[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
Albert-Ludwigs Universität Freiburg, Physikalisches Institut,
D-79104 Freiburg, Germany
We present predictions of the total cross sections for pair production of squarks and gluinos at the LHC, including the stop-antistop production process. Our calculation supplements full fixed-order NLO predictions with resummation of threshold logarithms and Coulomb singularities at next-to-leading logarithmic (NLL) accuracy, including bound-state effects. The numerical effect of higher-order Coulomb terms can be as big or larger than that of soft-gluon corrections. For a selection of benchmark points accessible with data from the 2010-2012 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 hard-scattering cross sections is typically reduced to the level.
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 final-state signatures, see  for a recent review. For squark- and gluino-pair 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 GeV-TeV [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 .
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 gluino-pair 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. Next-to-leading order (NLO) SUSY QCD corrections for production of squarks and gluinos were computed in  and implemented in the program PROSPINO [6, 7]. The corrections are large, up to of the tree-level 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 higher-order QCD corrections, and makes it desirable to include at least the dominant contributions beyond NLO. It is known that a non-negligible 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 centre-of-mass energy. In the threshold region the partonic cross section is dominated by soft-gluon emission off the initial- and final-state coloured particles and by Coulomb interactions of the two non-relativistic 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 centre-of-mass energy . Hence, in these regions the threshold-enhanced 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 next-to-leading (NLL) logarithmic accuracy have been presented in [14, 15, 16, 17, 18] using the Mellin-space resummation formalism developed by [19, 20, 21, 22]. Recently the same formalism has been extended to NNLL order  and applied to squark-antisquark production . These works do not resum Coulomb corrections to all orders, though the numerically dominant terms are accounted for at fixed order. All-order resummation of Coulomb contributions and bound-state effects for squark-gluino and gluino-gluino 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 squark-antisquark 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 squark-antisquark production , and NNLL resummation of hadroproduction . Contrary to the traditional Mellin-space formalism, in our approach, which is based on soft-collinear effective theory (SCET) and potential non-relativistic QCD (pNRQCD), resummation is performed directly in momentum space via renormalization-group evolution equations [33, 34, 35]. The combined soft-Coulomb effects have been found to be sizeable for the case of squark-antisquark production  and lead to a reduction of the scale uncertainty, as has been observed as well in .
In this work we extend the results given in  to the remaining production processes for squarks and gluinos at NLL accuracy, i.e. squark-squark, squark-gluino and gluino-gluino production. We also consider separately the production of pairs of stops, which requires the extension of the formalism presented in  to particles pair-produced 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 stop-antistop 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  and a comparison to results using the Mellin-space 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 momentum-space 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 pair-production processes of the form
where denote the incoming hadrons and , the two sparticles. The total hadronic cross sections for the processes (2.1) can be obtained by convoluting short-distance production cross sections for the partonic processes
with the parton luminosity functions :
where , with the average sparticle mass
The parton luminosity functions are defined from the parton density functions (PDFs) as
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 heavy-particle velocity. Our predictions include all corrections to the Born cross section of the schematic form
A resummation at NNLL accuracy in the counting , , which is beyond the scope of this paper but has recently been performed for top-pair production , 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 higher-order potentials, as well as the non-logarithmic one-loop hard corrections. The recent NNLL calculation of squark-antisquark production 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 momentum-space resummation formalism and our procedure to estimate the remaining theoretical uncertainty are discussed in 2.4.
2.1 LO and NLO results
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 light-flavour squarks () are always summed over. Furthermore, the ten scalars are assumed to be degenerate in mass, with the common light-flavour squark mass given by . In the following, the charge-conjugate subprocesses for squark-squark and gluino-squark productions will be included in our results. As input for the convolution (2.3) we will use the MSTW08 set of PDFs  at the appropriate perturbative order (the LO PDFs for Born-level 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 .
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 centre-of-mass energy. From the left-hand 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 squark-squark and squark-gluino production are by far the dominant channels over the full mass range considered. In the right-hand side plot, the relative contributions are shown as a function of the squark-gluino mass ratio for average mass TeV and it is seen that only for gluinos that are significantly lighter than squarks, gluino-pair production becomes the dominant production channel.
In Figure 2 we show the -factor
SUSY-QCD corrections for the various production processes as obtained
from PROSPINO .
