Soft gluon contributions to Drell-Yan and Higgs productions beyond - PowerPoint PPT Presentation

Soft part of NNLO Catani et al, Harlander and Kilgore Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab • Φ ab ( x ) becomes large when LHC ( S = (14 TeV)^2) x → x min = τ 1e+07 qqb gg (0.01) qg 1e+06 • Dominant contribution to Higgs production comes from the region when x → τ 100000 φ ab 10000 • It is sufficient if we know the partonic cross section when x → τ 1000 100 • x → τ is called soft limit . 10 0 0.001 0.002 0.003 0.004 0.005 0.006 x = Q 2 /S Gluon flux is largest at LHC - p. 5/24

Soft part of NNLO Catani et al, Harlander and Kilgore Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab • Φ ab ( x ) becomes large when LHC ( S = (14 TeV)^2) x → x min = τ 1e+07 qqb gg (0.01) qg 1e+06 • Dominant contribution to Higgs production comes from the region when x → τ 100000 φ ab 10000 • It is sufficient if we know the partonic cross section when x → τ 1000 100 • x → τ is called soft limit . 10 0 0.001 0.002 0.003 0.004 0.005 0.006 x = Q 2 /S • Expand the partonic cross section around x = τ . Gluon flux is largest at LHC - p. 5/24

Soft part Catani et al, Harlander and Kilgore - p. 6/24

Soft part Catani et al, Harlander and Kilgore • Expand the partonic cross section around x = τ or z = x τ = 1 . - p. 6/24

Soft part Catani et al, Harlander and Kilgore • Expand the partonic cross section around x = τ or z = x τ = 1 . ∞ z = x X σ ( z ) = C (0) ( z ) + (1 − z ) i C ( i ) d ˆ τ i =1 - p. 6/24

Soft part Catani et al, Harlander and Kilgore • Expand the partonic cross section around x = τ or z = x τ = 1 . ∞ z = x X σ ( z ) = C (0) ( z ) + (1 − z ) i C ( i ) d ˆ τ i =1 • C (0) : ∞ ! log k (1 − z ) C (0) = C (0) C ( k ) X δ (1 − z ) + 0 0 (1 − z ) k =0 + - p. 6/24

Soft part Catani et al, Harlander and Kilgore • Expand the partonic cross section around x = τ or z = x τ = 1 . ∞ z = x X σ ( z ) = C (0) ( z ) + (1 − z ) i C ( i ) d ˆ τ i =1 • C (0) : ∞ ! log k (1 − z ) C (0) = C (0) C ( k ) X δ (1 − z ) + 0 0 (1 − z ) k =0 + • C ( i ) will be pure constants such as ζ (2) , ζ (3) . 0 - p. 6/24

Soft part Catani et al, Harlander and Kilgore • Expand the partonic cross section around x = τ or z = x τ = 1 . ∞ z = x X σ ( z ) = C (0) ( z ) + (1 − z ) i C ( i ) d ˆ τ i =1 • C (0) : ∞ ! log k (1 − z ) C (0) = C (0) C ( k ) X δ (1 − z ) + 0 0 (1 − z ) k =0 + • C ( i ) will be pure constants such as ζ (2) , ζ (3) . 0 • Compute the entire cross section in the "soft limit". - p. 6/24

Soft part Catani et al, Harlander and Kilgore • Expand the partonic cross section around x = τ or z = x τ = 1 . ∞ z = x X σ ( z ) = C (0) ( z ) + (1 − z ) i C ( i ) d ˆ τ i =1 • C (0) : ∞ ! log k (1 − z ) C (0) = C (0) C ( k ) X δ (1 − z ) + 0 0 (1 − z ) k =0 + • C ( i ) will be pure constants such as ζ (2) , ζ (3) . 0 • Compute the entire cross section in the "soft limit". OR Extract from "Form factors and DGLAP kernels" using 1) Factorisation theorem 2) Renormalisation Group Invariance 3) Drell-Yan NNLO results - p. 6/24

Soft plus Virtual at N 3 LO and beyond VR - p. 7/24

Soft plus Virtual at N 3 LO and beyond VR Using "factorisation" of Virtual, Soft and Collinear: !˛ ˛ ∆ sv I,P ( z, q 2 , µ 2 R , µ 2 Ψ I P ( z, q 2 , µ 2 R , µ 2 F ) = C exp F , ε ) I = q, g n = 4 + ε ˛ ˛ ˛ ε =0 - p. 7/24

Soft plus Virtual at N 3 LO and beyond VR Using "factorisation" of Virtual, Soft and Collinear: !˛ ˛ ∆ sv I,P ( z, q 2 , µ 2 R , µ 2 Ψ I P ( z, q 2 , µ 2 R , µ 2 F ) = C exp F , ε ) I = q, g n = 4 + ε ˛ ˛ ˛ ε =0 ! ” 2 “ Z I (ˆ ˛ 2 Ψ I P ( z, q 2 , µ 2 R , µ 2 a s , µ 2 R , µ 2 , ε ) ˛ ˆ F I (ˆ a s , Q 2 , µ 2 , ε ) ˛ ˛ F , ε ) = ln + ln δ (1 − z ) +2 Φ I a s , q 2 , µ 2 , z, ε ) − 2 m C ln Γ II (ˆ a s , µ 2 , µ 2 P (ˆ F , z, ε ) - p. 7/24

