Hadronic production of top-squark pairs with electroweak NLO contributions
aa r X i v : . [ h e p - ph ] D ec Preprint typeset in JHEP style - HYPER VERSION
MPP-2007-149arXiv:0712.0287 [hep-ph]
Hadronic production of top-squark pairs withelectroweak NLO contributions
Wolfgang Hollik, Monika Koll´ar, and Maike K. Trenkel
Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, D-80805 M¨unchen, Germany
Abstract:
Presented are complete next-to-leading order electroweak (NLO EW) correc-tions to top-squark pair production at the Large Hadron Collider (LHC) within the MinimalSupersymmetric Standard Model (MSSM). At this order, also effects from the interferenceof EW and QCD contributions have to be taken into account. Moreover, photon-inducedtop-squark production is considered as an additional partonic channel, which arises fromthe non-zero photon density in the proton.PACS: 12.15.Lk, 13.85-t, 13.87.Ce, 14.80.Ly ontents
1. Introduction 12. Top-squark eigenstates and LO cross sections 23. Classification of EW NLO corrections 4
4. Numerical results 10
5. Conclusions 20Appendix 20A. Feynman diagrams 20
1. Introduction
Within supersymmetric theories top-squarks are the supersymmetric partners of the left-and right-handed top quarks. The two superpartners ˜ t L and ˜ t R , which belong to chiralsupermultiplets ˆ Q and ˆ T , in general mix to produce two mass eigenstates ˜ t and ˜ t . Inmany supersymmetric models the lighter mass eigenstate appears as the lightest coloredparticle [1], for reasons related to the large top Yukawa coupling. The large mixing inthe stop sector leads to a substantial splitting between the two mass eigenstates, and theevolution from the GUT scale to the electroweak scale yields low values for the stop masseswhen a universal scalar mass is assumed at the high scale [2]. The search for top-squarksis therefore of particular interest for the coming LHC experiments, where they would beprimarily produced in pairs via the strong interaction, with relatively large cross sections.Current experimental limits on top-squark pair production include searches performedat LEP [3] reviewed e.g in [4], and at the Tevatron, done by the CDF and DØ collaborationsin approximately 90 pb − of Run I data [5]. Extended searches have been done using Run IIdata samples by both CDF and DØ [6]. Limits on the top-squark mass, depending on the– 1 –ass of the lightest neutralino, are provided with the assumption that BR ( ˜ t → c ˜ χ ) =100 % in [7].Experimental searches for the top-squarks have also been done in ep collisions atHERA [8], where only single stop production could be kinematically accessed and henceconstraints have been derived essentially on the R-parity violating class of supersymmetricmodels.Concerning the theoretical predictions, QCD-based Born-level cross sections for theproduction of squarks and gluinos in hadron collisions have been calculated in [9]. Theyhave been improved by including NLO corrections in supersymmetric QCD (SUSY-QCD),worked out in [10] with the restriction to final state squarks of the first two generations,and for the stop sector in [11]. The production of top-squark pairs in hadronic collisions isdiagonal at lowest order at O ( α ). Electroweak (EW) contributions of O ( α ) are suppressedby two orders of magnitude. Also at O ( α ) the production mechanism is still diagonal.Non-diagonal production occurs at O ( α ), and the cross section is accordingly suppressed.Production of non-diagonal top-squark pairs can also proceed at O ( α ) mediated by Z -exchange through qq annihilation [12] as well as in e + e − annihilation [13].The LO cross section for diagonal top-squark pair production depends only on the massof the produced squarks. As a consequence, bounds on the production cross section caneasily be translated into lower bounds on the lightest stop mass. At NLO, the cross sectionbecomes considerably changed and dependent on other supersymmetric parameters, likemixing angles, gluino mass, masses of other squarks, etc., which enter through the higherorder terms. Once top-squarks are discovered, measurement of their masses and crosssections will provide important observables for testing and constraining the supersymmetricmodel.In the following, we study the NLO contributions to diagonal top-squark pair pro-duction that arise from the electroweak interaction within the Minimal SupersymmetricStandard Model (MSSM). We assume the MSSM with real parameters, R-parity conser-vation, and minimal flavor violation. The outline of our paper is as follows. In Section 2,we present analytical expressions for the partonic and hadronic LO cross sections. We alsointroduce some basic notations used throughout the paper. Section 3 is dedicated to theclassification of the NLO EW contributions into virtual and real corrections with the treat-ment of soft and collinear singularities, and photon-induced contributions. In Section 4,we give a list of input parameters and conventions, followed by our numerical results forthe hadronic cross sections and distributions for pp collisions at a center-of-mass energy √ S = 14 TeV at the LHC. We also investigate the application of kinematical cuts, and weanalyze the impact of varying the MSSM parameters.
