JHEP12(2008)087 ant. ) to the α 2 s α ( August 13, 2008 December 3, 2008 O December 19, 2008 November 24, 2008 ions and 25% in Received: Revised: Accepted: s and distributions at the in various supersymmetric Published: –proton collisions. Compared to titut) Published by IOP Publishing for SISSA [email protected] , ) electroweak terms, the NLO contributions are also signific 2 α + α s α ( O Beyond Standard Model, Supersymmetric Standard Model, NLO We present the complete NLO electroweak contribution of [email protected] ¨higrRingF¨ohringer 6, D-80805 Germany M¨unchen, E-mail: Max-Planck-Institut Physik f¨ur (Werner-Heisenberg-Ins SISSA 2008 c production of diagonal squark–anti-squark pairsthe in lowest-order proton W. Hollik and E. Mirabella

Computations, Hadronic Colliders. We discuss the LO andLHC NLO for electroweak the effects production inbenchmark of cross squarks scenarios. section different from LOdistributions. top and squarks, NLO can add upKeywords: to 10% in cross sect Squark–anti-squark pair production atthe the electroweak LHC: contribution Abstract: JHEP12(2008)087 9 9 8 5 6 4 5 1 12 16 11 24 25 35 dictive framework that ) 4 2 ics beyond the SM. The ing new scalar couplings α ndard Model (MSSM) [3 – ( ative version of the Higgs e of the main tasks at the ther verify the Higgs mech- O irect and direct experimental ectroweak scale. The realization es 10 ) and α s α ( O – 1 – ) corrections α 2 s α ( O fusion g ) annihilation with real gluon emission annihilation with real photon emission annihilation with electroweak and QCD loops 8 q ¯ ¯ ¯ (¯ q q q q q q q 3.6 3.5 3.4 3.1 Gluon fusion3.2 with electroweak loops Gluon fusion3.3 with real photon emission 3.7 Factorization of initial-state collinear singulariti 4.1 Different squark4.2 species Different SUSY scenarios 2.2 Tree-level electroweak contributions of 2.1 Squark pair production at leading order 4. Numerical Analysis 5. Conclusions A. Feynman diagrams B. Counter terms and renormalization constantsC. Bremsstrahlung integrals 32 3. Virtual and real 1. Introduction The exploration of electroweak symmetry breaking will be on Large Hadron Collider (LHC). Experimentsanism are of expected the to Standardconcept ei Model of (SM), supersymmetry ormechanism to [1, where detect 2] signals symmetry provides ofthat breaking a potentially phys occurs can promising become without altern strong,of introduc thus supersymmetry stabilizing in the terms el 5], of the has Minimal come Supersymmetric upallows Sta as to the make most precise promising predictions extension to of be the investigated SM, by a ind pre 2. Tree-level contributions to squark pair production 3 Contents 1. Introduction JHEP12(2008)087 ) t 6= (1.1) esses Q annihi- ( ¯ q q X oweak am- b ˜ aching values Q a ˜ Q t Higgs boson mass ), yielding terms of → ˜ t , it will be accessible o partonic processes of tion of squark pairs in % [16 – 18]. uncertainty of the order s the distinction between ease the cross section by ction of top-squark pairs, dence on the factorization tion of SUSY particles. In squark cascade decays into ns one can reconstruct the 31], they were found to be exchange, and also through X, PP O have to be taken into ac- ∗ n the proton. In this paper based on the following pecu- b he SM, in specific observables Z ak (EW) contributions consist , yielding an involved struture ss section by 40–50%. . ˜ uarks, Q troweak precision data [6] pro- ) ed to play an important role for pair production from a ), to the production of diagonal b ˜ 5 to 10 pb for masses of squarks l be copiously produced, and the ˜ . annihilation and gluon fusion, to- α Q ˜ t, 2 s ¯ q α q 6= → ( is different from ˜ O ˜ Q Q ( P P ) in supersymmetric QCD [19 – 23]. QCD 2 s α X ( ∗ a O – 2 – ˜ plus jets and possibly leptons that is well suited Q a -channel photon and T ˜ s -channel (if Q E t → P P ), for the processes 3 s annihilation. Moreover, photon–gluon induced parton proc α ( ¯ q q O ). Due to interference between the tree-level QCD and electr , the electroweak contributions can also become sizable, re ˜ t α s 6= α ( ˜ Q O 2 of the muon [9, 10] even better, and yields bounds on the ligh − ) and lead to a signature with missing For reliable predictions, electroweak contributions at NL Besides the QCD-based production mechanisms, there are als If supersymmetry (SUSY) is realized at the TeV scale or below The first prediction of the cross section for hadronic produc 2 g α 0 1 ( ˜ χ also contribute owing towe the present the non-zero NLO photon electroweak distribution contributions, i of gether with real photon andof gluon interference bremsstrahlung terms processes in count as well. Insignificant, with the effects up case to of 20%.of top-squark In loop general, pair contributions NLO production electrowe to [29 the – tree-level amplitudes for up to 20% [27]. They show significant differenceslarities. to top-squark production, squark–anti-squark pairs different from top- and bottom-sq lation [27, 28]. They proceed through neutralino/chargino exchange in the were calculated more thantypically 20 ten to years 30 laterand %, renormalization [24, and scale. 25]. theyperformed NLO substantially They in QCD reduce [26], corrections incr the are to depen also the positive produ and can increaseelectroweak the origin, cro like diagonal and non-diagonal squark vides an overall fit [7, 8]like with a quality at least as good as in t studies. The indirect access through virtual effects in elec O plitudes for with less tension than in the SM [11 – 13]. to direct experimental studiesparticular, at colored the particles LHC like throughhadronic squarks the production and produc of gluinos squark– wil anti-squarkSUSY pairs hunting. is expect The crossand section gluinos is below in 1 TeV theof and range a can from few be percent 0 even measuredq in with the a low statistical luminosity regime. Moreover, contributions at NLO, gluino and squark decays.mass Finally of the with squarks the up help to of 2 TeV decay with chai athe resolution early better 1980’s than was 10 done at lowest order to detect MSSM signals [14, 15]. The number of hard jets allow JHEP12(2008)087 ∗ b ˜ a,b X b ˜ (2.1) dσ ) and → α b ¯ s the tree- b α ( luino in the O is the isospin ) to the pro- ) in the strong ′ b α 2 s α Q a s , α -tagging, bottom- -channel diagrams ( b t α ( -squarks, mixing an- i O b ummarize the various O τ x of  er terms with specifica- i , where ation channel in (1.1). presented in section 4 and ∗ f f NLO can reach the same ) a ks of the first two genera- x ˜ Q ( he process (1.1) arising from a j es for getting to the hadronic zation scale [38]. A dedicated ), which we will include in our describes the structure of the ˜ f 2 in the proton (PDF). f the orders of the perturbative Q ulation of singular integrals are α i ght-handed tributions and enhanced Yukawa + can be produced via annhilation ( ). We will use the convention →  ∗ O 2 ′ a x τ α ) and the strategy of the calculation. ˜ Q ( Q at a given order  α ′ j O 2 s a production where no f X Q ) and ˜ α ) production the partonic process ∗ Q ( ) amplitudes. ˜ t α x ∗ s ( ˜ t α O i b ˜ ( α b f ˜ ) and and ( h O α – 3 – ∗ O s a x α ˜ dx Q ( -channel neutralino and chargino exchange are part a 1 t O τ ˜ ) and Q Z 2 s ), with the need of resummation [35]; other peculiarities α 2 s ( → ij α β δ ( O 1 Q O ), the squark pair 1+ Q 2 s α ( )= O τ ( production will not be treated here. Owing to ij ∗ -squark final states thus seems appropriate. in a quark doublet, yielding EW–QCD interference already at b dτ ˜ b dL b ˜ Q ) many types of interferences occur between amplitudes of pair through amplitudes that involve also the exchange of a g α 2 s Q α ( ) as well as between ) is the momentum distribution of the parton Q s x O α ( i ( f -channel, thus enhancing the relative weight of the annihil O level. At Electroweak tree diagrams with partner of t of a In leading order of the amplitudes for The outline of the paper is as follows. In section 2 we briefly s The case of • • • to denote the cross section for a partonic process expansion [36, 37] andextra the analysis appropriate for choice of the factori for massive initial-state partons are the proper counting o tree-level contributions to theNLO terms processes of (1.1). EW origin Section that 3 contribute at Evaluation of the EW effectssummarized and their in analysis section for 5. thetion LHC of A are renormalization, list and ofcollected technical Feynman in diagrams details the for and appendix. count the calc 2. Tree-level contributions to squarkIn pair this production sectiontree-level we amplitudes list at the order lowest-order cross sections for t and electroweak coupling constants.cross section The are parton given luminositi by the convolution gle renormalization [34], non-negligiblecouplings Higgs-boson for con large vaues of tan exhibits specific features, like mixing between left- and ri squarks can betions experimentally [32, distinguished 18, from 33]. the Moreover, in squar the case of discussion as well. occur at lowest order. Oursize analysis as shows the that tree-level the EW EW contributions of effects o These features make the calculation of the EW contributions cesses (1.1) more involved than in the case of where JHEP12(2008)087 , . . )  ∗ s a ( (2.5) (2.2) (2.3) (2.4) (2.7) (2.6) ˜ ∗ Q a a ˜ Q ˜ Q a ˜ → Q 1 , 1 γg → 0 k terms, can , . 2 gg dσ /S ) dσ τ ) ). In the notation ( ˜ 2 s 2 Q,a τ ( α γg m ree-level electroweak ( dτ ) gg e exchange of a gluino 2 O dL dτ -channel neutralino ex- = 4 α t dL ( + , 0 2 τ O  | in figure 11 of appendix A. dτ ∗ ∗ al expressions for these cross are the squared CM energies a a ˜ , respectively. Moreover, with ˜ 1 . , , 0 Q Q ased, of τ ) ) ) a a and lows, 2 2 2 ˜ Z ˜ τS Q Q k k k ) ( ( ( → → 2 = ∗ ∗ ∗ , α q ns can be found in refs. [19, 22, 23]. )+ ) differential cross sections for the a 0 a a s q 0 γg s 2 s s ˜ ˜ ˜ α , Q Q Q ( α ( dσ 2 ∗ ( ) ) ) ) a |M O 1 1 1 1 ˜ + Q O , since we treat the CKM matrix as unity. k k k k ∗ a ( q ( ( a ˜ Q a a a X ˜ is determined by − Q ˜ ˜ ˜ . Sand a → Q Q Q 0 2 1 0 , q ˜ Q p τ 2 q – 4 – πs → dt . 1 → , q → → dσ 1 q = ( 16 Q ) ) ) ) t 2 is obtained squaring the tree-level electroweak dia- τ 2 2 dσ p = ( = p denote the p ∗ (  ( q ( a q ∗ g q u, d, c, s q ) g ∗ ˜ a Q a ) ˜ τ dτ a ) Q ) ˜ ( 1 = Q , and the diagram with chargino exchange appears only if dL ˜ a 1 1 Q a q p ), photon–gluon fusion occurs as a further partonic process ˜ p p q Q Q q ( ˜ ( ( Q → α 2 , γ q q → dτ g s dτ 1 0 q → = , 0 dL α , 1 γg 1 0 ( 2 gg q  τ dσ O dσ Z dσ dτ 1 0 ) contributions to the process (1.1), involving electrowea , the threshold value τ 2 q Z X α ˜ Q,a ( and q O m X ∗ )= a S ˜ = Q ( a X ∗ originates from interference between the aforementioned t ˜ ∗ a Q ) and a ∗ ˜ being the SU(2) partner of the quark Q ˜ a α → Q 0 a ′ , s q ˜ a Q ˜ channel contributes only if 2 q q Q a ˜ α Q t , ( ˜ → Q dσ → LO Q , O As a new element at → 1 , LO q PP 1 ew q PP = ′ dσ change contributes only if of the hadronic processthe (1.1) squark and mass of the partonic subprocess The parton cross section grams depicted in figure 12 of appendix A. The diagram with q 2.1 Squark pair production at leadingThe leading-order order contribution to the process (1.1) is QCD b dσ The sum runs over the quarks mentioned above, the differential cross section reads as fol be written as follows, dσ diagrams and the tree-level QCDsections graphs of can figure be 11. found Analytic in ref. [27]. The respectively, which are obtainedExplicit from expressions the for Feynman these diagrams leading-orderOwing cross to sectio flavour conservation inin SUSY the QCD, the diagram with th The corresponding cross section, with partonic processes 2.2 Tree-level electroweak contributions of JHEP12(2008)087  g (3.2) (3.1) (3.3) ∗ a ˜ Q a . ˜ Q ) 3 → ) 1 k . , q ( [40, 41]. The 2 q q ˜ ) 4 Q,a ) dσ  2 , m q 2 } k ∗ 2 + ) a ( ∗ a γ ˜ ∗ Q − ∗ ˜ a ˜ a Q 2 Q,a of [39]; together with a ˜ a ˜ ˜ Q Q Q tu ˜ m Q FeynArts lution equations for the ( a ) → 1 ˜ , 1 Q → − those listed in [30] for the ew tu 2 . The presence of photons qg  , k → 1 gg γ ( 1 u ˜ , Q q ) electroweak virtual contri- ∗ ( a The corresponding contribu- dσ 2 q rther suppressed by an addi- a α ) 2 ˜ )+ near singularities. notation, M ˜ Q ection are of the type Q 2 s ) 2 rals is performed with support τ is the one-loop amplitude with udes from the diagrams in fig- ∗ dσ a ( s α a ˜ ) corrections to the process (1.1) Q ( ˜ ˜ 2 Q,a Q + qg ew → α + , a → O 2 s 1 ∗ 1 ) m , dτ ˜ Q a 2 α gg 2 dL ˜ ˜ ( Q 4 Q,a p → − M a ( dσ + t O ˜ 0 gg m q Q ( ( q + ) ∗ → 1 of the squark 1 , a q ∗ ˜ 2 q ˜ 4 Q,a {M p a Q ˜ Q ( ˜ a Q e m ˜ dσ a Q Re ˜  Q ˜ – 5 – Q → 2 1 ) , e → 1 2 τ qg , γ s , ( ) 2 gg 3 q X dσ tree diagrams. These loop diagrams do not depend q αα k ) dσ ( 2 2 dτ τ q  ( dL gg π ) πs ) dt qg 2 ) corrections τ dτ k ( 16 dτ α ( 1 0 dL = 16 [42, 43]. IR singularities are regularized by a small photon 2 s ∗ gg τ = a  2 Z α dτ | ˜ ∗ Q ∗ ( dL a a ( dτ ˜ ) Q ˜ amplitude (figure 11), and Q O 1 1 a . We therefore do not repeat them here. 0 luminosity entering (2.5) is built according to eq. (2.1). a dτ ˜ k t τ q ˜ Q ˜ ( Q 1 X Z 0 gg a τ → = γg 1 → , ˜ , and the electric charge + + Z Q 2 gg 2 ˜ 0 γg Q LoopTools ) = 2 dσ with respect to process (2.6), and thus negligible. k → |M s X fusion (2.4), given by the partonic cross section ) ∗ and − α 2 a p 1 ˜ Q gg ( X p a q ˜ Q ) = ( 1 → NLO , p u ( is the tree level ew Diagrams and corresponding amplitudes are generated using PP γ FormCalc 0 dσ mass, while quark masses are kept as regulators3.1 for Gluon the colli fusion with electroweak loops The first class of corrections entering eq. (3.1) are the butions to algebraic treatment and numericalof evaluation of loop integ We will not considertional this factor class of processes here; they are fu EW insertions in the QCD-based ure 13, contains the spin- and color-averaged squared tree-amplit Other bremsstrahlung contributions to the hadronic cross s M on the flavour of theparticular final case squark of and thus they are identical to with in the proton followsPDFs. from including The NLO photon QEDthe PDF effects gluon is in PDF, part the the of evo the publicly available PDF set In this section we describe the computation of the arising from loops andtions from to photon/gluon the bresmsstrahlung. hadronic cross section are expressed in obvious 3. Virtual and real JHEP12(2008)087 , ∗ a ˜ Q (3.4) a ˜ Q . These 0 → H 0 H proper counterterms non trivial check of the n and isolation of diver- integration to the region heme [30, 44, 34], where nd the two mixing angles fusion subprocess we have tary integral over the sin- ity of the soft singularities , oduction are loop diagrams le subtraction and of phase d results which are in good ) ider scenarios in which such over the photon phase space charged bosons owing to the 3 twise the singuilarities and is k are chosen to be the masses of rticular we use the expressions ( rence apply to processes involv- re also given in appendix B. o add and subtract an auxiliary r the subtracted cross section is re cancelled by including brems- where mixing can be neglected). in terms of the renormalization γ 1 ons according to eq. (3.1). 0, and cancels the corresponding ) 2 nt squarks from the same SU(2) doublet is k → ( ∗ 0 3 a k ˜ Q ) for 2. The integration over this region is thus 1 is sufficiently small. More details are given and have to be considered a contribution to k s/ ( s a δ i.e. – 6 – √ ˜ ˜ s Q,a Q δ m 2 = → ) ≥ ), 2 E p α 0 ( 2 s H g α ( ) m 1 can be done analytically in the eikonal approximation [46], O p ( E g vertex, with the heavy neutral MSSM Higgs boson ∆ 0 < H 3 production via gluon fusion with the subsequent decay k 0 H Notice that part of the virtual corrections to squark pair pr In order to get rid of the UV divergences we have to include the The phase space slicing technique restricts the phase space For the technical treatment of photon-momentum integratio Due to SU(2) invariance, mass renormalization of the differe 1 convergent and can begular performed region numerically. with The complemen correlated and has to be performed simultaneously. for the gluon-gluon- the two up-squarks, the(which, mass however of are one irrelevant for of theThe the light-quark actual two squarks expressions down-squarks, for a the renormalization constants a which is a good approximation if the cut for one-loop renormalization.constants Their can be explicit found expressions into appendix renormalize B. the Inthe the squark case independent sector of parameters only. the for gluon- a We squark use isospin the doublet on-shell sc terms become resonant when the process of in appendix C. Comparison between the two methods provides a computation. As illustrated innumerical agreement. figure 1, the two methods yiel rather than an electroweakresonances loop occur. correction. We will not cons 3.2 Gluon fusion with realThe photon IR singularities emission arising fromstrahlung virtual of photons in real (3.3) photons a at according to the diagrams depictedis in IR figure divergent 14. in The the integral soft-photon region, with a minimum photon energy ∆ gences we apply two differentspace procedures: slicing. the In methods the dipole offunction subtraction dipo to approach, one the has differentialeasy t cross enough to section be that integratedconvergent matches and analytically; poin the can integral be ove general done expression numerically. for Due these togiven functions in the ref. are universal [45]. available. Althoughing In the fermions formulae pa only, quoted they in this canuniversal refe structure be of generalized the to IR processes singularities. with virtual singularities when added to the virtual contributi JHEP12(2008)087 → u u 2200 2200 2200 2200 2000 2000 2000 2000 (first panel), 1800 1800 1800 1800 γ (GeV) (GeV) (GeV) (GeV) ∗ 1/2 L 1/2 1/2 1/2 ˜ u (s) (fourth panel), computed (s) (s) (s) 1600 L 1600 1600 1600 point [55]. ˜ u u ′ ∗ . The error bars represent the L → ˜ u 1400 1400 1400 1400 L gg ˜ u Dipole σ → 1200 he SPS1a 1200 1200 1200 − 0 ug 0 0 0 0.001 0.01 -0.001 -0.01 Slicing 0.0005

