Beyond the Standard Model physics
made easy with FEYNRULES
Benjamin Fuks
LPTHE / Sorbonne Université

Tutorial session @ Appi (Iwate )1 March 2023

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 1

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Outline1.

FEYNRULES in a nutshell

2. Implementing supersymmetric QCD in FEYNRULES
3. Using FEYNRULES with supersymmetric QCD model
4. Summary
5. Appendix: advanced model implementation techniques

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

A comprehensive approach to MC simulations[ Christensen, de Aquino, Degrande, Duhr, BF, Herquet, Maltoni & Schumann (EPJC`11) ]Tools connecting an idea to simulated collisions• Model building
Idea / Lagrangian

FEYNRULES / UFO

MatrixElements[ MG5_aMC ]Partonshowers
Hadronization

Detector
Reconstruction[ PYTHIA 8 ]Simulated collisions
Analysis[ MADANALYSIS 5 ]New physics made easy with FEYNRULES[ DELPHES ][ FASTJET ]• Hard scattering★ Feynman diagram and amplitude generation★ Monte Carlo integration★ Event generation• QCD environment★ Parton showering★ Hadronisation★ Underlying event• Detector simulation★ Simulation of the detector response★ Object reconstruction• Event analysis★ Signal/background analysis★ LHC recasting

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

FEYNRULES in a nutshell
What is FEYNRULES?
• A framework to develop new physics models• Automatic export to Monte Carlo event generators

phenomenological investigations of BSM models➟ Facilitate➟ Facilitate the confrontation of BSM models to data
• Validation of implementations with several Monte Carlo programmes

Main features• MATHEMATICA package• Core function: derives Feynman rules from a Lagrangian• Requirements: locality, Lorentz and gauge invariance• Supported elds: scalars, (two- and four-component) fermions,vectors (and ghosts), spin-3/2, tensors, super elds

Benjamin Fuks - 01.03.2023 - 4fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

From FEYNRULES to Monte Carlo tools...
[ Christensen, Duhr (CPC ’09); Alloul, Christensen, Degrande, Duhr, BF (CPC’14) ]Model
Particles ^Gaugesymmetries

TEX output

Parameters

FEYNRULES

Lagrangian

Feynman rules

CALCHEP

ALOHA

GOSAM

Superspace module

FEYNARTSHERWIG ++SHERPA

Mass diagonalisation

WHIZARD*

MADGRAPH5_aMC@NLO

Translationinterfaces

WHIZARD

MADANALYSIS 5

The UFO[ Degrande, Duhr, BF, Grellscheid, Mattelaer, Reiter (CPC ’12) ]* Whizard interface: Christensen, Duhr, BF, Reuter, Speckner (EPJC ’12)

[Duhr, BF (CPC ’11); BF (IJMPA ’12) ][Alloul et al. (EPJC ’13) ]Decay package[Alwall et al. (CPC`16) ]NLO module[Degrande (CPC`15); Frixione et al. (JHEP`19)]

Computational tools^ Support for spin 3/2: Christensen, de Aquino, Deutschmann, Duhr, BF, Garcia-Cely, Mattelaer, Mawatari, Oexl, Takaesu (EPJC ’13)

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 5

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Outline1.

FEYNRULES in a nutshell

2. Implementing supersymmetric QCD in FEYNRULES
3. Using FEYNRULES with supersymmetric QCD model
4. Summary
5. Appendix: advanced model implementation techniques

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 6

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Supersymmetric QCD: particle content

Particle content• Two matter supermultiplets in the fundamental representation of SU(3)c★ One massive Dirac fermion: a quark★ Two massive scalar elds: a left-handed and a right-handed squark• One (SU(3)c ) gauge supermultiplet★ One massive Majorana fermion: a gluino★ One massless gauge boson: the gluonfi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 7

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques(Broken) supersymmetric QCD: the model
The dynamics of the model is embedded in the Lagrangian
1
i¯
†
†
µ⌫
/ + Dµ q̃L/L=gµ⌫ g + g̃ Dg̃
Dµ q̃L + Dµ q̃RDµ q̃R + iq̄ Dq
4
2
1 ¯
2 †
2 †
mq̃L q̃L q̃L mq̃L q̃R q̃R mq q̄qmg̃ g̃g̃
2
ih
i
gs2 h
† a
† a
† a
† a
q̃L
T q̃L + q̃R
T q̃R
q̃L
T q̃L + q̃R
T q̃R
2 h
i
_
p
† a a
+ 2gs
q̃L
T g̃ PL q + q̄PL g̃ a T a q̃R + h.c.
R

• Kinetic terms for all elds ( rst line)• Mass terms for the squarks, quarks and gluino (second line)• Supersymmetric gauge interactions for all elds (last two lines)

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fi

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

A FEYNRULES model le
A FEYNRULES model le is an .fr le written in MATHEMATICA syntax
Declaration of the gauge group

Preamble★ Author information★ Abelian or not★ Model information★ Representation matrices★ Index de nitions★ Structure constants★ Coupling constant★ Gauge boson or vector super eld

Declaration of elds
★ Names, spins, PDG codes

Declaration of parameters★ Indices, quantum numbers★ External and internal★ Masses, widths★ Classes and class members★ Scalar and tensor

A Lagrangianfi

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fi

fi

fi

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Preamble: general information
An electronic signature for the model implementation• Important for traceability, documentation, contact with the authors, etc.
• Reference publications• Webpage information

