NNLO Jet Cross Sections by Subtraction. G´ abor Somogyi DESY Zeuthen HP2.3rd in collaboration with U. Aglietti, P. Bolzoni, V. Del Duca, C. Duhr, S.-O. Moch, Z. Tr´ ocs´ anyi
Motivation G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 2
Why jets at NNLO? Hadronic jets occur frequently in final states of high energy particle collisions. Because of large production cross sections, jet observables can be measured with high statistical accuracy; can be ideal for precision studies. Examples include measurements of: ◮ α s from jet rates and event shapes in e + e − → jets; ◮ gluon PDFs and α s from 2 + 1 jet production in DIS; ◮ PDFs in single jet inclusive, V + jet in pp (or p ¯ p ) collisions. Often, relevant observables measured with accuracy of a few % or better. Theoretical predictions with same level of accuracy necessary. This usually requires NNLO corrections. G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 3
What is a subtraction scheme? We know that IR singularities cancel according to the KLN theorem between real and virtual quantum corrections at the same order in perturbation theory, for sufficiently inclusive (IR safe) observables. Example (simple residuum subtraction) d σ R ( x ) = x − 1 − ǫ S ( x ) , � 1 d σ R ( x ) + σ V , σ = where S (0) = S 0 < ∞ , σ V = S 0 /ǫ + F . 0 Define the counterterm d σ R , A ( x ) = x − 1 − ǫ S 0 . Then � � � 1 � 1 � � σ V + d σ R ( x ) − d σ R , A ( x ) d σ R , A ( x ) σ = ǫ =0 + 0 0 ǫ =0 � 1 � S ( x ) − S 0 � � S 0 � ǫ + F − S 0 = + x 1+ ǫ ǫ 0 ǫ =0 ǫ =0 � 1 S ( x ) − S 0 = + F x 0 The last integral is finite, computable with standard numerical methods. G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 4
In a rigorous mathematical sense, the cancellation of both kinematical singularities and ǫ -poles must be local. I.e. the counterterm must have the following general properties ◮ must match the singularity structure of the real emission cross section pointwise, in d dimensions ◮ its integrated form must be combined with the virtual cross section explicitly, before phase space integration; ǫ -poles must cancel point by point G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 5
In a rigorous mathematical sense, the cancellation of both kinematical singularities and ǫ -poles must be local. I.e. the counterterm must have the following general properties ◮ must match the singularity structure of the real emission cross section pointwise, in d dimensions ◮ its integrated form must be combined with the virtual cross section explicitly, before phase space integration; ǫ -poles must cancel point by point The construction should be universal (i.e. process and observable independent) ◮ to avoid tedious adaptation to every specific problem ◮ the integration of counterterms can be performed once and for all ◮ the IR limits of QCD (squared) matrix elements are universal, so a general construction should be possible G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 5
In a rigorous mathematical sense, the cancellation of both kinematical singularities and ǫ -poles must be local. I.e. the counterterm must have the following general properties ◮ must match the singularity structure of the real emission cross section pointwise, in d dimensions ◮ its integrated form must be combined with the virtual cross section explicitly, before phase space integration; ǫ -poles must cancel point by point The construction should be universal (i.e. process and observable independent) ◮ to avoid tedious adaptation to every specific problem ◮ the integration of counterterms can be performed once and for all ◮ the IR limits of QCD (squared) matrix elements are universal, so a general construction should be possible Different specific choices of the counterterm correspond to different IR subtraction schemes (CS dipole, FKS, antenna,. . . ). G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 5
Why a new subtraction scheme at NNLO? ◮ Dipole subtraction (Catani, Seymour) ✔ fully local counterterms ✘ faces fundamental difficulties when going to NNLO ✔ explicit expressions including colour for a general process ◮ Antenna subtraction (Gehrmann-De Ridder, Gehrmann, Glover; Weinzierl) ◮ q ⊥ subtraction (Catani, Grazzini; Cieri, Ferrera, de Florian) ◮ Sector decomposition (Binoth, Heinrich; Anastasiou, Melnikov, Petriello) ◮ This scheme (Del Duca, GS, Tr´ ocs´ anyi) G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 6
