ITPUU12/05
SPIN12/04
FRPHENO2012004
arXiv:1202.2260 [hepph]
[0.2cm] July 2, 2012
NLL soft and Coulomb resummation for squark
[0.2cm] and gluino production at the LHC
P. Falgari, C. Schwinn, C. Wever
Institute for Theoretical Physics and Spinoza Institute,
Utrecht University, 3508 TD Utrecht, The Netherlands
AlbertLudwigs Universität Freiburg, Physikalisches Institut,
D79104 Freiburg, Germany
Abstract
We present predictions of the total cross sections for pair production of squarks and gluinos at the LHC, including the stopantistop production process. Our calculation supplements full fixedorder NLO predictions with resummation of threshold logarithms and Coulomb singularities at nexttoleading logarithmic (NLL) accuracy, including boundstate effects. The numerical effect of higherorder Coulomb terms can be as big or larger than that of softgluon corrections. For a selection of benchmark points accessible with data from the 20102012 LHC runs, resummation leads to an enhancement of the total inclusive squark and gluino production cross section in the  range. For individual production processes of gluinos, the corrections can be much larger. The theoretical uncertainty in the prediction of the hardscattering cross sections is typically reduced to the level.
1 Introduction
Despite the good agreement of the Standard Model (SM) with a wealth of experimental data, both empirical reasons (e.g. the observation of dark matter) and theoretical arguments (such as the naturalness problem and the desire for gauge coupling unification) point to physics beyond the SM. One of the most thoroughly studied extensions of the SM is Supersymmetry (SUSY). In particular the Minimal Supersymmetric Standard Model (MSSM) with parity conservation and superpartner masses at the TeV scale could provide a solution to the above issues. The search for SUSY at the TeV scale is therefore a central part of the physics program of the Large Hadron Collider (LHC) at CERN. In the context of the MSSM, and of any other parity conserving model, the supersymmetric partners of the SM particles are produced in pairs, and squarks and gluinos, coupling strongly to quarks and gluons, have typically the highest production rates. Experimental searches for SUSY have been performed at LEP, the Tevatron and the LHC in various finalstate signatures, see [1] for a recent review. For squark and gluinopair production the tightest bounds generically arise from jets+missing energy signatures, where the two LHC experiments have set lower bounds on the mass of squarks and gluinos of about GeVTeV [2, 3], depending on the precise underlying theoretical model assumed. Once upgraded to its nominal energy of TeV the LHC should be sensitive to squark and gluino masses of up to TeV [4].
SUSY searches and, if squarks and gluinos are discovered, the measurement of their properties rely on a precise theoretical understanding of the production mechanism and on accurate predictions of the observables used in the analysis. From the theoretical point of view, the simplest of such observables is the total production cross section, on which we will focus in this work. For QCD mediated processes, such as squark and gluinopair production, the Born cross section is notoriously affected by large theoretical uncertainties, such that the inclusion of at least the next term in the expansion in the strong coupling constant is mandatory for a reliable prediction. Nexttoleading order (NLO) SUSY QCD corrections for production of squarks and gluinos were computed in [5] and implemented in the program PROSPINO [6, 7]. The corrections are large, up to of the treelevel result, and lead to a significant reduction of the scale dependence of the cross section. Electroweak contributions were also investigated [8, 9, 10, 11, 12, 13], but found to be much smaller than the QCD contributions, less than of the Born result.
The size of the corrections raises the question of the magnitude of unknown higherorder QCD corrections, and makes it desirable to include at least the dominant contributions beyond NLO. It is known that a nonnegligible part of the full NLO corrections arises from the partonic threshold region, defined by the limit , with the average mass of the particles produced and the partonic centreofmass energy. In the threshold region the partonic cross section is dominated by softgluon emission off the initial and finalstate coloured particles and by Coulomb interactions of the two nonrelativistic heavy particles, which give rise to singular terms of the form and , respectively. These corrections can be resummed to all orders in , thus leading to improved predictions of the cross section and smaller theoretical uncertainties. Note that to obtain the total hadronic cross section, the partonic cross section is convoluted with parton luminosity functions. The convolution scans over regions where is not necessarily small, unless is close to the hadronic centreofmass energy . Hence, in these regions the thresholdenhanced terms cannot be expected a priori to give the dominant contribution to the cross section. However, one often finds after convoluting with the parton luminosity that the threshold contributions give a reasonable approximation to the total hadronic cross section (see Figures 3 and 5 below for the case of squark and gluino production), so resummation is also relevant for improving predictions of the hadronic cross section.
