Hadronic production of bottom-squark pairs with electroweak contributions

We present the complete computation of the tree-level and the next-to-leading order electroweak contributions to bottom-squark pair production at the LHC. The computation is performed within the minimal supersymmetric extension of the Standard Model. We discuss the numerical impact of these contributions in several supersymmetric scenarios.


Introduction
Supersymmetry (SUSY), is a well motivated Beyond the Standard Model (BSM) scenario. Electroweak precision data indicates that SUSY should be realized at the TeV scale or below. If this is the case, it will be accessible to direct experimental measurements at the Large Hadron Collider (LHC) through the production of SUSY particles. Monte Carlo simulations have shown the possibility of the discovery of TeV-scale SUSY with 1 fb −1 of integrated luminosity [1,2]. First measurements involving supersymmetry-sensitive variables have been performed by both ATLAS [3][4][5][6] and CMS [7] collaborations.
At the LHC, colored particles like the SUSY partners of quarks and gluons, i.e. squarks and gluinos, will be copiously produced. Theoretically, these processes are extensively studied within the Minimal Supersymmetric extension of the Standard Model (MSSM). The leading-order (LO) contributions, of O(α s ), have been known for a long time [8][9][10][11][12]. The next-to leading order (NLO) QCD corrections have been computed [13,14] and implemented in the public code PROSPINO [15]. They affect the LO predictions substantially and they reduce the scale dependence considerably. More recent is the estimation of the logarithmically enhanced next-to-next-to-leading order (NNLO) QCD contributions to squark hadroproduction, the resummation of the QCD Sudakov logarithms at the nextto-leading-logarithmic (NLL) accuracy, and the resummation of the leading Coulomb corrections [16][17][18][19][20][21][22]. Their contribution amounts up to 10% for squark and gluino masses of the order of 1 TeV. They further stabilize the prediction against scale variation.
The electroweak (EW) contributions have been computed for several processes producing colored SUSY particles. They exhibit an extremely rich and complicate pattern, in particular when the EW contributions appear already at tree level. The tree-level EW contributions are of O(α 2 + α s α). At the parton level, they arise from qq-annihilation, qq-scattering or photon-induced processes, depending on the final state considered. They are known [23][24][25][26][27][28] and they can increase the LO cross section by up to 20%. The impact of these contributions in the context of non-minimal flavor violation and explicit CP violation has been investigated as well [29,30]. The NLO EW corrections contribute at O(α 2 s α) and have been computed for stop-anti-stop [23,31], squark-(anti-)squark [28,32], gluino-squark [25], and gluino-gluino [33] production. Their size is comparable with that of the tree-level EW and NNLO QCD contributions, and their impact strongly depend on the SUSY scenario considered.
The production of third generation squarks is special. The non-negligible mixing in the stop and sbottom sector could lead to relatively low masses for the lightest bottom and top squarks, favoring their direct production at the LHC. Moreover b-tagging makes bottom-and top-squark production experimentally distinguishable from the production of the squarks of the first two generations [34][35][36]. A dedicated analysis looking for third generation squark production at the LHC is already available [6]. This kind of searches are particularly important in SUSY scenarios such as the ATLAS benchmark scenario SU 6, where inclusive searches with jets, missing transverse energy, and leptons are problematic [2].
In this paper we focus on the hadronic production of bottom-squark pairs substantially different from those for squark-anti-squark [32] and stop-anti-stop [23] production, and justify a specific investigation, which is reported in this paper.
The outline of the paper is as follows. In Section 2 we summarize the tree-level contributions to the processes (1.2). Section 3 describes the various partonic processes contributing at O(α 2 s α) and the strategy of the calculation. The numerical impact of the NLO EW contributions at the LHC with √ S = 14 TeV and √ S = 7 TeV is presented in Section 4. In Section 5 we discuss the numerical impact of the subleading bottom-squark pair production processes. The Feynman diagrams and the technical details of the renormalization of the sbottom sector are collected in the Appendix.

