NNLL-fast: predictions for coloured supersymmetric particle production at the LHC with threshold and Coulomb resummation

We present state-of-the art predictions for the production of supersymmetric squarks and gluinos at the Large Hadron Collider (LHC), including soft-gluon resummation up to next-to-next-to-leading logarithmic (NNLL) accuracy, the resummation of Coulomb corrections and the contribution from bound states. The NNLL corrections enhance the cross-section predictions and reduce the scale uncertainty to a level of 5-10%. The NNLL resummed cross-section predictions can be obtained from the computer code NNLL-fast, which also provides the scale uncertainty and the pdf and alpha_s error.


Introduction
One of the most important goals of the present phase of the Large Hadron Collider (LHC) programme is the search for supersymmetry (SUSY) [1][2][3][4][5][6].SUSY predicts a rich variety of new particles and a number of dark-matter candidates, while at the same time providing a natural solution to the hierarchy problem and facilitating gauge-coupling unification at a high energy scale.For SUSY to be natural, it has to be realised close to the electroweak scale, within the reach of the current Run II of the LHC with a centre-of-mass energy of √ S = 13 TeV.In particular, the partners of the coloured particles -squarks (q) and gluinos (g) -would be produced in copious amounts if their masses were around a few TeV or below.Experimental searches set a current lower mass limit on the coloured supersymmetric particles of around 1 TeV up to 1.8 TeV, depending on the specific SUSY model [7][8][9][10][11][12].
In the framework of the Minimal Supersymmetric Standard Model (MSSM) with R-parity conservation [13,14], supersymmetric particles (or short: sparticles) are always produced in pairs at collider experiments.In the following we therefore consider pairproduction of squarks and gluinos through the collision of two hadrons h 1 and h 2 .i.e. h 1 h 2 → q q * , q q (+ q * q * ) , gg , qg (+ q * g) + X and In the first set of production channels, charge-conjugated processes are combined and the chiralities of the squarks (q L , qR ) are understood to be summed over.Also the different squark flavours are understood to be summed over with the exception of the stops ( t), the superpartners of the third-generation top quarks.The production of stops is treated separately because of the potentially strong mixing between the weakly-interacting gauge eigenstates of the stops, which can lead to a large mass splitting between the lighter ( t1 ) and heavier ( t2 ) physical stop mass eigenstates [15] and to a stronger dependence on SUSY model parameters.
Accurate predictions for the production of squarks and gluinos are important to derive mass exclusion limits, and, should SUSY be realised in nature, to precisely measure the masses and properties of the sparticles.The next-to-leading order (NLO) corrections to total production cross sections and differential distributions have been known for some time, both within QCD [16][17][18][19][20][21][22][23][24][25] and including electroweak corrections [26][27][28][29][30][31][32][33].An important contribution to the total cross section originates from the kinematical regime where the production of the coloured sparticles proceeds close to the production threshold.In particular for high sparticle masses, the production cross sections receive a significant fraction of the NLO corrections from this kinematical regime due to soft-gluon emission off the coloured initial-and final-state particles.Furthermore, Coulombic gluon exchange between the slowly moving heavy final-state particles results in additional large contributions.
Taking into account the soft-gluon corrections to all orders in the strong coupling can be achieved by means of threshold resummation techniques in Mellin space [34][35][36][37][38][39].In this framework, corrections from soft gluons at next-to-leading logarithmic (NLL) accuracy have already been available for a while for all production processes of squarks and gluinos in the MSSM, including stops [40][41][42][43][44].Alternatively, the analysis of the soft-gluon effects can be performed in the framework of soft-collinear effective theory (SCET), which also allows an efficient treatment of Coulomb contributions [45,46].The present state-of-the-art is that threshold resummation corrections up to next-to-next-to-leading logarithmic (NNLL) accuracy have been implemented for gluinos and first-and second-generation squarks in [47][48][49][50][51][52][53], and recently also for stops in [54,55].
Coulombic gluon-exchange effects have first been studied for the case of top quark production at e + e − colliders [56][57][58].As they solely depend on the colour structure of the final-state particles, the methods derived for top production can straightforwardly be applied to other processes like the production of squarks and gluinos.While resummed Coulomb contributions involving a leading-order Coulomb interaction can be described by a Sommerfeld factor (see e.g.[56]), higher-order corrections to the Coulomb potential can be conveniently implemented through Green's functions in the non-relativistic QCD (NRQCD) approach [59][60][61].
In this work, we improve the NNLL predictions for all squark and gluino production cross sections obtained with the conventional Mellin-space technique by including resummation of Coulomb corrections and by adding effects from bound states.We analyse the impact of these types of corrections on the size of the cross section and its theoretical error.The effects of NNLL resummation on squark and gluino production processes are demonstrated at the LHC with √ S = 13 TeV, examining squark and gluino masses up to 3 TeV.Our results can be obtained from a new computer code, called NNLL-fast [62], which provides NNLL predictions and error estimates for these processes.
The structure of the paper is as follows.In section 2 we briefly review the NNLL resummation formalism in Mellin space, including in particular our treatment of the Coulomb and bound-state corrections.Numerical results are presented in section 3, where we also compare with results obtained within the SCET formalism.We conclude in section 4.