Resummation of soft logarithms for squark and gluino production at the nexttoleading (NLL) logarithmic accuracy have been presented in [14, 15, 16, 17, 18] using the Mellinspace resummation formalism developed by [19, 20, 21, 22]. Recently the same formalism has been extended to NNLL order [23] and applied to squarkantisquark production [24]. These works do not resum Coulomb corrections to all orders, though the numerically dominant terms are accounted for at fixed order. Allorder resummation of Coulomb contributions and boundstate effects for squarkgluino and gluinogluino production were on the other hand investigated in [25, 26, 27], without the inclusion of soft resummation. In [28, 29] partial NNLL resummation of soft logarithms has been used to construct approximated NNLO results for the squarkantisquark production cross section. Recently a new formalism for the combined resummation of soft and Coulomb corrections has been developed [30, 31], and applied to NLL resummation of squarkantisquark production [31], and NNLL resummation of hadroproduction [32]. Contrary to the traditional Mellinspace formalism, in our approach, which is based on softcollinear effective theory (SCET) and potential nonrelativistic QCD (pNRQCD), resummation is performed directly in momentum space via renormalizationgroup evolution equations [33, 34, 35]. The combined softCoulomb effects have been found to be sizeable for the case of squarkantisquark production [31] and lead to a reduction of the scale uncertainty, as has been observed as well in [24].
In this work we extend the results given in [31] to the remaining production processes for squarks and gluinos at NLL accuracy, i.e. squarksquark, squarkgluino and gluinogluino production. We also consider separately the production of pairs of stops, which requires the extension of the formalism presented in [31] to particles pairproduced in a wave state. The paper is organized as follows: in Section 2 we give an overview of squark and gluino production processes, set up the calculation and briefly review the resummation formalism we employ, listing the ingredients needed for NLL resummation. The validity of the formalism for wave induced processes, which is necessary for resummation of the stopantistop cross section, is established in Appendix A. Numerical results for the cross sections are presented in Section 3, including predictions for a representative set of the benchmark points proposed in [36] and a comparison to results using the Mellinspace formalism[16, 17]. Finally in Section 4 we present our conclusions and outlook. Explicit expressions for resummation functions appearing in the NLL cross sections are provided in Appendix B, while Appendix C contains some details on the scales used in the momentumspace resummation and on our method to estimate ambiguities in the resummation procedure.
2 NLL resummation for squark and gluino production
At hadron colliders, the dominant production channels for squarks and gluinos are pairproduction processes of the form
(2.1) 
where denote the incoming hadrons and , the two sparticles. The total hadronic cross sections for the processes (2.1) can be obtained by convoluting shortdistance production cross sections for the partonic processes
(2.2) 
with the parton luminosity functions :
(2.3) 
where , with the average sparticle mass
(2.4) 
The parton luminosity functions are defined from the parton density functions (PDFs) as
(2.5) 
We perform a NLL resummation of threshold logarithms and Coulomb corrections to the partonic cross section, counting both and as quantities of order one, where is the heavyparticle velocity. Our predictions include all corrections to the Born cross section of the schematic form
(2.6) 
A resummation at NNLL accuracy in the counting , , which is beyond the scope of this paper but has recently been performed for toppair production [32], would include in addition terms of the form and corrections of order relative to the NLL cross section, including NLO corrections to the Coulomb potential and other higherorder potentials, as well as the nonlogarithmic oneloop hard corrections. The recent NNLL calculation of squarkantisquark production[24] included the corrections of the type related to soft corrections and the hard corrections, but kept only the term in the sum over . In Section 2.1 we collect some facts about the production processes of gluinos and the superpartners of the light quarks at LO and NLO while the formalism employed for the NLL resummation is reviewed in Section 2.2. The production of stop pairs is included in 2.3, while details about the choice of the soft scale in the momentumspace resummation formalism and our procedure to estimate the remaining theoretical uncertainty are discussed in 2.4.
2.1 LO and NLO results
At leading order [37, 38, 39], the following partonic channels contribute to the production of lightflavour squarks and gluinos:
(2.7) 
where . At NLO further partonic processes contribute to the cross section. To keep the notation as simple as possible, in (2.1) we have suppressed the helicity and flavour indices of the squarks. It is understood that in the predictions for the cross sections presented below the contributions of the ten lightflavour squarks () are always summed over. Furthermore, the ten scalars are assumed to be degenerate in mass, with the common lightflavour squark mass given by . In the following, the chargeconjugate subprocesses for squarksquark and gluinosquark productions will be included in our results. As input for the convolution (2.3) we will use the MSTW08 set of PDFs [40] at the appropriate perturbative order (the LO PDFs for Bornlevel predictions and the NLO PDFs for the NLO and NLL results) and set the factorization scale to the average mass of the produced sparticles, . We use a set of PDFs with an improved accuracy at large provided to us by the MSTW collaboration that has also been employed for the NLL results in [18].