Tree-level cross section
In this section we describe the tree-level contributions to the process (1.2), which are of order O(α 2 s ), O(α s α), and O(α 2 ). We will conventionally denote the cross section (amplitude) of a partonic process X at a given order O(α a s α b ) as dσ a ,b X (M a, b X ). The parton luminosities are defined as is the parton distribution function (PDF) of the parton i inside the hadron A.

Tree-level QCD contributions
The leading-order cross section, of the order O(α 2 s ), is given by where τ 0 = 4m 2 bα /S is the production threshold. S andŝ = τ S are the squared center-ofmass (c.m.) energies of the hadronic and partonic processes, respectively. The sum runs over q = u, d, c, s. The three classes of partonic processes contributing are The corresponding partonic cross section can be obtained from the Feynman diagrams in Figure B.1. In terms of the Mandelstam variables, 4) the differential partonic cross section for a given subprocess ξξ ′ →b αb * α can be written as with the squared lowest order matrix element averaged (summed) over initial (final) state spin and color.

Tree-level EW contributions
The tree-level electroweak (EW) contributions, which are of the order O(α s α) and O(α 2 ), read as follows, The contributions of O(α 2 ) arise from the processes (2.3b) and (2.3c). The partonic cross sections, are obtained from the diagrams in Figure B.1. In the case of process (2.3c), the diagrams with t-channel gluino and neutralino exchange further allow for a non-vanishing QCD-EW interference term of O(α s α), (2.8) We approximate the CKM matrix V with the unity matrix, thus we consistently neglect the diagram depicted in Figure B.1(d). The latter diagram, together with the diagrams in Figure B.1(b), gives rise to O(α s α + α 2 ) contributions. These contributions are at least quadratic in |V cb | ≈ 10 · |V ub | ≈ 4 · 10 −2 and numerically negligible (see Section 4.3).
The first O(α s α) contribution in Eq. (2.6) arises from the photon-gluon induced process The corresponding partonic cross section can be obtained from the diagrams in Figure B.2 and reads as follows (2.10)

Next-to-leading order EW contributions
In this section we list the NLO EW corrections to the process (1.2). These contributions are of the order O(α 2 s α) and arise from virtual corrections and bremsstrahlung processes. Using an obvious notation, the corresponding contributions to the total cross section read as follows We do not consider the contributions arising from the bremsstrahlung processes As already pointed out in Ref. [32], they are suppressed because of the O(α) suppression of the photon PDF inside the proton. Moreover, these processes are further suppressed by an additional factor α s with respect to the process (2.9) and thus negligible. The O(α 2 s α) contributions of the partonic processes with a bottom quark in the initial state are neglected as well. The reason is twofold. First of all these contributions are suppressed by the bottom PDF with respect to the contributions in Eq. (3.1). In addition they have an additional factor α s with respect to the O(α s α + α 2 ) contributions of the process (2.3c), which turn out to be small (cf. Section 4.3).
The amplitudes are generated and algebraically simplified with support of FeynArts [40,41] and FormCalc [41,42], while the numerical evaluation of the one-loop integrals has been performed using LoopTools [42]. Infrared (IR) singularities are regularized giving a small mass λ γ and λ g to the photon and to the gluon, respectively. The mass of the light quarks is kept in order to regularize the collinear singularities.

Virtual corrections
The O(α 2 s α) contributions to the partonic process (2.3a) are given by where M 1, 0 is the tree-level amplitude while M 1, 1 is the one-loop amplitude obtained from the diagrams depicted in Figure B In order to cancel the UV divergences we need the O(α) renormalization of the wavefunction of the light quarks and of the sbottom sector. The field renormalization constants are fixed in the on-shell scheme, in analogy to that described in Ref. [32]. The renormalization of the sbottom sector has to be performed together with that of the stop sector. In order to avoid numerical instabilities and artificially big contributions from the counterterms, care has to be taken in choosing the renormalization scheme [37,38]. We use the "DR bottom-quark mass" scheme introduced in Ref. [37]. Since in particular regions of the MSSM parameter space this scheme can give rise to numerical instabilities, we have explicitly checked its reliability in the SUSY scenarios considered in this paper (cf. Section 4). The explicit expression of the renormalization constants in the "DR bottom-quark mass" scheme are collected in Appendix A.

