Threshold cross sections for Drell-Yan & Higgs productions in N 3 LO QCD

Transcription

1 Narayan Rana 20/10/2016 1/46 Threshold cross sections for Drell-Yan & Higgs productions in N 3 LO QCD Narayan Rana 20/10/2016 in collaboration with T. Ahmed, M. C. Kumar, M. Mahakhud, M. K. Mandal, P. Mathews & V. Ravindran

2 Narayan Rana 20/10/2016 2/46 Outline 1 Prologue 2 Beyond NNLO 3 The universal structure of QCD 4 Threshold framework 5 Result

3 Prologue Narayan Rana 20/10/2016 3/46 Outline 1 Prologue 2 Beyond NNLO 3 The universal structure of QCD 4 Threshold framework 5 Result

4 Prologue Why Drell-Yan and Higgs Drell-Yan Large production rates and clean signatures Standard candle for detector calibration Testing physics within and beyond the SM Higgs The Nobel particle The discovery put the SM in firm footing Is it the Higgs or a Higgs? Need detail info about it s quantum nature We need precise theoretical prediction Narayan Rana 20/10/2016 4/46

5 Prologue Narayan Rana 20/10/2016 5/46 State-of-the-art : DY production NNLO NNLO inclusive. Altarelli, Ellis, Martinelli (1979). Hamberg, Matsuura, van Neerven (1991). Harlander, Kilgore (2002) NNLO differential. Anastasiou, Dixon, Melnikov, Petriello (2003). Melnikov, Petriello (2006). Catani, Cieri, Ferrera, de Florian, Grazzini (2009)

6 Prologue Narayan Rana 20/10/2016 6/46 State-of-the-art : Higgs production in gluon fusion NNLO NNLO inclusive (large m t approximation). Harlander, Kilgore (2002). Anastasiou, Melnikov (2002). Ravindran, Smith, van Neerven (2003) NNLO inclusive (1/m t expansion). Marzani, Ball, Del Duca, Forte, Vicini (2008). Harlander, Ozeren (2009). Pak, Rogal, Steinhauser (2010) N 3 LO soft approximation (partial). Moch, Vogt (2005). Ravindran (2006) NNLO + NNLL resummation. de Florian, Grazzini (2012)

7 Beyond NNLO Narayan Rana 20/10/2016 7/46 Outline 1 Prologue 2 Beyond NNLO 3 The universal structure of QCD 4 Threshold framework 5 Result

8 Beyond NNLO Narayan Rana 20/10/2016 8/46 Going beyond NNLO: Threshold approximation σ I (τ, q 2 ) = 1 S a, b 1 τ dx ( x Φ ab(x) ŝ ˆσ ab I τ x, q2) τ = q2 S, I = g, q, b

9 Beyond NNLO Going beyond NNLO: Threshold approximation σ I (τ, q 2 ) = 1 S a, b 1 τ dx ( x Φ ab(x) ŝ ˆσ ab I τ x, q2) τ = q2 S, I = g, q, b Partonic flux Φ ab (x) becomes large when x τ or z = q2 ŝ = τ x 1 Dominant contributions come from the region z 1. For DY N(N)LO SV 0.95 N(N)LO Narayan Rana 20/10/2016 8/46

10 Beyond NNLO Going beyond NNLO: Threshold approximation σ I (τ, q 2 ) = 1 S a, b 1 τ dx ( x Φ ab(x) ŝ ˆσ ab I τ x, q2) τ = q2 S, I = g, q, b Partonic flux Φ ab (x) becomes large when x τ or z = q2 ŝ = τ x 1 Dominant contributions come from the region z 1. For DY N(N)LO SV 0.95 N(N)LO z 1 is called the soft limit. Expand the partonic cross section around z = 1. Narayan Rana 20/10/2016 8/46

11 Narayan Rana 20/10/2016 9/46 Beyond NNLO Expand the partonic cross section around z = 1 σ I (z) = sv (z) + (1 z) i H,(i) i=0

12 Narayan Rana 20/10/2016 9/46 Beyond NNLO Expand the partonic cross section around z = 1 σ I (z) = sv (z) + sv,δ δ(1 z) + (1 z) i H,(i) i=0 ( ) sv,(k) ln k (1 z) 1 z k=0 + ( ) ln k (1 z) D k (z) 1 z +

13 Narayan Rana 20/10/2016 9/46 Beyond NNLO Expand the partonic cross section around z = 1 σ I (z) = sv (z) + sv,δ δ(1 z) + (1 z) i H,(i) i=0 ( ) sv,(k) ln k (1 z) 1 z k=0 + ( ) ln k (1 z) D k (z) 1 z + Different methods Matrix element square and phase space integrals in the soft limit [Catani et al., Harlander and Kilgore] Form factors and DGLAP kernels [Moch and Vogt; Ravindran, Smith, van Neerven] Soft collinear effective theory [Becher and Neubert]

