Matchmakereft: automated tree-level and one-loop matching

The authors present a tool for one-loop matching that can also be used to calculate anomalous dimensions and perform changes of operator bases. The tool performs standard computational steps and relies on several other tools.

The program has a wide range of applicability and the paper is well written.

1. Application to a wide class of models that can be matched.

2. The program allows for an automatic calculation of one-loop anomalous
dimensions (ADMs) for a wide class of Effective Theories (not only
standard model degrees of freedom).

3. The program allows for a translation of different EFT bases.

4. Off-shell calculation allows for consistency checks that result
from (low-energy) gauge invariance.

Some computational steps are discussed in a potentially unnecessary
level of detail (tensor reduction, partial fractions, integration by
parts and Taylor expansion) while the relevant quantum field
theoretical subtleties, that could occur in the calculations, are not
discussed in much detail.

The weaknesses fall into two categories: Potential renormalisation
issues and tests/applicability.

A one-loop matching calculation is partially scheme dependent (if tree-level
is non-zero). In this context several questions arise:

1. the authors discuss evanescent operators and expand in external
momenta and masses, which leads to IR divergences that cancel with
the UV divergences and render all EFT loop matrix elements zero. If
an evanescent operator is generated at LO (maybe through
Fierzing), are the authors sure that all finite operators
renormalisations are considered in the EFT?

2. Additional finite renormalisations might be necessary in anomalous
theories, see e.g. hep-ph/9302240. Results are well known for
one-loop calculations with renormalisable interactions and the
authors should compare their matching results with the literature
for diagrams that involve traces of gamma_5. The situation becomes
more complicated when matching EFTs onto EFTs at one-loop level,
but I am not sure if matchmaker is applicable in these
scenarios. The authors should clarify for which models their code
is applicable.

3. Field renormalisations in matching calculations can be
non-trivial. Minimal subtraction is a non-decoupling scheme, while
decoupling is shown in MOM-schemes. This fixes the field
renormalisation condition in order to achieve decoupling by hand in
minimal subtraction. E.g. the hermitian and anti-hermitian field
renormalisations are relevant for canceling gauge dependence or for
the finite renormalisation of the CKM matrix. I did not see a
detailed discussion of these properties.

4. Specific models might require different regularisation schemes, such as
DRED vs DREG. Is this relevant for matchmaker?

5. When calculating ADMs of operators of different dimensions, one has
to be careful that the Wilson coefficient can involve different
powers of mu^epsilon so that each term in the Lagrangian has the
same unit of dimension for epsilon unequal zero. This is relevant when
calculating mu/d mu C. I understand that the tool is not primarily
developed for ADMs, but a word of caution and reference to the
literature might be useful.

The examples considered do not cover all possible applications of the
program. It appears that only models with extra fermions and scalars
have been considered.

6. If one would consider extra vectors, one would need to perform the
gauge fixing of the full gauge theory, before the symmetry breaking
relevant for generating masses for the additional vectors occurs. Hence,
it would be interesting to specify how the gauge is fixed for the
light and heavy fields, and also if other types of gauge fixings
are possible. Obviously, working with a non background gauge fixing
would change the required EOM vanishing operators. A discussion of
this issue would be useful. The authors might compare with the
literature for possible examples.

7. One should also consider additional examples, such as supersymmetric models, that have fermion flow "violation", possible finite normalisations,
or other subtleties and compare with results in the literature.

The program will be useful, the paper is well written and I recommend
the work for publication after the above points are considered.

The weaknesses 1 - 7 should be considered by the authors.

1- The paper presents a very powerful tool, automating a class of highly technical calculations. Matchmakereft will certainly be very useful for the high energy physics community, streamlining the process of searching for heavy new physics via EFT techniques.

2- The explanations of the calculations are very clear, with an appropriate amount of examples. The code examples are also well documented, and I was able to follow them all without any problems.

3- In presenting their tool, the authors have showcased several examples and validated the results against existing calculations in the literature, some performed by the authors themselves and others by independent groups. The small differences are openly discussed and hopefully, the results of the exchanges with the corresponding authors can be published in the final version of the paper or in a short update in the future.

4- Several new results are also presented as by-products of this work, that are self-contained and merit appreciation. These include the extension of the Green's basis with the evasescent operators, the capacity to compute renormalisation group equations of generic theories, and the possibility to use matchmakereft to translate between generic bases of dimension-6 operators.

1- In general, there is a significant amount of assumed knowledge required to be able to digest this paper. While this is normal for academic journals, it is mostly a question concerning the intended audience. To me, the level of detail is appropriate for experts in the field, who have already performed one-loop matching and RGE calculations by hand, as well as having had some exposure to the background field method. If the authors would like to expand their readership to EFT enthusiasts, who perhaps do not have as much experience on the one-loop side, for example. I think some sections would benefit from a small amount of additional detail and/or useful references.

