#### Rosetta: an operator basis translator for standard model effective field theory

Eur. Phys. J. C
ROSETTA: an operator basis translator for standard model effective field theory
Adam Falkowski 2
Benjamin Fuks 1
Kentarou Mawatari 0
Ken Mimasu 4
Francesco Riva 3
Verónica Sanz 4
0 Theoretische Natuurkunde and IIHE/ELEM, Vrije Universiteit Brussel, and International Solvay Institutes , Pleinlaan 2, 1050 Brussels , Belgium
1 Département Recherches Subatomiques, Institut Pluridisciplinaire Hubert Curien, Université de Strasbourg/CNRS-IN2 P3, 23 rue du Loess, 67037 Strasbourg , France
2 Laboratoire de Physique Théorique , Bat. 210 , Université Paris-Sud , 91405 Orsay , France
3 CERN, Theory Division , 1211 Geneva , Switzerland
4 Department of Physics and Astronomy, University of Sussex , Brighton BN1 9QH , UK
We introduce Rosetta, a program allowing for the translation between different bases of effective field theory operators. We present the main functions of the program and provide an example of usage. One of the Lagrangians which Rosetta can translate into has been implemented into FeynRules, which allows Rosetta to be interfaced into various high-energy physics programs such as Monte Carlo event generators. In addition to popular bases choices, such as the Warsaw and Strongly Interacting Light Higgs bases already implemented in the program, we also detail how to add new operator bases into the Rosetta package. In this way, phenomenological studies using an effective field theory framework can be straightforwardly performed.
1 Introduction
The start of a second LHC experimental era raises new hopes
to detect physics beyond the standard model (BSM). The
high energy of the experiment increases the chances of a
direct discovery of new physics resonances, while a
combination of high energy and high luminosity favors the
possible observation of new phenomena via standard model (SM)
precision tests. Interestingly the latter offers a
complementary and model-independent tool for BSM searches if the
results are interpreted in the context of an effective field
theory (EFT). The EFT indeed captures in a general way the
low-energy effects of heavy new physics from a bottom-up
perspective. More precisely, it systematically organizes
possible departures from the SM as an expansion in the energy
at which the processes of interest occur over the (high) new
physics scale, and simultaneously provides a dictionary to
interpret these departures in the context of explicit BSM
models.
Given the SM field content (including a single Higgs
doublet), assuming baryon and lepton number conservation,
flavor universality and a linear realization of the electroweak
symmetry, the leading effects implied by an EFT description
consist of dimension-six operators that are supplemented to
the SM Lagrangian. At this order, 59 (76 real) new
independent coefficients [
1,2
]1 capture all possible deformations
from the SM. Despite this large number of new free
parameters, important classes of observables (e.g., Higgs production
and decay or Z -pole observables) depend on a much smaller
subset of parameters [
4–9
]. Owing to that, the EFT approach
is not only useful for parameterizing BSM searches but is also
testable per se by looking at correlations among the expected
signatures.
Another important aspect of the EFT approach is the
choice of the operator basis, so that a given physical effect
could be modeled by different combinations of operators at
a fixed order in the EFT expansion. This well-known fact is
related to the possibility of redefining the SM fields in such
a way that the zeroth order Lagrangian in the EFT expansion
(i.e., the SM Lagrangian) is unaltered, while combinations of
the first-order operators (i.e., dimension-six operators)
proportional to the SM equations of motion can be eliminated
up to subleading higher-dimensional effects. For this reason,
different complete and non-redundant operator bases have
been proposed in the literature, sharing the same physical
predictions but having different advantages. The most
popular choices include the so-called Warsaw basis [
2
], SILH
(strongly interacting light Higgs) basis [
10,11
] and BSM
pri1 Relaxing flavor universality, the number of independent
dimensionsix operators grows to 2499 [3].
maries basis [
6, 12, 13
]. The Warsaw basis represents the first
set of non-redundant operators that has been proposed and is
particularly appropriate for comparisons with BSM theories
that modify the interactions of the SM fermions. In contrast,
the SILH basis has been designed to capture the effects of
universal theories where new physics mostly couples to the
SM bosons. Finally, the BSM primaries basis is more
suitable for a bottom-up approach since it is formulated in terms
of mass-eigenstates and has a more transparent connection to
measurable quantities, its operators being aligned with
physical observables.
Given these multiple viewpoints, it is cumbersome to
express the experimental results in a basis-independent
manner that can be readily interpreted in any of the
abovementioned frameworks. On the other hand, different bases
may be convenient for particular applications, either because
they facilitate the comparison with a given class of BSM
theories or simply because different experimental analyses look
more transparent in a specific basis. For instance, the
Warsaw basis contains an apparent blind direction with respect
to the electroweak precision tests [
6, 14
], which introduces
large theoretical correlations among all LEP constraints. As a
result, the bounds on the strength of the dimension-six
interactions appear less transparent [15]. The SILH basis has a
similar drawback yielding a correlation between LEP2 and
LHC constraints, while the downside of the BSM primaries
basis lies in the comparison with explicit BSM models that
is complicated. The Rosetta package that we present in this
paper has been designed to explicitly solve such problems by
allowing for a straightforward translation between different
EFT languages.
In addition to translating, another important goal of the
Rosetta program is to provide a platform for
communication with Monte Carlo event generators, no matter which
EFT basis is chosen. To achieve this, we have implemented
in Rosetta the Higgs basis for EFT operators that has been
recently designed by the LHC Higgs Cross Section
working group (LHCHXSWG) [
16
]. This proposal, built on the
BSM primaries basis (see Ref. [
13
]), combines two
ingredients. First, all possible operators of dimension up to six are
written in terms of the SM mass-eigenstates
cdii Oi Gaμ, Wμ±, Zμ, Aμ, h, t , b, ντ , τ, . . . , (1)
L
=
(mass)
i
Basis
Warsaw, SILH
BSM primaries, Higgs
Higgs/BSM characterisation
Underlying gauge symmetry
SU (3)C × SU (2)L × U (1)Y
SU (3)C × SU (2)L × U (1)Y
SU (3)C × U (1)E M
Fields used in the
Lagrangian
Gauge-eigenstates
Mass-eigenstates
Mass-eigenstates
where the operators Oi have a mass dimension ranging from
two to six. The dimensionless coefficients ci are then
suppressed by an appropriate power di of the high-energy scale
, with di = −2, . . . , 2. We refer the ensemble of operators
included in the resulting Lagrangian, which is in spirit very
similar to the Higgs characterisation Lagrangian of Ref. [
17
],
as the BSM characterisation (BSMC) Lagrangian. Due to
the lack of manifest SU (2)L × U (1)Y invariance, the BSMC
Lagrangian is associated with a larger number of
independent coefficients compared to the Warsaw, SILH or BSM
primaries bases. For this reason, the second ingredient
defining the Higgs basis consists of relations among the ci
coefficients that restore the full SU (3)C × SU (2)L × U (1)Y
symmetry. As summarized in Table 1, the BSM primaries and
Higgs Lagrangians are both of the form of L(mass), but they
additionally include constraints among the different Wilson
coefficients that render the Lagrangian invariant under the
electroweak symmetry. In contrast, the Warsaw and SILH
basis Lagrangians are directly written in terms of the SM
gauge-eigenstates,
L
(gauge)
=
i
ci
2 Oi
Gaμ, W μi, Bμ, , Q L , u R , dR , L L , eR ,
(2)
and are manifestly symmetric under the electroweak
symmetry group.
