FeynRules Tutorial
Celine Degrande (UIUC)
Olivier Mattelaer (UIUC)
TASI
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
© C. Degrande
C. Degrande
Output : vertices
cp3
The new FeynRules interface
Celine Degrande
The new FeynRules interface to FeynArts
Conclusion
For each tool, the right input
FeynRules outputs
FeynRules
outputs can be
used directly by
event generators
UFO : output with
the full information
used by several
generators
C. Degrande
cp3
The new FeynRules interface
Celine Degrande
of this tutorial is to implement a simple extension1 of the SM into FeynRules In the followin
ules
Tutorial of how to implement the model into FeynRules and how to obtain the corre
nstructions
es for MadGraph 5.
The
model
:
SM
extension
1. Introduction
+
(SM
Fields:
+1)
Z
SU
(3)
×
SU
(2)
×
U
(1)
2
Y
FeynRules Tutorial
2. The
model
he
model
φ
∼
(1,
0)
m of this tutorial is to implement
a 1,
simple
extension1 of the SM into FeynRules In the following
1
We
real
scalar
and φare
areunder
singlets
all SM
gaugeTheir
groups
wo add
real two
scalar
fields,
φ fields,
and φφ2 .1 They
singlets
all under
SM gauge
groups.
m
2 . They
d instructions of how 1to1implement
the model
into FeynRules and how to obtain the corresp
Their
mass
terms are
: 2 ∼ (1, 1, 0)
φ
files for
MadGraph
5.
2
2
e
m
m
1 2
2 2
µ
µ
2
−1
k
i
Lkin,scalar
=
1/2∂
φ
∂
φ
+
1/2∂
φ
∂
φ
−
φ
−
φ
−
m
2
2
µ 1
1
µ 2
2 m
l
1 m
2
12 φ1 φ2 .
1
1
U
∼
(3,
1,
2/3)
2
µ
µ
1 22
2 22
r
(2.1)
L
=
∂
φ
∂
φ
+
∂
φ
∂
φ
−
φ
−
φ
−
m
o
µ 1
1
µ 2
2
kin,scalar
12 φ1 φ2 .
1
2
t
2 and Φ2 , and2their masses M
2 1 and M
2 2 , respectively, and we wi
c
The
model
call mass
eigenstates
Φ
1
e
E ∼ (1, 1, −1)
V
}
. will call mass eigenstates Φ1 and Φ2 , and their masses M1 and M2 , respectively, and
We
d two
real
scalar
fields,fields,
φ1 and
φand
are singlets
under all
SM gauge
groups.
Their
mas
2 . They
dd
two
Dirac
fermion
U
E.
Their
SM
quantum
numbers
are
those
of
the
SM
u
we will assume M1 < M2 .
ely. These fields have mass terms
2
2
m
m
We add
two
Dirac
fermion
fields,
U
and
E.
Their
SM
quantum
numbers are those o
Lkin,scalar = 1/2∂µ φ1 ∂ µ φ1 + 1/2∂µ φ2 ∂ µ φ2 − 1 φ21 − 2 φ22 − m212 φ1 φ2 .
2 terms
2
he SM uR and eR , respectively.
These fields
mass
Ldirac,mass
= Mhave
U
U
+
M
EE
U
E
l call mass eigenstates Φ1 and Φ2 , and their masses M1 and M2 , respectively, and we will
M
2.
Ldirac,mass = MU U U + ME EE
(2.2)
eract
with scalars via
add two Dirac fermion fields, U and E. Their SM quantum numbers are those of the SM uR
!
!
ively.
These
fields
have
mass
terms
L
=
λ
φ
U
P
t
+
λ
φ
U
P
t
+
λ
φ
EP
e
+
λ
They interact
scalars
via
F Fwith
S
1 1
R
2 2
R
R
1 1
2 φ2 EPR e + H.c.,
Ldirac,mass = MU !U U + ME EE !
nd eLare
the SM top-quark and electron fields. Note that there is a Z symmetry under
(2.3)
F F S = λ1,i φ1 U PR ui + λ2,i φ2 U PR ui + λ1,i φ1 EPR li + λ2 φ2 EPR2li + h.c. ,
Celine
Degrande
added (φ
U, E) flip
must
be pairnteract
with
via sign, while all SM fields do not, so the new particles
1,2 ,scalars
Step 0
• Download FeynRules (beta 1.8) from
https://cp3.irmp.ucl.ac.be/projects/
madgraph/wiki/TASISchool13
• Copy the SM directory in feynrules/
models and rename it Tutorial
• Create a model file Tutorial.fr (text
file)
Celine Degrande
Step 1 : model information
M$ModelName = "Tutorial";
M$Information = {Authors -> {"C. Duhr"},
Version -> "1.0",
Date -> "27. 02. 2012",
Institutions -> {"ETH Zurich"},
Emails -> {"duhrc@itp.phys.ethz.ch"}
};
Celine Degrande
Step 2 : parameters
• 9 new external parameters : m1, m2,
m12, MU, ME, λ1, λ2, λ’1, λ’2
See Step 3
M$Parameters = {
...