14 The universal structure of QCD Narayan Rana 20/10/ /46 Outline 1 Prologue 2 Beyond NNLO 3 The universal structure of QCD 4 Threshold framework 5 Result

15 The universal structure of QCD Narayan Rana 20/10/ /46 QCD Divergences The higher order computations in QCD contain two types of divergences Ultraviolet Infrared

16 The universal structure of QCD Narayan Rana 20/10/ /46 QCD Divergences The higher order computations in QCD contain two types of divergences Ultraviolet Infrared Removed by UV renormalization

17 The universal structure of QCD Narayan Rana 20/10/ /46 QCD Divergences The higher order computations in QCD contain two types of divergences Ultraviolet Infrared Soft divergences (k 0 0) soft emissions + IR in virtual Collinear divergences (cos θ pk 1) sum over degenerate states 1 (p k) 2 = 1 2p 0 k 0 (1 cos θ pk )

18 The universal structure of QCD Narayan Rana 20/10/ /46 QCD Factorization The hadronic cross section following parton model S σ I (τ) = a,b f B a (τ) f B b (τ) }{{} bare PDFs ŝ ˆσ B ab (τ, ɛ c ) }{{} partonic cross section

19 The universal structure of QCD QCD Factorization The hadronic cross section following parton model S σ I (τ) = a,b f B a (τ) f B b (τ) }{{} bare PDFs ŝ ˆσ B ab (τ, ɛ c ) }{{} partonic cross section The bare partonic cross section ˆσ ab B (τ, ɛ c) contains collinear divergences. ˆσ ab B (τ, ɛ c ) = ( ) ˆσ B,V ab (τ, ɛ s, ɛ c ) + ˆσ B,R ab (τ, ɛ s, ɛ c ) deg. states ɛ s : soft gluon regulator ɛ c : collinear parton regulator Narayan Rana 20/10/ /46

20 The universal structure of QCD Narayan Rana 20/10/ /46 QCD Factorization The hadronic cross section following parton model S σ I (τ) = a,b f B a (τ) f B b (τ) }{{} bare PDFs ŝ ˆσ B ab (τ, ɛ c ) }{{} partonic cross section The bare partonic cross section ˆσ ab B (τ, ɛ c) contains collinear divergences. ˆσ ab B (τ, ɛ c ) = ( ) ˆσ B,V ab (τ, ɛ s, ɛ c ) + ˆσ B,R ab (τ, ɛ s, ɛ c ) deg. states ɛ s : soft gluon regulator ɛ c : collinear parton regulator KLN theorem KLN theorem : : Summing over the degenerate final states remove the soft divergences and final state collinear divergences. Initial state collinear singularity remains

21 The universal structure of QCD Narayan Rana 20/10/ /46 Collinear singularity factorizes ˆσ B ab (τ, ɛ c ) = cd Γ ca (τ, µ F, ɛ c ) Γ db (τ, µ F, ɛ c ) ˆσ I ab (τ, µ F ) In MS scheme f a (τ, µ F ) = Γ ac (τ, µ F, ɛ c ) f B c (τ)

22 The universal structure of QCD Narayan Rana 20/10/ /46 Collinear singularity factorizes ˆσ B ab (τ, ɛ c ) = cd Γ ca (τ, µ F, ɛ c ) Γ db (τ, µ F, ɛ c ) ˆσ I ab (τ, µ F ) In MS scheme f a (τ, µ F ) = Γ ac (τ, µ F, ɛ c ) f B c (τ) Renormalized version of parton model S σ I (τ) = ab f a (τ, µ F ) f b (τ, µ F ) ab (τ, µ F )

23 The universal structure of QCD Narayan Rana 20/10/ /46 Collinear singularity factorizes ˆσ B ab (τ, ɛ c ) = cd Γ ca (τ, µ F, ɛ c ) Γ db (τ, µ F, ɛ c ) ˆσ I ab (τ, µ F ) In MS scheme f a (τ, µ F ) = Γ ac (τ, µ F, ɛ c ) f B c (τ) Renormalized version of parton model S σ I (τ) = ab f a (τ, µ F ) f b (τ, µ F ) ab (τ, µ F ) f 1(x)... f n(x) = 1 dx dx nf 1(x 1)... f n(x n)δ(x x 1... x n)

24 Threshold framework Narayan Rana 20/10/ /46 Outline 1 Prologue 2 Beyond NNLO 3 The universal structure of QCD 4 Threshold framework 5 Result

25 Threshold framework Narayan Rana 20/10/ /46 ab (z, µ F ) can be expanded order by order in a s [ ] ab (z, µ F ) = δ(1 z) + a s (µ 2 R) a 11 δ(1 z) + a 12 D 0 + a 13 D 1 + R 1 (z) [ ] = + a 2 s(µ 2 R) a 21 δ(1 z) a 25 D 3 + R 2 (z) +...