Since the focus of this work is on higher-order 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 higher-order corrections as
The sum is over the irreducible colour representations appearing in the decomposition , where are the representations of the two final-state sparticles. The relevant decompositions for squark and gluino production are given by
The explicit basis tensors for the various representations have been constructed in  (see also [15, 16]), where it has been shown that an -channel colour basis based on the decompositions (2.9) is advantageous for the all-order summation of soft-gluon corrections. In Eq. (2.8) represents the tree-level cross section for a given process in colour channel , while the are colour-specific NLO scaling functions. The colour-separated 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 colour-averaged form . However, a simple formula is available for the threshold limit of the NLO scaling functions, containing all the threshold-enhanced contributions, for arbitrary colour representation :
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 initial-state 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:
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 one-loop contribution to the hard matching coefficient appearing
in Eq. (2.13) below, and represents the only process-specific
quantity in Eq. (2.10). It has been obtained recently
for squark-antisquark production and gluino-gluino
production [24, 26] but is not known yet
for the remaining production processes. The knowledge of
is required for NNLL
resummation , 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 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 threshold-enhanced contributions to the full
NLO corrections is at the level over the whole mass range
considered, with the exception of the squark-squark 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 squark-antisquark
production improves. For the singular
terms approximate the full corrections very well for all processes
apart from squark-squark production.
2.2 Soft-gluon and Coulomb resummation
Next, we briefly review the formalism for the combined resummation of soft- and Coulomb-gluon 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 squark-antisquark production in .
The combined soft-Coulomb resummation for the production of squarks and gluinos is based on a factorization of the hard-scattering total cross section for partonic subprocesses of the type (2.2). It can be shown that near the partonic threshold,
the partonic cross section factorizes into three contributions , a hard function , a soft function containing soft gluons to all orders, and a potential function summing Coulomb-gluon exchange:
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 , is chosen for the hard-scattering amplitudes. Independent of the sparticle type, the soft function then depends only on the colour representations of the initial-state partons and the irreducible representation of the sparticle pair appearing in the decompositions (2.9), in agreement with the picture that soft-gluon radiation is only sensitive to the total colour charge of the slowly moving sparticle pair . The formula (2.13) has been derived in  for -wave dominated production processes up to NNLL accuracy. This covers all production processes of squarks and gluinos, apart from quark-antiquark initiated stop-antistop 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 well-behaved higher-order corrections, is of the order of , while the natural scale for soft-gluon radiation is of the order of . We use the momentum-space 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 non-relativistic QCD [43, 44, 45].
For resummation at NLL accuracy, the leading-order hard and soft functions are required as fixed-order input to the evolution equations. The leading-order soft function is trivial, . The leading-order hard functions are obtained from the threshold limit of the Born cross section for the colour channel ,
the Born-cross sections in the threshold limit appear on the left-hand
side in (2.14), we keep the exact expressions in our
numerical implementation, so the hard functions in practice depend on
. This incorporates some higher-order terms in , albeit not
For the resummation of Coulomb corrections, we use results for the non-relativistic Coulomb Green’s function obtained for top-quark production at electron-positron 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
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
Their contribution to the -wave potential function is given by
For sufficiently broad squarks and gluinos with decay widths exceeding
the binding energy of the would-be 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 momentum-space obtained in , the NLL cross section is written as
Here the label jointly refers to the colour of the initial-state 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 . We have used the non-relativistic expression in the argument of the potential function, that is valid near the partonic threshold (2.12). This follows the default treatment of top-pair production in  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 squark-antisquark production ) 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  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:
Since contributions to the cross section from outside the threshold region can be numerically non-negligible, we match the NLL resummed cross section to the fixed-order NLO calculation by subtracting the NLO expansion of the NLL expression and adding back the full NLO corrections:
where is the fixed-order NLO cross section obtained in standard perturbation theory, as implemented in PROSPINO , and is the resummed cross section expanded to NLO, as given in . The total hadronic cross section at NLL is then obtained by convoluting (2.20) with the parton luminosity, as in (2.3).
2.3 Stop-antistop production
where only the initial states appearing at leading order have been shown. The NLO SUSY-QCD corrections have been computed in  and are implemented in PROSPINO . Contrary to the light-flavour squark case, in most scenarios the mixing of the two weak eigenstates , , and the mass difference of the resulting mass eigenstates, , , is non-negligible. Off-diagonal production of the mass eigenstates, e.g. , appears at NLO in SUSY-QCD, 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 top-quark 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 centre-of-mass 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 fixed-order 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 squark-antisquark pair, stop-pair production in the channel cannot be mediated by a -channel diagram like in Figure 4(a), again due to the extreme suppression of top-quark PDFs inside the proton. As a result, at LO in QCD a stop-antistop pair is produced in a -wave state in quark-antiquark collisions. As shown in Appendix A, the resummation formalism can be extended in a straightforward way to -initiated stop-antistop 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  (see also )
The bound-state contributions of the -wave Green’s function can be found in  but are not needed here, since only the repulsive colour-octet channel appears in our application to stop-pair production. By expanding (2.22) in the strong coupling constant one obtains the coefficients of the fixed-order Coulomb corrections. While the one-loop 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 leading-order hard functions for -wave production reads
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
In analogy to the -wave result (2.10), one can use the resummation formalism to obtain the threshold-enhanced one-loop scaling functions for -wave production in the colour channel from initial-state partons in the representations and :
In agreement with  one finds that the coefficient of the single logarithms related to initial-state 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 one-loop 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 squark-antisquark 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 bound-state formation. We thus choose the scale in to be
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:
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 short-distance scales and that logarithmic corrections to the hadronic cross section are correctly resummed. This was the method adopted in  for resummation of the squark-antisquark 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  a different approach was adopted. There a running soft scale,
was used in the interval , and replaced by a fixed soft scale
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 , 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 ,
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 power-suppressed 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 fixed-order NLO result. Thus, to reliably ascertain the residual uncertainty of the fixed-order 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 running-scale implementation uncertainties related to the choice of and are estimated according to the procedure given in  (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 power-suppressed 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 .
