Threshold improved $Z H$ production at the LHC

Dear Editor,
We thank the referee for providing feedback. Please find below our response.

"The authors write that ‘The fixed order results thus suffer from the large threshold logarithms arising from soft gluons emission.’ Yet, they do not explicitly provide any numbers to show the percentage contribution of these logarithms to the known complete results through N3LO. It would be nice to have a short discussion on that. The authors also state that ‘To obtain stable and accurate predictions, it is essential to resum these large soft-virtual (SV) logarithms to all orders.’ This is an oft-repeated but largely incorrect and meaningless statement. Numerous studies have shown that expansions of the resummed cross section provide quickly diminishing numbers at higher orders and, in fact, the differences in numerical results among various resummation prescriptions are often larger than contributions beyond N3LO (or even beyond NNLO). Hence, fixed-order expansions to high orders (at least one order higher than the exact results) is perfectly fine and is often more accurate."

Resummation plays a crucial role in collider phenomenology, providing reliable theoretical predictions where fixed-order calculations alone are insufficient. While fixed-order results offer valuable theoretical insights, they often fail to accurately describe experimental data unless computed to very high orders, which is typically impractical. Moreover, for many observables (such as transverse momentum distributions), even higher-order predictions (e.g. up to N3LO) may not adequately capture the correct behavior in certain kinematic regions, such as the low-pT regime. This limitation has been well established in numerous studies over the past several decades. In the context of threshold resummation for colorless final states, e.g. the Higgs boson production at the LHC, as given in Fig.2 and Table-3 of [1], the fixed order results change from 36.9 pb at NLO to 46.5 pb at NNLO for the scale choice of $\mu_0=m_H/2$, while the resummation prediction at NLO+NLL is 46.2 pb, making the resummation contribution quite sizeable. Similarly, the NNLL contributions are non-negligible compared to the $\mathcal{O}(\alpha_s^3) $ corrections. Thus, in the low Q region, due to the large parton fluxes, the resummation effects are non-negligible. For the case of $q\bar{q}$ initiated DY type ZH production process, in the high invariant mass region, see. Fig.8 of [2], it is shown that the N3LO results are about 30% of LO, while the resummation at NNLO+NNLL contributes about 31%. It is true that the difference between NLO+NLL and NNLO+NNLL is smaller compared to the one between NNLO and N3LO in the high invariant mass region, say Q = 3000 GeV. However, there is a reduction in the scale uncertainties as a result of resummation in the same Q-region, see the left panel of Fig.16 of [2]. For threshold resummation, the impact is generally less dramatic but still significant, particularly in improving fixed-order predictions when higher-order results are challenging to obtain, as is the case for gluon-fusion-induced ZH production discussed in the current work.

"In particular, the authors use the minimal prescription of their Ref. [38]. This prescription, while previously used by the authors and some other groups, has made seriously wrong predictions for the size of higher-order corrections (beyond NLO) for several processes. The NNLO soft-gluon corrections in [38] were more than an order of magnitude smaller than the exact NNLO corrections that were calculated much later. These matters, i.e. differences among prescriptions and, in particular, critiques of the minimal prescription, have been studied in numerous papers since then, and the authors should acknowledge the prescription dependence of their results. "

As with any theoretical framework (including fixed-order perturbation theory), resummation procedures may involve additional prescriptions. These can be viewed as tools to explore theoretical uncertainties rather than as drawbacks. The minimal prescription is one such widely-used standard approach, and alternative schemes (such as the Borel prescription) are also available, and their predictions differ only by subleading terms, which may nonetheless be relevant in specific regions of the distributions. Having said that, the requested minor changes are reasonable for us.

"1. Provide some discussion of the contribution of the soft-gluon logarithms at fixed order (either percentage or actual numbers)."

We have added the following sentence highlighting the contributions from SV logarithms on page-2, in the introduction:
Indeed, the threshold soft-virtual (SV) logarithms at NLO can contribute to 90 − 99% of the complete NLO results in the range Q = 350 − 2000 GeV.

"2. Delete or modify the sentence about ‘essential to resum ... to all orders’. "

We have rephrased the sentence accordingly on page-2, second paragraph: By resumming these soft-virtual (SV) logarithms to all orders, one obtains predictions that are stable and well-behaved across the relevant kinematic regions.

"3. State that the results are prescription dependent. "

We have added a sentence on this on page-3, after eq.(3): Other prescriptions, for instance the Borel prescription, may lead to differences with respect to the minimal prescription that are confined to subleading terms.

Sincerely,
Authors.

References
[1] Marco Bonvini, Simone Marzani, Claudio Muselli, and Luca Rottoli. On the Higgs cross
section at N3LO+N3LL and its uncertainty. JHEP, 08:105, 2016.

[2] Goutam Das, Chinmoy Dey, M. C. Kumar, and Kajal Samanta. Threshold enhanced cross
sections for colorless productions. Phys. Rev. D, 107(3):034038, 2023.