2. Top-squark eigenstates and LO cross sections
In the MSSM Lagrangian, mixing of the left- and right-handed top-squark eigenstates ˜ t L/R into mass eigenstates ˜ t / is induced by the trilinear Higgs-stop-stop coupling term A t and– 2 –he Higgs-mixing parameter µ . The top-squark mass matrix squared is given by [14] M = (cid:18) m t + A LL m t B LR m t B LR m t + C RR (cid:19) , (2.1)with m t denoting the top-quark mass and A LL = (cid:16) −
23 sin θ W (cid:17) m Z cos 2 β + m Q ,B LR = A t − µ cot β ,C RR = 23 sin θ W m Z cos 2 β + m U . (2.2)Here, tan β is the ratio of the vacuum expectation values of the two Higgs doublets and m ˜ Q , m ˜ U are the soft-breaking mass terms for left- and right-handed top-squarks, respec-tively.The top-squark mass eigenvalues are obtained by diagonalizing the mass matrix, U M U † = m t m t ! , U = (cid:18) cos θ ˜ t sin θ ˜ t − sin θ ˜ t cos θ ˜ t (cid:19) , (2.3) m t , = m t + 12 (cid:18) A LL + C RR ∓ q ( A LL − C RR ) + 4 m t B LR (cid:19) , (2.4)and the mixing angle θ ˜ t is determined bytan 2 θ ˜ t = 2 m t B LR A LL − C RR . (2.5)At hadron colliders, diagonal pairs of top-squarks can be produced at leading order inQCD in two classes of partonic subprocesses, gg → ˜ t ˜ t ∗ and ˜ t ˜ t ∗ ,q ¯ q → ˜ t ˜ t ∗ and ˜ t ˜ t ∗ , (2.6)where qq denotes representatively the contributing quark flavors. The corresponding Feyn-man diagrams for the example of ˜ t ˜ t ∗ production are shown in the appendix, Fig. A.1.As already mentioned, mixed pairs cannot be produced at lowest order since the g ˜ t ˜ t ∗ and gg ˜ t ˜ t ∗ vertices are diagonal in the chiral as well as in the mass basis.The differential partonic cross sections for the subprocesses, d ˆ σ gg,q ¯ q (ˆ s ) = 116 π ˆ s X(cid:12)(cid:12) M gg,q ¯ q (ˆ s, ˆ t, ˆ u ) (cid:12)(cid:12) d ˆ t , (2.7)can be expressed in terms of the squared and spin-averaged lowest-order matrix elements,as explicitly given by [10], X(cid:12)(cid:12) M gg (cid:12)(cid:12) = 14 · · π α s (cid:20) C (cid:18) − t r ˆ u r ˆ s (cid:19) − C K (cid:21) " − sm t i ˆ t r ˆ u r − ˆ sm t i ˆ t r ˆ u r ! , (2.8) X(cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = 14 · · π α s N C F ˆ t r ˆ u r − m t i ˆ s ˆ s , (2.9)– 3 –ith ˆ t r = ˆ t − m t i , ˆ u r = ˆ u − m t i , where ˆ s, ˆ t, ˆ u are the usual Mandelstam variables. i = 1 , SU (3) color factors are given by N = 3, C = N ( N −
1) = 24, C K = ( N − /N = 8 / C F = ( N − / (2 N ) = 4 / AB → ˜ t i ˜ t ∗ i , i =1, 2,is related to the partonic cross sections through dσ AB ( S ) = X a,b Z τ dτ d L ABab dτ d ˆ σ ab (ˆ s ) , (2.10)with τ = ˆ s/S , S (ˆ s ) being the hadronic (partonic) center-of-mass energy squared and τ = 4 m t i /S is the production threshold. The sum over a, b runs over all possible initialpartons. The parton luminosities are given by d L ABab dτ = 11 + δ ab Z τ dxx (cid:20) f a/A (cid:0) x, µ F (cid:1) f b/B (cid:16) τx , µ F (cid:17) + f b/A (cid:16) τx , µ F (cid:17) f a/B (cid:0) x, µ F (cid:1)(cid:21) , (2.11)where the parton distribution functions (PDFs) f a/A ( x, µ F ) parameterize the probabilityof finding a parton a inside a hadron A with fraction x of the hadron momentum at afactorization scale µ F .
3. Classification of EW NLO corrections
In the following we describe the calculation of EW contributions to top-squark pair pro-duction at NLO. For the treatment of the Feynman diagrams and corresponding ampli-tudes we make use of
FeynArts 3.2 [15] and
FormCalc 5.2 with
LoopTools 2.2 [16],based on Passarino-Veltman reduction techniques for the tensor loop integrals [17], whichwere further developed for 4-point integrals in [18]. Higgs properties are computed with
FeynHiggs 2.5.1 [19].The supersymmetric final state does not allow to separate the SM-like corrections fromthe superpartner contributions which are necessary for the cancellation of ultra-violet (UV)singularities. As the photino is not a mass eigenstate of the theory, it is also not possibleto split the EW corrections into a QED and a weak part, which is often the case in SMprocesses. In order to obtain a UV finite result, we have to deal with the complete set ofEW virtual corrections including photonic contributions. These are infrared (IR) singularand thus also the real photonic corrections have to be taken into account. In addition,a photon-induced subclass of corrections appears at NLO as an independent productionchannel.
The virtual corrections arise from self-energy, vertex, box, and counter-term diagrams.These are shown in the appendix, in Fig. A.3 for the qq annihilation and and in Fig. A.4 forthe gluon fusion channel, respectively. Getting an UV finite result requires renormalizationof the involved quarks and top-squarks. The renormalized quark and squark self-energiesare obtained from the unrenormalized initial quark self-energiesΣ q ( p/ ) = p/ω − Σ qL ( p ) + p/ω + Σ qR ( p ) + m q Σ qS ( p ) , (3.1)– 4 –ccording to ˆΣ qL ( p ) = Σ qL ( p ) + δZ qL , ˆΣ qR ( p ) = Σ qR ( p ) + δZ qR , (3.2)ˆΣ qS ( p ) = Σ qS ( p ) − (cid:0) δZ qL + δZ qR (cid:1) + δm q m q , and from the top-squark self-energies Σ ˜ t i ( k ) (for i =1, 2), according toˆΣ ˜ t i ( k ) = Σ ˜ t i ( k ) + k δZ ˜ t i − m t i δZ ˜ t i − δm t i , (3.3)with the renormalized quantities denoted by the symbol ˆΣ.The full set of virtual contributions is UV finite after including the proper counter-terms for self-energies, quark vertices, and squark triple and quartic vertices, as listed inthe following set of Feynman rules: g ~~~~ tttt iiii iδ Σ ˜ t i = i (cid:16) k δZ ˜ t i − m t i δZ ˜ t i − δm t i (cid:17) , (3.4) iδ Λ µ i = − ig s T c (cid:0) k + k ′ (cid:1) µ δZ ˜ t i , (3.5) g ~~g q q g tt ii iδ Λ SSV Vµ i = 12 ig s (cid:18) δ ab + d abc T c (cid:19) g µν δZ ˜ t i , (3.6) iδ Λ qµ = − ig s T c γ µ (cid:0) ω − δZ qL + ω + δZ qR (cid:1) , (3.7)where k , k ′ denote the momenta of top-squarks (in the