Feynman Diagrams

Lecture 2 April 25, 20138

Color sums

Up to 1/N

c

2

suppressed effects, squared

subamplitudes

have

definite color flow

important

for development of

parton

shower

Parton model says

to sum/average over final/initial colors

(as well as

helicities):

Insert:

and

do color sums to get:Slide9

Beyond Feynman Diagrams

Lecture 2 April 25, 20139

Spinor helicity formalismScattering amplitudes for massless plane waves of definite momentum: Lorentz 4-vectors

ki

m

k

i

2

=0

Natural to use Lorentz-invariant products

(invariant masses):

Take “square root”

of

4-vectors

k

i

m

(spin 1)

use Dirac (Weyl) spinors

ua(ki

) (spin ½)

But

for elementary particles with

spin

(

e.g.

all

except Higgs

!)

there is a better way:

q

,g

,

g

, all have 2

helicity

states,Slide10

Massless Dirac spinors

Positive and negative energy solutions to the massless Dirac equation,

are identical up to normalization.Chirality/helicity eigenstates areExplicitly, in the Dirac representationBeyond Feynman Diagrams

Lecture 2 April 25, 2013

10Slide11

Beyond Feynman Diagrams

Lecture 2 April 25, 201311

Spinor productsUse spinor products:

Instead of Lorentz products:

These are

complex square roots

of Lorentz

products (for

real

k

i

):

Identity Slide12

Beyond Feynman Diagrams

Lecture 2 April 25, 201312

~ Simplest Feynman diagram of all

add helicity information,

numeric labels

L

R

R

L

1

2

3

4

g

Fierz identity

helicity suppressed

as 1 || 3 or 2 || 4 Slide13

Beyond Feynman Diagrams

Lecture 2 April 25, 2013

13

Useful to rewrite answer

Crossing symmetry more manifest

if we switch to

all-outgoing

helicity

labels

(flip signs of incoming

helicities

)

1

+

2

-

3

+

4

-

useful

identities:

holomorphic

antiholomorphic

SchoutenSlide14

Beyond Feynman Diagrams

Lecture 2 April 25, 201314

Symmetries for all other helicity config’s

1+

2

-

3

+

4

-

1

-

2

+

3

+

4

-

C

1

-

2

+

3

-

4

+

P

1

+

2

-

3

-

4

+Slide15

Beyond Feynman Diagrams

Lecture 2 April 25, 201315

Unpolarized, helicity-summed cross sections

(the norm in QC

D

)Slide16

Beyond Feynman Diagrams

Lecture 2 April 25, 201316

Helicity formalism for massless vectorsobeys

(required transversality)

(bonus)

under

azimuthal

rotation about

k

i

axis,

helicity

+1/2

helicity -1/2

so

as required for helicity +1

Berends

,

Kleiss

, De

Causmaecker

,

Gastmans

, Wu (1981); De

Causmaecker

,

Gastmans

,

Troost

, Wu (1982);

Xu

, Zhang, Chang (1984);

Kleiss

, Stirling (1985); Gunion, Kunszt (1985)Slide17

Beyond Feynman Diagrams

Lecture 2 April 25, 201317

Next most famous pair of Feynman diagrams(to a higher-order QCD

person)

g

gSlide18

Beyond Feynman Diagrams

Lecture 2 April 25, 201318

(cont.)

Choose

to remove 2

nd

graph Slide19

Beyond Feynman Diagrams

Lecture 2 April 25, 201319

Properties of1. Soft gluon behavior

Universal “eikonal” factors

for emission of soft gluon

s

between two hard partons

a

and

b

Soft emission is from the

classical

chromoelectric

current

:

independent

of

parton

type

(

q

vs.

g) and helicity

– only depends on

momenta

of

a,b

, and color

charge:Slide20

Beyond Feynman Diagrams

Lecture 2 April 25, 201320

Properties of2. Collinear behavior

(cont.)

z

1-z

Universal collinear factors, or

splitting amplitudes

depend on

parton

type

and

helicity

Square root

o

f Altarelli-

Parisi

splitting

probablilitySlide21

Beyond Feynman Diagrams

Lecture 2 April 25, 201321

Simplest pure-gluonic amplitudes

Note

: helicity label assumes particle is outgoing; reverse if it’s incoming

Maximally helicity-violating (MHV) amplitudes:

Parke-Taylor formula (1986)

=

1

2

(i-1)

Strikingly, many vanish:Slide22

Beyond Feynman Diagrams

Lecture 2 April 25, 201322

MHV amplitudes with massless quarksGrisaru, Pendleton, van Nieuwenhuizen (1977);

Parke, Taylor (1985); Kunszt (1986); Nair (1988)

the MHV amplitudes:

Related to pure-gluon MHV amplitudes by a secret

supersymmetry

:

after stripping off color factors,

massless quarks

~

gluinos

Helicity conservation on fermion line

more vanishing ones:Slide23

Beyond Feynman Diagrams

Lecture 2 April 25, 201323

Properties of MHV amplitudes1. Soft limit

2.

G

luonic

collinear limits:

So

and

plus parity conjugatesSlide24

Beyond Feynman Diagrams

Lecture 2 April 25, 201324

Spinor Magic Spinor products precisely capture

square-root + phase behavior in collinear limit.

Excellent variables for

helicity amplitudes

scalars

gauge theory

angular momentum mismatchSlide25

Beyond Feynman Diagrams

Lecture 2 April 25, 201325

Utility of Complex Momenta

real (singular)

Makes

sense of most basic

process:

all 3 particles

massless

complex (nonsingular)

i

-

k

+

j

-

u

se conjugate kinematics for (++-):Slide26

Beyond Feynman Diagrams

Lecture 2 April 25, 201326

Tree-level “plasticity” BCFW recursion relationsBCFW consider a family of

on-shell amplitudes A

n

(

z

)

depending on a

complex

parameter

z

which shifts the

momenta

to

complex

values

For example, the

[

n,1

› shift:On-shell

condition: similarly,

Momentum conservation:Slide27

Beyond Feynman Diagrams

Lecture 2 April 25, 201327

Analyticity  recursion relations

Cauchy:

meromorphic

function,

each pole corresponds

to one factorization

Where are the poles? Require

on-shell intermediate state, Slide28

Beyond Feynman Diagrams

Lecture 2 April 25, 201328

Final formula

A

k

+1

and

A

n-k

+1

are

on-shell

color-ordered

tree

amplitudes with fewer legs,

evaluated with

2

momenta shifted by a

complex amount

Britto, Cachazo, Feng, hep-th/0412308Slide29

Beyond Feynman Diagrams

Lecture 2 April 25, 201329

To finish proof, show

Propagators:

Britto, Cachazo, Feng, Witten, hep-th/0501052

3-point vertices:

Polarization vectors:

Total:Slide30

Beyond Feynman Diagrams

Lecture 2 April 25, 201330

Apply the [n,1›

BCFW formula to the MHV amplitude

The generic diagram

vanishes

because 2 + 2 = 4 > 3

So one of the two tree

amplitudes is always

zero

The one exception is

k

= 2,

which is different because

MHV example

-

+Slide31

Beyond Feynman Diagrams

Lecture 2 April 25, 201331

For k = 2, we compute the value of z:

Kinematics are complex

collinear

The only term in the BCFW formula is:

MHV example (cont.)Slide32

Beyond Feynman Diagrams

Lecture 2 April 25, 201332

Using one confirmsThis proves the Parke-Taylor formula by induction on

n.

MHV

example

(cont.)Slide33

Beyond Feynman Diagrams

Lecture 2 April 25, 201333

Initial data

Parke-Taylor formulaSlide34

Beyond Feynman Diagrams

Lecture 2 April 25, 201334

A 6-gluon example

220 Feynman diagrams for gggggg

Helicity + color + MHV results + symmetries

3 BCF diagrams

related by symmetry Slide35

Beyond Feynman Diagrams

Lecture 2 April 25, 201335

The one diagram

Slide36

Beyond Feynman Diagrams

Lecture 2 April 25, 201336

Simpler

than form found in 1980sdespite (because of?) spurious singularities

Mangano, Parke, Xu (1988)

Simple final form

Relative simplicity

much

more striking for

n>6