To illustrate the relative magnitude of the various processes depending on the squark and gluino masses, the ratio of the total hadronic cross section for the processes (2.1) to the total inclusive cross section for squark and gluino production is shown in Figure 1 for the LHC with TeV centreofmass energy. From the lefthand side plot, showing the relative contributions of the various processes as a function of a common squark and gluino mass, it can be seen that squarksquark and squarkgluino production are by far the dominant channels over the full mass range considered. In the righthand side plot, the relative contributions are shown as a function of the squarkgluino mass ratio for average mass TeV and it is seen that only for gluinos that are significantly lighter than squarks, gluinopair production becomes the dominant production channel.
In Figure 2 we show the factor
for the
SUSYQCD corrections for the various production processes as obtained
from PROSPINO [6].
Since the focus of this work is on higherorder corrections that are enhanced in the threshold limit , we consider here the corresponding terms appearing at NLO. To this end, we decompose the partonic cross section into a complete colour basis, and parametrize the higherorder corrections as
(2.8) 
The sum is over the irreducible colour representations appearing in the decomposition , where are the representations of the two finalstate sparticles. The relevant decompositions for squark and gluino production are given by
(2.9)  
The explicit basis tensors for the various representations have been constructed in [30] (see also [15, 16]), where it has been shown that an channel colour basis based on the decompositions (2.9) is advantageous for the allorder summation of softgluon corrections. In Eq. (2.8) represents the treelevel cross section for a given process in colour channel , while the are colourspecific NLO scaling functions. The colourseparated Born cross sections for squark and gluino production are available in [15, 16, 17]. The NLO scaling functions, on the contrary, are only known numerically in their colouraveraged form [5]. However, a simple formula is available for the threshold limit of the NLO scaling functions, containing all the thresholdenhanced contributions, for arbitrary colour representation [41]:
(2.10)  
with the average mass of the two particles produced (2.4), while denotes the reduced mass, . , and are the Casimir invariants for the colour representations of the initialstate particles, and , and for the irreducible representation of the SUSY pair. The coefficients of the Coulomb potential for the production of heavy particles in representations and in the colour channel are given in terms of the quadratic Casimir operators for the various representations:
(2.11) 
where negative values correspond to an attractive Coulomb potential,
positive values to a repulsive one. The numerical values for the
representations relevant for squark and gluino production can be found
in [42, 31] and are collected in
Table 1. The coefficient is
the oneloop contribution to the hard matching coefficient appearing
in Eq. (2.13) below, and represents the only processspecific
quantity in Eq. (2.10). It has been obtained recently
for squarkantisquark production and gluinogluino
production [24, 26] but is not known yet
for the remaining production processes. The knowledge of
is required for NNLL
resummation [24], but not at NLL accuracy as
considered here, so the will be always set to
zero in the following. Using the Born cross sections for the different
colour channels [15, 16]
and (2.10) one can reproduce the threshold expansions of
the NLO corrections
in [5].
In Figure 3 we study to which extent the full NLO
corrections as obtained from PROSPINO are approximated by
the singular NLO corrections, obtained by dropping all constant terms
from (2.10), including terms, and convoluting
the resulting partonic cross section (2.8) with the
parton luminosities. For the Born cross sections
in (2.8) the exact
expressions, without use of the threshold approximation, have been
kept, but colour channels with a vanishing threshold limit of the
Born cross section at leading order in have been dropped.
For the case of degenerate squark and gluino masses it is seen that
the difference of the thresholdenhanced contributions to the full
NLO corrections is at the level over the whole mass range
considered, with the exception of the squarksquark production
channel where the threshold contributions account for only
of the full NLO corrections. For the
singular terms overestimate the corrections for the processes
involving gluinos, while the agreement for squarkantisquark
production improves. For the singular
terms approximate the full corrections very well for all processes
apart from squarksquark production.
2.2 Softgluon and Coulomb resummation
Next, we briefly review the formalism for the combined resummation of soft and Coulombgluon corrections [30, 31] and provide the relevant ingredients for squark and gluino production at NLL accuracy. We also discuss some features of our implementation that differ from that used previously for squarkantisquark production in [31].