direction of arrows), a , b , and c are the gluonic color indices, T c and d abc are the color factors (we skip the fermionicand sfermionic color indices), and ω ± = (1 ± γ ) / δm t i = Re Σ ˜ t i ( m t i ) , (3.8) δZ ˜ t i = − ddk Re Σ ˜ t i ( k ) (cid:12)(cid:12)(cid:12)(cid:12) k = m ti , (3.9) δZ qL,R = − Re Σ qL,R ( m q ) − m q ∂∂p Re (cid:2) Σ qL ( p ) + Σ qR ( p ) + 2Σ qS ( p ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) p = m q . (3.10)– 5 –here is no renormalization of the gluon field at O ( α ). Also, the strong couplingconstant does not need renormalization since UV singularities cancel in the sum of 3-and 4-point functions and their corresponding counter-terms from quark and squark fieldrenormalization (see Figs. A.3 and A.4 in the appendix).Loop diagrams involving virtual photons generate IR singularities. According to Bloch-Nordsieck [20], IR singular terms cancel against their counterparts in the real photoncorrections. To regularize the IR singularities we introduce a fictitious photon mass λ . Incase of external light quarks, also collinear singularities occur if a photon is radiated off amassless quark in the collinear limit. We therefore keep non-zero initial-state quark masses m q in the loop integrals. This gives rise to single and double logarithmic contributions ofquark masses. The double logarithms cancel in the sum of virtual and real corrections,single logarithms, however, survive and have to be treated by means of the factorization.In the gg fusion channel, IR singularities originate only from final-state photon ra-diation, and mass singularities do not occur. In the qq annihilation subprocess, the IRsingular structure is extended by the contributions related to the gluons which appear inthe 4-point UV finite loop integrals. There are two types of IR singular box contributions(Fig. A.3 c). The first group is formed by the gluon–photon box diagrams with two sourcesof IR singularities, one related to photons, the other to gluons. The second group consistsof the gluon– Z box diagrams with IR singularities originating from the gluons only. Thereis also an IR finite group of O ( αα s ) box diagrams which consists of gluino–neutralino loops(Fig. A.3 d). Owing to the photon-like appearance of the gluon in the box contributions,the gluonic IR singularities can be handled in analogy to the photon IR singularities. To compensate IR singularities in the virtual EW corrections, contributions with realphoton (Fig. A.5 a and c) and real gluon radiation are required. In case of gg fusion,only photon bremsstrahlung is needed, whereas in the qq annihilation channel, also gluonbremsstrahlung at the appropriate order O ( αα ) has to be taken into account (Fig. A.6)to cancel the IR singularities related to the gluon. The necessary contributions originatefrom the interference of QCD and EW Born level diagrams, which vanishes at LO. Notall of the interference terms contribute. Due to the color structure, only the interferencebetween initial and final state gluon radiation is non-zero.Including the EW–QCD interference in the real corrections does not yet lead to an IRfinite result. Also the IR singular QCD-mediated box corrections interfering with the O ( α )photon and Z -boson tree-level diagrams are needed. Besides the gluonic corrections thereare also the IR finite QCD-mediated box corrections, which contain gluinos in the loop.Interfered with the O ( α ) tree-level diagrams, these also give contributions of the respectiveorder of O ( αα ). The set of all O ( α ) diagrams is shown in Fig. A.7.So far we have mentioned only the IR singular bremsstrahlung contributions. However,there are also IR finite real corrections to both gluon fusion and qq annihilation processes.In addition to the photon radiation off the off-shell top-squark there are photon radiationcontributions originating from the quartic gluon–photon–squark–squark coupling. Thesecontributions do not have to be regularized since they are not singular (Fig. A.5 b and d).– 6 –he treatment of IR singular bremsstrahlung is done using the phase space slicingmethod. Imposing cut-offs ∆ E on the photon/gluon energy and ∆ θ on the angle betweenthe photon/gluon and radiating fermion, the photonic/gluonic phase space is split into softand collinear parts which contain singularities and a non-collinear, hard part which is freeof singularities and is integrated numerically. The sum of virtual and real contributions,each of them dependent on the cut-off parameters ∆ E and ∆ θ , has to provide a fullyindependent result. To ensure this we perform numerical checks.In the singular regions, the squared matrix elements for the radiative processes fac-torize into the lowest-order squared matrix elements and universal factors containing thesingularities. The soft-photon part of the radiative cross section in the qq annihilation channel d ˆ σ q ¯ qsoft,γ (ˆ s ) = απ (cid:16) e q δ insoft + e t δ finsoft + 2 e q e t δ intsoft (cid:17) d ˆ σ q ¯ q (ˆ s ) , (3.11)and in the gg fusion channel d ˆ σ ggsoft,γ (ˆ s ) = απ e t δ finsoft d ˆ σ gg (ˆ s ) , (3.12)can be expressed using universal factors, δ in,fin,intsoft , which refer to the initial state radiation,final state radiation or interference of initial and final state radiation, respectively. d ˆ σ q ¯ q,gg denote the corresponding partonic lowest order cross sections. The singular universal fac-tors, similar to those in [21], read as follows, δ insoft = (cid:20) ln δ s − ln λ ˆ s (cid:21) (cid:20) ln ˆ sm q − (cid:21) −