26 Threshold framework Narayan Rana 20/10/ /46 ab (z, µ F ) can be expanded order by order in a s [ ] ab (z, µ F ) = δ(1 z) + a s (µ 2 R) a 11 δ(1 z) + a 12 D 0 + a 13 D 1 + R 1 (z) [ ] = + a 2 s(µ 2 R) a 21 δ(1 z) a 25 D 3 + R 2 (z) +... Contributions from the soft distribution functions factorizes & exponentiates [Catani, Collins, Soper, Sterman]

27 Threshold framework Narayan Rana 20/10/ /46 ab (z, µ F ) can be expanded order by order in a s [ ] ab (z, µ F ) = δ(1 z) + a s (µ 2 R) a 11 δ(1 z) + a 12 D 0 + a 13 D 1 + R 1 (z) [ ] = + a 2 s(µ 2 R) a 21 δ(1 z) a 25 D 3 + R 2 (z) +... Contributions from the soft distribution functions factorizes & exponentiates [Catani, Collins, Soper, Sterman] Due to the factorization property of UV, Soft and Collinear terms, the complete threshold part ( sv ab ) of the partonic cross section, hence, acquires the following structure ( ) 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

28 Threshold framework Narayan Rana 20/10/ /46 ab (z, µ F ) can be expanded order by order in a s [ ] ab (z, µ F ) = δ(1 z) + a s (µ 2 R) a 11 δ(1 z) + a 12 D 0 + a 13 D 1 + R 1 (z) [ ] = + a 2 s(µ 2 R) a 21 δ(1 z) a 25 D 3 + R 2 (z) +... Contributions from the soft distribution functions factorizes & exponentiates [Catani, Collins, Soper, Sterman] Due to the factorization property of UV, Soft and Collinear terms, the complete threshold part ( sv ab ) of the partonic cross section, hence, acquires the following structure ( ) 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 Ce f(z) = δ(1 z) + 1 1! f(z) + 1 f(z) f(z) + 2!

29 Threshold framework Narayan Rana 20/10/ /46 The finite distribution Ψ ( Ψ I (z, q 2, µ 2 R, µ 2 F, ɛ) = ln ˆF I (â s, Q 2, µ 2, ɛ) 2 δ(1 z) ˆF I (â s, Q 2, µ 2, ɛ) is the bare form factor

30 Threshold framework Narayan Rana 20/10/ /46 The finite distribution Ψ ( Ψ I (z, q 2, µ 2 R, µ 2 F, ɛ) = ln ˆF I (â s, Q 2, µ 2, ɛ) 2 δ(1 z) [ 2δ(1 + ln Z I (â s, µ 2 R, µ 2, ɛ)] z) ˆF I (â s, Q 2, µ 2, ɛ) is the bare form factor Z I (â s, µ 2 R, µ2, ɛ) is the overall renormalization constant

31 Threshold framework Narayan Rana 20/10/ /46 The finite distribution Ψ ( Ψ I (z, q 2, µ 2 R, µ 2 F, ɛ) = ln ˆF I (â s, Q 2, µ 2, ɛ) 2 δ(1 z) [ 2δ(1 + ln Z I (â s, µ 2 R, µ 2, ɛ)] z) 2m C ln Γ II (â s, µ 2, µ 2 F, z, ɛ) ˆF I (â s, Q 2, µ 2, ɛ) is the bare form factor Z I (â s, µ 2 R, µ2, ɛ) is the overall renormalization constant Γ II (â s, µ 2, µ 2 F, z, ɛ) is the mass factorization kernel m = 1 for Drell-Yan & Higgs, m = 1 2 for DIS

32 Threshold framework Narayan Rana 20/10/ /46 The finite distribution Ψ ( Ψ I (z, q 2, µ 2 R, µ 2 F, ɛ) = ln ˆF I (â s, Q 2, µ 2, ɛ) 2 δ(1 z) [ 2δ(1 + ln Z I (â s, µ 2 R, µ 2, ɛ)] z) 2m C ln Γ II (â s, µ 2, µ 2 F, z, ɛ) ) + 2Φ I (â s, q 2, µ 2, z, ɛ) ˆF I (â s, Q 2, µ 2, ɛ) is the bare form factor Z I (â s, µ 2 R, µ2, ɛ) is the overall renormalization constant Γ II (â s, µ 2, µ 2 F, z, ɛ) is the mass factorization kernel Φ I (â s, q 2, µ 2, z, ɛ) is the soft distribution function m = 1 for Drell-Yan & Higgs, m = 1 2 for DIS

