Threshold Corrections To DY and Higgs at N 3 LO QCD

Transcription

1 Threshold Corrections To DY and Higgs at N 3 LO QCD Taushif Ahmed Institute of Mathematical Sciences, India April 12, 2016 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 1

2 Prologue : Processes of Interest Drell-Yan (DY) pp γ /Z : Xsection and rapidity distributions. q γ /Z q Higgs boson production through gg : Rapidity distributions. g H g Higgs boson production through b b : Xsection and rapidity distributions. b b H Such processes are very important for precision studies in SM. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 2

3 Prologue : State of the Art in Early 2014 DY Xsection : NNLO QCD Hamberg, Matsuura, Van Neerven ( 91) N 3 LO psv Moch, Vogt (2005); Ravindran ( 06) Higgs in gg Rapidity: NNLO QCD Anastasiou, Dixon, Melnikov, Petriello ( 03); Melnikov, Petriello ( 06) N 3 LO psv Ravindran, Smith, van Neerven ( 07) Xsection : NNLO QCD Harlander, Kilgore ( 02); Anastasiou, Melnikov ( 02); Ravindran, Smith, van Neerven ( 03) N 3 LO psv Moch, Vogt ( 05); Ravindran ( 06) Higgs in b b Rapidity : NNLO QCD Anastasiou, Melnikov, Petriello ( 04, 05) N 3 LO psv Ravindran, Smith, van Neerven ( 07) Xsection : NNLO QCD Harlander, Kilgore ( 03) N 3 LO psv Ravindran ( 06) Rapidity : NNLO QCD Buehler, Herzog, Lazopoulos, Mueller ( 12) N 3 LO psv Ravindran, Smith, van Neerven ( 07) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 3

4 Prologue : What next? Can we push the theoretical boundary little more? Testing SM with more accuracy to answer many unanswered questions Reducing the unphysical scale uncertainties First crucial step towards this direction Higgs production through gg fusion at threshold N 3 LO by Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger ( 14) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 4

5 Prologue : What next? Can we push the theoretical boundary little more? Testing SM with more accuracy to answer many unanswered questions Reducing the unphysical scale uncertainties First crucial step towards this direction Higgs production through gg fusion at threshold N 3 LO by Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger ( 14) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 4

6 Prologue : Spin-offs from Higgs Production at Threshold N 3 LO Recent results on threshold N 3 LO QCD corrections TA, Mahakhud, Mandal, Rana, Ravindran dσ/dq 2 of DY Phys. Rev. Lett. 113 (2014) dσ/dy of Higgs in gg fusion and d 2 σ/dq 2 /dy of DY Phys. Rev. Lett. 113 (2014) σ tot and dσ/dy of Higgs in b b annihilation JHEP 1410 (2014) 139 JHEP 1502 (2015) 131 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 5

7 Prologue : Spin-offs from Higgs Production at Threshold N 3 LO Recent results on threshold N 3 LO QCD corrections TA, Mahakhud, Mandal, Rana, Ravindran dσ/dq 2 of DY Phys. Rev. Lett. 113 (2014) Goal dσ/dy of Higgs in gg fusion and d 2 σ/dq 2 /dy of DY Phys. Rev. Lett. 113 (2014) σ tot and dσ/dy of Higgs in b b annihilation JHEP 1410 (2014) 139 JHEP 1502 (2015) 131 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 6

8 Plan of the Talk What is threshold corrections? Prescription of Computation Factorization of Soft Gluons Overall Operator Renormalization Form Factor Soft-Collinear Distribution Mass Factorization Master Formula Result of DY SV Xsection at N 3 LO Conclusions Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 7

9 Why threshold corrections? Basic calculational framework for inclusive production : QCD factorization theorem σ X ( τ, q 2 ) = 1 S ab 1 τ q2 S, q2 m 2 l + l for DY and m 2 H for Higgs τ dx x Φ ab (x) ab ( τ x, q2) Perturbatively calculable partonic cross section (CS) Non-perturbative partonic flux Φ ab (x) = ab ŝ ˆσ ab 1 x du ( x ) u f a (u) f b u Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 8

10 Why threshold corrections? Basic calculational framework for inclusive production : QCD factorization theorem σ X ( τ, q 2 ) = 1 S ab 1 τ q2 S, q2 m 2 l + l for DY and m 2 H for Higgs τ dx x Φ ab (x) ab ( τ x, q2) Perturbatively calculable partonic cross section (CS) Non-perturbative partonic flux Φ ab (x) = ab ŝ ˆσ ab 1 x du ( x ) u f a (u) f b u Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 8

11 Why Threshold... Φ ab (x) becomes large when x x min = τ enhances σ X x τ ŝ q 2 partonic COM energy is just enough to produce final state particle all the radiations are soft soft limit. In this region, Dominant contributions arise from Virtual and Soft gluon emission processes (SV). For Higgs and DY productions, substantial contribution come from this region. e.g. for DY 91% - 95% at NNLO Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 9

12 Threshold Expan : Soft-Plus-Virtual (SV) Splitting the partonic CS into singular & regular parts around z τ x = q2 ŝ = 1 sing (z) SV (z) = SV δ (z) = sing (z) + hard (z) δ(1 z)+ j=0 SV j D j D j ( ln j ) (1 z) 1 z + These two are the most singular terms when z 1, so dominant contribution comes from these. hard (z) : subleading and polynomial in ln(1 z). Plus distribution + is defined by its action on test function f(z) 1 1 dzd j (z)f(z) dz lnj (1 z) [f(z) f(1)] z Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 10

13 Goal Goal : Computation of SV (z) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 11

14 Various Methods to Compute SV Matrix elements square and phase space integrals in the soft limit. Form factors and DGLAP kernels : Catani et al; Harlander and Kilgore Factorization theorem Renormalization group invariance Sudakov resummation of form factors Soft Collinear effective theory Moch and Vogt; Ravindran, Smith and Van Neerven Becher and Neubert Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 12

15 Diagrams at O(a 3 s) for DY Triple virtual : 244 Double virtual : 13 1 Double virtual real : Real virtual squared : Double real virtual : Triple real squared: Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 13

16 Various Methods to Compute SV Matrix elements square and phase space integrals in the soft limit. Form factors and DGLAP kernels : Catani et al; Harlander and Kilgore Factorization theorem Renormalization group invariance Sudakov resummation of form factors Soft Collinear effective theory Moch and Vogt; Ravindran, Smith and Van Neerven Becher and Neubert Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 14

17 Diagrams at O(a 3 s) for DY Triple virtual : 244 Double virtual : 13 1 We will not compute these!! Symmetry Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 15

18 Diagrams at O(a 3 s) for DY Triple virtual : 244 Double virtual : 13 1 We will not compute these!! Symmetry Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 15