12 ln ˆ sm q + ln ˆ sm q − π ,δ finsoft = (cid:20) ln δ s − ln λ ˆ s (cid:21) (cid:20) ˆ s − m t i ˆ sβ ln (cid:18) β − β (cid:19) − (cid:21) + 1 β ln (cid:18) β − β (cid:19) − ˆ s − m t i ˆ sβ (cid:20) (cid:18) β β (cid:19) + 12 ln (cid:18) β − β (cid:19)(cid:21) ,δ intsoft = (cid:20) ln δ s − ln λ ˆ s (cid:21) ln (cid:18) − β cos θ β cos θ (cid:19) − Li (cid:18) − − β − β cos θ (cid:19) − Li (cid:18) − β − β cos θ (cid:19) + Li (cid:18) − − β β cos θ (cid:19) + Li (cid:18) − β β cos θ (cid:19) . (3.13)Here, e q and e t are the electric charges of the initial quark and of the top-squark, re-spectively, and we introduced δ s = 2∆ E/ √ ˆ s , where ∆ E is the slicing parameter for themaximum energy a soft photon may have. For application purposes, it is useful to expressEq. (3.13) in terms of Mandelstam invariants, ˆ t and ˆ u , using the relationsˆ t, ˆ u = m t i − ˆ s ∓ β cos θ ) , β = r − m ˜ t i ˆ s . (3.14)– 7 –he soft-gluon part for the qq channel can be written in a way similar to (3.11), butwith a different arrangement of the color matrices, d ˆ σ q ¯ qsoft,g (ˆ s ) = α s π δ intsoft h T aij T bji T alm T bml i × X (cid:16) f M q ¯ q ∗ ,g f M q ¯ q ,γ + f M q ¯ q ∗ ,g f M q ¯ q ,Z (cid:17) d ˆ t π ˆ s , (3.15)with f M denoting the “Born” matrix elements for g , γ and Z exchange where the colormatrices are factorized off. Explicitly, it can be written as follows, d ˆ σ q ¯ qsoft,g (ˆ s ) = α s π δ intsoft N C F " e q e t ˆ s + (cid:0) ( U i ) − e t sin θ W (cid:1)(cid:0) ǫ − e q sin θ W (cid:1) sin θ W cos θ W ˆ s (ˆ s − m Z ) × π αα s · h(cid:0) ˆ t − m t i (cid:1)(cid:0) ˆ u − m t i (cid:1) − m t i ˆ s i d ˆ t π ˆ s , (3.16)involving the top-squark mixing matrix of Eq. (2.4), and ǫ = ± Collinear singularities arise only from initial-state photon radiation in q ¯ q annihilation. Thecollinear part of the 2 → a withmomentum p a radiates off a photon with p γ = (1 − z ) p a , the parton momentum availablefor the hard process is reduced to zp a . Accordingly, the partonic energy of the total processinclusive photon radiation is ˜ s = ( p a + p b ) = ˜ τ S , and for the hard process the reducedpartonic energy is ˆ s = ( zp a + p b ) = τ S . The ’total’ and ’hard’ variables are thus relatedby ˆ s = z ˜ s and τ = z ˜ τ .Having defined these variables, the partonic cross section in the collinear cones can bewritten in the following way [22, 23] d ˆ σ coll (ˆ s ) = απ e q Z − δ s dz d ˆ σ q ¯ q (ˆ s ) κ coll ( z ) , with κ coll ( z ) = 12 P qq ( z ) (cid:20) ln (cid:18) ˜ sm q δ θ (cid:19) − (cid:21) + 12 (1 − z ) , (3.17)where P qq ( z ) = (1+ z ) / (1 − z ) is an Altarelli-Parisi splitting function [24] and δ θ is the cut-off parameter to define the collinear region by cos θ > − δ θ . The Born cross section refersto the hard scale ˆ s , whereas in the collinear factor the total energy ˜ s is the scale needed.In order to avoid an overlap with the soft region, the upper limit of the z -integration inEq. (3.17) is reduced from z = 1 to z = 1 − δ s .As already mentioned, after adding virtual and real corrections, the mass singularity inEq. (3.17) does not cancel and has to be absorbed into the (anti-)quark density functions.This can be formally achieved by a redefinition of the parton density functions (PDFs) at– 8 –LO QED as follows [22, 25, 26], f a/A ( x ) → f a/A ( x, µ F ) + f a/A ( x, µ F ) απ e q κ P DFsoft + απ e q Z − δ s x dzz f a/A (cid:16) xz , µ F (cid:17) κ P DFcoll ( z )(3.18)with κ P DFsoft = − δ s + ln δ s − ln (cid:18) µ F m q (cid:19) (cid:20)
34 + ln δ s (cid:21) + 14 λ sc (cid:20) π δ s − δ s (cid:21) ,κ P DFcoll ( z ) = 12 P qq ( z ) " ln (cid:18) m q (1 − z ) µ F (cid:19) + 1 − λ sc (cid:20) P qq ( z ) ln 1 − zz −
32 11 − z + 2 z + 3 (cid:21) . The QED factorization scheme dependent λ sc -parameter is λ sc = 0 in the M S -scheme and λ sc = 1 in the DIS scheme.At the hadronic level, we define the collinear part of the real corrections for the casewhere parton a radiates off a collinear photon, in the following way by use of Eq. (3.18), dσ coll ( S ) = απ e q Z dτ Z dxx Z − δ s x dzz d ˆ σ q ¯ q (ˆ s ) h κ coll ( z ) + κ P DFcoll ( z ) i × (cid:20) f a/A (cid:16) xz , µ F (cid:17) f b/B (cid:16) τx , µ F (cid:17) + f b/A (cid:16) τx , µ F (cid:17) f a/B (cid:16) xz , µ F (cid:17) (cid:21) , (3.19)where the lower limit of the z -integration is constrained to x , since the parton momentumfraction x/z has to be smaller than unity. The integral is free of any mass singularity, κ coll ( z ) + κ P DFcoll ( z ) = 12 P qq ( z ) ln (cid:18) ˆ sz (1 − z ) µ F δ θ (cid:19) + 12 (1 − z ) − λ sc (cid:20) P qq ( z ) ln 1 − zz −
32 11 − z + 2 z + 3 (cid:21) . (3.20)The κ P DFsoft -term in Eq. (3.18) cancels the mass singularities owing to soft photons thatremain in the sum of the virtual corrections and the soft correction factor δ insoft in Eq. (3.13). We also consider the photon-induced mechanisms of the top-squark pair production. Atthe hadronic level, these processes vanish at leading order owing to the non-existence of aphoton distribution inside the proton. At NLO in QED, however, a non-zero photon densityarises in the proton as a direct consequence of including higher order QED effects into theevolution of PDFs, leading thus to non-zero photon-induced hadronic contributions.Feynman diagrams corresponding to the photon–gluon partonic process are illustratedin Fig. A.2. Although these are contributions of different order, they are tree-level contribu-tions to the same hadronic final state and thus deserve a closer inspection. The differential– 9 –ross section for this subprocess is d ˆ σ gγ (ˆ s ) = 116 π ˆ s X(cid:12)(cid:12) M gγ (ˆ s, ˆ t r , ˆ u r ) (cid:12)(cid:12) d ˆ t , X(cid:12)(cid:12) M gγ (cid:12)(cid:12) = 14 · · π αα s e t N C F " − sm t i ˆ t r ˆ u r − ˆ sm t i ˆ t r ˆ u r ! , (3.21)expressed in terms of the reduced Mandelstam variables ˆ t r = ˆ t − m t i , ˆ u r = ˆ u − m t i . Thequark–photon partonic processes represent contributions of higher order and we do notinclude them in our discussion here.The photon density is part of the PDFs at NLO QED, which have become availableonly recently [27]; here we present the first study of these effects on the top-squark pairproduction.
