Jet cross sections at NNLO via local subtraction

Transcription

1 Jet cross sections at NNLO via local subtraction Gábor Somogyi DESY IKTP, TU Dresden, October Gábor Somogyi Jet cross sections at NNLO via local subtraction page

2 Differential NNLO A young and promising field in the LHC era less than a decade old starting from simple decays and single production processes moved to/moving towards complicated jet production and pair production processes at colliders already a significant impact on phenomenology at collider experiments 95% CL limit on σ/σ SM 0 CMS Preliminary, Combined, L int s = 7 TeV - =.-.7 fb Observed Expected ± σ Expected ± 2σ Higgs boson mass (GeV/c ) α s = 0.75 ± (exp)±0.005 (theo) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

3 Outline Introduction Basics of subtraction Local subtraction at NNLO The tedious part: integrating the counterterms Integrated approximate cross sections Outlook Gábor Somogyi Jet cross sections at NNLO via local subtraction page 3

4 Introduction Gábor Somogyi Jet cross sections at NNLO via local subtraction page 4

5 Motivation - why NNLO? Precision QCD requires computations beyond NLO in certain cases NLO corrections are large: Higgs production from gluon fusion in hadron collisions the main source of uncertainty in extracting info from data is due to NLO theory: α s measurements reliable error estimate is needed: precise measurement of parton luminosities In short, NNLO is relevant when NLO fails to do its job (Anastasiou, Dixon, Melnikov, Petriello, Phys. Rev. D69 (2004) ) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 5

6 Motivation - why jets at NNLO? Jets are essential analysis tools at LHC: precise understanding is needed status at LHC: looks good... (CMS Collaboration, Phys. Rev. Lett. 07 (20) 3200.) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 6

7 Motivation - why jets at NNLO? Jets are essential analysis tools at LHC: precise understanding is needed status at LHC: looks good...but have a closer look! (CMS Collaboration, Phys. Rev. Lett. 07 (20) 3200.) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 6

8 Motivation - why jets at NNLO? Jets are essential analysis tools at LHC: precise understanding is needed status at LHC: looks good...but have a closer look! jet energy scale uncertainty of 5-0 % warrants precision physics precision prediction for standard candles : inclusive jet, V + jet,... missing piece for precise determination of pdf s NLO is effectively LO: energy distribution inside jet cones, jet p asymmetry,... Gábor Somogyi Jet cross sections at NNLO via local subtraction page 6

9 Status Processes measured to few precent accuracy e + e 3j ep (2+)j pp j +X pp V pp V +j pp t t Processes with potentially large radiative corrections pp H pp H +j pp VV Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

10 Status Processes measured to few precent accuracy e + e 3j ep (2+)j pp j +X pp V pp V +j pp t t Processes with potentially large radiative corrections pp H pp H +j pp VV (VV = γγ) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

11 NNLO ingredients A generic m-jet cross section at NNLO involves Tree-level squared matrix elements with m +2 parton kinematics known from LO calculations doubly-real contribution (RR) One-loop squared matrix elements with m + parton kinematics usually known from NLO calculations real-virtual contribution (RV) Two-loop squared matrix elements with m parton kinematics known for all massless 2 2 processes doubly-virtual contribution (VV) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 8

12 NNLO ingredients A generic m-jet cross section at NNLO involves Tree-level squared matrix elements with m +2 parton kinematics known from LO calculations doubly-real contribution (RR) One-loop squared matrix elements with m + parton kinematics usually known from NLO calculations real-virtual contribution (RV) Two-loop squared matrix elements with m parton kinematics known for all massless 2 2 processes doubly-virtual contribution (VV) Assuming we know the relevant matrix elements, can we use those matrix elements to compute cross sections? Gábor Somogyi Jet cross sections at NNLO via local subtraction page 8

