Higgs decays to two leptons and a photon beyond leading order in the SMEFT

Sorry for the delay. (FWIW, I originally tried to submit it over a week ago.)

This paper presents a groundbreaking calculation of $h \to Z \gamma$ in the Standard Model Effective Field Theory (SMEFT). It is the first work to my knowledge to interfere a full one-loop SM amplitude with a claim full tree level, dimension-8 SMEFT amplitudes. My main concern is it appears not all the dimension-8 operators were included.

In particular, the so-called "Class 8" operators of Ref. [35] also contribute to this process. Specifically, the operators $Q_{WH^4D^2}^{(1)}$ and $Q_{BH^4D^2}^{(1)}$ generate an amplitude proportional to $(Z^\mu \partial^\nu h -Z^\nu \partial^\mu h) F_{\mu\nu}$. Similarly, the operators generate $Q_{WH^4D^2}^{(2)}$ and $Q_{BH^4D^2}^{(2)}$ generate an analogous amplitude but with a dual electromagnetic field strength.

These operators should be discussed in Section 4. Additionally, they should either also be included in the analysis of Sections 4.1 and 4.2 or a reason should be given as to why don't contribute.

There is a silver lining here. The inclusion of the Class 8 operators would provide additional motivation for studying the process $h \to Z \gamma$ as they do not contribute to $h \to \gamma gamma$ or $g g \to h$.

see PDF attached.

The paper considers the $H\to \ell\bar\ell\gamma$ decay in the SMEFT, and considers the contribution to the decay rate from interference of tree-level SMEFT amplitudes (up to dimension 8). In addition, the interference of the tree-level SMEFT amplitude with the one-loop SM amplitude (in the limit of $m_{\ell\to0}$ is considered.

The paper contains a combination of technical advancements (in computing SMEFT amplitudes up to dimension 8, and the corresponding implementation) and applies the machinery to the case of $H\to \ell\bar\ell\gamma$ decays (at the high-lump LHC). While the paper contains new information which is relevant and valuable for the field, I believe the presentation of the results lacks clarity and that the validity of certain assumptions made in the calculation to be poor or unjustified.

I would recommend the paper for publication after these major criticisms are addressed.

To assist the authors, I have made a list of comments below, which I into “technical comments on the presented calculation” as well as ”comments on the phenomenology”. I hope the comments are received with the aim, that they are constructive.

$\textbf{Technical comments on the presented calculation}$
Section 3.1
Large corrections induced in the presence of exact mass effects (e.g. +26% quoted for tau-leptons). Can the authors comment upon this? Naively one would expect that exact mass effects introduce power corrections of the form $m_\ell^2/Q^2$. For an inclusive decay rate, the scale Q would be the Higgs mass, meaning these corrections would be small. Instead, the large correction may indicate strong power-correction dependence on the energy cut of the photon, which is used to regulate the soft and collinear divergence. Probably the source of correction is not logarithmic, since it is small for the electron.

Section 3.2
Do I understand the statement of Eq. (14) that corrections involving both the yukawa coupling ($H\ell\bar\ell$) as well corrections from propagators and phase-space are ignored? In other words, the decay rate is considered by squaring those one-loop SM amplitudes which vanish in the strict limit that the lepton mass is set to zero (including the yukawa). This should be made clear if it is the case, by defining exactly the loop order of contributions to the decay rate which are considered [in fact only later at the start of Section 4 is that made somewhat clearer, which is not useful for a reader]

Before equation (17) it is stated that the one-loop process “does not contain IR divergences”. Inspecting those diagrams in figure (3), it is not clear why the photon cannot be emitted in either a soft and/or collinear configuration. Can the authors comment on this?

The results for Eq (17) indicate that the (one-loop)-squared pieces are numerically similar for muons (factor of three large for [one-loop]-squared pieces) and substantially smaller than the squared tree-level contribution for tau-leptons. How justified is it then to ignore those terms which are chirally suppressed but not loop suppressed (i.e. $SM^{0} * SM^{1}$? It seems not well justified for muons, and simply a bad approximation for taus.

While these comments/questions are for the SM part, which is not the main focus of the paper, as far as I can tell these assumptions are also carried across to the study of SMEFT contributions and should therefore be clarified.

Section 4.2
I find the notation $|M_{full}|^2$, i.e. “Full” to be a little misleading as a vast number of terms have been excluded.

As I understand, the expression given in Eq. (33) is an expansion in $1/\Lambda^2$ and in loop-order. But all amplitudes $M_{C}$ (which refer to specific cases of SMEFT insertions, not operator classes) are tree-level. Is this correct?

Can the authors comment further on the potential impact of M_{C} when considered beyond leading-order? In particular, it is not clear to me that the interference of two one-loop SMEFT amplitudes cannot give new kinematic structure. Or is there a reason why that is not the case.

As muon final states are considered, and the tree-level SM contribution is of the same order as the one-loop SM parts in the $m_{\ell}\to0$ limit, interference of $M_{C}$ amplitudes with the tree-level SM is surely necessary? Or have I missed an important point.

$\textbf{Comments on the phenomenology }$
With regards to the event rates in Table 3, expected with 3/ab integrated luminosity. How are these event rates estimated? This seems like a critical point for the paper, as it indicates whether the kinematic regions in the Dalitz plot will be feasibly accessed (and hence provide potentially useful information on the structure of new physics) in an experimental setup.

In particular, the muon final state is considered. This is presumably because the feasibility (i.e. efficiency, and background rejection) is higher for this channel. What efficiency (and justification) are assumed. If dressed leptons are considered (i.e. those reconstructed as a leptonic jet avoiding IRC unsafe configurations), could it also be necessary to consider $H\to \ell\bar\ell \gamma\gamma$ SMEFT contributions?