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Preamble: indices
Index dimensions• Fundamental SU(3)C indices for squarks and quarks: Colour, dimension 3• Adjoint SU(3)C indices for gluons and gluinos: Gluon, dimension 8• Lorentz and spin indices automatically handled

QCD ≡ special role in event generators ➙ special names• Colour, Sextet and Gluon for colour indices• G for the gluon eld• T for the fundamental representation matrices• f and d for the structure constants• G/gs and aS for the coupling constants

The style of the indices can be speci ed• Fundamental indices starting with m• Adjoint indices starting with a
Benjamin Fuks - 01.03.2023 - 11fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Gauge groups (1)Each element the group is declared in the M$GaugeGroups list
• One group ≣ one set of MATHEMATICA replacement rules★ We must only declare SU(3)C (name: SU3C)• One rule ≡ one group property★ Abelian: abelian or non-abelian group★ GaugeBoson: associated gauge boson★ CouplingConstant, StructureConstant: coupling and structure constants★ Representations: list of 2-tuples linking indices to representations
See the manual for more details on gauge groups

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Gauge groups (2)Advantages of a proper gauge group declaration• Writing Lagrangians more easily★ Covariant derivatives (DC[ eld, Lorentz index])★ Field strength tensors (FS[ eld, Lorentz index 1, Lorentz index 2])★ Useful for Lagrangian building in superspace (brie y covered in the last slides)• Example: gluon and gluino kinetic terms

1
i¯
†
†
µ⌫
/ + Dµ q̃L/L=gµ⌫ g + g̃ Dg̃
Dµ q̃L + Dµ q̃RDµ q̃R + iq̄ Dq
4
2
1 ¯
2 †
2 †
mq̃L q̃L q̃L mq̃L q̃R q̃R mq q̄qmg̃ g̃g̃
2
ih
i
gs2 h
† a
† a
† a
† a
q̃L T q̃L + q̃R T q̃Rq̃L T q̃L + q̃R T q̃R
2 h
i
p
† a a
+ 2gs
q̃L
T g̃ PL q + q̄PL g̃ a T a q̃R + h.c.

Benjamin Fuks - 01.03.2023 - 13
fl

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The gluon eld
Each eld is an element of the M$ClassesDescription list
• A declaration ≣ a set of MATHEMATICA replacement rules• In our SUSY-QCD model, we rst declare the SU(3)C gauge boson: G• One rule ≡ one property of the eld★ Vectoreld ➙ the label is V[1] (with V, and not F, S, R, T, etc.)★ ClassName: de nes a symbol for Lagrangians ➙ G★ Indices: the gluon in the adjoint representation➙ tagged as gauge boson of SU(3)c ➙ Gluon internally linked to the adjoint★ Others: vanishing mass and width, PDG code ≡ 21, self-conjugate
See the manual for more options for eld declarationsfi

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fi

fi

fi

fi

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

A Majorana gluino eld
A second element in the M$ClassesDescription list`

• Differences with the gluon eld declaration★ Four-component fermioniceld ➙ F[1] (starting with an F)★ Classname: two symbols for the Lagrangian ➙ go and gobar★ Mass/width: two symbols and associated numerical values

See the manual for more details on eld declarations

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fi

fi

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Weyl/Majorana gluino elds
Second option: two-component spinors• Extra options available★ Two-component and four-component fermions ➙ W[1] (with a W) and F[1]★ Weyl fermions unphysical ➙ linked to four-component fermions (WeylComponents)★ Several symbols de ned ➙ go and gobar; gow and gowbar★ Chirality to specify

See the manual for more details on eld declarations

Benjamin Fuks - 01.03.2023 - 16
fi

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The (top) quark eld
A third element in the M$ClassesDescription list

• Nothing special compared to the other elds
★ Fundamental QCD indices speci ed

New physics made easy with FEYNRULES
fi

fi

fi

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniquesMore on quarks: generation indicesThree generations of up-type quarks• Slight differences for a bunch of options (+ new options)★ ClassMembers: all the members of the class★ A generation index: Gen (FlavorIndex de nes the avour index)★ Mass, Width, PDG: one per class member (+ generic mass symbol)

New physics made easy with FEYNRULES
flfi

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Squarks
Extra elements in the M$ClassesDescription list
• Nothing special★ Scalar eld ➙ S[3]★ Classname ➙ pairs of symbols:

sq1, sq2, sq1bar and sq2bar★ Mass, width, Indices, etc. (standard)

Missing:squark mixingfi

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Squark mixing
Mixing implemented via physical and unphysical elds

• Squark elds mix as
✓ ◆ ✓
◆✓ ◆
q̃1
cos ✓ sin ✓
q̃L
=
q̃2
sin ✓ cos ✓q̃R★ Left and right-handed squarks unphysical ➙ removed from the Lagrangian and replaced★ De nitions ≡ the replacement rules (inversion of the above relation)➙ Rotations performed automatically➙ Lagrangian written in the gauge basis (easier with sqL / sqR)
Benjamin Fuks - 01.03.2023 - 20
fi

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Parameters
The model requires to implement three parameters• Masses and widths handled automatically