0.005 -0.0005 -0.005 1e-05 5e-06 1e-05 5e-06 -5e-06 -1e-05 -5e-06 -1e-05

σ

(pb) (pb) (pb) (pb) ∆ ∆ ∆ ∆ – 7 – 2200 2200 2200 2200 (third panel) and g ∗ 2000 2000 L 2000 2000 Dipole Dipole Dipole Dipole ˜ u L ˜ u 1800 1800 1800 1800 (GeV) → (GeV) (GeV) (GeV) 1/2 1/2 u 1/2 1/2 Phase Space Slicing Phase Space Slicing Phase Space Slicing Phase Space Phase Space Slicing (s) u (s) 1600 (s) (s) 1600 1600 1600 1400 1400 1400 1400 1200 1200 Lowest order partonic cross sections for the process 1200 1200 0 0 (second panel), 0 0 0.002 0.001 0.003 0.1 -0.01 -0.2 -0.3 -0.4 -0.1 γ 0.0005 0.0025 0.0015 0.08 0.06 0.04 0.02 0.12 -0.004 -0.006 -0.008 -0.002 -0.45 -0.15 -0.25 -0.35 -0.05 ∗

L

(pb) (pb) (pb) (pb) σ σ σ σ ˜ u L ˜ integration uncertainty. The SUSY parameters are those of t Figure 1: u with the two different methods. ∆ is defined as ∆ = JHEP12(2008)087 , d by (3.6) (3.5) s de- o } ∗ e heavy a ˜ Q dix A, in- a ons to the ˜ Q → qcd , q 1 q M ∗ a ˜ Q amplitude for photon a ˜ Q ∗ EW tree-level diagrams. → ew , q . also quark renormalization q 0 ft-photon region and in the y renormalization constants ) 3 ops. The virtual contributions e interference of tree-level and of the squark sector, we have k , and the quark–squark–gluino for renormalization and cancel- ( {M m subtraction” used in ref. [47]. s in higher orders, a finite differ- γ g Re ) 2 . For the non-standard loop contri- k s ( ial-state radiation remain and have to α + 2 . More details and the specification of ∗ s a } g ˜ ∗ Q a ˜ ) Q 1 a by the corresponding finite amount, which and ˜ k Q ( s s a → g g ew – 8 – annihilation at higher order is more involved and , q ˜ Q 1 q ¯ q q 0). Although IR singularities cancel in sufficiently M → ∗ annihilation channel, a → ) via supersymmetry. The strong coupling constant is ˜ Q ¯ 2 q i a are displayed in figures 15–17 of appendix A. They also s p q p ˜ ∗ ( g Q 3 q ew k is the one-loop amplitude arising from the EW corrections → , qcd , ) q 1 0 q 1 ew , p M 1 ( q ) in the {M M is encountered at one-loop order. Supersymmetry is restore α 2 s Re s α g whenever 2 ) is the amplitude related to the tree-level QCD (EW) diagram ( MS scheme, modified according to ref. [24] in order to decoupl n ew O , 0 and e.g. is the one-loop amplitude corresponding to the QCD correcti X s = M g ( ∗ 2 qcd a , can be obtained from the Feynman diagrams in figure 18 of appen ˜ , which is related to 1 Q qcd πs s , dt a 0 g ˜ Q M qcd 16 , annihilation with electroweak and QCD loops annihilation with real photon emission 1 → 1 M , ¯ ¯ q q 2 qq q q The diagrams entering M dσ The structure of the parton processes of requires a simultaneous treatment of electroweak andof QCD one-loop lo order to theloop partonic amplitudes, cross section is given by th 3.3 The corresponding cross sectioncollinear is region singular ( both in the IR so particles (top, gluino, squarks) from the running of to the QCD tree-levelFinally, diagrams and the QCD corrections to the inclusive observables, collinear singularities from init be absorbed via factorization in the PDFs. picted in figure 11 (12). QCD tree-level diagrams. where butions, this procedure is equivalentSince to the dimensional “zero-momentu regularization violatesence supersymmetry between ˆ contain the diagrams with countertermlation insertions of required UV divergences.can The be counterterms found and in theis appendix necessar needed. B. Besides squark renormalization, The diagrams in figure 19 of appendix A constitute the generic bremsstrahlung at the counterterms can be found in appendix B. 3.4 cluding the proper counterterms.to renormalize Besides also renormalization the gluino mass, the strong coupling shifting the renormalization constant for ˆ means an unsymmetric renormalization of ˆ coupling ˆ renormalized in the JHEP12(2008)087 (3.8) (3.7) as an γ ytically. ∗ L quark. For ˜ u L . ˜ u u ∗ → L ˜ u u L u (figure 20 b). This ˜ u arities when the gluon → rent has to be modified visualize the comparison annihilation, when added ocess ug , coming gluon and outgoing ¯ q g the two different methods . scribed above in section 3.2. given by (anti-)quark-gluon , ) q olor algebra. In figure 1 we 45] accordingly, following the ) ) 3 e Dipole Subtraction method on (see refs. [48 – 50] and ap- 3 o account also when using the son between the two methods 3 lae are given there for the case k in k ( k ( ( g e, we have to introduce a further q q re again absorbed into the PDFs. uark. The soft singularities cancel ed by mass regularization. ) region can be found in appendix C. ave to be absorbed in the PDFs by ) ) 2 2 2 k k ( k annihilation processes with real gluon ( ( ∗ ∗ a ∗ ¯ q a a ˜ q Q ˜ ˜ Q Q ) through interference between the graphs ) ) ) 1 α 1 1 k 2 s k ( k ( α ( a a ( – 9 – a ˜ Q ˜ ˜ Q O Q ) through the interference between the diagrams of α → 2 s → → ) α ) ( ) 2 2 2 p p O ( p ( ( q g g ) ) 1 ) 1 1 p p ( p ( ( q q q , the integral over the singular region can be performed anal c on the angle between the photon and the radiating quark/anti δ c δ fusion g ) annihilation with real gluon emission q ¯ q (¯ q q For applying the phase space slicing method, the eikonal cur The extraction of the singularities has been performed usin Finally, we have to take into account the class of example. 3.5 bremsstrahlung, along eq. (3.1), while remainingfactorization. collinear IR singularities h and collinear singularities can be treat due to colour correlationspendix after C the for details). emissionDipole Colour of Subtraction correlation the Method; has soft we to modified glu beguidelines the taken of formulae int of ref. ref. [48]. [ also for In gluon figure radiation, 1 with we good illustrate numerical the agreement. 3.6 compari A last class offusion, partonic processes at the considered order is sufficiently small of figure 20 a and figure 20 b. The cross section exhibits singul becomes soft or collinear toagainst the those initial-state from quark/antiq the virtual photon/gluon contributions described in section 3.2.collinear In cutoff phase space slicing, in this cas This IR finite class contributes at Explicit expressions can bebetween found the in two appendix methods C. in In the figure specific 1 case we of the partonic pr figure 21 a and figure 21 b. Mass singularities arise when the in (anti-)quark are collinear. TheseTheir collinear extraction divergences has been a performedExplicit using expressions the for two the methods cross de sectionThe in actual the collinear expression ofis the obtained subtraction from function the used formulaeof in in photon–quark ref. th [51]. splitting Since weshow those have the formu to agreement between consistently the redo two the methods c for the example from either EW-basedclass (figure contributes 20 to a) the or cross section QCD-based at Born diagrams JHEP12(2008)087 ) s  ( ) is # ) ) , ∗ ¯ ) q ) s (3.9) a ) δ (3.12) (3.10) (3.11) (3.13) # ˜ ads: Q ( z zs ) q, a ( ( . h ˜ ∗ Q zs 2 i a H ( ˜ → π 1 (= nd thus to Q 2 i ∗ + 3 , q αe π a a i 1 q z ˜ ˜ 2 Q Q 4 1 αe a dσ . The splitting → ˜ 1 Q i , + + 2 q + 1 q e soft q → 1 z # , q κ 1 q dσ  1 − , ) F 2 i initial-state collinear 2 F 1 + 1 z dσ 1 C ( µ m π s 3 2  − ! owed, the internal-state , α  coll q 2 2 q ) for parton 2 2 F ) 2 − ! κ ) x actorization scheme where ln z µ m 2 q ( 2 F z F  i ducing the width of the cor-  µ − m

2 F z f C ) via [53, 54] )+ π − ar singularities is to substitute 4 3 s es s µ F blems related to gauge invari-

z ln ( − (1 α ∗ 2 i + 1 a ln F s ˜ m x,µ Q π + (1 T )+  δ (  a s 2 i 2  ˜ 4 3 Q zs α f z ln ( ) ) ln ln → ∗ 0  + z , q z a ( 2 " q and the electric charge ( ˜ s Q )= + , s δ ii a qg z 1 2 s dσ α ˜ ( P  Q δ ln F 2 i ig ! 2 2 F → = 0 ) are defined as  ) , q C dz P P µ m π z 2 s q )= F ( 1 0 2 δ – 10 – + + z  x ( T ( + ln dσ , s 2 i Z coll q h δ 3 4 s ln κ 2 q ! δ 2  2 π αe ) z ) F z = 2 ) z αe τ π T ( z ( ), the lowest order PDF − ,H s 2 F ( and + ln s g + α H α 1 ii δ q C 1+ s ( ) 2 q 1+ln P δ 2 , dτ z soft q O π soft q − ( dL F ( κ αe κ  )= ig µ 2 ln  2 q are given by z P − F = ln F ( )+ π − ) αe C  ii H τ s z s 2 ,µ ( F P ) contribution to be added to eq. (3.1). This contribution re ( δ soft q α z x , while α κ qg π ,µ 2 s coll q  ( ( + and dτ /s z i ) x κ α ) dL 2 i ( f τ 2 q h  are defined in the usual way, ( +3ln ) in the definition of the quark–antiquark luminosity (2.1) a  π O g z ˜ αe q 2 Q,a dz αe 2 F " f ig q 3 s π P m dτ δ z 2 dz " " dL , dτ 1+ − x,µ ) corrections to partonic cross sections contain universal 1 ii 1 ( 1 0 dz  i x x τ α P dτ = (4 f ) s Z 2 s Z Z δ 1 0 0 F α τ − ( x + + 1 0 Z q )=9+ x O ) = X s In specific cases of SUSY parameters, when kinematically all The actual effect of the factorization of the initial colline ) by Z x, µ x δ q ( x ( ( i ( i X + + h i f f obtain a further singularities that can besingularities absorbed of into relative the order PDFs choosing a f functions where related to the experimentally accessible distribution gauginos can be on-shell.responding The gluino, poles neutralino, areance or regularized [52] chargino. by do intro not Potential occur pro here. 3.7 Factorization of initial-state collinearThe singulariti with the factorization scale and the functions f JHEP12(2008)087 - and and γ ∗ soft q a κ ˜ Q a ˜ Q → 1 , q x C]. The re- 2 q are put together dσ β 3%. The standard 004qed parton dis- + − 10 SU4 xercise [57]: the point 400 GeV are treated as massless, 160 GeV 200 GeV e difference between the ned with the help of the oints chosen for detector − e two different codes are uts given by the two codes essions of ) contributions as “the EW er of 2 st line of eq. (3.12) are can- α rized by light SUSY particles. ffects, we choose four different herefore we do not show the . 2 s [59] as a cross check. The pole ]. # ion scales are chosen as equal, α 0 ( + 10 ancelled against those in SU1 O + 1 70 GeV 350 GeV ! 2 ) SuSpect z ) and 2 , as can be inferred from the their analytic − q 2 F α ∗ ( µ a 5 + (1 ˜ Q O 2 q SPS5 a – 11 – 1000 GeV ˜ 300 GeV 150 GeV m Q ), − α →