We have implemented the mass basis Lagrangian L(mass)
into FeynRules [
18
]2 and tuned the output format of
Rosetta so that the translation maps an EFT Lagrangian given
in a specific basis to L(mass) and generates an output file
that is compatible with the FeynRules implementation. As
a consequence, any high-energy physics tool (and in
particular any Monte Carlo event generator) that is interfaced to
FeynRules can be employed within the context of any EFT
basis of operators that is included in Rosetta.
With the advent of automated next-to-leading order (NLO)
accurate Monte Carlo event generation software, it is
important that Rosetta remains flexible enough to eventually
provide compatibility with this new generation of tools. Recent
progress has been made on the theory side both in
implement2 Implementations of the Higgs characterisation [
17
] and the SILH
basis [
19
] Lagrangians are also available.
ing the linear dimension-six description discussed above in
the FeynRules framework [
20
] and in calculating the
renormalisation group (RG) evolution of the full set of operators
and their mutual mixing [
3,21–23
]. In the former case,
Rosetta can simply be extended to provide an output
compatible with the eventual NLO model implementation,
analogously to the BSMC Lagrangian. The latter case of evaluating
the RG running effects, while being a slightly separate issue,
highlights a key feature of our tool, given that the calculation
of these effects has only been performed in the original
Warsaw basis of Ref. [2]. The framework provided by Rosetta
allows for the application of these results in any desired basis.
Although the initial version of the software does not
explicitly deal with these effects, its translation functionality can
already be used in their context and we plan for future
versions to incorporate RG evolution of the SM EFT Wilson
coefficients.
The remainder of this paper is organized as follows. In
Sect. 2, we describe the basic functionalities of Rosetta
and how to make use of the program. Sect. 3 is dedicated to
an example of usage of Rosetta in which we focus on new
physics Higgs couplings to the SM bosons. We express them
in different bases and detail the output that is provided by
Rosetta. Our work is summarized in Sect. 4.
2 Rosetta
The aim of Rosetta is to provide a modular and flexible
package for EFT basis translation and communication with
event generation tools. The primary framework which
Rosetta has been designed to translate into is the
phenomenological effective Lagrangian, L(mass), which will be
explicitly defined in Sect. 3.1. The motivation for this choice lies
in the availability of an implementation within the
FeynRules framework [
18
], to be downloaded from the
FeynRules model repository [
24
], which ensures the link with
event generators and high-energy physics programs [
25,26
].
The most basic functionality of Rosetta is to map a
chosen set of input parameters (the Wilson coefficients in a
specific basis choice) onto the BSMC coefficients such that the
output can be employed within tools relying on a BSMC
basis description. As long as the input format respects the
conventions sketched in Sect. 2.2 and that are inspired by
the Supersymmetry Les Houches Accord (SLHA) [
27,28
],
the user may define his/her own map to the BSMC
coefficients (or to any other basis implementation) and proceed
with event generation using the accompanying FeynRules
implementation. This highlights one of the key features of
Rosetta, the possibility to easily define one’s own input
basis and directly use it in the context of many programs
via the translation functionality of Rosetta. The strength of
this approach is that it is much simpler than developing from
scratch new modules for existing tools in the context of a new
basis. To this end, Rosetta not only enables the translation
of an EFT basis into the BSMC Lagrangian, but also allows
for translations into any of the other bases included in the
package, i.e., currently the Higgs, Warsaw and SILH bases.
Translations between these three bases in any direction are
possible, so that the addition of a new basis by the user only
requires the specification of translation rules to any one of
the three core bases. One is subsequently able to indirectly
translate the new basis into any of the other two bases, as well
as into the BSMC Lagrangian. The details of how one can
implement a new basis in Rosetta are discussed in Sect. 2.4.
2.1 Getting started with Rosetta
The latest release of Rosetta can be obtained from
http://rosetta.hepforge.org
The package contains a Python executable named
translate, an information file named README and two
directories, a first folder (named Cards) collecting example
input files and a second folder (named Rosetta)
including the source code of Rosetta. The executable takes as
input an SLHA-style parameter file with the coefficients
of the dimension-six operators associated with a particular
basis. Information on the format of such an input file can
be found in Sect. 2.2. The execution of the translate
script from a shell yields the generation of an output
parameter file where all parameters are this time the coefficients of
the dimension-six operators associated with a specified new
basis, the default choice being the BSMC Lagrangian. The
tool can be used by typing in
./translate PARAMCARD.dat OPTIONS
where PARAMCARD.dat is the name of the SLHA-style
input file and OPTIONS stands for optional arguments. The
latter could consist of one or more of the following choices
that will modify the behavior of the program.
-h or –help
-o or –output
-s or –silent
-t or –target
This displays a help message and
exits the program.
This allows for the specification of
the name of the output file, that is by
default PARAMCARD_new.dat.
The program suppresses warnings
and takes the default answer to any
question that may have to be asked
to the user.
This allows for providing the name
of the basis into which the
translation occurs, the default being bsmc
-w or –overwrite
-e or –ehdecay
-f or –flavor
-d or –dependent
and the other acceptable choices
being higgs, silh or warsaw.
This allows the program to
overwrite any pre-existing output file.
This allows to use the interface
with the eHDecay program [
11
] for
the calculation of the Higgs boson
width and branching fractions. See
Sect. 2.5.2.