13 The problem - IR singularities Consider the NNLO correction to a generic m-jet observable σ NNLO = dσm+2 RR J m+2 + dσm+ RV J m+ + dσm VV Jm. m+2 m+ m Doubly-real dσ RR m+2j m+2 Tree MEs with m+2-parton kinematics kin. singularities as one or two partons unresolved: up to O(ǫ 4 ) poles from PS integration no explicit ǫ poles Real-virtual dσ RV m+ J m+ One-loop MEs with m +-parton kinematics kin. singularities as one parton unresolved: up to O(ǫ 2 ) poles from PS integration explicit ǫ poles up to O(ǫ 2 ) Doubly-virtual dσ VV m J m One- and two-loop MEs with m-parton kinematics kin. singularities screened by jet function: PS integration finite explicit ǫ poles up to O(ǫ 4 ) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 9

14 The problem - IR singularities Consider the NNLO correction to a generic m-jet observable σ NNLO = dσm+2 RR J m+2 + dσm+ RV J m+ + dσm VV Jm. m+2 m+ m THE KLN THEOREM Infrared singularities cancel between real and virtual quantum corrections at the same order in perturbation theory, for sufficiently inclusive (i.e. IR safe) observables. HOWEVER How to make this cancellation explicit, so that the various contributions can be computed numerically? Need a method to deal with implicit poles. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 9

15 Approaches Sector decomposition (Binoth, Heinrich; Anastasiou, Melnikov, Petriello; Czakon) extract ǫ poles of each contribution (RR, RV, VV) separately by expanding the integrand in distributions resulting expansion coefficients are finite multi-dimensional integrals, integrate numerically cancellation of poles numerical, depends on observable first method to yield physical results, but can it handle complicated final states? Subtraction (Catani, Grazzini; Cieri, Ferrera, de Florian; Gehrmann, Gehrmann-De Ridder, Glover; Weinzierl; Del Duca, Trócsányi, GS) rearrange the poles between real and virtual contributions by subtracting and adding back suitable approximate cross sections cancellation of explicit ǫ poles achieved analytically, remaining PS integrals are finite nice properties (generality, efficiency) expected form experience at NLO definition of subtraction terms is not unique, hence several approaches: q, antenna, local Gábor Somogyi Jet cross sections at NNLO via local subtraction page 0

16 Approaches Sector decomposition (Binoth, Heinrich, Anastasiou, Dixon, Melnikov, Petriello, Czakon) first method to yield physical cross sections cancellation of divergences fully numerical cancellation of poles also, and depends on jet function can it handle complicated final states? q subtraction (Catani, Grazzini, Cieri, Ferrera, de Florian, Tramontano) exploits universal behavior of q distribution at small q efficient and fully exclusive calculation limited scope: applicable only to production of massive colorless final states in hadron collisions Antenna subtraction (Gehrmann, Gehrmann-De Ridder, Glover, Heinrich, Weinzierl) successfully applied to e + e 2, 3j analytic integration of antennae over unresolved phase space is understood counterterms are nonlocal treatment of color is implicit cannot cut factorized phase space Gábor Somogyi Jet cross sections at NNLO via local subtraction page 0

17 Approaches Is the agreement between antenna implementations satisfactory? (Weinzierl) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 0

18 Why a new scheme? Goal: devise a subtraction scheme with general and explicit expressions, including color (view towards automation, color space notation is used) fully local counterterms, taking account of all color and spin correlations (mathematical rigor, efficiency) option to constrain subtractions to near singular regions (efficiency, important check) very algorithmic construction (valid at any order in perturbation theory) Gábor Somogyi Jet cross sections at NNLO via local subtraction page

19 Basics of subtraction Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

20 Subtraction - a caricature Want to evaluate (at ǫ 0) σ = dσ R (x)+σ V 0 where dσ R (x) = x ǫ R(x) R(0) = R 0 < σ V = R 0 /ǫ+v define the counterterm dσ R,A (x) = x ǫ R 0 use it to reshuffle singularities between R and V contributions [ ] [ ] σ = dσ R (x) dσ R,A (x) σ V + dσ R,A (x) 0 [ ] R(x) R0 = 0 x +ǫ R(x) R 0 = +V 0 x + ǫ=0 + ǫ=0 [ R0 ǫ +V R 0 ǫ 0 ] The last integral is finite, computable with standard numerical methods. ǫ=0 ǫ=0 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 3

