Single-top-quark production and decay at NNLO

This calculation is based on ref.¹. See also ref.² for the role of double-DIS scales and the relevancy for PDFs.

This process can be run by using process number 1610. The resulting histograms and cross-sections are printed for a strict fixed-order expansion as well as for a naive addition of all contributions. The fixed-order expansion assembles pieces according to the following formula. Please see ref.¹ for more details.

$$\begin{aligned} \mathrm{d}\sigma_\text{LO} & = \frac{1}{\Gamma_t^{(0)}} \mathrm{d}\sigma^{(0)}\otimes\mathrm{d}\Gamma_t^{(0)} \,,\\ \mathrm{d}\sigma_{\delta \text{NLO}} & = \frac{1}{\Gamma_t^{(0)}} \Bigg[ \mathrm{d}\sigma^{(1)}\otimes\mathrm{d}\Gamma_t^{(0)} + \mathrm{d}\sigma^{(0)}\otimes\left(\mathrm{d}\Gamma_t^{(1)} - \frac{\Gamma_t^{(1)}}{\Gamma_t^{(0)}} \mathrm{d}\Gamma_t^{(0)} \right) \Bigg] \,,\\ \mathrm{d}\sigma_{\delta \text{NNLO}} & = \frac{1}{\Gamma_t^{(0)}} \Bigg[ \mathrm{d}\sigma^{(2)}\otimes\mathrm{d}\Gamma_t^{(0)} + \mathrm{d}\sigma^{(1)}\otimes\left( \mathrm{d}\Gamma_t^{(1)} - \frac{\Gamma_t^{(1)}}{\Gamma_t^{(0)}}\mathrm{d}\Gamma_t^{(0)} \right) \\ & \qquad\qquad + \mathrm{d}\sigma^{(0)}\otimes \left( \mathrm{d}\Gamma_t^{(2)} - \frac{\Gamma_t^{(2)}}{\Gamma_t^{(0)}}\mathrm{d}\Gamma_t^{(0)} -\frac{\Gamma_t^{(1)}}{\Gamma_t^{(0)}} \left( \mathrm{d}\Gamma_t^{(1)} - \frac{\Gamma_t^{(1)}}{\Gamma_t^{(0)}} \mathrm{d}\Gamma_t^{(0)} \right) \right) \Bigg] \,. \end{aligned}$$

At each order a corresponding top-decay width is used throughout all parts. The NNLO width is obtained from ref.³ and at LO and NLO from ref.⁴. These widths agree with numerical results obtained from our calculation of course.

This process can be run with a fixed scale or with dynamic DIS (DDIS) scales by setting dynamicscale = DDIS, renscale = 1.0 and facscale = 1.0.

At NNLO there are several different contributions from vertex corrections on the light-quark line, heavy-quark line in production, and heavy-quark line in the top-quark decay. Additionally there are one-loop times one-loop interference contributions between all three contributions. These contributions can be separately enabled in the singletop block:

[singletop]
    nnlo_enable_light = .true.
    nnlo_enable_heavy_prod = .true.
    nnlo_enable_heavy_decay = .true.
    nnlo_enable_interf_lxh = .true.
    nnlo_enable_interf_lxd = .true.
    nnlo_enable_interf_hxd = .true.
    nnlo_fully_inclusive = .false.

For a fully inclusive calculation without decay the last setting has to be set to .true. and the decay and decay interference parts have to be removed. Additionally jet requirements must be lifted, see below.

When scale variation is enabled with DDIS scales then automatically also a variation around the fixed scale $\mu=m_t$ is calculated for comparison.

This process uses a fixed diagonal CKM matrix with $V_{ud}=V_{cs}=V_{tb}=1$. The setting removebr=.true. removes the $W\to \nu e$ branching ratio.

This process involves complicated phase-space integrals and we have pre-set the initial integration calls for precise differential cross-sections with fiducial cuts. The number of calls can be tuned overall with the multiplier setting integration%globalcallmult. For total fully inclusive cross-sections the number of calls can be reduced by a factor of ten by setting integration%globalcallmult = 0.1, for example.

For scale variation uncertainties and PDF uncertainties we recommend to start with the default number of calls and a larger number of warmup iterations integration%iterbatchwarmup=10, for example. For the warmup grid no scale variation or PDF uncertainties are calculated and this ensures a good Vegas integration grid that can be calculated fast. The setting integration%callboost modifies the number of calls for subsequent integration iterations after the warmup. For example setting it to 0.1 reduces the calls by a factor of ten. This is typically enough to compute the correlated uncertainties for a previously precisely determined central value.

