= A complete top-quark EFT implementation

Under the umbrella of the LHC TOP WG, common standards and prescriptions were established for the EFT interpretation of top-quark measurements at the LHC. They are summarized in the note at [https://arxiv.org/abs/1802.07237]. Details concerning the present UFO model implementation are provided in Appendix B.1.

EFT degrees of freedom that are natural for the description of top-quark processes were defined as linear combinations of Warsaw basis operator coefficients. They match the interference structure of the EFT with SM amplitudes and give a direct parametrization of the top-quark couplings to physical W and Z gauge bosons (see Appendices C, D, and E of the note for degrees of freedom definitions).

The CKM matrix is approximated as a unit matrix. Masses and Yukawa couplings of all fermions except for the top and bottom quarks are neglected by default. A U(2)q+u+d flavour symmetry is imposed on the first two generations of quarks (see Section 4 of the note for details). All operators of the Warsaw basis involving a top quark and satisfying this flavour assumption are included (four-quark, two-quark, and two-quark-two-lepton operators). Baryon and lepton number violating operators are not included. In total the model includes O(90) flavour-conserving degrees of freedom which have a {{{DIM6=1}}} coupling order.

Enhancing the U(2)q+u+d flavour symmetry to U(2)q×U(2)u×U(2)d (the baseline scenario considered as prescription) is done by neglecting the 10+10 CPV supplementary degrees of freedom. A further restriction to the ''top-philic'' can also be obtained following the constraints given in the note.

Top-quark FCNCs are also included by allowing one quark bilinear in each operator to break the U(2)q×U(2)u×U(2)d baseline flavour symmetry requirement and couple the third generation with either the first or the second. Operators with one light and one heavy quark, one light quark one heavy quark and two leptons, one light quark and three heavy quarks, three light and one heavy quark are included. The O(300) degrees of freedom are assigned a {{{FCNC=1}}} coupling order. Requiring {{{FCNC=0}}} at the generation level is required to not consider them.

Two versions of the models are provided {{{dim6top_LO_UFO}}} and {{{dim6top_LO_UFO_each_coupling_order}}}. In the second one, each degree of freedom is assigned an individual coupling order, like {{{DIM6_ctZ}}}, {{{DIM6_cQQ1}}}, or {{{FCNC_cqq11x3331}}}, in addition to the {{{DIM6}}} or {{{FCNC}}} one. This allows for the selection of individual degrees of freedom interferences at the generation level in {{{MG5_aMC@NLO}}}. The syntax:
{{{
> generate p p > t t~ FCNC=0 DIM6=1 DIM6^2==1 DIM6_ctZ^2==1
> generate p p > t t~ FCNC=0 DIM6=1 DIM6^2==2 DIM6_ctZ^2==1 DIM6_ctW^2==1
}}}
would for instance respectively retain only the interference between SM and {{{ctZ}}} amplitudes, and 
the interferences between {{{ctZ}}} and {{{ctW}}} amplitudes, in top pair production.
The {{{FCNC=0}}} specification excludes FCNC interactions, {{{DIM6=1}}} allows for at most one operator insertion at the amplitude level (to be removed to include multiple insertions too), {{{DIM6^2==n}}} specifies the overall EFT order of the selected squared amplitude, {{{DIM6_ctZ^2==n}}} specifies the order of the squared amplitude in the specific {{{ctZ}}} operator coefficient.

Only tree-level simulation in the unitary gauge is possible. Loop induced couplings of the Higgs boson to pairs of gluons, photons, or {{{Zγ}}} are not included.

A positive {{{QED=n}}} coupling order is also assigned to operators involving {{{n}}} Higgs doublet fields in the unbroken phase to compensate for the {{{QED=-1}}} coupling order of the Higgs vev which appears in the broken phase.

The bottom quark is massive by default. Switching to the five-flavour factorization scheme can be achieve by using a restriction card in which {{{MB}}} is set to {{{0}}}, or by redefining
{{{
> define p = p b b~
> define j = p
}}}
before process generation and setting {{{MB}}} to {{{0}}} in the {{{param_card}}} (setting {{{ymb}}} to {{{0}}} may also be desired, for consistency).