NNLL formalism in Mellin space: a brief review
We begin with the general formulation of the resummed total cross section in Mellin space for hadronic processes involving the production of a heavy particle pair.Subsequently we move on to discuss the resummation of Coulomb corrections and the inclusion of boundstate effects.

Resummation of threshold logarithms at NNLL
The inclusive hadronic cross section for the production of particles k and l, σ h 1 h 2 →kl , can be written in terms of the partonic cross section, σ ij→kl , in the following manner: Here {m 2 } denotes all the masses entering the calculation, i and j are the initial-state parton flavours, f i/h 1 and f j/h 2 are the parton distribution functions, µ is the common factorisation and renormalisation scale, x 1 and x 2 are the momentum fractions of the partons inside the hadrons h 1 and h 2 , and ρ and ρ are the hadronic and partonic threshold variables, respectively.The threshold for the production of two final-state particles k and l with masses m k and m l corresponds to a hadronic center-of-mass energy squared of S = (m k + m l ) 2 .Therefore we define the hadronic threshold variable ρ, measuring the distance from threshold in terms of a quadratic energy fraction, as

S ,
where m av = (m k + m l )/2 is the average mass of the final-state particles k and l.
In the threshold regime, the dominant contributions to the higher-order QCD corrections due to soft-gluon emission have the general form where s = x 1 x 2 S is the partonic center-of-mass energy squared and α s the strong coupling.We perform the resummation of the soft-gluon emission after taking the Mellin transform (indicated by a tilde) of the cross section: 3) The logarithmically enhanced terms now take the form of α n s log m N , m ≤ 2n, where the threshold limit β → 0 corresponds to N → ∞.The all-order summation of such logarithmic terms follows from the near-threshold factorisation of the cross section into functions that each capture the contributions of classes of radiation effects: hard, collinear and wide-angle soft radiation [34][35][36][37][38][39].Near threshold the resummed partonic cross section takes the form where we have introduced the hard scale Q 2 = 4m 2 av .The soft radiation is coherently sensitive to the colour structure of the hard process from which it is emitted [36-39, 63, 64].At threshold, the resulting colour matrices become diagonal to all orders by performing the calculation in the s-channel colour basis [40,41,65].The different contributions then correspond to different irreducible representations I. Correspondingly, σ(0) ij→kl,I in equation (2.4) are the colour decomposed leading-order (LO) cross sections.The effects from collinear radiation are summed into the functions ∆ i and ∆ j , and the wide-angle soft radiation is described by ∆ (s) ij→kl,I .The radiative factors can then be written as This exponent contains all the dependence on large logarithms L = log N .The leading logarithmic approximation (LL) is represented by the g 1 term alone, whereas the NLL approximation requires including the g 2 term in addition.Similarly, the g 3 term is needed for the NNLL approximation.The customary expressions for the g 1 and g 2 functions can be found in e.g.[41] and the one for the NNLL g 3 function in e.g.[47].The matching coefficients C ij→kl,I in (2.4) collect non-logarithmic terms as well as logarithmic terms of non-soft origin in the Mellin moments of the higher-order contributions.At NNLL accuracy the coefficients C ij→kl,I factorise into a part that contains the Coulomb corrections C Coul ij→kl,I and a part containing hard contributions C ij→kl,I [45]: ij→kl,I + . . . ) . (2.6)