This allows to specify the
treatment of the flavor structure
relevant for the fermionic operators,
the default being general and
the other acceptable choices being
universal and diagonal. See
Sect. 2.5.1.
This allows the program to also
write out any dependent parameters
calculated by the translation
function to the output file.
On run time, Rosetta starts by performing several checks
on the input parameters and verifies the consistency of the
input file with respect to the specifications of the
internal basis implementation. In this way, any missing SM
inputs (with respect to the requirements included in the
required_inputs and required_masses attributes
of the basis class, see Sect. 2.3) can be included using the
value provided in the Particle Data Group (PDG) review [
29
],
while any missing coefficient associated with an
operator that is present in the basis (and thus declared in the
independent attribute of the basis class, see Sect. 2.3)
can be included with a zero value.
Once the translation is achieved, Rosetta outputs a new
parameter file that is by default named PARAMCARD_new.
dat. This file contains the values of all parameters relevant
for the target basis and also includes the necessary
modifications to the input parameters, such as the W -boson mass that
may depend on some dimension-six operator coefficients.
2.2 Input files and their handling in Rosetta
Rosetta requires input parameters to be given under the
form of a file encoded in a format similar to the SLHA one
detailed in Refs. [
27,28
]. Parameters are grouped into blocks
and each parameter is identified inside its own block by one or
more integer numbers called counters. For instance, the SM
inputs necessary for the definition of the SILH basis would
read
BLOCK SMINPUTS # 1 +1.27916e+02 # aEWM1 2 +1.16638e-05 # Gf
3
4
25
+1.18400e-01 # aS
+9.11876e+01 # MZ
+1.25000e+02 # MH
where the different entries respectively correspond to the
inverse of the electromagnetic coupling constant (aEWM1),
the Fermi constant (Gf), the strong coupling constant (aS),
the Z -boson mass (MZ) and the Higgs boson mass (MH).
Inspired by the usual SLHA conventions, all masses are also
collected into a block called MASS where the counters
correspond to the PDG identifiers of the particles [
29
].
Furthermore, matrix quantities receive a block of their own with
counters specifying the position inside the matrix. In this
way, a single block would be needed to encode, for instance,
the cHud coefficients associated with the OHud operator of
the Warsaw basis that is defined by
OHud = −i u¯γ μd
.
In this expression, u and d denote the SU (2)L singlets of
right-handed up-type and down-type quark fields,
respectively, and and Dμ stand for the weak doublet of Higgs
fields and its gauge-covariant derivative. In flavor space, the
cHud coefficients take the form of a matrix, implemented in
the input file as
(3)
BLOCK WBxHud
1 1 0.1e+00 # cHud11
1 2 0.0e+00 # cHud12
1 3 0.0e+00 # cHud13
2 1 0.0e+00 # cHud21
2 2 0.1e+00 # cHud22
2 3 0.0e+00 # cHud23
3 1 0.0e+00 # cHud31
3 2 0.0e+00 # cHud32
3 3 0.1e+00 # cHud33
The block name contains information on the basis (WB) and
on the considered operator (Hud). Sample parameter files for
all core bases can be found in the Cards directory shipped
with the program. Within those files, we have adopted the
above block naming scheme. The name of each block starts
with two letters identifying the basis (BC, HB, SB and WB for
the BSMC, Higgs, SILH and Warsaw bases respectively) that
are followed by a separator (x), and ends with the name of the
considered coefficient as it is defined in the LHCHXSWG
proposal for an EFT basis choice [
16
]. In the case of EFT
operators independent of fermions, the related (non-matrix)
coefficients are collected in different blocks as a function
of the Lorentz structure of the operators. For instance, the
SBxV2H2 block will include all operators of the SILH basis
containing two occurrences of the Higgs field and two
occurrences of the gauge fields. Their ordering follows their order
of appearance in the LHCHXSWG proposal. The imaginary
parts of all parameters are stored in corresponding blocks
whose names are prefixed with the IM tag.
The Rosetta package contains built-in methods for
dealing with an SLHA-like structure, and these methods
have all been implemented in the Rosetta/SLHA.py
file. When an input file is read, the parser included in the
SLHA.py file recognizes the existing BLOCK and DECAY
structures of the input file and stores them as instances of
the SLHA.NamedBlock, SLHA.NamedMatrix and of
the SLHA.Decay classes. These are dictionary-like objects
that can be assigned, indexed and iterated over as
regular Python dictionaries. An SLHA.NamedBlock object
reflects the information embedded in an SLHA block so that
it possesses a name attribute and stores values associated
with integer keys as well as a mapping from the integer
keys to the parameter names. In this way, parameters can be
accessed by indexing either their integer key or their name.
Similarly, SLHA.NamedMatrix objects function
analogously but operate with a pair of indices for indexing. An
SLHA.Decay object contains an integer attribute PID that is
the PDG identifier of the particle whose decays are described
by the considered block, as well as a total attribute
allowing for the storage of the total width. Individual decay
channels are then indexed by tuples of PDG codes associated with
the decay products, and the stored values are the
branching fractions. Finally, the SLHA.py file also includes the
definition of the SLHA.Card class that serves as a
container for a collection of instances of the above objects. The
implementation of any basis in Rosetta therefore requires
the user to provide definitions for the blocks and
parameters to be specified in the input file that will be read into an
SLHA.Card instance belonging to that basis class. More
practical information and examples are given in Sects. 2.3
and 2.4.
Three special blocks named BASIS, SMINPUTS and
MASS must always be present. The first and only element of
the BASIS block refers the name of the basis into which
Rosetta must read the input file and informs the program on the
other blocks it should look for, based on the structure
specified in the implementation of that particular basis. This name
should be a single unique string with no spaces. The next two
mandatory blocks consist of conventional input blocks
specifying the values of the SM inputs and of the particle masses.
The set of required inputs will depend on the specifications in
the corresponding basis implementation. Moreover, the user
can optionally specify the value of the elements of the CKM
matrix by setting their real and imaginary parts within the
VCKM and IMVCKM blocks. If absent, the information of the
PDG review [
29
] is used by Rosetta. All extra blocks and
decay structures are stored, left unchanged and passed to the
output file unless the user demands to use the eHDecay
program, which will overwrite any existing decay information
on the Higgs.
