The Effective-Vector-Boson Approximation at the LHC

Transcription

1 The Effective-ector-Boson Approximation at the LHC Ramon interhalder INSTITUTE FOR THEORETICAL PHYSICS UNIERSITY OF HEIDELBERG Higgs Couplings 8th November th November 2017

2 Content 1 Introduction 2 ector-boson Scattering at the LHC 3 The Effective-ector-Boson Approximation 4 Logarithmic Electroweak Corrections 5 Conclusion 8th November 2017 Ramon interhalder 1 / 24

3 Content 1 Introduction 2 ector-boson Scattering at the LHC 3 The Effective-ector-Boson Approximation 4 Logarithmic Electroweak Corrections 5 Conclusion 8th November 2017 Ramon interhalder 2 / 24

4 Longitudinal -Boson Scattering The gauge-bosons ± and Z obtain their mass and longitudinal polarisation state through ESB. Z/γ + Z/γ + A( L L L L ) s t unitarity is violated! Mechanism of ESB must restore unitarity and regulate σ( L L L L ): 8th November 2017 Ramon interhalder 3 / 24

5 Longitudinal -Boson Scattering The gauge-bosons ± and Z obtain their mass and longitudinal polarisation state through ESB. Z/γ + Z/γ + + H + H s 2 A( L L L L ) s t + s MH 2 t 2 + t MH 2 unitarity is violated! Mechanism of ESB must restore unitarity and regulate σ( L L L L ): a light SM Higgs exactly cancels terms leading in energy 8th November 2017 Ramon interhalder 3 / 24

6 Unitarity iolation If any or all couplings are slightly modified the cancellation fails and unitarity is violated. Alboteanu, Kilian, Reuter 08 [arxiv: ] In order to restore unitarity new interactions or particles need to be introduced. Any deviations from the SM represent signals of new physics. 8th November 2017 Ramon interhalder 4 / 24

7 Unitarity iolation If any or all couplings are slightly modified the cancellation fails and unitarity is violated. Alboteanu, Kilian, Reuter 08 [arxiv: ] In order to restore unitarity new interactions or particles need to be introduced. Any deviations from the SM represent signals of new physics. BS is a key tool in probing the machanism of ESB. 8th November 2017 Ramon interhalder 4 / 24

8 Content 1 Introduction 2 ector-boson Scattering at the LHC 3 The Effective-ector-Boson Approximation 4 Logarithmic Electroweak Corrections 5 Conclusion 8th November 2017 Ramon interhalder 5 / 24

9 BS at Hadron Colliders BS is only a non-gauge-invariant subprocess in the production of jj final states. q q H H q q There are two main types of production modes at LO: E jj production: O(α 6 ) QCD jj production: O(α 4 α 2 s) 8th November 2017 Ramon interhalder 6 / 24

10 BS at Hadron Colliders BS is only a non-gauge-invariant subprocess in the production of jj final states. q q H H q q There are two main types of production modes at LO: E jj production: O(α 6 ) QCD jj production: O(α 4 α 2 s) and interferences: O(α 5 α s) (colour and kinematically suppressed) interference of both modes is negligible 8th November 2017 Ramon interhalder 6 / 24

11 Production Modes E jj production: O(α 6 ) BS diagrams q q non-bs E diagrams q q QCD jj production: O(α 4 α 2 s) gauge-invariantly separable: can be suppressed by cuts th November 2017 Ramon interhalder 7 / 24

12 Same-Sign BS est, Anger 15 final state -channel σ E [fb] σ QCD [fb] σ E /σ QCD l ± l ± νν jj ± ± : 1 l + l νν jj ±, ZZ : 35 l + l l ± ν jj ± Z : 20 l + l l + l jj ZZ : 70 most promising jj channel in terms of BS: same-sign ± ± jj channel 8th November 2017 Ramon interhalder 8 / 24