The combined softCoulomb resummation for the production of squarks and gluinos is based on a factorization of the hardscattering total cross section for partonic subprocesses of the type (2.2). It can be shown that near the partonic threshold,
(2.12) 
the partonic cross section factorizes into three contributions [31], a hard function , a soft function containing soft gluons to all orders, and a potential function summing Coulombgluon exchange:
(2.13) 
Here is the energy relative to the production threshold and the sum is over the colour representations (2.9). In (2.13) the channel colour basis mentioned above, that can be shown to diagonalize the soft function to all orders [30], is chosen for the hardscattering amplitudes. Independent of the sparticle type, the soft function then depends only on the colour representations of the initialstate partons and the irreducible representation of the sparticle pair appearing in the decompositions (2.9), in agreement with the picture that softgluon radiation is only sensitive to the total colour charge of the slowly moving sparticle pair [22]. The formula (2.13) has been derived in [31] for wave dominated production processes up to NNLL accuracy. This covers all production processes of squarks and gluinos, apart from quarkantiquark initiated stopantistop production, that proceeds through a wave. The applicability of the formalism to stop production is discussed in Section 2.3 and Appendix A.
It can be argued that the natural scale for the evaluation of the hard function in (2.13), leading to wellbehaved higherorder corrections, is of the order of , while the natural scale for softgluon radiation is of the order of . We use the momentumspace resummation formalism of [33, 34, 35] to evolve the soft and hard functions from their natural scales to the factorization scale used for the evaluation of the parton distribution functions, commonly taken to be of the order of . In this way, logarithms of are summed to all orders. The precise prescription for the choice of the soft scale adopted in our calculation is discussed in 2.4. The exchange of multiple Coulomb gluons can be summed up using the method of Coulomb Green’s functions in nonrelativistic QCD [43, 44, 45].
For resummation at NLL accuracy, the leadingorder hard and soft functions are required as fixedorder input to the evolution equations. The leadingorder soft function is trivial, . The leadingorder hard functions are obtained from the threshold limit of the Born cross section for the colour channel [31],
(2.14) 
Although
the Borncross sections in the threshold limit appear on the lefthand
side in (2.14), we keep the exact expressions in our
numerical implementation, so the hard functions in practice depend on
. This incorporates some higherorder terms in , albeit not
systematically.
For the resummation of Coulomb corrections, we use results for the nonrelativistic Coulomb Green’s function obtained for topquark production at electronpositron colliders and stop production [43, 46]. For positive values of and vanishing decay widths of the sparticles, the wave potential function is given by the Sommerfeld factor
(2.15) 
with the coefficients of the Coulomb potential given in (2.11). For an attractive potential, a series of bound states develops below threshold with energies
(2.16) 
Their contribution to the wave potential function is given by
(2.17) 
For sufficiently broad squarks and gluinos with decay widths exceeding
the binding energy of the wouldbe bound states,
, the bound state
poles are smeared out by the finite lifetimes.
The resummed cross section at NLL accuracy is obtained by inserting the potential function (2.15) and the solutions to the evolution equations of the hard and soft functions [49, 30] into the factorization formula (2.13). Using the solutions in momentumspace obtained in [35], the NLL cross section is written as
(2.18) 
Here the label jointly refers to the colour of the initialstate partons and the representation of the sparticle pair. The function contains single logarithms, while the resummation function sums the Sudakov double logarithms and . Explicit expressions up to NLL accuracy are given in Appendix B. For the function is negative and the factor in the resummed cross section (2.18) has to be understood in the distributional sense, as discussed in detail in [32]. We have used the nonrelativistic expression in the argument of the potential function, that is valid near the partonic threshold (2.12). This follows the default treatment of toppair production in [32] and leads to the customary expansion of the cross section in (see Eq. 2.10). We also perform this replacement in the definition of the hard functions (2.14). The difference between this default implementation and the results obtained by consistently keeping the expression (as in the previous results for squarkantisquark production [31]) will be used to estimate the effect of subleading terms in the cross section, as discussed in Section 2.4.
In order to assess the importance of the Coulomb corrections and to compare to the results of the Mellin approach [16] we will also present results without Coulomb resummation, obtained by inserting the trivial potential function into (2.18). In this approximation, that we will denote by NLL, a fully analytical expression for the resummed cross section can be obtained:
(2.19) 
Since contributions to the cross section from outside the threshold region can be numerically nonnegligible, we match the NLL resummed cross section to the fixedorder NLO calculation by subtracting the NLO expansion of the NLL expression and adding back the full NLO corrections:
(2.20) 
where is the fixedorder NLO cross section obtained in standard perturbation theory, as implemented in PROSPINO [6], and is the resummed cross section expanded to NLO, as given in [31]. The total hadronic cross section at NLL is then obtained by convoluting (2.20) with the parton luminosity, as in (2.3).