MM1 == {
ParameterType -> External,
Value -> 200},
...
};
Celine Degrande
Step 2 : parameters
• 9 new external parameters : m1, m2,
m12, MU, ME, λ1, λ2, λ’1, λ’2
See Step 3
M$Parameters = {
...
MM1 == {
ParameterType -> External,
Value -> 200},
...
};
InteractionOrder ->{NP, 1},
Celine Degrande
physical masses, we need to diagonalize the mass matrix
!
"
m21 m212
.
2
2
m12 m2
Step 2 : parameters
e denote•
the3 eigenvalues
MPe1 and MPe2.
internal by
parameters
: MIn1,addition,
M2, ϑ we need to
g the fields φi to the mass eigenstates Φi by,
!
" !
"!
"
φ1
− sin θ cos θ
Φ1
=
.
φ2
cos θ sin θ
Φ2
mass matrix is only two-dimensional, we can compute the eigenvalu
nd simplyInteraction
implement the analytical formulas into
FeynRules. The imp
Mass
nes as for eigenstates
the masses m1 , m2 , m12 , with the
only
differences
that
eigenstates
rType is Internal (as these parameters are dependent on the externa
is Internal
given by •
anParameterType
analytical expression (in
Mathematica syntax).
the implementation of the new coupling constants, which we will call
Value is a Mathematica expression
•
l external parameters, and thus the implementation follows exactly th
Degrande
the mass parameters, with only one modification: someCeline
matrix
eleme
ynRules Tutorial
Step 3 : fields
U
uv
E
ev
φ1
pi1
φ2
pi2
Φ1
p1
Φ2
p2
Table 1: Symbols used for the fields in the FeynRules implementation.
M$ClassesDescription = {
...
ClassesDescription
= { {
F[100] ==
...
-> uv,
F[100]ClassName
== {
ClassName
-> uv,
SelfConjugate
-> False,
SelfConjugate -> False,
Indices ->->{Index[Colour]},
{Index[Colour]},
Indices
QuantumNumbers
-> {Y -> 2/3, ->
Q -> {Y
2/3},-> 2/3, Q -> 2/3},
QuantumNumbers
Mass
-> {Muv, 500},
Mass -> ->{Muv,
Width
{Wuv,1} 500},
},
Width
->
{Wuv,1}
...
},
...
e meaning
of this definition is as follows: each particle class has a name of the form X[i], where X
}
Celine Degrande
ated to the spin of the field (See Tab. 2), and i is an integer that labels the classes. Note that i can
ynRules Tutorial
Step 3 : fields
U
uv
E
ev
φ1
pi1
φ2
pi2
Φ1
p1
Φ2
p2
Table 1: Symbols used for the fields in the FeynRules implementation.
M$ClassesDescription = {
...
ClassesDescription
= { {
F[100] ==
...
-> uv,
F[100]ClassName
== {
ClassName
-> uv,
SelfConjugate
-> False,
SelfConjugate -> False,
Indices ->->{Index[Colour]},
{Index[Colour]},
Indices
QuantumNumbers
-> {Y -> 2/3, ->
Q -> {Y
2/3},-> 2/3, Q -> 2/3},
QuantumNumbers
Mass
-> {Muv, 500},
Mass -> ->{Muv,
{MPe1, Internal}
Width
{Wuv,1} 500},
},
Width
->
{Wuv,1}
...
},
...
e meaning
of this definition is as follows: each particle class has a name of the form X[i], where X
}
Celine Degrande
ated to the spin of the field (See Tab. 2), and i is an integer that labels the classes. Note that i can
ynRules Tutorial
Step 3 : fields
U
uv
E
ev
φ1
pi1
φ2
pi2
Φ1
p1
Φ2
p2
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
-> {Index[Colour]},
Unphysical
->
True,
QuantumNumbers -> {Y -> 2/3, Q -> 2/3},
Mass
-> {Muv,
500},
Definitions
-> {pi1
-> - Sin[th] p1 +
Width
-> {Wuv,1}
Cos[th]
p2}
},
},...