13 Same-Sign BS est, Anger 15 final state -channel σ E [fb] σ QCD [fb] σ E /σ QCD l ± l ± νν jj ± ± : 1 l + l νν jj ±, ZZ : 35 l + l l ± ν jj ± Z : 20 l + l l + l jj ZZ : 70 most promising jj channel in terms of BS: same-sign ± ± jj channel no gg and gq induced contributions at tree-level 8th November 2017 Ramon interhalder 8 / 24

14 Same-Sign BS est, Anger 15 final state -channel σ E [fb] σ QCD [fb] σ E /σ QCD l ± l ± νν jj ± ± : 1 l + l νν jj ±, ZZ : 35 l + l l ± ν jj ± Z : 20 l + l l + l jj ZZ : 70 most promising jj channel in terms of BS: same-sign ± ± jj channel no gg and gq induced contributions at tree-level l ± l ± ν νjj final state is most sensitive to BS measurements 8th November 2017 Ramon interhalder 8 / 24

15 Same-Sign BS est, Anger 15 final state -channel σ E [fb] σ QCD [fb] σ E /σ QCD l ± l ± νν jj ± ± : 1 l + l νν jj ±, ZZ : 35 l + l l ± ν jj ± Z : 20 l + l l + l jj ZZ : 70 most promising jj channel in terms of BS: same-sign ± ± jj channel no gg and gq induced contributions at tree-level l ± l ± ν νjj final state is most sensitive to BS measurements However, BS diagrams cannot gauge-invariantly separated from non-bs E diagrams! 8th November 2017 Ramon interhalder 8 / 24

16 Same-Sign BS est, Anger 15 final state -channel σ E [fb] σ QCD [fb] σ E /σ QCD l ± l ± νν jj ± ± : 1 l + l νν jj ±, ZZ : 35 l + l l ± ν jj ± Z : 20 l + l l + l jj ZZ : 70 most promising jj channel in terms of BS: same-sign ± ± jj channel no gg and gq induced contributions at tree-level l ± l ± ν νjj final state is most sensitive to BS measurements However, BS diagrams cannot gauge-invariantly separated from non-bs E diagrams! Are there kinematic regions in which the genuine BS process is dominant? 8th November 2017 Ramon interhalder 8 / 24

17 Content 1 Introduction 2 ector-boson Scattering at the LHC 3 The Effective-ector-Boson Approximation 4 Logarithmic Electroweak Corrections 5 Conclusion 8th November 2017 Ramon interhalder 9 / 24

18 The Effective-ector-Boson Approximation In the EBA the -Boson is treated as constituent of the quark and assumed to be on-shell. Driven by logarithmic enhancements log 2 (M 2 /s) originating from collinear vector-boson emission off the quarks, in analogy to the eizsäcker-illiams approximation of QED 1. q 1 2 X q 1 eizsäcker 34, illiams 34 2 Kuss, Spiesberger 96 8th November 2017 Ramon interhalder 10 / 24

19 The Effective-ector-Boson Approximation In the EBA the -Boson is treated as constituent of the quark and assumed to be on-shell. Driven by logarithmic enhancements log 2 (M 2 /s) originating from collinear vector-boson emission off the quarks, in analogy to the eizsäcker-illiams approximation of QED 1. q 1 2 X q Factorization into probability densities P 1 q(z 1 ) and P 2 q (z 2) and a hard scattering process σ 1 2 X(xs) at reduced CM energy xs, with x = z 1 z 2. σ qq X(s) = 1, dz 1 dx P 1 q(z 1 )P 2 q x min z min z (x/z 1) σ 1 2 X(xs) 1 1 eizsäcker 34, illiams 34 2 Kuss, Spiesberger 96 8th November 2017 Ramon interhalder 10 / 24

20 The Effective-ector-Boson Approximation In the EBA the -Boson is treated as constituent of the quark and assumed to be on-shell. Driven by logarithmic enhancements log 2 (M 2 /s) originating from collinear vector-boson emission off the quarks, in analogy to the eizsäcker-illiams approximation of QED 1. q 1 2 X q Factorization into probability densities P 1 q(z 1 ) and P 2 q (z 2) and a hard scattering process σ 1 2 X(xs) at reduced CM energy xs, with x = z 1 z 2. σ qq X(s) = 1, dz 1 dx P 1 q(z 1 )P 2 q x min z min z (x/z 1) σ 1 2 X(xs) 1 However, naive convolution of single-vector-boson probabilities is not adequate to describe the two-vector-boson process sufficiently 2. 1 eizsäcker 34, illiams 34 2 Kuss, Spiesberger 96 8th November 2017 Ramon interhalder 10 / 24

