PACKAGE-X SYMBOL

# Projector

Projector["name"][{p,m}]

gives the projector for the off shell self-energy form factor associated with "".

Projector["name"][{p_{1},m_{1}},{p_{2},m_{2}}]

gives the projector for the on shell scalar/pseudoscalar density form factor associated with "".

Projector["name",μ][{p_{1},m_{1}},{p_{2},m_{2}}]

gives the projector for the on shell vector/axial-vector form factor associated with "" and index μ.

## Details and OptionsDetails and Options

- Projector is recognized as an object in spinor space by Spur.
- Projector must occupy a slot of Spur or DiracMatrix without any accompanying multiplicative or additive symbols.
- A product of Dirac matrices can generically be written as a sum of form factors. Multiply by the appropriate Projector, and apply Spur to extract the appropriate form factor.
- A List of "name"s can be supplied to generate a list of projections.
- Vertex form factor projections generate spurious kinematic singularities proportional to . If LoopIntegrate contains a vertex Projector in the numerator, LoopIntegrate (with the setting CancelAutomatic or False) attempts to remove the singularity. »
- The off shell self-energy matrix is parametrized as

and the corresponding projectors are available: -
"Kinetic" or "A" "Mass" or "B" "AxialKinetic" or "C" "ImaginaryMass" or "E" - In the chiral parametrization, the self-energy matrix is

and the corresponding projectors are available: -
"AL" "BL" "AR" "BR" - The parameter m in Projector["name"][{p,m}] only serves to set the normalization of , , and . Other non-zero choices (e.g. or ) can be made to obtain the desired normalization.
- The on shell vertex form factor projectors are given for space-like momentum transfer ():

, with .

For time-like vertex form factors (), a negative momentum should be supplied to Projector. » - The on shell scalar/pseudoscalar vertex matrix is parametrized as

,

and the corresponding projectors are available: -
"Scalar" or "S" "Pseudoscalar" or "P" - The on-shell vector/axial-vector vertex matrix is parametrized as

,

and the corresponding projectors are available: -
"Dirac" or "F1" "Pauli" or "F2" "F3" "Anapole" or "G1" "EDM" or "G2" "G3" - If () the parametrization becomes:

,

and additional projectors are available: -
"SachsElectric" "SachsMagnetic" - In the left/right-chiral parametrization, the vector vertex matrix is:

,

and the corresponding projectors "AL", "BL", "CL", "AR", "BR","CR" are available.