Resummation of Coulomb corrections
In our previous work [53,55] we have included the Coulomb corrections up to O(α 2 s ), i.e.
ij→kl,I .Here we perform an additional resummation of Coulomb corrections by employing the Coulomb Green's function obtained in the NRQCD framework.This results in the following expression for the cross section: with G(0, 0; E) being the Green's function at the origin and m av (1− ρ) being the energy relative to the production threshold.The free Green's function G free (0, 0; E) corresponds to freely propagating particles with no Coulomb interactions present.Note that in our approach we use the approximation that the produced sparticles are stable, i.e. we use the zero-width approximation Γ → 0. The effect of a non-zero decay width has been studied in [66] at NLL accuracy.For a moderate width, Γ/m 5%, the higher-order soft and Coulomb corrections are well described by the limit Γ → 0, and the remaining ambiguities due to finite-width effects are of similar size as the theoretical uncertainties of the threshold-resummed higher-order calculations.The Green's function up to a certain accuracy is obtained by solving the Schrödinger equation with a Coulomb potential up to that particular accuracy [61,67].Up to NLO the Coulomb potential is given by the following expression in coordinate space: with γ E = 0.57721... the Euler-Mascheroni constant, b 0 = (11 The general solution to the Schrödinger equation for two particles with unequal masses can be written in the following compact form [61,[67][68][69][70]: with Γ = (Γ k + Γ l )/2 the average decay width of the two particles, the free Green's function, and (1) . (2.12) Here in terms of the non-relativistic velocity The function 4 F 3 (1, 1, 1, 1; 2, 2, 1 − κ; 1) in eq.(2.12) is the generalised hypergeometric function 1 and the masses of the final-state particles enter the solution in the form of the reduced mass m red = m k m l /(m k + m l ).Setting Γ = 0 and rewriting E = 2m red v2 = √ ŝ−2m av , the non-relativistic velocity v can be related to the threshold variable β through its definition β = 1 − 4m 2 av /ŝ , resulting in v ≈ β m av /(2m red ) if we neglect thresholdsuppressed terms of O(β 4 ).
The partonic leading-order cross section σ ij→kl,I splits up close to threshold into a β-dependent part corresponding to the free Green's function and a β-independent hard function H (0) ij→kl,I : The proper incorporation of the Coulomb corrections is then achieved by a rescaling of the leading-order cross section by the factor [71]: for every colour channel I separately, giving rise to eq. (2.7).To further increase the accuracy, we supplement the Green's function at NLO with the log β corrections originating from two-loop non-Coulombic terms in the NRQCD potential [61,67,72].This is achieved by multiplying the free Green's function part and the LO Coulomb contribution g LO in eq.(2.10) by the ∆ nC factor presented in [51,52,72]. 2  The Green's functions presented in this section and in particular the near-threshold expansion eq.(2.15) and multiplication prescription eq.(2.16) are only valid for a final state produced with zero orbital angular momentum, corresponding to an s-wave state.Higher partial-wave contributions, such as p-wave states, require a modification of the Green's function (see e.g.[46] and references therein).Whereas cross sections for s-wave states are proportional to β close to threshold, p-wave states experience an additional phase-space suppression factor of β 2 relative to the s-wave states.For the analysis of squark and gluino production at NNLL accuracy presented here, the p-wave states only give a negligible contribution.They are thus treated at NLL accuracy which does not require the inclusion of the Coulomb terms.