2.3 Stopantistop production
Beside the channels listed in (2.1), in Section 3 we will also present predictions for stoppair production:
(2.21) 
where only the initial states appearing at leading order have been shown. The NLO SUSYQCD corrections have been computed in [7] and are implemented in PROSPINO [6]. Contrary to the lightflavour squark case, in most scenarios the mixing of the two weak eigenstates , , and the mass difference of the resulting mass eigenstates, , , is nonnegligible. Offdiagonal production of the mass eigenstates, e.g. , appears at NLO in SUSYQCD, and through electroweak contributions. It is therefore suppressed compared to diagonal production [7, 50] and will not be considered here. It must also be mentioned that because of the absence of a significant topquark component inside the nucleon the processes and first contribute to the cross section at NLO, and are thus numerically suppressed. The NLO factor for the process is shown in Figure 5 for a centreofmass energy of 7 TeV, and the mass range GeV. As can be seen, NLO corrections are in the range. The predictions for the second mass eigenstate differ only in the fixedorder NLO results, and for a given mass the numerical difference between the cross sections for and production is below for the mass range considered in this work. We therefore omit results for the process .
Contrary to the production of a light squarkantisquark pair, stoppair production in the channel cannot be mediated by a channel diagram like in Figure 4(a), again due to the extreme suppression of topquark PDFs inside the proton. As a result, at LO in QCD a stopantistop pair is produced in a wave state in quarkantiquark collisions. As shown in Appendix A, the resummation formalism can be extended in a straightforward way to initiated stopantistop production at NLL, and the only modification of (2.13) is the replacement of the potential function by that appropriate for wave processes. For stable particles, above threshold the result is given by [46] (see also [51])
(2.22) 
The boundstate contributions of the wave Green’s function can be found in [46] but are not needed here, since only the repulsive colouroctet channel appears in our application to stoppair production. By expanding (2.22) in the strong coupling constant one obtains the coefficients of the fixedorder Coulomb corrections. While the oneloop Coulomb correction agrees with that for wave production (2.10), the second Coulomb correction differs from the wave case. Due to the different normalization of the wave Green’s function, the definition of the leadingorder hard functions for wave production reads
(2.23) 
In addition to the the combined soft/Coulomb resummation, we will again consider the NLL approximation where the trivial wave potential function is used in the resummation formula, leading to the analytical result
(2.24) 
In analogy to the wave result (2.10), one can use the resummation formalism to obtain the thresholdenhanced oneloop scaling functions for wave production in the colour channel from initialstate partons in the representations and :
(2.25) 
In agreement with [17] one finds that the coefficient of the single logarithms related to initialstate radiation is multiplied by a factor of compared to the wave case while the double logarithm and the logarithms related to final state radiation proportional to are unchanged. In addition, the constant terms are different which is irrelevant at NLL accuracy but has to be taken into account if one aims to extract the oneloop hard function from a computation of the NLO cross section. In Figure 5 we study the accuracy of the threshold approximation defined by inserting the NLO singular terms, obtained from Eqs. (2.10) and (2.25) by dropping constant terms, into (2.8). The ratio of the singular NLO corrections to the full NLO corrections to the hadronic cross section obtained using PROSPINO is shown in Figure 5 (right plot). Analogously to squarkantisquark production, the threshold terms provide an excellent approximation of the full NLO result.
2.4 Scale choices
As explained in Section 2.2, the resummed partonic cross section (2.13) depends on a number of scales related to the factorization of hard, soft and Coulomb effects. The dependence on these scales would cancel in the exact result, but a residual dependence remains at a given logarithmic order. As already pointed out, our default choice for the factorization and hard scales are and , respectively. On the other hand, the resummation of all NLL effects related to Coulomb exchange requires that the scale in the potential function is chosen of the order of , which is the typical virtuality of Coulomb gluons. A more detailed analysis shows in fact that for an attractive Coulomb potential the Coulomb scale freezes when , due to boundstate formation. We thus choose the scale in to be
(2.26) 
Note that, for a repulsive potential, , no bound states arise, so that (2.26) is not completely justified. However in this case resummation of Coulomb corrections leads to small effects, and vanishes for small , so that the precise choice of in this limit has a negligible numerical impact on predictions of the cross section.
