Saddle point N 0 vs τ 5 p-p p-pbar Q = 100 GeV NNPDF 2.0 ( α s (m Z ) = 0.118) 4 N 0 3 2 1 0.001 0.01 0.1 τ N 0 � 2 ⇒ the log contribution is dominant Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 6
Saddle point N 0 vs τ 5 p-p p-pbar Q = 100 GeV NNPDF 2.0 ( α s (m Z ) = 0.118) 4 N 0 3 2 1 0.001 0.01 0.1 τ N 0 � 2 ⇒ the log contribution is dominant � 0 . 003 for pp colliders (LHC) τ � 0 . 02 for p ¯ p colliders (Tevatron) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 6
Relevance of resummation To summarize: Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 7
Relevance of resummation To summarize: resummation is relevant when log contribution is dominant (at hadron level) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 7
Relevance of resummation To summarize: resummation is relevant when log contribution is dominant (at hadron level) log contribution is dominant (at parton level) for N � 2 Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 7
Relevance of resummation To summarize: resummation is relevant when log contribution is dominant (at hadron level) log contribution is dominant (at parton level) for N � 2 the Mellin inversion integral is dominated by the saddle point N = N 0 Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 7
Relevance of resummation To summarize: resummation is relevant when log contribution is dominant (at hadron level) log contribution is dominant (at parton level) for N � 2 the Mellin inversion integral is dominated by the saddle point N = N 0 log contribution is dominant (at hadron level) when N 0 � 2 Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 7
Relevance of resummation To summarize: resummation is relevant when log contribution is dominant (at hadron level) log contribution is dominant (at parton level) for N � 2 the Mellin inversion integral is dominated by the saddle point N = N 0 log contribution is dominant (at hadron level) when N 0 � 2 resummation is relevant for � 0 . 003 for pp colliders (LHC) τ � 0 . 02 for p ¯ p colliders (Tevatron) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 7
Relevance of resummation To summarize: resummation is relevant when log contribution is dominant (at hadron level) log contribution is dominant (at parton level) for N � 2 the Mellin inversion integral is dominated by the saddle point N = N 0 log contribution is dominant (at hadron level) when N 0 � 2 resummation is relevant for � 0 . 003 for pp colliders (LHC) τ � 0 . 02 for p ¯ p colliders (Tevatron) Much smaller than expected! Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 7
Resummation ( L = 2 β 0 α s log 1 Resummation is performed in N –space N ) � 1 � σ res ( N ) = g 0 ( α s ) exp g 1 ( L ) + g 2 ( L ) + α s g 3 ( L ) + α 2 ˆ s g 4 ( L ) + . . . α s known up to g 4 (N 3 LL): S.Moch, J.A.M.Vermaseren, A.Vogt (hep-ph/0506288) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 8
Resummation ( L = 2 β 0 α s log 1 Resummation is performed in N –space N ) � 1 � σ res ( N ) = g 0 ( α s ) exp g 1 ( L ) + g 2 ( L ) + α s g 3 ( L ) + α 2 ˆ s g 4 ( L ) + . . . α s known up to g 4 (N 3 LL): S.Moch, J.A.M.Vermaseren, A.Vogt (hep-ph/0506288) 1 Branch cut due to the Landau singularity for N > N L = exp 2 β 0 α s N space N L Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 8
