Single top quark production at ultra-high energies

Among the most challenging problems for the modern physics of fundamental interactions is search for the physics beyond the Standard Model (SM) and the study of the deconfined quark–gluon matter under extreme conditions called also quark–gluon plasma (QGP) which can be created in subatomic particle collisions at high enough energies. The top quark ($t$) sector plays an important role in searches for new physics due to largest mass of $t$ among fundamental particles of the SM and consequent its enhanced sensitivity to hypothetical new heavy particles and interactions. At present a particles beyond the SM are not observed at the Large Hadron Collider (LHC) and this situation implies that there is a considerable energy gap between the SM particles and new physics. Due to the scale separation, various physics effects beyond SM (BSM) below the energy threshold of new physics particle can be characterized by the model-independent Effective Field Theory (EFT) framework. The study of top is crucially important for future development of EFT approach and constraint of its parameters. The uniqueness of the top quark is not only due to its heavy mass, but also due to the fact that it is the only quark that decays before it can hadronize. Thus the study of $t$ behavior in hot environment created at ultra-high energies opens a new ways for investigation of, in particular, a very early pre-equilibrium stages of space–time evolution of QGP. Measurements of interactions of ultra-high energy cosmic rays (UHECR), i.e. cosmic ray particles with initial laboratory energies larger than 0.1–1 EeV, with nuclei in the atmosphere allow the new unique possibilities for study of multiparticle production processes at energies (well) above not only the LHC range but future collider on Earth as well. Due to the air composition and main components of the UHECR the passage of UHECR particles through atmosphere can be considered as collision mostly small systems. It should be emphasized collisions at ultra-high energies can lead to creation of QGP even in light nuclear interactions Okorokov-PAN-82-838-2019 . Therefore the study of single $t$ production at ultra-high energies seems important for search for the signatures of physics BSM and possible creation of bubble of QGP in small system collisions.

Signle top quark is produced through electroweak (EW) interactions. The total inclusive single $t$ production cross section in (anti)proton–proton ($pp$, $\bar{p}p$) collision can be written, in particular, as follows

Here $\mu_{F,R}$ are the factorization and renormalization scales, $i,j$ run over all initial state partons contributed in the production channel under discussion, $\forall\,k=1,2:x_{k}$ is the fraction of the 4-momentum of incoming hadron carried out by the parton, $f_{i}(x_{k},\mu_{F})$ is the distribution function for (anti)parton $i$, $m_{W}$, $m_{t}$ are the mass of $W$ boson and $t$ quark, $s=x_{1}x_{2}s_{\scriptsize{p(\bar{p})p}}$ is squared partonic c.m. energy with $s_{\scriptsize{p(\bar{p})p}}$ is the square of the c.m. energy of the colliding particles, namely $p(\bar{p})$ here, $Q$ is the mass of virtual boson ($W$). The (anti)parton distribution functions (PDFs) are multiplied on the total partonic (short-distance) cross section $\hat{\sigma}_{ij}$ for the single $t$ production from partons $i,j$. Here following choice is used for the factorization and renormalization scales $\mu_{F}=\mu_{R}\equiv\mu$, $\mu=m_{t}$; and, as previously Okorokov-JPCS-1690-012006-2020 , the fixed value $x_{1}x_{2}=1/9$ is chosen in order to get the well-known relation between $e^{+}e^{-}$ and partonic process $s_{\scriptsize{e^{+}e^{-}}}=s$. One can note the information about PDFs is very limited and model–dependent at ultra-high energies $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}\geq 0.1$ PeV. That amplifies the uncertainties for hadronic cross section (1) significantly. Thus the partonic cross section $\hat{\sigma}_{ij}(m_{W},m_{t},s)$ is in the main quantity for study in the present work.

Taking into account the relative strengths of the EW interactions with quark mixing the following partonic subprocesses mostly contribute in the single $t$ production PRD-74-114012-2006

The partonic collision (2a) is the $s$–channel process considered in the present work. The subprocesses (2b) and (2c) correspond to the $t$–channel and type of partonic interaction (2d) is the $tW$ associated production.

Within the approach of vanishing all quark masses except $m_{t}$ ($\forall\,f\geq b:m_{f}\to 0$, $f$ is the quark flavor) the dominant contribution to the leading order (LO) partonic cross section for single $t$ production in the $s$–channel in SM is described by the formula PRD-83-034006-2011 ; JHEP-2002-131-2020

where $\forall\,x=W,t:\rho_{xs}=m_{x}^{2}/s$, $\beta_{xs}^{2}=1-\rho_{xs}$ in accordance with the common notation basis with top pair production Okorokov-JPCS-1690-012006-2020 ; $\alpha_{W}(\mu)$ is the SU(2) running coupling associated with $q\bar{q}W$ vertex and the constant $\alpha_{W}(\mu)$ is renormalized with $N_{f}=5$ active flavors; $\Pi_{V}\equiv|V_{ud}||V_{tb}|$ is the product of the CKM elements for $u\bar{d}$ and $t\bar{b}$ vertices.