21 The issue of locality In a rigorous mathematical sense, the cancellation of both kinematical singularities and ǫ-poles must be local. I.e. the subtraction term must have the following general properties it must match the singularity structure of (singly- and doubly-) real emissions pointwise, in d dimensions its integrated form must be combined with the (real- and doubly-) virtual cross section explicitly, before phase space integration; ǫ-poles must cancel point by point What about singular terms in the real emission cross section that cancel upon phase space integration (e.g. azimuthal correlations in gluon splitting)? they cancel upon integration in d dimensions, the corresponding four dimensional integrals are ill-defined it is mandatory to treat these terms, since naive numerical integration (in four dimensions) can give any result whatsoever however, can be treated with methods other than strict local subtraction, e.g. auxiliary phase space slicing (as in antenna subtraction) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 4

22 A more efficient subtraction scheme? Want to evaluate (at ǫ 0) σ = dσ R (x)+σ V where 0 dσ R (x) = x ǫ R(x) R(0) = R 0 < σ V = R 0 /ǫ+v define the counterterm dσ R,A (x) = x ǫ R 0 use it to reshuffle singularities between R and V contributions [ ] [ ] σ = dσ R (x) dσ R,A (x) σ V + dσ R,A (x) 0 [ ] R(x) R0 = 0 x +ǫ R(x) R 0 = +V 0 x + ǫ=0 + ǫ=0 [ R0 ǫ +V R 0 ǫ 0 ] The last integral is finite, computable with standard numerical methods. ǫ=0 ǫ=0 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 5

23 A more efficient subtraction scheme? Want to evaluate (at ǫ 0) σ = dσ R (x)+σ V where 0 dσ R (x) = x ǫ R(x) R(0) = R 0 < σ V = R 0 /ǫ+v define the counterterm to be nonzero only near singular region dσ R,A (x) = x ǫ R 0 Θ(x 0 x) use it to reshuffle singularities between R and V contributions [ ] [ ] σ = dσ R (x) dσ R,A (x) + σ V + dσ R,A (x) 0 ǫ=0 0 ǫ=0 [ ] [ R(x) R0 Θ(x 0 x) R0 = 0 x +ǫ + ǫ=0 ǫ +V R ] 0 ǫ +R 0logx 0 +O(ǫ ) ǫ=0 R(x) R 0 Θ(x 0 x) = +V +R 0 logx 0 0 x The last integral is finite, computable with standard numerical methods. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 5

24 A more efficient subtraction scheme? It is sufficient to perform subtraction only near the singular region gain in efficiency: subtraction term only needs to be computed over a fraction of phase space strong check: final result is independent of value of phase space cut analytical integration of subtraction term more difficult (extra scale involved) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 5

25 Local subtraction at NNLO Gábor Somogyi Jet cross sections at NNLO via local subtraction page 6

26 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

27 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m. dσ RR,A 2 m+2 regularizes the doubly-unresolved limits of dσ RR m+2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

28 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m. dσ RR,A 2 m+2 regularizes the doubly-unresolved limits of dσ RR m+2 2. dσ RR,A m+2 regularizes the singly-unresolved limits of dσ RR m+2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

29 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m. dσ RR,A 2 m+2 regularizes the doubly-unresolved limits of dσ RR m+2 2. dσ RR,A m+2 regularizes the singly-unresolved limits of dσ RR m+2 3. dσ RR,A 2 m+2 accounts for the overlap of dσ RR,A m+2 and dσ RR,A 2 m+2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

30 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m. dσ RR,A 2 m+2 regularizes the doubly-unresolved limits of dσ RR m+2 2. dσ RR,A m+2 regularizes the singly-unresolved limits of dσ RR m+2 3. dσ RR,A 2 m+2 accounts for the overlap of dσ RR,A m+2 and dσ RR,A 2 m+2 4. dσ RV,A m+ regularizes the singly-unresolved limits of dσ RV m+ Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