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 squark-squark, squark-antisquark and squark-gluino production and up to for gluino-pair production . 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  where the errors from variation of the hard and Coulomb scales, and of the soft scale for the fixed-scale implementation, had been incorporated into the scale uncertainty, while we consider them as resummation ambiguities. Additionally, in  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 fixed-order NLO result .
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 centre-of-mass energy of TeV (the situation for 14 TeV is qualitatively similar). Results for the stop-pair 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 squark-antisquark and gluino-pair production, and for squark-gluino production at larger masses. For squark-squark production the agreement is less satisfactory, especially for smaller masses. This is consistent with the observation from Figure 3 that the NLO corrections for squark-squark 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 fixed-scale implementation NLL is mostly contained inside the uncertainty band of the running-scale result, with the possible exception of the small-mass 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 light-flavour squarks and gluinos. In Section 3.2 we provide predictions for a selection of the benchmark points defined in . The results for stop-antistop 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 light-flavour 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 bound-state 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  and the associated strong coupling constant . Unless otherwise specified, the parameter , defined as
is set to one.
We start presenting results for the NLL -factor, defined as
where is our matched result for one of the NLL implementations defined in the beginning of this Section and the fixed-order NLO result obtained using PROSPINO. The NLL--factor for LHC with TeV centre-of-mass energy is plotted in Figure 7, for the four light-squark/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 fixed-order NLO results of up to in the upper mass range for gluino-gluino production at 7 TeV. The higher-order effects are smaller, but still sizeable, for the other three processes, due to the smaller colour charges involved in squark-antisquark, squark-squark and squark-gluino production. Furthermore, for a fixed SUSY mass the -factor decreases from 7 to 14 TeV, consistently with the expectation that at lower centre-of-mass 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 fixed-order 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 squark-squark production process, where the effect of Coulomb corrections is small. This particular behaviour originates from cancellations between the cross sections for same-flavour squark production, where the repulsive colour-sextet channel is numerically dominant and gives rise to negative corrections, and different-flavour squark production, where the corresponding term is positive, due to the dominance of the attractive colour-triplet channel.
For squark-antisquark, squark-gluino and gluino-gluino production, a significant portion of the total Coulomb and soft-Coulomb corrections originates from bound-state effects below threshold. These correspond to the difference between the NLL and NLL (dot-dashed purple) curves in the plots. For squark-antisquark and squark-gluino production bound-state corrections amount to of the fixed-order NLO cross section, whereas for gluino-gluino 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 single-process 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  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 light-flavour 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 squark-mass 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 squark-gluino and gluino-gluino 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 squark-gluino production rate compared to the squark-squark 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 squark-antisquark cross section for large gluino masses ().
As a result of including the threshold-enhanced higher-order 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 scale-variation 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 squark-antisquark and squark-squark production, where the error is reduced by a factor 2 or more in the large-mass region. The behaviour of NLL (dashed red) is more process-dependent, with basically no uncertainty reduction compared to the fixed-order NLO result for squark-antisquark production, and moderate effects for squark-gluino and gluino-gluino production. For squark-squark production (and, as a consequence of the dominance of squark-squark 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 squark-antisquark production by soft-Coulomb interference effects is consistent with recent NNLL studies in this channel  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 centre-of-mass 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 , 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  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 quasi-stable neutralino as next-to-lightest SUSY particle (NLSP), so a similar reach as for CMSSM-type 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  of TeV ( TeV) for at () and TeV ( TeV) for in a CMSSM scenario with , . The remaining families of benchmark scenarios introduced in  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 higher-order 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 light-flavour squark and gluino
production (including simultaneous soft-gluon and Coulomb resummation
as well as bound-state effects). Here denotes
the average mass of all squarks except the stops, following the setup
of [24, 53]. The low-scale mass parameters
have been generated using
SUSY-HIT  employing
SuSpect2.41  with the standard model
input GeV, , GeV.