2.3 Structure of Rosetta
Rosetta is a Python package containing the
implementation of a Basis class equipped with several utility functions
for reading, processing and writing SLHA-style parameter
files. Working implementations of bases are derived from this
class and only require a small amount of information
specifying the block structure of the EFT parameters, the required
SM inputs and a series of translation functions to other
existing basis implementations. In order to be able to define a new
basis class, we describe in this section the properties of the
Basis objects.
The Basis class has a number of intrinsic data members
that should be defined in order to get a working
implementation of an EFT basis. These consist of the independent,
required_inputs and required_masses attributes
already mentioned in Sect. 2.1, together with the name,
blocks and flavored members of the class.
name Unique string identifier for the
basis implementation, e.g., higgs,
bsmc, silh or warsaw for the
core bases shipped with the
package.
independent List of strings containing the names
of the independent EFT operator
coefficients of the basis. These are
expected to be present in the input
parameter file.
required_inputs Set of integers containing the SLHA
counters of the required SM inputs.
See Table 2 for the complete list of
those allowed in Rosetta.
required_masses Set of integers containing the PDG
identifiers of the particles whose
masses are required as input and
that are not included in required
_inputs.
blocks Dictionary with the non-matrix
SLHA block names as keys and lists
of coefficients stored in that block
as values.
flavored Dictionary with matrix SLHA block
names as keys. The values are other
dictionaries with the keywords
kind, domain and cname as
keys. This describes the properties
of the matrices.
In the case of the definition of a matrix block, the
selfexplanatory possible values for the keyword kind are
symmetric, hermitian and general and those related
to the keyword domain are real and complex. The
properties of the ensuing matrix object will depend on the choice
of these keywords. The name to be given to the individual
EFT coefficients are derived from the value of the keyword
cname. Conventionally, the real and imaginary components
are prefixed with the letters R and I respectively, while the
position (i, j ) in the matrix is referred to by a suffix ixj. A
complex parameter comes with a prefix C.
Once an input file is read, an instance of the SLHA.Card
class that can be accessed via the card member of the basis
class is created and populated with the information provided
as input. The content of the mandatory MASS and SMINPUTS
blocks is exported to data members of the basis class named
mass and inputs that can then be used for accessing the
SM parameters, while the CKM matrix is stored into the ckm
container of the basis class. In the Rosetta framework, the
EFT operator coefficients are implemented as elements of the
relevant basis class and can be accessed via standard Python
methods. For instance, all the coefficients associated with a
basis object named MyBasis could be printed, together with
their values, by coding
for k, v in MyBasis.items():
print k, v
In addition, a direct accessor to each EFT operator coefficient
is created from its name, which facilitates the
implementation of the translation functions that in general extensively
reference individual parameter values. This however assumes
that there are no duplicate parameter names in the SLHA-like
input file, which nevertheless leads to a program exception.
There are hence multiple ways to access a given parameter.
For example, a parameter A stored as the third element of
a block MyBlock that is part of the definition of a basis
MyBasis could be equally accessed as
Rosetta
name
aEWM1
Gf
aS
MZ
MB
MT
MTAU
MH
MyBasis[’A’]
MyBasis.card[’A’]
MyBasis.card.blocks[’MyBlock’][’A’]
MyBasis.card.blocks[’MyBlock’][3]
In the lines above, the parameter A is respectively accessed
from the MyBasis object, from the SLHA.Card instance
associated with the current basis and from the SLHA.
NamedBlock object associated with the MyBlock block
(using either the parameter name or the counter as an index).
2.4 Implementing a new basis
One of the important features of Rosetta is the intended
ease with which a user can define a new basis class to suit
his/her specific physics needs. In the context of an ultraviolet
complete model, he/she may be interested in the
phenomenological consequences of a particular high-scale scenario in
the EFT framework. Imagining that he/she has derived all
dimension-six Wilson coefficients in a particular basis,
Rosetta could be used to map these coefficients to the
FeynRules effective Lagrangian implementation in the
masseigenstate basis so that the collider phenomenology of such a
scenario could be investigated. This task is realized by
implementing a new basis in Rosetta and by connecting the new
basis input parameters to those of one of the existing core
basis implementations.
Alternatively, the user may have developed a particular
resource performing a useful analysis or calculation in a
nonstandard basis choice. The corresponding basis
implementation in Rosetta with a translation to one of the core bases
could then allow one to use this tool in the context of all
other existing basis implementations in Rosetta and
therefore greatly widen its scope. The eHDecay feature of
Rosetta is an example of this, as it works with a set of operators
corresponding to the SILH basis. Via Rosetta, eHDecay
is now available for calculations in the SILH, Warsaw and
Higgs bases, as well as in any additional basis that may be
implemented in the future.
In this section, we provide an example that outlines the
basic requirements for implementing a new basis in
Rosetta. We also refer the reader to the file Rosetta/
TemplateBasis.py which serves as a concrete toy
example that can be used as a template for creating a new
basis class as well as the core basis implementations for more
complete realizations.
All Rosetta basis classes inherit from the mother
class Basis implemented in the Rosetta/internal/
Basis.py file. This class contains all the machinery
necessary for reading, writing and translating so that a new
basis implementation solely demands the user to create a
new file that must be saved in the Rosetta directory and
that includes the declaration of a Basis subclass. The user
has then to define the class attributes described in Sect. 2.3.
First, it is essential that the name of the basis class matches
the filename in which it is saved minus the extension in
order to ensure a proper running of the program. Second, the
independent, blocks and flavored attributes of the
class define the input parameters of the basis and their desired
SLHA-like structure, while the required_inputs and
required_masses lists are specified according to the
needs of the translation functions that are planned to be
implemented. One can also specify a dependent attribute
to explicitly define a particular parameter as dependent. For
instance, the following code refers to the implementation of
a new basis class called MyBasis and has been included in
the file Rosetta/MyBasis.py.
