Threshold resummation for the production of four top quarks at the LHC

We compute the total cross section for $t\bar{t}t\bar{t}$ production at next-to-leading logarithmic (NLL$^{\prime}$) accuracy. This is the first time resummation is performed for a hadron-collider process with four colored particles in the final state. The calculation is matched to the next-to-leading order strong and electroweak corrections. The NLL$^{\prime}$ corrections enhance the total production rate by 15\%. The size of the theoretical error due to scale variation is reduced by more than a factor of two, bringing the theoretical error significantly below the current experimental uncertainty of the measurement.

The production of four top quarks, pp → t tt t, is one of the rarest Standard Model (SM) production processes currently accessible experimentally at the Large Hadron Collider (LHC).Its cross section is known to receive significant contributions in various SM extensions, hence an accurate measurement can set strong constraints on new physics models.Examples of such scenarios include supersymmetric theories, where the t tt t signal can be enhanced by squark and gluino decays [1,2], the production of a new heavy (pseudo)scalar boson in association with a t t pair [3][4][5], or pair production of scalar gluons [6][7][8][9].Moreover, the t tt t production rate is sensitive to the Yukawa coupling of the top quark, making it a useful process to further constrain the nature of Higgs-top quark interactions [10,11].When interpreted in the framework of an effective theory, a measurement of the t tt t production process places strong constraints on the four-fermion operator [12][13][14][15][16][17][18].
The ATLAS and CMS experiments have searched for the production of t tt t at the LHC operating at √ s = 13 TeV [19][20][21][22][23][24].In the latest ATLAS analysis [23] a cross section of σ t tt t = 24 ± 4(stat.)+5  −4 (syst.)fb is measured, whereas the recent combined analysis of CMS [24] reports a cross section of σ t tt t = 17 +5 −5 fb.Intriguingly, these obtained central values are above the SM prediction, which is calculated at the next-to-leading order (NLO) accuracy both in the strong (QCD) and electroweak (EW) coupling [25][26][27][28][29], with the ATLAS measurement consistent with the SM result only within 2σ.The NLO calculations carry a theoretical error due to scale variation of around 25%, which is comparable with the size of the individual errors of the latest ATLAS and CMS measurements.It is therefore of crucial importance to improve the precision of the theoretical predictions for the t tt t production, especially having in mind that the future analysis will involve much larger sets of LHC data and the precision of the measurement will increase substantially.
More than 90% of the full NLO result originates from pure QCD interactions.Currently, the calculation of the next-to-next-to-leading order (NNLO) QCD corrections remains out of reach.However, it is possible to systematically consider a part of higher-order QCD corrections originating from multiple soft gluon emissions.Given the very large partonic centre-of-mass (CM) energy √ ŝ needed to produce four top quarks, √ ŝ 700 GeV, the t tt t production at the LHC very often takes place close to production threshold, with any additional real radiation strongly suppressed.One can therefore expect that a large part of the higher-order corrections is due to soft emission and stems from the threshold region.Correspondingly, computing higher-order corrections of this type offers a promising way to improve the precision of the prediction.
Higher-order QCD corrections from soft gluon emission can be systematically taken into account using resummation, either in direct QCD or in the soft-collinear effective-field-theory framework.The resummation programme for processes involving multiple top quarks has been very successful over the recent years, leading to substantial improvements of theoretical precision for the calculation of the total production cross section for such processes, such as top-pair production [30][31][32][33][34][35][36][37][38][39] or t tH/Z/W ± /γ [40][41][42][43][44][45][46][47][48][49][50][51][52].However, in contrast to t tt t, these processes involve at most two coloured particles in the final state.To the best of our knowledge, resummation for processes involving a higher number of coloured particles has not been achieved before.
In this work, we perform for the first time the resummation of a process with 4 coloured particles at the Born level by applying direct QCD resummation methods in Mellin space to the process pp → t tt t.The calculations are carried out at the next-to-leading logarithmic (NLL) accuracy, and take into account constant O(α s ) nonlogarithmic contributions that do not vanish at threshold (referred to as NLL accuracy).