33 Threshold framework Narayan Rana 20/10/ /46 The finite distribution Ψ ( Ψ I (z, q 2, µ 2 R, µ 2 F, ɛ) = ln ˆF I (â s, Q 2, µ 2, ɛ) 2 δ(1 z) UV, C, S [ 2δ(1 + ln Z I (â s, µ 2 R, µ 2, ɛ)] z) UV, C, S 2m C ln Γ II (â s, µ 2, µ 2 F, z, ɛ) ) + 2Φ I (â s, q 2, µ 2, z, ɛ) UV,C, S UV,C, S ˆF I (â s, Q 2, µ 2, ɛ) is the bare form factor Z I (â s, µ 2 R, µ2, ɛ) is the overall renormalization constant Γ II (â s, µ 2, µ 2 F, z, ɛ) is the mass factorization kernel Φ I (â s, q 2, µ 2, z, ɛ) is the soft distribution function m = 1 for Drell-Yan & Higgs, m = 1 2 for DIS

34 Threshold framework Narayan Rana 20/10/ /46 Form Factor & Sudakov resummation Mueller, Collins, Sen, Sudakov The bare form factors ˆF I (â s, Q 2, µ 2, ɛ) of both fermionic and gluonic operators satisfy the following integro differential equation that follows form the gauge as well as the RG invariances. Q 2 d dq 2 ln ˆF I ( â s, Q 2, µ 2, ɛ ) = 1 ( ) )] [K I â s, µ2 R 2 µ 2, ɛ + G (â I s, Q2 µ 2, µ2 R R µ 2, ɛ all the poles in ɛ terms finite as ɛ 0

35 Threshold framework Narayan Rana 20/10/ /46 Form Factor & Sudakov resummation Mueller, Collins, Sen, Sudakov The bare form factors ˆF I (â s, Q 2, µ 2, ɛ) of both fermionic and gluonic operators satisfy the following integro differential equation that follows form the gauge as well as the RG invariances. Q 2 d dq 2 ln ˆF I ( â s, Q 2, µ 2, ɛ ) = 1 ( ) )] [K I â s, µ2 R 2 µ 2, ɛ + G (â I s, Q2 µ 2, µ2 R R µ 2, ɛ µ R independence of ˆF I d µ 2 R dµ 2 R K I = µ 2 R all the poles in ɛ terms finite as ɛ 0 d dµ 2 R G I = a i s(µ 2 R)A I i i=1 A I s are cusp anomalous dimensions

36 Threshold framework Narayan Rana 20/10/ /46 We expand K I & G I ( ) K I â s, µ2 R µ 2, ɛ G I ( â s, Q2 µ 2, µ2 R R µ 2, ɛ ( ) µ = â i 2 i ɛ 2 R s µ i=1 2 Sɛ i K I,(i) (ɛ) ) = a i s (Q2 ) G I i (ɛ) + â i s i=1 i=1 ( µ 2 R µ 2 ) i ɛ ( ) 2 i Q 2 ɛ 2 µ 2 1 Sɛ i KI,(i) (ɛ) R The solutions for K I,(i) s ( ) K I,(1) (ɛ) = 1 ɛ 2A I 1 ) ( ) K I,(2) (ɛ) = (2β 1 ɛ 2 0 A I A I 2 ɛ ( ) ( ) ( ) K I,(3) (ɛ) = 1 ɛ β2 0A I 1 + 1ɛ2 2 3 β 1A I β 0A I ɛ 3 AI 3

37 Threshold framework Narayan Rana 20/10/ /46 To solve the KG equation, we expand ln ˆF I as ln ˆF I (â s, Q 2, µ 2, ɛ) = i=1 â i s Vogt, Vermaseren, Moch, Ravindran ( Q 2 µ 2 ) i ɛ 2 S i ɛ I,(i) ˆL F (ɛ)

