Standard Model Effective Theory at OneLoop in QCD
Céline Degrande, Gauthier Durieux, Fabio Maltoni, Ken Mimasu, Eleni Vryonidou & Cen Zhang, arXiv:2008.11743
The implementation is based on the Warsaw basis of dimensionsix SMEFT operators, after canonical normalization. Electroweak input parameters are taken to be G_{F}, M_{Z}, M_{W}. The CKM matrix is approximated as a unit matrix, and a U(2)_{q} x U(2)_{u} x U(3)_{d} x (U(1)_{l} x U(1)_{e})^{3} flavor symmetry is enforced. It forbids all fermion masses and Yukawa couplings except that only of the top quark. The model therefore implements the fiveflavor scheme for PDFs.
A new coupling order, NP=2, is assigned to SMEFT interactions. The cutoff scale Lambda takes a default value of 1 TeV^{2} and can be modified along with the Wilson coefficients in the param_card. Operators definitions, normalisations and coefficient names in the UFO model are specified in definitions.pdf. The notations and normalizations of topquark operator coefficients comply with the LHC TOP WG standards of 1802.07237. Note however that the flavor symmetry enforced here is slightly more restrictive than the baseline assumption there (see the dim6top page for more information). This model has been validated at tree level against the dim6top implementation (see 1906.12310 and the comparison details).
Current implementation
UFO model: SMEFTatNLO_v1.0.tar.gz
The current implementation imposes CP conservation. In the quark sector, it focuses primarily on topquark interactions. The lightquark current operator, qqHDH, uuHDH, ddHDH, with coefficients cpq3i, cpqMi, cpu, cpd are however included. The triplegluon operator, with coefficient cG, is currently not available (see the loopcapable GGG implementation). Vertices including more than four scalars or four leptons are not included. Scalar and tensor QQll operators, with coefficients ctlS3, ctlT3, and cblS3, break our flavor symmetry assumption and are not available for oneloop computations. Topquark flavorchanging interactions, not compatible with the imposed flavor symmetry, are not included (see the loopcapable TopFCNC implementation).
Unlike prescribed by the LHC TOP WG, the top quark chromomagneticdipole operator coefficient ctG is normalized with a factor of the strong coupling, g_{S}. This normalization factor temporarily ensures compatibility with the 2.X.X series of MadGraph5_aMC@NLO but may be dropped in the future. As with every other appearance of this coupling in MadGraph5_aMC@NLO, its value is renormalisationgroup evolved to the QCD renormalization scale (set in the run_card).
Counterterms required for oneloop computations are currently included up to five points. The unitary gauge (default) is recommended when computing anomalous quarkloop amplitudes like ggZ, gggZ, ggZH and ggff.
MadGraph5_aMC@NLO does not evolve operator coefficients which are therefore kept at fixed scale mueft distinguished from the QCD renormalization scale MUR. We recommend to use fixed renormalization and factorization scales (in the run_card), and to set mueft equal to those (in the param_card).
The 3.0.3neworders development branch (tarball) of MG is required for NLO predictions involving fourfermion operators and (in general) H^{2}G^{2} with coefficient cpG not normalized with any power of g_{S}. It also allows for a better control over coupling orders and, in particular, for the separate computation of linear and quadratic EFT contributions at NLO, in fixed order mode. A branch allowing for the separate computation of different orders in eventgeneration mode (with matching to parton shower) is being validated. The 2.X.X series of MadGraph5_aMC@NLO can handle bosonic and twofermion operators at oneloop.
Version updates
The model version number can be found in the __version__ variable at the end of __init__.py.
 2018/12/20  v0.1: First version upload, 4F and c_{G} operators at LO pending validation; a few minor convention tweaks required to match dim6top exactly. decays.py missing.
 2019/04/03  v0.1: Added definitions.pdf document and uploaded a new version with a fix for restrict_default.dat
 2019/08/12  v0.1: Uploaded a new version matching dim6top operator conventions, also some bugfixes and gs normalisation for OtG
 2020/08/24  v1.0: Official release including notably fourquark operators at NLO.
Support
Please direct any questions to smeftatnlodev[at]cern[dot]ch.
Usage notes
Restriction cards
Because of the mixture of LO/NLO compatible operators included in the model, restriction cards must be used to access the SMEFT interactions.
Default loading of the model
> import model SMEFTatNLO
will load the pure SM without any effective operators.
The LO restriction card should be used when importing the model for LO generation:
> import model SMEFTatNLOLO
For NLO QCD generation, the NLO restriction card should be used when importing the model:
> import model SMEFTatNLONLO
This invokes a restricted set of operators for which the required counterterms are implemented.
Coupling orders
We recommend specifying the full QCD, QED and NP orders for process generation.
For example:
> generate p p > t t~ QCD=2 QED=0 NP=2 [QCD]
generates topquark pair production at NLO QCD, including the QCDinduced SM and the SMEFT contributions.
Excluding operators
We recommend avoiding setting values of Wilson coefficients to 0 when computing at NLO using MadGraph5_aMC@NLO.