Real corrections
The O(α 2 s α) contributions to the partonic processes with a photon in the final state, are obtained from the tree-level diagrams in Figure B.6. The phase space integration is divergent in the soft photon region, i.e. in the region p 0 5 → 0. In the case of the process (3.5b) further singularities arise in the collinear region, i.e. p 1,2 · p 5 → 0. IR and collinear singularities are treated using the phase-space slicing method. The description of the method and the relevant formulae are collected in Ref. [28].
The gluon bremsstrahlung process, contributes at O(α 2 s α) via the interference of QCD-based and EW-based Feynman diagrams depicted in Figure B.7. IR singularities of gluonic origin are treated in close analogy to the photonic case. Color correlations are taken into account using the formulae collected in Appendix B of Ref. [28]. Due to the color structure, the interference term of a QCD-based and an EW-based diagram vanishes if both gluons are emitted from an initial-state or a final-state particle.
Real quark radiation contributes at O(α 2 s α) as well, This IR-and collinear-finite set is given by the interference of QCD and EW tree-level diagrams (cf. Figure B.8). Only the interference from initial-state and final-state radiation contributes.
The IR singularities arising in the gg channel cancel in the sum of virtual corrections, process (3.3), and real photon radiation (3.5a). In the qq channel the sum of the virtual corrections (3.4) and of the contributions of real photon radiation (3.5b) and real gluon radiation (3.6) is IR finite. This sum is affected by universal collinear singularities of photonic origin that can be absorbed in the PDFs. This can be achieved by means of the following substitution [43], e q is the electric charge of quark q expressed in units of the positron charge, while with the splitting function P qq (z) = (1 + z 2 )/(1 − z). The factorization is performed in the DIS scheme. The replacement of the PDFs in Eq. (2.1) gives further contributions of O(α 2 s α) to the total cross section. As already mentioned, they cancel the collinear singularities affecting the O(α 2 s α) contributions (3.1).

Resummation in the b/b sector
The Higgs sector in the MSSM corresponds to a type-II two-Higgs doublet model, i.e. the down-type quarks couple to H 1 and the up-type quarks to H 2 . After spontaneous symmetry breaking, the up-(down-)type quarks get their mass by their coupling to the vacuum expectation value v 2 (v 1 ) of H 2 (H 1 ). At tree-level, the bottom-quark mass m b is related to the H 1 bb Yukawa coupling λ b via Radiative corrections induces and effective H 2 bb coupling that can significantly alter the tree-level relation (3.10) [39,[44][45][46][47]. This higher-order contributions do not decouple at low energies and are enhanced by a factor tan β = v 2 /v 1 . As shown in Ref. [39], the leading tan β enhanced terms can be resummed by using an appropriate effective bottom-quark Yukawa coupling. We follow Ref. [37] and use an effective Yukawa coupling defined as where ∆m b is given by Large logarithms from the running of the Yukawa coupling λ b at the renormalization scale µ R are resummed using the DR bottom-quark mass, (3.14) Σ fin. is the finite part of the scalar self-energies defined according to the Lorentz decomposition (A.7) of the Appendix A. The term proportional to ∆m b in the numerator of Eq. (3.11) has to be inserted to avoid double counting of the one-loop contributions of the resummed terms. Further tan β enhancement effects arise from three-point functions involving Higgsbottom vertices. The tan β-enhanced terms can be taken into account by modifying the Hbb coupling g Hbb . 1 The combined effect of the resummation in the relation between λ b and m b and of the resummation in the Higgs-bottom vertices is accounted for by performing the following substitutions, The coupling involving the Goldstone boson G 0 is not modified since the contribution from the vertex corrections exactly compensates the contribution of the bottom-Yukawa coupling resummation.