38 Threshold framework Narayan Rana 20/10/ /46 To solve the KG equation, we expand ln ˆF I as ln ˆF I (â s, Q 2, µ 2, ɛ) = The formal solution up to three loops i=1 â i s ˆL I,(1) (ɛ) = 1 { } F ɛ 2 2A I } {G I1 (ɛ) ɛ ˆL I,(2) (ɛ) = 1 { } F ɛ 3 β 0 A I { ɛ 2 ˆL I,(3) (ɛ) = 1 { F ɛ 4 8 } β0 2 AI { 2 9 ɛ ɛ 2 { Vogt, Vermaseren, Moch, Ravindran ( Q 2 µ 2 ) i ɛ 2 S i ɛ 1 } A I 2 β 0 GI 1 (ɛ) + 1 { 1 2 ɛ I,(i) ˆL F (ɛ) } G I 2 (ɛ) 2 β 1 A I β 0 A I } β0 2 GI 1 (ɛ) A I 3 1 β 1 G I 1 (ɛ) 4 } β 0 G I 2 (ɛ) + 1 { ɛ } G I 3 (ɛ) 3 A I s are maximally non-abelian A g i = C A C F A q i

39 Threshold framework Narayan Rana 20/10/ /46 To solve the KG equation, we expand ln ˆF I as ln ˆF I (â s, Q 2, µ 2, ɛ) = The formal solution up to three loops i=1 â i s ˆL I,(1) (ɛ) = 1 { } F ɛ 2 2A I } {G I1 (ɛ) ɛ ˆL I,(2) (ɛ) = 1 { } F ɛ 3 β 0 A I { ɛ 2 ˆL I,(3) (ɛ) = 1 { F ɛ 4 8 } β0 2 AI { 2 9 ɛ ɛ 2 { Vogt, Vermaseren, Moch, Ravindran ( Q 2 µ 2 ) i ɛ 2 S i ɛ 1 } A I 2 β 0 GI 1 (ɛ) + 1 { 1 2 ɛ I,(i) ˆL F (ɛ) } G I 2 (ɛ) 2 β 1 A I β 0 A I } β0 2 GI 1 (ɛ) A I 3 1 β 1 G I 1 (ɛ) 4 } β 0 G I 2 (ɛ) + 1 { ɛ All poles except the single one can be predicted. } G I 3 (ɛ) 3 A I s are maximally non-abelian A g i = C A C F A q i

40 Threshold framework Narayan Rana 20/10/ /46 Ravindran, Smith, van Neerven Explicit computation of two loop ˆF g and ˆF q in SU(N) reveals a structure of G I s G I i = 2(B I i γ I i) + f I i + C I i + k=1 ɛ k g I,(k) i C A C F

41 Threshold framework Narayan Rana 20/10/ /46 Ravindran, Smith, van Neerven Explicit computation of two loop ˆF g and ˆF q in SU(N) reveals a structure of G I s G I i = 2(B I i γ I i) + f I i + C I i + k=1 ɛ k g I,(k) i C A C F Collinear anomalous dimensions

42 Threshold framework Narayan Rana 20/10/ /46 Ravindran, Smith, van Neerven Explicit computation of two loop ˆF g and ˆF q in SU(N) reveals a structure of G I s G I i = 2(B I i γ I i) + f I i + C I i + k=1 ɛ k g I,(k) i C A C F UV anomalous dimensions

43 Threshold framework Narayan Rana 20/10/ /46 Ravindran, Smith, van Neerven Explicit computation of two loop ˆF g and ˆF q in SU(N) reveals a structure of G I s G I i = 2(B I i γ I i) + f I i + C I i + k=1 ɛ k g I,(k) i C A C F Soft anomalous dimensions f g i = C A C F f q i

44 Threshold framework Narayan Rana 20/10/ /46 Ravindran, Smith, van Neerven Explicit computation of two loop ˆF g and ˆF q in SU(N) reveals a structure of G I s G I i = 2(B I i γ I i) + f I i + C I i + k=1 ɛ k g I,(k) i C A C F Soft C I 1 = 0 C2 I = 2β 0g I,(1) 1 ( ) C3 I = 2β 1g I,(1) 1 2β 0 g I,(1) 2 + 2β 0g I,(2) 1 ( C4 I = 2β 2g I,(1) 1 2β 1 g I,(1) 2 + 4β 0g I,(2) 1 ) 2β 0 ( ) g I,(1) 3 + 2β 0g I,(2) 2 + 4β0g 2 I,(3) 1

45 Threshold framework Narayan Rana 20/10/ /46 Ravindran, Smith, van Neerven Explicit computation of two loop ˆF g and ˆF q in SU(N) reveals a structure of G I s C A C F C I 1 = 0 G I i = 2(B I i γ I i) + f I i + C I i + k=1 ɛ k g I,(k) i The single pole can be predicted C2 I = 2β 0g I,(1) 1 ( ) C3 I = 2β 1g I,(1) 1 2β 0 g I,(1) 2 + 2β 0g I,(2) 1 ( C4 I = 2β 2g I,(1) 1 2β 1 g I,(1) 2 + 4β 0g I,(2) 1 ) 2β 0 ( ) g I,(1) 3 + 2β 0g I,(2) 2 + 4β0g 2 I,(3) 1

46 Threshold framework Narayan Rana 20/10/ /46 DGLAP equation The DGLAP kernels satisfy RG equation d µ 2 F dµ 2 F Γ(z, µ 2 F, ɛ c ) = 1 2 P (z, µ2 F ) Γ(z, µ 2 F, ɛ c ) Both the kernels Γ and the splitting functions P can be expanded in a s. The RGE can be solved to get Γ i in terms of P i 1. The diagonal terms of the splitting functions have the following structure P (i) II (z) = 2 [ B I i+1δ(1 z) + A I i+1d 0 ] + P (i) II,reg (z) P (i) II,reg (z) are regular when z 1.