Soft plus Virtual at N 3 LO and beyond VR Using "factorisation" of Virtual, Soft and Collinear: !˛ ˛ ∆ sv I,P ( z, q 2 , µ 2 R , µ 2 Ψ I P ( z, q 2 , µ 2 R , µ 2 F ) = C exp F , ε ) I = q, g n = 4 + ε ˛ ˛ ˛ ε =0 ! ” 2 “ Z I (ˆ ˛ 2 Ψ I P ( z, q 2 , µ 2 R , µ 2 a s , µ 2 R , µ 2 , ε ) ˛ ˆ F I (ˆ a s , Q 2 , µ 2 , ε ) ˛ ˛ F , ε ) = ln + ln δ (1 − z ) +2 Φ I a s , q 2 , µ 2 , z, ε ) − 2 m C ln Γ II (ˆ a s , µ 2 , µ 2 P (ˆ F , z, ε ) • Z I (ˆ a s , µ 2 R , µ 2 , ε ) is operator renormalisation constant with µ is mass parameter in n = 4 + ε dimensional regularisation → N 3 LO a s , Q 2 , µ 2 , ε ) is the Form factor with Q 2 = − q 2 → N 3 LO ˆ F I (ˆ • • Φ I a s , q 2 , µ 2 , z, ε ) is the soft distribution function → NNLO level P (ˆ a s , µ 2 , µ 2 F , z, ε ) is mass factorisation kernel → N 3 LO • Γ II (ˆ - p. 7/24

Soft plus Virtual at N 3 LO and beyond VR Using "factorisation" of Virtual, Soft and Collinear: !˛ ˛ ∆ sv I,P ( z, q 2 , µ 2 R , µ 2 Ψ I P ( z, q 2 , µ 2 R , µ 2 F ) = C exp F , ε ) I = q, g n = 4 + ε ˛ ˛ ˛ ε =0 ! ” 2 “ Z I (ˆ ˛ 2 Ψ I P ( z, q 2 , µ 2 R , µ 2 a s , µ 2 R , µ 2 , ε ) ˛ ˆ F I (ˆ a s , Q 2 , µ 2 , ε ) ˛ ˛ F , ε ) = ln + ln δ (1 − z ) +2 Φ I a s , q 2 , µ 2 , z, ε ) − 2 m C ln Γ II (ˆ a s , µ 2 , µ 2 P (ˆ F , z, ε ) • Z I (ˆ a s , µ 2 R , µ 2 , ε ) is operator renormalisation constant with µ is mass parameter in n = 4 + ε dimensional regularisation → N 3 LO a s , Q 2 , µ 2 , ε ) is the Form factor with Q 2 = − q 2 → N 3 LO ˆ F I (ˆ • • Φ I a s , q 2 , µ 2 , z, ε ) is the soft distribution function → NNLO level P (ˆ a s , µ 2 , µ 2 F , z, ε ) is mass factorisation kernel → N 3 LO • Γ II (ˆ g 2 ˆ m = 1 s ˆ a s = for DIS , m = 1 for DY , Higgs 16 π 2 2 - p. 7/24

Sudakov Resummation for Form factors Vogt,Vermaseren,Moch,VR - p. 8/24

Sudakov Resummation for Form factors Vogt,Vermaseren,Moch,VR " ! ! # a s , µ 2 , µ 2 a s , Q 2 d 1 dQ 2 ln ˆ Q 2 a s , Q 2 , µ 2 , ε K I R + G I R F I ` ´ ˆ = ˆ µ 2 , ε ˆ µ 2 , ε µ 2 2 R « i ε ∞ „ Q 2 2 L I, ( i ) ln ˆ F I (ˆ a s , Q 2 , µ 2 , ε ) = X a i S i ε ˆ Solution : ˆ ( ε ) s F µ 2 i =1 - p. 8/24

Sudakov Resummation for Form factors Vogt,Vermaseren,Moch,VR " ! ! # a s , µ 2 , µ 2 a s , Q 2 d 1 dQ 2 ln ˆ Q 2 a s , Q 2 , µ 2 , ε K I R + G I R F I ` ´ ˆ = ˆ µ 2 , ε ˆ µ 2 , ε µ 2 2 R « i ε ∞ „ Q 2 2 L I, ( i ) ln ˆ F I (ˆ a s , Q 2 , µ 2 , ε ) = X a i S i ε ˆ Solution : ˆ ( ε ) s F µ 2 i =1 Formal solution upto 4 loops: ! ! 1 + 1 L I, (1) ˆ − 2 A I G I = 1 ( ε ) 1 F ε 2 ε ! ! 1 + 1 − 1 + 1 L I, (2) ˆ β 0 A I 2 A I 2 − β 0 G I 2 εG I = 1 ( ε ) 2 ( ε ) 1 F ε 3 ε 2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· - p. 8/24

