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Projector

Projector["name"][{p,m}]
gives the projector for the off shell self-energy form factor associated with "name".

Projector["name"][{p1,m1},{p2,m2}]
gives the projector for the on shell scalar/pseudoscalar density form factor associated with "name".

Projector["name",μ][{p1,m1},{p2,m2}]
gives the projector for the on shell vector/axial-vector form factor associated with "name" 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.

ExamplesExamplesopen allclose all

Basic Examples  (2)Basic Examples  (2)

Let :

In[1]:=
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Out[1]=

Apply Spur and insert Projector to extract the form factors:

In[2]:=
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Out[2]=
Out[2]=
Out[2]=
Out[2]=

Let :

In[1]:=
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Out[1]=

Apply Spur and insert Projector to extract the elastic form factors (be sure to include the on-shell conditions for and ):

In[2]:=
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Out[2]=
Out[2]=
In[3]:=
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Out[3]=