47 Threshold framework Narayan Rana 20/10/ /46 An ansatz for soft distribution function Ravindran The finiteness of Ψ I demands a integro differential equation for Φ I, similar to the Form factors q 2 d ( dq 2 ΦI â s, q 2, µ 2, z, ɛ ) = 1 2 ( ) )] [K I â s, µ2 R µ, z, ɛ + G (â I s, q2, µ2 R 2 µ 2 R µ, z, ɛ 2 The solutions for Φ I can be obtained similar to ln ˆF I, by expanding Φ I ( â s, q 2, µ 2, z, ɛ ) = i=1 â i s ( q 2 µ 2 ) i ɛ 2 S i ɛ ˆΦ I,(i) (z, ɛ) ˆΦ I,(i) (ɛ) = L I,(i) F (ɛ) ( A I δ(1 z)a I,G I (ɛ) G I ) (z,ɛ)

48 Threshold framework Narayan Rana 20/10/ /46 An ansatz for soft distribution function Ravindran The finiteness of Ψ I demands a integro differential equation for Φ I, similar to the Form factors q 2 d ( dq 2 ΦI â s, q 2, µ 2, z, ɛ ) = 1 2 ( ) )] [K I â s, µ2 R µ, z, ɛ + G (â I s, q2, µ2 R 2 µ 2 R µ, z, ɛ 2 The solutions for Φ I can be obtained similar to ln ˆF I, by expanding Φ I ( â s, q 2, µ 2, z, ɛ ) = i=1 â i s ( q 2 µ 2 ) i ɛ 2 S i ɛ ˆΦ I,(i) (z, ɛ) ˆΦ I,(i) (ɛ) = L I,(i) F (ɛ) ( A I δ(1 z)a I,G I (ɛ) G I ) (z,ɛ)

49 Threshold framework Narayan Rana 20/10/ /46 Instead, observed form of soft functions suggests to expand Φ I as Φ I (â s, q 2, µ 2, z, ɛ) = i=1 â i s ( q 2 (1 z) 2m µ 2 ) i ɛ 2 ( ) S i imɛ ɛ ˆφ I,(i) (ɛ) 1 z ˆφ I,(i) (ɛ) = L I,(i) F (ɛ) (A I A I,G I (ɛ) G I (ɛ)) 1 [ K I,(i) + G I,(i)] iɛ I,(i) I G are related to G through δ(1 z) and D j (1 z) 1+nɛ = 1 nɛ δ(1 z) + D 0 + ( nɛ)d 1 + ( nɛ)2 D

50 Threshold framework Narayan Rana 20/10/ /46 Expanding in a s i=1 â i s ( ) iɛ q 2 2 z S i ɛg I,(i) = µ 2 a i s(qz)g 2 I i(ɛ) q 2 z =q 2 (1 z) 2 i=1

51 Threshold framework Narayan Rana 20/10/ /46 Expanding in a s i=1 â i s ( ) iɛ q 2 2 z S i ɛg I,(i) = µ 2 a i s(qz)g 2 I i(ɛ) q 2 z =q 2 (1 z) 2 i=1 provides a structure of G I i G I i(ɛ) = f I i + C I i + k=1 ɛ k G I,k i where C I i = Ci I ( g I,(k) i ) G I,k i and G g,k i = C A C F G q,k i Φ g = C A C F Φ q

52 Threshold framework Narayan Rana 20/10/ /46 The threshold cross section

53 Result Narayan Rana 20/10/ /46 Outline 1 Prologue 2 Beyond NNLO 3 The universal structure of QCD 4 Threshold framework 5 Result