1 suggested by the SPA convention [55], the snowmass , s 2 qg ′ α ln ( , defined at the GUT scale, and tan ′ dσ " 0 O ) A z 37 ( . , + 0 and qq 10 q 300 GeV P SPS1a 70 GeV M ∗ 250 GeV a , − ˜ 2 Input parameters in the four benchmark scenarios. Q ) cancel in the sum of the real corrections and of the contribu / a z ) = 1 ˜ ( ) Q z ( Z M ) → 1 coll q , 2 µ κ 2 M qg coll q / 0 0 ( 1 κ A β M dσ [58], together with the program M Table 1: . sign( a parameter tan ˜ Q in the collinear regions [see eqs. (C.4) and (C.8) of appendi m g ∗ a = SPheno ˜ Q a F ˜ Q µ contribution”. We will refer to the sum of For the calculation of hadronic observables we use the MRST2 → 1 = • , q 2 q R model parameters are taken fromWe introduce ref. the [60]. following conventions: in table 1.program The MSSM input for the actual calculation is obtai in the virtual corrections.celled by those The of mass singularities in the la masses of theshown squarks in of table the 2.same-chirality first and generation Since obtained same-isospin themasses with quarks squarks of of th are the the degenerate, firstmasses squarks t provided two belonging by generations the to two codesgive the is rise second below to 1%. generation. a The differences different Th inp in the total cross section of the ord point SPS5 [56] characterized by light stops, and two of the p simulation in the ATLASSU1 in “Computing the System coannihilation region, Commissioning”The and e input the parameters point SU4 characte expressions in the collinear region [eq. (C.9) of appendixtribution C functions [39].µ Factorization and renormalizat 4. Numerical Analysis For the numerical evaluation and forbenchmark illustration scenarios: of the the point EW SPS1a e The singularities in tion (3.12), as can be easily checked using the analytic expr maining singularities of the real corrections are exactly c dσ JHEP12(2008)087 e = δ of the ), and 2 α inv ( M O . In the case ), L c α s α ( and ˜ O is the sum of the LO L ˜ handed squarks. For a d [59] are quoted inside the , he various SUSY scenarios. he “cumulative invariant NLO 6 3 9 5 2) 6) 5) 8) R in the low invariant mass ecreasing as the invariant ...... is a cancellation beetween O u e squarks are also the mass point. Since the masses of ubsection we will study the , ˜ SU4 quarks, hence the production re important in in the case of 412 420 413 427 L ′ (411 (418 (412 (425 ) correction is negative and of ponding NLO QCD corrections en flavour are distinguished by u 2 SuSpect α 7 3) 6 7) 0 6) 6 7) In table 3 we show the integrated + ...... α s SU1 739 765 738 769 α (736 (760 (734 (764 ( O 3 4) 5 5) 2 9) 5 8) ...... 48%. ) one. Since they have the same sign their effect – 12 – 660 681 659 685 SPS5 − α (657 (677 (656 (681 2 s ) contribution being the most important ones (see . α α R ( ′ 2 s ˜ Q 0) 9 4) 7 5) 1 6) 3 α O is a generic observable and ...... , ( L to denote the relative EW contribution, defined as O O ˜ [58]; those computed with Q 547 570 548 565 δ production the situation is different: (562 (545 (567 SPS1a (545 = L c a ˜ [25]; they are positive, weakly dependent on the flavour of th Q ˜ ˜ ), that is the cross section integrated up to the value ˜ ˜ d,L u,L d,R u,R , where SPheno inv m m m m LO M O ( / σ ) ) contributions. The overall 2 LO α PROSPINO ( O − O Pole masses (in GeV) of the squarks of the first generation in t NLO ) corrections are positive, ) and O ( in eq. (2.2) and the EW contribution. We will use the quantity α α As a general remark, the EW effects are always larger for left- Figure 2 shows the relative EW contribution (right part) in t 2 s s • α α ( ( O is enhanced. In the case of ˜ Table 2: of the production ofO the squarks of the first generation there also the discussion below). given chirality and generation, theup-type EW squarks. contributions For comparison are we mo alsousing estimate the the corres code the same order of magnitude as the Dependence on squark flavour and chirality. hadronic cross section for the diagonal pair production of ˜ the light quarks caneigenstates; be thus, neglected, in the the following,means weak the of eigenstates two their of squarks chiralities, th of a giv brackets for comparison. 4.1 Different squark species Electroweak interactions depend on the hypercharge of the s cross sections are flavourproduction and of chirality four dependent. squark In species, this focusing s on the SPS1a produced squarks, and of the order of 47 They are obtained using mass distribution” squark–antisquark invariant mass.region A the common NLO feature EW is contribution that is positive, rather steeply d JHEP12(2008)087 ¯ q 2 3 3 3 q gg gg − − − − and 10 ∗ 10 10 10 · · · · L channel u ∗ L 01) 01) 01) 01) fusion [ L . 8 . . . . u ˜ c gγ 0 0 0 0 4 ut of the L qγ ˜ c ± ± ± ± 65 82 11 89 . . . . ( 2 ( 2 ( 5 (22 annihilation channel n; for charm-squark is defined according to 2 3 2 2 q δ − − − q − rent squark species in the e-isospin squarks the 10 10 10 10 ich makes the · · ) terms are dominated by · · s in the case of e 3 for the various squark egion and become negative α 2 s ∗ section, one can understand e of the individual contribu- ced squark. 01) 07) 5 yed by the partonic processes 01) 01) L annihilation channels are neg- . . α . . ded charm-squarks it is of the . ˜ nnels. For each squark species, ( section is negative at the level d er than that for down-squarks. 0 0 5 0 rom the gluon fusion channel is ¯ 0 q squarks. he different channels in the high L O q − ˜ iven in pb. ± ± d ± ± L u 83 82 , wheras the 89 20 . . . . 3 7 inv ) and − − ( 3 ), we show the yields of the different orders (25 α ( ( M 2 s s 2000 GeV which corresponds to the total 2 2 2 α 2 α ( − − − ( − ≥ O O 10 10 10 10 · · · · inv – 13 – belong to the same isospin doublet. Indeed, in )] corrections are dominated by the ∗ M 01) 01) 0 ′ 01) L 01) α . . . . . ˜ 2 s u 0 0 7 Q 0 0 L α − ( ± ± ˜ u ± ± channel is independent on the squark chirality, deter- O and 54 05 31 39 . . . . ) [ production they are suppressed by the PDFs of the charm 3 1 gγ Q α ∗ − − s ( 2 (31 ( ( L α c 3 3 ( 2 3 L − − − − c O 10 10 , reaching the level of 15% for ˜ 10 10 · · · · , where inv ∗ ∗ a 05) 01) 6 03) R M 01) ˜ . . . . Q . production their contribution is negative and the largest o ˜ u 0 0 2 0 a 0 ∗ R ˜ − Q ± ± L ˜ ± u ± u 09 00 L → 83 42 . . . . u annihilation channels, which yield a negative contributio 3 9 ′ Total cross section for the diagonal pair production of diffe ¯ q − − Q ( 2 (36 ( ( q ′ ) ) scenario. Beside the LO contribution, of ) ) production. For up-squark pairs, the The contribution of the ′ α α 2 s 2 s 2 s ∗ The invariant mass distribution itself is displayed in figur α α α α L ( ( (%) Q, Q ( ( c δ O O L O O fusion] channel and thus positiveQ This shows the key role pla production, however, the the the mined only by the electric charge of the produced squarks, wh and strange quarks and hence the contributions from the ligible. As aof result 5% the for overall the contributionsame left-handed to order up-squarks, total of while cross magnitude for but the positive. left-han Table 3: part is negative. Lookinginvariant at mass the relative region, contributions which ofthe t corresponds origin to of the the differentc total behaviour cross of the NLO EW correction SPS1a annihilation channels. In contributing to the NLO EW corrections. Cross sections are g the case of contribution for up-squark pairOwing production four to times the bigg mass degeneracy between same-chirality and sam species, showing also the breakdownthe into EW the individual contributions cha arefor positive larger in values the of low invariant mass r fusion channel is independent on the generation of the produ section 4. mass increases, reaching the plateau at cross section. The lefttions arising part from of the figure variousalways 2 channels. positive shows and The the contribution dominates f relative at lower siz values of JHEP12(2008)087 t [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M . es of squark pairs, defined as ′ 2000 3000 4000 S1a 2000 3000 4000 2000 3000 4000 2000 3000 4000 nnels, the right ones show the complete 1

8 6 4 2 0 5 4 3 2 0 5 0 8 6 -1

-2 -4 -6 -2 -3 -5 15 10 16 14 12 10

[%] [%] [%] [%] δ δ δ δ L L R L ˜ ˜ c ˜ d ˜ u u – 14 – ) ) ) ) α α α α s s s s of the invariant mass of the squark-antisquark pair. The lef ) ) ) ) α α α α + + + + α α α α 2 2 2 2 2 s 2 s 2 s 2 s α α α α α α α α inv , O( , O( , O( , O( , O( , O( , O( , O( channel channel channel channel M q q q q q q q q γ γ γ γ g g g g gg channel q q qg channel qg channel gg channel q q qg channel qg channel gg channel q q gg channel q q [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M 2000 3000 4000 2000 3000 4000 2000 3000 4000 2000 3000 4000 Cumulative invariant mass distribution for different speci 5 0 8 6 4 2 0 8 6 4 2 0 6 4 2 0

-5 -2 -2 -4 -2 -4 15 10 14 12 10 10

the cross section integrated up to Figure 2: panels show the relative contributionsEW from contribution. the The various SUSY cha parameter point corresponds to SP -10

[%] [%] [%] [%] δ δ δ δ JHEP12(2008)087 rious [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M pairs, for the SUSY parameter 1500 2000 2500 3000 1500 2000 2500 3000 1500 2000 2500 3000 1500 2000 2500 3000 5 0 5 0 5 0 0

-5 -5 -5 15 10 10 20 10

-10 -15 -10 -20

[%] [%] [%] [%] δ δ δ δ L L R L ˜ ˜ c ˜ d ˜ u u – 15 – ) ) ) ) α α α α s s s s ) ) ) ) α α α α + + + + α α α α 2 2 2 2 2 s 2 s 2 s 2 s α α α α α α α α , O( , O( , O( , O( , O( , O( , O( , O( channel channel channel channel q q q q q q q q γ γ γ γ g g g g gg channel q q qg channel qg channel gg channel q q qg channel qg channel gg channel q q gg channel q q . The left panels show the relative contributions from the va ′ [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M Invariant mass distribution for different species of squark 1500 2000 2500 3000 1500 2000 2500 3000 1500 2000 2500 3000 1500 2000 2500 3000 -3 -3 -3 -3 10 10 10 10 × × × × 0 0 0 0 0.1

0.01

0.06 0.04 0.02 0.04 0.02 0.05 0.02

inv inv inv inv -0.01

channels, the right ones show the complete EW contribution. point corresponding to SPS1a Figure 3: -0.02 -0.05 -0.02 [pb/GeV] /dM d [pb/GeV] /dM d [pb/GeV] /dM d [pb/GeV] /dM d σ σ σ σ JHEP12(2008)087 w = → ¯ u ˜ u,R u fusion m gγ SY particles and , assuming a 2 and / increases. This 1 s increases with ˜ lly compensated u,R − ) gg m ˜ u,R al cross section, for f the light Higgs bo- m = NLO rrections are estimated an 20%. In the charm- he squarks. Again, the ned production process. e 5. For illustration, we ˜ Lσ d,R -channel gluino exchange strange PDFs. The total t EW corrections are shown cenarios mentioned above. m io NLO EW corrections are point. The values are also he NLO EW contributions = ( ′ he other parameters are kept contributions decreases as the lding more than 30% negative 50%. [64 – 66]. Moreover, each point f the stat channel with increasing squark squarks, with the corresponding δ − ¯ q L q u , setting ˜ u,R m displayed in the right panel is the fraction ξ is the dominant squark contribution to the annihilation allows a better understanding annihilation channel. ρ ¯ q ¯ q q To study the dependence of the NLO EW con- – 16 – q FeynHiggs 2.5.1 annihilation channel in the total cross section, at leading ¯ q 025, where ∆ . q 0 channel is subleading with respect to the < ¯ q case, originating from the PDF-enhanced parton process [25]; they are of the order of 45 | . As new feature, the LO EW contribution can be positive for lo q 03, which is the value at the SPS1a . ρ . T L 1 ˜ p d ∆ − channel becomes more and more important as | = 0 ¯ q ) contributions in the q ε α 2 s -channel chargino exchange. This positive part is practica α t PROSPINO fusion and the ( parameter. increases, while the relative yield of the NLO EW correction = 100 fb O . Each parameter point was checked to satisfy the bounds on SU ρ ′ gg ). The L ) with ˜ 2 s u,L ε α m ( through , which has been computed using O 0 ∗ (1 + The relative EW contributions are shown in figure 5 for the tot In table 4 we show the total cross section for the aforementio Figure 4 contains the transverse momentum distribution of t h L ˜ , especially for the d ˜ u,L L T ˜ channels due to thesum aforementioned of suppression the of EWalso charm contributions give is and an shown estimate in of the the formal right statistical panel uncertainty of figur order luminosity of each the each of the various squark types. The quantity electroweak m As a concrete example, wemasses consider listed the in production table of 2. ˜ 4.2 Different SUSY scenarios Here we discuss the electroweak effects in the different SUSY s The LO contribution and theseparately. different As orders one entering can the see NLO mass the of absolute value of the different diagrams. The increasing importance of masses. Especiallybecome for more left-handed important up- thansquark the and LO production down-squarks, ones, case t with effects of more th feature, already pointed out in ref. [19], is a consequence o of the particular role of the NLO corrections to the the mass of the producednegative squarks. and In of the the case order ofusing the of the SU1 10%. code scenar The corresponding NLO QCD co tributions on the mass of the squarks, we vary Dependence on the squark masses. by the NLO fullfills the condition EW effects are more pronouncedcontributions for for left-handed large chirality yie d p taken for the other generationsas as in well SPS1a as for the sleptons. T from LEP [61, 62]son and Tevatron [63], and the bound on the mass o JHEP12(2008)087 [GeV] [GeV] [GeV] [GeV] T T T T p p p p quark pairs. Notations and 0 500 1000 0 500 1000 1500 0 500 1000 1500 0 500 1000 1500 5 0 0 5 0 0

-5 -5 10 20

-10 -20 -30 -10 -20 -40

[%] [%] [%] [%] δ δ δ δ L L R L ˜ ˜ c ˜ d ˜ u u – 17 – ) ) ) ) α α α α s s s s ) ) ) ) α α α α + + + + α α α α 2 2 2 2 2 s 2 s 2 s 2 s α α α α α α α α , O( , O( , O( , O( , O( , O( , O( , O( channel channel channel channel q q q q q q q q γ γ γ γ g g g g gg channel q q qg channel qg channel gg channel q q qg channel q qg channel gg channel q q gg channel q [GeV] [GeV] [GeV] [GeV] T T T T p p p p Transverse momentum distribution for different species of s -3 -3 -3 -3 10 10 10 10 × × × × 0 500 1000 1500 0 500 1000 1500 0 500 1000 1500 0 500 1000 1500 0 0 0 0 Figure 4: input parameters as in 3.