from internal import Basis
class MyBasis(Basis.Basis):
name = ’mybasis’
independent=[’A’,’B’,’one’,’two’,’MYxMAT’]
dependent = [’Cmat3x3’]
blocks = {’letters’:[’A’,’B’,’C’],
’numbers’:[’one’,’two’,’three’]}
flavored = {’MYxMAT’:{’kind’:’hermitian’,
’domain’:’complex’,
’cname’:’mat’}}
required_inputs = {1,2,4}
required_masses = {24,25,6}
This snippet of code specifies the declaration of the basis
class MyBasis whose unique string identifier is given by
mybasis. The independent parameters to be read from an
input SLHA-like file are defined to be A, B, one and two
and are assumed to be organized into the two SLHA blocks
LETTERS and NUMBERS. A flavored matrix, MYxMAT, is
also present and deemed to be an independent input
parameter except for its (3,3) component that is explicitly included
within the dependent attribute of the basis class. The
translation methods to be implemented require the knowledge of
six SM masses and parameters that must be specified via the
required_inputs and required_masses attributes
of the basis class. In our case, the electroweak inputs α−1,
G F and m Z are connected to the SMINPUTS block of the
SLHA-like input structure, while the W -boson, Higgs boson
and top quark masses are connected to its MASS block. The
extra parameters C and three are dependent parameters that
the user has to define in terms of the independent and SM
parameters (see below). The non-SM part of an illustrative
input file could be
BLOCK BASIS
1 mybasis # input basis
BLOCK LETTERS 1 8.6e-2 # A 2 0.002 # B
BLOCK NUMBERS
1 1.5e-2 # one
2 2.8e-3 # two
BLOCK MYxMAT
1 1 3.4e-2 # Rmat1x1
1 2 7.8e-5 # Rmat1x2
1 3 5.2e-4 # Rmat1x3
2 2 5.6e-3 # Rmat2x2
2 3 3.3e-3 # Rmat2x3
BLOCK IMMYxMAT
1 2 9.9e-3 # Imat1x2
1 3 1.9e-4 # Imat1x3
2 3 4.6e-3 # Imat2x3
while its SM part would include the SMINPUTS and MASS
blocks with values for the six above-mentioned SM inputs, as
well as the two blocks related to the CKM matrix in the case
where one would be interested in using non-default values
for its elements. Only the relevant elements of MYxMAT need
be provided given that it is declared to be Hermitian, and the
(3,3) element is left unspecified as it is a dependent parameter.
The dependent parameters are evaluated via a method
named calculate_dependent() that must be
provided by the user. Continuing with the example above, we
include in the new basis class implementation the code
def calculate_dependent(self):
self[’C’]=(self[’A’]+self[’B’])/2.
self[’three’]=(self[’one’]-self[’two’])/2.
self[’MYxMAT’][
3,3
]=10.*self[’MYxMAT’][
2,2
]
This imposes that the C parameter is defined as the mean of
the A and B parameters, that the three parameter equals
half of the difference of the one and two parameters and
that the (3,3) entry of the MYxMAT matrix is equal to 10 times
the value of its (2,2) entry.
When executed, Rosetta begins with the reading of the
input file and next calls the calculate_dependent()
method for evaluating the remaining basis parameters.
Rosetta finally performs the translation to another basis by
using the translation methods defined by the user. Their
implementation requires the use of a translation
decorator with an argument that refers to the name of the target
basis and that must match a basis implementation contained
in the Rosetta directory. For example, we could link the
mybasis basis above to the Warsaw basis by implementing
@Basis.translation(’warsaw’)
def mytranslation(self, wbasis):
a_EW = 1./self.inputs[’aEWM1’]
wbasis[’cWW’] = a_EW*self[’C’]
wbasis[’WBxHpl’][
1,1
] = self[’two’]
return wbasis
Translation functions such as the mytranslation(...)
one above take an instance of the target basis class as their
only argument and return it after setting its parameter values.
Relations involving (matrix) parameters with a flavor
structure should be performed in a flavor general way, as discussed
in Sect. 2.5.1.
If modifications to the SM input parameters need to be
made (i.e., the mass and inputs attributes of the basis
class), the function modify_inputs() must be
implemented similarly to the calculate_dependent()
method. The following example defines a shift of the W
boson mass by the A parameter,
def modify_inputs(self):
self.mass[
24
] = self.mass[
24
] + self[’A’]
In general, the user-defined functions may require the
evaluation of parameters such as the weak and hypercharge
gauge couplings or the electroweak mixing angle. The choice
of relations (e.g., tree- or loop-level) to be used to
consistently derive these parameters from the inputs is left to
the user. In the core bases provided with Rosetta, the
calculate_inputs() method relies on tree-level
relations to deduce all the SM parameters.
Having defined a basis class according to these
specifications, Rosetta is able to detect the presence of the
basis implementation and to automatically construct
possible translation paths to other existing bases from the
userdefined translation functions. The recognition of the
implemented basis by Rosetta is also reflected in the help
message accompanying the translate script, the name of the
new basis appearing as one of the possibilities for the target
basis option.
2.5 Additional features
2.5.1 Flavor structure of the fermionic operators
In the general case, each matrix block of the input file includes
one entry for each possible flavor assignment of the
corresponding operator. The flavor option of the translate
executable introduced in Sect. 2.1 allows the user to make
assumptions on the flavor structure of the operators so that
Rosetta reads input files and generates output files with a
simplified block structure (unless the BSMC basis is used
as it requires all coefficients to be specified). Setting this
option will act on all of the matrix parameters declared
in the flavored attribute of a basis class
implementation. The flavor option can be fixed either to universal
where all matrices of operator coefficients are proportional
to the identity or to diagonal where only their
flavordiagonal elements are retained. In the universal case one
is allowed to define matrix blocks containing only the (1,1)
element while in the diagonal case, all three diagonal
elements must be provided. Sample input files can be found in
the Cards directory of the program. In the definitions of the
three core bases, the flavor-symmetry-breaking Yukawa-like
operators are normalized by the masses of the fermions such
that the universal flavor option will lead to a minimally
flavor-violating (MFV) structure where the physical effects
of the coefficients are scaled by the corresponding fermion
masses [
30
]. For example, in the Higgs basis, these
Yukawalike terms are written as:
mif m jf δyifj f¯i cos φi j − i γ5 sin φi j f j .
(4)
The corresponding normalizations are also used for the
Warsaw and SILH basis implementations in Rosetta to simplify
the translations and also the possibility of encoding MFV into
any EFT description. The same argument applies to the dipole
operators, O f V , as well as the OHud operator mentioned in
Sect. 2.2. The former set of operators breaks the flavor
symmetry in an identical way to the Yukawa-like operators and
will hence receive the same mif m jf normalization. In the
latter case, the flavor structure of the operator requires two
Yukawa insertions as it is composed of right-handed quarks
only. Moreover, being a charged-current operator, the MFV
construction requires the insertion of the CKM matrix such
that the operator is normalized as
OHud = −i miu mdj VCi jK M u¯i γ μd j
.
