Rosetta: an operator basis translator for Standard Model effective field theory
Adam Falkowski, Benjamin Fuks, Kentarou Mawatari, Ken Mimasu, Francesco Riva, Verónica sanz
RRosetta: an operator basis translator for Standard Modeleffective field theory
Adam Falkowski , Benjamin Fuks , Kentarou Mawatari , Ken Mimasu , Francesco Riva , Ver´onica Sanz Laboratoire de Physique Th´eorique, Bat. 210, Universit´e Paris-Sud, 91405 Orsay, France Institut Pluridisciplinaire Hubert Curien/D´epartement Recherches Subatomiques, Universit´e de Strasbourg/CNRS-IN2P3,23 rue du Loess, F-67037 Strasbourg, France Theoretische Natuurkunde and IIHE/ELEM, Vrije Universiteit Brussel, and International Solvay Institutes,Pleinlaan 2, B-1050 Brussels, Belgium Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK CERN, Theory Division, 1211 Geneva, Switzerland.
Abstract.
We introduce
Rosetta , a program allowing for the translation between different bases of ef-fective field theory operators. We present the main functions of the program and provide an example ofusage. 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 Carloevent generators. In addition to popular bases choices, such as the Warsaw and Strongly Interacting LightHiggs 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 bestraightforwardly performed.MCNET-15-25
The start of a second LHC experimental era raises new hopesto detect physics beyond the Standard Model (BSM). Thehigh energy of the experiment increases the chances of a di-rect discovery of new physics resonances, while a combina-tion of high energy and high luminosity favors the possibleobservation of new phenomena via Standard Model (SM) pre-cision tests. Interestingly the latter offers a complementary andmodel-independent tool for BSM searches if the results are in-terpreted in the context of an effective field theory (EFT). TheEFT indeed captures in a general way the low-energy effectsof heavy new physics from a bottom-up perspective. More pre-cisely, it systematically organizes possible departures from theSM as an expansion in the energy at which the processes ofinterest occur over the (high) new physics scale, and simulta-neously provides a dictionary to interpret these departures inthe context of explicit BSM models.Given the SM field content (including a single Higgs dou-blet), assuming baryon and lepton number conservation, flavoruniversality and a linear realization of the electroweak symme-try, the leading effects implied by an EFT description consistof dimension-six operators that are supplemented to the SMLagrangian. At this order, 59 (76 real) new independent coef-ficients [1, 2] capture all possible deformations from the SM.Despite this large number of new free parameters, importantclasses of observables ( e.g. , Higgs production and decay or Z -pole observables) depend on a much smaller subset of parame-ters [4–9]. Owing to that, the EFT approach is not only useful Relaxing flavor universality, the number of independentdimension-six operators grows to 2499 [3]. for parameterizing BSM searches but is also testable per se bylooking at correlations among the expected signatures.Another important aspect of the EFT approach is the choiceof the operator basis, so that a given physical effect could bemodeled by different combinations of operators at a fixed or-der in the EFT expansion. This well-known fact is related tothe possibility of redefining the SM fields in such a way thatthe zeroth order Lagrangian in the EFT expansion ( i.e. , theSM Lagrangian) is unaltered, while combinations of the first-order operators ( i.e. , dimension-six operators) proportional tothe SM equations of motion can be eliminated up to sublead-ing higher-dimensional effects. For this reason, different com-plete and non-redundant operator bases have been proposed inthe literature, sharing the same physical predictions but hav-ing different advantages. The most popular choices include theso-called
Warsaw basis [2],
SILH (strongly interacting lightHiggs) basis [10, 11] and
BSM primaries basis [6, 12, 13]. TheWarsaw basis represents the first set of non-redundant opera-tors that has been proposed and is particularly appropriate forcomparisons with BSM theories that modify the interactionsof the SM fermions. In contrast, the SILH basis has been de-signed to capture the effects of universal theories where newphysics mostly couples to the SM bosons. Finally, the BSM pri-maries basis is more suitable for a bottom-up approach sinceit is formulated in terms of mass-eigenstates and has a moretransparent connection to measurable quantities, its operatorsbeing aligned with physical observables.Given these multiple viewpoints, it is cumbersome to ex-press the experimental results in a basis-independent mannerthat can be readily interpreted in any of the above-mentionedframeworks. On the other hand, different bases may be conve- a r X i v : . [ h e p - ph ] D ec Falkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: RosettaBasis Underlying gauge symmetry Fields used in the LagrangianWarsaw, SILH SU (3) C × SU (2) L × U (1) Y Gauge-eigenstatesBSM primaries, Higgs SU (3) C × SU (2) L × U (1) Y Mass-eigenstatesHiggs/BSM characterisation SU (3) C × U (1) EM Mass-eigenstates
Table 1.
Main features of the different EFT basis choices discussed in this document.nient for particular applications, either because they facilitatethe comparison with a given class of BSM theories or simplybecause different experimental analyses look more transparentin a specific basis. For instance, the Warsaw basis contains anapparent blind direction with respect to the electroweak preci-sion tests [6,14], which introduces large theoretical correlationsamong all LEP constraints. As a result, the bounds on thestrength of the dimension-six interactions appear less trans-parent [15]. The SILH basis has a similar drawback yieldinga correlation between LEP2 and LHC constraints, while thedownside of the BSM primaries basis lies in the comparisonwith explicit BSM models that is complicated. The
Rosetta package that we present in this paper has been designed to ex-plicitly solve such problems by allowing for a straightforwardtranslation between different EFT languages.In addition to translating, another important goal of the
Rosetta program is to provide a platform for communicationwith Monte Carlo event generators, no matter which EFT basisis chosen. To achieve this, we have implemented in
Roset-ta the
Higgs basis for EFT operators that has been recentlydesigned by the LHC Higgs Cross Section working group (LHC-HXSWG) [16]. This proposal, built on the BSM primaries basis(see Ref. [13]), combines two ingredients. First, all possibleoperators of dimension up to six are written in terms of theSM mass-eigenstates ∆ L (mass) = (cid:88) i c i Λ d i O i ( G aµ , W ± µ , Z µ , A µ , h, t, b, ν τ , τ, . . . ) , (1)where the operators O i have a mass dimension ranging fromtwo to six. The dimensionless coefficients c i are then sup-pressed by an appropriate power d i of the high-energy scale Λ , with d i = − , · · · ,
2. We refer the ensemble of operatorsincluded in the resulting Lagrangian, which is in spirit verysimilar to the Higgs characterisation Lagrangian of Ref. [17],as the
BSM characterisation (BSMC) Lagrangian. Due to thelack of manifest SU (2) L × U (1) Y invariance, the BSMC La-grangian is associated with a larger number of independentcoefficients compared to the Warsaw, SILH or BSM primariesbases. For this reason, the second ingredient defining the Higgsbasis consists of relations among the c i coefficients that re-store the full SU (3) C × SU (2) L × U (1) Y symmetry. As sum-marized in Table 1, the BSM primaries and Higgs Lagrangiansare both of the form of ∆ L (mass) , but they additionally includeconstraints among the different Wilson coefficients that renderthe Lagrangian invariant under the electroweak symmetry. Incontrast, the Warsaw and SILH basis Lagrangians are directlywritten in terms of the SM gauge-eigenstates, ∆ L (gauge) = (cid:88) i (cid:48) c (cid:48) i Λ O i (cid:48) ( G aµ , W iµ , B µ , Φ, Q L , u R , d R , L L , e R ) , (2) and are manifestly symmetric under the electroweak symmetrygroup.We have implemented the mass basis Lagrangian ∆ L (mass) into FeynRules [18] and tuned the output format of Ro-setta so that the translation maps an EFT Lagrangian givenin a specific basis to ∆ L (mass) and generates an output filethat is compatible with the FeynRules implementation. As aconsequence, any high-energy physics tool (and in particularany Monte Carlo event generator) that is interfaced to