31 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m. dσ RR,A 2 m+2 regularizes the doubly-unresolved limits of dσ RR m+2 2. dσ RR,A m+2 regularizes the singly-unresolved limits of dσ RR m+2 3. dσ RR,A 2 m+2 accounts for the overlap of dσ RR,A m+2 and dσ RR,A 2 m+2 4. dσ RV,A m+ regularizes the singly-unresolved limits of dσ RV m+ 5. ( dσrr,a m+2 ) A regularizes the singly-unresolved limit of dσrr,a m+2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 7

32 Defining a subtraction scheme Strategy: IR limits are process independent and known. Start from defining the subtraction terms based on IR limit formulae they are trivially general, explicit and local done some time ago (2006) for colorless initial states 2. Worry about integrating them later since this is in principle a very narrowly defined problem, given. but in practice is very cumbersome, due to lack of technology Gábor Somogyi Jet cross sections at NNLO via local subtraction page 8

33 Defining a subtraction scheme The following three problems must be addressed. Matching of limits to avoid multiple subtraction in overlapping singular regions of PS. Easy at NLO: collinear limit + soft limit - collinear limit of soft limit. A M (0) m+ 2 = i [ 2 C ir +S r i r i r C ir S r ] M (0) m Extension of IR factorization formulae over full PS using momentum mappings that respect factorization and delicate structure of cancellations in all limits. {p} m+ r { p} m : dφ m+ ({p} m+ ;Q) = dφ m({ p} m ;Q)[dp,m ] {p} m+2 r,s { p} m : dφ m+2 ({p} m+2 ;Q) = dφ m({ p} m ;Q)[dp 2,m ] 3. Integration of the counterterms over the phase space of the unresolved parton(s). Gábor Somogyi Jet cross sections at NNLO via local subtraction page 8

34 The need for extension IR limit formulae are only well-defined in the strict limit. E.g. collinear: C ir is a symbolic operator that takes the p i p r limit C ir M (0) m+2 (p i,p r,...) 2 = 8πα sµ 2ǫ s ir ˆPfi f r (z i,z r,k ;ǫ) M (0) m+ (p ir,...) 2 soft: S r is a symbolic operator that takes the p r 0 limit S r M (0) m+2 (pr,...) 2 = 8πα sµ 2ǫ i,k 2 S ik(r) M (0) m+,(i,k) ( pr,...) 2 NOTICE momenta in factorized ME s on the r.h.s. conserve momentum and/or mass shell conditions only in the strict limit arguments of AP splitting functions, e.g. momentum fractions z i, z r and transverse momentum k are only defined in the strict limit HENCE must specify precisely momenta entering factorized ME s away from limit must define z i, z r and k away from limit Gábor Somogyi Jet cross sections at NNLO via local subtraction page 9

35 Defining a subtraction scheme 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. 2. Extension is delicate. E.g. counterterms for singly-unresolved real emission (unintegrated and integrated) must have universal IR limits. This is not guaranteed by QCD factorization. 3. Choosing the counterterms such that integration is (relatively) straightforward generally conflicts with the delicate cancellation of IR singularities. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 20

36 NNLO subtraction terms - an example MESSAGE Subtraction terms are defined completely explicitly for any number of jets. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

37 NNLO subtraction terms - an example Double collinear counterterm: among others, in dσ RR,A 2 m+2 we find C (0,0) ir;js ({p}) = (8παsµ2ǫ ) 2 s ir s js ( α ir α js ) d(m;ǫ) Θ(α 0 α ir α js ) M (0) m ({ p} (ir;js) ) ˆP fi f r (z r,i,z i,r,k,i,r ;ǫ)ˆp fj f s (z s,j,z j,s,k,j,s ;ǫ) M (0) m ({ p} (ir;js) ) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

38 NNLO subtraction terms - an example Double collinear counterterm: among others, in dσ RR,A 2 m+2 we find C (0,0) ir;js ({p}) = (8παsµ2ǫ ) 2 s ir s js ( α ir α js ) d(m;ǫ) Θ(α 0 α ir α js ) M (0) m ({ p} (ir;js) ) ˆP fi f r (z r,i,z i,r,k,i,r ;ǫ)ˆp fj f s (z s,j,z j,s,k,j,s ;ǫ) M (0) m ({ p} (ir;js) ) collinear poles: s irs js s kl = 2p k p l, k,l = i,r or j,s Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