1
i¯
†
†
µ⌫
/ + Dµ q̃L/L=gµ⌫ g + g̃ Dg̃
Dµ q̃L + Dµ q̃RDµ q̃R + iq̄ Dq
4
2
1 ¯
2 †
2 †
mq̃L q̃L q̃L mq̃L q̃R q̃R mq q̄qmg̃ g̃g̃
2
ih
i
gs2 h
† a
† a
† a
† a
q̃L T q̃L + q̃R T q̃Rq̃L T q̃L + q̃R T q̃R
2 h
i
_
p
† a a
+ 2gs
q̃L
T g̃ PL q + q̄PL g̃ a T a q̃R + h.c.
• The strong coupling explicit and implicit➙ gs needed➙ s as required by MC tools• Squark mixing angle to add
✓ ◆ ✓
q̃1
cos ✓
=
q̃2
sin ✓

𝜃

𝛼

New physics made easy with FEYNRULES

sin ✓
cos ✓

◆✓

q̃L
q̃R

◆
Benjamin Fuks - 01.03.2023 - 21

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Parameters: strong coupling
Parameters are declared as elements of the list M$Parameters• One declaration ≣ one set of MATHEMATICA replacement rules• Strong coupling: s and gs not independent★ Internal and External parameters➙ External ≣ free param. (numerical)➙ Internal ≣ dependent param (formula)★ InteractionOrder: speci c to some MC➙ more ef cient diagram generation★ ParameterName: QCD special

See the manual for more details on parameter declarations

fi

𝛼

fi

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Parameters: squark mixing

The squark mixing angle• Nothing special (standard external parameter)

More options (assuming matrix-form mixing)• Tensor parameters can be implemented too (Indices, Unitary)• Complex parameters can be implemented too (ComplexParameter)

See the manual for more details on parameter declarations

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Parameter organisation
Les Houches blocks to organise external parameters• Les Houches parameter card by MC programmes➙ BlockName + OrderBlock• Unspeci ed: the ‘FRBlock’ block

Completes implementation as an example• Many options (TeX, Description, BlockName, OrderBlock)𝛼fi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 24

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Outline1.

FEYNRULES in a nutshell

2. Implementing supersymmetric QCD in FEYNRULES
3. Using FEYNRULES with supersymmetric QCD model
4. Summary
5. Appendix: advanced model implementation techniques

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 25

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Getting started

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 26

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Starting a FEYNRULES MATHEMATICA session
Step 1: loading FEYNRULES into MATHEMATICA• The FEYNRULES path
• Loading FEYNRULES

The output:• Information on FEYNRULES, authors, version numbers, references, etc.
New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 27

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Loading the model implementation
Step 2: loading the SUSY-QCD implementation into FEYNRULES• Moving to the right directory• Loading the model itself

The output:• Preamble of the model are printed to the screen• More information from ModelInformation[ ]New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 28

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The vector
Lagrangian

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 29

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The SUSY-QCD vector Lagrangian
The gauge sector of the theory : one gauge supermultiplet• One massive Majorana fermion: a gluino• One massless gauge boson: the gluon

Dynamics embedded in the vector Lagrangian
Lvector =1i/gµ⌫ g µ⌫ + g̃¯Dg̃
4
2

1 ¯
mg̃ g̃g̃
2• Kinetic terms (both elds)• Mass terms for the gluino• Gauge interactions (both elds, through gauge-covariant objects)fi

fi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 30

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Implementing the vector Lagrangian
Textbook expression:
Lvector =1i/gµ⌫ g µ⌫ + g̃¯Dg̃
4
2

1 ¯
mg̃ g̃g̃
2

Strengths of a proper gauge-group implementation• Shortcut functions (DC, FS, etc.)• Very compact★ Indices understood (automatic handling)

Indices could be noted explicitly

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 31

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Checks of the implementation
FEYNRULES implementation

Textbook expression
Lvector =1i/gµ⌫ g µ⌫ + g̃¯Dg̃
4
2

1 ¯
mg̃ g̃g̃2

Printing the vector Lagrangian

The output:• Covariant derivatives and eld strength tensors automatically evaluated• FSU3C ≡ structure constants ➙ to be replaced in a second step by fabcfi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 32

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Expansion of the Lagrangian
All indices restored automatically• Kinetic terms (gluon and gluino)• Mass terms (gluino)• QCD interactions (with the proper structure constants)

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 33

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Checks: Hermiticity
FEYNRULES implementation

Textbook expression
Lvector =

1
i¯
µ⌫
/
gµ⌫ g + g̃ Dg̃
4
2

1 ¯
mg̃ g̃g̃
2

The Lagrangian must be Hermitian• Feynman rules of the Lagrangian minus its Hermitian conjugate ➙New physics made easy with FEYNRULES

-

†

𝓛

𝓛

Benjamin Fuks - 01.03.2023 - 34

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Checks: normalisation
FEYNRULES implementation

Textbook expression
Lvector =

1
i¯
µ⌫
/
gµ⌫ g + g̃ Dg̃
4
2

1 ¯
mg̃ g̃g̃2

Kinetic and mass terms canonically normalised• Normalisation of the quadratic terms (kinetic and mass terms)• Absence of non-diagonal quadratic terms• Other checks: CheckDiagonalQuadraticTerms, CheckDiagonalKineticTerms

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 35

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Checks: the mass spectrum
FEYNRULES gluino implementation
FEYNRULES Lagrangian implementation

Consistent mass information (numerically)• Masses extracted from the Lagrangian• Masses xed in particle declarationsfi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 36

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Feynman rules
Extract all N-point interactions from the Lagrangian (with N>2)

• Three vertices found in the Lagrangian★ One triple and quartic gluon interactions★ One gluino-gluon interaction