Feyn-Rules can be employed within the context of any EFT basisof operators that is included in
Rosetta .With the advent of automated next-to-leading order (NLO)accurate Monte Carlo event generation software, it is impor-tant that
Rosetta remains flexible enough to eventually pro-vide compatibility with this new generation of tools. Recentprogress has been made on the theory side both in implement-ing the linear dimension-six description discussed above in the
FeynRules framework [20] and in calculating the renormal-isation group (RG) evolution of the full set of operators andtheir mutual mixing [3, 21–23]. In the former case,
Rosetta can simply be extended to provide an output compatible withthe eventual NLO model implementation, analogously to theBSMC Lagrangian. The latter case of evaluating the RG run-ning effects, while being a slightly separate issue, highlightsa key feature of our tool, given that the calculation of theseeffects has only been performed in the original
Warsaw basisof Ref. [2]. The framework provided by
Rosetta allows forthe application of these results in any desired basis. Althoughthe initial version of the software does not explicitly deal withthese effects, its translation functionality can already be usedin their context and we plan for future versions to incorporateRG evolution of the SM EFT Wilson coefficients.The remainder of this paper is organized as follows. In Sec-tion 2, we describe the basic functionalities of
Rosetta andhow to make use of the program. Section 3 is dedicated to anexample of usage of
Rosetta in which we focus on new physicsHiggs couplings to the SM bosons. We express them in differ-ent bases and detail the output that is provided by
Rosetta .Our work is summarized in Section 4.
The aim of
Rosetta is to provide a modular and flexible pack-age for EFT basis translation and communication with eventgeneration tools. The primary framework which
Rosetta hasbeen designed to translate into is the phenomenological effec-tive Lagrangian, ∆ L (mass) , which will be explicitly defined inSection 3.1. The motivation for this choice lies in the avail-ability of an implementation within the FeynRules frame- Implementations of the Higgs characterisation [17] and theSILH basis [19] Lagrangians are also available.alkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosetta 3work [18], to be downloaded from the
FeynRules model repos-itory [24], which ensures the link with event generators andhigh-energy physics programs [25, 26].The most basic functionality of
Rosetta is to map a cho-sen set of input parameters (the Wilson coefficients in a spe-cific basis choice) onto the BSMC coefficients such that theoutput can be employed within tools relying on a BSMC ba-sis description. As long as the input format respects the con-ventions sketched in Section 2.2 and that are inspired by theSupersymmetry Les Houches Accord (SLHA) [27, 28], the usermay define his/her own map to the BSMC coefficients (or toany other basis implementation) and proceed with event gen-eration using the accompanying
FeynRules implementation.This highlights one of the key features of
Rosetta , the pos-sibility to easily define one’s own input basis and directly useit in the context of many programs via the translation func-tionality of
Rosetta . The strength of this approach is thatit is much simpler than developing from scratch new modulesfor existing tools in the context of a new basis. To this end,
Rosetta not only enables the translation of an EFT basis intothe BSMC Lagrangian, but also allows for translations into anyof the other bases included in the package, i.e. , currently theHiggs, Warsaw and SILH bases. Translations between thesethree bases in any direction are possible, so that the addi-tion of a new basis by the user only requires the specificationof translation rules to any one of the three core bases. One issubsequently able to indirectly translate the new basis into anyof 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 Section 2.4.
The latest release of
Rosetta can be obtained from http://rosetta.hepforge.org
The package contains a
Python executable named tran-slate , an information file named
README and two directories,a first folder (named
Cards ) collecting example input files anda second folder (named
Rosetta ) including the source codeof
Rosetta . The executable takes as input an SLHA-style pa-rameter file with the coefficients of the dimension-six operatorsassociated with a particular basis. Information on the formatof such an input file can be found in Section 2.2. The executionof the translate script from a shell yields the generation ofan output parameter file where all parameters are this timethe coefficients of the dimension-six operators associated witha specified new basis, the default choice being the BSMC La-grangian. The tool can be used by typing in ./translate PARAMCARD.dat OPTIONS where
PARAMCARD.dat is the name of the SLHA-style inputfile and
OPTIONS stands for optional arguments. The lattercould consist of one or more of the following choices that willmodify the behavior of the program. -h or --help This displays a help message and ex-its the program. -o or --output This allows for the specification ofthe name of the output file, that isby default
PARAMCARD new.dat . -s or --silent The program suppresses warningsand takes the default answer to anyquestion that may have to be askedto the user. -t or --target This allows for providing the nameof the basis into which the transla-tion occurs, the default being bsmc and the other acceptable choices be-ing higgs , silh or warsaw . -w or --overwrite This allows the program to over-write any pre-existing output file. -e or --ehdecay This allows to use the interface withthe e
HDecay program [11] for thecalculation of the Higgs boson widthand branching fractions. See Sec-tion 2.5.2. -f or --flavor This allows to specify the treatmentof the flavor structure relevant forthe fermionic operators, the defaultbeing general and the other ac-ceptable choices being universal and diagonal . See Section 2.5.1. -d or --dependent This allows the program to alsowrite out any dependent parameterscalculated by the translation func-tion to the output file.On run time,
Rosetta starts by performing several checkson the input parameters and verifies the consistency of the in-put file with respect to the specifications of the internal basisimplementation. In this way, any missing SM inputs (with re-spect to the requirements included in the required inputs and required masses attributes of the basis class, see Sec-tion 2.3) can be included using the value provided in the Par-ticle Data Group (PDG) review [29], while any missing coeffi-cient associated with an operator that is present in the basis(and thus declared in the independent attribute of the basisclass, see Section 2.3) can be included with a zero value.Once the translation is achieved,
Rosetta outputs a newparameter file that is by default named
PARAMCARD new.dat .This file contains the values of all parameters relevant for thetarget basis and also includes the necessary modifications tothe input parameters, such as the W -boson mass that maydepend on some dimension-six operator coefficients. Rosetta requires input parameters to be given under the formof a file encoded in a format similar to the SLHA one detailedin Refs. [27, 28]. Parameters are grouped into blocks and eachparameter is identified inside its own block by one or moreinteger numbers called counters. For instance, the SM inputsnecessary for the definition of the SILH basis would read
BLOCK SMINPUTS where the different entries respectively correspond to the in-verse of the electromagnetic coupling constant ( aEWM1 ), the Falkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: RosettaFermi constant ( Gf ), the strong coupling constant ( aS ), the Z -boson mass ( MZ ) and the Higgs boson mass ( MH ). Inspiredby the usual SLHA conventions, all masses are also collectedinto a block called MASS where the counters correspond tothe PDG identifiers of the particles [29]. Furthermore, matrixquantities receive a block of their own with counters specify-ing the position inside the matrix. In this way, a single blockwould be needed to encode, for instance, the c Hud coefficientsassociated with the O Hud operator of the Warsaw basis that isdefined by O Hud = − i (cid:2) ¯ uγ µ d (cid:3)(cid:2) ˜ Φ † D µ Φ (cid:3) . (3)In this expression, u and d denote the SU (2) L singlets of right-handed up-type and down-type quark fields, respectively, and Φ and D µ Φ stand for the weak doublet of Higgs fields and itsgauge-covariant derivative. In flavor space, the c Hud coefficientstake the form of a matrix, implemented in the input file as