According to the detailed discussion elsewhere Okorokov-JPCS-1690-012006-2020 , effects BSM can be described within EFT approach with the general form of Lagrangian $\mathcal{L}_{\scriptsize{\mbox{EFT}}}=\sum_{j=0}\mathcal{L}_{j}\Lambda^{-j}$, where $\mathcal{L}_{0}$ is the SM Lagrangian and $\mathcal{L}_{\scriptsize{\mbox{eff}}}=\sum_{j=1}\mathcal{L}_{j}\Lambda^{-j}$ – effective part containing the effects of new physics, $\Lambda$ is the energy scale of the possible physics BSM. The leading contributions arise at dimension six¹¹1Contributions from odd–dimensional operators lead to lepton- and baryon-number violation AP-335-21-2013 and are neglected in this work. In particular, operator of dimension five, called also Weinberg operator, can be interested for neutrino physics but it is insignificant for the energy range considered within the present work below Okorokov-JPCS-1690-012006-2020 ; PU-65-653-2022 . and can be parameterized in terms of Wilson coefficients $C_{k}^{(6)}$ of dimension–6 operators $O_{k}^{(6)}$ in the effective part $\mathcal{L}_{\scriptsize{\mbox{eff}}}=\mathcal{L}_{\scriptsize{\mbox{eff}}}^{(0)}+\mathcal{O}(\Lambda^{-4})$, $\mathcal{L}_{\scriptsize{\mbox{eff}}}^{(0)}=\sum_{k}\bigl{(}C_{k}^{(6)}\Lambda^{-2}{}^{+}O_{k}^{(6)}+\mbox{h.c.}\bigr{)}+\sum_{l}C_{l}^{(6)}\Lambda^{-2}O_{l}^{(6)}$, where the sum runs over all operators corresponding to the interaction processes under consideration and non-hermitian operators are denoted as ${}^{+}O$ JHEP-2002-131-2020 . The partonic subprocesses considered here involve $t$-quark and the lists of the dimension–6 operators for $t$ production can be found elsewhere PRD-83-034006-2011 ; JHEP-2002-131-2020 . In general the Wilson coefficients are free parameters by definition and are constrained by experimental measurements. Truncation of $\mathcal{L}_{\scriptsize{\mbox{eff}}}$ by only leading contributions, i.e. dimension–6 operators, results in the following general form of the modification of any measured observable $\mathcal{Z}$, in particular, cross section in terms of the Wilson coefficients JHEP-2002-131-2020 ; PU-65-653-2022 ; JHEP-2106-010-2021

where $\mathcal{Z}^{\scriptsize{\mbox{SM}}}$ is the SM prediction, the second term contains the contributions $\mathcal{Z}^{\scriptsize{\mbox{Int},(6)}}_{i}$ arising from the interference of a single dimension–6 operator with the SM, the third term arises from the interference of two diagrams containing one dimension–6 operator each²²2The term $\Lambda^{-4}$-order also appears as a consequence of the interference of the dimension–8 operator contributions with the SM ones, but this part does not included in (4) because of truncation of $\mathcal{L}_{\scriptsize{\mbox{eff}}}$ established above., represents non-linear effects of new physics only and, consequently, quantities $\mathcal{Z}^{\scriptsize{\mbox{BSM},(6)}}_{ij}$ are due to purely physics BSM.

As indicated above the dominant process is (2a) for the $s$–channel of single $t$ production through the EW interaction. The LO partonic single $t$ production cross section due to process (2a) within EFT with leading modification to SM process up to the $\Lambda^{-2}$-order terms is PRD-83-034006-2011 ; JHEP-2002-131-2020

where $C_{\scriptsize{\mbox{$\phi$Q}}}^{(3)}$, $C_{\scriptsize{\mbox{tW}}}$ and $C_{\scriptsize{\mbox{Qq}}}^{(3,1)}$ are the Wilson coefficients for dimension–6 operators in $\mathcal{L}_{\scriptsize{\mbox{eff}}}$.

