[an error occurred while processing this directive]
Package-X is hosted by Hepforge, IPPP Durham

LoopIntegrate

LoopIntegrate[num, k, {q0, m0}, {q1, m1}, ...]
expresses the one-loop tensor integral over integration variable k with numerator num and denominator factors in terms of Passarino-Veltman coefficient functions.

LoopIntegrate[num,k,{q0,m0,w0},{q0,m1,w1},...]
expresses the one-loop tensor integral in terms of Passarino-Veltman coefficients with weights w0, w1, .

Details and OptionsDetails and Options

  • LoopIntegrate generates a Lorentz covariant representation of the integral in terms of coefficient functions PVA, PVB, PVC, PVD, and PVX, which is valid for any value of the kinematic arguments and to all orders in ϵ.
  • Apply LoopRefine to convert PVA, PVB, PVC, and PVD into analytic expressions.
  • Denominator specifiers typically have the form {qn,mn} and represent a factor in the integrand, with qn and mn corresponding to internal momentum and mass variables.
  • Denominator specifiers may optionally take an additional argument {qn,mn,wn}, with the integer wn indicating the weight of that denominator factor . The default value for wn is 1. »
  • Denominators independent of integration variable k are factored out of the integral, and identical denominator factors are merged to form weighted denominators.
  • The number of unique denominator specifiers given to LoopIntegrate determines the topology of the corresponding one-loop graph:
  • LoopIntegrate[num,k,{k+p0, m0},{k+p1, m1},]
        
    LoopIntegrate[num,k,{k+p0, m0,w0},{k+p1, m1, w1},]
        
  • Although the canonical integration measure that arises in typical one-loop calculations is , LoopIntegrate gives results with the combination C_epsilon=(i e^(-gamma_(E)epsilon))/((4pi)^(d/2)) factored out and removed. To interpret results in terms of the canonical integration measure near 4 spacetime dimensions, (1) multiply the output of LoopRefine by , and (2) interpret poles near as follows:
  • LoopRefine output:implied expression:
    1/epsilon-gamma_(E)+ln(4pi)
    1/(epsilon^2)+1/epsilon(-gamma_(E)+ln(4pi))+(gamma_(E)^2)/2-gamma_(E)ln(4pi)+1/2ln^2(4pi)
  • The option Apart controls whether to carry out a partial fraction expansion of the integrand before making the Lorentz covariant decomposition. Possible values are:
  • False (default)Does not make a partial fraction expansion
    TrueMakes a partial fraction expansion
  • The option Cancel controls whether to cancel common factors in the numerator and denominator before making the Lorentz covariant decomposition. Possible values are:
  • Automatic (default)Does not cancel common factors if the numerator contains kinematic singularities, and cancels otherwise
    TrueCancels common factors
    FalseDoes not cancel common factors
  • The option Dimensionsn sets the dimension of integration measure to . Default setting is 4, and may be set to any even integer.
  • The option DiracAlgebra specifies whether to apply the Dirac algebra to simplify the numerator. Possible values are:
  • TrueApplies the Dirac algebra.
    FalseDoes not apply the Dirac algebra. Uses only linearity to decompose the integral.
  • With the setting DiracAlgebraTrue, LoopIntegrate observes the default option settings of FermionLineExpand, and can be changed with SetOptions. »
  • The option Organization specifies how the result should be organized. Possible values are:
  • Automatic (default)Chooses organization scheme automatically
    LTensorOrganizes result by Lorentz tensor structures
    FunctionOrganizes result by Passarino-Veltman functions
    NoneLeaves result unorganized (fastest)

ExamplesExamplesopen allclose all

Basic Examples  (4)Basic Examples  (4)

Compute a scalar one-loop integral with one propagator (tadpole diagram):
    

In[1]:=
Click for copyable input
Out[1]=

Apply LoopRefine to obtain an explicit expression:

In[2]:=
Click for copyable input
Out[2]=

To interpret the result, (1) restore an overall factor of , and (2) read as :
Therefore,

Massless tadpoles vanish in dimensional regularization:

In[1]:=
Click for copyable input
Out[1]=
Out[1]=

Compute a tensor bubble diagram with identical internal masses:
    

In[1]:=
Click for copyable input
Out[1]=
Out[1]=

Interpretation:

Apply HoldForm to check your input:

In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=