Sudakov Resummation for Form factors Vogt,Vermaseren,Moch,VR " ! ! # a s , µ 2 , µ 2 a s , Q 2 d 1 dQ 2 ln ˆ Q 2 a s , Q 2 , µ 2 , ε K I R + G I R F I ` ´ ˆ = ˆ µ 2 , ε ˆ µ 2 , ε µ 2 2 R « i ε ∞ „ Q 2 2 L I, ( i ) ln ˆ F I (ˆ a s , Q 2 , µ 2 , ε ) = X a i S i ε ˆ Solution : ˆ ( ε ) s F µ 2 i =1 Formal solution upto 4 loops: ! ! 1 + 1 L I, (1) ˆ − 2 A I G I = 1 ( ε ) 1 F ε 2 ε ! ! 1 + 1 − 1 + 1 L I, (2) ˆ β 0 A I 2 A I 2 − β 0 G I 2 εG I = 1 ( ε ) 2 ( ε ) 1 F ε 3 ε 2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· i = C A A g A q • A I are maximally non − abelian i = 1 , 2 , 3 . i C F - p. 8/24

Sudakov Resummation for Form factors Vogt,Vermaseren,Moch,VR " ! ! # a s , µ 2 , µ 2 a s , Q 2 d 1 dQ 2 ln ˆ Q 2 a s , Q 2 , µ 2 , ε K I R + G I R F I ` ´ ˆ = ˆ µ 2 , ε ˆ µ 2 , ε µ 2 2 R « i ε ∞ „ Q 2 2 L I, ( i ) ln ˆ F I (ˆ a s , Q 2 , µ 2 , ε ) = X a i S i ε ˆ Solution : ˆ ( ε ) s F µ 2 i =1 Formal solution upto 4 loops: ! ! 1 + 1 L I, (1) ˆ − 2 A I G I = 1 ( ε ) 1 F ε 2 ε ! ! 1 + 1 − 1 + 1 L I, (2) ˆ β 0 A I 2 A I 2 − β 0 G I 2 εG I = 1 ( ε ) 2 ( ε ) 1 F ε 3 ε 2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· i = C A A g A q • A I are maximally non − abelian i = 1 , 2 , 3 . i C F • Every order in ˆ a s , all the poles except the lowest one can be predicted from the previous order results using A and β function. - p. 8/24

New observation for single pole in ε VR,Smith,van Neerven - p. 9/24

New observation for single pole in ε VR,Smith,van Neerven F q and ˆ F g in SU ( N ) solves the single pole problem: Two loop results for ˆ - p. 9/24

New observation for single pole in ε VR,Smith,van Neerven F q and ˆ F g in SU ( N ) solves the single pole problem: Two loop results for ˆ G I s have interesting structure: ∞ ε k g I,k G I 2( B I 1 − γ I 1 ) + f I X 1 ( ε ) = 1 + 1 k =1 ∞ 2 − 2 β 0 g I, 1 ε k g I,k X G I 2( B I 2 − γ I 2 ) + f I 2 ( ε ) = + 1 2 k =1 - p. 9/24

New observation for single pole in ε VR,Smith,van Neerven F q and ˆ F g in SU ( N ) solves the single pole problem: Two loop results for ˆ G I s have interesting structure: ∞ ε k g I,k G I 2( B I 1 − γ I 1 ) + f I X 1 ( ε ) = 1 + 1 k =1 ∞ 2 − 2 β 0 g I, 1 ε k g I,k X G I 2( B I 2 − γ I 2 ) + f I 2 ( ε ) = + 1 2 k =1 B I i are δ (1 − z ) part of P II splitting functions. The new constants " f I 1 and f I 2 " satisfy i = C A f g f q i = 1 , 2 i C F - p. 9/24

New observation for single pole in ε VR,Smith,van Neerven F q and ˆ F g in SU ( N ) solves the single pole problem: Two loop results for ˆ G I s have interesting structure: ∞ ε k g I,k G I 2( B I 1 − γ I 1 ) + f I X 1 ( ε ) = 1 + 1 k =1 ∞ 2 − 2 β 0 g I, 1 ε k g I,k X G I 2( B I 2 − γ I 2 ) + f I 2 ( ε ) = + 1 2 k =1 B I i are δ (1 − z ) part of P II splitting functions. The new constants " f I 1 and f I 2 " satisfy i = C A f g f q i = 1 , 2 i C F Even the single pole can be predicted: G I i = 2( B I i − γ I i ) + f I i + · · · - p. 9/24

New observation for single pole in ε VR,Smith,van Neerven F q and ˆ F g in SU ( N ) solves the single pole problem: Two loop results for ˆ G I s have interesting structure: ∞ ε k g I,k G I 2( B I 1 − γ I 1 ) + f I X 1 ( ε ) = 1 + 1 k =1 ∞ 2 − 2 β 0 g I, 1 ε k g I,k X G I 2( B I 2 − γ I 2 ) + f I 2 ( ε ) = + 1 2 k =1 B I i are δ (1 − z ) part of P II splitting functions. The new constants " f I 1 and f I 2 " satisfy i = C A f g f q i = 1 , 2 i C F Even the single pole can be predicted: G I i = 2( B I i − γ I i ) + f I i + · · · Recent three loop result by Moch,Vermaseren, Vogt confirms our prediction: 3 = C A f g f q 3 C F - p. 9/24

New observation for single pole in ε VR,Smith,van Neerven F q and ˆ F g in SU ( N ) solves the single pole problem: Two loop results for ˆ G I s have interesting structure: ∞ ε k g I,k G I 2( B I 1 − γ I 1 ) + f I X 1 ( ε ) = 1 + 1 k =1 ∞ 2 − 2 β 0 g I, 1 ε k g I,k X G I 2( B I 2 − γ I 2 ) + f I 2 ( ε ) = + 1 2 k =1 B I i are δ (1 − z ) part of P II splitting functions. The new constants " f I 1 and f I 2 " satisfy i = C A f g f q i = 1 , 2 i C F Even the single pole can be predicted: G I i = 2( B I i − γ I i ) + f I i + · · · Recent three loop result by Moch,Vermaseren, Vogt confirms our prediction: 3 = C A f g f q 3 C F This completes the understanding of all the poles of the form factors. - p. 9/24