0.01

0.02 0.04 0.02 0.02 0.03 0.02

T T T

T -0.02 -0.02 -0.04 -0.02

[pb/GeV] /dp d [pb/GeV] /dp d [pb/GeV] /dp d [pb/GeV] /dp d σ σ σ σ JHEP12(2008)087 channel channel channel channel q q q q gg channel q gg channel q gg channel q gg channel q [GeV] [GeV] [GeV] [GeV] R R R R u u u c ~ ~ ~ ~ m m m m ntribution (left), indi- 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000

90 80 70 60 50 40 30 20 10 90 80 70 60 50 40 30 20 10 90 80 70 60 50 40 30 20 10 90 80 70 60 50 40 30 20 10

[%] [%] [%] [%] ξ ξ ξ ξ The panels in the right column show the L L R L ˜ ˜ c ˜ d ˜ u u [GeV] [GeV] [GeV] [GeV] R R R R u u u c ~ ~ ~ ~ – 18 – m m m m ) ) ) ) α α α α s s s s ) ) ) ) α α α α α α α α + + + + 2 s 2 s 2 s 2 s 2 2 2 2 α α α α α α α α , O( , O( , O( , O( , O( , O( , O( , O( channel channel channel channel q q q q q q q q γ γ γ γ qg channel gg channel q q qg channel gg channel q q qg channel gg channel q q gg channel q q qg channel g g g g 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 0 0 0 0 5 5 5 5

-5 -5 -5 -5 10

-10 -15 -20 -25 -30 -10 -15 -20 -25 -10 -15 -20 -25 -30 -10 -15 -20 -25 -30

[%] [%] [%] [%] δ δ δ δ [GeV] [GeV] [GeV] [GeV] R R R R Squark-mass dependence of the EW contributions. Total EW co u u u c ~ ~ ~ ~ m m m m stat stat stat stat δ δ δ δ total total total total 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 relative yield of the two channels that contribute at LO. vidual contributions from the various channels (central). Figure 5: 0 0 0 0

-5 -5 -5 20 10

-10 -20 -10 -15 -20 -25 -30 -35 -10 -15 -20 -25 -30 -35 -10 -15 -20 -25 -30 -35

[%] [%] [%] [%] δ δ δ δ JHEP12(2008)087 gγ 2 1 2 1 − − − − region the 10 10 10 10 ˜ g · annihilation · · · 3% for gluino m q q − 3 01) 01) 01) . 01) . . . . 4 0 0 0 0 SU4 − ± ± ± 7 to ± − uts on the transverse 16 e various cases. Again, 45 61 14 contributions from the . . . tical cuts are imposed, . ributions to the rapidity 1 2 annihilation channel con- − − (16 q ( ( ( 10 q NLO part is always of about on, again with the individual est part of the gluon channel he total cross section is larger n from o the contribution of the ur for all the chosen scenarios. rt dominates and renders the ibutions from 3 3 3 3 bsection 4.1. At LO, the gluon ion, while gluon fusion becomes − − − . − 5 . 10 10 10 10 2 · · ed in figure 7, has similar qualitative · · < annihilation, both the tree-level term , with the other parameters kept fixed 1 . | 01) 02) ˜ g 02) 01) ¯ . . q . y . production in different SUSY scenarios. | 0 0 q 10 m 0 0 SU1 ∗ − ± ± ± L ± ), are practically of the same size. , ˜ u α L 22 75 channel by excluding the region with a positive 77 2 s 73 . . . . u q 7 2 α – 19 – q ( − − 5. More important is the cut on the transverse ( 4 Finally we study the dependence of the EW con- (51 . ( ( O 2 , as displayed in figure 10. In the low ˜ g 150 GeV 3 2 2 > 3 m > − − − − | y T 10 10 | 10 10 p · · · · 9 03) 01) . 01) 01) . . . . 8 point. Again, the parameter range is in accordance with the 0 0 0 0 ′ − SPS5 ± ± ± ± 83 37 62 11 . . . . 4 1 − − ( 9 (10 ( ( Same as table 3 but focusing on ˜ ) ) ) ) ) and the NLO corrections α α 2 s 2 2 s 2 s α α α α α ( ( (%) ( ( δ + O O O O α In figure 8 we show the transverse momentum distribution in th This general situation is only slightly changed when kinema Figure 6 contains the cumulative invariant mass distributi s Table 4: distribution. As a result, the negative EW contribution to t α ( T than without cuts, as one can seeDependence from on figure 9. the gluinotribution as mass. a function of the mass of the gluino according to the SPS1a fusion channel does not depend on the gluino mass, while the phenomenological constraints described in the previous su momentum. It excludes the kinematicalcontribution region comes where from. the larg channel Moreover, and enhances this the influence cut of suppresses the als turn the EW contribution tothe the same negative size region; as thereby the the LO part. their shape depends only weakly on the scenario. as we findmomentum from and repeating on the our rapidity analysis of the for two an squarks, exemplary set of c p two channel contribute equally to thedominant production as cross the sect massvarious channels of are flat, the adding gluino up to increases. a total The EW contributio relative EW tribution decreases with increasing and the total EW contributions, whichAlso show the a differential similar invariant behavio mass distribution, display features in all scenarios.total At EW low contribution values, positive. the gluon At fusion larger pa values, the contr The cut on the rapiditydistribution is not are effective very because small the NLO for EW cont O masses between 500 and 2000 GeV. Thereby, in JHEP12(2008)087 [GeV] [GeV] [GeV] [GeV] inv inv inv inv in different SUSY scenarios. M M M M X ∗ L 2000 3000 4000 ˜ u L 2000 3000 4000 ˜ u 2000 2500 3000 3500 4000 → 1000 2000 3000 4000 P P 8 6 4 2 0 5 0 5 0 5 0

-2 -4 -5 -5 -5 ′ 12 10 15 10 20 15 10 15 10

-10 -10

[%] [%] [%] [%] δ δ δ δ SU1 SU4 SPS5 – 20 – SPS1a ) α s ) α + α 2 2 s α α , O( , O( channel q q γ g qg channel gg channel q q [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M 2000 3000 4000 Cumulative invariant mass distribution for 2000 3000 4000 2000 2500 3000 3500 4000 1000 2000 3000 4000 5 0 5 0 5 0 5 0

-5 -5 -5 -5 15 10 15 10 15 10 10

Figure 6: Notations as in figure 2. -10 -10 -10

[%] [%] [%] [%] δ δ δ δ JHEP12(2008)087 [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M 2000 2500 3000 1500 2000 2500 3000 in different SUSY scenarios. Notations X 1000 1500 2000 2500 1500 2000 2500 3000 ∗ L 0 0 0 0 ˜ u ′ 20 10 10 20 10 20 10 -10 -20 -10 -10 -20 -10 -20

L

˜

u [%] [%] [%] [%] δ δ δ δ → SU1 SU4 SPS5 – 21 – SPS1a P P ) α s ) α + α 2 2 s α α , O( , O( channel q q γ g qg channel gg channel q q [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M 2000 2500 3000 Invariant mass distribution for 1500 2000 2500 3000 -3 -3 -6 -3 1000 1500 2000 2500 10 10 10 10 1500 2000 2500 3000 × × × × 0 0 5 0 0 -5 0.1 0.4 0.2

-0.2 0.01

0.05 0.02

inv inv inv inv -0.01

Figure 7: as in figure 3. -0.05 [pb/GeV] /dM d [pb/GeV] /dM d [pb/GeV] /dM d [pb/GeV] /dM d σ σ σ σ JHEP12(2008)087 in different SUSY X ∗ L ˜ u [GeV] [GeV] [GeV] [GeV] L T T T T ˜ p p p p u → P P 0 500 1000 1500 0 500 1000 0 500 1000 1500 0 500 1000 1500 0 0 0 0

′ 20 10 20 20

-20 -40 -10 -20 -30 -20 -40 -20 -40

[%] [%] [%] [%] δ δ δ δ SU1 SU4 SPS5 – 22 – SPS1a ) α s ) α + α 2 2 s α α , O( , O( channel q q γ g qg channel gg channel q q [GeV] [GeV] [GeV] [GeV] T T T T p p p p Transverse momentum distribution of the process -3 -6 -6 -3 10 10 10 10 × × × × 0 500 1000 0 500 1000 1500 0 500 1000 1500 0 500 1000 1500 0 5 0 0 5 0 -5 -5 10 Figure 8: scenarios. Notations as in figure 4. 0.2 -10

-0.2

0.04 0.02

T T T

T -0.02 -0.04

[pb/GeV] /dp d [pb/GeV] /dp d [pb/GeV] /dp d [pb/GeV] /dp d σ σ σ σ JHEP12(2008)087 on 4.2. [GeV] [GeV] [GeV] [GeV] inv inv inv inv M M M M 2000 3000 4000 2000 3000 4000 2000 2500 3000 3500 4000 1000 2000 3000 4000 4 2 0 5 0 5 0 5 0

-2 -4 -6 -8 -5 -5 -5 ′ 10 10

-10 -10 -10

[%] [%] [%] [%] δ δ δ δ SU1 SU4 SPS5 – 23 – SPS1a ) α s ) α + α 2 2 s α α , O( , O( channel q q γ g q qg channel gg channel q [GeV] [GeV] [GeV] [GeV] inv inv inv inv Same as figure 6, but with the kinematical cuts defined in secti M M M M 2000 3000 4000 2000 3000 4000 Figure 9: 2000 2500 3000 3500 4000 1000 2000 3000 4000 5 0 5 0 5 0 5 0

-5 -5 -5 -5 15 10 15 10 15 10 10

-10 -10 -10

[%] [%] [%] [%] δ δ δ δ JHEP12(2008)087 - dividual (lower right) ibutions for each case [GeV] ~ g merical analysis for the arios and found that the on that the NNLO QCD m ations in cross sections and ant; about half is carried by ) α weak contributions in squark- ibutions. They show a strong s influence with the mass of the ) α ependent on the scenario. Also α + 2 s 2 sions, in combination with the α α . The upper panel shows the relative , O( channel , O( reby, the NLO terms are significant X q γ q ). g gg channel q q qg channel ∗ 2 L α ˜ u 500 1000 1500 2000 L + ˜ 0 5 u -5 α

-10 -15 -20 -25

s

→ [%] δ α ( P P O – 24 – ) NLO electroweak contributions to the production of flavour channel α q q gg channel 2 s α ( O ), can be expected to be of similar size. 4 s α [GeV] [GeV] ( ~ g ~ g m m O Gluino mass dependence of the total (lower left) and of the in stat δ total Moreover, we have investigated several SUSY benchmark scen We have performed an explicit study of the electroweak contr 500 1000 1500 2000 500 1000 1500 2000 0

-5

90 80 70 60 50 40 30 20 10 of the four squarkLHC. The species electroweak in effects thedistributions, can in first give particular rise for SU(2) left-handed to doublet, squarks.and sizeable have with The modific to a be nu dependence considered together on with the the squarksquarks. tree-level masses, contr increasing their relative behaviour of the electroweak contributionsthe is gluino-mass only dependence weakly is d weak.pair In production summary, can the electro reach 20–25%the in NLO size contributions. and As arecontributions, a thus of final signific remark we would like to menti diagonal squark–anti-squark pairselectroweak in LO proton–proton tree-level contributions colli of EW contributions to the total cross section for We have computed the 5. Conclusions yield of the two channels that contribute at LO. Figure 10: -10 -15 -20