21 Factorisation of the Amplitude q 1 q 1 µ p 1 p 1 µ = ν p 2 p 2 ν q 2 q 2 J µ 1 µ µ ν ν M µ ν 4l J ν 2 The full amplitude reads M qq = λ 1,λ 2,λ 3,λ 4 ( 1)λ1+λ2 K 1 K 2 K 3 K 4 M prod where we have introduced the abbreviations λ M prod 1 λ 2 M BS λ 1 λ 2 λ 3 λ 4 M decay λ 3 M decay λ, 4 im prod λ = J µ i i ε λ i,µ, K i = ki 2 M 2, i = 1, 2, im decay λ f = J µ f ε λ f,µ, K f = k 2 f M 2 + im Γ, f = 3, 4. 8th November 2017 Ramon interhalder 11 / 24

22 On-Shell Projection In order to guarantee gauge invariance of the amplitude we need to consider on-shell -boson scattering. M qq = λ 1,λ 2,λ 3,λ 4 ( 1)λ1+λ2 K 1 K 2 K 3 K 4 M prod λ M prod 1 λ 2 } {{ } off-shell BS decay M λ 1 λ 2 λ 3 λ 4 M λ 3 Mdecay λ 4 } {{ } on-shell here the on-shell momenta are defined as follows BS : Decay : p µ 3/4 = p µ 3/4 kµ 12 = s 2 (1, 0, 0, ±β), kµ 3/4 = s 2 (1, ±β sin θ, 0, ±β cos θ), M 2 2 k 3/4 p 3/4, p µ 3/4 = k µ 3/4 p µ 3/4, with s = 4 M 2 t u, β = 1 4M 2 / s, cos θ = 1 + 2t/ sβ 2. 8th November 2017 Ramon interhalder 12 / 24

23 On-Shell Projection In order to guarantee gauge invariance of the amplitude we need to consider on-shell -boson scattering. M qq = λ 1,λ 2,λ 3,λ 4 ( 1)λ1+λ2 K 1 K 2 K 3 K 4 M prod λ M prod 1 λ 2 } {{ } off-shell BS decay M λ 1 λ 2 λ 3 λ 4 M λ 3 Mdecay λ 4 } {{ } on-shell here the on-shell momenta are defined as follows BS : Decay : p µ 3/4 = p µ 3/4 kµ 12 = s 2 (1, 0, 0, ±β), kµ 3/4 = s 2 (1, ±β sin θ, 0, ±β cos θ), M 2 2 k 3/4 p 3/4, p µ 3/4 = k µ 3/4 p µ 3/4, with s = 4 M 2 t u, β = 1 4M 2 / s, cos θ = 1 + 2t/ sβ 2. hy do we keep t and u fixed? 8th November 2017 Ramon interhalder 12 / 24

24 On-Shell Projection In order to guarantee gauge invariance of the amplitude we need to consider on-shell -boson scattering. M qq = λ 1,λ 2,λ 3,λ 4 ( 1)λ1+λ2 K 1 K 2 K 3 K 4 M prod λ M prod 1 λ 2 } {{ } off-shell BS decay M λ 1 λ 2 λ 3 λ 4 M λ 3 Mdecay λ 4 } {{ } on-shell here the on-shell momenta are defined as follows BS : Decay : p µ 3/4 = p µ 3/4 kµ 12 = s 2 (1, 0, 0, ±β), kµ 3/4 = s 2 (1, ±β sin θ, 0, ±β cos θ), M 2 2 k 3/4 p 3/4, p µ 3/4 = k µ 3/4 p µ 3/4, with s = 4 M 2 t u, β = 1 4M 2 / s, cos θ = 1 + 2t/ sβ 2. hy do we keep t and u fixed? Appearance of a photon pole! 8th November 2017 Ramon interhalder 12 / 24