Mass factorisation using DGLAP kernel VR - p. 10/24

Mass factorisation using DGLAP kernel VR Due to the massless partons, collinear singularities appear in • the phase space of the real emission processes • loop integrals of the virtual corrections - p. 10/24

Mass factorisation using DGLAP kernel VR Due to the massless partons, collinear singularities appear in • the phase space of the real emission processes • loop integrals of the virtual corrections They are removed by Mass Factorisation by adding: a s , µ 2 , µ 2 − ln Γ(ˆ F , z, ε ) - p. 10/24

Mass factorisation using DGLAP kernel VR Due to the massless partons, collinear singularities appear in • the phase space of the real emission processes • loop integrals of the virtual corrections They are removed by Mass Factorisation by adding: a s , µ 2 , µ 2 − ln Γ(ˆ F , z, ε ) DGLAP kernels satisfy Renormalisation Group Equations: d F , ε ) = 1 µ 2 Γ( z, µ 2 z, µ 2 z, µ 2 ` ´ ` ´ ⊗ Γ 2 P F , ε . F F dµ 2 F - p. 10/24

Mass factorisation using DGLAP kernel VR Due to the massless partons, collinear singularities appear in • the phase space of the real emission processes • loop integrals of the virtual corrections They are removed by Mass Factorisation by adding: a s , µ 2 , µ 2 − ln Γ(ˆ F , z, ε ) DGLAP kernels satisfy Renormalisation Group Equations: d F , ε ) = 1 µ 2 Γ( z, µ 2 z, µ 2 z, µ 2 ` ´ ` ´ ⊗ Γ 2 P F , ε . F F dµ 2 F The diagonal terms of the splitting functions P ( i ) ( z ) have the following structure " # P ( i ) + P ( i ) B I i +1 δ (1 − z ) + A I i +1 D 0 II ( z ) = 2 reg,II ( z ) , - p. 10/24

Mass factorisation using DGLAP kernel VR Due to the massless partons, collinear singularities appear in • the phase space of the real emission processes • loop integrals of the virtual corrections They are removed by Mass Factorisation by adding: a s , µ 2 , µ 2 − ln Γ(ˆ F , z, ε ) DGLAP kernels satisfy Renormalisation Group Equations: d F , ε ) = 1 µ 2 Γ( z, µ 2 z, µ 2 z, µ 2 ` ´ ` ´ ⊗ Γ 2 P F , ε . F F dµ 2 F The diagonal terms of the splitting functions P ( i ) ( z ) have the following structure " # P ( i ) + P ( i ) B I i +1 δ (1 − z ) + A I i +1 D 0 II ( z ) = 2 reg,II ( z ) , „ 1 « P ( i ) D 0 = , reg,II are regular when z → 1 . 1 − z + We will be left with only maximally non-abelian constants A I i and f I i - p. 10/24

Finiteness of the Cross section VR - p. 11/24

Finiteness of the Cross section VR Observable ∆ I ( α s , Q 2 ) are finite: Infra − red safe - p. 11/24

Finiteness of the Cross section VR Observable ∆ I ( α s , Q 2 ) are finite: Infra − red safe The remaining poles after UV Operator Renormalisation( Z α s and Z I ) and Mass factorisation: 1 i th loop at ε i +1 Highest poles are not removed by renormalisation and factorisation - p. 11/24

Finiteness of the Cross section VR Observable ∆ I ( α s , Q 2 ) are finite: Infra − red safe The remaining poles after UV Operator Renormalisation( Z α s and Z I ) and Mass factorisation: 1 i th loop at ε i +1 Highest poles are not removed by renormalisation and factorisation • The structure of soft part should be "similar" to the Form Factors. • Hence using gauge invariance and RG invariance, we can propose " ! ! # a s , µ 2 , µ 2 a s , q 2 d 1 K I + G I q 2 dq 2 Φ I ` ˆ a s , q 2 , µ 2 , z, ε ´ R R = ˆ µ 2 , z, ε ˆ µ 2 , z, ε µ 2 2 R - p. 11/24

Finiteness of the Cross section VR Observable ∆ I ( α s , Q 2 ) are finite: Infra − red safe The remaining poles after UV Operator Renormalisation( Z α s and Z I ) and Mass factorisation: 1 i th loop at ε i +1 Highest poles are not removed by renormalisation and factorisation • The structure of soft part should be "similar" to the Form Factors. • Hence using gauge invariance and RG invariance, we can propose " ! ! # a s , µ 2 , µ 2 a s , q 2 d 1 K I + G I q 2 dq 2 Φ I ` ˆ a s , q 2 , µ 2 , z, ε ´ R R = ˆ µ 2 , z, ε ˆ µ 2 , z, ε µ 2 2 R RG invariance of Φ I implies: ! ! a s , µ 2 a s , q 2 , µ 2 d d K I G I = − A I ( a s ( µ 2 µ 2 R = − µ 2 R R )) δ (1 − z ) ˆ µ 2 , z, ε ˆ µ 2 , z, ε R R dµ 2 dµ 2 µ 2 R R R - p. 11/24

