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
Download Presentation The PPT/PDF document "Beyond Feynman Diagrams" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
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