39 NNLO subtraction terms - an example Double collinear counterterm: among others, in dσ RR,A 2 m+2 we find C (0,0) ir;js ({p}) = (8παsµ2ǫ ) 2 s ir s js ( α ir α js ) d(m;ǫ) Θ(α 0 α ir α js ) M (0) m ({ p} (ir;js) ) ˆP fi f r (z r,i,z i,r,k,i,r ;ǫ)ˆp fj f s (z s,j,z j,s,k,j,s ;ǫ) M (0) m ({ p} (ir;js) ) Altarelli-Parisi splitting functions: ˆPfi f r ˆPfj f s z k,l = y kq, k µ,k,l = ζ k,l p µ l ζ l,k p µ k y +ζ kl p µ kl, k,l = i,r or j,s (kl)q with ζ k,l = z k,l y kl α kl y (kl)q, ζ kl = y kl α kl y kl Q (z l,k z k,l ) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

40 NNLO subtraction terms - an example Double collinear counterterm: among others, in dσ RR,A 2 m+2 we find C (0,0) ir;js ({p}) = (8παsµ2ǫ ) 2 s ir s js ( α ir α js ) d(m;ǫ) Θ(α 0 α ir α js ) M (0) m ({ p} (ir;js) ) ˆP fi f r (z r,i,z i,r,k,i,r ;ǫ)ˆp fj f s (z s,j,z j,s,k,j,s ;ǫ) M (0) m ({ p} (ir;js) ) mapped momenta: { p} (ir;js) = { p,..., p ir,..., p js,..., p m+2} m p µ kl = α ir α js (p µ k +pµ l α kl Q µ ), k,l = i,r or j,s with p µ n = α ir α js p µ n, α kl = 2 n i,r,j,s [ ] y (kl)q y 2 4y (kl)q kl, k,l = i,r or j,s Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

41 NNLO subtraction terms - an example Double collinear counterterm: among others, in dσ RR,A 2 m+2 we find C (0,0) ir;js ({p}) = (8παsµ2ǫ ) 2 s ir s js ( α ir α js ) d(m;ǫ) Θ(α 0 α ir α js ) M (0) m ({ p} (ir;js) ) ˆP fi f r (z r,i,z i,r,k,i,r ;ǫ)ˆp fj f s (z s,j,z j,s,k,j,s ;ǫ) M (0) m ({ p} (ir;js) ) constrain subtraction to near singular region: Θ(α 0 α ir α js) 0 < α 0, α 0 = : subtract over full phase space Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

42 NNLO subtraction terms - an example Double collinear counterterm: among others, in dσ RR,A 2 m+2 we find C (0,0) ir;js ({p}) = (8παsµ2ǫ ) 2 s ir s js ( α ir α js ) d(m;ǫ) Θ(α 0 α ir α js ) M (0) m ({ p} (ir;js) ) ˆP fi f r (z r,i,z i,r,k,i,r ;ǫ)ˆp fj f s (z s,j,z j,s,k,j,s ;ǫ) M (0) m ({ p} (ir;js) ) make integrated counterterm m-independent: ( α ir α js) d(m;ǫ) d(m;ǫ) = 2m( ǫ) 2d 0, d 0 = D 0 +d ǫ, D 0 2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

43 NNLO subtraction terms - an example Double collinear counterterm: among others, in dσ RR,A 2 m+2 we find C (0,0) ir;js ({p}) = (8παsµ2ǫ ) 2 s ir s js ( α ir α js ) d(m;ǫ) Θ(α 0 α ir α js ) M (0) m ({ p} (ir;js) ) ˆP fi f r (z r,i,z i,r,k,i,r ;ǫ)ˆp fj f s (z s,j,z j,s,k,j,s ;ǫ) M (0) m ({ p} (ir;js) ) The complete approximate cross section is a sum of such terms where A 2 M (0) m+2 2 = r s r i r,s j i,r,s dσ RR,A 2 m+2 = dφ m[dp 2 ]A 2 M (0) m+2 2 { [ 6 C(0,0) irs + j i,r,s i r,s [ 2 C irs CS(0,0) ir;s + j i,r,s 2 C ir;js S(0,0) rs C irs CS ir;s S rs (0,0) 8 C(0,0) ir;js + ] 2 CS(0,0) ir;s + 2 S(0,0) rs 2 C ir;js CS(0,0) ir;s + 2 C irs S(0,0) rs +CS ir;s S rs (0,0) ]} Gábor Somogyi Jet cross sections at NNLO via local subtraction page 2