Resummation ( L = 2 β 0 α s log 1 Resummation is performed in N –space N ) � 1 � σ res ( N ) = g 0 ( α s ) exp g 1 ( L ) + g 2 ( L ) + α s g 3 ( L ) + α 2 ˆ s g 4 ( L ) + . . . α s known up to g 4 (N 3 LL): S.Moch, J.A.M.Vermaseren, A.Vogt (hep-ph/0506288) 1 Branch cut due to the Landau singularity for N > N L = exp 2 β 0 α s N space c N L Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 8
Resummation ( L = 2 β 0 α s log 1 Resummation is performed in N –space N ) � 1 � σ res ( N ) = g 0 ( α s ) exp g 1 ( L ) + g 2 ( L ) + α s g 3 ( L ) + α 2 ˆ s g 4 ( L ) + . . . α s known up to g 4 (N 3 LL): S.Moch, J.A.M.Vermaseren, A.Vogt (hep-ph/0506288) 1 Branch cut due to the Landau singularity for N > N L = exp 2 β 0 α s N space The Mellin inverse does not exist c N L Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 8
Minimal prescription S.Catani, M.L.Mangano, P.Nason, L.Trentadue (hep-ph/9604351) � c + i ∞ 1 dN τ − N L ( N ) ˆ σ res ( N ) σ MP ( τ ) = 2 πi c − i ∞ 1 with c < N L = exp 2 β 0 α s , as in the figure. N space c N L Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 9
Minimal prescription S.Catani, M.L.Mangano, P.Nason, L.Trentadue (hep-ph/9604351) � c + i ∞ 1 dN τ − N L ( N ) ˆ σ res ( N ) σ MP ( τ ) = 2 πi c − i ∞ 1 with c < N L = exp 2 β 0 α s , as in the figure. Good properties: N space well defined for all τ < 1 c N L Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 9
Minimal prescription S.Catani, M.L.Mangano, P.Nason, L.Trentadue (hep-ph/9604351) � c + i ∞ 1 dN τ − N L ( N ) ˆ σ res ( N ) σ MP ( τ ) = 2 πi c − i ∞ 1 with c < N L = exp 2 β 0 α s , as in the figure. Good properties: N space well defined for all τ < 1 exact for invertible functions c N L Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 9
Minimal prescription S.Catani, M.L.Mangano, P.Nason, L.Trentadue (hep-ph/9604351) � c + i ∞ 1 dN τ − N L ( N ) ˆ σ res ( N ) σ MP ( τ ) = 2 πi c − i ∞ 1 with c < N L = exp 2 β 0 α s , as in the figure. Good properties: N space well defined for all τ < 1 exact for invertible functions c N L asymptotic to the original divergent series Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 9
Minimal prescription S.Catani, M.L.Mangano, P.Nason, L.Trentadue (hep-ph/9604351) � c + i ∞ 1 dN τ − N L ( N ) ˆ σ res ( N ) σ MP ( τ ) = 2 πi c − i ∞ 1 with c < N L = exp 2 β 0 α s , as in the figure. Good properties: N space well defined for all τ < 1 exact for invertible functions c N L asymptotic to the original divergent series But... a non-physical region of the parton cross-section contributes Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 9
Minimal prescription S.Catani, M.L.Mangano, P.Nason, L.Trentadue (hep-ph/9604351) � c + i ∞ 1 dN τ − N L ( N ) ˆ σ res ( N ) σ MP ( τ ) = 2 πi c − i ∞ 1 with c < N L = exp 2 β 0 α s , as in the figure. Good properties: N space well defined for all τ < 1 exact for invertible functions c N L asymptotic to the original divergent series But... a non-physical region of the parton cross-section contributes difficult numerical implementation Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 9
Minimal prescription: non-physical contribution � + ∞ dz � τ � σ MP ( τ ) = z L ˆ σ MP ( z ) z τ The integral extends to + ∞ , not to 1 ! Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 10