19 Various Methods to Compute SV Matrix elements square and phase space integrals in the soft limit. Form factors and DGLAP kernels : Catani et al; Harlander and Kilgore Factorization theorem Renormalization group invariance Sudakov resummation of form factors Soft Collinear effective theory Moch and Vogt; Ravindran, Smith and Van Neerven Becher and Neubert Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 16

20 Various Methods to Compute SV Matrix elements square and phase space integrals in the soft limit. Catani et al; Harlander and Kilgore Form factors and DGLAP kernels : Factorization theorem Renormalization group invariance Sudakov resummation of form factors Soft Collinear effective theory Moch and Vogt; Ravindran, Smith and Van Neerven Becher and Neubert Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 17

21 The Prescription Master Formula Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 18

22 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 19

23 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 19

24 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 19

25 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 19

26 Factorization of Soft Gluons Soft distribution factorizes as exponentiation ( z, q 2) = S ( z, q 2, µ 2 ) [ R δ (1 z) + a s R1 (z, q 2 ) + a 2 R ] s 2 (z, q 2 ) and S ( ( ) z, q 2, µ 2 ) R = C exp Φ(z, q 2, µ 2 R), Φ is a finite distribution Defn: Ce f(z) = δ(1 z) + 1 1! f(z) + 1 f(z) f(z) + 2! is Mellin convolution. Drop all regular functions. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 20

27 Factorization of Soft Gluons Soft distribution factorizes as exponentiation ( z, q 2) = S ( z, q 2, µ 2 ) [ R δ (1 z) + a s R1 (z, q 2 ) + a 2 R ] s 2 (z, q 2 ) and S ( ( ) z, q 2, µ 2 ) R = C exp Φ(z, q 2, µ 2 R), Φ is a finite distribution Defn: Ce f(z) = δ(1 z) + 1 1! f(z) + 1 f(z) f(z) + 2! is Mellin convolution. Drop all regular functions. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 20

28 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 I ˆF ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall operator ren. const. required for 1. effective Lagrangian of Higgs in gg 2. Yukawa coupling for Higgs in b b and Z I = 1 for DY. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 21

29 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 I ˆF ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall operator ren. const. required for 1. effective Lagrangian of Higgs in gg 2. Yukawa coupling for Higgs in b b and Z I = 1 for DY. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 21

30 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 I ˆF ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall operator ren. const. required for 1. effective Lagrangian of Higgs in gg 2. Yukawa coupling for Higgs in b b and Z I = 1 for DY. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 21

31 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 I ˆF ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall operator ren. const. required for 1. effective Lagrangian of Higgs in gg 2. Yukawa coupling for Higgs in b b and Z I = 1 for DY. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 21

32 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 22

33 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 22

34 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 22

35 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 22

36 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 22

37 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 22

38 Factorization of Collinear Singularities ˆ SV,I ( z, q 2, µ 2 ) R = Γ T (z, µ 2 F ) SV,I ( z, q 2, µ 2 R, µ 2 ) F Γ(z, µ 2 F ) UV, Soft & Collinear finite Kernels Γ absorb all the initial state collinear singularities. Only diagonal elements contribute to threshold limit. Substituting this we arrive at the master formula to compute SV CS. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 23

39 Factorization of Collinear Singularities ˆ SV,I ( z, q 2, µ 2 ) R = Γ T (z, µ 2 F ) SV,I ( z, q 2, µ 2 R, µ 2 ) F Γ(z, µ 2 F ) UV, Soft & Collinear finite Kernels Γ absorb all the initial state collinear singularities. Only diagonal elements contribute to threshold limit. Substituting this we arrive at the master formula to compute SV CS. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 23

40 Master Formula for SV CS SV,I ( z, q 2, µ 2 R, µ 2 F ) = C exp ( Ψ I ( z, q 2, µ 2 R, µ 2 F, ɛ )) ɛ=0 where ( [ 2 Ψ I = ln Z I (â s, µ 2 R, µ 2, ɛ)] + ln ˆF I (â s, Q 2, µ 2, ɛ) 2) δ(1 z) + 2Φ I ( â s, q 2, µ 2, z, ɛ ) 2C ln Γ II (âs, µ 2, µ 2 F, z, ɛ ) Ravindran Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 24

41 Ingredients for SV CS 1. Logarithm of the overall operator renormalization ln Z I 2. Logarithm of the form factor ln ˆF I 3. Soft distribution Φ I 4. Logarithm of the mass factorization kernel ln Γ II Goal boils down to compute these quantities. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 25

42 Ingredients for SV CS 1. Logarithm of the overall operator renormalization ln Z I 2. Logarithm of the form factor ln ˆF I 3. Soft distribution Φ I 4. Logarithm of the mass factorization kernel ln Γ II Goal boils down to compute these quantities. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 25

43 The Prescription Master Formula How do we compute these? Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 26

44 Ingredient 1 : Overall Operator Renormalization ln Z I It satisfies the RGE d µ 2 R dµ 2 R ln Z I (â s, µ 2 R, µ 2, ɛ) = a i s(µ 2 R)γi 1 I γi I s are anomalous dimensions. γ I are available up to O(a 3 s) for I = b, g and γ q = 0 (q b). Solution of this RGE provides ln Z I. OOR is not required for DY Z I = 1. i=1 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 27

45 Ingredient 2 : Form factor ln ˆF I ln ˆF I 2 can be computed from available results up to 3-loop. Instead, we have followed an alternative way. The reason would be clear soon. Gauge and RG invariance implies KG eqn Q 2 d dq 2 ln F ˆI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, ɛ + G (â ] I s, Q2 µ 2, µ2 R R µ 2, ɛ Ashoke Sen, Mueller, Collins, Magnea K I contains all the poles in ɛ where, ST dimensions d = 4 + ɛ. G I collects finite terms at ɛ 0. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 28

46 Ingredient 2... RG invariance of ˆF implies ) µ 2 d R K (â I s, µ2 R µ 2, ɛ = µ 2 R dµ 2 R d dµ 2 R ( ) G I â s, Q2, µ2 R µ 2, ɛ = A I ( a s (µ 2 ) R µ 2 R A I s are finite, called cusp anomalous dimensions. A I s are maximally non-abelian C A = N, C F = (N 2 1)/2N A g = C A C F A q Solve by expanding K I, G I and A I in powers of a s ( µ 2 R ). Substitute these K I & G I in KG eqn to solve for ln ˆF I to order by order. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 29

47 Ingredient 2... RG invariance of ˆF implies ) µ 2 d R K (â I s, µ2 R µ 2, ɛ = µ 2 R dµ 2 R d dµ 2 R ( ) G I â s, Q2, µ2 R µ 2, ɛ = A I ( a s (µ 2 ) R µ 2 R A I s are finite, called cusp anomalous dimensions. A I s are maximally non-abelian C A = N, C F = (N 2 1)/2N A g = C A C F A q Solve by expanding K I, G I and A I in powers of a s ( µ 2 R ). Substitute these K I & G I in KG eqn to solve for ln ˆF I to order by order. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 29