54 Result Narayan Rana 20/10/ /46 The available ingredients 1 Form factors up to 3 loops ˆF q and ˆF g [Moch et al.; Baikov et al.; Gehrman et al.;] ˆF b [Gehrman, Kara] 2 Renormalization constant up to 3 loops Z q = 1 Z g [Chetyrkin, Kniehl, Steinhauser] Z b [van Ritbergen, Vermaseren, Larin; Czakon] 3 Splitting functions up to third order P (i) (z) [Moch, Vermaseren, Vogt] 4 Soft distribution function up to 2 loops G I,k i [de Florian, Mazzitelli] sv I,3 = sv I,3 δ δ(1 z) + sv I,3 D0 D 0 + sv I,3 D1 D sv I,4 = sv I,4 δ δ(1 z) + sv I,4 D0 D 0 + sv I,4 D1 D 1 + sv I,4 D2 D

55 Result Narayan Rana 20/10/ /46 The available ingredients 1 Form factors up to 3 loops ˆF q and ˆF g [Moch et al.; Baikov et al.; Gehrman et al.;] ˆF b [Gehrman, Kara] 2 Renormalization constant up to 3 loops Z q = 1 Z g [Chetyrkin, Kniehl, Steinhauser] Z b [van Ritbergen, Vermaseren, Larin; Czakon] 3 Splitting functions up to third order P (i) (z) [Moch, Vermaseren, Vogt] 4 Soft distribution function up to 2 loops G I,k i [de Florian, Mazzitelli] sv I,3 = sv I,3 δ δ(1 z) + sv I,3 D0 D 0 + sv I,3 D1 D sv I,4 = sv I,4 δ δ(1 z) + sv I,4 D0 D 0 + sv I,4 D1 D 1 + sv I,4 D2 D

56 Result Ahmed, Mahakhud, NR, Ravindran The complete threshold N 3 LO for Higgs production provides sv g,3 δ. We extract G g,1 3. Using the maximally non abelian nature of the soft distribution function, we obtain G q,1 3. Narayan Rana 20/10/ /46

57 Result Ahmed, Mahakhud, NR, Ravindran The complete threshold N 3 LO for Higgs production provides sv g,3 δ. We extract G g,1 3. Using the maximally non abelian nature of the soft distribution function, we obtain G q,1 3. Narayan Rana 20/10/ /46

58 Result Narayan Rana 20/10/ /46 Ahmed, Mahakhud, NR, Ravindran The complete threshold N 3 LO for Higgs production provides sv g,3 δ. We extract G g,1 3. Using the maximally non abelian nature of the soft distribution function, we obtain G q,1 3. G I,1 3 { 2 ( = C I C A ζ 2 + ζ 2 + ζ 2 ζ ζ ζ 3 ζ ( + C A n f ζ 2 ζ 2 ζ ζ ζ ζ ) ( C F n f ζ 2 88 ζ2 ζ ζ ζ ζ ) ( + n f ζ 2 ζ ζ ) } ζ ) 8748

59 Result Narayan Rana 20/10/ /46 The new results sv I,3 = sv I,3 δ δ(1 z) + sv I,3 D0 D 0 + sv I,3 D1 D sv I,4 = sv I,4 δ δ(1 z) + sv I,4 D0 D 0 + sv I,4 D1 D 1 + sv I,4 D2 D SV ( q,3 δ = C2 A C F ζ 2 + ζ 2 ζ 2 ζ ζ ζ 3 + ζ ζ ) C A CF 2 ( ζ 2 ζ 2 + ζ 2 ζ ζ ζ 3 ζ ζ ) ( + C A C F n f ζ 2 + ζ 2 ζ ζ ζ 3 8 ζ ) C F 3 ( ζ 2 + ζ ζ2 ζ ζ ζ ζ ζ ) C F 2 ( n f ζ 2 ζ 2 ζ ζ ζ ζ ) ( N 2 4 ) ( + C F n f,v 4 2 ζ ζ ζ ) ζ N C F n 2 ( f ζ 2 + ζ ζ )

60 Result Narayan Rana 20/10/ /46 The new results sv I,3 = sv I,3 δ δ(1 z) + sv I,3 D0 D 0 + sv I,3 D1 D sv I,4 = sv I,4 δ δ(1 z) + sv I,4 D0 D 0 + sv I,4 D1 D 1 + sv I,4 D2 D A I 4 and f I 4 computed by Padé approximation. SV ( q,3 δ = C2 A C F ζ 2 + ζ 2 ζ 2 ζ ζ ζ 3 + ζ ζ ) C A CF 2 ( ζ 2 ζ 2 + ζ 2 ζ ζ ζ 3 ζ ζ ) ( + C A C F n f ζ 2 + ζ 2 ζ ζ ζ 3 8 ζ ) C F 3 ( ζ 2 + ζ ζ2 ζ ζ ζ ζ ζ ) C F 2 ( n f ζ 2 ζ 2 ζ ζ ζ ζ ) ( N 2 4 ) ( + C F n f,v 4 2 ζ ζ ζ ) ζ N C F n 2 ( f ζ 2 + ζ ζ )