(5)
Since this particular operator is unique and maps to a
single operator in all of the other core bases, the corresponding
translations remain unaffected. This normalization is
however not the same as the one chosen in Ref. [
16
]. Users should
therefore bear in mind these normalizations which have been
chosen to single out operators that explicity break the flavor
symmetry of the Lagrangian. That being said, they are merely
a convenient way for the user to implement MFV and can be
worked around if the user so desires.
Rosetta recognizes coefficients by their names so that
the naming of the elements of the matrix coefficients must
respect the conventions described in the previous section
for their real (an R prefix) and imaginary (an I prefix)
parts, and for their position (i, j ) inside the matrix (an
ixj suffix). Implementing translations from flavored
parameters should ideally always be done in the most general
case such that the various flavor options work correctly.
To this aim, basic matrix algebra operations have been
implemeted in the internal/matrices.py module
of Rosetta. The available functions are matrix_mult,
matrix_add, matrix_sub and matrix_eq and
correspond to matrix multiplication, addition, subtraction and
assignment respectively. They can be used to assign
values to a matrix SLHA block according to the result of a
specific operation between two other matrix SLHA blocks.
These functions require two mandatory arguments for the
objects between which the operation should be performed
and a third optional argument specifying the matrix block
to which the result of the operation should be assigned.
For instance, matrix_mult(M1,M2,M3) assigns to the
matrix M3 the result of the multiplication of the matrices
M1 and M2. If the M3 argument is omitted, a generic matrix
object is returned such that matrix utility functions can be
combined. The matrix_eq(M1,M2) method is the only
exception. It takes two mandatory arguments M1 and M2 and
allows for the assignment of the elements of the M1 matrix
to the M2 matrix.
A concrete example can be found in the calculate_
dependent() function included in the Higgs basis
implementation. The deviations of the W -boson couplings to the
weak doublet of left-handed quark fields δgLWq are related to
those of the Z -boson couplings to the individual left-handed
up-type and down-type quarks δgLZu and δgLZd via the CKM
matrix VCKM,
δgLWq = δgLZu · VCKM − VCKM · δgLZd .
(6)
The Rosetta implementation of this relation makes use of
a combination of the matrix_mult and matrix_sub
method,
matrix_sub(
matrix_mult(HB[’HBxdGLzu’], HB.ckm),
matrix_mult(HB.ckm, HB[’HBxdGLzd’]),
HB[’HBxdGLwq’] )
where HB is an instance of the Higgs basis class. The third
argument of the matrix_sub method allows one to assign
the result of the matrix subtraction to the elements of the
HBxdGLwq matrix block. Matrix blocks also come with the
T() and dag() methods for transposing and Hermitian
conjugation.
2.5.2 Interface to eHDecay
In order to calculate dimension-six operator contributions
to the Higgs boson width and branching ratios, Rosetta
includes an interface to the eHDecay program [
11
]. It can
be switched on by executing the translate script with the
eHDecay option (see Sect. 2.1). In order to use this feature,
the path to a local installation of eHDecay on the user
system should be specified in the Rosetta/config.txt
file, next to the eHDECAY_dir keyword, and a
(possibly indirect) translation linking the basis of interest to the
SILH basis should exist. If so, the translation will be
performed, eHDecay will be run internally and an SLHA decay
block for the Higgs boson will be appended to the output
file.
Since the SILH basis description adopted in eHDecay
assumes the MFV paradigm, an additional layer of translation
is internally performed by Rosetta to render its internal
SILH basis implementation MFV-compliant. Details can be
found in Sect. 3.4.
3 Mapping different EFT basis choices
In this section, we discuss the BSMC Lagrangian containing
redundant parameters that is the default basis which
Rosetta has been designed to translate into. We explain the
relations with the non-redundant Higgs, Warsaw and SILH
bases and focus on a subset of operators connected to single
Higgs production at the LHC to provide examples of usage
of Rosetta.
3.1 The BSMC Lagrangian and the Higgs basis
To study the Higgs boson properties in detail at the next
LHC runs, the LHCHXSWG has proposed a
parameterization of anomalous interactions of the SM fermions, gauge
bosons and the Higgs boson allowing both for a transparent
linking to physical observables and for an easy
implementation in Monte-Carlo event generators [
16
]. The framework
is that of a general effective Lagrangian defined in the
masseigenstate basis, where all kinetic terms are canonically
normalized and diagonal, and where all mass terms are
diagonal. Moreover, the tree-level relations between the gauge
couplings and the usual electroweak input parameters (G F ,
α(0), m Z ) are the same as in the SM. In such a frame, i.e.,
in the BSMC Lagrangian, the coefficients of the interaction
terms in the Lagrangian are related in an intuitive way to
quantities observable in experiment, and any parameter in
the effective Lagrangian can be measured.
The BSMC Lagrangian captures all physics effects that
may arise in the presence of lepton-number and
baryonnumber conserving dimension-six operators beyond the SM.
However, it is more general than a basis defined before
electroweak symmetry breaking as it contains more free
parameters. This is because the SU (3)C × SU (2)L × U (1)Y gauge
symmetry linearly realized at the level of an operator basis
implies relations between different couplings of the
effective Lagrangian defined after electroweak symmetry
breaking. The latter indeed only respects the SU (3)C × U (1)E M
symmetry. In particular, the charged and neutral gauge boson
interactions are related, as are those with zero, one and two
Higgs bosons. These relations are not implemented at the
level of the BSMC Lagrangian so that it may be used to
study more general theories such as when the electroweak
symmetry is non-linearly realized or when some operators
of dimension greater than six are included.
The Higgs basis has been proposed as a convenient
parameterization of another non-redundant dimension-six EFT
basis. In this approach, the relations (that hold in any
nonredundant dimension-six basis of EFT operators) between
different couplings of the BSMC Lagrangian required by
a linearly realized SU (2)L × U (1)Y local symmetry are
enforced. Furthermore, the Higgs basis has been defined by
choosing a specific subset of independent parameters from
all couplings of the BSMC Lagrangian. The choice of the
independent couplings is motivated by their direct
connection to observables constrained by electroweak precision and
Higgs studies. This approach is similar to the one introduced
in Ref. [
6
], except that a different subset of couplings has
been chosen, and the number of independent couplings is
the same as for any basis of non-redundant dimension-six
operators. Moreover, there exists a linear one-to-one
invertible transformation between the independent couplings of
the Higgs basis and the Wilson coefficients in any basis. The
remaining BSMC Lagrangian couplings are all dependent
parameters that can be expressed in terms of the independent
ones.