Numerical results
In this section we perform a detailed numerical analysis for diagonal sbottom-pair production at NLO EW. We stick to the notation introduced in Ref. [28]. The leading order cross section, the tree-level EW and the NLO EW contributions to the cross section are labeled by respectively. ∆σ EW = ∆σ tree EW + ∆σ NLO EW will be referred to as the EW contribution. The total sum of the LO cross section with the EW contributions is denoted by σ NLO = σ Born + ∆σ EW . Relative EW contributions are defined by In distributions δ denotes the relative EW contribution defined as where O is a generic observable and O NLO is the sum of the Born and the EW contributions.

Input parameters
The Standard Model input parameters are chosen in correspondence with [48,49], The strong coupling constant α s has been defined in the MS scheme using the two-loop renormalization group equation with five active flavors.
For the numerical analysis we consider the mSUGRA scenarios SPS1a ′ and SPS4. The first one is a "typical" SUSY scenario proposed by the SPA convention for comparison with other calculations [50]. The scenario SPS4 is characterized by a large value of tan β. Within this scenario we study the dependence of the total cross section on the squark masses and on tan β. The third scenario considered is the GMSB scenario SPS8.
The particle spectrum is determined following the procedure described in Ref. [28]. Starting from GUT-scale parameters, cf. Table 1, we use the program Softsusy [51] to evolve the soft-breaking parameters down to the SUSY scale M SUSY . In accordance to the SPA convention, a common SUSY scale M SUSY = 1 TeV has been chosen. To get the right mixing in the sbottom sector, we first compute the low energy masses and use these to calculate the effective bottom-quark mass m DR,eff b , Eq. (3.11). This mass is then used in the bottom-squark mass matrix to calculate the sbottom-mass eigenstates. The lighter of the two bottom-squarks is taken as the dependent squark. Its mass is therefore fixed by SU(2) invariance. Table 2 collects the shift ∆m b , cf. Eq. (3.12) and the effective bottomquark mass together with the on-shell mass of the bottom squarks, of the gluino and of the lightest neutralino and chargino.
Unless otherwise stated, the results presented in this section are computed setting the hadronic center of mass energy to √ S = 14 TeV and using the MRST2004QED parton distribution functions [52]. The factorization and renormalization scales are set to a common value, µ = µ R = µ F = mb α , i.e. to the mass of the produced bottom squark.   production. All masses are given in GeV. Table 3 shows the hadronic cross section for diagonal bottom-squark production within the three considered scenarios for √ S = 14 TeV. As expected, the total cross section is dominated by the LO QCD contribution of O(α 2 s ). The contribution of the photon-induced process is independent of the mixing angle. In each scenario, its yield relative to the leading order cross section is similar for the two processes considered. Although formally suppressed by a factor α s , the NLO EW corrections are typically bigger than the tree-level EW contributions. In the SPS1a ′ (SPS4) scenario the tree level and NLO EW contributions are more important in case of b 1b * 1 (b 2b * 2 ) production. This can be explained by the chirality dependence of the SU(2) coupling and by the fact that in the SPS1a ′ (SPS4) scenariob 1 (b 2 ) is mostly left-handed. In the SPS8 scenario the bottom squarks are twice as heavy as in the aforementioned scenarios, thus the Born cross section is about two orders of magnitude smaller. Further, the mixing between left-and right-handed squarks is more important and the sbottom masses are nearly degenerate. These features partially soften the differences among the tree-level EW contributions tob 1b * 1 production and the ones tob 2b * 2 production. 2 Huge cancellations between the qq and the gg channel amplify the dependence of the NLO EW contribution on the production process considered. As a result the NLO EW contributions to b 1 b * 1 production and the ones to b 2 b * 2 production have opposite sign, the latter being three times bigger than the former. Summing up the various contributions, the relative yield in the scenario considered is below 2%. Table 4 collects the hadronic cross section for √ S = 7 TeV. The leading order total  Table 3: Hadronic cross section for diagonalb αb * α production at the 14 TeV LHC within three different scenarios. Shown are the LO cross section, the tree-level EW as well as NLO EW contributions and the relative corrections as defined in the text. The numbers in brackets refer to the integration uncertainty in the last digit and are omitted if the uncertainty is at least one order of magnitude smaller than the quoted precision. All cross sections are given in femtobarn (fb). cross sections are reduced proportionally to the mass of the produced squark. They amount to 1 − 10% of their value at √ S = 14 TeV. In all the scenarios considered, the contribution of the photon-induced channel is enhanced with respect to the 14 TeV case. In the SPS1a ′ (SPS4) scenario the impact of the NLO EW contributions to the production of the mostly left-handed bottom-squark,b 1b * 1 (b 2b * 2 ), is reduced. In contrast, the NLO EW contributions become more important in case of the production of the mostly right-handed sbottom. In the SPS8 scenario the EW contributions of the various channels are enhanced. In particular the NLO EW contributions are positive for both production processes. This is a consequence of the enhancement of the NLO EW contributions to the gg channel at √ S = 7 TeV (cf. Section 4.4).