The choice of the soft scale presents some subtleties. The exponentiation of all NLL terms in the partonic cross section would require a choice . However a running scale leads to strong oscillations of the cross section for small , due to the prefactor in (2.18), amplified by the factor and terms in the function , and eventually hits the Landau pole of the strong coupling constant when . To overcome these problems two different approaches have been used in the literature:
Fixed : In [33, 34, 35] the choice of a fixed soft scale was advocated. Such a scale is determined from the minimization of the oneloop soft corrections to the hadronic cross section,
(2.27) 
In this approach threshold logarithms are resummed in an average sense and not locally at the level of the partonic cross section. However one can argue that, for threshold dominated processes, the choice (2.27) preserves the hierarchy between the soft and shortdistance scales and that logarithmic corrections to the hadronic cross section are correctly resummed. This was the method adopted in [31] for resummation of the squarkantisquark production cross section. The explicit value of the scales determined with the minimization procedure (2.27) are given in Eq. (C.1) in Appendix C.
Running : For the NNLL resummation of soft effects in production presented in [32] a different approach was adopted. There a running soft scale,
(2.28) 
was used in the interval , and replaced by a fixed soft scale
(2.29) 
below the cutoff. With this scale choice, logarithms of are exponentiated locally in the partonic cross section in the large region, where the use of a fixed
soft scale cannot be a priori justified.
On the other hand if is not too big, in the lower interval the hadronic cross section
is in fact dominated by logarithms of , as can be explicitly checked by convoluting the partonic
cross section with toy parton luminosities [35], so that the use of a fixed scale once again correctly resums the dominant
logarithms. The precise value of is chosen through the prescription described in [32],
which is reviewed in Appendix C. The default choice for the prefactor adopted here is . We have observed that,
for the SUSY processes considered here, the NLL expression and its NNLO expansion are generally stable against variations of for this choice.
The two possible choices of the soft scale just discussed are one of the ambiguities associated with threshold resummation. Others are related to the choice of the hard and Coulomb scales and to powersuppressed terms which are not controlled by resummation. Additionally, one has to consider the ambiguity arising from the choice of the factorization scale . The latter clearly also applies to the fixedorder NLO result. Thus, to reliably ascertain the residual uncertainty of the fixedorder and resummed results we present in Section 3, we adopt the following procedure:

Scale uncertainty: for both the NLO and NLL result the factorization scale is varied between half and twice the default value, i.e. . For the NLL result, this is done keeping the other scales , and the parameters and fixed.

Resummation uncertainty: both hard and Coulomb scales are varied between half and twice the default values, i.e. and , where is the solution of the implicit equation (2.26). In addition, for the NLL implementation with a fixed soft scale, is varied between half and twice its default value, while for the runningscale implementation uncertainties related to the choice of and are estimated according to the procedure given in [32] (and reviewed in Appendix C). Finally, as anticipated below (2.18), we take the difference in parametrizing the resummed cross section in terms of or as a measure of the effect of powersuppressed terms. All the scales and the parameters and are varied one at the time keeping the other fixed to their central values, and the resulting errors are summed in quadrature.

PDF uncertainty: we estimate the error due to uncertainties in the PDFs using the confidence level eigenvector set of the MSTW08NLO PDFs [40].
An additional source of error arises from the uncertainty on the determination. This effect has been found to be of the order of for the NLO cross sections of squarksquark, squarkantisquark and squarkgluino production and up to for gluinopair production [18]. We expect a similar uncertainty of the NLL results.
In the following we will often refer to the sum in quadrature of scale and resummation uncertainty as “total theoretical uncertainty”. Note that the terminology adopted here differs slightly from the one used for production in [32] where the errors from variation of the hard and Coulomb scales, and of the soft scale for the fixedscale implementation, had been incorporated into the scale uncertainty, while we consider them as resummation ambiguities. Additionally, in [32] independent and simultaneous variations of the factorization and renormalization scale have been considered, whereas in this work we identify the factorization and renormalization scales and vary them as one scale, i.e. . This is the default procedure implemented in the numerical code PROSPINO used for the computation of the fixedorder NLO result [6].
It is interesting to study how the choice of a fixed or running soft scale affects the NLL resummed cross section, especially in view of the uncertainties just discussed. In Figure 6 we plot the NLL factor, defined in Eq. (3.2), as a function of a common SUSY mass for the four processes listed in (2.1), for a centreofmass energy of TeV (the situation for 14 TeV is qualitatively similar). Results for the stoppair production process and the total SUSY production rate are also shown. The thick lines represent the central values for the two implementations, whereas the bands (delimited by thinner lines) correspond to the resummation uncertainty as defined above. The central values are in good agreement for squarkantisquark and gluinopair production, and for squarkgluino production at larger masses. For squarksquark production the agreement is less satisfactory, especially for smaller masses. This is consistent with the observation from Figure 3 that the NLO corrections for squarksquark production are not as dominated by the threshold contributions as those for the other processes. In all cases, however, the two different NLL predictions are consistent with each other once the uncertainty associated with the resummation procedure is taken into account. It can also be seen that the uncertainty band for the fixedscale implementation NLL is mostly contained inside the uncertainty band of the runningscale result, with the possible exception of the smallmass region. In light of this, in Section 3 we will take the matched NLO/NLL result, Eq. (2.20), with a running soft scale, Eqs. (2.28) and (2.29), as our default and best prediction.