25 The Photon Pole + k 1 k k 2 γ k 4 + M γ,t 1 t M γ,u 1 ũ t = sβ2 θ 0 (1 cos θ) 0, 2 ũ = sβ2 θ π (1 + cos θ) 0 2 Physically a consequence of the of the infinite range of the Coulomb potential. Appears also in Møller-, Bhabha-, and in Rutherford scattering. either cut on scattering angle θ or fix invariants t and u in the on-shell projection. 8th November 2017 Ramon interhalder 13 / 24

26 The Photon Pole + k 1 k k 2 γ k 4 + M γ,t 1 t M γ,u 1 ũ t = sβ2 θ 0 (1 cos θ) 0, 2 ũ = sβ2 θ π (1 + cos θ) 0 2 Physically a consequence of the of the infinite range of the Coulomb potential. Appears also in Møller-, Bhabha-, and in Rutherford scattering. either cut on scattering angle θ or fix invariants t and u in the on-shell projection. Fix t and u invariants to avoid photon pole: t = t, ũ = u. 8th November 2017 Ramon interhalder 13 / 24

27 The Photon Pole + k 1 k k 2 γ k 4 + M γ,t 1 t M γ,u 1 ũ t = sβ2 θ 0 (1 cos θ) 0, 2 ũ = sβ2 θ π (1 + cos θ) 0 2 Physically a consequence of the of the infinite range of the Coulomb potential. Appears also in Møller-, Bhabha-, and in Rutherford scattering. either cut on scattering angle θ or fix invariants t and u in the on-shell projection. Fix t and u invariants to avoid photon pole: t = t, ũ = u. Note: Photon pole lies outside the physical phase space of the full 2 6 process! 8th November 2017 Ramon interhalder 13 / 24

28 Numerical Results - Phase Space Cuts e first use standard BS cuts 3 : Jet recombination: D = 0.7, k T algorithm. Cuts on jets: p T,j > 20 Ge, y j < 4.5, M jj > 600 Ge, y j1 y j2 < 0, y jj > 4. Cuts on leptons: p T,l > 20 Ge, y l < 2.5, R jl > 0.4, y jmin < y l < y jmax, R ll > Denner, Hosekova, Kallweit 12 [arxiv: ] 8th November 2017 Ramon interhalder 14 / 24

29 Numerical Results - EBA I pp νee + νµµ + jj (@14 Te) pp νee + νµµ + jj (@14 Te) dσ [fb/ge] dpt,lmax EBA LO dσ [fb/ge] dpt,miss EBA LO δ [%] δ EBA [Ge] pt,lmax δ [%] δ EBA pt,miss [Ge] s [Te] σlo [fb] σ EBA [fb] δ EBA (2) (2) 34.0 % (3) (5) 43.6 % 8th November 2017 Ramon interhalder 15 / 24

30 Numerical Results - EBA I pp νee + νµµ + jj (@100 Te) pp νee + νµµ + jj (@100 Te) [fb/ge] dσ dpt,lmax 10 2 EBA dσ [fb/ge] dpt,miss 10 2 EBA 10 3 LO 10 3 LO δ [%] δ EBA [Ge] pt,lmax δ [%] δ EBA pt,miss [Ge] s [Te] σlo [fb] σ EBA [fb] δ EBA (2) (2) 34.0 % (3) (5) 43.6 % 8th November 2017 Ramon interhalder 15 / 24

31 Phase Space Cuts - Reloaded e first use standard BS cuts: Jet recombination: D = 0.7, k T algorithm. Cuts on jets: p T,j > 20 Ge, y j < 4.5, M jj > 600 Ge, y j1 y j2 < 0, y jj > 4. Cuts on leptons: p T,l > 20 Ge, y l < 2.5, R jl > 0.4, y jmin < y l < y jmax, R ll > th November 2017 Ramon interhalder 16 / 24

