Namespace containing the main simplification method for MARTY and all its dependencies. More...

Data Structures
struct	FermionEOMData
	Helper structure for the application of Dirac equation (equations of motions for spin 1/2) More...

Enumerations
enum	Mode { FeynmanRule, Amplitude, SquaredAmplitude, WilsonCoefficient }
	Mode for the main simplification routine specifying what calculation is done to adapt the simplification procedure.

Functions
void	simplify (csl::Expr &expr, std::vector< mty::QuantumField > const &insertions, std::vector< csl::Tensor > const &momenta, mty::FeynOptions const &options, Mode mode)
	Main simplification routine of MARTY, used for amplitudes, squared amplitude and Wilson coefficients. More...

color::ColorSpace const *	inColorSpace (csl::Index const &index)
	Returns the colorspace associated to an index (nullptr if the index does not live in any ColorSpace). More...

color::ColorSpace const *	isColorStructure (csl::IndexStructure const &structure)
	Returns the adjoint colorspace for a generator, nullptr for another structure.o. More...

color::ColorSpace const *	isColorStructure (csl::Expr const &expr)
	Returns a colorspace if the expression is a color structure. More...

csl::IndexStructure	colorStructure (csl::Expr const &node, color::ColorSpace const *color)
	Returns all color indices found in an expression lying in a particular color space. More...

bool	factorIndicial (csl::Expr &res)
	Factors indicial expressions in a bigger expression. More...

csl::Expr	colorAbbreviation (std::vector< csl::Expr > const &exprs, std::vector< color::ColorSpace const *> const &spaces)
	Abbreviates color structure in expressions under the generic name "Color". More...

bool	expandInProd (csl::Expr &prod, std::function< csl::IndexStructure(csl::Expr const &)> const &structureGetter)
	Expands indicial expressions in a way avoiding unnecessary terms, in particular not expanding expressions that have no common index. More...

bool	expandColorIndices (csl::Expr &res)
	Expands expressions containing color indices to allow all algebra simplifications. More...

bool	simplifyColorWeights (csl::Expr &expr)
	Calculates color traces in an expression. More...

bool	findColorAbbreviation (csl::Expr &expr)
	Abbreviates color structures in an expression under the generic name "Color". More...

bool	expandMinkoStructures (csl::Expr &expr)
	Expands Minkowski structures to contract all possible indices. More...

bool	expandMinkoMetric (csl::Expr &expr)
	Expands Minkowski metrics to contract all possible indices. More...

bool	expandMinkoEpsilon (csl::Expr &expr)
	Expands Minkowski epsilon symbols to contract all possible indices. More...

bool	simplifyEpsilonInProd (csl::Expr &prod, csl::IndexStructure &indices)
	Simplifies the contraction of an epsilon tensor with a symmetric structure index. More...

bool	simplifyEpsilon (csl::Expr &expr)
	Simplifies epsilon contractions with symmetric indices in an expression. More...

bool	expandGammaMatrices (csl::Expr &expr)
	Expands gamma matrices in an expression to be able to simplify fermion chains (and traces). More...

bool	simplifyFermionChains (csl::Expr &expr)
	Simplifies fermion chains using the mty::FermionChain utility. More...

void	expandForFermionOrdering (csl::Expr &expr)
	Forces the ordering of external fermions by making sure that all Dirac structures are expanded. More...

void	expandMomentaExperimental (csl::Expr &res, std::vector< csl::Tensor > const &momenta)
	Expands expressions containing momenta to contract all indices, typically replacing \( p_i^\mu\cdot p_{j\mu} \mapsto s_{ij} \). More...

void	reduceTensorIntegrals (csl::Expr &expr)
	Replaces the one-loop momentum integrals by their reduced form depending on scalar integrals. More...

std::pair< csl::Expr, csl::Expr >	getMomentumReplacement (std::vector< mty::QuantumField > const &insertions, std::vector< csl::Tensor > const &momenta, size_t posReplaced)
	Creates the relevant replacement (using momentum conservation) to replace one particular momentum. More...

