This example notebook shows how to create the Universal Feynman Output [arXiv:1108.2040] [arXiv:2304.09883] for the d=8 SMEFT with the Warsaw [arXiv:1008.4884] and Murphy [arXiv:2005.00059] bases. As the full d=8 file is usually too large to be handled properly by Monte Carlo event generators such as MadGraph [arXiv:1106.0522], we provide functionality to select the Wilson coefficients of interest and assign flavor symmetry assumptions. For a general tutorial on Feynman rules and the UFO export see "SMEFT: Feynman Rules and Universal Feynman Output (UFO)".

In the first step we need to ensure the Matchete package is loaded in the Kernel:

We load the d=8 Lagrangian both in the unbroken and broken phase. The model file "SMEFT_D8_brokenPhase" (only available i) already contains the result of the symmetry breaking obtained by loading the model file "SMEFT_D8+breaking" and applying ToBrokenPhase and ImplementVacuumConditions. For more details on the implementation of spontaneous symmetry breaking patterns see the tutorials "SM EWSB" and "Symmetry Breaking". By default, the broken Lagrangian does not contain any Goldstone fields. Goldstone fields can be included by running the symmetry breaking from scratch with ImplementVacuumConditions[ToBrokenPhase[LoadModel["SMEFT_D8+breaking"], GoldstoneBoson->True]].

Note however, that including for Goldstones for the complete d=8 SMEFT consumes a large amount of memory (>100GB!). It is thus recommended to only keep Goldstones in the d=8 SMEFT, if the majority of the Wilson coefficients have been set to zero before calling ToBrokenPhase.

Note that loading this file from the model database can crash due to its size. In this case, either retry or download it manually.

Select the Wilson coefficients of interest and set the rest to zero. The Wilson coefficients are presorted in lists in the following subsection.

Lists of SMEFT couplings

Two helper functions to select or remove couplings from the Lagrangian.

The required Wilson coefficients can now be selected with repeated usage of the two helper functions above and the predefined lists of coefficients. An example selection that only keeps d=6 operators and d=8 operators with four field strengths and removes all baryon number violating terms looks like:

We explicitly introduce gauge fixing terms because once the input shifts are done, it is no longer possible to automatically derive the gauge fixing from the Lagrangian. We have to use unitary gauge since the Goldstones have already been dropped from the Lagrangian:

Electroweak parameters get SMEFT corrections due to the choice of the electroweak input scheme. The input scheme relations are implemented in detail in the parameter card, however we need to define shift variables per mass dimension to truncate the EFT expansion consistently. For each electroweak parameter we substitute a relation, where is of EFT order and is of EFT order . The truncation in the end is done per mass dimension to reduce unnecessary complexity of the substitutions.

It can be helpful to define new couplings for the Yukawa interactions of the broken phase. This is not necessary in principle, but it allows, for example, to define a massless bottom quark, which has a non-zero Yukawa coupling:

To ensure the new Yukawas are not replaced into the mass terms, we subtract off the free Lagrangian before making the replacement and then add it again eventually.

We also remove any generated mass terms for neutrinos as they are defined as massless in the field definitions:

It is useful to define coupling orders for the UFO files to control which interaction should be considered in simulations. To that end, various different coupling orders ("QED", "QCD", "NP", ...) are defined below using DefineCouplingOrder:

This assigns "QED" order 1 to M, M, M, yM, yM, yM, ℯ, ℊY, and ℊL, order -1 to vT, and order 2 to λ (the order is the inverse of the exponent provided above). In addition, every EFT Wilson coefficient is assigned "QED" order 2 as well. The only QCD parameter gs is assigned "QCD" order 1. Furthermore, two insertions of gs are assigned the same hierarchy as one "QED" order. Every d=6 EFT Wilson coefficient is assigned "NP" order 1 with hierarchy 99 and every d=8 Wilson coefficient is assigned "NP" order 2. Therefore, the selection of a maximum "NP" order during event generation consistently truncates the EFT expansion at the amplitude level. Furthermore, we assign "QED" order 1 to the inverse BSM scale 1/Λ to avoid negative "QED" coupling order.

It can also be helpful to define an individual coupling order for every d=8 operator class to better control which coefficients should be included in simulations. This can be achieved using:

For finer control, further coupling order s can be defined as desired.

To further reduce the degrees of freedom, Flavor symmetries can be implemented following the tutorial "SMEFT: Feynman Rules and Universal Feynman Output (UFO)".

A template for the parameter card in form of a JSON file can be obtained using the function DefaultParamCard:

This generates a JSON file named "model_parameters.json" in the directory given by DirectoryUFO, which contains a minimal parametrization of all couplings in terms of a set of non-redundant parameters, exploiting the known symmetries and conjugation properties of the couplings.

Note that while this file can in principle be used to directly generate the UFO files. However, all non-redundant parameters are initialized with the value 0, which is most likely inconvenient for any application. Therefore, the parameter card file should be edited manually to include the SM input values, the definitions of the electroweak scheme shifts, the masses and widths of all particles, etc.

Note that specifying the option DefaultValues->True for DefaultParamCard only works for simple models that do not affect the SSB breaking pattern and can therefore not be used for the SMEFT.

In a next step, the UFO files for the SMEFT can be generated using the ExportUFO routine.

Here, the option Gauge is used to specify that the UFO should be exported in unitary gauge, whereas OutputDirectory determines where the UFO files are saved and InputFile should point to the parameter card generated before. If no InputFile is specified, the default parameter card will be used and a UI will open that allows to manually edit the parameters and inputs.
Note that for large models such as the SMEFT, it is not advisable to use this UI as it is unwieldy and rather slow. Note also that in order to use a different gauge than unitary, it is required to run the symmetry breaking with the option GoldstonesTrue, as explained in the beginning of this guide. The option Legs specifies the maximum number of external legs. Note that including more legs severely increases the computational complexity and the size of the resulting UFO.