beyond feynman diagrams lecture 2 lance dixon academic training lectures cern april 24-26, 2013
TRANSCRIPT
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Beyond Feynman DiagramsLecture 2
Lance Dixon Academic Training Lectures
CERNApril 24-26, 2013
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Lecture 2 April 25, 2013 2Beyond Feynman Diagrams
Modern methods for trees
1. Color organization (briefly)2. Spinor variables3. Simple examples4. Factorization properties5. BCFW (on-shell) recursion relations
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Lecture 2 April 25, 2013 3Beyond Feynman Diagrams
How to organizegauge theory amplitudes
• Avoid 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 constants
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Lecture 2 April 25, 2013 4Beyond Feynman Diagrams
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(Nc): + all non-cyclic permutations
Color
Use definition:
+ normalization:
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Lecture 2 April 25, 2013 5Beyond Feynman Diagrams
Double-line picture (’t Hooft)
• In limit of large number of colors Nc, a gluon is always a combination of a color and a different anti-color.
• Gluon tree amplitudes dressed by lines carrying color indices, 1,2,3,…,Nc.
• 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/ Nc
2
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Lecture 2 April 25, 2013 6Beyond Feynman Diagrams
Trace-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 si,i+1 , …
For n-gluon tree amplitudes, the color decomposition is
momentacolor helicities
color-ordered subamplitude only depends on momenta.Compute separately for each cyclicly inequivalent helicity configuration
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Lecture 2 April 25, 2013 7
Far fewer factorization channels with color ordering
Beyond Feynman Diagrams
1 1
2 3
k
k+1
n
only n
3 k+1
k+2
…
…n-1
16
nk+2
2
4
( nk )
partitions
…
…
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Lecture 2 April 25, 2013 8Beyond Feynman Diagrams
Color sums
Up to 1/Nc2 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:
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Lecture 2 April 25, 2013 9Beyond Feynman Diagrams
Spinor helicity formalism
Scattering amplitudes for massless plane waves of definite momentum: Lorentz 4-vectors ki
m ki2=0
Natural to use Lorentz-invariant products (invariant masses):
Take “square root” of 4-vectors ki 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,
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Lecture 2 April 25, 2013 10
Massless Dirac spinors• Positive and negative energy solutions to the massless
Dirac equation,
are identical up to normalization.• Chirality/helicity eigenstates are
• Explicitly, in the Dirac representation
Beyond Feynman Diagrams
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Lecture 2 April 25, 2013 11Beyond Feynman Diagrams
Spinor products
Use spinor products:
Instead of Lorentz products:
These are complex square roots of Lorentz products (for real ki):
Identity
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Lecture 2 April 25, 2013 12Beyond Feynman Diagrams
~ Simplest Feynman diagram of all
add helicity information, numeric labels
L
R R
L1
2 3
4
g
Fierz identity
helicity suppressedas 1 || 3 or 2 || 4
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Lecture 2 April 25, 2013 13Beyond Feynman Diagrams
Useful to rewrite answerCrossing symmetry more manifestif we switch to all-outgoing helicity labels (flip signs of incoming helicities)
1+
2- 3+
4-
useful identities:
“holomorphic”
“antiholomorphic”
Schouten
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Lecture 2 April 25, 2013 14Beyond Feynman Diagrams
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+
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Lecture 2 April 25, 2013 15Beyond Feynman Diagrams
Unpolarized, helicity-summed cross sections
(the norm in QCD)
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Lecture 2 April 25, 2013 16Beyond Feynman Diagrams
Helicity formalism for massless vectors
obeys (required transversality)
(bonus)
under azimuthal rotation about ki 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)
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Lecture 2 April 25, 2013 17Beyond Feynman Diagrams
Next most famous pair of Feynman diagrams
(to a higher-order QCD person)
g
g
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Lecture 2 April 25, 2013 18Beyond Feynman Diagrams
(cont.)
Choose
to remove 2nd graph
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Lecture 2 April 25, 2013 19Beyond Feynman Diagrams
Properties of
1. Soft gluon behavior
Universal “eikonal” factors for emission of soft gluon sbetween 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:
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Lecture 2 April 25, 2013 20Beyond Feynman Diagrams
Properties of
2. Collinear behavior
(cont.)
z
1-z
Universal collinear factors, or splitting amplitudes depend on parton type and helicity
Square rootof Altarelli-Parisi splitting probablility
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Lecture 2 April 25, 2013 21Beyond Feynman Diagrams
Simplest pure-gluonic amplitudesNote: 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:
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Lecture 2 April 25, 2013 22Beyond Feynman Diagrams
MHV amplitudes with massless quarks
Grisaru, 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:
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Lecture 2 April 25, 2013 23Beyond Feynman Diagrams
Properties of MHV amplitudes1. Soft limit
2. Gluonic collinear limits:
So
and plus parity conjugates
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Lecture 2 April 25, 2013 24Beyond Feynman Diagrams
Spinor Magic
Spinor products precisely capture square-root + phase behavior in collinear limit. Excellent variables for helicity amplitudes
scalars 0
gauge theory
angular momentum mismatch
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Lecture 2 April 25, 2013 25Beyond Feynman Diagrams
Utility of Complex Momenta
real (singular)
• Makes sense of most basic process: all 3 particles massless
complex (nonsingular)
i-
k+
j-
use conjugate kinematics for (++-):
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Lecture 2 April 25, 2013 26Beyond Feynman Diagrams
Tree-level “plasticity” BCFW recursion relations
• BCFW consider a family of on-shell amplitudes An(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:
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Lecture 2 April 25, 2013 27Beyond Feynman Diagrams
Analyticity recursion relations
Cauchy:
meromorphic function,each pole correspondsto one factorization
Where are the poles? Requireon-shell intermediate state,
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Lecture 2 April 25, 2013 28Beyond Feynman Diagrams
Final formula
Ak+1 and An-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/0412308
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Lecture 2 April 25, 2013 29Beyond Feynman Diagrams
To finish proof, show
Propagators:
Britto, Cachazo, Feng, Witten, hep-th/0501052
3-point vertices:
Polarization vectors:
Total:
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Lecture 2 April 25, 2013 30Beyond Feynman Diagrams
• 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
-
+
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Lecture 2 April 25, 2013 31Beyond Feynman Diagrams
• For k = 2, we compute the value of z:
• Kinematics are complex collinear
• The only term in the BCFW formula is:
MHV example (cont.)
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Lecture 2 April 25, 2013 32Beyond Feynman Diagrams
• Using
one confirms
• This proves the Parke-Taylor formula by induction on n.
MHV example (cont.)
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Lecture 2 April 25, 2013 33Beyond Feynman Diagrams
Initial dataParke-Taylor formula
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Lecture 2 April 25, 2013 34Beyond Feynman Diagrams
A 6-gluon example
220 Feynman diagrams for gggggg
Helicity + color + MHV results + symmetries
3 BCF diagrams
related by symmetry
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Lecture 2 April 25, 2013 35Beyond Feynman Diagrams
The one diagram
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Lecture 2 April 25, 2013 36Beyond Feynman Diagrams
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