void	replaceMomentum (csl::Expr &init, std::vector< mty::QuantumField > const &insertions, std::vector< csl::Tensor > const &momenta, size_t posReplaced)
	Applies the momentum conservation by replacing one of the external momenta by the combination of the others. More...

void	simplifyImpulsions (csl::Expr &init, std::vector< mty::QuantumField > const &insertions, std::vector< csl::Tensor > const &momenta)
	Applies the momentum conservation by replacing one of the external momenta by the combination of the others. More...

void	applyEOM (csl::Expr &ampl, std::vector< FermionEOMData > &onShellFermions)
	Applies the equations of motions in an expression. More...

void	applyEOM (csl::Expr &ampl, std::vector< mty::QuantumField > const &insertions, std::vector< csl::Tensor > const &momenta)
	Applies the equations of motion in an expression. More...

void	addLocalTerms (csl::Expr &res)
	Adds the local terms in an expression. More...

bool	maybeSimplified (csl::Expr const &expr)
	Helper function that tells if an expression may be simplified in amplitudes. More...

void	findExternalAbbreviation (csl::Expr &expr)
	Abbreviates external legs in the expression using the generic name "EXT". More...

void	abbreviateIntegral (csl::Expr &res)
	Abbreviate scalar integrals (or combinations of them) into abbreviations with the generic name "INT". More...

void	abbreviateAll (csl::Expr &res)
	Abbreviates all the sub-expressions that are relevant. More...

bool	findAbbreviations (csl::Expr &res)
	Abbreviates all the sub-expressions that are relevant considering constant factors only. More...

void	applyDerivativesInStructure (csl::Expr &expr)
	Searches in the expression derived fields and applies the derivatives in the fields themselves. More...

void	suppressDiracDelta (csl::Expr &expr, csl::Expr const &PSum)
	Removes a dirac delta sub-expression. More...

bool	mayBeSimplified (csl::Expr const &expr)

static bool	isMomentum (csl::Expr &tensor)

bool	commonIndex (std::vector< size_t > const &A, std::vector< size_t > const &B)

void	mergeIn (std::vector< size_t > &target, std::vector< size_t > const &toCopy)

std::vector< std::vector< size_t > >	findStructureMatches (std::vector< csl::IndexStructure > const &structures)

static bool	hasCommonIndex (csl::Expr const &A, csl::Expr const &B, csl::Space const *space=nullptr)

static bool	compareFields (QuantumField const &A, QuantumField const &B)

static int	getDMinkoPower (csl::Expr const &expr)

Detailed Description

Namespace containing the main simplification method for MARTY and all its dependencies.

Function Documentation

◆ abbreviateAll()

void mty::simpli::abbreviateAll ( csl::Expr & res )

Abbreviates all the sub-expressions that are relevant.

This function is more powerful than findAbbreviations() as it will also find external abbreviations ("EXT") and momentum abbreviations ("P")

Parameters

res	Expression to abbreviate.

◆ abbreviateIntegral()

void mty::simpli::abbreviateIntegral ( csl::Expr & res )

Abbreviate scalar integrals (or combinations of them) into abbreviations with the generic name "INT".

Parameters

res	Expression to abbreviate.

◆ addLocalTerms()

void mty::simpli::addLocalTerms ( csl::Expr & res )

Adds the local terms in an expression.

This function is defined in dimensionalRegularization.cpp but the declaration is reminded. Local terms are added when, in dim. reg., the Minkowki space dimension \( D = 4 - 2\epsilon \) is multiplied by a divergent integral \( I = \frac{a}{\epsilon} + b + \mathcal{O}(\epsilon)\), as the epsilon terms cancel each other to produce another constant term. In this simple case we obtain (removing the purely divergent term):

\[ D\cdot I = (4 - 2\epsilon)(\frac{a}{\epsilon} + b) = 4b + Local, \]

with

\[ Local = -2a. \]

Warning: This uses a simple algorithm to parse the total \( D \) factors mutiplying integrals but works only for non-factored expressions. This is fine as long as, in the main simplification chain, the amplitude is factored after this function is called.