At NNLO the default value for $\tau_\text{cut}$ is $10^{-3}$, which is the value used for all the plots in our publication. We find that cutoff effects are negligible at the sub-permille level for this choice. We strongly recommend to not change this value.

Using the plotting routine with b-quark tagging

The calculation has been set up with b-quark tagging capabilities that can be accessed in both the gencuts_user.f90 routine and the plotting routine nplotter_singletop_new.f90. The plotting routine is prepared to generate all histograms shown in our publication in ref.¹. By default the top-quark is reconstructed using the leading b-quark jet and the exact W-boson momentum, but any reconstruction algorithm can easily be implemented.

We have added the gencuts_user.f90 file as used for the plots in our paper ¹ in src/User/docs/gencuts_user_singletop_nnlo.f90 that can be used as a guide on how to access the b-quark tagging in the gencuts_user routine.

See also nplotter_ktopanom.f (used for the NLO off-shell calculation in ref.⁵ for a reconstruction of the W-boson. It is based on requiring an on-shell W-boson and selecting the solution for the neutrino z-component that gives the closest on-shell top-quark mass by adding the leading b-quark jet.

Calculating fully inclusive cross-sections

When calculating a fully inclusive cross-section without top-quark decay please set `zerowidth = .true., removebr = .true. in the general section of the input file; inclusive = .true, ptjetmin = 0.0, etajetmax = 99.0 in the basicjets section; makecuts = .false. in the cuts section; also set nnlo_enable_heavy_decay = .false. and nnlo_enable_interf_lxd = .false., nnlo_enable_interf_hxd = .false. and nnlo_fully_inclusive = .true. in the singletop section.

These settings ensure that neither the decay nor any production times decay interference contributions are included. The last setting makes sure that only the right pieces in the fixed-order expansion of the cross-section are included. It also ensures that the b-quark from the top-quark decay is not jet-tagged and just integrated over.

Notes on runtimes and demo files

Running the provided input file input_singletop_nnlo_Tevatron_total.ini with -integration%globalcallmult=0.1 and without histograms takes about 4-5 CPU days. So depending on the number of cores, this can be run on a single desktop within a few hours.

Running the input file input_singletop_nnlo_LHC_fiducial.ini with the default set of calls and histograms takes about 3 CPU months (about 3 wall-time hours on our cluster with 45 nodes). For the fiducial cross-section (without precise histograms) a setting of -integration%globallcallmult=0.2 can also be used.

Note that -extra%nohistograms = .true. has been set in these demonstration files, so no further histograms from nplotter_singletop_new.f90 are generated.

The input file input_singletop_nnlo_LHC_fiducial.ini together with the file src/User/docs/gencuts_user_singletop_nnlo.f90 replacing src/User/gencuts_user.f90 reproduces the fiducial cross-sections in ref.¹ table 6.

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1. John Campbell, Tobias Neumann, and Zack Sullivan. Single-top-quark production in the $t$-channel at NNLO. JHEP, 02:040, 2021. arXiv:2012.01574, doi:10.1007/JHEP02$2021$040. ↩↩↩↩↩

2. John Campbell, Tobias Neumann, and Zack Sullivan. Testing parton distribution functions with t-channel single-top-quark production. Phys. Rev. D, 104$9$:094042, 2021. arXiv:2109.10448, doi:10.1103/PhysRevD.104.094042. ↩

3. Ian Richard Blokland, Andrzej Czarnecki, Maciej Slusarczyk, and Fyodor Tkachov. Next-to-next-to-leading order calculations for heavy-to-light decays. Phys. Rev. D, 71:054004, 2005. [Erratum: Phys.Rev.D 79, 019901 $2009$]. arXiv:hep-ph/0503039, doi:10.1103/PhysRevD.71.054004. ↩

4. Andrzej Czarnecki. QCD corrections to the decay t —\ensuremath > W b in dimensional regularization. Phys. Lett. B, 252:467–470, 1990. doi:10.1016/0370-2693$90$90571-M. ↩

5. Tobias Neumann and Zack Edward Sullivan. Off-Shell Single-Top-Quark Production in the Standard Model Effective Field Theory. JHEP, 06:022, 2019. arXiv:1903.11023, doi:10.1007/JHEP06$2019$022. ↩