[%] [%] δ ξ JHEP12(2008)087 . a ˜ u γ, Z a ˜ u = . . ∗ 0 ∗ a a a ˜ u ˜ V u ˜ u g a g a ˜ u ˜ u → → . gg ∗ a a γg a ˜ u ˜ u In the following we will ˜ u a a ˜ u a ˜ u ˜ u g g a a a → iscussions, and Maike Trenkel and for ions to the different channels ˜ ˜ u u ˜ u q ∗ q a ˜ i a u a ˜ ˜ χ a u ˜ u d g ˜ d γ u a ˜ u → g q q a a g a a ˜ ˜ u u ˜ ˜ u u g i 0 a ˜ χ ˜ a u – 25 – g u γ u ˜ u a ˜ u a ˜ u a g ˜ u a g ˜ a u ˜ u Tree-level EW diagrams for 0 V g a q a γ ˜ u ˜ u q ˜ g Lowest-order diagrams for photon–gluon fusion Tree-level QCD diagrams for u u Figure 12: ) to denote all the neutral (charged) Higgs bosons, while a S ˜ ( u a 0 ˜ u S g Figure 13: Figure 11: q q for cross checking part of the results. A. Feynman diagrams In this appendix genericare diagrams shown. for We the chooseuse various the the contribut up-squark label case as a specific example. Acknowledgments We thank Stefan Dittmaier and Stefano Pozzorini for useful d JHEP12(2008)087 a a W a W a a a a ˜ ˜ γ a u u a ˜ a u ˜ ˜ ˜ u u u ˜ u ˜ ˜ u u ˜ u γ a s i s i ˜ u 0 ˜ ˜ d d a g s i V ˜ ˜ g u d a a g ˜ ˜ u u g g g q q q q q g a q q . ˜ u γ a ∗ ˜ u a 0 ˜ u 0 g a a a V a a V a a a ˜ ˜ u a u ˜ a γ γ u ˜ u ˜ u ˜ ˜ ˜ a a u u u ˜ ˜ u u ˜ ˜ u u a q ˜ a a u → 0 q ˜ ˜ u u g a V g a a ˜ u g gg ˜ ˜ u u a g g g g q q ˜ q q q u q a q normalization constants in the ˜ u g a ˜ u a a a S a a ˜ ˜ ˜ u q u u a a γ ˜ ˜ u u a a u ˜ ˜ u u γ q ˜ ˜ u a u g ˜ i i s s u 0 i ˜ ˜ d d ˜ g ′ ′ ˜ χ a a q q g ˜ ˜ u u a a . The diagrams with counter terms can be g g s g ˜ ˜ u u g q q q ˜ ∗ q q q a q a W 0 ˜ u g ˜ u a V ˜ u a q ˜ 0 q q u γ – 26 – a a → S a a a ˜ ˜ u u a a a ˜ ˜ u u ˜ u u ˜ ˜ ˜ u u u γ q ). a g ˜ q u a α s s 0 i ( ˜ ˜ ˜ u u u ˜ g γ a a ˜ a χ g ˜ q q a ˜ u u O ˜ u g ˜ a u a g g s ˜ u a g ˜ 0 u q q q ˜ q g q q ˜ u 0 g V a q V ˜ u g g q q q a i ˜ u a i s a a ˜ ˜ χ a ˜ u d a γ ˜ ˜ u a u a ˜ u γ a a ˜ u ˜ ˜ u u γ ˜ u ˜ u a a a g i i ˜ ˜ a u ˜ Tree-level diagrams for real photon emission in u a u d d s s ˜ a u ˜ u ′ ′ ˜ ˜ ˜ u g q q a W u i 0 g g ˜ ˜ g u ˜ χ g g a q g q ˜ q i q u g g q s ˜ g χ q q q ˜ q a One-loop EW diagrams for ˜ u 0 i a γ a Figure 14: a a ˜ a ˜ χ u a a ˜ ˜ ˜ u u u a a a γ a ˜ u ˜ a ˜ u u ˜ ˜ ˜ a u γ ˜ u u u ˜ a u a a ˜ g u ˜ ˜ u u ˜ u u u u a 0 i ˜ g ˜ s 0 s ˜ u χ g ˜ ˜ g q q V g g s g g q g g q q ˜ q q q q 0 i g q ˜ q χ Figure 15: computed according to thecounter terms Feynman rules have to in be appendix evaluated B. at The re JHEP12(2008)087 a a a a ˜ ˜ u ˜ ˜ u u u ˜ ˜ g g s u u u ˜ d . The dia- ∗ d d d d W W a ˜ u a ˜ u ). s → α a a a a ( g d ˜ ˜ ˜ ˜ u u u u S d O s t a s d ˜ ˜ d ˜ d u i ˜ g d d ˜ d d χ a a e evaluated at a g to the Feynman rules in appendix B. ˜ ˜ a ˜ u u u W a ˜ u ˜ u s g u ˜ d i ˜ χ d d ˜ g d d d – 27 – a a a g ˜ a u ˜ ˜ u u ˜ u u a a ˜ g ˜ ˜ u u a i a ˜ ˜ u ˜ s χ u ˜ d d i d d ˜ d χ i ˜ χ d d a a a ˜ a u ˜ ˜ u u W a ˜ u i ˜ u a ˜ a χ t s ˜ u ˜ u g ˜ ˜ d d d d ˜ g i d d d ˜ χ d d One-loop EW diagrams that enter only in the case of the proces a a a a i a i ˜ ˜ ˜ a ˜ ˜ u ˜ u ˜ u u u χ χ ˜ u i u ˜ χ a ˜ d d d g ˜ u s ˜ d d g g d d d d d grams containing the counter terms canThe be renormalization computed accordin constants in the counter terms have to b Figure 16: JHEP12(2008)087 a ˜ u a a a g a a a ˜ g ˜ ˜ u a a u u ˜ ˜ a u u ˜ u ˜ ˜ ˜ u u u 0 i u u . The ˜ χ 0 a a ∗ ˜ ˜ g g 0 a ˜ ˜ a u u V 0 i ˜ u ˜ V u ˜ s s χ a ˜ ˜ i u u 0 u u u u u u ˜ ˜ u u u χ u u → u ), the other ones a a u ˜ u α ˜ u ( 0 a a a a s a ˜ ˜ a a ˜ u u ˜ a u u V O i s s i ˜ u 0 ˜ ˜ ˜ u u ˜ ˜ u d d V ˜ g 0 a a ˜ ˜ W W u u V ˜ g u u i i d d u u ˜ g u u u u u u u g u ppendix B. The renormalization a ˜ u e evaluated at a a a a a a 0 i i 0 a ˜ ˜ ˜ a a ˜ u u ˜ ˜ u u u ˜ u ˜ a a χ χ ˜ u 0 ˜ ˜ u u ˜ ˜ u u V ˜ g 0 0 a u u u s s ˜ u V V ˜ g ˜ ˜ u u u u u u a g a g u u u ˜ u u u u ˜ u ˜ u u g ˜ g a u u ˜ u – 28 – a a a a a a a ˜ ˜ ˜ a a ˜ u u ˜ u ˜ u u u ˜ ˜ a a g g ˜ u ˜ ˜ 0 u u ˜ 0 i ˜ u u S ˜ χ a g g u u u s t ˜ u a 0 i a ˜ ˜ u u ˜ ˜ u ˜ χ u u u u u 0 0 u u u u u u ˜ u u g V V i 0 ˜ χ u u g a a a a a a a a 0 ˜ ˜ u u g ˜ ˜ ˜ a a ˜ u u ˜ ˜ u u u u S i i ˜ ˜ u u d d a a a a ˜ g i i t s a ˜ u ˜ ˜ u ˜ u u u ˜ ˜ ˜ ˜ ˜ χ u χ u u ˜ g 0 i s i s V ˜ ˜ d d u u ˜ i 0 g ˜ g u u u u ˜ u u u χ u u u One-loop EW diagrams that enter only in the case of the proces a a a 0 a ˜ ˜ g a a a u a ˜ ˜ u u a ˜ a u ˜ ˜ ˜ ˜ a a a u u u u V ˜ u ˜ ˜ u ˜ ˜ u u u u u ). ˜ u u g 0 i i s 0 a u 0 g ˜ ˜ ˜ χ u χ α ˜ i 0 g 0 ( V ˜ s s χ V ˜ ˜ u ˜ u u u u g u O u u u u u u u u Figure 17: at diagrams in theconstants last in row the contain quark–squark–gluino the counter counter term terms have to listed b in a JHEP12(2008)087 a ˜ u a a a ˜ be u a a g a a a a a a ˜ ˜ a u u ˜ a ˜ u a a u ˜ ˜ ˜ ˜ u u ˜ u u ˜ u g u ˜ u ˜ u ˜ ˜ u u ˜ g g a g i i s s g g g g ˜ u ˜ ˜ u a d d ˜ ˜ g g g ˜ u g g u g u u q u q q q q u u u q q q q a a ˜ u ˜ u g a a a a a a a a ˜ ˜ u u a ˜ a a a u ˜ ˜ ˜ u u u ˜ u ˜ u g u ˜ u ˜ ˜ ˜ u u u g u g a a ˜ g u s i i s ˜ ˜ ˜ ˜ ˜ ˜ ˜ u u g g g g g g ˜ ˜ u u ˜ ˜ g g g g s ˜ g a ˜ u u u g u s s ˜ q q q q u u u ˜ u q ˜ q u a q q q ˜ u . These diagrams interfere with g ∗ . The renormalization constants a a a a ˜ g u ˜ ˜ u u ˜ u a a u a a a a a ˜ ˜ q u u ). a a g ˜ a a a u ˜ ˜ ˜ u u s u i a ˜ g u s ˜ ˜ u u ˜ ˜ ˜ ˜ q u u u d ˜ u g g α ˜ ˜ g g ˜ ˜ g g ( a a i s → ˜ ˜ g g g ˜ ˜ ˜ d u u a ˜ g g O g q ˜ u g g i u q g ˜ u u u d g a u u q q u q ˜ u q q q q q q g a a a ˜ q u ˜ ˜ u u – 29 – a a q ˜ g a a a a ˜ ˜ u u a a ˜ a a a u ˜ ˜ s u u i ˜ ˜ g g u ˜ ˜ u u ˜ ˜ ˜ ˜ u u u d g g g a ˜ ˜ u u g g ˜ i i i s u a u u d d ˜ ˜ u u ˜ g a q q g g ˜ g ˜ i u g u u u d g q q u u u q g q q g q q q ) contributions. The diagrams containing counter terms can α q 2 s a q a α ˜ u ˜ u ( a a a a ˜ u a a a ˜ u a O ˜ ˜ a a a a u u i s ˜ ˜ u ˜ a u u g ˜ a u ˜ ˜ ˜ ˜ ˜ u u u u u ˜ u ˜ ˜ g a u g a a ˜ ˜ g g a ˜ u g