According to the mostly used approach, the present work is focused on CP–conserving extensions of the SM, assuming that all Wilson coefficients are real and therefore neglecting CP–violating interactions JHEP-2002-131-2020 . Based on the available estimations, the following ranges are used for the Wilson coefficients $C_{\scriptsize{\mbox{$\phi$Q}}}^{(3)}\in[-1.145;0.740]$, $C_{\scriptsize{\mbox{tW}}}\in[-0.313;0.123]$, $C_{\scriptsize{\mbox{Qq}}}^{(3,1)}\in[-0.163;0.296]$ in units of $(\mbox{TeV}/\Lambda)^{2}$ in the present work and these ranges correspond to the 95% confidence level from the global (marginalized) fit using linear in the $\Lambda^{-2}$ EFT calculations JHEP-2111-089-2021 for self-consistency with (4) and (II). The corresponding median values calculated as simple average of the boundary values are $\langle C_{\scriptsize{\mbox{$\phi$Q}}}^{(3)}\rangle=-0.2\pm 0.9$, $\langle C_{\scriptsize{\mbox{tW}}}\rangle=-0.10\pm 0.22$, $\langle C_{\scriptsize{\mbox{Qq}}}^{(3,1)}\rangle=0.07\pm 0.23$ in units of $(\mbox{TeV}/\Lambda)^{2}$.

The energy range for protons in laboratory reference system considered in the present paper is $E_{p}=10^{17}$–$10^{21}$ eV. This range includes the energy domain corresponded to the Greisen–Zatsepin–Kuzmin (GZK) limit Greisen-PRL-16-748-1966 and somewhat expands it, taking into account, on the one hand, both possible uncertainties of theoretical estimations for the limit values for UHECR and experimental results, namely, measurements of several events with $E_{p}>10^{20}$ eV and the absence of UHECR particle flux attenuation up to $E_{p}\sim 10^{20.5}$ eV Okorokov-PAN-81-508-2018 and, on the other hand, the energies corresponding to the nominal value $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}=14$ TeV of the commissioned LHC as well as to the parameters for the main international projects high energy LHC (HE–LHC) with the nominal value $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}=27$ TeV and Future Circular Collider (FCC) with $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}=100$ TeV. Therefore the estimations below can be useful for both the UHECR physics and the collider experiments.

For sufficiently high collision energies at which $\sqrt{\rho_{ts}}\ll 1$, the following limiting relations can be used $\forall\,x=W,t:\rho_{xs}\to 0$, $\beta_{xs}\to 1$ and, as consequence³³3The condition $\rho_{ts}\leq\rho_{ts}^{\tiny{\mbox{max}}}$ can be used for estimating the onset of the domain in which high-energy approach is valid, where $\rho_{ts}^{\tiny{\mbox{max}}}$ is some empirical number. The choice $\rho_{ts}^{\tiny{\mbox{max}}}=10^{-2}$ allows $\sqrt{\smash[b]{s_{\tiny{\mbox{min}}}}}=10m_{t}\approx 1.72$ TeV which corresponds to the $\sqrt{\smash[b]{s_{\scriptsize{pp}}^{\tiny{\mbox{min}}}}}\approx 5.2$ TeV. Thus one can expect that the asymptotic relation (6) will be valid for energy range under consideration here.,

Thus the relative contribution of leading modification to SM process growths with the increase of collision energy and can be dominant for the LO partonic cross section for single $t$ production due to process (2a) at finite value of the Wilson coefficient in (6) at large $s\gg m_{t}^{2}$.

It should be emphasized the following important features of the calculations within EFT. At present all available estimations obtained from individual and global fits of experimental data have a (very) large uncertainties and coincide with null within errors for the Wilson coefficients in (II), (6). As consequence the contribution of the leading modification to SM process, strictly speaking, agrees with null for the single $t$ production in $s$–channel (2a) at available accuracy of measurements. The terms $\propto C_{\scriptsize{\mbox{Qq}}}^{(3,1)}$ have an opposite signs for $s$–channel (2a) and for $t$–channels (2b,2c). Therefore it is expected that the contributions of these channels will cancel or, at least, decrease significantly the term $\propto C_{\scriptsize{\mbox{Qq}}}^{(3,1)}$ in the total LO partonic single $t$ production cross section within EFT with leading modification to SM process. The quantitative verification is in the progress for this qualitative expectation. Therefore the consideration below will be focused on the cross section estimations with median values of the Wilson coefficients taking into account the above clarifications.