44 NNLO subtraction terms - general features Based on universal IR limit formulae Altarelli-Parisi splitting functions, soft currents (tree and one-loop, triple AP functions) simple and general procedure for matching of limits using physical gauge extension based on momentum mappings that can be generalized to any number of unresolved partons Fully local in color spin space no need to consider the color decomposition of real emission ME s azimuthal correlations correctly taken into account in gluon splitting can check explicitly that the ratio of the sum of counterterms to the real emission cross section tends to unity in any IR limit Straightforward to constrain subtractions to near singular regions gain in efficiency independence of physical results on phase space cut is strong check Given completely explicitly for any process with non colored initial state Gábor Somogyi Jet cross sections at NNLO via local subtraction page 22

45 The tedious part: integrating the counterterms Gábor Somogyi Jet cross sections at NNLO via local subtraction page 23

46 Basic setup Momentum mappings used to define the counterterms {p} n+p R { p} n implement exact momentum conservation recoil distributed democratically (can be generalized to any p) different collinear and soft mappings (R labels precise limit) exact factorization of phase space dφ n+p({p};q) = dφ n({ p} (R) n ;Q)[dp (R) p,n] Counterterms are products (in color and spin space) of factorized ME s independent of variables in [dp (R) p,n] singular factors (AP functions, soft currents), to be integrated over [dp (R) p,n] Strategy for computing the integrals explicit parametrization of factorized phase space leads to parametric integral representations evaluate the parametric integrals Gábor Somogyi Jet cross sections at NNLO via local subtraction page 24

47 Types of integrated counterterms NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m Gábor Somogyi Jet cross sections at NNLO via local subtraction page 25

48 Types of integrated counterterms NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + tree-level and one-loop singly-unresolved integrals [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m Gábor Somogyi Jet cross sections at NNLO via local subtraction page 25

49 Types of integrated counterterms NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + tree-level and one-loop singly-unresolved integrals tree-level iterated singly-unresolved integrals [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσrr,a A m+2 J m Gábor Somogyi Jet cross sections at NNLO via local subtraction page 25

50 Types of integrated counterterms NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + tree-level and one-loop singly-unresolved integrals tree-level iterated singly-unresolved integrals tree-level doubly-unresolved integrals [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m Gábor Somogyi Jet cross sections at NNLO via local subtraction page 25

51 Types of integrated counterterms NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + tree-level and one-loop singly-unresolved integrals tree-level iterated singly-unresolved integrals tree-level doubly-unresolved integrals [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m Gábor Somogyi Jet cross sections at NNLO via local subtraction page 25

52 Phase space integrals - an example MESSAGE The integral is (very) difficult, but the result is numerically (very) simple. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 26

53 Phase space integrals - an example Abelian soft-double soft counterterm: among many others, in dσ RR,A 2 m+2 we find ( ) S t S (0) ab rt = (8παsµ 2ǫ ) 2 8 S îˆk (ˆr)S jl(t) M (0) m,(i,k)(j,l) ({ p}) 2 i,j,k,l ( y tq ) d 0 m( ǫ) ( yˆrq ) d 0 m( ǫ) Θ(y 0 y tq )Θ(y 0 yˆrq ) The set of m momenta, { p}, is obtained by an iterated mapping which leads to an exact factorization of phase space S {p} t Sˆr m+2 {ˆp} m+ { p} : dφ m+2({p};q) = dφ m({ p};q)[d p,m][dp,m+] R S R K Ŝ K m + 2 m + 2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 26