Minimal prescription: non-physical contribution � + ∞ dz � τ � σ MP ( τ ) = z L σ MP ( z ) ˆ z τ The integral extends to + ∞ , not to 1 ! σ MP ( z > 1) suppressed by powers of Λ ˆ Q , but huge oscillations near z = 1 1 Q = 8 GeV 0.5 σ MP (z) 0 ^ -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 10
Minimal prescription: non-physical contribution � + ∞ dz � τ � σ MP ( τ ) = z L ˆ σ MP ( z ) z τ The integral extends to + ∞ , not to 1 ! σ MP ( z > 1) suppressed by powers of Λ ˆ Q , but huge oscillations near z = 1 1 Q = 8 GeV The MP is more conveniently used 0.5 in the N –space formulation σ MP (z) 0 ^ -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 10
Minimal prescription: non-physical contribution � + ∞ dz � τ � σ MP ( τ ) = z L ˆ σ MP ( z ) z τ The integral extends to + ∞ , not to 1 ! σ MP ( z > 1) suppressed by powers of Λ ˆ Q , but huge oscillations near z = 1 1 Q = 8 GeV The MP is more conveniently used 0.5 in the N –space formulation σ MP (z) 0 ^ Need for L ( N ) , for values of N where the Mellin transform of L ( x ) -0.5 does not converge -1 0 0.2 0.4 0.6 0.8 1 1.2 z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 10
Borel prescription (1) σ res ( N ) ˆ Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ log k 1 σ res ( N ) = � α k ˆ h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Treat the divergent series M − 1 (ˆ σ res ( N )) with Borel method: ⋆ ⋆ S.Forte, G.Ridolfi, J.Rojo, M.Ubiali (hep-ph/0601048); R.Abbate, SF, GR (hep-ph/0707.2452); MB, SF, GR (hep-ph/0807.3830); MB, SF, GR (coming soon) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Treat the divergent series M − 1 (ˆ σ res ( N )) with Borel method: ⋆ ∞ � b k k =1 ⋆ S.Forte, G.Ridolfi, J.Rojo, M.Ubiali (hep-ph/0601048); R.Abbate, SF, GR (hep-ph/0707.2452); MB, SF, GR (hep-ph/0807.3830); MB, SF, GR (coming soon) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Treat the divergent series M − 1 (ˆ σ res ( N )) with Borel method: ⋆ � 1 � + ∞ ∞ � dw e − w w k � b k k ! 0 k =1 ⋆ S.Forte, G.Ridolfi, J.Rojo, M.Ubiali (hep-ph/0601048); R.Abbate, SF, GR (hep-ph/0707.2452); MB, SF, GR (hep-ph/0807.3830); MB, SF, GR (coming soon) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Treat the divergent series M − 1 (ˆ σ res ( N )) with Borel method: ⋆ � 1 � + ∞ � + ∞ ∞ ∞ � b k dw e − w w k Borel � � dw e − w k ! w k b k = k ! 0 0 k =1 k =1 ⋆ S.Forte, G.Ridolfi, J.Rojo, M.Ubiali (hep-ph/0601048); R.Abbate, SF, GR (hep-ph/0707.2452); MB, SF, GR (hep-ph/0807.3830); MB, SF, GR (coming soon) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Treat the divergent series M − 1 (ˆ σ res ( N )) with Borel method: ⋆ � 1 � + ∞ � + ∞ ∞ ∞ � b k dw e − w w k Borel � � dw e − w k ! w k b k = k ! 0 0 k =1 k =1 the inner sum converges ⋆ S.Forte, G.Ridolfi, J.Rojo, M.Ubiali (hep-ph/0601048); R.Abbate, SF, GR (hep-ph/0707.2452); MB, SF, GR (hep-ph/0807.3830); MB, SF, GR (coming soon) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Treat the divergent series M − 1 (ˆ σ res ( N )) with Borel method: ⋆ � 1 � + ∞ � + ∞ ∞ ∞ � b k dw e − w w k Borel � � dw e − w k ! w k b k = k ! 0 0 k =1 k =1 the inner sum converges the integral diverges (the series is not Borel-summable) ⋆ S.Forte, G.Ridolfi, J.Rojo, M.Ubiali (hep-ph/0601048); R.Abbate, SF, GR (hep-ph/0707.2452); MB, SF, GR (hep-ph/0807.3830); MB, SF, GR (coming soon) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (1) ∞ � log k 1 � M − 1 � σ res ( N ) � α k M − 1 � ˆ = h k (¯ α ) ¯ , α = 2 β 0 α s ¯ N k =1 Treat the divergent series M − 1 (ˆ σ res ( N )) with Borel method: ⋆ � 1 � + ∞ � + ∞ ∞ ∞ � b k dw e − w w k Borel � � dw e − w k ! w k b k = k ! 