i i ˜ u g ˜ u g g g g ˜ u u s s u u ˜ ˜ g g ˜ ˜ q q ˜ g i u g g g u u u g u u ˜ q g q q q q q u u q q q ˜ g u g u q q q a a One loop QCD diagrams for the process ˜ u ˜ u a a a ˜ a a u ˜ u a a a a a ˜ a u ˜ ˜ a a a a a u u i s ˜ ˜ u ˜ u u g ˜ ˜ a u u ˜ ˜ u ˜ ˜ ˜ ˜ ˜ g u u u u u u u ˜ g u ˜ ˜ g g s t u g g a ˜ ˜ ˜ q ˜ ˜ q g g g g ˜ u u u ˜ ˜ ˜ g g g g ˜ ˜ g g ˜ g s i ˜ u g u u u u u u u s ˜ g q q q ˜ q q u u q q q q s q ˜ q q Figure 18: those of figure 12 yielding computed according to theappearing in Feynman the rules counter listed terms in have to appendix be B evaluated at JHEP12(2008)087 a ˜ u a a a a a g a g g ˜ γ u a ˜ u ˜ ˜ u u ˜ ˜ u u ˜ u g i ˜ 0 i g q d u u ˜ χ ˜ χ q q d d u u u u a a n diagrams. They ˜ ˜ u u a a a a a g a g γ ˜ ˜ ˜ ˜ ˜ u u u u u g 0 ˜ u a g g ˜ q u V i q q ˜ χ q q d q q d q q g a ˜ u a a ˜ u ˜ u ˜ g g u u a a a a a a a g ˜ ˜ ˜ ˜ ˜ u u u u u γ g a ˜ 0 u ˜ u ˜ g u a q q V . The last four diagrams contribute only ˜ u γ i g d ∗ ˜ χ a a d d g ˜ q ˜ u q q q q a u ˜ u a q ˜ u ˜ g u → γ (a) (b) u u a q a g – 30 – ˜ u ˜ q a u a g a a g ˜ u a ˜ u ˜ ˜ u u a a ˜ u a ˜ u ˜ a u g ˜ u ˜ u a γ 0 a a g ˜ 0 i u a ˜ ˜ q g u V u ˜ χ ˜ a u a a u u ˜ u ˜ u q ˜ u q q ˜ ˜ g g q q u u u u a a a ˜ u ˜ u ˜ u a a a γ a g ˜ u g ˜ ˜ u u a ˜ u g ˜ a u g a γ a g ˜ u a ˜ u ˜ a a u ˜ u g 0 i 0 ˜ ˜ u u u g ˜ χ V ) through QCD–EW interference u u ˜ ˜ g g u u q ) real photon emission in q q α 2 s q α u u u u q q 2 s α ( α ( O Diagrams for gluon bremsstrahlung from QCD (a) and EW (b) Bor O a a a ˜ ˜ ˜ g u u u a a a a γ g g a a a ˜ ˜ ˜ a a ˜ a u γ u u u g g ˜ ˜ ˜ a u u u a a ˜ ˜ ˜ u u u a ˜ u . ˜ ˜ u u ˜ u 0 u g g i ˜ ˜ ˜ i 0 g g g V ˜ χ ˜ χ = u u d d u u u u q q q q q q q if contribute at Figure 20: Figure 19: JHEP12(2008)087 1 ˜ u d 1 1 1 u 1 q ˜ ˜ ˜ u u u 1 ˜ u ˜ g i u ˜ χ d 0 i q 1 ˜ ˜ χ u q g g u g d 1 ˜ u 1 1 1 1 ls. Their interference 1 q d ˜ ˜ ˜ u u u q ˜ 0 u ˜ u g g V i q d ˜ χ q g d g q g 1 u 1 ˜ u ˜ u 1 ˜ g ˜ u 1 1 1 d q q g u 1 ˜ ˜ u u 1 1 ˜ u ˜ u ˜ ˜ u u i 1 1 1 0 g ˜ χ ˜ ˜ ˜ u u u V 1 u q q g g g d 1 ˜ u ˜ u ˜ g ˜ g (a) (b) g u – 31 – 1 1 1 u 1 1 1 ˜ ˜ u u ˜ u q q 0 ˜ ˜ ˜ u u u 0 1 i 1 1 ˜ ˜ ˜ ˜ u g V u u χ u 1 ˜ u 1 ˜ u ˜ g q q g g g u u g 1 1 u ˜ ˜ u u u 1 1 1 ˜ ˜ ˜ u u u q q 1 ˜ u 0 i 0 g 1 1 u ˜ ˜ χ u ˜ u V q u u ). ˜ g α u g g g 2 s q q g u u α ( O QCD (a) and EW (b) Born diagrams for quark gluon fusion channe 1 1 d 1 1 u 1 1 1 1 1 1 q q u ˜ ˜ ˜ ˜ u u u u 0 ˜ u ˜ ˜ ˜ ˜ ˜ i u u u 1 u u ˜ ˜ V u g χ ˜ i 0 g 1 u ˜ ˜ χ u g g u u g g g q q d contributes at Figure 21: JHEP12(2008)087 C T  γ, Z (B.2) (B.1) γ, Z + = ω = µ 0 0 γ V can be dis- V  . qR ˜ 2 Q,a δZ , for the strong s , + g µ δm g )  , ′ + k δZ ]  ˜ Q,a ˜ g + + + ren ω ˜ δZ 2 Q,a G k µ δZ µ 2 ( ) 1 2 ′ γ 1 2 m δZ coupling k ˜ ized fields, the first two generations, we quarks, squarks, gluons, and Q,a  qR = µν + es and propagators in the one- 1+ + 1+ g t for our squark-pair production δZ δZ k )  ] −  ) ( bare A q ω ˜ C ( aR 2 Q,a ˜ µ T ˜ ren g ren Q,a δ T V γ + ) A ) 2 ˜ g C  Q C ( =Ψ 1 =Φ , m qL + δZ V C + ) ˆ g f − C ˜ δZ + ˜ bare g bare Q,a ω + δZ + + µ Ψ Φ G 2 g 2 γ δZ , , C aL – 32 – 1 qL δZ δ   C (1 + + ) + δ . The Feynman rules for the counter terms can be G ˜ δZ qa Q 3 1 ren ) s ˜ G ( ( q g 2 Q,a δZ V ( δZ − δZ ˜ 2 1 L, R 1 2 = ˆ V − C δZ Q,a [ (  C s = [ s ie δZ bare 1+ s ig 1+ 2 a s ie ig − ˆ g   − − ig − = ren µ , = ren qa = = = ) G g = =Ψ δZ a ˜ , and for the squark masses, which are defined according to Q a a 0 a a a s g bare µ ˜ ˜ ˜ ˜ bare qa V Q Q ˜ Q Q g Q mixing, and weak eigenstates are also mass eigenstates that (1 + G Ψ ren s R - g L 0 g g q = q V g q q bare s g Vertex counter terms involving gauge bosons: • expressed in terms of the field renormalization constants of gluinos, defined from the relation between bare and renormal The actual expressions of the countertermsprocesses that are are given relevan below. Yukawa coupling ˆ B. Counter terms and renormalization constants Here we list the counter terms for renormalization of vertic loop amplitudes for squark-paircan production. neglect For squarks of together with the renormalization constants for the strong tinguished by their chiralities JHEP12(2008)087 C fixed C (B.3) T T − ] ] − − + + . ω − ω ω ω W , θ aL aR δ aL q aR δ δ e ) δ ) ) W W ) c s QL fermions. Moreover, ] QL ] + QR + ] − QR + + − δZ − the structure constants ˜ − ω δZ δZ ω ω ω 2 Q,a δZ ω + )= + + q ˜ g aR + aL aR aL ( δm ˜ ˜ g g aL δ δ δ ABC δ ˜ g Z ] ) δ + ) ) δZ ) f − ˜ ) g δZ δZ L δZ G + ′ QR QL QR QL ˜ + + δm ˆ g Q Q,a + , C δZ ˆ ˆ g g δZ δZ δZ δZ ) ˆ g −
δZ δZ 2 δZ + + ˜ δZ ) δZ g + + p 2 W δZ + s ˜ µν + 2 q δZ ˜ ˜ 2 Q,a ˜ ˜ Q,a Q,a + 2 + 2 g ) Q,a Q,a e ˜ + 2 ˜ g Q,a ˜ m − Q,a ˜ ˜ δZ − δZ δZ δZ m Q,a ˜ Q,a − ν δZ 3 q Q,a ( p δZ ) ( ) ( I 2 ) ( ) ( − ) δZ δZ µ ′ p /p Q Q δZ Q Q p W ( ( ( ( Q [( ( ( [ ( – 33 – ( [( s ( i W i i 2 2 + ∗ − s ∗ + − B s [ ( g 1 √ A A 2 A A s i [ [ ie W g √ c − ie − i ie are the color matrices and )= = = = q C ( T = Z = = = = − are sine and cosine of the electroweak mixing angle , C q e W c a a i ˜ 0 0 a a a a g ˜ i i ˜ g g ˜ g Q ˜ ˜ χ )= ˜ ˜ ˜ ˜ ˜ ˜ χ χ Q Q Q Q Q q ( and are the momenta of the squark and the antisquark, and they are γ ± ′ W C k s a Q Q Q Q ˜ ˜ g g Q’ Q and Self energy counter terms: of the color group.we We define omit the color indices of fermions and s according to the arrow. where k Vertex counter terms involving gauginos: • • JHEP12(2008)087 ] 1 ∗ i V nded (B.8) (B.4) (B.7) (B.5) (B.6) is fixed )= , ′ β o , Q ) ( o ,R ) B , ′ [ ˜ Q 2  ˜ 2 Q,a 1 for tan ∗ 2 i is the self energy ∗ m i W ( m β U s ( N ,R δt ˜ epicted in the quark ′ Q,a Q ven by ˜ ˜ I Q Q,a + rescription of ref. [67]; )= , ′ Σ Σ ω + β Q n n ( 1 ∗ δt i aL by on-shell conditions (see W B ) can be expressed in terms δ c β ′ Re N Re ) s ), and Σ , L Q 1 6 ˜ s 3 β ′ = = ( c