Why a new subtraction scheme at NNLO? ◮ Dipole subtraction (Catani, Seymour) ◮ Antenna subtraction (Gehrmann-De Ridder, Gehrmann, Glover; Weinzierl) ✔ successfully applied to e + e − → 3 ✘ counterterms not fully local jets ✘ cannot constrain subtractions ✔ complete analytical integration near singular regions of antennae tractable ◮ q ⊥ subtraction (Catani, Grazzini; Cieri, Ferrera, de Florian) ◮ Sector decomposition (Binoth, Heinrich; Anastasiou, Melnikov, Petriello) ◮ This scheme (Del Duca, GS, Tr´ ocs´ anyi) G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 6
Why a new subtraction scheme at NNLO? ◮ Dipole subtraction (Catani, Seymour) ◮ Antenna subtraction (Gehrmann-De Ridder, Gehrmann, Glover; Weinzierl) ◮ q ⊥ subtraction (Catani, Grazzini; Cieri, Ferrera, de Florian) ✔ exploits universal behaviour of ✘ applicable only to the production q ⊥ distribution at small q ⊥ of colourless final states in hadron collisions ✔ numerically efficient implementation possible ◮ Sector decomposition (Binoth, Heinrich; Anastasiou, Melnikov, Petriello) ◮ This scheme (Del Duca, GS, Tr´ ocs´ anyi) G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 6
Why a new subtraction scheme at NNLO? ◮ Dipole subtraction (Catani, Seymour) ◮ Antenna subtraction (Gehrmann-De Ridder, Gehrmann, Glover; Weinzierl) ◮ q ⊥ subtraction (Catani, Grazzini; Cieri, Ferrera, de Florian) ◮ Sector decomposition (Binoth, Heinrich; Anastasiou, Melnikov, Petriello) ✔ dispenses with the subtraction ✘ cancellation of ǫ -poles numerical method, but conceptually very ✘ can it handle complicated final simple states? ✔ first method to yield physical cross sections ◮ This scheme (Del Duca, GS, Tr´ ocs´ anyi) G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 6
Why a new subtraction scheme at NNLO? ◮ Dipole subtraction (Catani, Seymour) ◮ Antenna subtraction (Gehrmann-De Ridder, Gehrmann, Glover; Weinzierl) ◮ q ⊥ subtraction (Catani, Grazzini; Cieri, Ferrera, de Florian) ◮ Sector decomposition (Binoth, Heinrich; Anastasiou, Melnikov, Petriello) ◮ This scheme (Del Duca, GS, Tr´ ocs´ anyi) ✔ fully local counterterms ✘ analytical integration of (efficiency, mathematical rigour) counterterms requires computing many new high dimensional ✔ explicit expressions including integrals, but can be done once colour (colour space notation of and for all dipole subtraction used) ✔ very algorithmic construction (in principle valid at N n LO) ✔ option to constrain subtraction near singular regions (efficiency, important check) G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 6
Subtraction at NNLO G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 7
What is needed to define a subtraction scheme? To define a subtraction scheme, three problems must be addressed 1. Matching of limits: the known IR factorization formulae must be written in such a way, that the overlapping soft/collinear singularities can be disentangled in order to avoid multiple subtraction. � � � � � 1 m +1 | 2 = A 1 |M (0) |M (0) m +1 | 2 2 C ir + S r − C ir S r i i � = r i � = r 2. Extension over PS: the IR factorization formulae valid in the strict soft/collinear limits have to be defined over the full PS. This requires the introduction of appropriate mappings of momenta that respect factorization and the (delicate) cancellation of IR singularities r { p } m +1 − → { ˜ p } m : d φ m +1 ( { p } m +1 ; Q ) = d φ m ( { ˜ p } m ; Q )[ d p 1 , m ] r , s { p } m +2 − → { ˜ p } m : d φ m +2 ( { p } m +2 ; Q ) = d φ m ( { ˜ p } m ; Q )[ d p 2 , m ] 3. Integration: the counterterms have to be integrated over the phase space of the unresolved parton(s). G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 8
Specific issues at NNLO ◮ Matching is cumbersome if done in a brute force way. However, an efficient solution that works at any order in PT is known. ◮ Extension is very delicate. Among other constraints, the counterterms for singly-unresolved real emission must have universal IR limits, which is not guaranteed by QCD factorization. ◮ Choosing the counterterms such that integration is (relatively) easy generally conflicts with the delicate cancellations in the various limits. G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 9
The NNLO cross section Consider the NNLO correction to a generic m -jet observable � � � σ NNLO = d σ RR d σ RV d σ VV m +2 J m +2 + m +1 J m +1 + m J m . m +2 m +1 m G´ abor Somogyi | NNLO Jet Cross Sections by Subtraction | HP2.3rd | page 10
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