0 0 k =1 k =1 the inner sum converges the integral diverges (the series is not Borel-summable) proposed solution: cut-off C in the integral ⋆ S.Forte, G.Ridolfi, J.Rojo, M.Ubiali (hep-ph/0601048); R.Abbate, SF, GR (hep-ph/0707.2452); MB, SF, GR (hep-ph/0807.3830); MB, SF, GR (coming soon) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 11
Borel prescription (2) � C 1 � dξ � log ξ − 1 1 � dw � w � α e − w α Σ σ BP ( z, C ) = ˆ ¯ 2 πi Γ( ξ + 1) z ¯ ξ 0 C + α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Borel prescription (2) � C 1 � dξ � log ξ − 1 1 � dw � w � α e − w α Σ σ BP ( z, C ) = ˆ ¯ 2 πi Γ( ξ + 1) z ¯ ξ 0 C + α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Remarks resummed expression at parton level → easier numerical implementation Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Borel prescription (2) � C 1 � dξ � log ξ − 1 1 � dw � w � α e − w α Σ σ BP ( z, C ) = ˆ ¯ 2 πi Γ( ξ + 1) z ¯ ξ 0 C + α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Remarks resummed expression at parton level → easier numerical implementation asymptotic to the original divergent series Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Borel prescription (2) � C 1 � dξ � log ξ − 1 1 � dw � w � α e − w α Σ σ BP ( z, C ) = ˆ ¯ 2 πi Γ( ξ + 1) z ¯ ξ 0 C + α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Remarks resummed expression at parton level → easier numerical implementation asymptotic to the original divergent series parameter C to estimate ambiguity Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Borel prescription (2) � C 1 � dξ � log ξ − 1 1 � dw � w � α e − w α Σ ˆ σ BP ( z, C ) = ¯ 2 πi Γ( ξ + 1) z ¯ ξ 0 C + α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Remarks resummed expression at parton level → easier numerical implementation asymptotic to the original divergent series parameter C to estimate ambiguity � C/ 2 cut-off related to the inclusion of higher-twist terms e − C � Λ 2 α ≃ ¯ Q 2 Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Borel prescription (2) � C 1 � dξ � log ξ − 1 1 � dw � w � α e − w α Σ σ BP ( z, C ) = ˆ ¯ 2 πi Γ( ξ + 1) z ¯ ξ 0 C + α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Remarks resummed expression at parton level → easier numerical implementation asymptotic to the original divergent series parameter C to estimate ambiguity � C/ 2 cut-off related to the inclusion of higher-twist terms e − C � Λ 2 α ≃ ¯ Q 2 z dependence under control: log k log 1 z log 1 z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Borel prescription (2) � C 1 � dξ dw � w � � (1 − z ) ξ − 1 � α e − w α Σ σ BP ( z, C ) = ˆ ¯ 2 πi Γ( ξ + 1) ¯ ξ + 0 C α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Remarks resummed expression at parton level → easier numerical implementation asymptotic to the original divergent series parameter C to estimate ambiguity � C/ 2 cut-off related to the inclusion of higher-twist terms e − C � Λ 2 α ≃ ¯ Q 2 z dependence under control: log k log 1 log k (1 − z ) z log 1 1 − z z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Borel prescription (2) � C � (1 − z ) ξ − 1 � � w � 1 � dξ dw α e − w + α Σ σ BP ( z, C ) = ˆ √ z ξ ¯ 2 πi Γ( ξ + 1) ¯ ξ C 0 α log 1 σ res ( N ) where Σ(¯ N ) ≡ ˆ Remarks resummed expression at parton level → easier numerical implementation asymptotic to the original divergent series parameter C to estimate ambiguity � C/ 2 cut-off related to the inclusion of higher-twist terms e − C � Λ 2 α ≃ ¯ Q 2 z dependence under control: log k 1 − z log k log 1 log k (1 − z ) √ z z log 1 1 − z 1 − z z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 12