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 37

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The triple gluon vertex
Literatureh

gs fabc (k1k2 ) ⌘ ↵ + (k3k1 ) ⌘ ↵ + (k2k3 ) ↵ ⌘i[Incoming momenta]

FEYNRULES• Colour: index ai related to the ith particle(a = style for adjoint SU(3)C indices)• Spin: index

i = Lorentz index of the ith (vector) particle𝝁

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 38

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The quartic gluon vertex
Literature1

2

igs2 f a1 a2 b f ba3 a4 (⌘ µ1 µ4 ⌘ µ2 µ3

34⌘ µ1 µ3 ⌘ µ2 µ4 )+ igs2 f a1 a3 b f ba2 a4 (⌘ µ1 µ4 ⌘ µ2 µ3⌘ µ1 µ2 ⌘ µ3 µ4 )+ igs2 f a1 a4 b f ba2 a3 (⌘ µ1 µ3 ⌘ µ2 µ4⌘ µ1 µ2 ⌘ µ3 µ4 )FEYNRULES• Colour: ai related to the ith particle (Gluon$1 summed over)• Spin:i related to the ith (vector) particle𝝁

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 39

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Gluon and gluino interactions
Literature2
1

g s fa1 a2 a3 µ1
3

FEYNRULES• Colour: ai related to the ith particle• Spin:i related to the ith vector; si related to the ith fermion𝝁

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 40

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The FeynmanRules function
The function FeynmanRules has many options• Restriction on the interactions to display: MaxParticles, MaxCanonicalDimension, etc.
• Selection of speci c particles: Free, Contains, etc.
• ScreenOutput: displaying the vertices to the screen or not• FlavorExpand: perform a avour expansion (otherwise classes used)

See the manual for more details on the function FeynmanRulesflfi

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The matter
Lagrangian

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 42

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The SUSY-QCD matter sector
Matter elds
• Two matter supermultiplets in the fundamental representation of SU(3)c
✓ ◆ ✓
◆✓ ◆
★ One massive Dirac fermion: a quarkq̃1
cos ✓ sin ✓q̃L=★ Two mixing massive scalar elds: two squarkq̃2
sin ✓ cos ✓q̃R★ Gauge coupling to the SU(3)c gauge supermultiplet

Lagrangian:††/Lmatter = Dµ q̃LDµ q̃L + Dµ q̃RDµ q̃R + iq̄ Dq
m2q̃i q̃i† q̃i mq q̄q
ih
i
gs2 h
† a
† a
† a
† a
q̃L T q̃L + q̃R T q̃Rq̃L T q̃L + q̃R T q̃R
2 h
i
_
p
† a a
+ 2gs
q̃L
T g̃ PL q + q̄PL g̃ a T a q̃R + h.c.

• Mixed use of mass / gauge bases (makes it easier to implement)• Kinetic terms ( rst three terms) and D-terms (second line) in the gauge basis• Mass terms (end of rst line) in the mass basis• SUSY gauge quark-squark-gluino interactions (fourth line) in the gauge basis

fi

fi

fi

fi

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Kinetic and gauge interactions
Kinetic and gauge interactions, as well as mass terms††/
Lmatter = Dµ q̃LDµ q̃L + Dµ q̃RDµ q̃R + iq̄ Dq
m2q̃i q̃i† q̃i mq q̄q
ih
i
gs2 h
† a
† a
† a
† a
q̃L
T q̃L + q̃R
T q̃R
q̃L
T q̃L + q̃R
T q̃R
2 h in FEYNRULES follows the textbook
The implementation
i expression
p
† a a
a
a
+
2g
q̃
T
g̃
P
q
+
q̄P
g̃
T
q̃R + h.c.
s
LL
• Shortcut functions (DC)L• Very compact implementation• Repeated indices summed

Gauge eigenstates are used
Mass eigenstates are used

New physics made easy with FEYNRULES

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FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Kinetic term Feynman rules
Feynman rules• Field rotations have been performed automatically★ No more gauge eigenstates (sqL, sqR)
• All squark and quark QCD interactions derived★ The 3-point qqg interaction★ Two 3-point squark-gluon interactions★ Two 4-point squark-gluon interactions
New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 45

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

D-terms
D-terms (in the gauge eigenbasis)
Lmatter =

gs2 h
2

† a
† a
q̃L
T q̃L + q̃R
T q̃R

ih

† a
† a
q̃L
T q̃L + q̃R
T q̃R

i

The FEYNRULES implementation follows the textbook expression• Repeated indices (cc1, cc2, cc3, cc4) summed• A single index can only be used twice

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 46

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

D-term Feynman rules
D-terms (in the gauge eigenbasis)
Lmatter =

gs2 h
2

† a
† a
q̃L
T q̃L + q̃R
T q̃R

ih

† a
† a
q̃L
T q̃L + q̃R
T q̃R

i

Feynman rules computation• All nine vertices automatically derived from the (very compact) Lagrangian
New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 47

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

SUSY-gauge squark-quark-gluino couplings††/
Lmatter = Dµ q̃LDµ q̃L + Dµ q̃RDµ q̃R + iq̄ Dq
m2q̃i q̃i† q̃i mq q̄q
ih
i
2h
g
† a
† a interactions† a (in the† gauges gluino-quark-squark
The SUSY gaugeq̃L
T q̃L + q̃R
T q̃R
q̃L
T q̃L + q̃R
T a q̃R basis)
2 h
_
i
p
† a a
+ 2gs
q̃L
T g̃ PL q + q̄PL g̃ a T a q̃R + h.c.
Lmatter =The FEYNRULES again following the textbook expression• All indices explicit (scalars not included in fermion chains)• ProjM ≡ left-handed chirality projector (ProjP right-handed)• The dot connects the elements of a fermion chain★ Remark: ProjM(s1,s2) is a scalar object