3 Numerical results
In this section we present numerical results for the cross sections of the five SUSY processes introduced in Section 2.1 and 2.3. In Section 3.1 we discuss the impact of the NLL soft and Coulomb corrections on the central value of the total cross sections and the uncertainties for the production of lightflavour squarks and gluinos. In Section 3.2 we provide predictions for a selection of the benchmark points defined in [36]. The results for stopantistop production are presented in 3.3. In order to facilitate the use of our results, the arXiv submission of this paper includes grids with predictions for the LHC with and TeV, for lightflavour squark and gluino masses from GeV and stop masses from GeV ( GeV and GeV, respectively, for TeV). We also provide a Mathematica file containing interpolations of the cross sections with an accuracy that is typically better than , and at worst for almost degenerate masses close to the edges of the grid, GeV and GeV.
3.1 Squark and gluino production at NLL
To illustrate how different classes of corrections contribute to the total cross section, we introduce three different NLL implementations:

NLL: as above, but without the inclusion of boundstate effects.
The three NLL approximations defined above are always matched to the exact NLO results computed with PROSPINO, according to (2.20). As input for the convolution with the parton luminosity functions, Eq. (2.3), we adopt the MSTW08NLO PDF set [40] and the associated strong coupling constant . Unless otherwise specified, the parameter , defined as
(3.1) 
is set to one.
We start presenting results for the NLL factor, defined as
(3.2) 
where is our matched result for one of the NLL implementations defined in the beginning of this Section and the fixedorder NLO result obtained using PROSPINO. The NLLfactor for LHC with TeV centreofmass energy is plotted in Figure 7, for the four lightsquark/gluino production processes and the mass range GeV. The results for TeV and the mass range GeV are given in Figure 8. The NLL corrections for our default implementation (solid blue lines) can be large, with corrections to the fixedorder NLO results of up to in the upper mass range for gluinogluino production at 7 TeV. The higherorder effects are smaller, but still sizeable, for the other three processes, due to the smaller colour charges involved in squarkantisquark, squarksquark and squarkgluino production. Furthermore, for a fixed SUSY mass the factor decreases from 7 to 14 TeV, consistently with the expectation that at lower centreofmass energies the threshold region plays a more prominent role.
The effect of including Coulomb resummation and its interference with soft resummation is on average as large as (or even larger than) the effect of pure soft and hard corrections, as can be seen comparing our default implementation NLL with NLL (dashed red lines). Pure soft contributions beyond amount to of the fixedorder NLO result, depending on the mass and process considered, whereas pure Coulomb effects and interference of soft and Coulomb corrections can amount to up to . An exception to this is the squarksquark production process, where the effect of Coulomb corrections is small. This particular behaviour originates from cancellations between the cross sections for sameflavour squark production, where the repulsive coloursextet channel is numerically dominant and gives rise to negative corrections, and differentflavour squark production, where the corresponding term is positive, due to the dominance of the attractive colourtriplet channel.
For squarkantisquark, squarkgluino and gluinogluino production, a significant portion of the total Coulomb and softCoulomb corrections originates from boundstate effects below threshold. These correspond to the difference between the NLL and NLL (dotdashed purple) curves in the plots. For squarkantisquark and squarkgluino production boundstate corrections amount to of the fixedorder NLO cross section, whereas for gluinogluino production they can be as large as .
Figure 9 shows the NLL factor for the total SUSY production rate at the TeV LHC as a contour plot in the plane. The dependence of the total resummed cross section arises from an interplay of the dependence of the singleprocess cross sections and of the relative dominance of the four subprocesses for a given . The largest factor is obtained for TeV and TeV, with corrections of to the NLO cross section. The plot shows also the recent exclusion limit published by the ATLAS collaboration in [2] assuming a simplified model of a massless neutralino, a gluino octet and degenerate squarks of the first two generations, while all the other supersymmetric particles, including stops and sbottoms, are decoupled by giving them a mass of TeV. The limits are therefore not directly comparable to our results which treat the sbottom as degenerate with the lightflavour squarks, but are shown here as an indication of the current LHC reach. We do not attempt to estimate how resummation would affect the determination of this limit. However, one can observe that in the large squarkmass region the exclusion limit crosses regions with a factor bigger than , where resummation effects on the limit extraction might be relevant.