Benchmark results for the linear and quadratic dimension-six EFT dependences of total rates are provided in Tables 10-21 of the note, for processes like {{{ pp→tt̅, tt̅bb̅, tt̅tt̅, tt̅e⁺ν, tt̅e⁺e⁻, tt̅γ, tt̅h }}} and {{{ pp→tj, te⁻ν, tje⁺e⁻, tjγ, tjh}}}.

Please refer to [https://arxiv.org/abs/1802.07237] for further details and when using this model.

Contact persons are Gauthier Durieux and Cen Zhang.

Feynrule files: [attachment:dim6top.m], [attachment:dim6top.fr], [attachment:dim6top_each_coupling_order.fr]

UFO model files: [attachment:dim6top_LO_UFO.tar.gz], [attachment:dim6top_LO_UFO_each_coupling_order.tar.gz]

Changelog:
- 2018-08-23: added back {{{DIM6}}} coupling orders to the {{{dim6top_LO_UFO_each_coupling_order}}} version (thanks to Jay Howarth)
- 2018-06-06: corrected the Lorentz structures of the operators relative to {{{cbtud1}}} and {{{cbtud8}}} coefficients from scalar to vector (thanks to Céline Degrande)
- 2018-05-28: moved the definition of 'ctA' below that of 'sw' and 'cw' in parameters.py, to avoid "MadGraph5Error : Unable to evaluate mdl_ctA = (mdl_ctW-mdl_cw*mdl_ctZ)/mdl_sw: raise error: name 'mdl_cw' is not defined" (thanks to Marcel Vos)
- 2018-05-21: corrected the Lorentz structure of Oledq operator from tensor to scalar (thanks to Ken Mimasu)

==== Note on NLO QCD
For NLO QCD simulation, the model authors recommend using the [wiki:SMEFTatNLO SMEFT@NLO] model which is mostly compatible with the baseline prescription of the LHC TOP WG. Few differences between the {{{dim6top}}} and [wiki:SMEFTatNLO SMEFT@NLO] (August 2019 version) models are to be noted:
- [wiki:SMEFTatNLO SMEFT@NLO] includes Higgs and electroweak operators but does not include top-quark FCNCs.
- [wiki:SMEFTatNLO SMEFT@NLO] is moreover restricted to CP-conserving operator coefficients.
- [wiki:SMEFTatNLO SMEFT@NLO] imposes an enhanced U(2)q×U(2)u×U(3)d flavour symmetry in the quark sector, such that the right-handed bottom quark is treated on the same footing as other light down-type quarks. This is required for consistency with the five-flavour scheme at NLO QCD, where {{{MB=0}}}.
- [wiki:SMEFTatNLO SMEFT@NLO] normalizes the top dipole operator (with coefficient {{{ctG}}}) with a factor of the strong coupling constant {{{G}}} for compatibility with MG 2.x series at NLO in QCD. The 3.x series does not requires it. This implies that {{{ctG_dim6top = G * ctG_SMEFTatNLO}}}. Note that {{{G}}} runs.

Altogether, due to the enhanced quark flavour symmetry and to CP-conservation, [SMEFTatNLO] only includes a subset of 45 top-quark operator coefficients:
cQQ1, cQQ8, cQd1, cQd8, cQe1, cQe2, cQe3, cQl31, cQl32, cQl33, cQlM1, cQlM2, cQlM3, cQq11, cQq13, cQq81, cQq83, cQt1, cQt8, cQu1, cQu8, cblS3, cdp, cpQ3, cpQM, cpt, ctG, ctW, ctZ, ctd1, ctd8, cte1, cte2, cte3, ctl1, ctl2, ctl3, ctlS3, ctlT3, ctp, ctq1, ctq8, ctt1, ctu1, ctu8
and equates cQb1=cQd1, cQb8=cQd8, ctb1=ctd1, ctb8=ctd8.

A high-precision comparison with {{{dim6top}}} has been performed at leading order following the prescription of [https://arxiv.org/abs/1906.12310 1906.12310], with the [https://bazaar.launchpad.net/~rwgtdim6/mg5amcnlo/plugin_eft_contrib/revision/ plugin_eft_contrib] of MG.

The [wiki:SMEFTatNLO SMEFT@NLO] version dated from August 2019 does not include an implementation of four-fermion operators at NLO in QCD. Its validation is ongoing.