BLOCK WBxHud1 1 0.1e+00
The block name contains information on the basis ( WB ) andon the considered operator ( Hud ). Sample parameter files forall core bases can be found in the
Cards directory shippedwith the program. Within those files, we have adopted theabove block naming scheme. The name of each block startswith two letters identifying the basis ( BC , HB , SB and WB forthe BSMC, Higgs, SILH and Warsaw bases respectively) thatare followed by a separator ( x ), and ends with the name ofthe considered coefficient as it is defined in the LHCHXSWGproposal for an EFT basis choice [16]. In the case of EFToperators independent of fermions, the related (non-matrix)coefficients are collected in different blocks as a function of theLorentz structure of the operators. For instance, the SBxV2H2 block will include all operators of the SILH basis containing twooccurrences of the Higgs field and two occurrences of the gaugefields. Their ordering follows their order of appearance in theLHCHXSWG proposal. The imaginary parts of all parametersare stored in corresponding blocks whose names are prefixedwith the IM tag.The Rosetta package contains built-in methods for deal-ing with an SLHA-like structure, and these methods have allbeen implemented in the
Rosetta/SLHA.py file. When aninput file is read, the parser included in the
SLHA.py file rec-ognizes the existing
BLOCK and
DECAY structures of the inputfile and stores them as instances of the
SLHA.NamedBlock , SLHA.NamedMatrix and of the
SLHA.Decay classes. Theseare dictionary-like objects that can be assigned, indexed and it-erated over as regular
Python dictionaries. An
SLHA.Named-Block object reflects the information embedded in an SLHAblock so that it possesses a name attribute and stores valuesassociated with integer keys as well as a mapping from theinteger keys to the parameter names. In this way, parameterscan be accessed by indexing either their integer key or theirname. Similarly,
SLHA.NamedMatrix objects function anal-ogously but operate with a pair of indices for indexing. An Counter Parameter
Rosetta name1 The inverse of the elec-tromagnetic coupling con-stant α − aEWM1 G F Gf α s aS Z -boson mass m Z MZ m b MB m t MT m τ MTAU
25 The Higgs boson mass m H MH Table 2.
Identifying counters of the
SMINPUTS block withthe SM parameters allowed to be used within
Rosetta . Thisgeneralizes the SLHA standards where the Higgs mass is ig-nored [28]. The internal names used by
Rosetta are also given.
SLHA.Decay object contains an integer attribute
PID that isthe PDG identifier of the particle whose decays are describedby the considered block, as well as a total attribute allowingfor the storage of the total width. Individual decay channelsare then indexed by tuples of PDG codes associated with thedecay products, and the stored values are the branching frac-tions. Finally, the
SLHA.py file also includes the definition ofthe
SLHA.Card class that serves as a container for a collectionof instances of the above objects. The implementation of anybasis in
Rosetta therefore requires the user to provide defini-tions for the blocks and parameters to be specified in the inputfile that will be read into an
SLHA.Card instance belonging tothat basis class. More practical information and examples aregiven in Sections 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
Roset-ta must read the input file and informs the program on theother blocks it should look for, based on the structure speci-fied in the implementation of that particular basis. This nameshould be a single unique string with no spaces. The next twomandatory blocks consist of conventional input blocks speci-fying the values of the SM inputs and of the particle masses.The set of required inputs will depend on the specificationsin the corresponding basis implementation. Moreover, the usercan optionally specify the value of the elements of the CKMmatrix by setting their real and imaginary parts within the
VCKM and
IMVCKM blocks. If absent, the information of thePDG review [29] is used by
Rosetta . All extra blocks anddecay structures are stored, left unchanged and passed to theoutput file unless the user demands to use the e
HDecay pro-gram, which will overwrite any existing decay information onthe Higgs.
Rosetta is a
Python package containing the implementationof a
Basis class equipped with several utility functions forreading, processing and writing SLHA-style parameter files.Working implementations of bases are derived from this classalkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosetta 5and only require a small amount of information specifying theblock structure of the EFT parameters, the required SM inputsand a series of translation functions to other existing basisimplementations. In order to be able to define a new basisclass, we describe in this section the properties of the
Basis objects.The
Basis class has a number of intrinsic data membersthat should be defined in order to get a working implemen-tation of an EFT basis. These consist of the independent , required inputs and required masses attributes alreadymentioned in Section 2.1, together with the name , blocks and flavored members of the class. name Unique string identifier for the basisimplementation, e.g. , higgs , bsmc , silh or warsaw for the core basesshipped with the package. independent List of strings containing the names ofthe independent EFT operator coeffi-cients of the basis. These are expectedto be present in the input parameterfile. required inputs
Set of integers containing the SLHAcounters of the required SM inputs.See Table 2 for the complete list ofthose allowed in
Rosetta . required masses Set of integers containing thePDG identifiers of the particleswhose masses are required as in-put and that are not included in required inputs . blocks Dictionary with the non-matrixSLHA block names as keys and listsof coefficients stored in that block asvalues. flavored
Dictionary with matrix SLHA blocknames as keys. The values are otherdictionaries with the keywords kind , domain and cname as keys. This de-scribes the properties of the matrices.In the case of the definition of a matrix block, the self-explana-tory 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 ensuingmatrix object will depend on the choice of these keywords. Thename to be given to the individual EFT coefficients are derivedfrom the value of the keyword cname . Conventionally, the realand imaginary components are prefixed with the letters R and I respectively, while the position ( i, j ) in the matrix is referredto 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 basisclass is created and populated with the information providedas 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 theSM parameters, while the CKM matrix is stored into the ckm container of the basis class. In the
Rosetta framework, theEFT operator coefficients are implemented as elements of therelevant basis class and can be accessed via standard
Python methods. For instance, all the coefficients associated with abasis object named
MyBasis could be printed, together withtheir values, by coding for k, v in MyBasis.items():print k, v
In addition, a direct accessor to each EFT operator coefficientis created from its name, which facilitates the implementationof the translation functions that in general extensively refer-ence individual parameter values. This however assumes thatthere are no duplicate parameter names in the SLHA-like in-put file, which nevertheless leads to a program exception. Thereare hence multiple ways to access a given parameter. For ex-ample, 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
MyBasis[’A’]MyBasis.card[’A’]MyBasis.card.blocks[’MyBlock’][’A’]MyBasis.card.blocks[’MyBlock’][3]
In the lines above, the parameter A is respectively accessedfrom the MyBasis object, from the
SLHA.Card instance asso-ciated with the current basis and from the
SLHA.NamedBlock object associated with the
MyBlock block (using either the pa-rameter name or the counter as an index).