Operators should either be removed explicitly with restriction cards or set to a very small nonzero value, e.g., 1e5
Plugin for bquark Yukawa coupling and operator (ymb and cbp)
A pluginlike modification to the model including the bbh (SM+SMEFT), bbhh and bbhhh interactions has been implemented to account for the Higgs coupling to bottom quarks. It can only be used at LO. A configuration.py file is included in the UFO model with a bottomYukawa flag set to False by default. Setting it to True restores the SM & SMEFT bottom Yukawa parameters (ymb and cbp), the bbh(h)(h) vertices, and corresponding couplings. The bottom mass parameters, MB, is not restored which has a percent effect on the h > b b~ partial width. The corresponding Goldstone boson interactions are not included, such that the extended model can only be used in unitary gauge (default).
Generation recipes for validated processes
Among many others, the following processes are supported at the oneloop level. Gauge invariance (see help check in MadGraph5_aMC@NLO) and pole cancellation have been checked explicitly for those. For complicated processes and in case of doubts, please contact the authors. Widths should be set to zero to ensure gauge invariance.
QCD
> p p > j j QED=0 QCD=2 NP=2 [QCD]
Drell Yan
> p p > mu+ mu QCD=0 QED=2 NP=2 [QCD] > p p > mu+ vm QCD=0 QED=2 NP=2 [QCD] > p p > W+ j $$ t QCD=1 QED=1 NP=2 [QCD] > p p > W j $$ t~ QCD=1 QED=1 NP=2 [QCD] > p p > Z j QCD=1 QED=1 NP=2 [QCD]
Multiboson production
quarkinitiated
> p p > W+ W QED=2 QCD=0 NP=2 [QCD] > p p > W+ Z QED=2 QCD=0 NP=2 [QCD] > p p > Z Z QED=2 QCD=0 NP=2 [QCD]
loopinduced
> g g > W+ W QED=2 QCD=2 NP=2 [QCD] > g g > Z Z QED=2 QCD=2 NP=2 [QCD] > g g > W+ W Z QED=3 QCD=2 NP=2 [QCD] > g g > Z Z Z QED=3 QCD=2 NP=2 [QCD]
Higgs production
loopinduced
> g g > H QED=1 QCD=2 NP=2 [QCD] > g g > H H QED=2 QCD=2 NP=2 [QCD] > g g > H H H QED=3 QCD=2 NP=2 [QCD] > g g > H j QED=1 QCD=3 NP=2 [QCD]
Top quark production
> e+ e > t t~ QED=2 QCD=0 NP=2 [QCD] > p p > t t~ QED=0 QCD=2 NP=2 [QCD] > p p > t t~ h QED=1 QCD=2 NP=2 [QCD] > p p > t t~ Z QED=1 QCD=2 NP=2 [QCD] > p p > t t~ W+ QED=1 QCD=2 NP=2 [QCD] > p p > t W $$ t~ QED=1 QCD=1 NP=2 [QCD] > p p > t W j $$ t~ QED=1 QCD=2 NP=2 [QCD] > p p > t j $$ W QED=2 QCD=0 NP=2 [QCD] > p p > t h j $$ W QED=3 QCD=0 NP=2 [QCD] > p p > t Z j $$ W QED=3 QCD=0 NP=2 [QCD] > p p > t a j $$ W QED=3 QCD=0 NP=2 [QCD]
When generating one of the last four processes (tj,thj,tZj,taj) with the cQq83 operator coefficient, all loops including a gluon have to be allowed. This can be achieved with the following modification of MadGraph5_aMC@NLO:
=== modified file 'madgraph/loop/loop_diagram_generation.py'  madgraph/loop/loop_diagram_generation.py 20200311 09:28:14 +0000 +++ madgraph/loop/loop_diagram_generation.py 20200403 21:08:18 +0000 @@ 384,7 +384,7 @@ # By default the user filter does nothing if filter is not set, # if you want to turn it on and edit it by hand, then set the # variable edit_filter_manually to True  edit_filter_manually = False + edit_filter_manually = True if not edit_filter_manually and filter in [None,'None']: return if isinstance(filter,str) and filter.lower() == 'true': @@ 415,6 +415,10 @@ raise InvalidCmd("The userdefined filter '%s' did not"%filter+ " returned the following error:\n > %s"%str(e)) + # requires a gluon to run in all loops + if 21 not in diag.get_loop_lines_pdgs(): + valid_diag = False + # if any([abs(pdg) not in range(1,7) for pdg in diag.get_loop_lines_pdgs()]): # valid_diag = False @@ 538,7 +542,7 @@ if valid_diag: newloopselection.append(diag)  self['loop_diagrams']=newloopselection + #self['loop_diagrams']=newloopselection # To monitor what are the diagrams filtered, simply comment the line # directly above and uncomment the two directly below. # self['loop_diagrams'] = base_objects.DiagramList(
Analytic validation
The following loop computations of amplitudes relevant for several processes have been crosschecked analytically:
 ttbar: tt, gg, ggg, gtt, ggtt
 single top/decay: tbW, 4f
 ttV: ttV, ggV, gggV, gttV
 ttH: ggh, gggh, htt, ghtt
Attachments (2)

definitions.pdf
(196.7 KB) 
added by kmimasu 4 weeks ago.
Notation, conventions, field redefinitions & input scheme

SMEFTatNLO_v1.0.tar.gz
(167.9 KB) 
added by kmimasu 11 days ago.
v1.0 uploaded 2020/09/09
Download all attachments as: .zip