4. Numerical results
For the numerical discussion we focus on the production of light top-squark pairs ˜ t ∗ ˜ t inproton–proton collisions for LHC energies. We present the results in terms of the followinghadronic observables: the integrated cross section, σ , the differential cross section withrespect to the (photon inclusive) invariant mass of the top-squark pair, ( dσ/dM inv ), thedifferential cross sections with respect to the transverse momentum, ( dσ/dp T ), to therapidity, ( dσ/dy ), and to the pseudo-rapidity, ( dσ/dη ), of one of the final state top-squarks.For getting experimentally more realistic results for the cross sections we also apply typicalsets of kinematical cuts. A study of the dependence on the various SUSY parameters isgiven towards the end of this section.The NLO differential cross section at the hadron level is combined from the contribut-ing partonic cross sections by convolution and summation as follows, dσ pp ( S ) = Z τ dτ ( X i d L ppq i ¯ q i dτ d ˆ σ q i ¯ q i (ˆ s ) + d L ppgg dτ d ˆ σ gg (ˆ s ) + d L ppgγ dτ d ˆ σ gγ (ˆ s ) ) , (4.1)where d ˆ σ q i ¯ q i and d ˆ σ gg represent full one-loop results, including complete virtual and realcorrections, and d ˆ σ gγ is given in Eq. (3.21). The respective parton luminosities referto Eq. (2.11).One has to take care of the fact that each top-squark observed in the laboratorysystem under a certain angle θ can originate from two different constellations at partonlevel: parton a ( b ) out of hadron A ( B ) and vice-versa, corresponding to θ → ( π − θ ).Both parton level configurations have to be added correctly for hadronic distributions (forexplicit formulas see e. g. [28]). Note that the two boost factors β relating the two partoniccenter-of-mass (c. m. ) systems with the laboratory system differ by a relative sign, as dothe rapidity and the pseudo-rapidity of each particle.Assuming that the forward-scattered parton a carries the momentum fraction x ofhadron A and the backward-scattered parton b the momentum fraction τ /x of hadron B ,– 10 –he boost factor β is given by β = x − τ /xx + τ /x . (4.2)The rapidity of one of the final state top-squarks in the laboratory system, y ( ≡ y ˜ t ∗ ), isrelated to the rapidity in the partonic c. m. frame, y cm = artanh( p cmz /E cm ), via a Lorentztransformation, y = y cm − artanh( − β ) = y cm + 12 ln x τ . (4.3)The pseudo-rapidity η is related to η cm = − ln(tan θ cm /
2) in the c.m. frame via η = arsinh s m t p T + cosh η cm (cid:18) x √ τ − √ τx (cid:19) + 12 sinh η cm (cid:18) √ τx + x √ τ (cid:19) , (4.4)which can be derived using the representation p = (cid:16)q m t + p T cosh η, , p T , p T sinh η (cid:17) (4.5)for the top-squark momentum p ≡ p ˜ t ∗ . Since the final state particles are massive, rapidityand pseudo-rapidity do not coincide; in the limit m → η = y . Our Standard Model input parameters are chosen in correspondance with [29], M Z = 91 . , M W = 80 .
403 GeV ,α − = 137 . , α ( M Z ) − = 127 . , G F = 1 . × − GeV − , (4.6) m t = 172 . , m b = 4 . , m b ( m b ) = 4 . . All lepton and all other quark masses are set to zero unless where they are used for regu-larization. As a reference we consider the SPA SUSY parameter point SPS 1a’ [29], unlessstated otherwise. The current value of the top-quark mass, m t = 170 . ± . m ˜ t by 0 . ≈ µ F = µ R = 2 m ˜ t . In Table 1 we show results for the cross section for top-squark pair production at theLHC within four different scenarios, chosen out of the SPS benchmark scenarios of theminimal SUGRA type [29, 31]. The integrated hadronic cross sections at leading order, σ LO , the absolute size of the EW corrections corresponding to the difference between theLO and NLO cross sections, ∆ σ NLO , and the relative corrections, δ , given as the ratio ofNLO corrections to the respective LO contributions, are presented for the gg fusion, the– 11 –cenario channel σ LO [fb] ∆ σ NLO [fb] δ = ∆ σ NLO σ LO SPS 1a qq
222 (+0.985) − . − . m ˜ t = 376 . gg − . − . gγ total 1666 3.90 023 %SPS 1a’ qq
439 (+1.88) − . − . m ˜ t = 322 . gg − . − . gγ total 3731 32.3 0 . %SPS 2 qq . − . × − − . m ˜ t = 1005 . gg − . × − − . gγ . × − total 4.14 3 . × − . %SPS 5 qq − . m ˜ t = 203 . gg . gγ total 34860 891 2 . % Table 1:
Numerical results for the integrated cross sections for light top-squark pair productionat the LHC within different SPS scenarios [29, 31]. qq annihilation, and the gγ fusion channel separately. The gγ channel contributes onlyat NLO. For the qq channel, also the numbers for the O ( α ) pure electroweak Born levelcontributions are given in brackets. These are typically smaller by one order of magnitudecompared to the EW NLO corrections.In scenarios where the top-squark ˜ t is of intermediate or high mass (as SPS 1a, SPS 1a’,and SPS 2) the NLO contributions are below 1%. The corrections to the qq and the gg channels are negative, whereas the gγ contribution is always positive and of the same sizeas the other corrections or even larger. The situation is different in scenarios where thetop-squark is very light, i.e. lighter than half of m H , the mass of the heavier neutral Higgsboson H , where a large fraction of the squarks appears through production and decay of H particles. This is the case in the SPS 5 scenario [ m ˜ t = 204 GeV, m H = 694 GeV andΓ( H ) = 9 . FeynHiggs [19]]. The electroweak contributions in the gg channel are positive and slightly larger than the gγ fusion contribution.The interplay of the three production channels is illustrated in Fig. 1 where the absoluteEW contributions ∆ σ per channel are shown as distributions with respect to p T , M inv , y , or η . Owing to the alternating signs, compensations occur where in particular the gγ channelplays an important role.For realistic experimental analyses, cuts on the kinematically allowed phase space of– 12 – [GeV] T p [f b / G e V ] T / dp sD d -0.100.10.20.3 gg fusion channelsqq fusion g gall SPS1a’, LHC [GeV] inv M