One of the important features of
Rosetta is the intended easewith which a user can define a new basis class to suit his/herspecific physics needs. In the context of an ultraviolet com-plete model, he/she may be interested in the phenomenologi-cal consequences of a particular high-scale scenario in the EFTframework. Imagining that he/she has derived all dimension-six Wilson coefficients in a particular basis,
Rosetta could beused to map these coefficients to the
FeynRules effective La-grangian implementation in the mass-eigenstate basis so thatthe collider phenomenology of such a scenario could be inves-tigated. This task is realized by implementing a new basis in
Rosetta and by connecting the new basis input parametersto those of one of the existing core basis implementations.Alternatively, the user may have developed a particularresource performing a useful analysis or calculation in a non-standard basis choice. The corresponding basis implementationin
Rosetta with a translation to one of the core bases couldthen allow one to use this tool in the context of all other ex-isting basis implementations in
Rosetta and therefore greatlywiden its scope. The e
HDecay feature of
Rosetta is an ex-ample of this, as it works with a set of operators correspondingto the SILH basis. Via
Rosetta , e
HDecay is now available forcalculations in the SILH, Warsaw and Higgs bases, as well asin any additional basis that may be implemented in the future.In this section, we provide an example that outlines the ba-sic requirements for implementing a new basis in
Rosetta . Wealso refer the reader to the file
Rosetta/TemplateBasis.py which serves as a concrete toy example that can be used as atemplate for creating a new basis class as well as the core basisimplementations 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 read-ing, writing and translating so that a new basis implementa-tion solely demands the user to create a new file that must be Falkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosettasaved in the
Rosetta directory and that includes the decla-ration of a
Basis subclass. The user has then to define theclass attributes described in Section 2.3. First, it is essentialthat the name of the basis class matches the filename in whichit is saved minus the extension in order to ensure a properrunning of the program. Second, the independent , blocks and flavored attributes of the class define the input param-eters of the basis and their desired SLHA-like structure, whilethe required inputs and required masses lists are speci-fied according to the needs of the translation functions that areplanned to be implemented. One can also specify a dependent attribute to explicitly define a particular parameter as depen-dent. For instance, the following code refers to the implementa-tion of a new basis class called MyBasis and has been includedin the file
Rosetta/MyBasis.py . from internal import Basisclass 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 as-sumed to be organized into the two SLHA blocks LETTERS and
NUMBERS . A flavored matrix,
MYxMAT , is also present anddeemed to be an independent input parameter except for its(3,3) component that is explicitly included within the depen-dent attribute of the basis class. The translation methods tobe implemented require the knowledge of six SM masses andparameters that must be specified via the required inputs and required masses attributes of the basis class. In ourcase, the electroweak inputs α − , G F and m Z are connectedto the SMINPUTS block of the SLHA-like input structure, whilethe W -boson, Higgs boson and top quark masses are connectedto its MASS block. The extra parameters C and three are de-pendent parameters that the user has to define in terms of theindependent and SM parameters (see below). The non-SM partof an illustrative input file could be BLOCK BASIS1 mybasis while its SM part would include the
SMINPUTS and
MASS blocks with values for the six above-mentioned SM inputs, aswell as the two blocks related to the CKM matrix in the casewhere one would be interested in using non-default values forits elements. Only the relevant elements of
MYxMAT need beprovided 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 methodnamed calculate dependent() that must be provided bythe user. Continuing with the example above, we include in thenew 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 ofthe A and B parameters, that the three parameter equals halfof the difference of the one and two parameters and that the(3,3) entry of the MYxMAT matrix is equal to 10 times the valueof its (2,2) entry.When executed,
Rosetta begins with the reading of the in-put file and next calls the calculate dependent() methodfor evaluating the remaining basis parameters.
Rosetta fi-nally performs the translation to another basis by using thetranslation methods defined by the user. Their implementa-tion requires the use of a translation decorator with anargument that refers to the name of the target basis and thatmust match a basis implementation contained in the
Rosetta directory. For example, we could link the mybasis basis aboveto 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(...) oneabove take an instance of the target basis class as their onlyargument and return it after setting its parameter values. Re-lations involving (matrix) parameters with a flavor structureshould be performed in a flavor general way, as discussed inSection 2.5.1.If modifications to the SM input parameters need to bemade ( i.e. , the mass and inputs attributes of the basis class),the function modify inputs() must be implemented simi-larly to the calculate dependent() method. The followingexample defines a shift of the W -boson mass by the A param-eter, def modify_inputs(self):self.mass[24] = self.mass[24] + self[’A’] In general, the user-defined functions may require the evalu-ation of parameters such as the weak and hypercharge gaugecouplings or the electroweak mixing angle. The choice of rela-tions ( e.g. , tree- or loop-level) to be used to consistently derivethese parameters from the inputs is left to the user. In the corebases provided with
Rosetta , the calculate inputs() alkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosetta 7method relies on tree-level relations to deduce all the SM pa-rameters.Having defined a basis class according to these specifica-tions,
Rosetta is able to detect the presence of the basis imple-mentation and to automatically construct possible translationpaths to other existing bases from the user-defined translationfunctions. The recognition of the implemented basis by
Ro-setta is also reflected in the help message accompanying the translate script, the name of the new basis appearing as oneof the possibilities for the target basis option.