e meaning of this definition is as follows: each particle class has a name of the form X[i], where X
Celine Degrande
ated to the spin of the field (See Tab. 2), and i is an integer that labels the classes. Note that i can
Step 4 : Lagrangian
$FeynRulesPath =
SetDirectory["~TASI/feynrules"];
<< FeynRules`
SetDirectory[ $FeynRulesPath <> "/Models/
Tutorial"]
FR$Parallel = False;
LoadModel["SM.fr", "Tutorial.fr"]
LoadRestriction["DiagonalCKM.rst",
"Massless.rst"]
Celine Degrande
r the new scalars can easily be implemented by using the symbols for t
meters defined in the model file, as well as the symbol for the space-t
.., mu]. As an example, we have
Step 4 : Lagrangian
1
1 2 2
µ
∂µ φ1 ∂ φ1 − m1 φ1
2
2
1/2 del[pi1, mu] del[pi1, mu] - 1/2 MM1^2 pi1^2
1/2 del[pi1, mu]del[pi1, mu] - 1/2 MM1^2 pi1^2
or the fermions can be implemented in a similar way. However,
SM gauge group, we have to use the covariant derivative DC rather
hermore, we have to use a “.” instead of an ordinary multiplication
ure of the fermions into account. As an example, we have
i Ūγ µ Dµ U − MU ŪU
I uvbar.Ga[mu].DC[uv, mu] - Muv uvbar.uv
e FeynRules symbol for the Dirac matrix γ µ . Finally, the Yukawa
same way as the kinetic terms for the fermions, e.g.,
λ1 φ1 Ū P+ u
Celine Degrande
meters defined in the model file, as well as the symbol for the space-ti
r the new
can easilywe
be have
implemented by using the symbols for t
...,
mu]. scalars
As an example,
meters defined in the model
file,
as
well
as
the
symbol
for
the
space-t
1
1 2 2
µ
∂µ φhave
.., mu]. As an example, we
1 ∂ φ1 − m1 φ1
2
2
1/2 del[pi1, mu]
mu]
- 1/2 MM1^2 pi1^2
1 del[pi1,
1
µ
2 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 have
to be
useimplemented
a “.” instead in
of an
ordinary
multiplication
or the fermions
can
a similar
way.
However, i
ature
of thegroup,
fermions
As covariant
an example,
we have DC rather
SM gauge
we into
haveaccount.
to use the
derivative
Step 4 : Lagrangian
µ 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.uv
I uvbar.Ga[mu].DC[uv,
mu]
Muv
uvbar.uv
µ
i
Ūγ
MU ŪU
µU −
he FeynRules symbol for the D
Dirac
matrix
γ µ . Finally, the Yukawa
mu]
- Muv e.g.,
uvbar.uv
e same wayIasuvbar.Ga[mu].DC[uv,
the kinetic terms for the
fermions,
µ
e FeynRules symbol for the λ
Dirac
matrix
γ
. Finally, the Yukawa
1 φ1 Ū P+ u
same way as the kinetic
terms
the fermions, e.g.,
lam1
pi1 for
uvbar.ProjP.u
Ū P+ProjP
u
λ1 φ1 and
he u quark field defined in SM.fr
denotes the
right
chiral
Celine
Degrande
meters defined in the model file, as well as the symbol for the space-ti
r the new
can easilywe
be have
implemented by using the symbols for t
...,
mu]. scalars
As an example,
meters defined in the model
file,
as
well
as
the
symbol
for
the
space-t
1
1 2 2
µ
∂µ φhave
.., mu]. As an example, we
1 ∂ φ1 − m1 φ1
2
2
1/2 del[pi1, mu]
mu]
- 1/2 MM1^2 pi1^2
1 del[pi1,
1
µ
2 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 have
to be
useimplemented
a “.” instead in
of an
ordinary
multiplication
or the fermions
can
a similar
way.
However, i
ature
of thegroup,
fermions
As covariant
an example,
we have DC rather
SM gauge
we into
haveaccount.
to use the
derivative
Step 4 : Lagrangian
µ 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.uv
µ
I
uvbar.Ga[mu].DC[uv,
mu]
Muv
uvbar.uv
ynRules symbol for the
Dirac
matrix
γ
.
Finally,
the
Yuk
µ
µ
MU ŪU
µU −
he FeynRules symbol fori Ūγ
the D
Dirac
matrix
γ . Finally, the Yukawa
ee way
as
the
kinetic
terms
for
the
fermions,
e.g.,
I
uvbar.Ga[mu].DC[uv,
mu]
Muv
uvbar.uv
same way as the kinetic terms for the fermions, e.g.,
λ
φ
Ū
P
t
µ
1
1
+
e FeynRules symbol for the λ
Dirac
matrix
γ
. Finally, the Yukawa
1 φ1 Ū P+ u
lam1
pi1
uvbar.ProjP.t
same way as the
kinetic
terms
for
the
fermions, e.g.,
lam1
pi1
uvbar.ProjP.u
Lint:=lam1 pi1
uvbar.ProjP.t
HC[Lint]
uark
field
defined
in
SM.fr
ProjPdenotes
denotes
the
right
Ū P+ProjP
u
λ1 φ1and
he u quark field defined in SM.fr
and
the
right
chiral
Celine
Degrande
Step 5 : run FeynRules
vertices = FeynmanRules[ LNew ];
CheckMassSpectrum[ LNew ]
ComputeWidths[vertices];
PartialWidth[ {uv, t, p1} ]
TotWidth[ {uv, t, p1} ]
BranchingRatio[ {uv, t, p1}]
SetDirectory[“~/TASI/madgraph5/models”];
WriteUFO[ LSM + LNew ];
Celine Degrande