I. METHODOLOGY
Soft-gluon corrections get logarithmically large at the absolute production threshold when √ ŝ approaches M ≡ 4m t , with m t the mass of the top quark.This corre-sponds to the limit ρ → 1 of the partonic threshold variable, ρ ≡ M 2 /ŝ.The theory of 2 → 4 threshold resummation builds on the theory resummation for 2 → 2 processes [31,[53][54][55].We work in Mellin space, where the hadronic cross section σ t tt t(N ) is the Mellin transform w.r.t. the variable ρ ≡ M 2 /s of the hadronic cross section in momentum space Here we use f i to denote parton distribution functions (PDFs), µ F the factorisation scale, and x 1,2 the momentum fraction of the two colliding partons i, j.Only two partonic channels contribute at leading order (LO), ij = {q q, gg}.The cross section σij→t tt t(N ) is a purely perturbative function that obeys a refactorisation in the soft and collinear limits into functions containing information on particular modes of dynamics.Correspondingly, one can identify a soft function S, containing corrections originating from soft gluon radiation, a collinear / jet function for each initial-state leg ∆ i , containing corrections from collinear gluon radiation.All terms that are non-logarithmic in the soft-gluon limit are collected in the hard function H.These functions are defined at the cross section level, i.e. they include the necessary phasespace integrals.The factorisation in Mellin space takes the form suppressing the dependence of the various ingredients on the factorisation and renormalisation scales.As the jet and soft functions both capture soft-collinear enhancements, care must be taken to subtract the overlap contributions.In practice, this is done by dividing out the eikonal jet functions J i from the soft function.This results in a new soft-collinear subtracted soft function that is denoted by S, and related to the full soft function as The soft and hard functions are generally matrices in colour space, as indicated by their bold font, and colourconnected, indicated by the ⊗-symbol.We now briefly go over the definition for each of the ingredients in Eq. ( 3).
The hard function H ij→t tt t in Eq. ( 3) obeys the perturbative expansion At the NLL accuracy we need H ij→t tt t, defined as a matrix in colour space with an element IJ where we sum (average) over final(initial)-state colour and polarization degrees of freedom.The Born phase space is denoted by Φ B .The object is the colour-stripped amplitude projected to the colourvector c I , with |A (0) the amplitude in the corresponding colour basis, while A (0) † J is its complex conjugate projected to c † J .We obtain the squared matrix elements numerically from aMC@NLO [26,56], after selecting a suitable colour basis as discussed below.
For the incoming jet functions we use the well-known expressions that can be found in e.g.[64][65][66], which are a function of λ = α s b 0 ln N with N ≡ N e γ E .The soft function is given by [31,53] with the evolution matrix written as a path-ordered exponential and Γ ij→t tt t(α s (q 2 )) the soft anomalous dimension matrix.To achieve NLL( ) resummation we need to know the one-loop contribution Γ (1) ij→t tt t in Eq. ( 9).This object consists of a kinematic part and a colour-mixing part, which accounts for the change in colour of the hard system, i.e.