Solution to (Soft)Sudakov Equation VR - p. 12/24

Solution to (Soft)Sudakov Equation VR Infra-red safeness of the cross section implies I = − A I A - p. 12/24

Solution to (Soft)Sudakov Equation VR Infra-red safeness of the cross section implies I = − A I A Solution to (soft) Sudakov equation: „ q 2 « i ε ∞ 2 Φ I ` a s , q 2 , µ 2 , z, ε X a i S i ε ˆ Φ I, ( i ) ( z, ε ) ´ ˆ = ˆ s µ 2 i =1 - p. 12/24

Solution to (Soft)Sudakov Equation VR Infra-red safeness of the cross section implies I = − A I A Solution to (soft) Sudakov equation: „ q 2 « i ε ∞ 2 Φ I ` a s , q 2 , µ 2 , z, ε X a i S i ε ˆ Φ I, ( i ) ( z, ε ) ´ ˆ = ˆ s µ 2 i =1 where ! A I → − δ (1 − z ) A I , G I ( ε ) → G I ( z, ε ) L I, ( i ) Φ I, ( i ) ( z, ε ) = ˆ ˆ ( ε ) F - p. 12/24

Solution to (Soft)Sudakov Equation VR Infra-red safeness of the cross section implies I = − A I A Solution to (soft) Sudakov equation: „ q 2 « i ε ∞ 2 Φ I ` a s , q 2 , µ 2 , z, ε X a i S i ε ˆ Φ I, ( i ) ( z, ε ) ´ ˆ = ˆ s µ 2 i =1 where ! A I → − δ (1 − z ) A I , G I ( ε ) → G I ( z, ε ) L I, ( i ) Φ I, ( i ) ( z, ε ) = ˆ ˆ ( ε ) F Most general solution: Φ I (ˆ a s , q 2 , µ 2 , z, ε ) Φ I (ˆ a s , q 2 (1 − z ) 2 m , µ 2 , ε ) = „ q 2 (1 − z ) 2 m « i ε ∞ „ i m ε « 2 X a i S i φ I, ( i ) ( ε ) ˆ = ˆ s ε µ 2 2(1 − z ) i =1 - p. 12/24

Solution to (Soft)Sudakov Equation VR Infra-red safeness of the cross section implies I = − A I A Solution to (soft) Sudakov equation: „ q 2 « i ε ∞ 2 Φ I ` a s , q 2 , µ 2 , z, ε X a i S i ε ˆ Φ I, ( i ) ( z, ε ) ´ ˆ = ˆ s µ 2 i =1 where ! A I → − δ (1 − z ) A I , G I ( ε ) → G I ( z, ε ) L I, ( i ) Φ I, ( i ) ( z, ε ) = ˆ ˆ ( ε ) F Most general solution: Φ I (ˆ a s , q 2 , µ 2 , z, ε ) Φ I (ˆ a s , q 2 (1 − z ) 2 m , µ 2 , ε ) = „ q 2 (1 − z ) 2 m « i ε ∞ „ i m ε « 2 X a i S i φ I, ( i ) ( ε ) ˆ = ˆ s ε µ 2 2(1 − z ) i =1 All the poles in ε are predictable. - p. 12/24

Universal Soft part VR - p. 13/24

Universal Soft part VR Single pole in ε : ∞ I I, ( k ) − f I X ε k G G 1 ( ε ) = 1 + 1 k =1 ∞ I I, (1) I, ( k ) X − f I ε k G G 2 − 2 β 0 G 2 ( ε ) = + 1 2 k =1 · · · · · · · · · · · · · · · · · · - p. 13/24

Universal Soft part VR Single pole in ε : ∞ I I, ( k ) − f I X ε k G G 1 ( ε ) = 1 + 1 k =1 ∞ I I, (1) I, ( k ) X − f I ε k G G 2 − 2 β 0 G 2 ( ε ) = + 1 2 k =1 · · · · · · · · · · · · · · · · · · Maximally non-abelian: i ( ε ) = C A g q G G i ( ε ) i = 1 , 2 , 3 C F - p. 13/24

Universal Soft part VR Single pole in ε : ∞ I I, ( k ) − f I X ε k G G 1 ( ε ) = 1 + 1 k =1 ∞ I I, (1) I, ( k ) X − f I ε k G G 2 − 2 β 0 G 2 ( ε ) = + 1 2 k =1 · · · · · · · · · · · · · · · · · · Maximally non-abelian: i ( ε ) = C A g q G G i ( ε ) i = 1 , 2 , 3 C F Soft part of the any cross section are independent of spin,colour,flavour or other quantum numbers. - p. 13/24

Universal Soft part VR Single pole in ε : ∞ I I, ( k ) − f I X ε k G G 1 ( ε ) = 1 + 1 k =1 ∞ I I, (1) I, ( k ) X − f I ε k G G 2 − 2 β 0 G 2 ( ε ) = + 1 2 k =1 · · · · · · · · · · · · · · · · · · Maximally non-abelian: i ( ε ) = C A g q G G i ( ε ) i = 1 , 2 , 3 C F Soft part of the any cross section are independent of spin,colour,flavour or other quantum numbers. a s , q 2 , z, µ 2 , ε ) = C F Φ q (ˆ Φ g (ˆ a s , q 2 , z, µ 2 , ε ) C A - p. 13/24