Parameters

res	Expression in which local terms are added.

◆ applyDerivativesInStructure()

void mty::simpli::applyDerivativesInStructure ( csl::Expr & expr )

Searches in the expression derived fields and applies the derivatives in the fields themselves.

QuantumField objects may carry an additional derivative indicial structure, that is used here to get simple quantum field objects (no derivative) while keeping the information about the Minkowski derivatives to use in the quantum calculation. This is equivalent to the analytical replacement / definition:

\[ \partial _\mu \Phi _A(X) \equiv \Phi_{A;\mu}(X) \]

Parameters

expr	Expression in which the the function is applied.

◆ applyEOM() [1/2]

void mty::simpli::applyEOM	(	csl::Expr &	ampl,
		std::vector< FermionEOMData > &	onShellFermions
	)

Applies the equations of motions in an expression.

This function uses the SGL (Symbolic Gamma-matrices Library), a module of MARTY specialized in this purpose, to apply the EOM in fermion currents.

Parameters

ampl	Expression in which the EOM must be applied.
onShellFermions	On-shell fermions.

◆ applyEOM() [2/2]

void mty::simpli::applyEOM	(	csl::Expr &	ampl,
		std::vector< mty::QuantumField > const &	insertions,
		std::vector< csl::Tensor > const &	momenta
	)

Applies the equations of motion in an expression.

This function detects all the on-shell fermions for the process and call the other overload of applyEOM() to effectively apply the Dirac equation.

Parameters

ampl	Expression in which the EOM must be applied.
insertions	Field insertions for the process.
momenta	External momenta corresponding to the field insertions.

◆ colorAbbreviation()

csl::Expr mty::simpli::colorAbbreviation	(	std::vector< csl::Expr > const &	exprs,
		std::vector< color::ColorSpace const *> const &	spaces
	)

Abbreviates color structure in expressions under the generic name "Color".

For all different terms this function tries to simplify first the expression before abbreviating it. If after simplification there is still color indices the result is encapsulated into an abbreviation, otherwise it is left as is.

Parameters

exprs	Different expressions containing color structures.
spaces	Color spaces corresponding to the different expressions ( must be the same size as exprs). )

Returns: The product of all abbreviated expressions in exprs.

◆ colorStructure()

csl::IndexStructure mty::simpli::colorStructure	(	csl::Expr const &	node,
		color::ColorSpace const *	color
	)

Returns all color indices found in an expression lying in a particular color space.

Parameters

node	Expression to explore.
color	Color space for which this function searches indices.

Returns: The IndexStructure containing all found indices.

◆ expandColorIndices()

bool mty::simpli::expandColorIndices ( csl::Expr & res )

Expands expressions containing color indices to allow all algebra simplifications.

Parameters

res	Expression in which color indexed structures must be expanded.

Returns: True if something has been expanded.; False else.

◆ expandForFermionOrdering()

void mty::simpli::expandForFermionOrdering ( csl::Expr & expr )

Forces the ordering of external fermions by making sure that all Dirac structures are expanded.

Parameters

expr	Expression to expand.

◆ expandGammaMatrices()

bool mty::simpli::expandGammaMatrices ( csl::Expr & expr )

Expands gamma matrices in an expression to be able to simplify fermion chains (and traces).

Parameters

expr	Expression to simplify.

Returns: True if a simplification has been found.; False else.

◆ expandInProd()

bool mty::simpli::expandInProd	(	csl::Expr &	prod,
		std::function< csl::IndexStructure(csl::Expr const &)> const &	structureGetter
	)

Expands indicial expressions in a way avoiding unnecessary terms, in particular not expanding expressions that have no common index.

This function takes as parameter another function that must return, for any expression, the index structure relevant for expansion (e.g. returning only Minkowski indices if the purpose is to expand Minkowski metric terms). The expansion will be performed between two sub-expressions if they have at least one common index by this mean.