Inclusion of bound-state contributions
A very interesting feature of the Coulomb interactions is the possibility of bound-state formation at energies below the production threshold of the particles.The Green's function then develops single poles of the form [75] G where |ψ n (0)| 2 corresponds to the wave function at the origin of the bound-state system in the n th level with bound-state energy E n < 0. In the narrow-width approximation Γ → 0, the imaginary part of the Green's function turns into a sum of sharp δ-peaks: Only taking into account the LO Coulomb potential, the bound-state energies can be computed by analysing eq.(2.10): the digamma function ψ (0) (1 − κ) is singular for κ = n, with n ≥ 1 a positive integer.Rewriting κ into E gives: The residues of Im G(E) at the bound-state energies precisely give the value of the wave function at the origin, i.e.
Therefore, the imaginary part of the bound-state Green's function for a LO Coulomb potential becomes In this work we consider the energy levels and residues of a LO Coulomb potential3 and suppress the interplay with the soft-gluon corrections, arriving at with the luminosity function While the integration range τ ∈ [ρ, 1] corresponds to the region above threshold where the definitions of eqs.(2.10) and (2.11) can be used, we are now interested in the integration range τ ∈ [0, ρ] that denotes the region below threshold.Firstly, we apply eq.(2.15) to rewrite the cross section close to threshold and subsequently replace Im G(0, 0; E) with eq.(2.21) to obtain Rewriting the delta function with τ n = (E n + 2m av ) 2 /S, the integration over τ can be performed trivially.Additionally, we improve the description of the partonic cross section near threshold by replacing the flux factor at threshold 1/ŝ thr ≈ 1/(4m 2 av ) with the exact flux factor 1/ŝ, resulting in It should be noted that in the sum over the colour channels I, only channels corresponding to an attractive Coulomb potential, i.e.D ij→kl,I > 0, are summed over.
The bound-state contributions below threshold are then added to the NNLL predictions above threshold including Coulomb resummation, matched in the Mellin-space formalism to the approximated next-to-next-to-leading order cross section NNLO Approx .The latter is constructed by adding the near-threshold approximation of the NNLO correction [72] to the full NLO result [18].As the final result, we thus obtain σ (NNLL matched, Coul+BS) To evaluate the inverse Mellin transform in (2.27) we adopt the "minimal prescription" of reference [76] for the integration contour CT.