Q  c, ) B , 2 (˜ ,R 0 W 2 ′ ˜ also gauge invariant. Furthermore, , 2 Q,a δZ 1 2 A

˜ ) . The vertices involving neutralinos 2 Q √ M ) ˜ d m + Q 2 ( − W u, Z + 4 ˜ 0 M self energy in dimensional reduction. As Q,a ( div A 2 )= W Z T W δZ Σ Q 0 ( , δm  ( Σ ) δM , δm A ′ −  ,a ′ Q β W self energy. ˜ ( Re 2 Q 2 s ˜ – 34 – 2 Q,a ∗ 2 c 2 β Re m m B c W = − = Z of the chargino sector: 2 = , A 1 ie 2 p 1 p | i | M − 2 V W ˜ 2 2 Q,L ) N W ) ) c ) Q 2 = δM 2 δm e p and p ( β 2 2 ( 2 = 1 = ,a U ′ δt etc. for abbreviation). The counter term √ ∂p ˜ ˜ ∂p Q,a Q ,L θ ′ Σ Σ ˜ 2 Q ∂ ∂ )= denotes the SU(2) partner of ′ Q ( ( δm ( = sin Q + θ i a Re Re ˜ s A χ ˜ . Due to SU(2) invariance the mass counter term of the left-ha Q − − , a is the mixing matrix of the neutralinos. θ ˜ Q = = ij ) is either of the two SU(2) doublets (˜ ,a N ′ ′ ˜ denotes the divergent part of the is the transverse part of the Q,a ˜ Q’ ˜ Q Q = cos θ δZ T W div ˜ δZ Q, c mass counter term appears in (B.7), in the on-shell scheme gi DR scheme and can be written in the following way [68, 69], of the mixing matrices contain the quantities The Feynman rules involving Majoranain particles particular follow the the fermionline. p flow As is usual, fixed according to the arrow d for up [down] type quarks. where W where Σ The renormalization constants of the squark sector are fixed also ref. [44, 34]), pointed out in [70], this process-independentthe condition is where Σ where ( of the squark (where down-type squark is a dependent quantity, in the JHEP12(2008)087 ) . (B.9) L, R (B.14) (B.13) (B.12) (B.11) (B.10) ˜ g 2 = m a = ( 2 p | 2 q  m
) =   2 2 p p  | ( : 2 t 2 g  . ˜ gS µ
m ) ) 2 δZ 2  p p ( ( ln , but since the gluon only qS qS )+2Σ 1 3 ∗ ed via on-shell conditons as rdingly, the renormalization 2 Σ a he self energy, op order can be achieved by q p rticular care. As mentioned in stant of the gluon in eq. (B.1), Q + ( r gluons radiated. In the phase a onstants by on-shell conditions, m ˜ Q ppear in the phase space integra- ! oducing a mismatch between the ˜ gR MS scheme decoupling the heavy