Γ
(1) where T k are colour operators.The explicit expression for Γ ij→t tt t,IJ depends on a choice of basis tensors represented by c I (and c † J for the complex conjugate) for the underlying hard scattering process ij → t tt t.The kinematic part, Γ kl , is given by the residue of the UV-divergent part of the one-loop eikonal contributions [30,67,68].
The matrix S ij→t tt t in Eq. ( 8) represents the boundary condition for the solution of the renormalisation group equation at µ R = M/ N from which Eq. ( 8) follows.Like H, it obeys a perturbative expansion which reads The lowest-order contribution S (0) ij→t tt t is given by the trace of the colour basis vectors for the underlying hard process.For NLL resummation we also need the firstorder correction S (1) ij→t tt t, which is calculated analytically by considering the eikonal corrections to S (0) ij→t tt t.
The major difficulty in the resummed calculations for the t tt t production cross section stems from the complicated colour structure of the underlying hard process, involving six coloured particles.The colour structure of the q q → t tt t process is The decomposition into irreducible representations reads For the gg channel we have and in terms of irreducible representations From this we infer that the q q colour space is 6dimensional, whereas the gg one is 14-dimensional, directly translating into the dimensions of the soft anomalous dimension matrices of Eq. (10).
Moreover, the one-loop soft anomalous dimension matrices Γ (1) ij→t tt t are in general not diagonal.Solving Eq. ( 9) in terms of standard exponential functions requires changing the colour bases to R where Γ (1) ij→t tt t,R is diagonal [53].We find such orthonormal bases using the technique outlined in Ref. [69].The resulting oneloop soft anomalous dimension matrices for N c = 3 in the threshold limit become 1 2Re[Γ q q→t tt t,R ] = diag (0, 0, −3, −3, −3, −3) , (16a) 1 Their full forms will be given in Ref. [70].The values above are the negative values of the quadratic Casimir invariants for the irreducible representations in which the colour structure of the final state can be decomposed in SU(3).This result corresponds to a physical picture where the soft gluon is only sensitive to the total colour charge of a system at threshold, and constitutes a strong check of our calculations.We have also verified that the virtual corrections obtained from Mad-Loop, rewritten in the new basis R, are consistently 0 for the base vector corresponding to a representation whose dimension is zero for N c = 3, which is another important consistency check of our work.With this, the contribution of the soft-collinearsubtracted soft function in Mellin space reads ij→t tt t,R after subtracting the soft-collinear contributions [54].Note that the hard function in Eq. ( 3) also needs to be written in terms of the colour tensor basis R, requiring us to transform from the trace-basis used in aMC@NLO to our new basis.
The last step to calculate a physical cross section in momentum space involves taking the inverse Mellin transform of the N -space expression σ f.o.+res , where 'res' denotes LL, NLL or NLL .To retain the full available information from the perturbative calculation, we match the resummed result to the fixed-order cross section σ f.o., leading to the 'f.o.+res' accuracy.To avoid double-counting, the resummed result is expanded up to O(α n s ) (denoted as σres ij→t tt t(N ) O(α n s ) ), with n = 4 for f.o.=LO or n = 5 for f.o.=NLO.The inverse Mellin transform in Eq. ( 18) relies on the so-called Minimal Prescription [71] and is evaluated numerically on a contour C parameterised by C MP and φ MP as with y ∈ [0, ∞).We calculated results for various values C MP and φ MP to verify the independence of the result on the choice of the contour.