Higgs productions from Drell-Yan beyond NNLO Universal soft function: VR - p. 14/24

Higgs productions from Drell-Yan beyond NNLO Universal soft function: VR a s , q 2 , z, µ 2 , ε ) = C A Φ g (ˆ Φ q (ˆ a s , q 2 , z, µ 2 , ε ) C F a s , q 2 , z, ε ) , Gluon form factor F g and operator renormalisation • From Drell-Yan Φ q (ˆ constant Z g and DGLAP kernel Γ gg , we can compute soft plus virtual part of σ ( g + g → Higgs ) without explicitly calculating the soft part of Higgs production. - p. 14/24

Higgs productions from Drell-Yan beyond NNLO Universal soft function: VR a s , q 2 , z, µ 2 , ε ) = C A Φ g (ˆ Φ q (ˆ a s , q 2 , z, µ 2 , ε ) C F a s , q 2 , z, ε ) , Gluon form factor F g and operator renormalisation • From Drell-Yan Φ q (ˆ constant Z g and DGLAP kernel Γ gg , we can compute soft plus virtual part of σ ( g + g → Higgs ) without explicitly calculating the soft part of Higgs production. • Our NNLO predictions agrees with the results by Catani et al, Harlander and Kilgore. No need for explicit computation of soft contributions for the Higgs production. - p. 14/24

Higgs productions from Drell-Yan beyond NNLO Universal soft function: VR a s , q 2 , z, µ 2 , ε ) = C A Φ g (ˆ Φ q (ˆ a s , q 2 , z, µ 2 , ε ) C F a s , q 2 , z, ε ) , Gluon form factor F g and operator renormalisation • From Drell-Yan Φ q (ˆ constant Z g and DGLAP kernel Γ gg , we can compute soft plus virtual part of σ ( g + g → Higgs ) without explicitly calculating the soft part of Higgs production. • Our NNLO predictions agrees with the results by Catani et al, Harlander and Kilgore. No need for explicit computation of soft contributions for the Higgs production. • Our N 3 LO predictions (without δ (1 − z ) part) for soft plus virtual contributions to Drell-Yan and Higgs productions agree with the results of Moch and Vogt - p. 14/24

Higgs productions from Drell-Yan beyond NNLO Universal soft function: VR a s , q 2 , z, µ 2 , ε ) = C A Φ g (ˆ Φ q (ˆ a s , q 2 , z, µ 2 , ε ) C F a s , q 2 , z, ε ) , Gluon form factor F g and operator renormalisation • From Drell-Yan Φ q (ˆ constant Z g and DGLAP kernel Γ gg , we can compute soft plus virtual part of σ ( g + g → Higgs ) without explicitly calculating the soft part of Higgs production. • Our NNLO predictions agrees with the results by Catani et al, Harlander and Kilgore. No need for explicit computation of soft contributions for the Higgs production. • Our N 3 LO predictions (without δ (1 − z ) part) for soft plus virtual contributions to Drell-Yan and Higgs productions agree with the results of Moch and Vogt • The scalar form factor F S = < P | ψψ | P > can be predicted at three loop from the known three loop A i , B i , f i and γ m i . - p. 14/24

Higgs productions from Drell-Yan beyond NNLO Universal soft function: VR a s , q 2 , z, µ 2 , ε ) = C A Φ g (ˆ Φ q (ˆ a s , q 2 , z, µ 2 , ε ) C F a s , q 2 , z, ε ) , Gluon form factor F g and operator renormalisation • From Drell-Yan Φ q (ˆ constant Z g and DGLAP kernel Γ gg , we can compute soft plus virtual part of σ ( g + g → Higgs ) without explicitly calculating the soft part of Higgs production. • Our NNLO predictions agrees with the results by Catani et al, Harlander and Kilgore. No need for explicit computation of soft contributions for the Higgs production. • Our N 3 LO predictions (without δ (1 − z ) part) for soft plus virtual contributions to Drell-Yan and Higgs productions agree with the results of Moch and Vogt • The scalar form factor F S = < P | ψψ | P > can be predicted at three loop from the known three loop A i , B i , f i and γ m i . • Soft plus Virtual part Higgs production through bottom quark fusion σ ( b + b → Higgs ) can be predicted upto N 3 LO (without δ (1 − z ) ). - p. 14/24

Higgs productions from Drell-Yan beyond NNLO Universal soft function: VR a s , q 2 , z, µ 2 , ε ) = C A Φ g (ˆ Φ q (ˆ a s , q 2 , z, µ 2 , ε ) C F a s , q 2 , z, ε ) , Gluon form factor F g and operator renormalisation • From Drell-Yan Φ q (ˆ constant Z g and DGLAP kernel Γ gg , we can compute soft plus virtual part of σ ( g + g → Higgs ) without explicitly calculating the soft part of Higgs production. • Our NNLO predictions agrees with the results by Catani et al, Harlander and Kilgore. No need for explicit computation of soft contributions for the Higgs production. • Our N 3 LO predictions (without δ (1 − z ) part) for soft plus virtual contributions to Drell-Yan and Higgs productions agree with the results of Moch and Vogt • The scalar form factor F S = < P | ψψ | P > can be predicted at three loop from the known three loop A i , B i , f i and γ m i . • Soft plus Virtual part Higgs production through bottom quark fusion σ ( b + b → Higgs ) can be predicted upto N 3 LO (without δ (1 − z ) ). • Our NNLO predictions agrees with the results of by Harlander and Kilgore. - p. 14/24