48 Ingredient 2... Solution of ln ˆF I The solution up to 3 loops are: ( ) ln ˆF I (â s, Q 2, µ 2, ɛ) = â i Q 2 i ɛ 2 i s S i=1 µ 2 ɛ I,(i) ˆL (ɛ) F with ˆL I,(1) (ɛ) = 1 { } F ɛ 2 2A I } {G I1 ɛ (ɛ), Moch, Vogt, Vermaseren; Ravindran; Magnea ˆL I,(2) (ɛ) = 1 { } F ɛ 3 β 0 A I { ɛ 2 12 } AI2 β 0G I1 (ɛ) + 1 { } 1 ɛ 2 GI 2 (ɛ), ˆL I,(3) (ɛ) = 1 { F ɛ 4 8 } 9 β2 0 AI { 2 ɛ 3 9 β 1A I β 0A I } 3 β2 0 GI 1 (ɛ) + 1 { ɛ 2 29 AI3 13 β 1G I1 (ɛ) 43 } β 0G I2 (ɛ) + 1 { } 1 ɛ 3 GI 3 (ɛ) At every order, all the poles except 1 ɛ previous order. can be predicted from Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 30

49 Ingredient 2... Solution of ln ˆF I The solution up to 3 loops are: ( ) ln ˆF I (â s, Q 2, µ 2, ɛ) = â i Q 2 i ɛ 2 i s S i=1 µ 2 ɛ I,(i) ˆL (ɛ) F with ˆL I,(1) (ɛ) = 1 { } F ɛ 2 2A I } {G I1 ɛ (ɛ), Moch, Vogt, Vermaseren; Ravindran; Magnea ˆL I,(2) (ɛ) = 1 { } F ɛ 3 β 0 A I { ɛ 2 12 } AI2 β 0G I1 (ɛ) + 1 { } 1 ɛ 2 GI 2 (ɛ), ˆL I,(3) (ɛ) = 1 { F ɛ 4 8 } 9 β2 0 AI { 2 ɛ 3 9 β 1A I β 0A I } 3 β2 0 GI 1 (ɛ) + 1 { ɛ 2 29 AI3 13 β 1G I1 (ɛ) 43 } β 0G I2 (ɛ) + 1 { } 1 ɛ 3 GI 3 (ɛ) At every order, all the poles except 1 ɛ previous order. can be predicted from Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 30

50 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 31

51 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 31

52 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 31

53 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 31

54 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 31

55 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 31

56 Ingredient 2... Single Pole Also Predictable 3-loop results for the ˆF I by Moch et al, Gehrmann et al. confirm this conclusion ( ) G I 3(ɛ) = 2(B3 I γ3) I + f3 I 2β 1 g I,1 1 2β 0 g I, β 0 g I,2 1 + ɛ k g I,k 3 with k=1 f I s are universal. f g 3 = C A C F f q 3 Explicit computation of FF gives ɛ dependent g I,k i. This understanding is crucial to get soft part. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 32

57 Ingredient 2... Single Pole Also Predictable 3-loop results for the ˆF I by Moch et al, Gehrmann et al. confirm this conclusion ( ) G I 3(ɛ) = 2(B3 I γ3) I + f3 I 2β 1 g I,1 1 2β 0 g I, β 0 g I,2 1 + ɛ k g I,k 3 with k=1 f I s are universal. f g 3 = C A C F f q 3 Explicit computation of FF gives ɛ dependent g I,k i. This understanding is crucial to get soft part. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 32

58 Ingredient 3 : Mass Factorization ln Γ DGLAP kernel Γ satisfies the RGE d µ 2 F dµ 2 F Γ(z, µ 2 F, ɛ) = 1 2 P ( z, µ 2 ) ( F Γ z, µ 2 F, ɛ ) P s are Altarelli Parisi splitting functions (matrix valued). Diagonal parts of the kernel ONLY contribute to SV limit. By solving we get Γ ln Γ Γ is available up to O(a 4 s). Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 33

59 Where are we? ln[z I ] 2 ln ˆF I 2 ln Γ II?? Φ I Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 34

60 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 35

61 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 36

62 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 37

63 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Φ I Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 38

64 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Φ I Demanding finiteness of partonic CS Φ I must have soft as well as collinear contributions. Now onward we will call Φ I as Soft-Collinear distribution. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 39

65 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Φ I Demanding finiteness of partonic CS Φ I must have soft as well as collinear contributions. Now onward we will call Φ I as Soft-Collinear distribution. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 39

66 Ingredient 4... Ansatz for Soft-Collinear Φ I Demand of 1. finiteness of SV,I (z) at ɛ 0 and 2. the RGE d µ 2 R dµ 2 R Φ I ( â s, q 2, µ 2, z, ɛ ) = 0 can be achieved if we make an ansatz that Φ I satisfies KG type DE q 2 d dq 2 ΦI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, z, ɛ + G (â ] I s, q2 µ 2, µ2 R R µ 2, z, ɛ K I contains all the poles in ɛ. G I contains finite terms in ɛ 0. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 40

67 Ingredient 4... Ansatz for Soft-Collinear Φ I Demand of 1. finiteness of SV,I (z) at ɛ 0 and 2. the RGE d µ 2 R dµ 2 R Φ I ( â s, q 2, µ 2, z, ɛ ) = 0 can be achieved if we make an ansatz that Φ I satisfies KG type DE q 2 d dq 2 ΦI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, z, ɛ + G (â ] I s, q2 µ 2, µ2 R R µ 2, z, ɛ K I contains all the poles in ɛ. G I contains finite terms in ɛ 0. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 40

68 Ingredient 4... Ansatz for Soft-Collinear Φ I Demand of 1. finiteness of SV (z) at ɛ 0 and 2. the RGE d µ 2 R dµ 2 R Φ I ( â s, q 2, µ 2, z, ɛ ) = 0 can be achieved if we make an ansatz that Φ satisfies KG type DE q 2 d dq 2 ΦI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, z, ɛ + G (â ] I s, q2 µ 2, µ2 R R µ 2, z, ɛ K I contains all the poles in ɛ. G I contains finite terms in ɛ 0. The solutions of KG consistent with our demands Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 41