Note: Dev note : This function could now probably be replaced by the much more powerful and general csl::DeepPartialExpand() that can be tuned as will. This function is now used only in expandColorIndices().

Parameters

prod	Prod in which the expansion must take place.
structureGetter	Function that must return the index structure of an expression relevant for the expansion.

Returns: True if something has been expanded; False else.

◆ expandMinkoEpsilon()

bool mty::simpli::expandMinkoEpsilon ( csl::Expr & expr )

Expands Minkowski epsilon symbols to contract all possible indices.

This function only expand sub-terms that allow to contract one epsilon with another tensor that has a common index. All other terms are let in sums, not expanded.

Note: Dev note : This is using the emitter / receiver pattern of csl::DeepPartialExpand(): to use without moderation !

Parameters

expr	Expression in which Minkowski epsilon terms must be expanded.

Returns: True if something has been expanded.; False else.

◆ expandMinkoMetric()

bool mty::simpli::expandMinkoMetric ( csl::Expr & expr )

Expands Minkowski metrics to contract all possible indices.

This function only expand sub-terms that allow to contract one metric with another tensor that has a common index. All other terms are let in sums, not expanded.

Note: Dev note : This is using the emitter / receiver pattern of csl::DeepPartialExpand(): to use without moderation !

Parameters

expr	Expression in which Minkowski metric terms must be expanded.

Returns: True if something has been expanded.; False else.

◆ expandMinkoStructures()

bool mty::simpli::expandMinkoStructures ( csl::Expr & expr )

Expands Minkowski structures to contract all possible indices.

This function calls expandMinkoMetric() and expandMinkoEpsilon() successively.

Parameters

expr	Expression in which Minkowski structures must be expanded.

Returns: True if something has been expanded.; False else.

◆ expandMomentaExperimental()

void mty::simpli::expandMomentaExperimental	(	csl::Expr &	res,
		std::vector< csl::Tensor > const &	momenta
	)

Expands expressions containing momenta to contract all indices, typically replacing \( p_i^\mu\cdot p_{j\mu} \mapsto s_{ij} \).

Parameters

res	Expression to expand.
momenta	Set of momenta involved.

◆ factorIndicial()

bool mty::simpli::factorIndicial ( csl::Expr & res )

Factors indicial expressions in a bigger expression.

This function is there mostly to compensate the lack of such a general function in CSL, that factors indicial tensors. This function is yet not very general and can probably not be used reliably to this purpose in other contexts.

Parameters

res	Expression to factor

Returns: True if something has been changed (factored).; False else.

◆ findAbbreviations()

bool mty::simpli::findAbbreviations ( csl::Expr & res )

Abbreviates all the sub-expressions that are relevant considering constant factors only.

This function is simply searches for combination of factors than can be grouped into a unique abbreviation "Ab".

Parameters

res	Expression to abbreviate.

◆ findColorAbbreviation()

bool mty::simpli::findColorAbbreviation ( csl::Expr & expr )

Abbreviates color structures in an expression under the generic name "Color".

This function gathers all colorstructures of possibly different vector spaces and then calls colorAbbreviation() to effectively determine an abbreviation for each different space.

Parameters

expr	Expression to abbreviate.

Returns: True if the expression has been abbreviated in some way.; False else.

◆ findExternalAbbreviation()

void mty::simpli::findExternalAbbreviation ( csl::Expr & expr )

Abbreviates external legs in the expression using the generic name "EXT".

Note: Dev Note : This function works only for external fields of type PolarizationField. Could probably be extended very easily to QuantumField objects using the following condition
dynamic_cast<mty::QuantumField const*>(sub[i].get())

Parameters

expr	Expression ni which the abbreviation will be applied.

◆ getMomentumReplacement()

std::pair< csl::Expr, csl::Expr > mty::simpli::getMomentumReplacement	(	std::vector< mty::QuantumField > const &	insertions,
		std::vector< csl::Tensor > const &	momenta,
		size_t	posReplaced
	)

Creates the relevant replacement (using momentum conservation) to replace one particular momentum.