Parameter scan
The impact of tan β and of the sbottom masses have been studied performing a parameter scan on these parameters. In this scan, the soft breaking parameters M L , Mb R , and Mt    In all scenarios considered, we have verified the smallness of the bottom-initiated treelevel contributions, justifying our procedure of neglecting the O(α 2 s α) contributions to this channel. In Figure 1 we show the relevance of the various production channels at treelevel. Forb 1b * 1 andb 2b * 2 production the gluon fusion channel contributes 70 − 90% of the total cross section. The remaining 10 − 30% of the total cross section is given by the qq channel, its relative yield increasing with tan β. The contribution of the bb channel is at the permille level, in accordance with the analysis in Ref. [21]. 3 Owing to the small yield of the bb channel at tree-level, we will safely neglect the NLO EW contributions to this channel.
The reliability of the renormalization scheme in the scenario considered has been verified by investigating the behavior of the dependent parameters since these can potentially get large finite shifts by their renormalization constants. In the case of m 2 b 1 and At we have checked that the finite part of their renormalization constant is smaller than the parameter itself. The mixing angle renormalization constant enters only the counterterm of theb * 1b 2 andb * 2b 1 quadratic terms via the combination   We have checked that the finite part of δY b is smaller than the mass of the bottom squarks. Figure 2 shows the ratios as a function of tan β for the various values of m squark . The value of r 1 , r 2 , and r 3 is below 0.2 in all the scenarios. We have explicitly checked the impact of the O(α s α + α 2 ) contributions arising from the diagram depicted in Figure B.1(d). In the region of the parameter space considered in this paper these contributions are negligible. Indeed its numerical value is six orders of magnitudes smaller than the Born cross section, justifying the approximation of identifying the CKM matrix with the unit matrix.
The results of the scan are collected in Figure 3. The qq contribution is the sum of the tree-level EW and of the NLO EW contributions from the qq annihilation channel. The peaks in the corrections correspond to neutralino, chargino, or sfermion thresholds. These unphysical singularities affect the self-energy of the produced sbottom and can be Figure 3: Relative yield of the EW contribution forb 1b * 1 production (left panels) and forb 2b * 2 production (right panels) as a function of tan β. Shown are the various production channels as well as the combinded effect. The soft breaking parameter m squark is defined according to Section 4.3 and is expressed in GeV. tan β is varied between 10 and 50.
regularized by taking into account the finite widths of the unstable particle [54]. The curves in Figure 3 exhibit a step-function like behavior in the region where the mass of the produced sbottom is above threshold. Both, forb 1b * 1 and forb 2b * 2 production the behavior of the various contributions strongly depend on the size of tan β. In the following we will distinguish between the low and the high tan β region, which are separated by the threshold.
The case ofb 1b * 1 production is shown in the left panels of Figure 3. The EW contributions of the qq and the gg channel have substantially the same tan β dependence close to threshold. In the low tan β region the contributions from the gγ channel cancel against the one coming from the qq annihilation. The overall effect of the EW contribution is below 1% of the tree-level QCD contribution. In the high tan β region the leading contributions come from the gg channel and are only partly canceled by the other channels. The EW contributions are of the order of a few percents. In the m squark = 900 GeV case partial cancellations between the gγ and the qq channels further suppress the EW contributions. In this case they are below 1%. The EW contributions tob 2b * 2 production are depicted in the right panels of Figure 3 and exhibit similar features. The NLO EW contributions are more pronounced since the corrections from the gg channel are more important. They are of the order of several percents, e.g. 5% for m squark = 600 GeV and tan β ≥ 45.
It is worth noticing that the only practical effect of the resummation in the b/b sector is to change the value of the mass of the bottom squark. Indeed the impact of the effective Yukawa couplings in the computation of the amplitudes is negligible. This can be inferred from