500 1000 1500 2000 [f b / G e V ] i n v / d M sD d -0.100.10.20.3 gg fusion channelsqq fusion g gall SPS1a’, LHC y -3 -2 -1 0 1 2 3 / d y [f b ] sD d -505101520 gg fusion channelsqq fusion g gall SPS1a’, LHC h -4 -2 0 2 4 [f b ] h / d sD d -5051015 gg fusion channelsqq fusion g gall SPS1a’, LHC
Figure 1:
Comparison of EW NLO contributions from the various parton channels, for the dis-tributions of transverse momentum p T (˜ t ∗ ), invariant mass of the stop pair, rapidity y (˜ t ∗ ), andpseudo-rapidity η (˜ t ∗ ) (from upper left to lower right). y and η are given in the laboratory frame.For gg fusion and qq annihilation, ∆ denotes the difference between NLO and LO distributions(∆ σ ≡ ∆ σ NLO ), for gγ one has ∆ σ ≡ σ gγ . the top-squarks have to be applied. They can be realized by a lower cut on the transversemomenta of the final-state particles to focus on high- p T jets. Moreover, detectability ofthe final state particles requires a minimal angle between the particles and the beam axis.Therefore, we set a cut on the pseudo-rapidity of the top-squarks restricting the scatteringangle θ to a central region. Two exemplary sets of cuts are applied in the following figures(Figs. 2 – 5), cuts 1: p T ≥
150 GeV and | η | ≤ . . ◦ ≤ θ ≤ . ◦ ) , cuts 2: p T ≥
250 GeV and | η | ≤ . . The differential cross sections and the influence of cuts are the content of Figs. 2 and 3.Displayed are the hadronic cross sections at NLO, differential with respect to p T , M inv andto y , η , respectively. Both the full (unconstrained) distributions and the distributionswith cuts applied are shown. The reduction of the integrated cross section owing to theapplication of cuts is summarized in Table 2.– 13 –
500 1000 1500 [f b / G e V ] T / dp s d no cuts cuts 1 cuts 2 gg fusion [f b / G e V ] T / dp s d no cuts cuts 1 cuts 2 channelsqq [GeV] T p [f b / G e V ] T / dp s d no cuts cuts 1 cuts 2 = 14 TeV (pp)SSPS 1a’, fusion g g
500 1000 1500 2000 2500 [f b / G e V ] i n v / d M s d no cuts cuts 1 cuts 2 gg fusion
500 1000 1500 2000 2500 [f b / G e V ] i n v / d M s d no cuts cuts 1 cuts 2 channelsqq [GeV] inv M
500 1000 1500 2000 2500 [f b / G e V ] i n v / d M s d no cuts cuts 1 cuts 2 = 14 TeV (pp)SSPS 1a’, fusion g g Figure 2:
Comparison of EW NLO differential hadronic cross sections (solid lines) and the dis-tributions where kinematical cuts on the final top-squarks are applied for all three productionchannels, gg fusion (upper red plots), qq channels (middle blue plots), and gγ fusion (lower greenplots). Cuts 1 (dashed lines): p T ≥
150 GeV , | η | ≤ . , cuts 2 (dotted lines): p T ≥
250 GeV , | η | ≤ . . Distributions with respect to the transverse momentum p T (˜ t ) (left) and the invariantmass of the stop pair (right) are shown for ˜ t ∗ ˜ t pair production at the LHC within the SPS 1a’scenario. The application of cuts reduces the gg and gγ channels strongly, cutting off the peakof the p T -distributions. The reduction is less pronounced in the qq channels where the p T -distribution is harder. The p T -cuts also shift the threshold of the invariant mass distri-butions towards higher values affecting again mainly the gg and gγ channels in height andshape. The situation for the rapidity distribution is similar. In the qq channel, the harder p T -distribution goes along with a narrower η -distribution, as shown in the right panels ofFig. 3. Most of the top-squarks produced via qq annihilation can be found in the centralregion. In contrast, top-squarks from gg or gγ fusion are often produced in the strong– 14 – / d y [f b ] s d no cuts cuts 1 cuts 2 gg fusion -3 -2 -1 0 1 2 3 / d y [f b ] s d no cuts cuts 1 cuts 2 channelsqq y -3 -2 -1 0 1 2 3 / d y [f b ] s d no cuts cuts 1 cuts 2 = 14 TeV (pp)SSPS 1a’, fusion g g -4 -2 0 2 4 [f b ] h / d s d no cuts cuts 1 cuts 2 gg fusion -4 -2 0 2 4 [f b ] h / d s d no cuts cuts 1 cuts 2 channelsqq h -4 -2 0 2 4 [f b ] h / d s d no cuts cuts 1 cuts 2 = 14 TeV (pp)SSPS 1a’, fusion g g Figure 3:
Same as Fig. 2, but with respect to the rapidity y (˜ t ∗ ) (left) and the pseudo-rapidity η (˜ t ∗ ) (right). forward (or backward) direction, and the application of cuts on the pseudo-rapidity thusreduces the number of gg or gγ based events significantly.In order to illustrate the numerical impact of the NLO contributions on the LO crosssection, we show in Fig. 4 K factors K = σ NLO /σ LO for the gg and the qq channel,respectively, as distributions with respect to p T and M inv . The application of cuts influencesthe K factors only at low values of p T and M inv . The EW corrections in the p T -distributionreach typically −
10% in the gg channel, and −
20% in the qq channel, for large values of p T . In the invariant mass distributions, they are somewhat smaller, but still sizeable, atthe 10% level for large M inv . The large effects at high p T and M inv are dominated by thedouble logarithmic contributions arising from virtual W and Z bosons in loop diagrams.The small peaks visible in the gg invariant mass distribution correspond to two-particlethresholds related to ˜ b ∗ ˜ b , ˜ b ∗ ˜ b , and ˜ t ∗ ˜ t pairs in gg vertex and box diagrams, illustrated– 15 –hannel full result p T <
150 GeV p T <
250 GeV(SPS 1a’) at NLO [fb] & | η | < . | η | < . gg − − qq
427 373 ( − − gγ − − Table 2:
Integrated hadronic cross section at NLO within the SPS 1a’ scenario for the differentproduction channels. Comparison of the full (unconstrained) results and cross sections where cutson the pseudo-rapidities η and on the transverse momenta p T of the outgoing top-squarks areapplied. The relative changes compared to the full results are given in brackets. in Fig. A.4 [in the SPS 1a’ scenario, the masses of the involved squarks are m ˜ b = 460 . m ˜ b = 514 . m ˜ t = 569 . p T -distribution, around300 GeV, but they are smeared out and much less pronounced.Fig. 5 shows total K factors, defined as K = ( σ NLOgg + σ NLOqq + σ LOgγ ) / ( σ LOgg + σ LOqq ). It isobvious that, although small for the total cross section, the EW higher order contributionscannot be neglected for differential distributions where, in the high- p T and high- M inv range,they are of the same order of magnitude as the SUSY-QCD corrections [10]. K f ac t o r = 14 TeV (pp)SSPS 1a’, gg fusion [GeV] T p K f ac t o r = 14 TeV (pp)SSPS 1a’, no cuts cuts 1 cuts 2 channelsqq K f ac t o r = 14 TeV (pp)SSPS 1a’, gg fusion [GeV] inv M K f ac t o r = 14 TeV (pp)SSPS 1a’, channelsqq Figure 4:
Same as Fig. 2, but shown are the K factors, K = σ NLO /σ LO , for gg fusion (upperplots) and qq channels (lower plots). – 16 – [GeV] T p K f ac t o r = 14 TeV (pp)SSPS 1a’, g + gqgg + q [GeV] inv M K f ac t o r = 14 TeV (pp)SSPS 1a’, g + gqgg + q Figure 5:
Same as Fig. 2, but shown is the total K factor, K = ( σ NLOgg + σ NLOqq + σ LOgγ ) / ( σ LOgg + σ LOqq ). In order to study the dependence of the EW contributions on the various SUSY parametersin more detail, we consider the ratio of the NLO contribution in each channel to the com-bined gg + qq Born cross section, δ tot = ∆ σ NLO { gg, qq, gγ } /σ LOtot . We focus on those parametersthat determine the top-squark mass, cf. Eq. (2.4), and vary each quantity out of the set m ˜ Q , m ˜ U , tan β , A t , or µ around its SPS 1a’ value while keeping all other parametersfixed to those of the default SPS 1a’ scenario. The results are displayed in the left panelsof Figs. 6 – 10. Simultaneously, we show the mass of the light top-squark ˜ t as a functionof the varied parameter in the respective right panels (black solid lines). The parameterconfiguration of the SPS 1a’ scenario is marked by a vertical gray dotted line in all thefigures.We find the following general behaviors. The gγ contributions are from tree level dia-grams and the only relevant parameter is thus the top-squark mass m ˜ t . In all scenarios,the gγ fusion channel is as important as the EW corrections to the qq and gg processes. The qq corrections, being practically always negative, involve many different SUSY particles inthe loops, although the relative corrections show only small variations. The gg contribu-tions are more sensitive to the considered SUSY parameters. The plots show striking peaks(some of them are also visible in qq annihilation), which correspond to threshold effects andcan be explained by the SUSY particle masses displayed at the right panels of Figs. 6 – 10.They occur in the Higgs-exchange diagrams when m ˜ t = m H / m ˜ t equals the sumof masses of a neutralino and the top-quark (green dash-dotted lines) or of a chargino andthe bottom-quark (blue dashed lines). The chargino-induced peaks are less pronouncedthan those from neutralinos and not visible in Fig. 7 and Fig. 10.Outside of such singular parameter configurations, over a wide range of SUSY pa-rameters, the combined EW contributions to top-squark pair production are only weaklyparameter dependent. – 17 – [GeV] Q~m(
200 400 600 800 1000 [ % ] t o t d -2-1012 gg fusion channelsqq fusion g g = 14 TeV (pp)S ) Q~SPS 1a’, varying m( ) [GeV] Q~m(
200 400 600 800 1000 s p a r t i c l e m ass [ G e V ] t~ m( )/2 m(H t ) + m c~ m( t ) + m c~ m( t ) + m c~ m( b ) + m – c~ m( ) Q~SPS 1a’, varying m(
Figure 6:
Left: Relative EW corrections as a function of the soft-breaking parameter m ˜ Q foreach of the indicated channels compared to the combined ( gg + qq ) LO cross section in the SPS 1a’scenario where m ˜ Q is varied around the SPS1a’ value (gray dotted line). Right: Mass of ˜ t , halfof the mass of H , sums of the masses of the top-quark and ˜ χ , ˜ χ , and ˜ χ , respectively, and sumof the masses of the bottom-quark and ˜ χ ± as a function of m ˜ Q . All other parameters are chosenaccording to the SPS1a’ scenario. ) [GeV] U~m(
200 400 600 800 1000 [ % ] t o t d -2-1012 gg fusion channelsqq fusion g g = 14 TeV (pp)S ) U~SPS 1a’, varying m( ) U~m(
200 400 600 800 1000 s p a r t i c l e m ass [ G e V ] t~ m( )/2 m(H t ) + m c~ m( t ) + m c~ m( t ) + m c~ m( b ) + m – c~ m( ) U~SPS 1a’, varying m(
Figure 7:
Same as Fig. 6, but for variation of the soft-breaking parameter m ˜ U . – 18 – tan [ % ] t o t d -2-1012 gg fusion channelsqq fusion g g = 14 TeV (pp)S b SPS1a’, varying tan b tan s p a r t i c l e m ass [ G e V ] t~ m( )/2 m(H t ) + m c~ m( t ) + m c~ m( t ) + m c~ m( b ) + m – c~ m( b SPS1a’, varying tan
Figure 8:
Same as Fig. 6, but for variation of tan β . [GeV] t A -1000 -500 0 500 1000 [ % ] t o t d -8-6-4-202 gg fusion channelsqq fusion g g = 14 TeV (pp)S t SPS 1a’, varying A [GeV] t A -1000 -500 0 500 1000 s p a r t i c l e m ass [ G e V ] t~ m( )/2 m(H t ) + m c~ m( t ) + m c~ m( t ) + m c~ m( b ) + m – c~ m( t SPS 1a’, varying A
Figure 9:
Same as Fig. 7, but for variation of trilinear coupling parameter A t . [GeV] m [ % ] t o t d -8-6-4-202 gg fusion channelsqq fusion g g = 14 TeV (pp)S m SPS 1a’, varying [GeV] m s p a r t i c l e m ass [ G e V ] t~ m( )/2 m(H t ) + m c~ m( t ) + m c~ m( t ) + m c~ m( b ) + m – c~ m( m SPS 1a’, varying
Figure 10:
Same as Fig. 6, but for variation of the Higgs parameter µ . – 19 – . Conclusions We have completed the NLO calculation for the ˜ t ˜ t ∗ production at hadron colliders byproviding the complete EW corrections at the one-loop level.To obtain a consistent and IR-finite result, we have considered the interference termsbetween QCD and EW NLO terms for both virtual and real contributions. Also, a newclass of photon-induced partonic processes of ˜ t ˜ t ∗ production occurs, which was found toyield considerable contributions, comparable in size to the corrections to qq annihilationand gg fusion or even larger.In total, the NLO EW contributions reach in size the 10-20% level in the p T andinvariant-mass distributions and are thus significant. Outside singular parameter config-urations associated with thresholds, the dependence on the MSSM parameters is rathersmooth.Recently, a preprint appeared on the same topic [32], where the authors consider virtualcorrections and the soft part of the real corrections, both for the gg fusion channel; thehard part of the real corrections, as well as the contributions from the other channels aremissing. The numerical results can therefore not directly be compared with ours at thisstage. Acknowledgments
The authors want to thank Tilman Plehn for helpful discussions and Edoardo Mirabellafor cross-checking parts of the results.
