Vector boson production and decay in hadron collisions: q T resummation at NNLL accuracy Giancarlo Ferrera Milan University & INFN Milan In collaboration with: G. Bozzi, S. Catani, D. de Florian & M. Grazzini HP2.5 – Florence – Sept. 5th 2014
Motivations The Drell–Yan process [Drell,Yan(’70)] is a benchmark process in hadron collider physics. Its study is well motivated: Large production rates and clean experimental signatures. Constraints for fits of PDFs. q T spectrum: important for M W measurement and Beyond the Standard Model analyses. Test of perturbative QCD predictions. The above reasons and precise experimental data demands for accurate theoretical predictions ⇒ computation of higher-order QCD corrections. Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 2/16
Motivations The Drell–Yan process [Drell,Yan(’70)] is a benchmark process in hadron collider physics. Its study is well motivated: Large production rates and clean experimental signatures. Constraints for fits of PDFs. q T spectrum: important for M W measurement and Beyond the Standard Model analyses. Test of perturbative QCD predictions. The above reasons and precise experimental data demands for accurate theoretical predictions ⇒ computation of higher-order QCD corrections. Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 2/16
h 1 ( x 1 ,µ 2 F ) h 1 ( p 1 ) f a The Drell–Yan q T distribution / . . > ℓ 1 a ( x 1 p 1 ) V ( M ) ℓ 2 h 1 (p 1 ) + h 2 (p 2 ) → V(M) + X → ℓ 1 + ℓ 2 + X σ ˆ ab V = γ ∗ , Z 0 , W ± ℓ 1 ℓ 2 = ℓ + ℓ − , ℓν ℓ where and . . � b ( x 2 p 2 ) X . > . h 2 ( p 2 ) h 2 ( x 2 ,µ 2 pQCD factorization formula: f b F ) / � 1 � 1 � d σ F ) d ˆ σ ab h 1 ( x 1 , µ 2 h 2 ( x 2 , µ 2 s ; α S ,µ 2 R ,µ 2 ( q T , M , s )= dx 1 dx 2 f a F ) f b ( q T , M , ˆ F ) . / / dq 2 dq 2 0 0 T T a , b Standard fixed-order perturbative expansions ( Q T ≪ 1): � � � Q 2 c 12 log 2 M 2 + c 11 log M 2 d ˆ σ q ¯ T q dq 2 ∼ 1 + α S + c 10 T dq 2 Q 2 Q 2 0 T T T � � c 24 log 4 M 2 + · · · + c 21 log M 2 + α 2 + O ( α 3 + c 20 S ) S Q 2 Q 2 T T Fixed order calculation reliable only for q T ∼ M For q T → 0 , α n S log m ( M 2 / q 2 T ) ≫ 1: need for resummation of logarithmic corrections. Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 3/16
h 1 ( x 1 ,µ 2 F ) h 1 ( p 1 ) f a The Drell–Yan q T distribution / . . > ℓ 1 a ( x 1 p 1 ) V ( M ) ℓ 2 h 1 (p 1 ) + h 2 (p 2 ) → V(M) + X → ℓ 1 + ℓ 2 + X σ ˆ ab V = γ ∗ , Z 0 , W ± ℓ 1 ℓ 2 = ℓ + ℓ − , ℓν ℓ where and . . � b ( x 2 p 2 ) X . > . h 2 ( p 2 ) h 2 ( x 2 ,µ 2 pQCD factorization formula: f b F ) / � 1 � 1 � d σ F ) d ˆ σ ab h 1 ( x 1 , µ 2 h 2 ( x 2 , µ 2 s ; α S ,µ 2 R ,µ 2 ( q T , M , s )= dx 1 dx 2 f a F ) f b ( q T , M , ˆ F ) . / / dq 2 dq 2 0 0 T T a , b Standard fixed-order perturbative expansions ( Q T ≪ 1): � � � Q 2 c 12 log 2 M 2 + c 11 log M 2 d ˆ σ q ¯ T q dq 2 ∼ 1 + α S + c 10 T dq 2 Q 2 Q 2 0 T T T � � c 24 log 4 M 2 + · · · + c 21 log M 2 + α 2 + O ( α 3 + c 20 S ) S Q 2 Q 2 T T Fixed order calculation reliable only for q T ∼ M For q T → 0 , α n S log m ( M 2 / q 2 T ) ≫ 1: need for resummation of logarithmic corrections. Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 3/16
h 1 ( x 1 ,µ 2 F ) h 1 ( p 1 ) f a The Drell–Yan q T distribution / . . > ℓ 1 V ( M ) a ( x 1 p 1 ) ℓ 2 h 1 (p 1 ) + h 2 (p 2 ) → V(M) + X → ℓ 1 + ℓ 2 + X σ ˆ ab V = γ ∗ , Z 0 , W ± ℓ 1 ℓ 2 = ℓ + ℓ − , ℓν ℓ where and . . � b ( x 2 p 2 ) X . > . h 2 ( p 2 ) h 2 ( x 2 ,µ 2 pQCD factorization formula: f b F ) / � 1 � 1 � d σ F ) d ˆ σ ab h 1 ( x 1 , µ 2 h 2 ( x 2 , µ 2 s ; α S ,µ 2 R ,µ 2 ( q T , M , s )= dx 1 dx 2 f a F ) f b ( q T , M , ˆ F ) . / / dq 2 dq 2 T 0 0 T a , b Standard fixed-order perturbative expansions ( Q T ≪ 1): � Q 2 � � c 12 log 2 M 2 + c 11 log M 2 d ˆ σ q ¯ T q dq 2 ∼ 1 + α S + c 10 T dq 2 Q 2 Q 2 0 T T T � � c 24 log 4 M 2 + · · · + c 21 log M 2 + α 2 + O ( α 3 + c 20 S ) S Q 2 Q 2 T T Fixed order calculation reliable only for q T ∼ M For q T → 0 , α n S log m ( M 2 / q 2 T ) ≫ 1: need for resummation of logarithmic corrections. Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 3/16
