Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Perturbation theory of computing QCD jet cross sections beyond NLO accuracy Zoltán Trócsányi University of Debrecen and Institute of Nuclear Research in collaboration with G. Somogyi, V. Del Duca, Z. Nagy October 4, 2007
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Outline Introduction 1 Standard motivation Less standard motivation pQCD computation of jet cross sections 2 Perturbative expansion NLO correction Extension to NNLO 3 Structure of NNLO subtraction Naive generalization of NLO subtraction fails NLO subtraction revisited 4 Separation of collinear and purely-soft subtractions NLO subtraction with new phase-space mappings NLO subtraction with fixed helicities Summary 5 Extra slides 6
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Standard motivation Precision QCD sometimes requires computations beyond NLO to reduce the dependence on unphysical renormalization and factorization scales if the NLO corrections are large (can be more than 100 %), such as Higgs production in hadron collisions the main source of uncertainty in experimental results is due to theory, such as α s measurements the NLO computation is effectively LO, such as energy distribution inside jet cones reliable error estimate is needed, such as, precise measurement of parton luminosity, but rather always . . .
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes . . . can we compute real radiation fast enough?
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes . . . can we compute real radiation fast enough? current approaches to fixed-order and parton shower computations have mutually exclusive elements, which may hamper their combination
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes . . . can we compute real radiation fast enough? current approaches to fixed-order and parton shower computations have mutually exclusive elements, which may hamper their combination . . . understanding NNLO helps further development of parton showers
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides The perturbative expansion at NNLO accuracy σ = σ LO + σ NLO + σ NNLO + . . .
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides The perturbative expansion at NNLO accuracy σ = σ LO + σ NLO + σ NNLO + . . . Consider e + e − → m jet production LO b m m b b σ LO = d φ m |M ( 0 ) m d σ B m | 2 J m R R m =
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides The perturbative expansion at NNLO accuracy σ = σ LO + σ NLO + σ NNLO + . . . Consider e + e − → m jet production LO b m m b b σ LO = d φ m |M ( 0 ) m d σ B m | 2 J m R R m = b b NLO b r b m m m +1 m +1 b b b b b σ NLO = R m + 1 d σ R R m d σ V m + 1 + m d φ m + 1 |M ( 0 ) d φ m 2Re �M ( 1 ) m |M ( 0 ) R m + 1 | 2 J m + 1 + R = m � J m
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides The perturbative expansion at NNLO accuracy σ = σ LO + σ NLO + σ NNLO + . . . Consider e + e − → m jet production LO b m m b b σ LO = d φ m |M ( 0 ) R m d σ B R m | 2 J m m = b b NLO b r b m m m +1 m +1 b b b b b σ NLO = R m + 1 d σ R R m d σ V m + 1 + m d φ m + 1 |M ( 0 ) d φ m 2Re �M ( 1 ) m |M ( 0 ) m + 1 | 2 J m + 1 + R R = m � J m b b r b b b NNLO r b b m +2 m +2 b m m m m m +1 m +1 m +1 s b b b b b b b b b b σ NNLO = m + 2 d σ RR m + 1 d σ RV m d σ VV R m + 2 + R + R = m + 1 m d φ m + 2 |M ( 0 ) d φ m + 1 2Re �M ( 1 ) m + 1 |M ( 0 ) m + 2 | 2 J m + 2 + = R R m + 1 � J m + 1 + h i |M ( 1 ) | 2 + 2Re �M ( 2 ) |M ( 0 ) + R d φ m � J m m m m
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides The perturbative expansion at NNLO accuracy σ = σ LO + σ NLO + σ NNLO + . . . Consider e + e − → m jet production LO b m m b b σ LO = d φ m |M ( 0 ) R m d σ B R m | 2 J m m = b b NLO b r b m m m +1 m +1 b b b b b σ NLO = R m + 1 d σ R R m d σ V m + 1 + m d φ m + 1 |M ( 0 ) d φ m 2Re �M ( 1 ) m |M ( 0 ) m + 1 | 2 J m + 1 + R R = m � J m b b r b b b NNLO r b b m +2 m +2 b m m m m m +1 m +1 m +1 s b b b b b b b b b b σ NNLO = d σ RR d σ RV d σ VV R m + 2 + R + R = m + 1 m d φ m + 2 |M ( 0 ) d φ m + 1 2Re �M ( 1 ) m + 1 |M ( 0 ) m + 2 | 2 J m + 2 + = R R m + 1 � J m + 1 + h i |M ( 1 ) | 2 + 2Re �M ( 2 ) |M ( 0 ) + R d φ m � J m m m m
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides b b r b b m m m m +1 m +1 m +1 b b b b b � � d d k d φ ( k ) 1 � �� � divergent in d = 4! Process independent methods (phase space slicing, residuum, dipole or antennae subtraction) use regularized integrals in d = 4 − 2 ǫ dimensions universal soft- and collinear factorization of QCD (squared) matrix elements C ir is a symbolic operator that takes the collinear limit m + 1 ( p i , p r , . . . ) | 2 ∝ 1 C ir |M ( 0 ) s ir �M ( 0 ) m ( p ir , . . . ) | ˆ P ( 0 ) ir |M ( 0 ) m ( p ir , . . . ) � S r is a symbolic operator that takes the soft limit s ik m + 1 ( p r , . . . ) | 2 ∝ S r |M ( 0 ) X s ir s kr �M ( 0 ) m ( . . . ) | T i T k |M ( 0 ) m ( . . . ) � i , k i � = k
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides b b r b b m +1 m +1 m +1 m m m b b b b b � � d d k d φ ( k ) 1 � �� � divergent in d = 4! Process independent methods (phase space slicing, residuum, dipole or antennae subtraction) use regularized integrals in d = 4 − 2 ǫ dimensions universal soft- and collinear factorization of QCD (squared) matrix elements to construct approximate cross section to regularize real emissions: � � � � � �� d σ R � � d σ A � � σ NLO = d σ V + d σ A ε = 0 − + ε = 0 m + 1 m 1 ε = 0 � � d σ NLO d σ NLO ≡ m + 1 + m m + 1 m
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Construction of the subtraction terms at NLO The collinear and soft regions overlap: C ir S r
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Construction of the subtraction terms at NLO The collinear and soft regions overlap: C ir S r The candidate subtraction term. . . � � � � 1 m + 1 | 2 ? A 1 |M ( 0 ) |M ( 0 ) m + 1 | 2 = 2 C ir r i � = r
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Construction of the subtraction terms at NLO The collinear and soft regions overlap: − C ir S r or C ir S r − S r C ir ? The candidate subtraction term. . . � � � � 1 m + 1 | 2 ? A 1 |M ( 0 ) |M ( 0 ) m + 1 | 2 = 2 C ir + S r r i � = r . . . has the correct singularity structure but performs double subtraction in the regions of phase space where the limits overlap
Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides Construction of the subtraction terms at NLO The collinear and soft regions overlap: − C ir S r C ir S r The candidate subtraction term. . . � � � �� � � 1 m + 1 | 2 = A 1 |M ( 0 ) |M ( 0 ) m + 1 | 2 2 C ir + S r − C ir S r r i � = r i � = r . . . is now free of double subtractions . . . but only defined in the strict collinear and/or soft limits
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