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 48

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Feynman
rules
from
the
D-terms
†
†
†
µ
µ
2
/

Lmatter = Dµ q̃L D q̃L + Dµ q̃R D q̃R + iq̄ Dq mq̃i q̃i q̃i mq q̄q
ih
i
gs2 h
† a
† a
† a
† a
q̃
T
q̃
+
q̃
T
q̃
q̃
T
q̃
+
q̃
q̃R basis)
The SUSY gauge gluino-quark-squark
L
Rinteractions
L(in theLRL
R T gauge
2 h
i
_
p
† a a
+ 2gs
q̃L
T g̃ PL q + q̄PL g̃ a T a q̃R + h.c.
Lmatter =Feynman rules computation• Including the Hermitian conjugate pieces with HC

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 49

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

To phenomenology

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 50

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

From FEYNRULES to phenomenology
The SUSY-QCD model has been implemented into FEYNRULES

The Feynman rules have been extracted (and checked)
Ready to export the model to MC tools• CALCHEP / COMPHEP• FEYNARTS / FORMCALC• UFO: MADGRAPH5_AMC@NLO / SHERPA / HERWIG++ / WHIZARD

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 51

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Limitations and neprints
Particle / parameter names• The strong interaction has a special role
★ Name for the gluon eld, the coupling constant, etc.
★ Which parameter is internal/external★ The numerical value of s at the Z-pole• For some generators, the electroweak interaction has also a special role
★ Name for the Fermi coupling, the Z-boson mass
★ Which parameter is external/internal★ At which scale must the numerical values be given

Colour structures: up to octets in FEYNRULES, not in MC generators• Interfaces discard non-supported vertices

Lorentz structures/spins: up to spin-2 in FEYNRULES, not in MC generators• Interfaces discard non-supported vertices

Benjamin Fuks - 01.03.2023 - 52
fi

𝛼

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Outline1.

FEYNRULES in a nutshell

2. Implementing supersymmetric QCD in FEYNRULES
3. Using FEYNRULES with supersymmetric QCD model
4. Summary
5. Appendix: advanced model implementation techniques

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 53

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Advanced techniques• More about UFOs
• Extension / restriction of existing models• The superspace module of FEYNRULES• Mass diagonalisation• Two-body decays• Next-to-leading order module

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 54

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Summary
The quest for new physics at the LHC• Relies on event generators for background and signal modelling• FEYNRULES facilitates the implementation of new physics models

FEYNRULES: http://feynrules.irmp.ucl.ac.be• Straightforward implementation of new physics model in Monte Carlo tools★ Interfaces to many programs• FEYNRULES shipped with its own computational modules★ A superspace module★ A decay package★ A mass diagonalisation module (ASPERGE)

Try it on with yourfavorite model!

★ An NLO module

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 55

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Outline1.

FEYNRULES in a nutshell

2. Implementing supersymmetric QCD in FEYNRULES
3. Using FEYNRULES with supersymmetric QCD model
4. Summary
5. Appendix: advanced model implementation techniques

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 56

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Advanced techniques for FEYNRULES implementation✦ More about the UFO✦ Extension / restriction of existing models✦ The superspace module of FEYNRULES✦ Mass diagonalization✦ Two-body decays✦ Next-to-leading order module

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 57

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The UFO

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 58

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

A step further: the Universal FEYNRULES Output✦ The UFO in a nutshell[ Degrande, Duhr, BF, Grellscheid, Mattelaer, Reiter (CPC ’12) ][ Degrande, Duhr, BF, Hirschi, Mattelaer, Shao et al. (in prep.) ]✤ UFO ≣ Universal FEYNRULES output★ Universal as not tied to any speci c Monte Carlo program✤ Consists of a set of PYTHON les to be linked to any code
✤ It contains all the model information★ Allows the models to contain generic color and Lorentz structures✤ Can be employed for next-to-leading order calculations✦ The UFO is now a standard and used by many other programs
ALOHA

GOSAM

HERWIG ++MADGRAPH5_aMC@NLO

New physics made easy with FEYNRULES

WHIZARD

MADANALYSIS 5SHERPALANHEP

SARAH

fi

fi

Benjamin Fuks - 01.03.2023 - 59

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The UFO in practice✦ The UFO is a set of PYTHON les✤ Factorization of the information: particles, interactions, propagation,

parameters, NLO, etc.

✦ Example
Parameters

Interactions

NLO

New physics made easy with FEYNRULES

Propagators

Particlesfi

Benjamin Fuks - 01.03.2023 - 60

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Particles✦ Particles are stored in the particles.py le✤ Instances of the particle class✤ Attributes: particle spin, color representation, mass, width, PDG code, etc.
✤ Antiparticles automatically derived

New physics made easy with FEYNRULES
fi

Benjamin Fuks - 01.03.2023 - 61

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Parameters✦ Parameters are stored in the parameters.py le✤ Instances of the parameter class✤ External parameters are organized following a Les Houches-like structure(blocks and counters)✤ PYTHON-compliant formula for the internal parameters