32 Numerical Results - Upper p T,j Cut e now require p T,j < 150 Ge. s [Te] σlo [fb] σ EBA [fb] δ EBA (1) 0.817(2) 41.1 % (2) 6.38(3) 32.6 % 8th November 2017 Ramon interhalder 17 / 24

33 Numerical Results - Upper p T,j Cut e now require p T,j < 150 Ge. pp νee + νµµ + jj (@14 Te) pp νee + νµµ + jj (@14 Te) dσ [fb/ge] dpt,lmax LO EBA dσ [fb/ge] dpt,miss LO EBA δ [%] 40 0 δ EBA δ [%] δ EBA [Ge] pt,lmax pt,miss [Ge] s [Te] σlo [fb] σ EBA [fb] δ EBA (1) 0.817(2) 41.1 % (2) 6.38(3) 32.6 % 8th November 2017 Ramon interhalder 17 / 24

34 Numerical Results - Upper p T,j Cut e now require p T,j < 150 Ge. pp νee + νµµ + jj (@100 Te) pp νee + νµµ + jj (@100 Te) dσ [fb/ge] dpt,lmax LO EBA dσ [fb/ge] dpt,miss LO EBA δ [%] δ EBA pt,lmax [Ge] δ [%] δ EBA pt,miss [Ge] s [Te] σlo [fb] σ EBA [fb] δ EBA (1) 0.817(2) 41.1 % (2) 6.38(3) 32.6 % 8th November 2017 Ramon interhalder 17 / 24

35 Numerical Results - Upper p T,j Cut e now require p T,j < 100 Ge and p T,l > 100 Ge. pp νee + νµµ + jj (@14 Te) pp νee + νµµ + jj (@14 Te) [fb/ge] dσ dpt,lmax LO EBA dσ [fb/ge] dpt,miss LO EBA δ [%] δ EBA [Ge] pt,lmax δ [%] δ EBA pt,miss [Ge] s [Te] σlo [fb] σ EBA [fb] δ EBA (4) (3) 3.5 % (6) (5) 3.0 % 8th November 2017 Ramon interhalder 18 / 24

36 Numerical Results - Upper p T,j Cut e now require p T,j < 100 Ge and p T,l > 100 Ge. pp νee + νµµ + jj (@100 Te) pp νee + νµµ + jj (@100 Te) [fb/ge] dσ dpt,lmax 10 4 LO EBA dσ [fb/ge] dpt,miss LO EBA δ [%] δ EBA [Ge] pt,lmax δ [%] δ EBA pt,miss [Ge] s [Te] σlo [fb] σ EBA [fb] δ EBA (4) (3) 3.5 % (6) (5) 3.0 % 8th November 2017 Ramon interhalder 18 / 24

37 Content 1 Introduction 2 ector-boson Scattering at the LHC 3 The Effective-ector-Boson Approximation 4 Logarithmic Electroweak Corrections 5 Conclusion 8th November 2017 Ramon interhalder 19 / 24

38 Logarithmic Electroweak Corrections Typical one-loop Feynman diagrams that lead to Sudakov logarithms: n k=1 l<k a=a,z, ± k l For high energies, i.e. s M 2, logarithms become dominant and are of significant size α 4πs 2 log 2 w s M 2 6.6%, α 4πs 2 log w s M 2 1.3%, at s = 1 Te. for BS at the LHC we can approximate the E corrections by considering only these logarithms! In LA the correction factorizes and the correction factor can be decomposed δm i 1...i n = M i 1...i n 0 δ i 1 i 1...i n in, δ = δ LSC + δ C + δ PR SSC + δ, }{{}}{{} = δ LL = δ NLL 8th November 2017 Ramon interhalder 20 / 24

39 Radiative Corrections to BS q 1 q 3 e neglect non-factorizable corrections. e neglect corrections to the emission and decay process. q 2 q 4 8th November 2017 Ramon interhalder 21 / 24