Numerical results
In this section we present numerical results for the NNLL resummed cross sections matched to the approximated NNLO results for pair production of squarks and gluinos at the LHC with √ S = 13 TeV.As already mentioned, all flavours of final-state squarks are included and summed over, except for stops, which are treated separately because of the large mixing effects and the mass splitting in the stop sector.We sum over squarks with both chiralities (q L and qR ), which are taken as mass degenerate.All light-flavour squarks are also assumed to be mass degenerate.The QCD coupling α s and the parton distribution functions at NLO and NNLO are defined in the MS scheme with five active flavours, and a top-quark mass of m t = 173.2GeV [77] is used.At NLO and beyond, the stop cross section depends not only on the stop mass, but also on the masses of the gluino and the other squark flavours, and on the stop mixing angle, all of which enter through loop contributions.We present the stop cross sections in a simplified SUSY scenario, where only the lighter stop mass eigenstate and the gluino are accessible at the LHC, while all other squark flavours are decoupled.In the stop cross sections presented below we set the gluino mass equal to the stop mass.The dependence of the cross section on the stop mixing angle θ t is small, with changes of typically less than 5%; for definiteness we have set sin 2θ t = 0.669 for the numerical evaluation [55].
As our default choice for NNLL and approximated NNLO calculations, we use the NNLO PDF4LHC15_mc set of parton distribution functions (pdfs) [78].The NLO and NLL results presented for reference are obtained using the NLO PDF4LHC15_mc pdf sets [78].Both at NLO and NNLO the value of α s (M Z ) = 0.118 is used.In order to use standard parametrizations of pdfs in x-space, we employ the method introduced in reference [79].We note that the impact of threshold-improved parton distributions on the squark and gluino cross-section predictions has been studied in [80] at NLL accuracy.
In the numerical calculations the renormalisation and factorisation scales are taken to be equal µ = µ R = µ F = m av .The relevant (Bohr) scale for bound-state effects is defined as twice the Bohr radius r B of the corresponding bound states [45,46,68,71,81].For a LO Coulomb potential it amounts to where D ij→kl,I denotes the colour factor of the Coulomb potential defined in eq.(2.9), and m red = m k m l /(m k + m l ) is the reduced mass of the final-state particles.The QCD coupling α s is evaluated at the Bohr scale itself, requiring eq.(3.1) to be solved iteratively.
It should be noted that the Bohr scale depends on the colour channel.Eq. (3.1) defines the scale that is used for the bound-state contributions, eq.(2.26).We note that in eq.(2.26), the α s factors of H (0) ij→kl,I , stemming from the off-shell interactions, are evaluated at the usual value of the renormalisation scale µ R close to the typical hard scale ∼ m av of the process, whereas the α s factors originating from the Green's function are evaluated at the Bohr scale as defined in eq.(3.1) for the attractive colour channels with D ij→kl,I > 0.
As argued in [45,46], from momentum power-counting of the Coulomb gluons and by analysing the scale dependence of the terms in eq.(2.12), the natural scale of Coulomb interactions turns out to be This choice of µ C minimises the effect of a residual higher-order scale dependence from a truncation of the perturbative expansion of α s .The Coulomb scale appears both explicitly in the Green's function and implicitly in the factors of α s for the Coulomb contributions above threshold in eqs.(2.27) and (2.7).Such a scale choice also leads to the inclusion of some of the mixed soft log(β) and Coulomb 1/β terms such as α s log(β) × α s /β, which are formally of NLL accuracy [45].The scale choice of eq.(3.2) needs to be bounded from below, since, besides hitting the Landau pole when β is integrated over, bound-state effects become important in the vicinity of the threshold.Therefore, we use the Bohr scale of eq.(3.1) to bound the Coulomb scale from below.While bound states only arise for an attractive Coulomb potential, D ij→kl,I > 0, we also use the Bohr scale as a lower bound for a repulsive potential.This is not completely justified, but leads to negligible effects for the total cross section above threshold [46].The Coulomb scale is consequently chosen to be which ensures that, as soon as the bound-state effects become relevant, the Bohr scale is used.We stress that eq.(3.1) is used for the contributions below threshold, while eq.( 3.3) applies to the contributions above threshold.
In the following discussion we present predictions for the LHC squark and gluino cross sections for a center-of-mass energy of 13 TeV, at various levels of theoretical accuracy: • The NLO cross sections [18], denoted as σ NLO .
• The NLL cross sections matched to NLO results, based on the calculations presented in [40][41][42], and corresponding to the output of the NLL-fast numerical package.They are denoted as σ NLO+NLL .
• The NNLL matched cross sections including Coulomb corrections up to O(α 2 s ), but without bound-state contributions, which are excluded by setting σ BS h 1 h 2 →kl = 0.These cross sections are referred to as "NNLO Approx +NNLL" in the plots.This level of accuracy corresponds to the one discussed in detail in [53].The NNLO Approx +NNLL accuracy, as detailed in Eq. (2.27), applies to the s-wave channels.The contributions from the β 2 -suppressed p-wave channels are taken into account at NLO+NLL accuracy.Both the s-wave and the p-wave contributions have been convoluted with the PDF4LHC15_mc NNLO parton distribution functions.In our previous work [53], we have checked that using NNLO pdfs instead of NLO pdfs for the suppressed p-wave contributions leads to a negligible modification of the full result.
• The NNLL matched cross sections including resummed Coulomb corrections, but without bound-state contributions, corresponding to eq. (2.27) with σ BS h 1 h 2 →kl = 0.These cross sections are referred to as "NNLO Approx +NNLL+Coul" in the plots.
Apart from the NLO cross sections, which were calculated using the publicly available PROSPINO code [82], all our results were obtained using two independent computer codes.with increasing final-state particle masses, as discussed in [53,55].The contributions from Coulomb resummation and bound states are also positive and result in a slight further increase of the K-factors.
The squark-antisquark and gluino-pair production cross sections computed with some replicas of the NNLO PDF4LHC15_mc pdf sets become negative for squark and gluino masses beyond about 2 TeV.Following the prescription discussed in [80] we set the cor-      responding cross sections to zero before evaluating the central cross section prediction from all replicas.The occurrence of negative cross sections is most pronounced for squarkantisquark production and leads to a decrease of the corresponding K-factor at large squark masses, see upper right panel in figure 3. The occurrence of negative cross section predictions and the resulting peculiar behaviour of the squark-antisquark K-factor are a further indication that the accuracy of cross section predictions for heavy new particles is currently limited by the pdf uncertainty.We finally investigate the scale dependence of the different squark and gluino production processes, see figure 4. The squark and gluino mass are both set to 1.2 TeV, and we vary the renormalization and factorization scales by a factor of five about the central value µ 0 = m q = m g, while the Coulomb scale is fixed to its default value eq.(3.3). 4 Figure 4 shows that the scale dependence in general decreases when including NNLL resummation.The NNLL scale dependence of gluino-pair production, however, is not improved compared to NLO+NLL, which can be understood from an intricate interplay of various effects, as discussed in reference [53].The inclusion of Coulomb resummation and bound-state contributions only has a small impact on the overall scale uncertainty.