2- On this topic, the assumed technical knowledge of at least Mathematica & FeynRules is also significant. Although this is, of course, unavoidable since the program makes heavy use of these two codes, it may be worthwhile to add a disclaimer of sorts, that using matchmaker requires a working knowledge of both, and preferably some experience in creating models in FeynRules.

Dear Editors,

In manuscript 202201_00006, the authors present MatchMakerEFT (MM), an automated tool for matching renormalizable models of heavy new physics to low energy EFTs by integrating out the heavy states at the one-loop level. As mentioned in my comments on the strength of the paper, the tool is extremely powerful, making use of state-of-the art codes to streamline the calculations. It will undoubtedly benefit our community significantly.

The paper is well written and the matching procedure is explained quite well,. modulo my comments on the level of detail above. I will mention some specific sections/concepts that could be expanded upon in my list of points that the authors may wish to consider.

Having installed the program and run the examples in the paper, I can see that the tool works as advertised. See below for some specific question/issues.

In summary, I do not hesitate to recommend this article for publication in SciPost physics.

The comments below are mainly stylistic, or involve optional changes that I think may improve the discussion or make it more accessible to non-experts. Hopefully my questions, if not too naive, will help the authors add the extra bit of detail that would have avoided my confusion.

1. There are some grammatical and typing errors in the manuscript, including missing prepositions. Further proofreading is recommended to iron these out.

2. The process of calculating the amplitudes and performing the off-shell matching of the two theories at a particular scale is mostly described in words. The exception to this is the nice explanation of the way in which the hard region expansion to the loop amplitudes are obtained. I certainly think it would have been nice to see an explicit example that applies the MM procedure step-by-step, to help the reader to understand exactly what is being done, so that they can better use the tool for their own applications. Perhaps a subset of the exercise given in Appendix C might be a good place to include this. For example, the renormalisation procedure is not explicitly explained, even though I would have though it necessary to obtain amplitudes in both theories.

3. In section 2.1, the procedure is summarised as "computing... the hard-region contribution to the one-light-particle-irreducible relevant amplitudes at tree and one-loop level in the UV theory and equating it to the tree level contribution in the EFT". Does this mean that, on UV theory side, the amplitudes are calculated up to one loop, while on the EFT side, they are kept at tree level? My understanding was that, in general, one should compute both sides to a given order in perturbation theory to achieve the matching.

4. Section 2.2 in the "integration by parts" bullet. The authors mention setting scaleless integrals to zero, unless RGE mode is activated. This part could do with a little more explanation on why this is done.

5. Right after, the identity for reducing the integrals to tadpoles is given. This kind of function (multiple powers of the same propagator) doesn't normally appear in a typical loop integral. I assume they arise from the hard region expansion, perhaps the authors could state this.

6. Section 2.4 line 3 typo:
Mathematica match -> Mathematica to match

7. In the same section, the large redundancy of the BFM is mentioned as a source of cross checks that MM performs. This procedure could also be described in more detail, I did not really get the gist of what is being done here.

8. The model creation section is very well presented. I had two very small comments:
a) The input form for the SparseArrays to specify the generators, structure functions etc. is very opaque. Is it possible to define them in the more intuitive & readable way? e.g. SparseArray[ {index}->value, ... ]
b) In 3.2.1 the BFM implementation is described, without much reference to the method itself i.e. assuming the reader is completely familiar with it. Perhaps the authors could point to som references here in order to put into context the Lagrangian for the gauge fields defined at the bottom of page 10. The last term, in particular, I am not familiar with as somebody who has never worked with the BFM.

9. Section 3.3:
a) on the need to include all operators of dimension equal to or less than those of interest, the possibility of it still working out if the matching failures occur in sectors that are not of interest is a bit vague. What is meant by "sector" Perhaps an example would help here.
b) Refer to the section in which the proposed solution for the treatment of gamma^5 is discussed

10. Section 3.4. I would refer to the protected words as "keywords". Furthermore, I find the use of dollar signs to be a nice convention to define protected keywords in mathematica modules that minimise the probability of accidental clashes. For example, one could prefix all internal MM-specific variables with "MM$", so "MM$iCPV" etc.

11. 4.1, page 13:
matching of matchmakereft -> matching of a matchmakereft

12. When trying the CLI, that no messgae is printed when MM fails to create the model. I had to look into the log files to see that some syntax error was present in the .fr file. It would be useful to raise an error that is visible in the CLI when this occurs.

13. Section 6.2, Page 17:
a) Greeen -> Green
b) M_{ij} and arbitrary -> M_{ij} an arbitrary

14. Appendix B on gamma^5. Would it be more accurate to add "at one-loop" to "Limiting ourselves, at first, to matching renormalizable UV models to the SMEFT,..."

15. Can the authors expand on the connection of this issue with the question of anomalies in the SMEFT, given the recent handful of papers discussing this points?