New physics made easy with FEYNRULES
fi

Benjamin Fuks - 01.03.2023 - 62

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Interactions: generalities✦ Vertices decomposed in a spin x color basis (coupling strengths ≡ coordinates)✤ Example: the quartic gluon vertex can be written as
igs2 f a1 a2 b f ba3 a4 (⌘ µ1 µ4 ⌘ µ2 µ3⌘ µ1 µ3 ⌘ µ2 µ4 )+ igs2 f a1 a3 b f ba2 a4 (⌘ µ1 µ4 ⌘ µ2 µ3+ igs2 f a1 a4 b f ba2 a3 (⌘ µ1 µ3 ⌘ µ2 µ4⌘µ1 µ2 µ3 µ4⌘)⌘ µ1 µ2 ⌘ µ3 µ4 )f a1 a2 b f ba3 a4 , f a1 a3 b f ba2 a4 , f a1 a4 b f ba2 a3
0 2
10 µµ µµigs
00
⌘ 1 4⌘ 2 3⇥ @ 0 igs2 0 A @ ⌘ µ1 µ4 ⌘ µ2 µ300 igs2⌘ µ1 µ3 ⌘ µ2 µ4★ 3 elements for the color basis★ 3 elements for the spin (Lorentz structure) basis★ 9 coordinates (6 are zero)1⌘ µ1 µ3 ⌘ µ2 µ4⌘ µ1 µ2 ⌘ µ3 µ4 A
⌘ µ1 µ2 ⌘ µ3 µ4✤ Several les are used for the storage of the informationfi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 63

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Example: the quartic gluon vertex✦ General information in vertex.py★ lorentz ≡ spin basis(in lorentz.py; common to all vertices)★ color ≡ color basis★ couplings ≡ coordinates(in couplings.py; common to all vertices)f a1 a2 b f ba3 a4 , f a1 a3 b f ba2 a4 , f a1 a4 b f ba2 a3
0 2
10 µµ µµigs
00
⌘ 1 4⌘ 2 3⇥ @ 0 igs2 0 A @ ⌘ µ1 µ4 ⌘ µ2 µ300 igs2⌘ µ1 µ3 ⌘ µ2 µ4µ1 µ3 µ2 µ4

1⌘⌘µ1 µ2 µ3 µ4 A⌘⌘
µ1 µ2 µ3 µ4⌘⌘✦ Lorentz structures: straightforward implementations in lorentz.py✦ Couplings: straightforward implementations in couplings.py
Coupling orders: for selecting diagrams

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 64

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Extending models

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 65

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Merging and extending models (1)✦ Many BSM models of interest are simple extensions of another model✤ FEYNRULES allows one to start from a given model★ Add new particles, parameters, Lagrangian terms★ Modify existing particles, parameters, Lagrangian terms★ Remove some particles, parameters, Lagrangian terms✦ Simpli ed models✤ Special cases very relevant for LHC physics: Simpli ed Model Spectra★ The Standard Model + one or two new particles★ Often inspired by the MSSM or dark matter models★ Ex.: the SM + lightest stop and neutralino + relevant subset of MSSM interactions

Benjamin Fuks - 01.03.2023 - 66fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Merging and extending models (2)✦ Merged FEYNRULES model contains two .fr les✤ The parent model implementation✤ One extra le with the modi cations✤ They must be loaded together (the parent model rst)
★ No need to re-implement what is common (gauge groups, etc.)✦ One can start from the models available on the FEYNRULES database
http://feynrules.irmp.ucl.ac.befi

fi

Benjamin Fuks - 01.03.2023 - 67fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The FEYNRULES model database✦ O(100) models are available online✤ Simple extensions of the Standard Model★ Simpli ed model spectra★ Four generation models★ Vector-like quarks★ Two-Higgs-Doublet Models, Hidden Abelian Higgs★ etc.
✤ Supersymmetric models★ MSSM with and without R-parity★ The NMSSM★ R-symmetric supersymmetric models★ Left-right supersymmetric models✤ Extra-dimensional models★ Universal extra-dimensions★ Large extra-dimensions★ Heidi, Minimal Higgsless models★ Randall-Sundrum✤ Strongly coupled and effective eld theories★ Technicolor★ Models with dimension-six and dimension-eight operators★ etc.
Benjamin Fuks - 01.03.2023 - 68fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Restricting model implementations✦ Many BSM models of interest are subset of other models✤ Equivalent to the parent model, with some parameters set to 0 or 1
★ Example 1: the massless version of a model (massless light quarks in the SM)★ Example 2: a mixing matrix set to the identity (no-CKM matrix in the SM)✤ FEYNRULES allows one to start from a given model★ Write the restrictions under the form of a list of MATHEMATICA replacement rules➢ M$Restrictions★ Read them into FEYNRULES★ Apply them before the computation of any Feynman rule

✦ The output Feynman rules (and thus Monte Carlo model les)✤ Are free from the restricted parameters✤ Smaller les, more ef ciency at the MC level★ Ex.: the MSSM has more than 10000 vertices; its avor-conserving version ~1000

fi

Benjamin Fuks - 01.03.2023 - 69
fl

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Example of restrictions✦ One practical example: the Standard Model without CKM-mixing

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 70

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Superspace module

See the manual for more details on the superspace module

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 71

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Super eld declaration✦ A module dedicated to calculations in superspace✤ Super eld declaration and links to the component elds
A left-handed squark super eld★ Component elds tobe declared too★ Vector super elds can bebe linked to gauge groups