Differential distributions
Even though the EW contributions seem to have a rather small impact on the total cross section, they can become important in specific phase space regions. In Figure 5 and 6 we consider differential distributions with respect to three different kinematical variables in the case ofb 1b * 1 andb 2b * 2 production, respectively. We focus on the the SPS1a ′ scenario and on the 14 TeV LHC. The left panels show to total EW contributions to the differential cross section. The tree-level EW contribution and the NLO EW contribution of the various production channels are depicted as well. The right panels show the impact of the EW contributions relative to the tree-level QCD cross section for each production channel. In contrast to the left panels, in the right panels the qq contribution is given by the sum of the tree-level and NLO EW contributions. Figure 5(a) refers to the transverse momentum distribution of the sbottom with highest p T . Close to the threshold, i.e. in the region p T < 300 GeV, the contribution of the gg channel is positive. Far from the threshold, this contribution becomes negative and relatively more important. In the low p T region the two contributions from the qq channel are different in sign, and their partial cancellation reduces the overall effect of this channel. In the high p T region the qq channel increases its importance. The photon induced channel peaks at low p T and it is almost proportional to the LO QCD cross section, i.e. its relative yield is constant in p T . As expected, the bb channel is irrelevant in the whole p T region. The total EW contributions have a small positive yield of the order of 1 − 2% in the low p T region, while for p T > 500 GeV the cross section is altered by 5 − 10%. It is interesting to note that a lower cut p T min on the transverse momentum can significantly rise the impact of the EW contributions. For instance the cut p T min = 320 GeV would discard the positive yield of the gg channel in the low p T region. As a consequence the relative yield of the EW contribution to the total cross section would become of the order of −3.2%.
The invariant mass distribution is displayed in Figure 5(b). The EW contributions exhibit the same high-energy behavior they have in the case of the p T distribution. In this energy region they alter the leading-order prediction up to 10%. The peaks in the gg channel correspond tob 2 andt 2 thresholds. Although in the low invariant mass region no remarkable cancellations occur among the various channels, the overall positive contribution is small, of the order of 2%.
In order to study how the EW contributions to the total cross section are altered by a lower cut M inv,min on the invariant mass, we consider σ(M inv,min ) defined as the total cross section integrated from the value M inv,min of the invariant mass. The upper-left part of Figure 7 shows the relative yield of the total EW contributions to σ(M inv,min ), together with the breakdown into the individual channels. The lower cut M inv,min excludes the region where the EW contributions are positive. Therefore the EW contributions decreases as M inv,min increases, while their relative impact increases. For instance, in the region M inv,min ≥ 1500 GeV the relative yield of the EW contributions