In the general case, each matrix block of the input file includesone entry for each possible flavor assignment of the correspond-ing operator. The flavor option of the translate executableintroduced in Section 2.1 allows the user to make assumptionson the flavor structure of the operators so that
Rosetta readsinput files and generates output files with a simplified blockstructure (unless the BSMC basis is used as it requires all co-efficients to be specified). Setting this option will act on all ofthe matrix parameters declared in the flavored attribute ofa basis class implementation. The flavor option can be fixed ei-ther to universal where all matrices of operator coefficientsare proportional to the identity or to diagonal where onlytheir flavor-diagonal 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 ele-ments must be provided. Sample input files can be found in the
Cards directory of the program. In the definitions of the threecore bases, the flavor-symmetry-breaking Yukawa-like opera-tors are normalized by the masses of the fermions such thatthe universal flavor option will lead to a minimally flavor-violating (MFV) structure where the physical effects of the co-efficients are scaled by the corresponding fermion masses [30].For example, in the Higgs basis, these Yukawa-like terms arewritten as: ∆ L Yuk = (cid:113) m if m jf δy ijf ¯ f i (cid:16) cos φ ij − iγ sin φ ij (cid:17) f j . (4)The corresponding normalizations are also used for the War-saw and SILH basis implementations in Rosetta to simplifythe translations and also the possibility of encoding MFV intoany EFT description. The same argument applies to the dipoleoperators, O fV , as well as the O Hud operator mentioned inSection 2.2. The former set of operators breaks the flavor sym-metry in an identical way to the Yukawa-like operators and willhence receive the same (cid:113) m if m jf normalization. In the lattercase, the flavor structure of the operator requires two Yukawainsertions as it is composed of right-handed quarks only. More-over, being a charged-current operator, the MFV constructionrequires the insertion of the CKM matrix such that the oper-ator is normalized as O Hud = − im iu m jd V ijCKM (cid:2) ¯ u i γ µ d j (cid:3)(cid:2) ˜ Φ † D µ Φ (cid:3) . (5)Since this particular operator is unique and maps to a singleoperator in all of the other core bases, the corresponding trans-lations remain unaffected. This normalization is however notthe same as the one chosen in Ref. [16]. Users should therefore bear in mind these normalizations which have been chosen tosingle out operators that explicity break the flavor symmetryof the Lagrangian. That being said, they are merely a conve-nient way for the user to implement MFV and can be workedaround if the user so desires. Rosetta recognizes coefficients by their names so that thenaming of the elements of the matrix coefficients must respectthe conventions described in the previous section for their real(an R prefix) and imaginary (an I prefix) parts, and for theirposition ( i, j ) inside the matrix (an ixj suffix). Implementingtranslations from flavored parameters should ideally always bedone in the most general case such that the various flavor op-tions work correctly. To this aim, basic matrix algebra oper-ations 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 tomatrix multiplication, addition, subtraction and assignmentrespectively. They can be used to assign values to a matrixSLHA block according to the result of a specific operation be-tween two other matrix SLHA blocks. These functions requiretwo mandatory arguments for the objects between which theoperation should be performed and a third optional argumentspecifying the matrix block to which the result of the operationshould be assigned. For instance, matrix mult(M1,M2,M3) assigns to the matrix M3 the result of the multiplication of thematrices M1 and M2 . If the M3 argument is omitted, a genericmatrix object is returned such that matrix utility functions canbe combined. The matrix eq(M1,M2) method is the only ex-ception. It takes two mandatory arguments M1 and M2 andallows for the assignment of the elements of the M1 matrix tothe M2 matrix.A concrete example can be found in the calculate de-pendent() function included in the Higgs basis implementa-tion. The deviations of the W -boson couplings to the weak dou-blet of left-handed quark fields δg W q L are related to those of the Z -boson couplings to the individual left-handed up-type anddown-type quarks δg Z u L and δg Z d L via the CKM matrix V CKM , δg W q L = δg Z u L · V CKM − V CKM · δg Z d L . (6)The Rosetta implementation of this relation makes use of acombination 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 argu-ment of the matrix sub method allows one to assign the resultof the matrix subtraction to the elements of the HBxdGLwq ma-trix block. Matrix blocks also come with the
T() and dag() methods for transposing and Hermitian conjugation.
HDecay
In order to calculate dimension-six operator contributions tothe Higgs boson width and branching ratios,
Rosetta includesan interface to the e
HDecay program [11]. It can be switchedon by executing the translate script with the e
HDecay op-tion (see Section 2.1). In order to use this feature, the pathto a local installation of e
HDecay on the user system should Falkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosettabe specified in the
Rosetta/config.txt file, next to the eHDECAY dir keyword, and a (possibly indirect) translationlinking the basis of interest to the SILH basis should exist. Ifso, the translation will be performed, e
HDecay will be runinternally and an SLHA decay block for the Higgs boson willbe appended to the output file.Since the SILH basis description adopted in e
HDecay as-sumes the MFV paradigm, an additional layer of translation isinternally performed by
Rosetta to render its internal SILHbasis implementation MFV-compliant. Details can be found inSection 3.4.
In this section, we discuss the BSMC Lagrangian containing re-dundant parameters that is the default basis which
Rosetta has been designed to translate into. We explain the relationswith the non-redundant Higgs, Warsaw and SILH bases andfocus on a subset of operators connected to single Higgs pro-duction at the LHC to provide examples of usage of