16. Page 28: More than one scalar fields -> more than one scalar field

17. Appendix C: I tried to run the scalar theory example using my MM installation and found it to run very smoothly. The following is alist of things I encountered that may help with debugging or better documenting.
a) Running:
match_model_to_eft two_scalars_MM one_scalar_MM
I encountered the following mathematica warning:
Part::partd: Part specification $Failed[[1]] is longer than depth of object.

b) I was not able to reconcile the results in MatchingResults.dat with the equations (79) onwards. For example, for the tree level matching result i get the following:
{
alpha2mass -> mL^2,
alpha4 -> lambda0 - (3*kappa^2)/MH^2,
alpha4kin -> 1,
alpha6 -> (45*kappa^2*lambda2)/MH^4,
alpha6Rhat -> 0,
alpha6Rtilde -> (4*kappa^2)/MH^4
}

which doesn't match with (79)-(82).

c) I also noted the appearance in invepsilonbar in the one-loop results. This suggests the presence of poles in the matching result and the possibility is generally not discussed in the article. Could the authors explain their appearance, if they are expected or not, and if they are expected, how should they be interpreted and related to the renormalization.

d) Equations (96) & (97), there is a factor 2 w.r.t the definition of beta(aV) defined in (94)

e) Below Eq (112), the equation referenced is (93) but it seems to be that it should be (92).

18. Appendix D, Green Basis.
I was confused by the sentence that begins "We should stress that in our case, redundant and evanescent operators..." coudl the authors clarify what is meant?

19. Page 37: can not -> cannot

1 - The authors present a very powerful tool to perform matching up to the one-loop level.

2 - Several examples are provided and thoroughly explained.

3 - Results obtained with Matchmaker have been compared with previous literature and the authors are very honest about (minor) disagreements found.

1 - Information on how to use the outputs provided by Matchmakereft for debugging could be extended.

The authors introduce an automated tool to compute matching onto effective field theories up to the one-loop level. The manuscript is well written and provides future users of Matchmakereft with a comprehensive manual. The python package Matchmakereft can easily be downloaded using conda or pip and the installation process as well as dependencies are well explained. Examples in the manual and those provided with the package make it easy to understand the programme's syntax.

I highly recommend the publication of the manuscript in SciPost physics after the following (mostly editorial) changes have been made:

1 - Downloading the programme via conda currently only seems to work on linux systems. This is most probably because only the path https://conda.anaconda.org/matchmakers/linux-64 exists and there is no */matchmakers/osx-64 equivalent. This should be fixed or explained in the manuscript.

2 - 'ee[x__]' could be added to the list of protected words.

3 - The structure and content of the file MatchingProblems.dat is currently not explained. I encourage the authors to provide more information on this file so that it can be used for debugging. On their GitLab website, the authors write that the file lists amplitudes "admittedly in a very cryptic way". So I also encourage the authors to make this output more user friendly. An example output with explanations could be added to the manual.

4 - The vector-like lepton example model file runs with error messages due to the definition of gamma5 in d dimensions. It would be good to point this out explicitly in sec. 6.2.

5 - There is little information provided on the models which come with the Matchmakereft installation. It would be good to add an explanation of these files and how to run them. For instance, it would be good to explicitly mention that the file UnbrokenSM_BFM.fr has to be loaded alongside the BSM models.

6 - When running the minimal example in appendix C with the sample code provided, most but not all of the Wilson coefficients agree with those in Eqs. (79)-(93):
- Eq. (80): The second term is not present when running the sample code.
- Eq. (81): The first two terms are not present when running the sample code.
- Eq. (84): The "1+" seems to be the tree-level contribution but only the one-loop pieces are listed here according to the text.
- Eq. (92): Pieces suppressed by eight powers of the heavy scale MH^-8 are not listed here. If these are removed because kappa^4 mL^4/MH^8 ~ mL^4/MH^4 while all other pieces are only suppressed by MH^-2, it would be good to comment on this.


7 - typos and other minor issues:
- The abbreviation BFM is used in the abstract without introducing it.
- It would be good to use either flavor or flavour consistently.
- p.2 last line: "at which experiment is performed" -> "at which experiments are performed" or "at which the experiment is performed"
- p.3, first paragraph: "references there in" -> "references therein"
- p.6, third line of sec 2.3, missing ED: "is perform" -> "is performed"
- p.6, sec. 2.4, missing word: "Matchmaker then uses Mathematica [to] match"
- p.7, sec. 3.1, first bullet point, additional words: "expects the Lagrangian of the to be defined" (remove "of the")
- p.8, section 3.2, first paragraph: "wigh" -> "with"
- p.15, first line of sec. 4.2, missing word: "provides an interactive experience to [the] user"
- p.26, A.3: I recommend to replace "(s)he" with the gender neutral "they"
- p.28, top paragraph: The W in "Wilson coefficients" is capitalised elsewhere in the paper. A lowercase W is used here three times.
- p.32, missing S: "Feynrule" -> "FeynRules" (also apart from appendix C a texttt environment was used for FeynRules).