Academic Articles & Supplements

Spinors and Conformal Correlators

Marc Gillioz

Author

Marc Gillioz

Title

Spinors and Conformal Correlators

Description

Computation of conformal 3-point functions using the spinor-helicity formalism

Spinors and Conformal Correlators

Marc Gillioz

Current institution: SISSA
Date: December 2021

In this notebook I compute and solve conformal Ward identitiesfor 2- and 3-point correlation functions in the momentum-spacerepresentation in 3 space-time dimensions using the spinor formalism.The first part of the notebook defines rules for spinors, vectors, and tensors.We define in particular a pair of complex-conjugated spinors (

λ,

) thatserves as a basis in spinor space.The second part focuses on the 2-point function of a primary operatorwith

spinor indices (i.e.

is a Weyl fermion):



···

(-p)

···

(p)

We define the momentum

in terms of the two spinors

λ,

,compute the Ward identities and give a solutionexpressed in terms of polarization tensors of definite helicity.The third part focuses on the 3-point function of two scalar primary operatorsand a generic operator with

spinor indices



···

)[

(

)

(

)]

We define the momentum

and

in terms of the two spinors

λ,

and of two scalar quantities

and

, compute the Ward identities andgive a solution in terms of polarization tensors.

Rules for spinors

Notation for spinors


Scalar product


Contraction of indices


Pair of complex-conjugate spinors


Vectors


Orthonormalized basis of vectors


Polarization spinors and tensors


Comparison with traceless symmetric polarizations


Spinor derivative


Spin matrix


Differential operators


2-point functions

Differential operators


Conformal Ward identities


Scalar 2-point function


Spinning 2-point function


Vanishing in case of distinct scaling dimensions or spin


3-point functions

Appell
F
4
differential operators and hypergeometric functions


Differential operators


Conformal Ward identities


Scalar case


Spinning case


General solution


Examples


Solution at
x=0
or
y=0


Cite this as: Marc Gillioz, "Spinors and Conformal Correlators" from the Notebook Archive (2022), https://notebookarchive.org/2022-01-8sgqac5

Spinors and Conformal Correlators

Rules for spinors

Notation for spinors

Scalar product

Contraction of indices

Pair of complex-conjugate spinors

Vectors

Orthonormalized basis of vectors

Polarization spinors and tensors

Comparison with traceless symmetric polarizations

Spinor derivative

Spin matrix

Differential operators

2-point functions

Differential operators

Conformal Ward identities

Scalar 2-point function

Spinning 2-point function

Vanishing in case of distinct scaling dimensions or spin

3-point functions

Appell F4 differential operators and hypergeometric functions

Differential operators

Conformal Ward identities

Scalar case

Spinning case

General solution

Examples

Solution at x=0 or y=0

Notation for spinors


Scalar product


Contraction of indices


Pair of complex-conjugate spinors


Vectors


Orthonormalized basis of vectors


Polarization spinors and tensors


Comparison with traceless symmetric polarizations


Spinor derivative


Spin matrix


Differential operators


Differential operators


Conformal Ward identities


Scalar 2-point function


Spinning 2-point function


Vanishing in case of distinct scaling dimensions or spin


Appell
F
4
differential operators and hypergeometric functions


Differential operators


Conformal Ward identities


Scalar case


Spinning case


General solution


Examples


Solution at
x=0
or
y=0