Hadro production in e + e − annihilation from DIS Bl¨ umlein and VR - p. 15/24

Hadro production in e + e − annihilation from DIS Bl¨ umlein and VR • The scaling variable in DIS is q 2 − q 2 > 0 x Bj = − 2 p · q - p. 15/24

Hadro production in e + e − annihilation from DIS Bl¨ umlein and VR • The scaling variable in DIS is q 2 − q 2 > 0 x Bj = − 2 p · q • The scaling variable in hadro production is x ee = 2 p · q q 2 > 0 q 2 - p. 15/24

Hadro production in e + e − annihilation from DIS Bl¨ umlein and VR • The scaling variable in DIS is q 2 − q 2 > 0 x Bj = − 2 p · q • The scaling variable in hadro production is x ee = 2 p · q q 2 > 0 q 2 • Drell-Levy-Yan showed that these two processes are related by crossing relation. - p. 15/24

Hadro production in e + e − annihilation from DIS Bl¨ umlein and VR • The scaling variable in DIS is q 2 − q 2 > 0 x Bj = − 2 p · q • The scaling variable in hadro production is x ee = 2 p · q q 2 > 0 q 2 • Drell-Levy-Yan showed that these two processes are related by crossing relation. • Gribov-Lipatov relation in the soft limit: a s , Q 2 , µ 2 , x Bj , ε ) a s , q 2 , µ 2 , x ee , ε ) Φ DIS (ˆ = Φ ee (ˆ ˜ P II ( x Bj ) = P II ( x ee ) Distributions - p. 15/24

Hadro production in e + e − annihilation from DIS Bl¨ umlein and VR • The scaling variable in DIS is q 2 − q 2 > 0 x Bj = − 2 p · q • The scaling variable in hadro production is x ee = 2 p · q q 2 > 0 q 2 • Drell-Levy-Yan showed that these two processes are related by crossing relation. • Gribov-Lipatov relation in the soft limit: a s , Q 2 , µ 2 , x Bj , ε ) a s , q 2 , µ 2 , x ee , ε ) Φ DIS (ˆ = Φ ee (ˆ ˜ P II ( x Bj ) = P II ( x ee ) Distributions • From DIS results, we can predict soft plus virtual part of the coefficient functions for hadro production in e + e − annihilation upto three loop level. C (3) ,sv ( α s , z ) New result ee - p. 15/24

Hadro production in e + e − annihilation from DIS Bl¨ umlein and VR • The scaling variable in DIS is q 2 − q 2 > 0 x Bj = − 2 p · q • The scaling variable in hadro production is x ee = 2 p · q q 2 > 0 q 2 • Drell-Levy-Yan showed that these two processes are related by crossing relation. • Gribov-Lipatov relation in the soft limit: a s , Q 2 , µ 2 , x Bj , ε ) a s , q 2 , µ 2 , x ee , ε ) Φ DIS (ˆ = Φ ee (ˆ ˜ P II ( x Bj ) = P II ( x ee ) Distributions • From DIS results, we can predict soft plus virtual part of the coefficient functions for hadro production in e + e − annihilation upto three loop level. C (3) ,sv ( α s , z ) New result ee - p. 15/24

Threshold Resummation VR • Alternate derivation for the threshold resummation formula in z space for both DY and DIS: ( Z q 2 (1 − z ) 2 m δ P dλ 2 m Φ I a s , q 2 , µ 2 , z, ε ) a s ( λ 2 ) ` ´ P (ˆ = λ 2 A I 1 − z µ 2 R )! I q 2 (1 − z ) 2 m δ P ` ` ´ ´ + G a s , ε P + „ q 2 δ P « i ε ∞ 2 φ I, ( i ) X a i S i ε ˆ + δ (1 − z ) ˆ ( ε ) s P µ 2 i =1 ! i ε ∞ µ 2 2 „ m « I, ( i ) ( ε ) X a i R S i + ˆ ε K s µ 2 1 − z + i =1 - p. 16/24

Threshold Resummation VR • Alternate derivation for the threshold resummation formula in z space for both DY and DIS: ( Z q 2 (1 − z ) 2 m δ P dλ 2 m Φ I a s , q 2 , µ 2 , z, ε ) a s ( λ 2 ) ` ´ P (ˆ = λ 2 A I 1 − z µ 2 R )! I q 2 (1 − z ) 2 m δ P ` ` ´ ´ + G a s , ε P + „ q 2 δ P « i ε ∞ 2 φ I, ( i ) X a i S i ε ˆ + δ (1 − z ) ˆ ( ε ) s P µ 2 i =1 ! i ε ∞ µ 2 2 „ m « I, ( i ) ( ε ) X a i R S i + ˆ ε K s µ 2 1 − z + i =1 I • The threshold exponents D I i for DY and B I i for DIS are related to G P ( ε = 0) . I P ( ε = 0) upto three loop gives D I i and B I • G i for i = 1 , 2 , 3 - p. 16/24