The two elements returned in a pair can be used directly in a csl replacement function on a particular expression as presented in the following sample code

auto [p, momentumEquivalent] = getMomentumReplacement(args...);

csl::Replace(amplitude, p, momentumEquivalent);

Parameters

insertions	Insertions for the process.
momenta	Set of external momenta of the process.
posReplaced	Position of the momentum (in momenta) that must be replaced by the relevant combination of the others.

Returns: A pair containing the momentum replacement to forward to csl::Replace().

◆ inColorSpace()

color::ColorSpace const * mty::simpli::inColorSpace ( csl::Index const & index )

Returns the colorspace associated to an index (nullptr if the index does not live in any ColorSpace).

Parameters

index Index form which the space is obtained.

Returns: The pointer to const ColorSpace if the index lives in such a vector space.; nullptr else

◆ isColorStructure() [1/2]

color::ColorSpace const * mty::simpli::isColorStructure ( csl::IndexStructure const & structure )

Returns the adjoint colorspace for a generator, nullptr for another structure.o.

All indices of the structure must lie in a color space otherwise it is not a generator and this function returns nullptr. This allows to select only the objects that may want to simplify with each other !

Parameters

structure The index structure of the tensor to test.

Returns: The adjoint colorspace if the indexstructure is the one of a group generator.; nullptr else.

See also: inColorSpace()

◆ isColorStructure() [2/2]

color::ColorSpace const * mty::simpli::isColorStructure ( csl::Expr const & expr )

Returns a colorspace if the expression is a color structure.

This function should not be used for mixed color / non color expressions. In a mixed case, the behaviour is not defined. Otherwise, a colorspace will be returned independently of the complexity of the expression (if it is a color structure).

Parameters

tensor Expression to test.

Returns: The color space associated to the expression if there is one; nullptr else.

◆ maybeSimplified()

bool mty::simpli::maybeSimplified ( csl::Expr const & expr )

Helper function that tells if an expression may be simplified in amplitudes.

If this function returns false, the main simplification routines can remove the expression from the amplitude to pass all simplification steps and put it back at the end to save unnecessary calculations.

Parameters

expr	Expression to check if it may be simplified.

Returns: True if the expression must not be discarded.; False else.

◆ reduceTensorIntegrals()

void mty::simpli::reduceTensorIntegrals ( csl::Expr & expr )

Replaces the one-loop momentum integrals by their reduced form depending on scalar integrals.

The scalar integrals are later evaluated numerically using the LoopTools library.

Parameters

expr	Expression to simplify.

◆ replaceMomentum()

void mty::simpli::replaceMomentum	(	csl::Expr &	init,
		std::vector< mty::QuantumField > const &	insertions,
		std::vector< csl::Tensor > const &	momenta,
		size_t	posReplaced
	)

Applies the momentum conservation by replacing one of the external momenta by the combination of the others.

Parameters

init	Expression in which the momentum conservation is applied.
insertions	Insertions for the process.
momenta	Set of external momenta of the process.
posReplaced	Position of the momentum (in momenta) that must be replaced by the relevant combination of the others.

See also: getMomentumReplacement()

◆ simplify()

void mty::simpli::simplify	(	csl::Expr &	expr,
		std::vector< mty::QuantumField > const &	insertions,
		std::vector< csl::Tensor > const &	momenta,
		mty::FeynOptions const &	options,
		Mode	mode
	)

Main simplification routine of MARTY, used for amplitudes, squared amplitude and Wilson coefficients.

This function for now is used for all calculations in MARTY with minor differences (handled through the mode given as a parameter). It calls directly or indirectly all the other functions in this file to simplify Dirac algebra, Group algebra, Equations of motions, momentum integrals, introducing abbreviations etc.

Parameters

expr	Expression to simplify.
insertions	Insertions of the process.
momenta	External momenta corresponding to the field insertions.
mode	Simplification mode specifying if the expression is an amplitude, squared amplitude or Wilson coefficient.