)+2Σ → qcd on the energy of the emitted pho- , q 2 2 )+ )) ˜ 2 Q,a 1 q p 2 ˜ 2 g s µ . ( p m )+Σ ( m √ M s π . ( 2 s qR

α 3 δ p g qR ( ˜ gS ln Σ + δZ to be different from ˜ gL + = 2 g 1 )+Σ s 12 Σ 2 /pω g E = 2 p δZ ( 2 )+2Σ – 35 – ˜ Q,a ˜ G 2 g X ∂ = )+ qL ∂p 2 m ˆ g + δZ Σ p (  ( ! δZ ˜ gR 2 qL ˜ 2 g 2 Re ∂ is renormalized in the Σ µ ∂p ˜ 2 g m on the angle between the photon/gluon and the radiating s − is cancelled in the sum of self energy and vertex counter g  m c

. In order to restore supersymmetry in physical amplitudes, )+Σ δ /pω s ˜ 2 g ). The treatment of UV divergences in dimensional regular- G − g Re π m 2 q

( δZ ) )= m ˜ 2 g 2 ˜ gL ∆+ln p − m ( 3 2 ( q

(Σ + ln(4 )   Σ ˜ g 2 q ˜ gL E s π Σ γ m m α 4 (   − − in (B.2) is given by [24] qa s Re Re Σ = /ǫ g  1 2 − g ) it enters only the one-loop amplitude = = Re δZ α 2 s ˜ ˜ g g − α ( δZ = δm O qa At For completeness we quote also the field renormalization con where ∆ = 2 ization violates supersymmetry at thestrong one-loop Yukawa coupling level, and intr The field renormalization constants of the quarks are obtain follows [71], δZ particles (top, gluino andconstant for squarks) from its runnning. Acco with the scalar coefficients in the Lorentz decomposition of t ton (gluon), and an angle cut appears in internal lines, The renormalization of the strongsection coupling deserves 3.3 some the pa strong coupling cancellation of this extra term is required, which at one-lo modifying the renormalization constant for ˆ terms. C. Bremsstrahlung integrals Here we list the IR andtion collinearly of singular the integrals that bremsstrahlung a processes,space with slicing either method, photons o cuts are imposed: ∆ Also in the gluino sector we determine the renormalization c JHEP12(2008)087 ; d on . , ) !# ) (C.3) (C.1) (C.2) (C.4) ). zs  t ( on that β β ∗ /s − ∗ a a ˜ − st Q ˜ ˜ Q ) 2 Q,a a ˜ 1 1+ a 2 Q,a ˜ β Q ˜ m Q  m → 0 ( 2 → , (4 q 0 2 , q 2 q 3 2 q π ln (1 + − ˜ Q,a 2 and a IR-singular dσ 1 4 1 dσ + m ∗  i 2 a − # q ) ) ˜ z Q  ically, regularizing the  ∗ = a FI a β β 1+ − δ β ˜ ˜ Q β Q β

a − − r region can be written in 2 ˜ 2 Q light quarks. With the help 1+ 1 → 1+ → IF and h is integrated numerically. 0  q ,  or function. ntegration can be expressed as δ ) + (1 2 2 gg q follows, ( + Li 2 z ns the factorized expressions for ˜ ( Q ln ! . dσ Li e ) qq u q t ) − P e −  − ! → FF  1 2 q δ st t 2 )+ s ) , ˜ − 2 Q,a m 2 E − β  FF λ m

 2 q F δ − ( 2 q 4∆ s δ 2 2 c ( m − (1  ln m ˜ ˜ Q 2 2 Q,a sδ  4 " – 36 – e F ln 2 m δ  1 2 π α ( 2 ) cross section for ln  ˜ 2 Q 2 s − ln + otherwise. Furthermore, e β β 2 1 α  ( 1+ 3 1 =  −  − β β 2 O

γ − )+ 1+ 1 ∗ 2 2 2 E 3 a − II , π dz  = ˜ λ Q δ Li 1 1+ by the substitution s a q ! δ 4∆ " This process is affected by IR singularities only. Integrate ln The differential cross section integrated over the soft regi − ˜ e  − Q 2 q − 2  2 1 IF s 1 0 I  → δ ln 1 m . x .  δ − , 2 ln ( 2 Z gg γ γ 1 β and

2 q ! 2 E ! ∗ ∗ 2 q e ! . The phase space is thus split into a soft and a collinear regi 2 λ a dσ a π c h ) 2 + 2 δ t ˜ ˜ ˜ ) 4∆ αe 2 Q,a Q Q u, c π α + ln  a a −  − m ˜ ˜ 2 2 Q,a s 2 Q,a − ˜ = ˜ 1 2  Q Q st ln 2 )= E 2 m q m ˜ s = 2 Q,a λ − 2 q (  2 E if − γ s → → θ > γ 4∆ 2 q m m λ ∗ ∗ ( t s 3 2 a a

 ( m 4∆ q ˜ ˜ Q Q q gg 2 a β a  

= 1+ ˜ ˜ Q Q q

e = = ln → → 1 1 2 , q , q 2 q 2 q = ln = ln = ln The integration over the soft region can be performed analyt The differential cross section integrated over the collinea F can be obtained from δ FF I is the infinitesimal mass regularizing the IR divergencies, dσ dσ δ II δ IF FI +2Li δ δ Process over the soft region, the differential cross section reads as λ can be expressed in terms of the are singular and a complementary non-singular region, whic singularities by small massesof for explicit the formulae photon for (gluon) thethe and IR cross the integrals section [71, given 72],a below. one convolution obtai In of the the collinear lowest-order cross region, section theProcess and i a radiat quark via cos δ where terms of a convolution integral, where factor, JHEP12(2008)087 , ∗ , ˜ a σ ) ) ˜ d Q ian- a zs zs (C.7) (C.6) (C.8) (C.5) ˜ ( Q ( ∗ ∗ a → and a 1 , q ˜ ˜ Q Q 1 q ¯ , σ a a is different , , d ˜ ˜ Q  Q dσ ∗ ˜ o o σ → ] ] a → 1 1 d , q 3 3 , q ˜ i Q 1 q 1 q ,c ,c a 4 2 ˜ IF Q , dσ ,c ,c ∗ ∗ dσ δ 2 4 a a ∗ → o qcd (t) , but only together ˜ ˜  ,c ,c − a , q  Q Q ith this notation we ∗ ) ∗ 4 4 0 q ) ˜ a a a Q c c a z ˜ [ [ ˜ ˜ z a Q Q Q ˜ FF ˜ a Q M Q δ − → → ˜ − a ew ew 4 Q , , q q c → ˜ ew (t) + 0 0 q q 2 Q , q → C (1 1 c 0 q , q r region can be written in z II ual indices. On top, average 1 q linear region. In that region T M M a gluon as a colored particle δ 1 → ˜ σ M ∗ ∗ c  treated by mass regularization ) + (1 into color factors and reduced d ) from eq. (3.10). 4 3 q 54], as follows, C   c c z i z ] ] 1 oft region can also be expressed q ∗ 2 N ( ) + 2 3 3 ( T lar color structure, c n be taken into account following a z ,c ,c δ qq FF ˜ qq ( + 2 1 Q 3 + δ P c P a ,c ,c qg ∗ 1 ∗ ∗ These processes exhibit singularities 1 2 ˜  FI a c a a + P Q ,c ,c δ 1 ˜ δ ˜ ˜ Q Q Q 1 1 a  c c  . a a II [ [ − + ˜ → ˜ ˜ 2 δ Q c q Q Q 1 N δ ∗ ∗ q  → → → a 2 qcd (s) qcd qcd + a q , q , , q q ) ˜ 2 q + Q 2 q 2 0 q c q q 0 0 ˜ z Q a F m FI ˜ m a sδ Q 4 δ M C M M − 4 -channel diagrams in figure 11: ˜ – 37 – 4 h → Q t  ew (s) c  , q 3 (1 0 q n n C . − c s ln ∗ +2 T ∗ → a a  M 1 ˜ Re Re  Q c ˜ 4  Q - and 2 a c 2 2 C F ln a c 3 s ˜ qg Q δ ˜ c Q T  2 2 δ → dz 0  1 1 2 → , q 1 N N c s , +2 q 2 q δ 1 1 q I dz 2 2 c C and − ¯ δ σ The singularities affecting this radiative process are Abel δ X 0 1 dσ 0 1 2 d x πs πs x q dt dt i  Z = = Z ∗ . F 16 16 a ] ] g IF F F 4 4 . C ˜ ∗ δ π T Q ,c ,c π a = = C s Q 2 and the quark splitting function 3 3 a 2 s − nh ˜ = 3. In order to specify the color-modified “cross sections” α ∗ ∗ ,c ,c Q ∗ ˜ α ∗ s Q a a 2 2 = a π a ) contributions to the cross section for /s a ˜ ˜ ˜ ,c ,c α Q Q FI 2 N Q ) instead of ˜ q ˜ α Q 1 1 a a δ a Q c c a s ∗ ˜ ˜ [ [ ˜ h − → ) = Q Q Q )= ˜ ˜ a Q α 2 Q,a s s ˜ ( → → ( Q → 1 1 and ( → ew − = → , , q q , q a qcd q m g , q qg 1 1 q q O 0 q ∗ ˜ g 3 4 ∗ q 0 Q q a ¯ ˜ ∗ σ σ a are the color matrices in the fundamental representation. W ˜ a q ˜ Q → d d 1 Q M = ˜ , q a M Q C a = (4 1 q ˜ a ˜ Q Q F 0 T ˜ Q → x dσ C 1 → 1 , → The differential cross section integrated over the collinea , q 1 2 qg , q 2 q 2 q dσ dσ dσ terms of a convolution integral similar to eq. (C.4), from zero only if we first separate the tree-level amplitudes for like, similar to the case ofas photon well. radiation, The and differential thus canin cross be section terms integrated of over the s when the final (anti-)quark is emmitted off the gluon in the col the differential cross section can be written, in analogy to [ where color summation has to beover performed the over initial each helicities pair is of eq assumed. Owing to the particu where with can write for the color-rearranged contributions, with matrix elements, according to the Processes with with a rearrangement of theleads color to color structure. correlations The inthe emission the eikonal prescription of current, of which ref. ca [48], yielding the result Process JHEP12(2008)087 B ) zs ( (C.9) ng B 70 ∗ a ˜ JHEP Q ]. a ]. ]. , ˜ B 46 Q Phys. Rev. ]. (1984) 1. Riv. Nuovo → , Phys. Lett. 1 , q constraints on ]. , , 1 q (2008) 042008 on 110 dσ Nucl. Phys. , ) ) 110 z Phys. Lett. The Supersymmetric , − hep-ex/0208001 hep-ex/0401008 (1 arXiv:0707.3447 Electroweak Working Group hep-ex/0509008 z Phys. Rept. ]. , hep-ph/0412214 eiglein, roweak Precision Data Electroweak Working Group, Physics interplay of the LHC and ) + 2 LEP , Precision electroweak z ( LEP qg (2007) 87 [ and ]. et al. P (2006) 257 [ J. Phys. Conf. Ser. (2002) 101804 [ (2004) 161802 [ ! , (2006) 265 [ 2 c 89 δ 92 427 Measurement of the negative muon anomalous Measurement of the Positive Muon Anomalous B 657 2 OPAL hep-ex/0612034 , ) , 2 q , 425 z Electroweak precision observables in the minimal ]. m L3 − 4 , – 38 – et al. et al. Squark and gluino production with jets (1 collaborations, s hep-ph/0410364

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