69 Ingredient 4... Solution of Φ I Φ I (â s, q 2, µ 2, z, ɛ) = where This implies ˆφ I,(i) (ɛ) = i=1 â i s I,(i) ˆL F (ɛ) G I 1 (ɛ) = f I 1 + k=1 ( q 2 (1 z) 2 ( ɛ k G I,k 1 G I 2 (ɛ) = f I 2 2β 0G I,1 1 + µ 2 ) i ɛ ( ) 2 S i iɛ ɛ ˆφ I,(i) (ɛ) 1 z A I A I, G I (ɛ) G I (ɛ) ɛ k G I,k 2 k=1 ( ) G I 3 (ɛ) = f 3 I 2β 1G I,1 1 2β 0 G I, β 0 G I,2 1 + ɛ k G I,k 3 k=1 Explicit computation is required to get ɛ dependent cof ḠI,k i. ) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 42

70 Ingredient 4... Solution of Φ I Φ I (â s, q 2, µ 2, z, ɛ) = where This implies ˆφ I,(i) (ɛ) = i=1 â i s I,(i) ˆL F (ɛ) G I 1 (ɛ) = f I 1 + k=1 ( q 2 (1 z) 2 ( ɛ k G I,k 1 G I 2 (ɛ) = f I 2 2β 0G I,1 1 + µ 2 ) i ɛ ( ) 2 S i iɛ ɛ ˆφ I,(i) (ɛ) 1 z A I A I, G I (ɛ) G I (ɛ) ɛ k G I,k 2 k=1 ( ) G I 3 (ɛ) = f 3 I 2β 1G I,1 1 2β 0 G I, β 0 G I,2 1 + ɛ k G I,k 3 k=1 Explicit computation is required to get ɛ dependent cof ḠI,k i. ) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 42

71 Ingredient 4... Crucial Property of Φ I Explicit computation shows up to O(a 2 s). Φ q = C F Φ g and Ḡ q,k i = C F Ḡ g,k i C A C A soft-collinear distributions are universal Φ I = C I Φ with C I = { C A C F for Higgs in gg for DY and for Higgs in b b Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 43

72 Ingredient 4... Crucial Property of Φ I Explicit computation shows up to O(a 2 s). Φ q = C F Φ g and Ḡ q,k i = C F Ḡ g,k i C A C A soft-collinear distributions are universal Φ I = C I Φ with C I = { C A C F for Higgs in gg for DY and for Higgs in b b Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 43

73 Where are we? ln[z I ] 2 ln ˆF I 2 ln Γ II Φ I Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 44

74 The Prescription Master Formula How do we compute these? What about DY at N 3 LO? Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 45

75 Results : SV XSection for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q,(3). ln[z q ] 2 up to O(a 3 s) ln ˆF q 2 up to O(a 3 s) ln Γ qq up to O(a 3 s) Φ q up to O(a 2 s) BUT? O(a 3 s) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 46

76 Results : SV XSection for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q,(3). ln[z q ] 2 up to O(a 3 s) ln ˆF q 2 up to O(a 3 s) ln Γ qq up to O(a 3 s) Φ q up to O(a 2 s) BUT? O(a 3 s) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 46

77 Results : SV XSection for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q,(3). ln[z q ] 2 up to O(a 3 s) ln ˆF q 2 up to O(a 3 s) ln Γ qq up to O(a 3 s) Φ q up to O(a 2 s) BUT? O(a 3 s) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 46

78 Results : SV XSection for DY at N 3 LO... Recently, complete SV XSec at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Recall Φ q = C F C A Φ g was established up to O(a 2 s) by explicit computation. We conjectured the relation to be hold true even at O(a 3 s). Using this we got Φ q Ḡq,1 3. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 47

79 Results : SV XSection for DY at N 3 LO... Recently, complete SV XSec at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Recall Φ q = C F C A Φ g was established up to O(a 2 s) by explicit computation. We conjectured the relation to be hold true even at O(a 3 s). Using this we got Φ q Ḡq,1 3. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 47

80 Results : SV XSection for DY at N 3 LO... Recently, complete SV XSec at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Recall Φ q = C F C A Φ g was established up to O(a 2 s) by explicit computation. We conjectured the relation to be hold true even at O(a 3 s). Using this we got Φ q Ḡq,1 3. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 47

81 Results : SV XSection for DY at N 3 LO... This gives the ONLY missing part to achieve δ(1 z) part. In addition, now D 7,... D 1 are available at N 4 LO exactly. TA, Mahakhud, Rana, Ravindran; de Florian, Mazzitelli, Moch, Vogt Later the result was reconfirmed by two groups : Catani, Cieri, de Florian, Ferrera, Frazzini; Li, von Manteuffel, Schabinger, Zhu This in turn establishes our conjecture at O(a 3 s). Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 48

82 Results : Numerical Implications At 14 TeV LHC with µ F = µ R = Q = m l + l [ Q 2 dσ dq ]δ [ 2 j Q 2 dσ Most of the contributions dq ]D 2 j from D j s are cancelled against δ(1 z) making N 3 LO SV more subleading. Q (GeV) δ (nb) D (nb) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 49

83 Results : Numerical Implications... Correction at N 3 LO SV is very very small Good newz! Q (GeV) NNLO (nb) N 3 LO SV (nb) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 50

84 Results : Numerical Implications... Scale dependence reduces NLO NNLO (i) R Q = 20 GeV 3 N LO SV Q = 200 GeV (i) R µ /Q R [ R (i) ] Q 2 d σ (i) (µ 2 dq 2 R ) / [ ] Q 2 d σ (i) (Q 2 ) and µ dq 2 F = Q Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 51

85 Summary A systematic way of computing threshold corrections to inclusive Higgs and DY productions in pqcd is prescribed. Underlying principles : factorization of soft and collinear divergences, RG invariance, gauge invariance and Sudakov resummation. δ(1 z) part of N 3 LO SV DY Xsection is computed for the first time. Importantly, the calculation is done even without performing the explicit computation of the real emission diagrams for DY at O(a 3 s). Necessary information arising from real emission processes is extracted from the computation of Higgs boson in gg at N 3 LO threshold. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 52

86 Summary A systematic way of computing threshold corrections to inclusive Higgs and DY productions in pqcd is prescribed. Underlying principles : factorization of soft and collinear divergences, RG invariance, gauge invariance and Sudakov resummation. δ(1 z) part of N 3 LO SV DY Xsection is computed for the first time. Importantly, the calculation is done even without performing the explicit computation of the real emission diagrams for DY at O(a 3 s). Necessary information arising from real emission processes is extracted from the computation of Higgs boson in gg at N 3 LO threshold. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 52

87 Summary... We find that the impact of the δ(1 z) contribution is quite large to the pure N 3 LO SV correction. This method has been later employed to compute some other inclusive and exclusive observables at threshold by us in Phys. Rev. Lett. 113 (2014) JHEP 1410 (2014) 139 JHEP 1502 (2015) 131 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 53

88 Other Areas of Interest RG Improved Higgs Boson Production to N 3 LO in QCD TA, Das, Kumar, Rana, Ravindran arxiv: Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 54

