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Twistors and Perturbative QCD
Yosuke Imamura
The Univ. of Tokyo
String Theory and Quantum Field Theory
Aug.19-23, 2005 at YITP
tree-level Yang-Mills
1. Twistor space (1960 s~70 s)
2. Scattering amplitudes (1970 s~80 s)
3. Twistor amplitudes (2003~04)
4. MHV diagrams (2004~05)
Plan of this talk
A new method of computing scattering amplitudes
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1. Twistor space
References:
R. Penrose
Twistor AlgebraJ.Math.Phys. 8 (1967) 345
R. Penrose
Twistor theory: An approach to the quantization of fields and space-timePhys.Rept. C6 (1972) 241
Textbooks:
R.S.Ward, R.O.Wells,Jr
Twistor Geometry and Field TheoryCambridge University Press
S. A. Huggett, K. P. Tod
An Introduction to Twistor TheoryCambridge University Press
A brief introduction to twistor theory
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Weyl tensor
Gravitino field strength
Maxwel field strength
Weyl fermion
One vector index = a pair of undotted and dotted spinor indices
1.1 Plane waves
Irreducible decomposition
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Equation of motion
These can be solved as follows
(4) Solutions of the equations of motion are
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Let s change the order in solving equations of motion
(2) Instead of taking plane wave for orbital part,
we fix the spin part first as follows.
(3) The equation of motion (transversality equation) gives
(4) This can be solved by
expanded by functions
Another way to solve the equation
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1.2 Twistor space
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Twistor space Momentum spaceFourier tr.
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Comments
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2. Scattering amplitudes
References:
S. J. Parke, T. R. Taylor
Perturbative QCD utilizing extended supersymmetryPLB157(1985)81
M. T. Grisaru, H. N. Pendelton
Some properties of scattering amplitudes in supersymmetric theoriesNPB124(1977)81
S. J. Parke, T. R. Taylor
Amplitude for n-gluon scatteringPRL56(1986)2459
A holomorphic structure in tree levelgluon scattering amplitudes
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External lines are labeled in color-ordering
1
2
3
n-1
n
2.1 Color ordering
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2.2 3 and 4-particle amplitudes
Due to the momentum conservation,
notations
(``holomorphic )(``anti-holomorphic )
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4-particle amplitudes
Amplitudes vanish unless total helicity = 0 (helicity conservation)
(Problem 17.3 (b) in Peskin & Schroeder)
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Such amplitudes are called Maximally Helicity Violating amplitudes.
2.3 Holomorphy
If all or all but one helicities are the same, the amplitude vanishes.
(The three particle amplitudes are exceptions.)
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3-particle
4-particle
5
-particle
(holomorphic)
(anti-holomorphic)
Helicity violation
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Explicit form ofMHV amplitudes is known.
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3. Twistor amplitudes
Reference:
E. Witten
Perturbative gauge theory as a string theory in twistor spaceCommun.Math.Phys. 252 (2004) 189-258, hep-th/0312171
Duality between supersymmetric Yang-Millsand string theory in the twistor space
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3.1 Fourier transformation
With this in mind, let us carry out the Fourier tr.
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A local theory in the twistor space
(holomorphic Chern-Simons)
Full Yang-Mills ??? in the twistor space
duality
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Fourier tr. of the MHV amplitude
This is a non-local interaction in the twistor space.
Witten proposed an interpretation of this amplitude.
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Corr. func. on D5D1-braneModuli
integral
3.2 Wittens proposal (Full YM = B-model in the twistor space)
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Comments
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4. MHV diagrams
References:
F. Cachazo, P. Svrcek, E. Witten
MHV Vertices And Tree Amplitudes In Gauge TheoryJHEP 0409 (2004) 006, hep-th/0403047
R. Britto, F. Cachazo, B. Feng
New Recursion Relations for Tree Amplitudes of GluonsNucl.Phys. B715 (2005) 499-522, hep-th/0412308
R. Britto, F. Cachazo, B. Feng, E. Witten
Direct Proof Of Tree-Level Recursion Relation In Yang-Mills TheoryPhys.Rev.Lett. 94 (2005) 181602, hep-th/0501052
A new method for computation ofscattering amplitudes
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This gives new ``Feynman rules .
4.1 MHV diagrams
MHV
MHV
MHV
In order to use the MHV amplitudes as vertices, we have to define a rule togive spinor variables for arbitrary (off-shell) momenta.
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Example
1
23
4 1
23
4
MHV MHV
MHV
MHV
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# of negative helicities
# of external lines# of Feynman diagrams
# of MHV diagrams
MHV diagrams drastically simplify the computation of
scattering amplitudes.
The # of MHV diagrams contributing an amplitude is much smaller
than the # of Feynman diagrams.
MHV diagrams are efficient especially for small k.
Even for k~n/2, the number of MHV diag, is much smaller than that of
Feynman diag.
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4.2 pole structureAn amplitude becomes singularwhen apropagatorin a Feynman
diagram becomes on-shell.
The residue for the pole is the product of two amplitudes connected by
the on-shell propagator.
Factorization formula
Feynman diagram
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There are two kinds of singularities in MHV diagrams.
This structure is reproduced by the MHV diagrams correctly.
Singularities in MHV propagators Singularities in MHV vertices
These singularities correctly reproduce the physical singularities inFeynman diagrams.
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4.3 BCF recursion relation
(Feynman diag.)
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All residues can be determined by the factorization formula
We can also show that
(Feynman diag.)
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BCF recursive relation:
If we have 3-pt amplitudes, we can construct an arbitrary tree amplitude
using this relation recursively.
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4.4 Proof of the MHV formula
BCF recursion relation shows that
the on-shell amplitudes are uniquely determined by the pole structure.
In order to prove the MHV formula, we have only to show that
the MHV formula gives the correct pole structure.
We have already shown that MHV diagrams correctly reproduce the
pole structure of tree-level amplitudes.
MHV formula is proven!
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5. Conclusions
Although the twistor string theory itself has not been established,
it inspired the new method to compute scattering amplitudes.
At the tree level, it was proven that the MHV diagrams give
correct scattering amplitudes.
The proof is based on the BCF recursion relation, which does not
depend on string theory.
MHV diagrams drastically simplify the computation of
scattering amplitudes.
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Generalization
The BCF recursion relation is generalized for graviton and
charged scalar fields.
J. Bedford, A. Brandhuber, B. Spence, G. Travaglini
A recursion relation for gravity amplitudesNucl.Phys. B721 (2005) 98-110, hep-th/0502146
A. Brandhuber, B. Spence, G. Travaglini
One-Loop Gauge Theory Amplitudes in N=4 Super Yang-Mills
from MHV VerticesNucl.Phys. B706 (2005) 150-180, hep-th/0407214
MHV diagrams correctly give one-loop MHV amplitudes.
J.-B. Wu, C.-J. Zhu
MHV Vertices and Scattering Amplitudes in Gauge TheoryJHEP 0407 (2004) 032, hep-th/0406085