II. NUMERICAL RESULTS
The phenomenological studies reported in this letter are performed using the central member of the LUXqed plus PDF4LHC15 nnlo 100 PDF set [72,73] for both the pure QCD results and the QCD + EW results.This PDF set is based on the PDF4LHC15 PDF set [74][75][76][77] and includes the photon content of the proton, needed for the calculation of the EW corrections.We use the α s value corresponding to the PDF set, take the mass of the top quark m t = 172.5 GeV (unless stated otherwise) and choose the central factorisation and renormalisation scale µ F,0 = µ R,0 = 2m t .The theoretical uncertainty is estimated by varying µ R and µ F using a 7-point scale variation.To this end, we consider the minimal and maximum cross section values calculated for , µ F µ F,0 7−point ∈{(0.5, 0.5), (0.5, 1), (1, 0.5), (1, 1), (1, 2), (2, 1), (2, 2)} .( The fixed-order results are obtained using aMC@NLO [26,56]. Since our calculation concerns pure QCD corrections, we present the LO and NLO QCD results for comparison.However, our final resummation-improved cross section incorporates the NLO(QCD+EW) result, where the electroweak corrections are included up to O(α 2 ) [28]. 2 We show our results for N resummation, but did confirm that those for N -resummation show qualitatively the same behaviour.We defer a detailed discussion of the subtle differences between N and N resummation to an upcoming publication [70].
In Fig. 1 we show the scale dependence of various fixed-order and matched resummed results for σ t tt t under the assumption µ R = µ F .While the NLL corrections only moderately improve the scale dependence of the NLO QCD cross section, the scale sensitivity of the NLO+NLL result is dramatically reduced.NLL contributions increase the σ t tt t predictions by 16% w.r.t. the pure NLO QCD result, and by 15% w.r.t. the complete NLO (QCD+EW) result, see the reported K NLL factors in Table I.These corrections are more than twice the size of the previously calculated complete EW effects at NLO.
Next we examine the reduction of the theoretical error of the resummation-improved cross section using the 7-point method.In Table I we quote the central values of the NLO, NLO(QCD+EW), NLO+NLL and NLO(QCD+EW)+NLL cross sections together with the corresponding error due to scale variation.This information is graphically represented in Fig. 2. We see that the 7-point method scale error gets smaller with increasing accuracy of the calculations.Remarkably, the scale error of the NLO+NLL predictions is reduced compared to NLO predictions by more than a factor of 2. Including the PDF uncertainty of ±6.9%, our state-of-the-art 2 In the notation of Ref. [28], we include up to (N)LO 3 .6 TeV, the central scale value of µ0 = 2mt and mt = 172.5 GeV.The number in parenthesis indicates the statistical uncertainty on the last digit whereas the percentage error indicates the 7-point scale uncertainty, obtained using the variations indicated in Eq. (20).The K NLL factor is the ratio of the resummation-improved cross section at NLO+NLL to the NLO cross section.In Table I  which is an increase of 18.3% w.r.t. the obtained cross section for √ s = 13 TeV.
We have also studied the effect of varying the value of the top mass in the window of [170 − 175] GeV.The resulting predictions are shown in Fig. 3 for √ s = 13 TeV.We observe that the correction stemming from soft-gluon resummation is flat under variation of the top quark mass.

III. CONCLUSION
In this letter, we have obtained predictions for the total cross section of the four top production process at NLO+NLL accuracy, including electroweak corrections for the fixed-order prediction.This is the first time that the framework of threshold resummation has been applied to a 2 → 4 process containing six coloured partons at leading order.We present our results both at a collider energy of 13 and 13.6 TeV, and vary the top mass in the window of 170−175 GeV.Setting m t = 172.5 GeV and √ s = 13.6 TeV, we find the total cross section σ NLO(QCD+EW)+NLL t tt t = 15.8 +1.5% −11.6% fb, where the indicated error is estimated using the 7-point scale uncertainty.When compared to the NLO(QCD+EW)-only prediction, σ NLO(QCD+EW) t tt t = 13.8 +22.6% −22.9% fb, we find that the central value is increased with a K-factor of 1.15.The uncertainty stemming from scale variation is reduced by more than a factor of two.Including the PDF error in quadrature we reduce the total theoretical uncertainty from (+23.6%, −23.9%) at NLO(QCD+EW) to (+6.8%, −13.4%) at NLO(QCD + EW)+ NLL , which lies comfortably below the current experimental uncertainty.These predictions will play an important role in stress-testing the SM, especially in view of the latest experimental results obtained for t tt t production.

FIG. 3 .
FIG.3.Cross section for the pp → t tt t process with √ s = 13 TeV for different values of mt.Shown are the LO, NLO and NLO+NLL' predictions (QCD + EW).The bands indicates the scale uncertainty calculated using the 7-point method, where the central scale is taken to be µ0 = 2mt.
tt t enters formally at next-tonext-to-leading logarithmic accuracy but can be used to supplement the NLL expressions, resulting in NLL precision.It consists of virtual one-loop corrections, V ij→t tt t that are not yet captured by the initial-state jet functions ∆ i , i.e.

TABLE I .
Fixed and resummed-and-matched total cross sections in fb for pp → t tt t with we also report the obtained cross section for the LHC CM energy of 13.6 TeV.Including the scale uncertainty of ±6.7% we obtain