FeynRules Tutorial
Celine Degrande (IPPP, Durham)
Olivier Mattelaer (IPPP, Durham)
Lund 2014

an rules to various matrix element
Introduction
From FeynRules to FeynArts so far
The new FeynRules interface to FeynArts
Conclusion

FeynRules
Welcome in the FeynRules era

with current public version

Input : model.fr
mpHep

mCalcega© C. Degrande
C. Degrande

Output : vertices

cp3

The new FeynRules interface

Celine Degrandeules to various matrix element
The new FeynRules interface to FeynArts
Conclusion

For each tool, the right inputFeynRules outputs

h current public version:

epalc

FeynRules
outputs can beused directly byevent generators

UFO : output withthe full information
used by severalgenerators
C. Degrande

cp3

The new FeynRules interface

Celine DegrandeV Comparison with pseudo-experimental data (detector level).

The model : SM extension

ules
TutorialThesimulation tools that will be used in the tutorial are: FeynRules (model implemeon), MadGraph5 aMC@NLO (matrix element generation at LO and at NLO), Pythia/H
1.arton-shower/hadronization),
Introduction
Delphes (detector simulation). The pheno study is done inadAnalysis
5 framework.
+Z2 (SM Fields: +1)
SU (3)SU (2) U (1)Y

2. The model

FeynRules Tutorial

Let us consider the SM Lagrangian start by adding
two real scalar fields, 1 and 2 . T
1m of this tutorial is to implementa 1,
simpleextension of the SM into
FeynRules In the following
⇥
(1,0)
11
e singlets
underallSM gaugegroups.
Theirmassare
terms
are : under all SM gauge groups
Weadd
tworealscalar
fields,⇥and⇥.
Theysinglets
12d instructions of how to implement the model into FeynRules and how to obtain the corres
1:Theirmassterms
are⇥(1,1,0)22
files for MadGraph
5.
2
11mme
µµ
2
1 21
2 2
Lkin,scalar
=@@+
@@m.
kµ
1
1µ2
2
12
1212i
22
l12(3, 1,µ 2/3)12
m21 2 m22 2U
⇥2µr
(2.1L
=⌃⇥⌃
⇥
+
⌃⇥⌃⇥⇥⇥moµ 11µ 22kin,scalar
12 ⇥1 ⇥2 .
12t2
222Thec mass eigenstates
e will modelcalland,andtheirmasses
MandM2 , respectively, and we121}

eE ⇥ (1, 1, 1)
Vsume
M2 . eigenstates
1 < mass
WewillMcalland, and their masses M and M , respectively, and1d two real scalar fields, φ1 and φ12 . They2are singlets under all SM
gauge2 groups. Their mas
two Diracfields, U and E. Their SM quantum numbers are those of
weWewilladd
assumeM1 < fermion
M2 .
22M uR, respectively.
Thesefields
havemasstermsmm
WeandaddeRtwo
Dirac fermion
fields,UandE.
TheirSMquantum
are those o2 2 numbers
1 22µµLkin,scalar = 1/2∂µ φ1 ∂ φ1 + 1/2∂µ φ2 ∂ φ2 −φ1 −φ2 − m12 φ1 φ2 .
2 terms 2
he SM uR and eR , respectively.
Thesefields
have
mass
Ldirac,mass = MU U U + ME EE
l call mass eigenstates Φ1 and Φ2 , and their masses M1 and M2 , respectively,
and we will+ Kinetic/M
2 . interact with scalars via Ldirac,mass = MU U U + ME EE
heygauge term (2.2
add two Dirac fermion fields, U and E. Their SM quantum numbers are those of the SM uR
00ively. interact
TheseLfields
havemassterms
Theywithscalars
via=UPu+UPu+EPl+2 2
1 1RRRFFS
1 12 2 EPR l + h.c. ,Ldirac,mass = MU U U + ME EE
L=
⇥U
Pu+
U PR uiand+ charged
h.c. ,there(2.3here uFi Fandli 1,iare 1the SM
is a2,i ⇥2 quarkS
R i up-type
R li + fields.
R li +that
1,i ⇥1 EPlepton
2 ⇥2 EPNoteCeline Degrande
mmetrywith
underwhich, U, E) flip sign, while all SM fields do not, sonteract
scalars
viaall fields we added (• Download FeynRules 2.0 from https://feynrules.irmp.ucl.ac.be• Copy the SM directory in feynrules/
models and rename it Tutorial• Create a model le Tutorial.fr (textle)

Celine Degrandefifi

Step 0Step 1 : model information
M$ModelName = "Tutorial";
M$Information = {Authors -> {"C.
Degrande"},Version -> "1.0",
Date -> "21. 07. 2014",
Institutions -> {“IPPP Durham"},
Emails ->{“celine.degrande@durham.ac.uk“}};

Celine DegrandeStep 2 : parameters• 9 new external parameters : m1, m2,m12, MU, ME, λ1, λ2, λ’1, λ’2
See Step 3
M$Parameters = {...
MM1 == {ParameterType -> External,Value -> 200},...
};
InteractionOrder ->{NP, 1},
Celine Degrandephysical masses, we need to diagonalize the mass matrix!
"m21 m212.
22m12 m2

Step 2 : parameterse denote•the3eigenvalues
MPe1 and MPe2.
internal byparameters: MIn1,addition,M2, ϑ we need tog the fields φi to the mass eigenstates Φi by,!
" !
"!
"φ1− sin θ cos θΦ1=.
φ2cos θ sin θΦ2mass matrix is only two-dimensional, we can compute the eigenvalu
nd simplyInteraction
implement the analytical formulas intoFeynRules. The imp
Mass
nes as for eigenstates
the masses m1 , m2 , m12 , with theonly
diﬀerencesthat
eigenstatesrType is Internal (as these parameters are dependent on the externais Internal
given by •anParameterType
analytical expression (inMathematica syntax).