The BSMC Lagrangian is displayed in Ref. [
16
], up to
four-fermion terms and interactions involving five or more
fields. Here, to illustrate the relationship between the BSMC
and other bases, we focus on a part of the Lagrangian
describing the C P-even interactions of the Higgs boson with two SM
gauge bosons. After denoting by Gaμ, Wμ±, Zμ, Aμ and h the
gluon, the W -boson, the Z -boson, the photon and the Higgs
boson fields, respectively, the relevant part of the Lagrangian
reads
h
Lh = v 2δcwm2W Wμ+W −μ + δcz m2Z Zμ Z μ
g2 g2
+ cgg 4s Gaμν Gaμν + cww 2
g2 gg
+ czz 4cθ2 Zμν Z μν + czγ 2
Wμ+ν W −μν
Zμν Aμν
+ cγ γ
4
g 2c2
θ Aμν Aμν +cw g2 Wμ−∂ν W +μν +h.c.
+ cz g2 Zμ∂ν Z μν + cγ gg Zμ∂ν Aμν .
(7)
In our notation, cθ (sθ ) stands for the cosine (sine) of the
electroweak mixing angle, v for the vacuum expectation value of
the neutral component of the Higgs doublet , and gs , g and
g are the strong, weak and hypercharge coupling constants.
Moreover, we have introduced the field strength tensors of
the gauge bosons that we define as
Vμν = ∂μVν − ∂ν Vμ for V = W ±, Z and A,
Gaμν = ∂μGaν − ∂ν Gaμ + gs f a bcGbμGcν ,
(8)
in which f a bc are the structure constants of SU (3)C .
The Lagrangian above contains ten real coupling
parameters that are all independent in the BSMC picture. However,
if Lh originates from an EFT with dimension-six operators,
only six of these couplings are independent and the
remaining four can be expressed in terms of these six and of the
SM parameters. In the Higgs basis, δcz , cgg, czz , czγ , cγ γ
and cz are chosen as independent parameters and the four
remaining couplings are calculated as
δcw = δcz + 4δm,
cww = czz + 2sθ2czγ + sθ cγ γ ,
4
g2cz
cw
=
cγ
=
+ g 2czz − (g2 − g 2)sθ2czγ − g2sθ4cγ γ ,
g2 − g 2
2g2cz +(g2 +g 2)czz −(g2 −g 2)czγ −g2sθ2cγ γ .
g2 −g 2
In the first of these relations, δm denotes the shift of the
W -boson mass that is possibly induced by the presence of
higher-dimensional operators and that we normalize as
Lmass = 2δm
It includes, in addition to the blocks above, the SM
parameters as well as vanishing values for all other EFT coefficients.
In order to export this setup to the BSMC Lagrangian, we use
Rosetta by typing in a shell
./translate HiggsBasis.dat
Rosetta first calculates all dependent coefficients and next
generates an output file named HiggsBasis_new.dat
given in the framework of the BSMC Lagrangian. This file
contains in particular values for the four δcw, cww, cw and
cγ dependent parameters, the corresponding output block
being, according to Eq. (9),
In addition, the HiggsBasis_new.dat file also includes
extra non-vanishing coefficients that are linked to the six
independent parameters δcz , cgg, cγ γ , czγ , czz and cz by
gauge invariance. For instance, a di-Higgs coupling to two
gluonic field strength tensors is present,
cz
cγ
BLOCK BCxhh 4 1.00000e-01 # cgg2
3.2 The Warsaw basis
The ten interaction terms of the Lh Lagrangian introduced
in Sect. 3.1 can be seen as generated by six independent
operators of the Warsaw basis,
W
Lh
1 gs2
= v2 cGG 4
+ cW B gg
+ cH ∂μ
†
†σi
†
∂μ
†
+cT
g2
Gaμν Gaμν + cW W 4
W μiν Bμν + cB B 4
g 2
†←→Dμ
†
†
W μiν Wiμν
Bμν Bμν
†←→Dμ
In this expression, we have introduced the Pauli matrices σi ,
the Hermitian derivative operator,
†←→Dμ
=
†(Dμ ) − (Dμ
†) ,
the gauge-covariant derivative and the hypercharge and weak
field strength tensors
Dμ
i i
= ∂μ − 2 gσk Wμk − 2 g Bμ
,
W μiν = ∂μWνi − ∂ν W μi + g i jk Wμj Wνk ,
Bμν = ∂μ Bν − ∂ν Bμ.
.
(11)
(12)
(13)
cgg = cGG ,
The six Wilson coefficients cGG , cW W , cW B , cB B , cH and
cT appearing in LWh are related to the ten couplings in the
effective Lagrangian Lh as
,
czγ =
g2cW W − 2(g2 − g 2)cW B − g 2cB B
g2 + g 2
,
cγ γ = cW W + cB B − 4cW B ,
2
cw = g2 − g 2 g 2cW B − cT + δv ,
2
= − g2 cT − δv ,
2
= g2 − g 2 (g2 + g 2)cW B − 2cT + 2δv .
Here, δv is defined by
we obtain an output file where several non-zero EFT
coefficients can be found. The numerical value of those on which
we focus here can be extracted from the generated file,
(14)
(15)
(16)
BLOCK BASIS
1 warsaw # translated basis BLOCK WBxH4D2 # 1 -1.98704e-01 # cH
3 These operators contribute to the muon decay at the tree level. Taking
this into account leads to a shift between the measured Fermi constant
and the vacuum expectation value of the Higgs field, which motivates
the notation δv.
2 +1.18790e-02 # cT
BLOCK WBxV2H2 #
1 +1.00000e-01 # cGG
2 +1.52190e-01 # cWW
3 +5.45943e-03 # cBB
4 +1.44124e-02 # cWB
In our benchmark scenario, the δv shift is vanishing.
3.3 SILH basis
We now consider the case where all operators included in the
Lh Lagrangian of Sect. 3.1 are induced by a set of operators
of the SILH basis,
S 1 g2 g 2
Lh = v2 sGG 4s † Gaμν Gaμν + sB B 4
† Bμν Bμν
†←→Dμ
∂ν Bμν
ig
+ sW 2
.