Given the large effect of resummation, especially for squarkgluino and gluinogluino production, it is interesting to study how the relative contribution of the four production processes to the total SUSY production rate is modified by the inclusion of NLL corrections. This is shown in Figure 10. The qualitative behaviour of the relative contribution of the four different processes is very similar to the LO result (Figure 1). However at large masses one can notice an enhancement of the squarkgluino production rate compared to the squarksquark channel (left plot), as one would expect from the larger NLL factor for the first processes. For a fixed average squark and gluino mass of 1.2 TeV (right plot) the relative ratios are basically unchanged for moderate values of , though one observes a significant enhancement of the squarkantisquark cross section for large gluino masses ().
As a result of including the thresholdenhanced higherorder corrections, one expects that the uncertainty due to missing perturbative corrections is reduced compared to the NLO results. While for NLO the theoretical uncertainty arises from scalevariation only, the total theoretical error of the NLL results is obtained by adding scale and resummation uncertainties in quadrature, as defined in Section 2.4. The uncertainty bands for NLO, NLL and NLL approximations are shown in Figure 11 for the LHC with 7 TeV and in Figure 12 for the LHC with 14 TeV . In all plots the cross sections are normalized to unity at the central values of the scales and other input parameters. It is evident that the combined resummation of soft and Coulomb effects (NLL, solid blue) generally leads to a significant reduction of theoretical uncertainties compared to the NLO result (dotted black), especially in squarkantisquark and squarksquark production, where the error is reduced by a factor 2 or more in the largemass region. The behaviour of NLL (dashed red) is more processdependent, with basically no uncertainty reduction compared to the fixedorder NLO result for squarkantisquark production, and moderate effects for squarkgluino and gluinogluino production. For squarksquark production (and, as a consequence of the dominance of squarksquark production, for the total SUSY production rate) the uncertainties of NLL and NLL are very similar, due to the smallness of Coulomb effects in this particular production channel. The large reduction of the scale dependence for squarkantisquark production by softCoulomb interference effects is consistent with recent NNLL studies in this channel [24] that include the first Coulomb correction.
3.2 Benchmark points for SUSY searches at LHC
In addition to the grid files provided with the arXiv submission of this paper, we here present numerical predictions for some benchmark points at the LHC with TeV centreofmass energy, in order to illustrate the effect of our NLL results on the production cross sections. We employ the sets of benchmark points defined in [36], that are compatible with recent LHC bounds and other data such as , but not necessarily with constraints from the anomalous magnetic moment of the muon, or from the dark matter relic abundance. We consider the seven lines in the constrained MSSM (CMSSM) parameter space defined in [36] and one line for the minimal gauge mediated SUSY breaking (mGMSB) scenario. For each line we selected one benchmark point expected to be relevant for of data and a second point relevant for , where we naively extrapolate the reach of the LHC data [2, 3] that exclude CMSSM benchmark points and simplified models with total SUSY production cross sections of the order of pb for and pb for . The mGMSB scenario we have selected has a quasistable neutralino as nexttolightest SUSY particle (NLSP), so a similar reach as for CMSSMtype scenarios are expected. Since only the squark and gluino masses are relevant for the production cross sections, we have chosen points with a reasonable spread of masses and mass ratios, covering the range of average sparticle masses, , and the mass ratios . This mass range is also compatible with the estimated discovery reach [52] of TeV ( TeV) for at () and TeV ( TeV) for in a CMSSM scenario with , . The remaining families of benchmark scenarios introduced in [36] tend to have very similar mass ratios as our selected points. Therefore the relative contributions of the different production channels and the effect of the higherorder QCD corrections will be similar, although the decay chains and the resulting collider signatures can be very different. For some scenarios lighter squarks and gluinos than the ones considered here might still be allowed, for instance in GMSB with a stau NLSP. Predictions for such scenarios can be obtained by an interpolation of the grid files provided with the arXiv submission of this paper.
The SUSY breaking parameters and the resulting mass
spectrum of the coloured SUSY particles for the selected points is
shown in Tables 2, 3
and 4 together with our best NLL predictions for the
total cross section for lightflavour squark and gluino
production (including simultaneous softgluon and Coulomb resummation
as well as boundstate effects). Here denotes
the average mass of all squarks except the stops, following the setup
of [24, 53]. The lowscale mass parameters
have been generated using
SUSYHIT [54] employing
SuSpect2.41 [55] with the standard model
input GeV, , GeV.
(pb), TeV  
Point  NLO  NLL  
10.1.3  150  600  1357  1209  
10.1.4  162.5  650  1461  1300  
10.2.2  225  550  1255  1130 