89 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order a n s (µ 2 R ) lnk (µ 2 R /m2 H ). Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 55

90 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order a n s (µ 2 R ) lnk (µ 2 R /m2 H ). Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 55

91 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order a n s (µ 2 R ) lnk (µ 2 R /m2 H ). Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 55

92 Large Scale Uncertainties Figure: Higgs boson production in gg fusion Duhr, Talk at RADCOR 15 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 56

93 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order a n s (µ 2 R ) lnk (µ 2 R /m2 H ). Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 57

94 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order a n s (µ 2 R ) lnk (µ 2 R /m2 H ). Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 57

95 Our Proposal to Improve µ R Dependence Our proposal : Perform resummations by including RG accessible µ R dependent logarithms of all orders. Consequence : Quite remarkable! Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 58

96 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 59

97 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 59

98 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 59

99 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 59

100 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 59

101 What are we resumming?... Collecting the highest logs to all orders σ (0) Σ (S, m2 H, µ 2 R) n=0 Collecting the next-to-highest logs to all orders σ (1) Σ (S, m2 H, µ 2 R) n=1 σ (n) n (a s L R ) n σ (n) n 1 (a sl R ) n 1 and so on. We re-write the Xsection in terms of these as Goal : Compute σ (i) Σ σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) = a 2 s(µ 2 R) a i s(µ 2 R)σ (i) Σ (S, m2 H, µ 2 R) i=0 for i = 0, 1, 2, 3 for gg H. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 60

102 Our Proposal to Improve µ R Dependence Our proposal : Perform resummations by including RG accessible µ R dependent logarithms of all orders. Consequence : Quite remarkable! µ = m F H MSTW TeV LHC LO NNLO NLO 3 N LO FO RESUM 50 [pb] σ µ / m R H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 61

103 Our Proposal to Improve µ R Dependence FO NNLO RESUM NNLO 3 FO N LO 3 RESUM N LO [ pb ] σ µ F µ R = m H = [m H /10,10m H ] MSTW s [ TeV ] Figure: Dependence of µ R in both the fixed order and resummed Xsections on S Demerit : Method fails to improve the central value of Xsection, unlike other resummation. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 62

104 Other Areas of Interest... Two-loop QCD corrections to Higgs b + b + g amplitude TA, Mahakhud, Mathews, Rana, Ravindran JHEP 1408 (2014) 075 Theory : SM. Bottom quark mass VFS scheme. No of diagrams : 251 Mostly in-house codes written in FORM and Mathematica. For IBP, LI identities LiteRed by Lee. Agrees with the universal IR-pole structures of QCD. Result will be used for Higgs + jet production through b b at NNLO QCD in hadron colliders. Analytically computed. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 63

105 Other Areas of Interest... Two-Loop QCD Correction to massive spin-2 resonance 3 gluons TA, Mahakhud, Mathews, Rana, Ravindran JHEP 1405 (2014) 107 Spin-2 couples to SM minimally through SM energy-momentum tensor. No of diagrams : 2362 Mostly in-house codes written in FORM and Mathematica. For IBP, LI identities LiteRed by Lee. Explicit verification of the universal IR-pole structure of QCD when spin-2 presents. Result will be used for spin-2 + jet production at NNLO QCD in hadron colliders. Analytically computed. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 64

106 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 65

107 Extra Slides Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 66

108 Extra Slide Extra 1 : RG Resum in Details Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 67

109 Understanding The Source Of The Problem The inclusive cross section of Higgs production σ H (S, m 2 H) = σ 0 a 2 s(µ 2 R) dx 1 dx 2 f a (x 1, µ 2 F )f b (x 2, µ 2 F ) a,b CH 2 ( as (µ 2 R) ) ( ) τ H ab, m 2 x 1 x H, µ 2 R, µ 2 F 2 a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) RG invariance wrt µ R of the observable µ 2 R d dµ 2 R σ H = 0 The Solution [ σ(µ 2 R) = σ(µ 2 0) exp µ 2 R µ 2 0 ] dµ 2 2 β(a s (µ 2 )) µ 2 a s (µ 2 ) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 68

110 Understanding The Source Of The Problem The inclusive cross section of Higgs production σ H (S, m 2 H) = σ 0 a 2 s(µ 2 R) dx 1 dx 2 f a (x 1, µ 2 F )f b (x 2, µ 2 F ) a,b CH 2 ( as (µ 2 R) ) ( ) τ H ab, m 2 x 1 x H, µ 2 R, µ 2 F 2 a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) RG invariance wrt µ R of the observable µ 2 R d dµ 2 R σ H = 0 The Solution [ σ(µ 2 R) = σ(µ 2 0) exp µ 2 R µ 2 0 ] dµ 2 2 β(a s (µ 2 )) µ 2 a s (µ 2 ) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 68

111 Understanding The Source Of The Problem The inclusive cross section of Higgs production σ H (S, m 2 H) = σ 0 a 2 s(µ 2 R) dx 1 dx 2 f a (x 1, µ 2 F )f b (x 2, µ 2 F ) a,b CH 2 ( as (µ 2 R) ) ( ) τ H ab, m 2 x 1 x H, µ 2 R, µ 2 F 2 a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) RG invariance wrt µ R of the observable µ 2 R d dµ 2 R σ H = 0 The Solution [ σ(µ 2 R) = σ(µ 2 0) exp µ 2 R µ 2 0 ] dµ 2 2 β(a s (µ 2 )) µ 2 a s (µ 2 ) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 68

112 Understanding The Source Of The Problem Considering µ 0 as central scale m H and using naive evolution of a s, the solution σ(µ 2 R) = ( ) µ 2 where, L R ln R. m 2 H The RG invariance dictates R n,n m = 1 (n m) n=0 k=0 n a n s (µ 2 R) R n,k L k R m (n i + 1)β i R n i 1,n m 1 i=0 i.e. the coefficients of the logarithms R n,k (0 < k n) can be expressed in terms of the lower order ones R n 1,0. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 69

113 Understanding The Source Of The Problem Considering µ 0 as central scale m H and using naive evolution of a s, the solution σ(µ 2 R) = ( ) µ 2 where, L R ln R. m 2 H The RG invariance dictates R n,n m = 1 (n m) n=0 k=0 n a n s (µ 2 R) R n,k L k R m (n i + 1)β i R n i 1,n m 1 i=0 i.e. the coefficients of the logarithms R n,k (0 < k n) can be expressed in terms of the lower order ones R n 1,0. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 69

114 Understanding The Source Of The Problem Coefficients of the highest logarithms at n th order in a s grows as (n + 1)a n s β n 0 R 0,0 can be potentially large contributions! fixed order predictions become unreliable RG invariance can be used to resum these large logarithms to all orders. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 70