Idea of (analytic) resummation Idea of large logs (Sudakov) resummation: reorganize the perturbative M 2 / q 2 expansion by all-order summation ( L = log( T )). α S L 2 α S L · · · · · · · · · O ( α S ) α 2 S L 4 α 2 S L 3 α 2 S L 2 α 2 O ( α 2 · · · S ) S L · · · · · · · · · · · · · · · · · · α n S L 2 n α n S L 2 n − 1 α n S L 2 n − 2 O ( α n · · · · · · S ) · · · · · · · · · · · · dominant logs next-to-dominant logs Ratio of two successive rows O ( α S L 2 ): fixed order expansion valid when α S L 2 ≪ 1. Ratio of two successive columns O (1 / L ): resummed expansion valid when 1 / L ≪ 1 i.e. when α S L 2 ∼ 1 (and α S ≪ 1). Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 4/16
Idea of (analytic) resummation Idea of large logs (Sudakov) resummation: reorganize the perturbative M 2 / q 2 expansion by all-order summation ( L = log( T )). α S L 2 α S L · · · · · · · · · O ( α S ) α 2 S L 4 α 2 S L 3 α 2 S L 2 α 2 O ( α 2 · · · S ) S L · · · · · · · · · · · · · · · · · · α n S L 2 n α n S L 2 n − 1 α n S L 2 n − 2 O ( α n · · · · · · S ) · · · · · · · · · · · · dominant logs next-to-dominant logs Ratio of two successive rows O ( α S L 2 ): fixed order expansion valid when α S L 2 ≪ 1. Ratio of two successive columns O (1 / L ): resummed expansion valid when 1 / L ≪ 1 i.e. when α S L 2 ∼ 1 (and α S ≪ 1). Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 4/16
Sudakov resummation is feasible when we have dynamics AND kinematics factorization ⇒ exponentiation. Dynamics factorization: general propriety of QCD matrix element for soft emissions based on colour coherence. It is the analogous of the independent multiple soft-photon emission is QED: n � dw n ( q 1 , . . . , q n ) ≃ 1 dw i ( q i ) n ! i =1 Kinematics factorization: not valid in general. For q T distribution of DY process it holds in the impact parameter space (Fourier transform). � � � n n n � � � d 2 q T exp( − i b · q T ) δ q T − q T j = exp( − i b · q T j ) = exp( − i b · q T j ) . j =1 j =1 j =1 Exponentiation holds in the impact parameter space. Results have then to be transformed back to the physical space. Resummed result can then be properly combined with the fixed order result ( matching ) to have a good control of both the kinematical regions: q T ≪ M and q T ∼ M . Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 5/16
Sudakov resummation is feasible when we have dynamics AND kinematics factorization ⇒ exponentiation. Dynamics factorization: general propriety of QCD matrix element for soft emissions based on colour coherence. It is the analogous of the independent multiple soft-photon emission is QED: n � dw n ( q 1 , . . . , q n ) ≃ 1 dw i ( q i ) n ! i =1 Kinematics factorization: not valid in general. For q T distribution of DY process it holds in the impact parameter space (Fourier transform). � � � n n n � � � d 2 q T exp( − i b · q T ) δ q T − q T j = exp( − i b · q T j ) = exp( − i b · q T j ) . j =1 j =1 j =1 Exponentiation holds in the impact parameter space. Results have then to be transformed back to the physical space. Resummed result can then be properly combined with the fixed order result ( matching ) to have a good control of both the kinematical regions: q T ≪ M and q T ∼ M . Giancarlo Ferrera – Universit` a & INFN Milano HP2.5 – Florence – 5/9/2014 Drell–Yan production and decay: q T resummation at NNLL accuracy 5/16
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