One-Loop Numerator Contractions

This module records the explicit one-loop numerator contractions for the gauge, ghost, and fermion sectors, using the tensor-reduction layer to attach the expected color-factor coefficients to scalar master integrals.

The current implementation captures the standard one-loop beta-function weights:

• gluon loop: $(5/3) C_A$
• ghost loop: $2 C_A$
• fermion loop: $-(4/3) T_F n_f$

These are then converted into PrimitivePoleAssumptions for the existing one-loop beta assembly theorem.

noncomputable def EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.gluonLoopReduction (cf : ColorFactors) : PerturbationTheory.DimensionalRegularization.TensorReduction.RankTwoReductionResult

Tensor-reduced gluon-loop numerator contraction.

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def EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.ghostLoopReduction (cf : ColorFactors) : PerturbationTheory.DimensionalRegularization.TensorReduction.RankTwoReductionResult

Tensor-reduced ghost-loop numerator contraction.

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noncomputable def EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.quarkLoopReduction (cf : ColorFactors) : PerturbationTheory.DimensionalRegularization.TensorReduction.RankTwoReductionResult

Tensor-reduced fermion-loop numerator contraction.

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noncomputable def EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.primitivePoleWeightsOfContractions (cf : ColorFactors) : OneLoopBetaFromScalars.PrimitivePoleWeights

Primitive pole weights read off from the standard one-loop numerator contractions.

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noncomputable def EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.primitivePolesOfContractions (cf : ColorFactors) : OneLoopBeta.PrimitivePoles

Primitive poles obtained from the standard one-loop numerator contractions.

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theorem EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.primitivePolesOfContractions_gluon (cf : ColorFactors) : (primitivePolesOfContractions cf).gluon = 5 / 3 * cf.cA

The contraction-derived gluon pole equals the expected coefficient.

theorem EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.primitivePolesOfContractions_ghost (cf : ColorFactors) : (primitivePolesOfContractions cf).ghost = 2 * cf.cA

The contraction-derived ghost pole equals the expected coefficient.

theorem EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.primitivePolesOfContractions_quark (cf : ColorFactors) : (primitivePolesOfContractions cf).quark = -(4 / 3) * cf.tF * cf.nF

The contraction-derived quark pole equals the expected coefficient.

theorem EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.primitivePoleAssumptions_of_standardContractions (cf : ColorFactors) : OneLoopBeta.PrimitivePoleAssumptions cf (primitivePolesOfContractions cf)

Standard one-loop numerator contractions discharge the primitive pole contracts.

theorem EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.beta0_from_standardContractions (cf : ColorFactors) (ms : Renormalization.MSLikeRenormalizationData) (hLink : OneLoopBeta.OneLoopRenormalizationLink ms (primitivePolesOfContractions cf)) : beta0 cf = Renormalization.beta0FromCouplingPole ms.couplingPole

Beta-function theorem from the standard one-loop numerator contractions.

theorem EpsilonEridani.QFT.QCD.OneLoopNumeratorContractions.beta0_suN_from_standardContractions (nC nF : ℝ) (ms : Renormalization.MSLikeRenormalizationData) (hLink : OneLoopBeta.OneLoopRenormalizationLink ms (primitivePolesOfContractions (suNColorFactors nC nF))) : beta0 (suNColorFactors nC nF) = Renormalization.beta0FromCouplingPole ms.couplingPole

SU(N) specialization of the numerator-contraction theorem.