115 Understanding The Source Of The Problem Coefficients of the highest logarithms at n th order in a s grows as (n + 1)a n s β n 0 R 0,0 can be potentially large contributions! fixed order predictions become unreliable RG invariance can be used to resum these large logarithms to all orders. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 70

116 The Remedy : Our Prescription We extend the approach by Ahmady et. al. to resum these large contributions. We rewrite the solution σ(µ 2 R ) as σ(µ 2 R) = a m s (µ 2 R) m=0 m=0 R n,n m (a s L R ) n m n=m a m s (µ 2 R) σ (m) ( Σ as (µ 2 ) R)L R, σ (m) Σ resums a s(µ 2 R )L R to all orders. Our Goal : Determine σ (m) Σ s. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 71

117 The Remedy : Our Prescription We extend the approach by Ahmady et. al. to resum these large contributions. We rewrite the solution σ(µ 2 R ) as σ(µ 2 R) = a m s (µ 2 R) m=0 m=0 R n,n m (a s L R ) n m n=m a m s (µ 2 R) σ (m) ( Σ as (µ 2 ) R)L R, σ (m) Σ resums a s(µ 2 R )L R to all orders. Our Goal : Determine σ (m) Σ s. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 71

118 The Remedy : Our Prescription Using the recursion relation we show σ (m) Σ satisfies [ ω d ] dω + (m + 2) σ (m) Σ m [ = Θ m 1 η i (1 ω) d ] dω (m i + 2) σ (m i) Σ i=1 η i β i /β 0 and ω 1 β 0 a s (µ 2 R )L R Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 72

119 The Remedy : The Solutions σ (0) Σ = 1 { } ω 2 R 0,0, σ (1) Σ = 1 { } ω 3 R 1,0 2η 1 R 0,0 ln(ω), σ (2) Σ = 1 { ( ) } 2 ω 3 2R 0,0 η 1 η2 + 1 { 2 ω 4 R 2,0 + 2R 0,0 (η ) 2 η 1 + ln(ω) 2 + 3η 1 R0,0 ln 2 } (ω), ( 2η 1 2 R0,0 3η 1 R 1,0 ) σ (3) Σ = 1 { ( ) } 3 ω 3 R 0,0 η 1 + 2η1 η 2 η { ( ) ( ) 3 2 ω 4 R 0,0 2η 1 2η1 η 2 + R 1,0 3η 1 3η2 3 + R 0,0 (6η ) } 1 η 2 6η 1 ln(ω) + 1 { ( 3 ) ( 2 ) ω 5 R 3,0 + R 0,0 η 3 η 1 + R 1,0 3η 2 3η 1 ) ) ln(ω) (R 0,0 (6η 1 8η1 η 2 3η 1 R1,0 4η 1 R 2,0 + ln 2 ) 3 2 (ω) (7η 1 R0,0 + 6η 1 R1,0 3 4η 1 R0,0 ln 3 } (ω), σ (4) Σ = Big expression We have resummed only µ R dependent logarithms. µ F has been chosen to some specific value m H µ F dependence remains unchanged. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 73

120 The Remedy : The Solutions σ (0) Σ = 1 { } ω 2 R 0,0, σ (1) Σ = 1 { } ω 3 R 1,0 2η 1 R 0,0 ln(ω), σ (2) Σ = 1 { ( ) } 2 ω 3 2R 0,0 η 1 η2 + 1 { 2 ω 4 R 2,0 + 2R 0,0 (η ) 2 η 1 + ln(ω) 2 + 3η 1 R0,0 ln 2 } (ω), ( 2η 1 2 R0,0 3η 1 R 1,0 ) σ (3) Σ = 1 { ( ) } 3 ω 3 R 0,0 η 1 + 2η1 η 2 η { ( ) ( ) 3 2 ω 4 R 0,0 2η 1 2η1 η 2 + R 1,0 3η 1 3η2 3 + R 0,0 (6η ) } 1 η 2 6η 1 ln(ω) + 1 { ( 3 ) ( 2 ) ω 5 R 3,0 + R 0,0 η 3 η 1 + R 1,0 3η 2 3η 1 ) ) ln(ω) (R 0,0 (6η 1 8η1 η 2 3η 1 R1,0 4η 1 R 2,0 + ln 2 ) 3 2 (ω) (7η 1 R0,0 + 6η 1 R1,0 3 4η 1 R0,0 ln 3 } (ω), σ (4) Σ = Big expression We have resummed only µ R dependent logarithms. µ F has been chosen to some specific value m H µ F dependence remains unchanged. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 73

121 The Remedy : Numerical Implications (a) Result is almost µ R independent for a wide range of µ R [0.1m H, 10m H ] LO NLO NNLO N 3 LO FO (%) RESUM (%) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 74

122 The Remedy : Numerical Implications µ = m F H MSTW TeV LHC LO NNLO NLO 3 N LO 50 FO RESUM [pb] 40 σ µ / m R H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 75

123 The Remedy : Numerical Implications 60 FO NNLO RESUM NNLO 3 FO N LO 3 RESUM N LO [ pb ] σ µ F µ R = m H = [m H /10,10m H ] MSTW s [ TeV ] (b) Cross section is always positive and reliable. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 76

124 The Remedy : Numerical Implications 60 FO NNLO RESUM NNLO 3 FO N LO 3 RESUM N LO [ pb ] σ µ F µ R = m H = [m H /10,10m H ] MSTW s [ TeV ] (b) Cross section is always positive and reliable. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 76

125 Extra Slide Extra 2 : What are we resumming? Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 77

126 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 78

127 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 78

128 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 78

129 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 78

130 What are we resumming? The inclusive cross section of Higgs production Structure of perturbation theory : σ = σ (0) 0 σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) + a s ( µ 2 R ) { σ (1) 0 + σ (1) 1 L R + a 2 s + a 3 s +... } ( µ 2 R ) { σ (2) 0 + σ (2) 1 L R + σ (2) 2 L2 R ( µ 2 R ) { σ (3) 0 + σ (3) 1 L R + σ (3) 2 L2 R + σ (3) 3 L3 R } } where ( ) µ 2 L R ln R m 2 H Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 78

131 What are we resumming?... Collecting the highest logs to all orders σ (0) Σ (S, m2 H, µ 2 R) n=0 Collecting the next-to-highest logs to all orders σ (1) Σ (S, m2 H, µ 2 R) n=1 σ (n) n (a s L R ) n σ (n) n 1 (a sl R ) n 1 and so on. We re-write the Xsection in terms of these as Goal : Compute σ (i) Σ σ H (S, m 2 H) = a 2 s(µ 2 R) σ(s, m 2 H, µ 2 R) = a 2 s(µ 2 R) a i s(µ 2 R)σ (i) Σ (S, m2 H, µ 2 R) i=0 for i = 0, 1, 2, 3 for gg H. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 79