Rosetta . To study the Higgs boson properties in detail at the next LHCruns, the LHCHXSWG has proposed a parameterization ofanomalous interactions of the SM fermions, gauge bosons andthe Higgs boson allowing both for a transparent linking tophysical observables and for an easy implementation in Monte-Carlo event generators [16]. The framework is that of a gen-eral effective Lagrangian defined in the mass-eigenstate basis,where all kinetic terms are canonically normalized and diag-onal, and where all mass terms are diagonal. Moreover, thetree-level relations between the gauge couplings and the usualelectroweak input parameters ( G F , α (0), m Z ) are the same asin the SM. In such a frame, i.e. , in the BSMC Lagrangian,the coefficients of the interaction terms in the Lagrangian arerelated in an intuitive way to quantities observable in exper-iment, and any parameter in the effective Lagrangian can bemeasured.The BSMC Lagrangian captures all physics effects thatmay arise in the presence of lepton-number and baryon-numberconserving dimension-six operators beyond the SM. However,it is more general than a basis defined before electroweak sym-metry breaking as it contains more free parameters. This isbecause the SU (3) C × SU (2) L × U (1) Y gauge symmetry lin-early realized at the level of an operator basis implies rela-tions between different couplings of the effective Lagrangiandefined after electroweak symmetry breaking. The latter in-deed only respects the SU (3) C × U (1) EM symmetry. In par-ticular, the charged and neutral gauge boson interactions arerelated, as are those with zero, one and two Higgs bosons.These relations are not implemented at the level of the BSMCLagrangian so that it may be used to study more general the-ories such as when the electroweak symmetry is non-linearlyrealized or when some operators of dimension greater than sixare included.The Higgs basis has been proposed as a convenient pa-rameterization of another non-redundant dimension-six EFTbasis. In this approach, the relations (that hold in any non-redundant dimension-six basis of EFT operators) between dif- ferent couplings of the BSMC Lagrangian required by a lin-early realized SU (2) L × U (1) Y local symmetry are enforced.Furthermore, the Higgs basis has been defined by choosing aspecific subset of independent parameters from all couplingsof the BSMC Lagrangian. The choice of the independent cou-plings is motivated by their direct connection to observablesconstrained by electroweak precision and Higgs studies. Thisapproach is similar to the one introduced in Ref. [6], exceptthat a different subset of couplings has been chosen, and thenumber of independent couplings is the same as for any basisof non-redundant dimension-six operators. Moreover, there ex-ists a linear one-to-one invertible transformation between theindependent couplings of the Higgs basis and the Wilson coeffi-cients in any basis. The remaining BSMC Lagrangian couplingsare all dependent parameters that can be expressed in termsof 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 andother bases, we focus on a part of the Lagrangian describ-ing the CP -even interactions of the Higgs boson with two SMgauge bosons. After denoting by G aµ , W ± µ , Z µ , A µ and h thegluon, the W -boson, the Z -boson, the photon and the Higgsboson fields, respectively, the relevant part of the Lagrangianreads ∆ L h = hv (cid:104) δc w m W W + µ W − µ + δc z m Z Z µ Z µ + c gg g s G aµν G µνa + c ww g W + µν W − µν + c zz g c θ Z µν Z µν + c zγ gg (cid:48) Z µν A µν + c γγ g (cid:48) c θ A µν A µν + c w (cid:3) g (cid:0) W − µ ∂ ν W + µν + h . c . (cid:1) + c z (cid:3) g Z µ ∂ ν Z µν + c γ (cid:3) gg (cid:48) Z µ ∂ ν A µν (cid:105) . (7)In our notation, c θ ( s θ ) stands for the cosine (sine) of the elec-troweak mixing angle, v for the vacuum expectation value ofthe neutral component of the Higgs doublet Φ , and g s , g and g (cid:48) are the strong, weak and hypercharge coupling constants.Moreover, we have introduced the field strength tensors of thegauge bosons that we define as V µν = ∂ µ V ν − ∂ ν V µ for V = W ± , Z and A ,G aµν = ∂ µ G aν − ∂ ν G aµ + g s f abc G bµ G cν , (8)in which f abc are the structure constants of SU (3) C .The Lagrangian above contains ten real coupling parame-ters that are all independent in the BSMC picture. However,if ∆ L h originates from an EFT with dimension-six operators,only six of these couplings are independent and the remain-ing four can be expressed in terms of these six and of the SMparameters. In the Higgs basis, δc z , c gg , c zz , c zγ , c γγ and c z (cid:3) are chosen as independent parameters and the four remainingcouplings are calculated as δc w = δc z + 4 δm ,c ww = c zz + 2 s θ c zγ + s θ c γγ ,c w (cid:3) = g c z (cid:3) + g (cid:48) c zz − ( g − g (cid:48) ) s θ c zγ − g s θ c γγ g − g (cid:48) ,c γ (cid:3) = 2 g c z (cid:3) + ( g + g (cid:48) ) c zz − ( g − g (cid:48) ) c zγ − g s θ c γγ g − g (cid:48) . (9)alkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosetta 9In 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 ∆ L mass = 2 δm g v W + µ W − µ . (10)An input file (that we name HiggsBasis.dat in our ex-ample) describing this part of the Higgs basis Lagrangian wouldbe of the form
BLOCK BASIS1 higgs
It includes, in addition to the blocks above, the SM parametersas well as vanishing values for all other EFT coefficients. Inorder 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 nextgenerates an output file named
HiggsBasis new.dat givenin the framework of the BSMC Lagrangian. This file contains inparticular values for the four δc w , c ww , c w (cid:3) and c γ (cid:3) dependentparameters, the corresponding output block being, accordingto Eq. (9), BLOCK BCxh1 +1.00000e-01
In addition, the
HiggsBasis new.dat file also includes extranon-vanishing coefficients that are linked to the six indepen-dent parameters δc z , c gg , c γγ , c zγ , c zz and c z (cid:3) by gauge in-variance. For instance, a di-Higgs coupling to two gluonic fieldstrength tensors is present, BLOCK BCxhh4 1.00000e-01
The ten interaction terms of the ∆ L h Lagrangian introducedin Section 3.1 can be seen as generated by six independentoperators of the Warsaw basis, ∆ L W h = 1 v (cid:104) c GG g s Φ † Φ G aµν G µνa + c WW g Φ † Φ W iµν W µνi + c WB gg (cid:48) Φ † σ i Φ W iµν B µν + c BB g (cid:48) Φ † Φ B µν B µν + c H ∂ µ (cid:2) Φ † Φ (cid:3) ∂ µ (cid:2) Φ † Φ (cid:3) + c T (cid:2) Φ † ←→ D µ Φ (cid:3)(cid:2) Φ † ←→ D µ Φ (cid:3)(cid:105) . (11) In this expression, we have introduced the Pauli matrices σ i ,the Hermitian derivative operator, Φ † ←→ D µ Φ = Φ † ( D µ Φ ) − ( D µ Φ † ) Φ , (12)the gauge-covariant derivative and the hypercharge and weakfield strength tensors D µ Φ = (cid:16) ∂ µ − i gσ k W kµ − i g (cid:48) B µ (cid:17) Φ ,W iµν = ∂ µ W iν − ∂ ν W iµ + g(cid:15) ijk W jµ W kν ,B µν = ∂ µ B ν − ∂ ν B µ . (13)The six Wilson coefficients c GG , c WW , c WB , c BB , c H and c T appearing in ∆ L W h are related to the ten couplings in theeffective Lagrangian ∆ L h as δc w = − c H − g g (cid:48) g − g (cid:48) c WB + 4 g g − g (cid:48) c T − g + g (cid:48) g − g (cid:48) δv ,δc z = − c H − δv ,c gg = c GG ,c ww = c WW ,c zz = g c WW + 4 g g (cid:48) c WB + g (cid:48) c BB ( g + g (cid:48) ) ,c zγ = g c WW − g − g (cid:48) ) c WB − g (cid:48) c BB g + g (cid:48) , (14) c γγ = c WW + c BB − c WB ,c w (cid:3) = 2 g − g (cid:48) (cid:2) g (cid:48) c WB − c T + δv (cid:3) ,c z (cid:3) = − g (cid:2) c T − δv (cid:3) ,c γ (cid:3) = 2 g − g (cid:48) (cid:2) ( g + g (cid:48) ) c WB − c T + 2 δv (cid:3) . Here, δv is defined by δv = 12 (cid:104) ( c (cid:48) H(cid:96) ) + ( c (cid:48) H(cid:96) ) (cid:105) −
14 ( c (cid:96)(cid:96) ) (15)and summarizes the dependence on the additional Warsaw ba-sis operators, iv c (cid:48) H(cid:96) (cid:2) ¯ (cid:96)σ i γ µ (cid:96) (cid:3)(cid:2) Φ † σ i ←→ D µ Φ (cid:3) + 1 v c (cid:96)(cid:96) (cid:2) ¯ (cid:96)γ µ (cid:96) (cid:3)(cid:2) ¯ (cid:96)γ µ (cid:96) (cid:3) . (16)Starting from the Higgs basis example of Section 3.1 whereall independent parameters are fixed to 0.1, we employ Ro-setta to invert the relations of this section and calculate thenumerical values of the Warsaw basis coefficients included in ∆ L W h that would yield the same ∆ L h Lagrangian. Typing ina shell ./translate HiggsBasis.dat -t warsaw we obtain an output file where several non-zero EFT coeffi-cients can be found. The numerical value of those on which wefocus here can be extracted from the generated file, These operators contribute to the muon decay at the treelevel. Taking this into account leads to a shift between themeasured Fermi constant and the vacuum expectation value ofthe Higgs field, which motivates the notation δv .0 Falkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosetta BLOCK BASIS1 warsaw