Threshold Resummation VR • Alternate derivation for the threshold resummation formula in z space for both DY and DIS: ( Z q 2 (1 − z ) 2 m δ P dλ 2 m Φ I a s , q 2 , µ 2 , z, ε ) a s ( λ 2 ) ` ´ P (ˆ = λ 2 A I 1 − z µ 2 R )! I q 2 (1 − z ) 2 m δ P ` ` ´ ´ + G a s , ε P + „ q 2 δ P « i ε ∞ 2 φ I, ( i ) X a i S i ε ˆ + δ (1 − z ) ˆ ( ε ) s P µ 2 i =1 ! i ε ∞ µ 2 2 „ m « I, ( i ) ( ε ) X a i R S i + ˆ ε K s µ 2 1 − z + i =1 I • The threshold exponents D I i for DY and B I i for DIS are related to G P ( ε = 0) . I P ( ε = 0) upto three loop gives D I i and B I • G i for i = 1 , 2 , 3 “ ” 2Φ I • Expansion of C e P leads to soft part of the cross section. • Fixed order N 3 LO soft plus virtual cross sections can be computed(except δ (1 − z ) ) - p. 16/24

Soft plus Virtual part at N 3 LO for Higgs Production Moch, Vogt and VR - p. 17/24

Soft plus Virtual part at N 3 LO for Higgs Production Moch, Vogt and VR Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab - p. 17/24

Soft plus Virtual part at N 3 LO for Higgs Production Moch, Vogt and VR Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab • Finite terms in F I and Φ I at 3-loop are still missing - p. 17/24

Soft plus Virtual part at N 3 LO for Higgs Production Moch, Vogt and VR Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab • Finite terms in F I and Φ I at 3-loop are still missing • We can not predict δ (1 − z ) part at 3-loop. - p. 17/24

Soft plus Virtual part at N 3 LO for Higgs Production Moch, Vogt and VR Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab • Finite terms in F I and Φ I at 3-loop are still missing • We can not predict δ (1 − z ) part at 3-loop. • At 3-loop we can predict all D j j = 5 , 4 , 3 , 2 , 1 , 0 - p. 17/24

Soft plus Virtual part at N 3 LO for Higgs Production Moch, Vogt and VR Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab • Finite terms in F I and Φ I at 3-loop are still missing • We can not predict δ (1 − z ) part at 3-loop. • At 3-loop we can predict all D j j = 5 , 4 , 3 , 2 , 1 , 0 • At 4-loop, we can predict only D j j = 7 , 6 , 5 , 4 , 3 , 2 - p. 17/24

Soft plus Virtual part at N 3 LO for Higgs Production Moch, Vogt and VR Z 1 τ = m 2 dx σ ab “ τ ” 2 S dσ P 1 P 2 ( τ, m h ) = X h x Φ ab ( x ) 2ˆ s d ˆ x, m h S τ ab 70 σ (pb) LHC(14 TeV) • Finite terms in F I and Φ I at 3-loop are still N 4 LO(pSV) 60 N 3 LO(pSV) missing NNLO NLO LO 50 • We can not predict δ (1 − z ) part at 3-loop. 40 • At 3-loop we can predict all 30 D j j = 5 , 4 , 3 , 2 , 1 , 0 20 • At 4-loop, we can predict only 10 D j j = 7 , 6 , 5 , 4 , 3 , 2 100 150 200 250 m H • They contribute bulk of the cross section Gluon flux is largest at LHC - p. 17/24

Scale variation at N 3 LO for Higgs production N = σ N i LO ( µ ) σ N i LO ( µ 0 ) - p. 18/24

Scale variation at N 3 LO for Higgs production N = σ N i LO ( µ ) σ N i LO ( µ 0 ) σ (pp → H+X) [pb]  s = 14 TeV √ 10 NNLO NLO LO Harlander 1 100 120 140 160 180 200 220 240 260 280 300 M H [GeV] - p. 18/24

Scale variation at N 3 LO for Higgs production N = σ N i LO ( µ ) σ N i LO ( µ 0 ) σ (pp → H+X) [pb]  s = 14 TeV √ 1.4 R-Ratio LHC(14 TeV) LO NLO NNLO N 3 LO(pSV) 1.2 N 4 LO(pSV) 10 1 NNLO NLO LO Harlander 0.8 1 0.5 1 1.5 2 100 120 140 160 180 200 220 240 260 280 300 µ / µ 0 M H [GeV] - p. 18/24

Scale variation at N 3 LO for Higgs production N = σ N i LO ( µ ) σ N i LO ( µ 0 ) σ (pp → H+X) [pb]  s = 14 TeV √ 1.4 R-Ratio LHC(14 TeV) LO NLO NNLO N 3 LO(pSV) 1.2 N 4 LO(pSV) 10 1 NNLO NLO LO Harlander 0.8 1 0.5 1 1.5 2 100 120 140 160 180 200 220 240 260 280 300 µ / µ 0 M H [GeV] • Scale uncertainity improves a lot • Perturbative QCD works at LHC - p. 18/24

Soft distribution for rapidity VR,Smith and van Neerven Using RGE and Factorisation: - p. 19/24

Soft distribution for rapidity VR,Smith and van Neerven Using RGE and Factorisation: Φ I a s , q 2 , µ 2 , z 1 , z 2 , ε ) Φ I d,f inite + Φ I d (ˆ = d,singular - p. 19/24