132 Extra Slide Extra 3 : SV Xsection of Higgs in b b Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 80

133 Result 2 : SV CS for Higgs in b b at N 3 LO SV,(3) for Higgs production in b b annihilation All the D i and partial δ(1 z) contributions were known. We completed the full computation of δ(1 z) part. All the required quantities were available except Ḡb,1 3 from Φ b and g b,1 3 from 3-loop form factor ˆF b. Being flavor independent Ḡ b,1 3 = Ḡq,1 3 which is available from our DY result. g b,1 3 is extracted from recent result of 3-loop Hb b form factor by Gehrmann et. al. This completes the calculation of threshold N 3 LO corrections to Higgs production in b b annihilation. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 81

134 Result 2 : SV CS for Higgs in b b at N 3 LO SV,(3) for Higgs production in b b annihilation All the D i and partial δ(1 z) contributions were known. We completed the full computation of δ(1 z) part. All the required quantities were available except Ḡb,1 3 from Φ b and g b,1 3 from 3-loop form factor ˆF b. Being flavor independent Ḡ b,1 3 = Ḡq,1 3 which is available from our DY result. g b,1 3 is extracted from recent result of 3-loop Hb b form factor by Gehrmann et. al. This completes the calculation of threshold N 3 LO corrections to Higgs production in b b annihilation. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 81

135 Result 2 : SV CS for Higgs in b b at N 3 LO SV,(3) for Higgs production in b b annihilation All the D i and partial δ(1 z) contributions were known. We completed the full computation of δ(1 z) part. All the required quantities were available except Ḡb,1 3 from Φ b and g b,1 3 from 3-loop form factor ˆF b. Being flavor independent Ḡ b,1 3 = Ḡq,1 3 which is available from our DY result. g b,1 3 is extracted from recent result of 3-loop Hb b form factor by Gehrmann et. al. This completes the calculation of threshold N 3 LO corrections to Higgs production in b b annihilation. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 81

136 Result 2 : SV CS for Higgs in b b at N 3 LO SV,(3) for Higgs production in b b annihilation All the D i and partial δ(1 z) contributions were known. We completed the full computation of δ(1 z) part. All the required quantities were available except Ḡb,1 3 from Φ b and g b,1 3 from 3-loop form factor ˆF b. Being flavor independent Ḡ b,1 3 = Ḡq,1 3 which is available from our DY result. g b,1 3 is extracted from recent result of 3-loop Hb b form factor by Gehrmann et. al. This completes the calculation of threshold N 3 LO corrections to Higgs production in b b annihilation. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 81

137 Result 2 : SV CS for Higgs in b b at N 3 LO SV,(3) for Higgs production in b b annihilation All the D i and partial δ(1 z) contributions were known. We completed the full computation of δ(1 z) part. All the required quantities were available except Ḡb,1 3 from Φ b and g b,1 3 from 3-loop form factor ˆF b. Being flavor independent Ḡ b,1 3 = Ḡq,1 3 which is available from our DY result. g b,1 3 is extracted from recent result of 3-loop Hb b form factor by Gehrmann et. al. This completes the calculation of threshold N 3 LO corrections to Higgs production in b b annihilation. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 81

138 Result 2 : SV CS for Higgs in b b at N 3 LO SV,(3) for Higgs production in b b annihilation All the D i and partial δ(1 z) contributions were known. We completed the full computation of δ(1 z) part. All the required quantities were available except Ḡb,1 3 from Φ b and g b,1 3 from 3-loop form factor ˆF b. Being flavor independent Ḡ b,1 3 = Ḡq,1 3 which is available from our DY result. g b,1 3 is extracted from recent result of 3-loop Hb b form factor by Gehrmann et. al. This completes the calculation of threshold N 3 LO corrections to Higgs production in b b annihilation. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 81

139 Extra Slide Extra 4 : Fixing soft-collinear distribution Φ I Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 82

140 Fixing Forms of Φ I SV,I = C exp(ψ I ) with [ Ψ I ( = ln Z I ) ] 2 I 2 + ln ˆF δ(1 z) + 2Φ I 2C ln Γ II Considering only poles at O(a s) with µ R = µ F in d = 4 + ɛ ( ln Z I,(1)) 2 = as(µ 2 F ) 4γI 0 ɛ ( ) ln ˆF I,(1) 2 = a s(µ 2 q 2 ɛ/2 [ F ) µ 2 4AI 1 ɛ F ɛ [ 2C ln Γ II,(1) = 2a s(µ 2 2B I F ) 1 δ(1 z) + 2AI 1 ɛ ɛ D 0 ( ) ] Collecting the coefficient of a s(µ 2 F [neglecting ) ln q 2 /µ 2 F terms Ψ I,(1) = a s(µ 2 F ) 4γI 0 4AI 1 ɛ ɛ 2 [{ = a s(µ 2 F ) 4AI 1 ɛ 2 + 2f I 1 ɛ + 1 ɛ } ( 2f I 1 + 4BI 1 4γI 0 ] δ(1 z) 4AI 1 ɛ D 0 + 2Φ I ( 2f I ) ] 1 + 4BI 1 4γI 0 ] ) 4B1 I δ(1 z) δ(1 z) + 4AI 1 ɛ ɛ D Demand 1 : Hence to cancel all the poles, we must have at O(a s) [{ 2Φ I,(1) poles = a s(µ 2 4A I F ) 1 ɛ 2 2f } ] I 1 δ(1 z) + 4AI 1 ɛ ɛ D 0 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 83

141 Fixing Forms of Φ I SV,I = C exp(ψ I ) with [ Ψ I ( = ln Z I ) ] 2 I 2 + ln ˆF δ(1 z) + 2Φ I 2C ln Γ II Considering only poles at O(a s) with µ R = µ F in d = 4 + ɛ ( ln Z I,(1)) 2 = as(µ 2 F ) 4γI 0 ɛ ( ) ln ˆF I,(1) 2 = a s(µ 2 q 2 ɛ/2 [ F ) µ 2 4AI 1 ɛ F ɛ [ 2C ln Γ II,(1) = 2a s(µ 2 2B I F ) 1 δ(1 z) + 2AI 1 ɛ ɛ D 0 ( ) ] Collecting the coefficient of a s(µ 2 F [neglecting ) ln q 2 /µ 2 F terms Ψ I,(1) = a s(µ 2 F ) 4γI 0 4AI 1 ɛ ɛ 2 [{ = a s(µ 2 F ) 4AI 1 ɛ 2 + 2f I 1 ɛ + 1 ɛ } ( 2f I 1 + 4BI 1 4γI 0 ] δ(1 z) 4AI 1 ɛ D 0 + 2Φ I ( 2f I ) ] 1 + 4BI 1 4γI 0 ] ) 4B1 I δ(1 z) δ(1 z) + 4AI 1 ɛ ɛ D Demand 1 : Hence to cancel all the poles, we must have at O(a s) [{ 2Φ I,(1) poles = a s(µ 2 4A I F ) 1 ɛ 2 2f } ] I 1 δ(1 z) + 4AI 1 ɛ ɛ D 0 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 83