In our benchmark scenario, the δv shift is vanishing. We now consider the case where all operators included in the ∆ L h Lagrangian of Section 3.1 are induced by a set of operatorsof the SILH basis, ∆ L S h = 1 v (cid:104) s GG g s Φ † Φ G aµν G µνa + s BB g (cid:48) Φ † Φ B µν B µν + s W ig (cid:2) Φ † σ i ←→ D µ Φ (cid:3) ∂ ν W µνi + s B ig (cid:48) (cid:2) Φ † ←→ D µ Φ (cid:3) ∂ ν B µν + i s HW g (cid:2) D µ Φ † σ i D ν Φ (cid:3) W µνi + i s HB g (cid:48) (cid:2) D µ Φ † D ν Φ (cid:3) B µν + s W D µ W iµν D ρ W ρνi + s B ∂ µ B µν ∂ ρ B ρν + s H ∂ µ (cid:2) Φ † Φ (cid:3) ∂ µ (cid:2) Φ † Φ (cid:3) + s T (cid:2) Φ † ←→ D µ Φ (cid:3)(cid:2) Φ † ←→ D µ Φ (cid:3) + i s (cid:48) H(cid:96) (cid:2) ¯ (cid:96)σ i γ µ (cid:96) (cid:3)(cid:2) Φ † σ i ←→ D µ Φ (cid:3)(cid:105) . (17)The ten Wilson coefficients included in ∆ L h can be rewrit-ten in terms of the eleven parameters appearing in ∆ L S h as δc w = − s H − g g (cid:48) ( s W + s B + s W + s B ) g − g (cid:48) − g g − g (cid:48) s T + 3 g + g (cid:48) g − g (cid:48) δv ,δc z = − s H − δv ,c gg = s GG ,c ww = − s WW ,c zz = − g s HW + g (cid:48) s HB − g (cid:48) s θ s BB g + g (cid:48) ,c zγ = − s HW − s HB − s θ s BB , (18) c γγ = s BB ,c w (cid:3) = 12 s HW + g ( s W + s W ) + g (cid:48) ( s B + s B ) − s T + 4 δv g − g (cid:48) ) ,c z (cid:3) = g ( s W + s W + s HW )+ g (cid:48) ( s B + s B + s HB ) − s T +4 δv g ,c γ (cid:3) = s HW − s HB g ( s W + s W )+ g (cid:48) ( s B + s B ) − s T +4 δvg − g (cid:48) , where δv = ( s (cid:48) H(cid:96) ) . Rosetta can be used to extract thenumerical values of the independent SILH parameters by in-verting the above relations. Adopting the benchmark scenarioof Section 3.1 where all the relevant Higgs basis independentparameters 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 fromthe generated output file,
BLOCK BASIS1 silh
An important difference between the definitions of the SILHand Warsaw bases provided in the LHCHXSWG proposal [16]and their original descriptions lies in the forms of the Yukawa-like operators, ∆ L Yuk = ( c f ) ij v (cid:0) Φ † Φ (cid:1) ¯ F iL Φf jR , (19)where F L and f R denote a generic weak doublet of left-handedfermions and a generic weak singlet of right-handed fermionsrespectively. In the original Warsaw basis definition, these Yu-kawa-like operators take the above form. In the LHCHXSWGproposal (on which Rosetta is based), these operators havebeen redefined in a way allowing one to decouple their contri-butions to the fermion masses (that are extracted from appro-priate measurements and thus fixed), as well as to simplify theimplementation of MFV, ∆ L (cid:48) Yuk = − √ m i m j v ( c (cid:48) f ) ij v (cid:16) Φ † Φ − v (cid:17) ¯ F (cid:48) iL Φf (cid:48) jR , (20)where the primes denote fields taken in the mass eigenbasis.The Wilson coefficients c f and c (cid:48) f are related by unitary trans-formations U L and U R that map the field gauge eigenbasis tothe mass eigenbasis with the would-be mass modifications ab-sorbed into the diagonalized Yukawa matrices Y Df , c (cid:48) f = v √ m i m j U † L c f U R and Y Df = U † L Y f U R + c f . (21)In the original SILH basis description, an additional assump-tion of minimal flavor violation is included, such that the flavorstructure is taken aligned with the Yukawa matrices, ∆ L MFVYuk = ( Y f ) ij c MFV f v (cid:0) Φ † Φ (cid:1) ¯ F iL Φf jR . (22)The Wilson coefficients c MFV f are therefore proportional to theidentity matrix in flavor space and are thus universal. Thanksto the convenient normalizations, they are now trivially relatedlinearly to those of the Warsaw and SILH basis descriptions ofthe LHCHXSWG proposal by( c f ) ii = vm i Y Df c MFV f = √ c MFV f . (23)alkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosetta 11These relations are used internally for the e HDecay in-terface of
Rosetta , which takes SILH basis input parame-ters assuming the MFV convention of Eq. (22). In order toconsistently use e
HDecay , Rosetta translates these coeffi-cients from the alternative version of the SILH basis detailedin Ref. [16]. As a consequence, a general flavor structure can-not be employed when making use of the e
HDecay interface.Although it is in principle possible to input different values forthe c MFV c , c MFV b , c MFV t , c MFV µ and c MFV τ parameters (referred inRef. [11] as ¯ c c , etc. ) when running e HDecay on its own, largedeviations from non-universality in these coefficients consist ofa significant departure from the MFV paradigm and shouldnot be used for complete consistency within
Rosetta . In this paper, we have introduced the
Rosetta package, a
Python program dedicated to the translation of a given EFTbasis of independent operators to other viable basis choices.We have also included, in this document, technical details sothat users can easily extend the program and implement theirown choices of EFT operator basis.Currently, the program allows the user to translate bench-marks designed in the Higgs, SILH and Warsaw bases into anyof these three bases. In addition, the code also expresses anyscenario in terms of the BSMC Lagrangian of EFT operators, abasis that has been defined from the Higgs basis after ignoringall relations among the operators that are induced by a linearrealization of the electroweak symmetry. A
FeynRules imple-mentation allows
Rosetta to be linked to other high-energyphysics tools. The relations among the different Wilson coeffi-cients that hold in the context of the Higgs, SILH and Warsawbases of independent operators have been implemented into
Rosetta so that they are preserved when a setup is exportedto the BSMC Lagrangian by the program. This scheme hasthe strength to be easily generalizable to study different setupsproviding a description of the Higgs boson properties, such asthose with a non-linearly realized electroweak symmetry or in-cluding higher-dimensional operators beyond dimension six.In the future, we believe that translations from one basis toanother will allow for broadening the scope and the use of pastcalculations very relevant for precision Higgs physics. Alongthese lines, higher-order calculations in QCD performed in theBSMC Lagrangian [31–33] could be used within any given EFTlanguage, and the renormalization group running of the Wilsoncoefficients, that has been calculated in the SILH basis [34,35] and in the Warsaw bases [3, 21, 22], could be exported todifferent bases too.
Acknowledgments
The authors are grateful to Fabio Maltoni for lively interac-tions during all phases of this project and would also like tothank Christophe Grojean, Alex Pomarol and Michael Trottfor useful comments. This work has been partially supportedby the FP7 Marie Curie Initial Training Network MCnetITN(PITN-GA-2012-315877), by the Belgian Federal Science Pol-icy Office through the Interuniversity Attraction Pole P7/37,by the Strategic Research Program ‘High Energy Physics’ andthe Research Council of the Vrije Universiteit Brussel, by the Swiss National Science Foundation under the Ambizione grantPZ00P2 136932, and by the Theory-LHC-France initiative ofthe CNRS (INP and IN2P3).