Benjamin Fuks - 01.03.2023 - 72
fi

fi

fi

fi

fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Supersymmetric model implementation✦ Supersymmetric model implementation✤ Declaration of the model gauge group✤ Declaration of all elds and super elds
✤ Declaration of all model parameters✤ Writing the Lagrangian (in a simpli ed way)

✦ Supersymmetric Lagrangian in superspace are very compact
1
1
↵
↵
˙
L= e
+
Tr(W
W
)
+
Tr(
W̄
W̄
)|✓¯2
↵ |✓ 2↵˙|✓2 ✓¯22
2
16g ⌧R16g ⌧R+ W ( )|✓2 + W ? ( † )|✓¯2 + Lsoft†

2gV✤ First line: kinetic and gauge interaction terms for all elds

➢ Model independent can be automated✤ Second line: superpotential and supersymmetry breaking Lagrangian➢ Model dependent to be provided✤ Series expansion in terms of component elds
✤ Automatic derivation of supersymmetric Lagrangians✤ Solving the equations of motion of the unphysical elds

fi

fi

fi

fi

Benjamin Fuks - 01.03.2023 - 73fi

fi

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Implementing supersymmetric Lagrangians✦ Supersymmetric Lagrangian in superspace are very compact
L=

1
1
↵
↵
˙
Tr(W
W
)
+
Tr(
W̄
W̄
)|✓¯2
↵
↵
˙
|
2
2
✓2
16g ⌧R16g ⌧R+ W ( )|✓2 + W ? ( † )|✓¯2 + Lsoft
†

e 2gV

|✓2 ✓¯2 +✤ First line: kinetic and gauge interaction terms for all elds
★ Model independent
can be automated (CSFKineticTerms / VSFKineticTerms)★ Dedicated methods to access the components of a super eld (Theta2Component, etc.)✤ Second line: superpotential and supersymmetry-breaking terms★ Model dependent
to be provided by the user (SuperW and LSoft)

New physics made easy with FEYNRULES
fi

fi

Benjamin Fuks - 01.03.2023 - 74

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Implementing supersymmetric Lagrangians✦ The implemented Lagrangian above must be post-processed✤ Solving the equation of motion for the auxiliary elds
✤ Inserting the solution back into the Lagrangian★ Automated (SolveEqMotionF and SolveEqMotionD)✤ Replacement of Weyl spinors in terms of Majorana and Dirac spinors★Automated (WeylToDirac)✤ Rotation to the mass basis★Standard FEYNRULES function (ExpandIndices)

New physics made easy with FEYNRULES
fi

Benjamin Fuks - 01.03.2023 - 75

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Mass matrix
diagonalization

See the manual for more details on the ASPERGE module

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 76

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Mass matrices and their diagonalization✦ The problematics of the mass matrices✤ Lagrangians are usually easily written in the gauge basis✤ The included mass matrices are thus in general non-diagonal➢ diagonalization required✤ The gauge basis must be rotated to the mass basis where the mass matrices are diagonal✤ This diagonalization cannot in general be achieved analytically✦ The ASPERGE package of FEYNRULES✤ A module allowing one to extract the mass matrices from the Lagrangian✤ A generator of C++ code ➢ numerical diagonalization of all mass matrices(the generated code can be used in a standalone way)
✤ See: Alloul, D’Hondt, De Causmaecker, BF, Rausch de Traubenberg [ EPJC 73 (2013) 2325 ]New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 77

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Example of the Z-boson/photon mixing✦ Example: the Z-boson and photon in the Standard Model✤ Each mixing is declared as a set of replacement rules (in M$MixingsDescription)✤ Each rule represent a property of the mixing relation

✓

Aµ
Zµ

◆

= Uw

✓

Bµ
Wµ3

◆

✤ ASPERGE can compute the mass matrices:✤ ASPERGE can generate its standalone C++ version

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 78

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

DecaysSee the manual for more details on the decay module

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 79

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Two-body decays✦ The problematics of the decay widths and branching ratios✤ Some MC tools need decay tables (widths and branching ratios) to decay particles✤ Widths and branching ratios are not independent quantities➢ need to be calculated✤ Some Monte Carlo tools compute these quantities on the y➢ the procedure is repeated each time it is needed✤ FEYNRULES offers a way to include analytical information on the two-body decay

New physics made easy with FEYNRULES
fl

Benjamin Fuks - 01.03.2023 - 80

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Two-body decays✦ Two-body decays in general✤ Two-body decays can be directly read from three-point vertices ( V )p
Z(M 2 , m21 , m22 ) a1 a2 a3 `1 `01 `2 `02 `3 `03 ⇤ a1 a2 a3
1
2
d N |M1!2 | =V`1 `2 `3 P1 P2 P3 (V )`0 `0 `0
1!2 =
3
1 2 3
2|M |S
16 ⇡ S |M |★ Partial width for the decay of a particle of mass M to two particles of masses m1 and m2★ Includes a symmetry factor S and P denotes the polarization tensor of each particle✦ The decay module of FEYNRULES✤ FEYNRULES makes use of MATHEMATICA to compute all partial widths of the model★ Ignores open and closed channels ➢ benchmark independent★ The information is exported to the UFO (used, e.g, by MADWIDTH)

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 81

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Running the decay module of FEYNRULES✦Automatic decay width computations✤ All two-body decay widths can be easily computed from the Lagrangian✤ Many functions available for analytical calculations➢ PartialWidth, TotWidth, BranchingRatio✤ The numerical value provided for the particle widths can be updated accordingly➢ UpdateWidths✤ See: Alwall, Duhr, BF, Mattelaer, Öztürk, Shen [ CPC (2015) ]New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 82