δ EW exceeds −5%. In this region, δ EW is amplified by a factor of seven with respect to its value in the case of the fully inclusive cross section, cf. Table 3. The cross section is reduced by a factor of five for M inv,min = 1500 GeV (upper-right panel of Figure 7). Figure 5(c) shows the pseudo-rapidity distribution, where always the squark with the higher absolute value of the pseudo-rapidity η is considered. The gap for zero rapidity is a consequence of this definition. The NLO EW contributions peak at |η| = 1 and dominate the EW contribution at this region. The contribution is negative for small values of |η| The total effect on the LO QCD cross section is up to 2%. Figure 6 shows the differential distributions forb 2b * 2 production in the transverse momentum (a), in the invariant mass (b), and in the pseudo-rapidity (c). In contrast tob 1b * 1 production, the threshold behavior initiated byt 2 is mild and hardly visible in both, the transverse momentum and the invariant mass distributions. The EW contribution is small and its relative yield stays below 5%, even in the high energy region. This is expected since in the SPS1a ′ scenariob 2 is mostly right-handed. Interestingly, the contributions from the qq channel and the gγ channel almost cancel in most parts of the phase-space. Therefore the EW contributions are well approximated by the gg channel corrections. The lower panels of Figure 7 show σ(M inv,min ) in the case ofb 2b * 2 production. In this scenario the EW contributions tob 2b * 2 production can be safely neglected for each value of M inv,min . Even in the case M inv,min = 2 TeV the EW contributions change the Born cross section only by an amount of the order of 2%.
In Figure 8 we consider the differential transverse momentum distributions forb 1b * 1 production in the SPS8 scenario for 14 TeV (upper panels) and 7 TeV (lower panels). It is worth analyzing the EW corrections to the gg channel in this scenario. The sbottom mass is heavier in the SPS8 scenario than in the SPS1a ′ scenario. Therefore the gluons producing a sbottom-anti-sbottom pair have a bigger typical momentum fraction x in the SPS8 scenario than in the SPS1a ′ scenario. Since the gluon PDF falls off rapidly at high x values, the negative EW contributions of the gg channel in the high p T region are strongly suppressed in the SPS8 scenario. In the √ S = 7 TeV case the typical value of x it is even bigger and this phenomenon is enhanced. In the SPS8 scenario the cancellation between the positive low p T corrections and the negative high p T ones is less effective. Therefore, as mentioned in Section 4.2, the overall (positive) gg channel contributions get relatively enhanced in the √ S = 7 TeV case. (c) η distribution Figure 5: Differential distributions for the transverse momentum p T , the invariant mass M inv and the pseudo-rapidity η forb 1b * 1 production at the 14 TeV LHC within the SPS1a ′ scenario. Shown are the tree-level and NLO EW cross section contributions for the various production channels (left) and the impact of the NLO EW contributions relative to the LO QCD cross section (right). In the left panels the tree-level and NLO EW contributions for the qq channel are plotted separately. In the right panels they are treated inclusively. (c) η distribution