the implementationofthenewcoupling
constants,
which
wewillcall
Value is a Mathematica expression•l external parameters, and thus the implementation follows exactly thDegrande
the mass parameters, with only one modification: someCelinematrixeleme

Step 3 : elds

ynRules TutorialUuv

Eev

φ1pi1

φ2pi2

Φ1p1

Φ2p2

Table 1: Symbols used for the fields in the FeynRules implementation.

fifi

M$ClassesDescription = {...
Id#:Unique(CheckinSM.fr)
F[100] ==ClassesDescription= { {...
ClassName-> uv,
F[100]=={
S SelfConjugateClassName-> uv, -> False, De ned in SM.fr
SelfConjugate -> False,Indices->-> {Index[Colour]},{Index[Colour]},
IndicesQuantumNumbers -> {Y -> 2/3,-> 2/3},
QuantumNumbers-> Q {Y-> 2/3, Q -> 2/3},
Mass-> {Muv, 500},
Mass500},{MPe1, Internal}
Width -> {Muv,-> {Wuv,1}},
Width-> {Wuv,1}
...
},
...
e meaningof this definition is as follows: each particle class has a name of the Celine
form X[i],where X}
DegrandeStep 3 : elds

ynRules Tutorial

Uuv

Eev

φ1
pi1

φ2pi2

Φ1p1

Φ2p2

Table 1: Symbols used for the fields in the FeynRules implementation.

S[100] == {ClassesDescription = {ClassName-> pi1,...
F[100] == {SelfConjugate-> True,ClassName-> uv,
Indices-> ->{},
SelfConjugate
False,
Indices->
Unphysical->{Index[Colour]},
True,
QuantumNumbers -> {Y -> 2/3, Q -> 2/3},
Definitions
Mass-> ->{Muv,{pi1500}, -> - Sin[th] p1 +Width
Cos[th]p2}-> {Wuv,1}},},...

fie meaning of this definition is as follows: each particle class has a name of the Celineform X[i],where X
DegrandeStep 4 : Lagrangian$FeynRulesPath =SetDirectory["~/feynrules"];<< FeynRules`
SetDirectory[ $FeynRulesPath <> "/Models/Tutorial"]
LoadModel["SM.fr", "Tutorial.fr"]
LoadRestriction["DiagonalCKM.rst","Massless.rst"]

Celine Degrandemeters defined in the model file, as well as the symbol for the space-tir the newcan easilywe
be have
implemented by using the symbols for t...,mu]. scalars
As an example,
meters defined in the modelfile,aswellasthesymbolforthespace-t11 2 2µ∂µ φhave., mu]. As an example, we1 ∂ φ1 − m1 φ12
2
1/2 del[pi1, mu]mu]- 1/2 MM1^2 pi1^2
1 del[pi1,1µ2 2∂µ φ1 ∂ φ1 − m1 φ1
for the fermions can be2 implemented2in a similar way. However, a
1/2
del[pi1,mu]
del[pi1,mu]
1/2
MM1^2
pi1^2
SM gauge group, we have to use the covariant derivative DC rather t
1/2
del[pi1,mu]del[pi1,mu]
1/2
MM1^2
pi1^2
rthermore,
we haveto beuseimplementeda “.” instead inof anordinaryway.
multiplicationor the fermionscana similar
However, iaturegaugeof thegroup,
fermions
As covariant
an example,
we have DC ratherSM
we intohaveaccount.
to use thederivative

Step 4 : Lagrangianµ instead of an ordinary multiplication
hermore, we have to useiaŪ“.”γ Dµ U − MU Ū U
ure of the fermions
into account. As an
we have
I uvbar.Ga[mu].DC[uv,mu]example,- Muv uvbar.uvµI
uvbar.Ga[mu].DC[uv,mu]Muv
uvbar.uv
nRules symbol for the Diracmatrix.
Finally,theYuµDµ U −matrix
MU ŪUhe FeynRules symbol fori Ūγ
the Diracγ µ . Finally, the Yukawa
wayas
thekineticterms
forthefermions,e.g.,
Fermions
antimu]- Muv e.g.,
uvbar.uv
e same wayI asuvbar.Ga[mu].DC[uv,the kinetic terms for thefermions,commute⇥⇤ŪPtµ De ned in SM.fr
11+ŪPu
φe FeynRules symbol for the λDiracmatrix
γ. Finally, the Yukawa
1 1+
lam1
pi1
uvbar.ProjP.t
lam1
pi1pi1
uvbar.ProjP.usame way as the
kineticterms
forthefermions, e.g.,
Lint:=lam1
uvbar.ProjP.tfiHC[Lint]
heu quarkdefinedSM.fr
andtherightchiraluark
field fielddefinedin inSM.fr
denotes
theright
CelineDegrandeŪ P+ProjP
uProjPdenotes
λ1 φ1 andStep 5 : run FeynRules
vertices = FeynmanRules[ LNew ];CheckMassSpectrum[ LNew ]ComputeWidths[vertices];
PartialWidth[ {uv, t, p1} ]TotWidth[ uv ]BranchingRatio[ {uv, t, p1}]
SetDirectory[“~/mg5amcnlo/models”];
WriteUFO[ LSM + LNew ];Celine Degrande