◆ simplifyColorWeights()

bool mty::simpli::simplifyColorWeights ( csl::Expr & expr )

Calculates color traces in an expression.

Parameters

expr	Expression to simplify.

Returns: True if a simplification has taken place.; Fasle else.

◆ simplifyEpsilon()

bool mty::simpli::simplifyEpsilon ( csl::Expr & expr )

Simplifies epsilon contractions with symmetric indices in an expression.

This function detects all epsilon tensors in a product and calls simplifyEpsilonInProd() to simplify it if it is contracted with symmetric indices.

Parameters

expr	Expression to simplify.

Returns: True if a simplification has been found.; False else.

◆ simplifyEpsilonInProd()

bool mty::simpli::simplifyEpsilonInProd	(	csl::Expr &	prod,
		csl::IndexStructure &	indices
	)

Simplifies the contraction of an epsilon tensor with a symmetric structure index.

This function is fully general and tries to swap indices that are contracted with the epsilon tensor to detect a symmetry. If a symmetry is detected, then the result is zero.

Parameters

prod	Product to simplify ni which the epsilon tensor has been removed.
indices	Indices of the initial epsilon tensor.

Returns: True if a simplification has been found (the result is zero).; False else.

◆ simplifyFermionChains()

bool mty::simpli::simplifyFermionChains ( csl::Expr & expr )

Simplifies fermion chains using the mty::FermionChain utility.

Parameters

expr	Expression to simplify.

Returns: True if a simplification has been found.; False else.

◆ simplifyImpulsions()

void mty::simpli::simplifyImpulsions	(	csl::Expr &	init,
		std::vector< mty::QuantumField > const &	insertions,
		std::vector< csl::Tensor > const &	momenta
	)

Applies the momentum conservation by replacing one of the external momenta by the combination of the others.

The momentum that is to be replaced is chosen automatically by this function following an ordering rule for the external fields. To maximize the possible simplifications, the momentum that are kept are those of on-shell fermions (because they can later be simplified later on through the Dirac equation). This function calls replaceMomentum().

Parameters

init	Expression in which the momentum conservation is applied.
insertions	Insertions for the process.
momenta	Set of external momenta of the process.

◆ suppressDiracDelta()

void mty::simpli::suppressDiracDelta	(	csl::Expr &	expr,
		csl::Expr const &	PSum
	)

Removes a dirac delta sub-expression.

This function is used at the end of a quantum calculation to remove the \( \delta ^{(4)}(\sum _i p_i) \) (with \( p_i \) external momenta) that appears naturally but does not enter the final amplitude result. The momentum sum in the dirac delta to remove must be provided.

Parameters

expr	Expression in which
PSum	Momentum sum appearing in the dirac delta to remove.

Data Structures

Enumerations

Functions

Detailed Description

Function Documentation

◆ abbreviateAll()

◆ abbreviateIntegral()

◆ addLocalTerms()

◆ applyDerivativesInStructure()

◆ applyEOM() [1/2]

◆ applyEOM() [2/2]

◆ colorAbbreviation()

◆ colorStructure()

◆ expandColorIndices()

◆ expandForFermionOrdering()

◆ expandGammaMatrices()

◆ expandInProd()

◆ expandMinkoEpsilon()

◆ expandMinkoMetric()

◆ expandMinkoStructures()

◆ expandMomentaExperimental()

◆ factorIndicial()

◆ findAbbreviations()

◆ findColorAbbreviation()

◆ findExternalAbbreviation()

◆ getMomentumReplacement()

◆ inColorSpace()

◆ isColorStructure() [1/2]

◆ isColorStructure() [2/2]

◆ maybeSimplified()

◆ reduceTensorIntegrals()

◆ replaceMomentum()

◆ simplify()

◆ simplifyColorWeights()

◆ simplifyEpsilon()

◆ simplifyEpsilonInProd()

◆ simplifyFermionChains()

◆ simplifyImpulsions()

◆ suppressDiracDelta()