Other processes leading to bottom-squark pair production
In this section we describe the remaining processes of (1.1) leading to a pair of bottomsquarks in the final state. The processes are non-diagonal sbottom-anti-sbottom production and (anti-)sbottom-(anti-)sbottom production, The leading-order cross sections, of the order O(α 2 s ), are given by where τ 0 (α, β) ≡ (mb α + mb β ) 2 /S is the production threshold. The parton luminosities dL ij /dτ are defined according to Eq. (2.1). The tree-level EW contributions are of the order O(α s α) and O(α 2 ) and read as follows, The numerical impact of these processes on bottom-squark pair production at tree-level is rather small. In Figure 9 we show the relative yield of diagonal sbottom-anti-sbottom production, Eq. (1.2), and of all the processes described in this section as a function of the soft breaking parameter m squark . Owing to the smallness of the (anti-)bottom PDF, the tree-level contribution of the processes (5.1) is below 1%. We therefore do not include NLO EW corrections; they are expected to be even smaller and thus can be safely neglected.

Conclusions
We have studied the EW contributions to b-squark pair production at the LHC within the MSSM. The tree-level EW contributions from both the qq-annihilation and the gluon fusion channels have been supplemented by the contribution from the photon-induced channel, not included in previous discussions. We have presented the first complete computation of the NLO EW contributions to diagonal sbottom-anti-sbottom production; together with the QCD corrections they complete the NLO analysis of the class of squark-antisquark production processes. Renormalization of theb-sector is shown in detail, with a check of the reliability of the adopted renormalization scheme in the numerical analysis. The potentially large corrections to the bottom Yukawa couplings have been resummed, introducing effective couplings. In the scenarios considered, the main effect is to change the value of the masses of the bottom squarks.
The EW contributions to the total cross section are strongly scenario dependent. However, in all the scenarios considered they are of the order of few percents of the LO contribution in inclusive cross sections. Their size is partly due to strong cancellations among different channels. The EW contributions of different channels peak in different regions of the phase space. Therefore, the EW contributions are enhanced whenever kinematical cuts are applied. For similar reasons the impact of the EW contributions is more important in differential distributions, in particular in the high-energy region. In the SPS1a ′ scenario, and in the case ofb 1b * 1 production, they can even exceed 10% of the LO contributions for invariant mass and transverse momentum distributions.

A. Renormalization of the stop and sbottom sector
In the MSSM the kinetic terms in the stop and sbottom sector read as follows with the squared-mass matrix M L , Mq ,R , Aq are soft breaking parameters, while µ is the supersymmetric Higgs mass parameter. m q , e q , T 3 q are the mass, the charge and the isospin of the quark q, respectively. s w is the sine of the weak mixing angle, while λt = cot β and λb = tan β. The matrix (A.2) is symmetric and can be diagonalised by an orthogonal matrix Rq such that In the rotated basis the squared-mass matrix is diagonal The rotation matrix Rq can be parametrized in terms of a mixing angle θq Rq = cos θq sin θq − sin θq cos θq or Rq = − sin θq cos θq cos θq sin θq , depending on the sign of the determinant of Rq. The mixing angle θq and the trilinear coupling Aq are related via Because of SU(2)-invariance the eigenvalues of M 2 t and M 2 b are connected. In particular they have to satisfy the relation Definition of the "DR bottom-quark mass" renormalization scheme In this scheme the independent parameters are chosen to be The squark mass squared and the top quark mass are defined in the on-shell (OS) scheme, while the bottom quark mass and the trilinear coupling Ab are fixed using the DR prescription, +δµ tan β + µδ tan β.
δµ and δ tan β appearing in Eq. (A.6) are defined in the DR scheme. Σ div. is the divergent part of the scalar self energies defined according to the following Lorentz decomposition, The stop mixing angle is defined according to .
In the "DR bottom-quark mass" scheme At, θb and m 2 b 1 are dependent quantities. Their counterterms read as follows δAt = 1 m t Rt 1,1 Rt 1,2 δm 2 t 1 − δm 2 t 2 + ξt Rt 1,1 Rt 2,2 − Rt 1,2 Rt 2,1 δθt(m 2 −(At − µ cot β)δm t + δµ cot β − µ cot 2 βδ tan β, (A.9) In the case ofbb * production only the diagonal entries of the matrix δZq are needed. They are defined as The wavefunction renormalization constants of the bottom-quark read as follows In the processes considered,b 1 is an external particle and its mass has to be defined on-shell. Therefore we set the value of theb 1 mass to its OS value, obtained using the following relation The renormalization constant of m 2 −ig s T c (δZ qL γ µ ω − + δZ qR γ µ ω + ).

B. Feynman diagrams
In this Appendix we list the Feynman diagrams relevant forb αb * α production at tree-level and at next-to leading order electroweak. S (S ± ) denotes the neutral (charged) scalar Higgs and Goldstone bosons. V = γ, Z.