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Snippet of the UFO output✦The information is (by default) employed by the UFO interface✤ Can be turned of: (AddDecays → False)✤ The UFO contains an extra le decays.py✤ This le can be used by MC codes✤ Example of the Standard Model UFO: the top quarkfifi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 83

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

NLO

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 84

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Higher-order corrections (in QCD)✦ NLO calculations matched to parton shower (for BSM) are automated✤ Model-dependent parts of calculations (on top of the tree-level information)★ Counterterms★ Finite pieces of the loop-integrals

UFO @ NLO
[ Degrande, Duhr, BF, Hirschi, Mattelaer, Shao et al. (in prep.) ]✤ Model independent contributions★ Subtraction of the divergences★ Matching to the parton showers

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 85

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Recap’ on NLO calculations✦ Contributions to an NLO result in QCD✤ Three ingredients: the Born, virtual loop and real emission contributions
N LO =

Z

d4

nB

Born

+

Z

d4

n

Z

loop

dd ` V

Virtuals: one extra powerof αs and divergent

+

Z

d4

n+1 R

Reals: one extra powerof αs and divergent

Extra informationis needed

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 86

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Virtual contributions✦ Loop diagram calculations✤ Calculations to be done in d=4-2 dimensions

m-point diagram with n external momenta
k1★ Divergences made explicit (1/ 2, 1/ )k2

kn

✤ Rewriting loop integrals with scalar integrals
Z

X
N (`)
d
d `
=
ai
D0 D1 · · · Dm 1

Z

1
d `
D i 0 D i1 · · ·d★ Involves integrals with up to four denominators➢ The decomposition basis is nite
➢ Can be computed once and for all

l + k1 = l + p1

l

D0

k3

D1
D2 l + k1 + k2 + k3 = l + p2

Dm 1

k4

D3
l + k1 + . . . + k6 = l + p3

k5
k6

★ The reduction is the process-dependent part𝞮Benjamin Fuks - 01.03.2023 - 87
𝞮

fi

𝞮

New physics made easy with FEYNRULESFEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

The rational terms (R1 and R2)✦ The loop momentum lives in a d-dimensional space✤ Reduction to be done in d dimensionsZ˜N (`, `)
with `¯ = ` + `˜d `D̄0 D̄1 · · · D̄m 1
d

D-dim

4-dim

(-2 )-dim✤ Numerical methods works in 4 dimensions: need to be compensated!

✦ The R1 terms originates from the denominators✤ Connected to the internal propagators✦ The R2 terms originates from the numerator✤ Can be seen as extra diagrams with special Feynman rules𝞮

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 88

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

R1 terms✦ The R1 terms originates from the denominators
✓

1
1
=
1
D
D̄

◆
2
˜
`
D̄

✤ These extra pieces can be calculated generically (3 integrals in total)

2
2
˜
i⇡
pi pj ) 2
d¯ `
2
2
d `
=
mi + mj+ O(")
2
2
D̄i D̄j
Z
`˜2
i⇡ 2
d¯
d `
=
+ O(")
2
D̄i D̄j D̄k
Z
`˜2
i⇡ 2
d¯
d `
=
+ O(")
6
D̄i D̄j D̄k D̄lZ✤ The denominator structure is already known at the reduction time
✤ The R1 coef cients are extracted during the reductionfi

New physics made easy with FEYNRULES

Benjamin Fuks - 01.03.2023 - 89

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

R2 terms✦ The R2 terms originates from the numerator¯ = N (`) + Ñ (`,˜ `, ")N̄ (`)
D-dim(-2 )-dim

4-dim

1R2 ⌘ lim"!0 (2⇡)4Z

˜ `, ")Ñ (`,d `D̄0 D̄1 · · · D̄m 1
d¯✤ Practically, we isolate the epsilon part
✤ There is only a nite set of loops for which it does not vanish✦ They can be re-expressed in terms of R2 Feynman rules

∝

fi

𝞮

New physics made easy with FEYNRULES

Z

`˜2
d `
=
D̄i D̄j D̄kd¯i⇡ 2+ O(")
2

↵ µ
i e
2⇡

Benjamin Fuks - 01.03.2023 - 90

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

R2 Feynman rules✦ The R2 are process dependent and model-dependent (like Feynman rules)✤ In a renormalizable theory, there is a nite number of them
✤ They can be derived from the sole knowledge of the bare Lagrangian[ Ossala, Papadopoulos, Pittau (JHEP’08) ]✦ The R2 calculation can be automated and performed once and for all✤ Development of the NLOCT package (extension of FEYNRULES)✤ Computation, for any model, of all R2 and UV counterterms★ In the on-shell and MSbar schemes✤ Inclusion of the output in the UFO

New physics made easy with FEYNRULES[ Degrande (CPC’15) ]fi

Benjamin Fuks - 01.03.2023 - 91

FEYNRULES in a nutshell

Model implementation

Using FEYNRULES

Summary

Advanced techniques

Automated NLO simulations with MG5_AMC✦ A comprehensive approach to Monte Carlo simulations at the NLO in QCD
UFO

Idea / Lagrangian
FEYNRULES

NLOCT

New physics made easy with FEYNRULESMADGRAPH5

aMC@NLO

ShowerHadronization

Detector
Reconstruction

Simulatedcollisions

Event
analysisAnalysis codes

Automatic
matching

Benjamin Fuks - 01.03.2023 - 92