Comparison with fixed order: Drell-Yan q ¯ q at NLO ( » log(1 − z ) ) − log √ z „ π 2 α s – 1 − z − 1 + z log 1 − z « π 4 C F + 12 − 1 δ (1 − z ) √ z 1 − z 2 + NLO full 40 30 20 10 0 0 2 4 6 8 10 12 14 N Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 13
Comparison with fixed order: Drell-Yan q ¯ q at NLO ( » log(1 − z ) ) − log √ z „ π 2 α s – 1 − z − 1 + z log 1 − z « π 4 C F + 12 − 1 δ (1 − z ) √ z 1 − z 2 + NLO full Borel 40 30 20 10 0 0 2 4 6 8 10 12 14 N Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 13
Comparison with fixed order: Drell-Yan q ¯ q at NLO ( » log(1 − z ) ) − log √ z „ π 2 α s – 1 − z − 1 + z log 1 − z « π 4 C F + 12 − 1 δ (1 − z ) √ z 1 − z 2 + » log log 1 – z log 1 z + NLO full Borel 40 Minimal 30 20 10 0 0 2 4 6 8 10 12 14 N Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 13
Comparison with fixed order: Drell-Yan q ¯ q at NLO ( » log(1 − z ) ) − log √ z „ π 2 α s – 1 − z − 1 + z log 1 − z « π 4 C F + 12 − 1 δ (1 − z ) √ z 1 − z 2 + » log log 1 – z log 1 z + NLO full Borel 40 Minimal Borel improved 30 20 10 0 0 2 4 6 8 10 12 14 N Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 13
Comparison with fixed order: Drell-Yan q ¯ q at NNLO 700 NNLO full Borel improved 600 Minimal 500 400 300 200 100 0 0 2 4 6 8 10 12 14 N Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 14
Comparison with fixed order: Drell-Yan q ¯ q at NNLO 700 NNLO full Borel improved 600 Minimal 500 400 300 200 100 0 0 2 4 6 8 10 12 14 N log k (1 − z ) → log k N Discrepancy due to terms like ⇒ subleading N Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 14
State of the art the BP can produce the same logs of MP indistinguishable at hadron level for τ ≪ 1 (always in phenomenological applications) BP has an easier and faster numerical implementation Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 15
State of the art the BP can produce the same logs of MP indistinguishable at hadron level for τ ≪ 1 (always in phenomenological applications) BP has an easier and faster numerical implementation the BP can produce “more physical” logs include some classes of subleading terms better small– z behaviour Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 15
State of the art the BP can produce the same logs of MP indistinguishable at hadron level for τ ≪ 1 (always in phenomenological applications) BP has an easier and faster numerical implementation the BP can produce “more physical” logs include some classes of subleading terms better small– z behaviour there are subleading terms which are important log k (1 − z ) and similar and which are not included in the resummed expressions Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 15