40 Radiative Corrections to BS q 1 q 3 e neglect non-factorizable corrections. e neglect corrections to the emission and decay process. e consider only corrections to the subprocess. q 2 q 4 Hence, using the EBA the correction can be written as δm EBA qq = λ 1,λ 2,λ 3,λ 4 ( 1)λ1+λ2 M prod K 1 K 2 K 3 K 4 BS δ M λ 1 λ 2 λ 3 λ 4 λ M prod 1 λ 2 decay M λ Mdecay 3 λ. 4 8th November 2017 Ramon interhalder 21 / 24

41 Numerical Results - E Logarithms pp νee + νµµ + jj (@14 Te) pp νee + νµµ + jj (@14 Te) dσ [fb/ge] dpt,lmax LL EBA LL+NLL dσ [fb/ge] dpt,miss LL EBA LL+NLL δ [%] δ LL+NLL δ LL LLtrans [Ge] pt,lmax δll trans δ [%] δ LL+NLL δ LL LLtrans δll trans pt,miss [Ge] s [Te] δ LL trans δ LL δ LL+NLL % 35.1 % 13.6 % % 37.5 % 13.8 % 8th November 2017 Ramon interhalder 22 / 24

42 Numerical Results - E Logarithms pp νee + νµµ + jj (@100 Te) pp νee + νµµ + jj (@100 Te) [fb/ge] dσ dpt,lmax δ [%] LL δ LL+NLL δ LL EBA LL+NLL LLtrans [Ge] pt,lmax δ LL trans dσ [fb/ge] dpt,miss δ [%] LL δ LL+NLL δ LL EBA LL+NLL δll trans LLtrans pt,miss [Ge] s [Te] δ LL trans δ LL δ LL+NLL % 35.1 % 13.6 % % 37.5 % 13.8 % 8th November 2017 Ramon interhalder 22 / 24

43 Content 1 Introduction 2 ector-boson Scattering at the LHC 3 The Effective-ector-Boson Approximation 4 Logarithmic Electroweak Corrections 5 Conclusion 8th November 2017 Ramon interhalder 23 / 24

44 Conclusion BS in general EBA ector Boson Scattering is a key tool in probing the mechanism of ESB. At the LHC all contributing processes need to be fully understood kinematical cuts are required to suppress background contributions The EBA may quantify the dominant region of BS The photon pole only appears as an artefact in a bad on-shell projection. For standard BS cuts the approximation is not applicable as the collinearity of the bosons is not ensured. If the collinearity is ensured the relative difference only amounts to 3.5 %. the EBA yields an appropriate approximation in certain kinematic regions! Logarithmic E Corrections Both LL and NLL need to be taken into account. The NLO E correction amounts to 13.6%, which is compatible with the literature. Biedermann, Denner, Pellen 2017 [arxiv: ] 8th November 2017 Ramon interhalder 24 / 24

45 QCD Background Suppression Cuts 8th November 2017 Ramon interhalder 24 / 24

46 The SU(2) model e consider a non-abelian SU(2) model in which no photon field appears. SU(2) U(1) Y : D µ = µ igi i i µ + ig Y 2 Bµ, g 0 (g = g SM = g SU(2) = e/s w) SU(2) : D µ = µ igi i i µ. + k1 k3 + + k1 k3 + Z γ + k2 k k1 k3 3 k2 + k4 + M Z,t c2 w s 2 w M 3,t 1 s 2 w 1 t M 2 Z 1 t M 2 M γ,t 1 t + k2 k4 + no photon diagrams no photon pole! 8th November 2017 Ramon interhalder 24 / 24

47 Numerical Results - SU(2) model pp νee + νµµ + jj (@14 Te) pp νee + νµµ + jj (@14 Te) 10 2 EBASM 10 2 EBASM dσ [fb/ge] dpt,lmax EBASU2 dσ [fb/ge] dpt,miss EBASU2 δ [%] δ 40 SU [Ge] pt,lmax δ [%] δ SU pt,miss [Ge] s [Te] σeba [fb] σ SU2 EBA [fb] δsu (2) (2) % (5) (5) % 8th November 2017 Ramon interhalder 24 / 24