142 Fixing Forms of Φ I SV,I = C exp(ψ I ) with [ Ψ I ( = ln Z I ) ] 2 I 2 + ln ˆF δ(1 z) + 2Φ I 2C ln Γ II Considering only poles at O(a s) with µ R = µ F in d = 4 + ɛ ( ln Z I,(1)) 2 = as(µ 2 F ) 4γI 0 ɛ ( ) ln ˆF I,(1) 2 = a s(µ 2 q 2 ɛ/2 [ F ) µ 2 4AI 1 ɛ F ɛ [ 2C ln Γ II,(1) = 2a s(µ 2 2B I F ) 1 δ(1 z) + 2AI 1 ɛ ɛ D 0 ( ) ] Collecting the coefficient of a s(µ 2 F [neglecting ) ln q 2 /µ 2 F terms Ψ I,(1) = a s(µ 2 F ) 4γI 0 4AI 1 ɛ ɛ 2 [{ = a s(µ 2 F ) 4AI 1 ɛ 2 + 2f I 1 ɛ + 1 ɛ } ( 2f I 1 + 4BI 1 4γI 0 ] δ(1 z) 4AI 1 ɛ D 0 + 2Φ I ( 2f I ) ] 1 + 4BI 1 4γI 0 ] ) 4B1 I δ(1 z) δ(1 z) + 4AI 1 ɛ ɛ D Demand 1 : Hence to cancel all the poles, we must have at O(a s) [{ 2Φ I,(1) poles = a s(µ 2 4A I F ) 1 ɛ 2 2f } ] I 1 δ(1 z) + 4AI 1 ɛ ɛ D 0 Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 83

143 Fixing Form of Φ I... Demand 2 : Also, Φ I has to be RG-invariant i.e. µ 2 R d dµ 2 Φ I = 0. R We make an ansatz that if Φ I satisfies KG-type DE, then we can accomplish this q 2 d ( dq 2 ΦI â s, q 2, µ 2 ), z, ɛ = 1 ( [K I â µ2 R s, 2 µ 2, z, ɛ ) + G I ( ) ] â q2 s, µ 2, µ2 R µ R 2, z, ɛ using Demand 2 µ 2 d R dµ 2 K I = µ 2 d R dµ R 2 G I X I R The solution with ( ) Φ I = â i q 2 iɛ/2 s i=1 µ 2 S i ɛ ΦI,(i) (z, ɛ) Φ I,(i) (z, ɛ) = ˆL I,(i) ( A I X I, G I G I ) (z, ɛ). So 2Φ I,(1) (z, ɛ) = 1 ( ɛ 2 4X I ) ɛ GI 1 (z, ɛ)) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 84

144 Fixing Form of Φ I... Demand 2 : Also, Φ I has to be RG-invariant i.e. µ 2 R d dµ 2 Φ I = 0. R We make an ansatz that if Φ I satisfies KG-type DE, then we can accomplish this q 2 d ( dq 2 ΦI â s, q 2, µ 2 ), z, ɛ = 1 ( [K I â µ2 R s, 2 µ 2, z, ɛ ) + G I ( ) ] â q2 s, µ 2, µ2 R µ R 2, z, ɛ using Demand 2 µ 2 d R dµ 2 K I = µ 2 d R dµ R 2 G I X I R The solution with ( ) Φ I = â i q 2 iɛ/2 s i=1 µ 2 S i ɛ ΦI,(i) (z, ɛ) Φ I,(i) (z, ɛ) = ˆL I,(i) ( A I X I, G I G I ) (z, ɛ). So 2Φ I,(1) (z, ɛ) = 1 ( ɛ 2 4X I ) ɛ GI 1 (z, ɛ)) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 84

145 Fixing Form of Φ I... where, ( Φ I = â i q 2 s i=1 µ 2 = a i s i=1 i=1 ( µ 2 F ) iɛ/2 S i ɛ ΦI,(i) (z, ɛ) ) ( q 2 µ 2 a i s ΦI,(i) (z, ɛ) R At O(a s(µ 2 F )) ΦI,(1) (z, ɛ) = Φ I,(1) (z, ɛ) R ) iɛ/2 Z i as ΦI,(i) (z, ɛ) X I = a i s (µ2 F )XI i i=1 G I (z, ɛ)) = a i s (µ2 F )GI i (z, ɛ)) i=1 Hence, X I 1 = AI 1δ(1 z) G I 1 (z, ɛ)) = f I 1 δ(1 z) + 2AI 1 D 0 + Explicit computation is required to determine O(ɛ) terms g I,k 1 (z). k=1 ɛ k g I,k 1 (z) Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 85

146 Fixing Form of Φ I... Alternative Method : We say our demands can be fulfilled if we assume the solution of KG Eq. ( ) Φ I = â i q 2 iɛ/2 s i=1 µ 2 S i ɛ ΦI,(i) (z, ɛ) with { } Φ I,(i) 1 (z, ɛ) iɛ [(1 z) 2] iɛ/2 φ I,(i) (ɛ) 1 z { (iɛ) j+1 = δ(1 z) + D j }φ I,(i) (ɛ) j=0 j! RGE invarinace of Φ I µ 2 d R dµ 2 K I = µ 2 d R dµ R 2 G I Y I R The solution φ I,(i) (ɛ) = ˆL I,(i) ( A I Y I, G I G I ) (ɛ). So 2Φ I,(1) (z, ɛ) = { 1 ( ɛ 2 4Y I ) { ɛ GI 1 }δ(1 (ɛ)) z) + 4Y 1 I ɛ } ɛ GI 1 (ɛ) ɛ j+1 D j j=0 j! Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 86

147 Fixing Form of Φ I... where Y I = a i s (µ2 F )XI i i=1 G I (ɛ) = a i s (µ2 F )GI i (ɛ) i=1 Hence, Y I 1 = AI 1 G I 1 (ɛ) = f I 1 + k=1 ɛ k G I,k 1 Explicit computation is required to determine the O(ɛ) terms G I,k 1. The methodology holds at every order. Both methods are equivalent. We have followed the 2nd one to perform our computations. Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 87

148 Extra Slide Extra 5 : Higgs N 3 LO Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 88

149 Band Plot for Higgs Boson Figure: Higgs boson production in gg fusion Anastasiou, Duhr, Dulat, Herzog, Mistlberger Threshold Corrections To DY and Higgs at N 3 LO QCD Bergische Universitat Wuppertal 89