AppendixA. Feynman diagrams
We show here generic Feynman Diagrams for the pair production of lighter top-squark at O ( αα s ). Diagrams for ˜ t ∗ ˜ t production can be constructed in complete analogy. The q ¯ q annihilation channels are exemplified by u ¯ u annihilation. Furthermore, the label S refersto all neutral Higgs (and Goldstone) bosons h , H , A , G , and the label S to all chargedHiggs (and Goldstone) bosons H ± , G ± . – 20 – ug ug gg~~ ~ ~~~ ~~~g gtt t ttt tttg
11 1 111 111
Figure A.1:
Feynman diagrams for top-squark pair production at the Born level via gg fusion(left) and qq annihilation (right), here shown for u -quarks. As in the following figures, diagramswith crossed final states are not shown explicitely. g g g g ~~ ~ ~~ ~~ ~tt t tt tt tg g
11 1 11 11 1
Figure A.2:
Feynman diagrams for gluon-photon fusion.(a)(b)(c) (d)
Figure A.3:
Feynman diagrams for virtual corrections to top-squark pair production via q ¯ q an-nihilation (here for u -quarks). The label S refers to all neutral Higgs bosons h , H , A , G , thelabel S to all charged Higgs bosons H ± , G ± . (a) counter-term diagrams, (b) vertex corrections,(c) IR singular box diagrams, (d) IR finite box diagram. – 21 – ~~ ~~ ~~ ~ ~~ ~ ~~ ~g g g g gg gg ggtt tt tt t tt t tt t
11 11 11 1 11 1 11 1 (a) ~ ~~ ~~~~~~~~ ~~g gg g h,h,h, S /S / HHH SS g t bt b q qq qqqqqqt b cc t b t bt b cc t b ~~ ~~ ~~ ~~ ~~~ ~~ ~~ ~ ~ ~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~~~~~~~ ~~~~~~~~ ~~~ ~~~~ ~~ ~ ~ ~ ~ ~ ~~ ~ ~~ ~ si sisi sisisi sisisiii0 ii0 g g gg g / / / Z/ W / Z/ W / Z/ W / Z/ W / Z/ W// /// // // // //// /// / ////// tt tt tt tt ttt bt bt b t t bt t bt t b t t b ttttttt tttttttt ttt tttt tt t t b t t bt t bt b
11 11 11 11 11s ss ss s 1 s s1 s s1 s s 1 s s 1111111 11111111 111 1111 11 1 s s 1 s s1 s ss s gg ggg g g g g ggg gg gg gg ggg gg gggg g (b) ~~S , S ,S, S t b cc ~ ~~~ ~~ ~~~~ ~~ ~~~ ~ ~~~~~ ~ ~ ~~~~ ii0 g / g , Z, W / Z/ W, / ///q ltt tt tttt tt ttt b ttttt t b tttt si si11 11 1111 11 11s s 11111 s s 1111 gggg gg gggg (c) ~~ ~~ ~~~g gg g~~S / S /S S tt ~~gg cc t b qq qqt b qt b ~~ ~~~~~~ ~~ ~~ ~~ ~~ ~ ~ ~ ~ ~~~~ ~ ~~ ~ ~~ ~~ ~~ ~ ~~ ~ ~~ ~ ~ ~ ~ ~~ ~ ~~ ~ ~ ~ ~~ ~ ~ tt ii0 sisi sisisi11 g gg g gg / / Z/ W / Z/ W / Z/ W / Z/ W / Z/ Wg gg/// // // //////// // // //// ///// tt tttttt tt tt tt bt b t t b tttt t bt t bt bt bt t bt t bt t b t t bt t bt b t t bt t b
11 111111 11 11 1s ss s 1 s s 1111 s s1 s ss ss s1 s s1 s s 1 s s 1 s s1 s ss s 1 s s1 s s ggg ggg gggg (d)
Figure A.4:
Feynman diagrams for virtual corrections to top-squark pair production via gg fusion,diagrams with crossed final states are not explicitely shown. The label S refers to all neutral Higgsbosons h , H , A , G , the label S to all charged Higgs bosons H ± , G ± . (a) counter-term diagrams,(b) vertex corrections, (c) self-energy corrections, (d) box diagrams. – 22 – t~t ~t ~t~t~t ~t~t~t
11 1 111 111 gg g g g g g gg g (a) g g ~t~t ~t~t~t~t ~t~t~t
11 1111 111 g g gg g gg g (b) ~ ~~ ~~t tt tt u uu uu g g g g ~~~tt tu u g g
11 1 (c) (d)
Figure A.5:
Feynman diagrams for real photon radiation. (a) IR divergent – (b) IR finite contribu-tions for the gg channel; (c) IR divergent – (d) IR finite contributions for the q ¯ q channels. Feynmandiagrams with photon radiation off the other quark or squark and with crossed final states are notshown explicitely. ~~ ~ ~~ ~ ~~ ~~tt t tt t tt tt/ Z / Z g g
11 1 11 1 11 11 u u u uuu u u uugg g g g g
Figure A.6:
Feynman diagrams for gluon bremsstrahlung from the QCD and EW Born diagrams(radiation from upper legs is not explicitly shown). Only interference terms between initial andfinal state gluon radiation are non-vanishing. uu u uut u / Zuu g u gg ~~~ ~ ~~~ ~~~ ~~ ~~~ uuu t ttt ttg gg ttt sss 1 111 11111 Figure A.7:
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