61 Result Narayan Rana 20/10/ /46 DY : total cross section Stable convergence in perturbation The δ contribution is almost equal and opposite in sign to the sum of the contributions from the D i s Q (GeV) δ N 3 LO D N 3 LO NNLO (SV) NNLO N 3 LO (SV) N 3 LO SV

62 Result DY : scale variation 1.15 NLO 1.1 NNLO Q = 20 GeV 3 N LO SV Reduction in scale dependence Note : Significant increase in scale uncertainties for low µ R Due to presence of large logarithms Resummation improve the scenario (i) R (i) R Q = 200 GeV µ /Q R Narayan Rana 20/10/ /46

63 Result DY : scale variation 1.15 NLO 1.1 NNLO Q = 20 GeV 3 N LO SV Reduction in scale dependence Note : Significant increase in scale uncertainties for low µ R Due to presence of large logarithms Resummation improve the scenario (i) R (i) R Q = 200 GeV µ /Q R Narayan Rana 20/10/ /46

64 Result DY : scale variation 1.15 NLO 1.1 NNLO Q = 20 GeV 3 N LO SV Reduction in scale dependence Note : Significant increase in scale uncertainties for low µ R Due to presence of large logarithms Resummation improve the scenario (i) R (i) R Q = 200 GeV µ /Q R Narayan Rana 20/10/ /46

65 Result DY : scale variation 1.15 NLO 1.1 NNLO Q = 20 GeV 3 N LO SV Reduction in scale dependence Note : Significant increase in scale uncertainties for low µ R Due to presence of large logarithms Resummation improve the scenario (i) R (i) R Q = 200 GeV µ /Q R Narayan Rana 20/10/ /46

66 Result Narayan Rana 20/10/ /46 Higgs production in bottom quark fusion Ahmed, NR, Ravindran 0.7 LO NLO NNLO 3 N LO SV 1.7 LO NLO NNLO 3 N LO SV 0.6 σ [pb] 0.5 σ [pb] µ F = M H /4 µ R = M H µ /m R H µ /m F H

67 Result Narayan Rana 20/10/ /46 Pseudo-scalar Higgs production in gluon fusion Ahmed, Kumar, Mathews, NR, Ravindran dσ (pp A) (pb) LHC 13 TeV MSTW 2008 µ R = µ F = m A K-factors dσ (pp A) (pb) LHC 13 TeV MSTW2008 µ R = µ F = m A LO NLO NNLO N 3 LO sv 1.8 K (1) K (2) 1.6 K (3) m A (GeV) m A (GeV)

68 Result Narayan Rana 20/10/ /46 Pseudo-scalar Higgs production in gluon fusion Ahmed, Kumar, Mathews, NR, Ravindran dσ (pp A) (pb) LHC 13 TeV MSTW2008 µ = µ R = µ F m A = 200 GeV dσ (pp A) (pb) LHC 13 TeV MSTW2008 µ F = m A = 200 GeV LO NLO NNLO N 3 LO (sv) 10 0 LO NLO NNLO N 3 LO (sv) µ / m A µ R / m A

69 Result Narayan Rana 20/10/ /46 Rapidity 3 LO sv Ahmed, Mandal, NR, Ravindran In the large N limit, ˆφ I,(n) d = Γ(1 + nɛ) I,(n) ˆφ Γ 2 (1 + n ɛ 2 ) DY production Higgs production in gluon fusion Higgs production in bottom quark fusion

70 Result Narayan Rana 20/10/ /46 Remarks (1) Factorization of UV, soft and collinear contributions (2) Universal structure of the Form factor singularities (3) Soft distribution is maximally non abelian (4) Inclusive cross section and rapidity distributions can be obtained up to threshold in N 3 LO

71 Result Narayan Rana 20/10/ /46 Remarks (1) Factorization of UV, soft and collinear contributions (2) Universal structure of the Form factor singularities (3) Soft distribution is maximally non abelian (4) Inclusive cross section and rapidity distributions can be obtained up to threshold in N 3 LO

72 Result Narayan Rana 20/10/ /46 Remarks (1) Factorization of UV, soft and collinear contributions (2) Universal structure of the Form factor singularities (3) Soft distribution is maximally non abelian (4) Inclusive cross section and rapidity distributions can be obtained up to threshold in N 3 LO

73 Result Narayan Rana 20/10/ /46 Remarks (1) Factorization of UV, soft and collinear contributions (2) Universal structure of the Form factor singularities (3) Soft distribution is maximally non abelian (4) Inclusive cross section and rapidity distributions can be obtained up to threshold in N 3 LO

74 Result Narayan Rana 20/10/ /46 Thanks for your patience