State of the art the BP can produce the same logs of MP indistinguishable at hadron level for τ ≪ 1 (always in phenomenological applications) BP has an easier and faster numerical implementation the BP can produce “more physical” logs include some classes of subleading terms better small– z behaviour there are subleading terms which are important log k (1 − z ) and similar and which are not included in the resummed expressions the difference in the included subleading terms is useful to estimate the importance of these terms Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 15
Impact in phenomenology: rapidity distributions (1) � 1 � 1 � � 1 dσ dx 1 dx 2 τ , Y − 1 2 log x 1 dQ 2 dY = f 1 ( x 1 ) f 2 ( x 2 ) C τ x 1 x 2 x 1 x 2 x 2 √ τe Y √ τe − Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 16
Impact in phenomenology: rapidity distributions (1) � 1 � 1 � � 1 dσ dx 1 dx 2 τ , Y − 1 2 log x 1 dQ 2 dY = f 1 ( x 1 ) f 2 ( x 2 ) C τ x 1 x 2 x 1 x 2 x 2 √ τe Y √ τe − Y Fourier transform of C ( z, y ) wrt y � + ∞ ˜ dy C ( z, y ) e iMy C ( z, M ) = −∞ Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 16
Impact in phenomenology: rapidity distributions (1) � 1 � 1 � � 1 dσ dx 1 dx 2 τ , Y − 1 2 log x 1 dQ 2 dY = f 1 ( x 1 ) f 2 ( x 2 ) C τ x 1 x 2 x 1 x 2 x 2 √ τe Y √ τe − Y Fourier transform of C ( z, y ) wrt y � − log √ z ˜ dy C ( z, y ) e iMy C ( z, M ) = log √ z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 16
Impact in phenomenology: rapidity distributions (1) � 1 � 1 � � 1 dσ dx 1 dx 2 τ , Y − 1 2 log x 1 dQ 2 dY = f 1 ( x 1 ) f 2 ( x 2 ) C τ x 1 x 2 x 1 x 2 x 2 √ τe Y √ τe − Y Fourier transform of C ( z, y ) wrt y � − log √ z ˜ C ( z, M ) = dy C ( z, y ) [1 + O ( y )] log √ z Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 16
Impact in phenomenology: rapidity distributions (1) � 1 � 1 � � 1 dσ dx 1 dx 2 τ , Y − 1 2 log x 1 dQ 2 dY = f 1 ( x 1 ) f 2 ( x 2 ) C τ x 1 x 2 x 1 x 2 x 2 √ τe Y √ τe − Y Fourier transform of C ( z, y ) wrt y � − log √ z ˜ C ( z, M ) = dy C ( z, y ) [1 + O ( y )] log √ z Since | log z | ≃ 1 − z we have ˜ C ( z, M ) = C ( z ) [1 + O (1 − z )] Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 16
Impact in phenomenology: rapidity distributions (1) � 1 � 1 � � 1 dσ dx 1 dx 2 τ , Y − 1 2 log x 1 dQ 2 dY = f 1 ( x 1 ) f 2 ( x 2 ) C τ x 1 x 2 x 1 x 2 x 2 √ τe Y √ τe − Y Fourier transform of C ( z, y ) wrt y � − log √ z ˜ C ( z, M ) = dy C ( z, y ) [1 + O ( y )] log √ z Since | log z | ≃ 1 − z we have ˜ C ( z, M ) = C ( z ) [1 + O (1 − z )] or, back to y space, C ( z, y ) = C ( z ) δ ( y ) [1 + O (1 − z )] Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 16
Impact in phenomenology: rapidity distributions (2) After changing variables we get the compact expression � 1 dσ res �� τ � �� τ � 1 dz z C res ( z ) f 1 z e Y z e − Y dQ 2 dY = f 2 τ τe 2 | Y | Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 17
Impact in phenomenology: rapidity distributions (2) After changing variables we get the compact expression � 1 dσ res �� τ � �� τ � 1 dz z C res ( z ) f 1 z e Y z e − Y dQ 2 dY = f 2 τ τe 2 | Y | depends on C res ( z ) = M − 1 [ˆ σ res ( N )] , the well-known rapidity-integrated resummed coefficient Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 17
Impact in phenomenology: rapidity distributions (2) After changing variables we get the compact expression � 1 dσ res �� τ � �� τ � 1 dz z C res ( z ) f 1 z e Y z e − Y dQ 2 dY = f 2 τ τe 2 | Y | depends on C res ( z ) = M − 1 [ˆ σ res ( N )] , the well-known rapidity-integrated resummed coefficient has the form of a convolution product → both Borel and minimal prescriptions are applicable! Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 17