†←→Dμ
†←→Dμ
δcw = −sH −
3g2 + g 2
+ g2 − g 2 δv,
δcz = −sH − 3δv,
cgg = sGG ,
The ten Wilson coefficients included in Lh can be
rewritten in terms of the eleven parameters appearing in LSh as
g2g 2(sW +sB +s2W +s2B )
g2 − g 2
4g2
− g2 − g 2 sT
czγ = −
cγ γ = sB B ,
1
cw = 2 sH W +
cww = −sW W ,
g2sH W + g 2sH B − g 2sθ2sB B ,
czz = − g2 + g 2
sH W − sH B 2
2 − sθ sB B ,
g2(sW + s2W ) + g 2(sB + s2B ) − 4sT + 4δv
2(g2 − g 2)
cz =
cγ
=
g2(sW +s2W +sH W )+g 2(sB +s2B +sH B )−4sT +4δv
2g2
,
sH W −sH B
2
+
g2(sW +s2W )+g 2(sB +s2B )−4sT +4δv
g2 − g 2
where δv = 21 (sH )22. Rosetta can be used to extract the
numerical values of the independent SILH parameters by
inverting the above relations. Adopting the benchmark
scenario of Sect. 3.1 where all the relevant Higgs basis
independent parameters have been fixed to 0.1, we type in a shell
./translate HiggsBasis.dat -t silh
,
,
so that we can extract all the required SILH coefficients from
the generated output file,
BLOCK BASIS
1 silh # translated basis
BLOCK SBxH4D2 #
1 -1.00000e-01 # sH
2 +0.00000e+00 # sT
BLOCK SBxV2H2 #
1 +1.00000e-01 # sGG
2 +1.00000e-01 # sBB
3 +4.65203e-01 # sW
4 -4.65203e-01 # sB
5 -1.52190e-01 # sHW
6 +9.45406e-02 # sHB
7 +0.00000e+00 # s2W
8 +0.00000e+00 # s2B
3.4 Yukawa-like operators
An important difference between the definitions of the
SILH and Warsaw bases provided in the LHCHXSWG
proposal [
16
] and their original descriptions lies in the forms of
the Yukawa-like operators,
(19)
where FL and f R denote a generic weak doublet of
lefthanded fermions and a generic weak singlet of right-handed
fermions respectively. In the original Warsaw basis
definition, these Yukawa-like operators take the above form. In
the LHCHXSWG proposal (on which Rosetta is based),
these operators have been redefined in a way allowing one to
decouple their contributions to the fermion masses (that are
extracted from appropriate measurements and thus fixed), as
well as to simplify the implementation of MFV,
LYuk = −
√mi m j (c f )i j
v
v2
†
v2
− 2
F¯Li f Rj ,
(20)
where the primes denote fields taken in the mass
eigenbasis. The Wilson coefficients c f and c f are related by unitary
transformations UL and UR that map the field gauge
eigenbasis to the mass eigenbasis with the would-be mass
modifications absorbed into the diagonalized Yukawa matrices
Y fD,
c f = √mvi m j UL†c f UR and Y fD = UL† Y f UR + c2f . (21)
In the original SILH basis description, an additional
assumption of minimal flavor violation is included, such that the
flavor structure is taken aligned with the Yukawa matrices,
MFV
LYuk = (Y f )i j
cMFV
f
The Wilson coefficients c MfFV are therefore proportional to
the identity matrix in flavor space and are thus universal.
Thanks to the convenient normalizations, they are now
trivially related linearly to those of the Warsaw and SILH basis
descriptions of the LHCHXSWG proposal by
v
(c f )ii = mi Y fDc MfFV =
√
2c MfFV.
These relations are used internally for the eHDecay
interface of Rosetta, which takes SILH basis input parameters
assuming the MFV convention of Eq. (22). In order to
consistently use eHDecay, Rosetta translates these coefficients
from the alternative version of the SILH basis detailed in
Ref. [
16
]. As a consequence, a general flavor structure
cannot be employed when making use of the eHDecay interface.
Although it is in principle possible to input different
values for the ccMFV, cbMFV, ctMFV, cμMFV and cτMFV parameters
(referred in Ref. [
11
] as c¯c, etc.) when running eHDecay
on its own, large deviations from non-universality in these
coefficients consist of a significant departure from the MFV
paradigm and should not be used for complete consistency
within Rosetta.
(22)
(23)
4 Summary
In this paper, we have introduced the Rosetta package, a
Python program dedicated to the translation of a given EFT
basis of independent operators to other viable basis choices.
We have also included, in this document, technical details so
that users can easily extend the program and implement their
own choices of EFT operator basis.
Currently, the program allows the user to translate
benchmarks designed in the Higgs, SILH and Warsaw bases into
any of these three bases. In addition, the code also expresses
any scenario in terms of the BSMC Lagrangian of EFT
operators, a basis that has been defined from the Higgs basis after
ignoring all relations among the operators that are induced by
a linear realization of the electroweak symmetry. A
FeynRules implementation allows Rosetta to be linked to other
high-energy physics tools. The relations among the
different Wilson coefficients that hold in the context of the Higgs,
SILH and Warsaw bases of independent operators have been
implemented into Rosetta so that they are preserved when
a setup is exported to the BSMC Lagrangian by the program.
This scheme has the strength to be easily generalizable to
study different setups providing a description of the Higgs
boson properties, such as those with a non-linearly
realized electroweak symmetry or including higher-dimensional
operators beyond dimension six.
In the future, we believe that translations from one basis to
another will allow for broadening the scope and the use of past
calculations very relevant for precision Higgs physics. Along
these lines, higher-order calculations in QCD performed in
the BSMC Lagrangian [
31–33
] could be used within any
given EFT language, and the renormalization group running
of the Wilson coefficients, that has been calculated in the
SILH basis [
34,35
] and in the Warsaw bases [
3,21,22
], could
be exported to different bases too.
Acknowledgments The authors are grateful to Fabio Maltoni for
lively interactions during all phases of this project and would also like to
thank Christophe Grojean, Alex Pomarol and Michael Trott for useful
comments. This work has been partially supported by the FP7 Marie
Curie Initial Training Network MCnetITN (PITN-GA-2012-315877),
by the Belgian Federal Science Policy Office through the Interuniversity
Attraction Pole P7/37, by the Strategic Research Program ‘High Energy
Physics’ and the Research Council of the Vrije Universiteit Brussel,
by the Swiss National Science Foundation under the Ambizione Grant
PZ00P2 136932, and by the Theory-LHC-France initiative of the CNRS
(INP and IN2P3).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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