Comparison with SCET results
The joint resummation of threshold and Coulomb corrections has first been considered in the framework of SCET [45,46].It is instructive to see how well the two independent approaches of threshold resummation agree, and to study the significance of the subleading contributions, which may differ in the two formalisms.
In this section we compare our results with those of ref. [52], which we denote "NNLL (SCET)", corresponding to the currently best result within the SCET formalism for squark and gluino production."NNLL (SCET)" includes the soft exponentials up to NNLL, oneloop hard-matching coefficients, Coulomb resummation with a NLO Coulomb potential, bound states, and matching to NNLO Approx .We also show "NNLL fixed−C (SCET)" predictions, which correspond to "NNLL (SCET)" with the Coulomb contributions expanded up to O(α 2 s ).The results of [52] are only available for the production of light-flavour squarks and gluinos.The results for stop-pair production presented in [54] do not include the resummation of the Coulomb corrections and have been compared to the corresponding results in the Mellin-space formalism in our earlier work [55].
The SCET results have been read off the plots in [52].For the sake of comparison we present the cross-sections at a collider energy of √ S = 8 TeV and use the MSTW2008 pdf set, with NLO pdfs for results at NLO and NLO+NLL accuracy, and NNLO pdfs for results at NNLO Approx +NNLL accuracy.We will show our predictions for the NNLO Approx +NNLL, NNLO Approx +NNLL+Coul and NNLO Approx +NNLL+Coul+BS accuracies.Comparing our predictions in the Mellin-space formalism with the SCET formalism, NNLO Approx +NNLL+Coul+BS corresponds to NNLL (SCET) and NNLO Approx +NNLL approximately corresponds to NNLL fixed−C (SCET).This is not an exact correspondence, as there are the following differences between our predictions and those from [52]: • The expressions in the SCET formalism depend on more scales that can be varied independently.In particular the soft scale can be chosen independently, while it is fixed to the value Q/N in the Mellin-space formalism.
• The SCET framework allows one to convolute the bound-state contributions with the soft function as described in [83].Furthermore, the authors of [52] also include boundstate effects from a NLO Coulomb potential, causing a small shift in the bound-state energies and residues, while we only include bound-state effects originating from a LO potential.
Notwithstanding these differences, the NNLO Approx +NNLL+ Coul+BS and NNLL (SCET) predictions agree very well, typically within a few percent, see figure 5. Larger corrections included in the fixed-order expressions are calculated at the scale µ.
Comparison of the K-factors for squark and gluino production as obtained in the Mellinspace approach adopted in the present paper, compared results obtained in soft-collinear effective theory (SCET).The cross sections correspond to a collider energy of √ S = 8 TeV and the MSTW2008 NNLO PDF sets.The SCET results are taken from the corresponding plots in [52].differences of 10-20% are observed only for very high sparticle masses in the gg and q q * channels.Comparing "NNLO Approx +NNLL+Coul+BS" and "NNLL (SCET)" with "NNLO Approx +NNLL" and "NNLL fixed−C (SCET)", respectively, we see that the contributions from Coulomb resummation and bound states are in general positive and increase the K-factors.An exception is observed in the q q-channel within the SCET formalism, where the inclusion of Coulomb resummation and bound state effects reduce the size of the higher-order corrections.The results, however, agree within the theoretical scale uncertainty.
The scale dependence of the cross sections, shown in figure 4 for our Mellin-space calculation, is not available for the SCET results.We can, however, quantify and compare the overall scale uncertainty of the cross section predictions, as obtained from varying all scales simultaneously by a factor of two about the central scale.This uncertainty is shown in figure 6 for the various predictions within the Mellin-space and SCET formalisms, respectively.Note that we have adopted unequal squark and gluino masses for the gg, q q * , and q q channels to enable a comparison with the SCET results of [52].
Let us first discuss the scale uncertainty for the various Mellin-space predictions.
In general we find that the NNLO Approx +NNLL scale uncertainty is smaller than at NLO+NLL.Exceptions are found for very large sparticle masses in the gg-channel as mentioned before and discussed in detail in [53].The inclusion of Coulomb resummation and bound-state contributions has a small effect on the overall scale uncertainty, which can be seen by comparing the NNLO Approx +NNLL and NNLO Approx +NNLL+Coul+BS predictions in figure 6.Note that a slight increase in the scale dependence from the inclusion of Coulomb and bound state effects can be explained by the dependence on the Bohr and Coulomb scales, which are not present in our predictions at NLO+NLL and NNLO Approx +NNLL accuracy.Finally, comparing our Mellin-space results to SCET, we observe a reasonable agreement in the estimate of the scale uncertainty, with our results falling within the NNLL (SCET) uncertainty band.This is not unexpected, as the additional variation of the soft scale, which is present in SCET, will in general lead to a larger scale uncertainty.
The results presented in figures 5 and 6 show that the Mellin-space and SCET methods for threshold resummation lead to comparable results, in particular for the best predictions in the two approaches, NNLO Approx +NNLL+Coul+BS and NNLL (SCET), respectively, which include Coulomb resummation and bound-state effects.(σ ± ∆σ µ )/σ (q q) m q (GeV) The relative scale uncertainty of the cross section prediction as obtained in the Mellinspace approach adopted in the present paper, compared with results obtained in soft-collinear effective theory (SCET).All scales have been varied by a factor of two about the central values.
The cross sections correspond to a collider energy of √ S = 8 TeV and the MSTW2008 NLO and NNLO PDF sets at NLL and NNLL accuracy, respectively.The SCET results are taken from the corresponding plots in [52].
and provides the theoretical basis to interpret current and future searches for supersymmetry at the LHC.
Note added: While finalising this work we became aware of Ref. [84], which addresses the joint resummation of threshold and Coulomb corrections for squark and gluino production at the LHC in the soft-collinear effective theory, and which provides a more comprehensive and detailed account of the work presented in Ref. [51].
the leading coefficient of the β-function for the running of α s , a 1 = (31 C A − 20 T F n f )/9, and µ C the Coulomb scale which will be specified in section 3.In these coefficients C A = N c is the number of colours, T F = 1/2 is the Dynkin index of the fundamental representation of SU (N c ), and n f = 5 is the number of light quark flavours.The colour factor D ij→kl,I of the QCD Coulomb potential, with the index I denoting the colour configuration of the final state, is related to the colour operators T a k , T a l of the final-state particles.It can be expressed in terms of the quadratic Casimir invariants C 2 (R I ), C k and C l of the particle pair in the colour representation R I and its constituent particles:

Figure 2 .
Figure 2. K-factors for squark and gluino production at the LHC with √ S = 13 TeV at NNLO Approx +NNLL accuracy, including Coulomb resummation and bound state effects.The Kfactors are evaluated with respect to the NLO cross sections and displayed as a function of the squark and gluino masses.

Figure 3 .
Figure 3. K-factors for squark and gluino production at the LHC with √ S = 13 TeV at various levels of theoretical accuracy.The K-factors are evaluated with respect to the NLO cross sections and displayed as a function of the common squark and gluino mass.

1 +Figure 4 .
Figure 4. Scale dependence of the squark and gluino cross section at the LHC with √ S = 13 TeV for squark and gluino masses of 1.2 TeV.Shown is the dependence of the cross section on the renormalization and factorization scales, with the Coulomb scale kept fixed, for various levels of theoretical accuracy.

Figure 6 .
Figure6.The relative scale uncertainty of the cross section prediction as obtained in the Mellinspace approach adopted in the present paper, compared with results obtained in soft-collinear effective theory (SCET).All scales have been varied by a factor of two about the central values.The cross sections correspond to a collider energy of √ S = 8 TeV and the MSTW2008 NLO and NNLO PDF sets at NLL and NNLL accuracy, respectively.The SCET results are taken from the corresponding plots in[52].