Impact in phenomenology: rapidity distributions (2) After changing variables we get the compact expression � 1 dσ res �� τ � �� τ � 1 dz z C res ( z ) f 1 z e Y z e − Y dQ 2 dY = f 2 τ τe 2 | Y | depends on C res ( z ) = M − 1 [ˆ σ res ( N )] , the well-known rapidity-integrated resummed coefficient has the form of a convolution product → both Borel and minimal prescriptions are applicable! Results at NNLO + NNLL Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 17
Impact in phenomenology: rapidity distributions (2) After changing variables we get the compact expression � 1 dσ res �� τ � �� τ � 1 dz z C res ( z ) f 1 z e Y z e − Y dQ 2 dY = f 2 τ τe 2 | Y | depends on C res ( z ) = M − 1 [ˆ σ res ( N )] , the well-known rapidity-integrated resummed coefficient has the form of a convolution product → both Borel and minimal prescriptions are applicable! Results at NNLO + NNLL C++ code: NNLO: C.Anastasiou, L.Dixon, K.Melnikov, F.Petriello (hep-ph/0312266) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 17
Impact in phenomenology: rapidity distributions (2) After changing variables we get the compact expression � 1 dσ res �� τ � �� τ � 1 dz z C res ( z ) f 1 z e Y z e − Y dQ 2 dY = f 2 τ τe 2 | Y | depends on C res ( z ) = M − 1 [ˆ σ res ( N )] , the well-known rapidity-integrated resummed coefficient has the form of a convolution product → both Borel and minimal prescriptions are applicable! Results at NNLO + NNLL C++ code: NNLO: C.Anastasiou, L.Dixon, K.Melnikov, F.Petriello (hep-ph/0312266) extension with NNLL resummation (Borel and minimal) Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 17
Impact in phenomenology: rapidity distributions (2) After changing variables we get the compact expression � 1 dσ res �� τ � �� τ � 1 dz z C res ( z ) f 1 z e Y z e − Y dQ 2 dY = f 2 τ τe 2 | Y | depends on C res ( z ) = M − 1 [ˆ σ res ( N )] , the well-known rapidity-integrated resummed coefficient has the form of a convolution product → both Borel and minimal prescriptions are applicable! Results at NNLO + NNLL C++ code: NNLO: C.Anastasiou, L.Dixon, K.Melnikov, F.Petriello (hep-ph/0312266) extension with NNLL resummation (Borel and minimal) interface to LHAPDF library Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 17
W asymmetry at Tevatron with NNPDF2.0 W asymmetry. Collider: ppbar 0.8 NNLO √ s = 1.96 TeV Borel NNLL+NNLO Q = µ R = µ F = M W Minimal NNLL+NNLO 0.7 τ = 0.00168 CDF data (1 fb -1 ) 0.6 0.5 A W 0.4 0.3 0.2 0.1 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 0 0.5 1 1.5 2 2.5 3 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 18
Rapidity distribution: DY ( 8 GeV) at NuSea τ ≃ 0 . 04 6 LO √ s = 38.76 GeV NLO M = 8 GeV 5 Q = 8 GeV NNLO 5 τ = 0.04260 Borel LL+LO 0.5 < µ R /Q < 2 Borel NLL+NLO Borel NNLL+NNLO 0.5 < µ F /Q < 2 4 E866 data 4 d σ /dQ/dY [pb/GeV] 3 3 2 2 1 1 0 � 1.5 � 1.0 � 0.5 0.0 0.5 1.0 1.5 0 -1.5 -1 -0.5 0 0.5 1 1.5 Y Y M.Bonvini, S.Forte, G.Ridolfi T.Becher, M.Neubert, G.Xu preliminary (hep-ph/0710.0680) NNPDF2.0 MRST04NNLO Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 19
