Renormalisation group effects in SMEFT for di-Higgs production

We thank the referees for the careful reading of the manuscript and for the constructive comments.
We have addressed the points raised by the referees in the following way (we first repeat the comments of the referee):

1) As written, the default scheme on pg. 9 of the text and used as the reference scheme in Section 4 seems to set \muEFT=\muR
in the NLO SMEFT matrix elements, with \muR varied, while in the Wilson coefficients Ci=Ci(\mu0) is kept static (i.e. do not depend on \muR).
If that were the case, then the cross section would have explicit $\mu$R dependence at NLO, instead of at NNLO and higher, and the scale variation of the cross section with \muR would not be a reliable estimate of missing NNLO terms, as is usually assumed to be the case.
If this is so, the authors should point it out in the text; if instead I misunderstand the scheme, the authors should change the wording so that the wrong interpretation is not possible.

--> The referee is right with these considerations. We added some text about the shortcomings of this choice in the second paragraph of chapter~4.

2) When performing an NLO calculation in SMEFT (as in the SM), one would normally like to see that 1) scale uncertainties decrease at NLO compared to LO, and 2) the NLO results lie within the uncertainty band of LO (at least for Wilson coefficients connected to the LO result through RG running). The presentation in Section 4 makes it hard to see quantitatively whether this happens, as LO and NLO are never shown on the same plot. To make clear the behavior of the perturbative series, I would suggest that the authors present numbers for the total cross section at LO and NLO in SMEFT, including scale variations in at least one benchmark scenario.

--> We included an appendix B where we investigate the scale uncertainties for \sigma_dim6, considering the part which is proportional to $C_{tH}$. We compare the results at LO and NLO QCD for the case of a SMEFT input scale $\mu_0=200GeV for both the $m_{hh}$-differential distribution and the total cross section.

In addition to the points above, I also have a few minor comments:
- pg. 3: weakly coupling UV theories $\to$ weakly coupled UV theories
--> Done

- Eq. (2.5): the running of the Yukawa coupling $y_t$ also receives SMEFT QCD contributions -- explain why they are irrelevant.
--> We renormalise the canonical mass term of SMEFT at dimension-6,
i.e. $m_t = \frac{v}{\sqrt{2}}\left(y_t^{SM}-\frac{v^2}{2}C_{tH}\right)$ on-shell, hence the Yukawa interaction is split into
$y_t = \frac{m_t}{v}\left(1+v^2(C_{H\Box}-\frac{1}{4}C_{HD})\right) -\frac{v^2}{\sqrt{2}}C_{tH}^*$.
The running of the part proportional to on-shell $m_t$ would not be QCD-induced but induced by electroweak loops, whereas the running of $C_{tH}$ is included in our calculation.

- Eq. (2.11) and elsewhere: I do not see a definition for v: what EW input scheme is being used?

--> With v we denote the true VEV of the Higgs field which is a combination of dim-4 parameters $\mu$, $\lambda$ and the dim-6 parameter $C_H$.
Since we use SMEFT only at dim-6 level, in the formulas where the dim-4 part multiplies a dim-6 Wilson coefficient, we use the full $v$ part where the difference is of ${\cal O}(\Lambda^{-4})$.
To determine the numerical value of $v$, we use $\{m_W,m_Z,G_F\}$ as input and do not consider SMEFT contributions to its determination. Other schemes would only make a difference once we included higher order electroweak corrections.

- pg 5: should reword so that the paragraph doesn't end with "guiding principles:" (one normally doesn't end a paragraph with a colon).
--> We have changed the sentence to ``Our selection of relevant RGE terms follows from a few guiding principles, as explained below.''

- pg 6: It is written ``we employ a decoupling of heavy particles like the top-quark and the Higgs boson from the RGE, as their logarithmic contributions should not be of high impact". However, only logs \mu0/\muEFT are being resummed by RG evolution, and in practice
\mu0~\muEFT is not far from $m_t$ or $m_h$, so are the logarithms related to decoupling really smaller than others being considered?

--> We made the selection of contributions to be included in the QCD corrections having in mind predictions at current collider energies.
For the strong coupling $g_s$, no running of heavy particles is included even though $C_{HG}$ would introduce a contribution at dim-6 and one-loop level.
However the latter contribution would come in through a Higgs-loop.
Similarly, we do not include the term in the SMEFT RGE that would introduce a contribution from $C_{HG}$ to $C_{tG}$.
We agree with the referee that a study of the effect of these neglected contributions would shed more light on the validity of this reasoning.
However, considering the current bounds for these two operators, it is likely that the effect should be rather negligible.

- pg 8: desireable $\to$ desirable --> Done