For numerical calculations all masses and CKM elements are from PTEP-2022-083C01-2022 . Figure shows the energy dependence of LO partonic cross sections within EFT for the channel (2a), where the dashed line corresponds to the contribution from electroweak part of the SM (k=EW), the dotted line – to the term from the new physics effects (k=eff) and the solid line is the sum for the single $t$ production in the framework of the EFT (k=EFT). As emphasized above the curves for k=eff and EFT are deduced for median value of $C_{\scriptsize{\mbox{$\phi$Q}}}^{(3)}$, $C_{\scriptsize{\mbox{tW}}}$ and $C_{\scriptsize{\mbox{Qq}}}^{(3,1)}$. In the inner panel the LO partonic cross sections are shown for the $s$–channel at energies $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}=1.8-14$ TeV which some lower than the range under consideration for completeness information. This range covers from the measurements at Tevatron ($\sqrt{\smash[b]{s_{\scriptsize{pp}}}}=1.8-1.96$ TeV) up to nominal $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}$ of the LHC and adjoins to the energy domain under study. In accordance with (II) the terms $\hat{\sigma}^{\scriptsize{\mbox{(0),EW}}}_{u\bar{d}\to t\bar{b}}$ and $\hat{\sigma}^{\scriptsize{\mbox{(0),eff}}}_{u\bar{d}\to t\bar{b}}$ show the opposite dependence on $s_{\scriptsize{pp}}$ in functional sense: the LO partonic cross section for channel (2a) decreases with the increase of the collision energy as $\propto s_{\scriptsize{pp}}^{-1}$ at qualitative level within SM whereas the contribution of the leading modification to SM process increases with collision energy up to the $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}\simeq 10$ TeV and then it is almost flat. The behavior of $\hat{\sigma}^{\scriptsize{\mbox{(0),EFT}}}_{u\bar{d}\to t\bar{b}}(s_{\scriptsize{pp}})$ at $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}\simeq 5-10$ TeV confirms the qualitative estimation deduced above for the low boundary in $s_{\scriptsize{pp}}$ of the domain of validity for the high-energy approach (6). The SM mechanism gives the dominate contribution in single $t$ production in $s$–channel in the energy range from Tevatron up to the $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}\approx 3$ TeV and excess of the contribution due to the physics BSM over SM growth at higher energies rapidly. With taking into account the important clarifications made above the cross section due to effective part of the Lagrangian exceeds the corresponding parameter from SM significantly for median values of the Wilson coefficients at $\sqrt{\smash[b]{s_{\scriptsize{pp}}}}=14$ TeV already (Fig., main panel). The flat behavior of $\hat{\sigma}^{\scriptsize{\mbox{(0),EFT}}}_{u\bar{d}\to t\bar{b}}(s_{\scriptsize{pp}})$ agrees with (6) and implies validity of the high-energy asymptotic approach in full main energy domain ($\sqrt{\smash[b]{s_{\scriptsize{pp}}}}\geq 13.7$ TeV) under study.

The uncertainty from uncalculated higher orders in the perturbative expansion is estimated by varying the $\mu_{F,R}$ independently around the central scale choice, $\mu$. Usually, the following intervals are considered for estimation of the theoretical uncertainty JHEP-0910-042-2009

Also values of the SM–parameters $m_{W}$, $m_{t}$, $V_{ud}$, $V_{tb}$ have errors and contribute to the uncertainties of the cross sections $\Delta\hat{\sigma}^{\scriptsize{\mbox{(0),k}}}_{u\bar{d}\to t\bar{b}}$ (k=EW, eff, EFT). Detailed analysis results in the conclusion that the contributions of all these uncertainty sources are neglected with respect to the errors due to (very) large uncertainties of the Wilson coefficients and, consequently, $\Delta\hat{\sigma}^{\scriptsize{\mbox{(0),EFT}}}_{u\bar{d}\to t\bar{b}}\approx\Delta\hat{\sigma}^{\scriptsize{\mbox{(0),eff}}}_{u\bar{d}\to t\bar{b}}$. Because of relative errors for $\langle C_{\scriptsize{\mbox{$\phi$Q}}}^{(3)}\rangle$, $\langle C_{\scriptsize{\mbox{tW}}}\rangle$, $\langle C_{\scriptsize{\mbox{Qq}}}^{(3,1)}\rangle$ larger than 1 there are no uncertainty bands in Figure for clearness and only curves obtained for median values of Wilson coefficients are shown as described above.

Summarizing the foregoing, one can draw the following conclusions.

The partonic cross section for single $t$ production is considered for the $s$–channel at LO level within both the SM and the EFT in ultra-high energy range. The EFT approach takes into account the dimension–6 operators.

The LO partonic cross sections differ significantly for SM and EFT with median values of the Wilson coefficients for single $t$ production in the $s$–channel at nominal LHC energy already. The LO partonic cross section for single $t$ production in the $s$–channel within EFT is almost flat at the level on order 0.15 pb in the energy domain under consideration, i.e. up to the highest energies $\mathcal{O}(1~{}\mbox{PeV})$.

The work is in the progress for other partonic processes ($t$–channel and $tW$ production) as well as for estimations of hadronic cross sections for single $t$ production at ultra-high energies.

This work was supported in part within the National Research Nuclear University MEPhI Program ”Priority 2030”.