48 Numerical Results - SU(2) model pp νee + νµµ + jj (@100 Te) pp νee + νµµ + jj (@100 Te) 10 1 EBASM 10 1 EBASM [fb/ge] dσ dpt,lmax 10 2 EBASU2 dσ [fb/ge] dpt,miss 10 2 EBASU δ [%] δ SU [Ge] pt,lmax δ [%] 0 10 δ 20 SU pt,miss [Ge] s [Te] σeba [fb] σ SU2 EBA [fb] δsu (2) (2) % (5) (5) % 8th November 2017 Ramon interhalder 24 / 24

49 Numerical Results - s fixed pp νee + νµµ + jj (@14 Te) pp νee + νµµ + jj (@14 Te) dσ [fb/ge] dpt,lmax EBAtu-fixed EBAs-fixed LO dσ [fb/ge] dpt,miss EBAtu-fixed EBAs-fixed LO δ [%] δ s-fixed δ tu-fixed [Ge] pt,lmax δ [%] δ s-fixed δ tu-fixed pt,miss [Ge] s [Te] σlo [fb] σeba s-fixed [fb] σtu-fixed EBA [fb] δs-fixed δ tu-fixed (2) (2) (2) 14.7 % 1.2 % (3) (5) (5) 87.9 % 28.2 % 8th November 2017 Ramon interhalder 24 / 24

50 Numerical Results - s fixed pp νee + νµµ + jj (@100 Te) pp νee + νµµ + jj (@100 Te) 10 1 EBAtu-fixed 10 1 EBAtu-fixed [fb/ge] dσ dpt,lmax 10 2 EBAs-fixed dσ [fb/ge] dpt,miss 10 2 EBAs-fixed 10 3 LO 10 3 LO δ [%] δ s-fixed δ tu-fixed [Ge] pt,lmax δ [%] δ s-fixed δtu-fixed pt,miss [Ge] s [Te] σlo [fb] σeba s-fixed [fb] σtu-fixed EBA [fb] δs-fixed δ tu-fixed (2) (2) (2) 14.7 % 1.2 % (3) (5) (5) 87.9 % 28.2 % 8th November 2017 Ramon interhalder 24 / 24

51 Numerical Results - Helicity Configurations pp νee + νµµ + jj (@14 Te) pp νee + νµµ + jj (@14 Te) 10 2 All 10 2 All dσ [fb/ge] dpt,lmax Trans Mixed Long dσ [fb/ge] dpt,miss Trans Mixed Long pt,lmax [Ge] pt,miss [Ge] s [Te] σeba [fb] σ trans EBA [fb] σmixed EBA [fb] σlong EBA [fb] (2) (2) (8) (3) (5) (5) (9) (2) 8th November 2017 Ramon interhalder 24 / 24

52 Numerical Results - Helicity Configurations pp νee + νµµ + jj (@100 Te) pp νee + νµµ + jj (@100 Te) 10 1 All 10 1 All 10 2 Trans 10 2 Trans [fb/ge] dσ dpt,lmax Long Mixed dσ [fb/ge] dpt,miss Long Mixed pt,lmax [Ge] pt,miss [Ge] s [Te] σeba [fb] σ trans EBA [fb] σmixed EBA [fb] σlong EBA [fb] (2) (2) (8) (3) (5) (5) (9) (2) 8th November 2017 Ramon interhalder 24 / 24

53 Complex-Shift Procedure e set the bosons on-shell by consistently shifting external momenta into the complex plane. 4 pȧb i = pȧi p B i, pb i = p B i + z if p B f, i = 1, 2, f=3 2 pȧb f = pȧf pb f, pȧf = pȧf + z if pȧi, f = 3, 4, i=1 hich is fixed by 4 on-shell conditions M 2 = k 2 i = ( p i p i )2 = 2 p i p i = 2p i p i 4 f=3 M 2 = k 2 f = ( p f + p f )2 = 2 p f p f = 2p f p f + 2 i=1 z if p i p i p f p i, i = 1, 2, z if p i p f p f p f, f = 3, 4. leads to inconsistent results which are off by several magnitudes or do not even converge. 8th November 2017 Ramon interhalder 24 / 24