References
1. W. Buchmuller and D. Wyler,
Effective LagrangianAnalysis of New Interactions and Flavor Conservation , Nucl.Phys.
B268 (1986) 621.2. B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek,
Dimension-Six Terms in the Standard Model Lagrangian , JHEP (2010) 085, [ arXiv:1008.4884 ].3. R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott,
Renormalization Group Evolution of the Standard ModelDimension Six Operators III: Gauge CouplingDependence and Phenomenology , JHEP (2014) 159,[ arXiv:1312.2014 ].4. A. Pomarol and F. Riva,
Towards the Ultimate SM Fit toClose in on Higgs Physics , JHEP (2014) 151,[ arXiv:1308.2803 ].5. J. Ellis, V. Sanz, and T. You,
Complete Higgs SectorConstraints on Dimension-6 Operators , JHEP (2014) 036, [ arXiv:1404.3667 ].6. R. S. Gupta, A. Pomarol, and F. Riva,
BSM PrimaryEffects , Phys.Rev.
D91 (2015), no. 3 035001,[ arXiv:1405.0181 ].7. J. Ellis, V. Sanz, and T. You,
The Effective StandardModel after LHC Run I , JHEP (2015) 157,[ arXiv:1410.7703 ].8. A. Falkowski and F. Riva,
Model-independent precisionconstraints on dimension-6 operators , JHEP (2015)039, [ arXiv:1411.0669 ].9. A. Efrati, A. Falkowski, and Y. Soreq,
Electroweakconstraints on flavorful effective theories , JHEP (2015) 018, [ arXiv:1503.07872 ].10. G. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, The Strongly-Interacting Light Higgs , JHEP (2007)045, [ hep-ph/0703164 ].11. R. Contino, M. Ghezzi, C. Grojean, M. Muhlleitner, andM. Spira,
Effective Lagrangian for a light Higgs-likescalar , JHEP (2013) 035, [ arXiv:1303.3876 ].12. E. Masso,
An Effective Guide to Beyond the StandardModel Physics , JHEP (2014) 128,[ arXiv:1406.6376 ].13. A. Pomarol,
Higgs Physics , in , 2014. arXiv:1412.4410 .14. C. Grojean, W. Skiba, and J. Terning,
Disguising theoblique parameters , Phys.Rev.
D73 (2006) 075008,[ hep-ph/0602154 ].15. Z. Han and W. Skiba,
Effective theory analysis ofprecision electroweak data , Phys.Rev.
D71 (2005) 075009,[ hep-ph/0412166 ].16. M. Duehrssen-Debling, A. T. Mendes, A. Falkowski, andG. Isidori,
Higgs Basis: Proposal for an EFT basis choicefor LHC HXSWG , .17. P. Artoisenet, P. de Aquino, F. Demartin, R. Frederix,S. Frixione, et al.,
A framework for Higgscharacterisation , JHEP (2013) 043,[ arXiv:1306.6464 ].2 Falkowski, Fuks, Mawatari, Mimasu, Riva and Sanz: Rosetta18. A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, andB. Fuks,
FeynRules 2.0 - A complete toolbox for tree-levelphenomenology , Comput. Phys. Commun. (2014)2250–2300, [ arXiv:1310.1921 ].19. A. Alloul, B. Fuks, and V. Sanz,
Phenomenology of theHiggs Effective Lagrangian via FEYNRULES , JHEP (2014) 110, [ arXiv:1310.5150 ].20. C. Degrande, B. Fuks, K. Mawatari, K. Mimasu, andV. Sanz. in preparation.21. E. E. Jenkins, A. V. Manohar, and M. Trott,
Renormalization Group Evolution of the Standard ModelDimension Six Operators I: Formalism and lambdaDependence , JHEP (2013) 087, [ arXiv:1308.2627 ].22. E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution of the Standard ModelDimension Six Operators II: Yukawa Dependence , JHEP (2014) 035, [ arXiv:1310.4838 ].23. B. Henning, X. Lu, and H. Murayama, How to use theStandard Model effective field theory , arXiv:1412.1837 .24. http://feynrules.irmp.ucl.ac.be/wiki/BSMCharacterisation .25. N. D. Christensen, P. de Aquino, C. Degrande, C. Duhr,B. Fuks, M. Herquet, F. Maltoni, and S. Schumann, AComprehensive approach to new physics simulations , Eur.Phys. J.
C71 (2011) 1541, [ arXiv:0906.2474 ].26. C. Degrande, C. Duhr, B. Fuks, D. Grellscheid,O. Mattelaer, et al.,
UFO - The Universal FeynRulesOutput , Comput.Phys.Commun. (2012) 1201–1214,[ arXiv:1108.2040 ].27. P. Z. Skands, B. Allanach, H. Baer, C. Balazs,G. Belanger, et al.,
SUSY Les Houches accord:Interfacing SUSY spectrum calculators, decay packages,and event generators , JHEP (2004) 036,[ hep-ph/0311123 ].28. B. C. Allanach et al.,
SUSY Les Houches Accord 2 , Comput. Phys. Commun. (2009) 8–25,[ arXiv:0801.0045 ].29.
Particle Data Group
Collaboration, K. Olive et al.,
Review of Particle Physics , Chin.Phys.
C38 (2014)090001.30. G. D’Ambrosio, G. F. Giudice, G. Isidori, andA. Strumia,
Minimal flavor violation: An Effective fieldtheory approach , Nucl. Phys.
B645 (2002) 155–187,[ hep-ph/0207036 ].31. F. Maltoni, K. Mawatari, and M. Zaro,
Higgscharacterisation via vector-boson fusion and associatedproduction: NLO and parton-shower effects , Eur. Phys. J.
C74 (2014), no. 1 2710, [ arXiv:1311.1829 ].32. F. Demartin, F. Maltoni, K. Mawatari, B. Page, andM. Zaro,
Higgs characterisation at NLO in QCD: CPproperties of the top-quark Yukawa interaction , Eur.Phys. J.
C74 (2014), no. 9 3065, [ arXiv:1407.5089 ].33. F. Demartin, F. Maltoni, K. Mawatari, and M. Zaro,
Higgs production in association with a single top quark atthe LHC , Eur. Phys. J.
C75 (2015), no. 6 267,[ arXiv:1504.00611 ].34. J. Elias-Mir, J. R. Espinosa, E. Masso, and A. Pomarol,
Renormalization of dimension-six operators relevant forthe Higgs decays h → γγ, γZ , JHEP (2013) 033,[ arXiv:1302.5661 ].35. J. Elias-Miro, J. R. Espinosa, E. Masso, and A. Pomarol, Higgs windows to new physics through d=6 operators: constraints and one-loop anomalous dimensions , JHEP (2013) 066, [ arXiv:1308.1879arXiv:1308.1879