Rapidity distribution: DY ( 8 GeV) at NuSea τ ≃ 0 . 04 6 LO √ s = 38.76 GeV NLO M = 8 GeV 5 Q = 8 GeV NNLO 5 τ = 0.04260 Minimal LL+LO 0.5 < µ R /Q < 2 Minimal NLL+NLO Minimal NNLL+NNLO 0.5 < µ F /Q < 2 4 E866 data 4 d σ /dQ/dY [pb/GeV] 3 3 2 2 1 1 0 � 1.5 � 1.0 � 0.5 0.0 0.5 1.0 1.5 0 -1.5 -1 -0.5 0 0.5 1 1.5 Y Y M.Bonvini, S.Forte, G.Ridolfi T.Becher, M.Neubert, G.Xu preliminary (hep-ph/0710.0680) NNPDF2.0 MRST04NNLO Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 19
Rapidity distribution: DY ( 1 TeV) at LHC with NNPDF2.0 DY rapidity distribution. Collider: pp Subprocess: Z+gamma 3.5e-06 LO √ s = 7.00 TeV NLO Q = 1000 GeV NNLO 3e-06 τ = 0.02041 Borel LL+LO 0.5 < µ R /Q < 2 Borel NLL+NLO 0.5 < µ F /Q < 2 Borel NNLL+NNLO 2.5e-06 d σ /dQ/dY [pb/GeV] 2e-06 1.5e-06 1e-06 5e-07 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 -1.5 -1 -0.5 0 0.5 1 1.5 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 20
Rapidity distribution: DY ( 1 TeV) at LHC with NNPDF2.0 DY rapidity distribution. Collider: pp Subprocess: Z+gamma 3.5e-06 LO √ s = 7.00 TeV NLO Q = 1000 GeV NNLO 3e-06 τ = 0.02041 Minimal LL+LO 0.5 < µ R /Q < 2 Minimal NLL+NLO 0.5 < µ F /Q < 2 Minimal NNLL+NNLO 2.5e-06 d σ /dQ/dY [pb/GeV] 2e-06 1.5e-06 1e-06 5e-07 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 -1.5 -1 -0.5 0 0.5 1 1.5 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 20
Rapidity distribution: Z at LHC with NNPDF2.0 DY rapidity distribution. Collider: pp Subprocess: Z+gamma 45 LO √ s = 7.00 TeV NLO Q = M Z 40 NNLO τ = 0.00017 Borel LL+LO 0.5 < µ R /Q < 2 Borel NLL+NLO 35 0.5 < µ F /Q < 2 Borel NNLL+NNLO 30 d σ /dQ/dY [pb/GeV] 25 20 15 10 5 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 -4 -3 -2 -1 0 1 2 3 4 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 21
Rapidity distribution: Z at LHC with NNPDF2.0 DY rapidity distribution. Collider: pp Subprocess: Z+gamma 45 LO √ s = 7.00 TeV NLO Q = M Z 40 NNLO τ = 0.00017 Minimal LL+LO 0.5 < µ R /Q < 2 Minimal NLL+NLO 35 0.5 < µ F /Q < 2 Minimal NNLL+NNLO 30 d σ /dQ/dY [pb/GeV] 25 20 15 10 5 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 -4 -3 -2 -1 0 1 2 3 4 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 21
Rapidity distribution: W + at LHC with NNPDF2.0 DY rapidity distribution. Collider: pp Subprocess: W+ 300 LO √ s = 7.00 TeV NLO Q = M W NNLO τ = 0.00013 Borel LL+LO 250 0.5 < µ R /Q < 2 Borel NLL+NLO 0.5 < µ F /Q < 2 Borel NNLL+NNLO 200 d σ /dQ/dY [pb/GeV] 150 100 50 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 -4 -3 -2 -1 0 1 2 3 4 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 22
Rapidity distribution: W + at LHC with NNPDF2.0 DY rapidity distribution. Collider: pp Subprocess: W+ 300 LO √ s = 7.00 TeV NLO Q = M W NNLO τ = 0.00013 Minimal LL+LO 250 0.5 < µ R /Q < 2 Minimal NLL+NLO 0.5 < µ F /Q < 2 Minimal NNLL+NNLO 200 d σ /dQ/dY [pb/GeV] 150 100 50 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 -4 -3 -2 -1 0 1 2 3 4 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 22
Rapidity distribution: W − at LHC with NNPDF2.0 DY rapidity distribution. Collider: pp Subprocess: W- 250 LO √ s = 7.00 TeV NLO Q = M W NNLO τ = 0.00013 Borel LL+LO 0.5 < µ R /Q < 2 200 Borel NLL+NLO 0.5 < µ F /Q < 2 Borel NNLL+NNLO d σ /dQ/dY [pb/GeV] 150 100 50 M.Bonvini, S.Forte, G.Ridolfi - preliminary 0 -4 -3 -2 -1 0 1 2 3 4 Y Marco Bonvini Threshold resummation for Drell-Yan production: theory and phenomenology 23
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