54 Phase space integrals - an example Abelian soft-double soft counterterm: among many others, in dσ RR,A 2 m+2 we find ( ) S t S (0) ab rt = (8παsµ 2ǫ ) 2 8 S îˆk (ˆr)S jl(t) M (0) m,(i,k)(j,l) ({ p}) 2 i,j,k,l ( y tq ) d 0 m( ǫ) ( yˆrq ) d 0 m( ǫ) Θ(y 0 y tq )Θ(y 0 yˆrq ) The set of m momenta, { p}, is obtained by an iterated mapping which leads to an exact factorization of phase space S {p} t Sˆr m+2 {ˆp} m+ { p} : dφ m+2({p};q) = dφ m({ p};q)[d p,m][dp,m+] Then we must compute [ ( ) [d p,m][dp,m+]s t S (0) αs µ 2 ǫ ] 2 rt 2π Sǫ [S Q 2 t S (0) rt ] ikjl M (0) m,(i,k)(j,l) ({ p}) 2 i,k,j,l where [S t S (0) rt ] ikjl [S t S (0) rt ] ikjl (p i,p k,p j,p l,ǫ,y 0,d 0) is a kinematics dependent function. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 26

55 Abelian soft-double soft integral For simplicity, consider the terms in the sum where j = i and l = k: [S t S (0) rt ] ikik. Kinematical dependence is through cosχ ik = (p i,p k ), we set cosχ ik = 2Y ik,q. Using angles and energies in some specific Lorentz frame to parametrize the factorized phase space measures, [d p,m] and [dp,m+], we find that [S t S (0) rt ] ikik is proportional to I () S (Y ik,q;ǫ,y 0,d 0 ) = ( ǫ) B y0 ( 2ǫ,d 4Γ4 0 +) y0 πγ 2 Y ik,q dy y 2ǫ ( y) d 0 +ǫ ( ǫ) ǫ 0 d(cosϑ)(sinϑ) 2ǫ d(cosϕ)(sinϕ) 2ǫ[ f(ϑ,ϕ;0) ] [ f(ϑ,ϕ;y ik,q ) ] [ Y(y,ϑ,ϕ;Y ik,q ) ] ǫ ( 2 F ǫ, ǫ, ǫ, Y(y,ϑ,ϕ;Yik,Q ) ) where f(ϑ,ϕ;y ik,q ) = 2 Y ik,q ( Y ik,q )sinϑcosϕ ( 2Y ik,q )χcosϑ Y(y,ϑ,ϕ;χ) = 4( y)y ik,q [2( y)+y f(ϑ,ϕ;0)][2( y)+y f(ϑ,ϕ;y ik,q )] Gábor Somogyi Jet cross sections at NNLO via local subtraction page 27

56 Abelian soft-double soft integral This integral is equal to I () S (Y ik,q;ǫ,y 0,d 0) = [ ] ǫ 2 ln(y 4 ik,q )+Σ(y 0,D 0)+Σ(y 0,D 0 ) ǫ 3 +O(ǫ 2 ) where D 0 = d 0 ǫ=0 and the dependence on the PS cut parameter, y 0, enters in Σ(z,N) = lnz N ( z) k k= k Higher order expansion coefficients computed numerically (y 0 =, D 0 = 3) Order: ǫ Order: ǫ Order: ǫ I () (Yik,Q;ǫ,y0 =,d 0 = 3) I () (Yik,Q;ǫ,y0 =,d 0 = 3) I () (Yik,Q;ǫ,y0 =,d 0 = 3) Yik,Q Yik,Q Yik,Q Gábor Somogyi Jet cross sections at NNLO via local subtraction page 28

57 Analytical vs. numerical As a matter of principle Rigorous proof of cancellation of IR poles requires poles of integrated counterterms in analytical form. Analytical forms are fast and accurate compared to numerical ones. However Analytical results show (in all cases where they are available) that integrated counterterms are smooth functions of kinematic variables. Hence Numerical forms of integrated counterterms are sufficient for practical purposes. Final results can be conveniently given by interpolating tables or approximating functions computed once and for all. Thus, efficient implementation is possible even if the full analytical calculation is not feasible or practical (e.g. finite parts of integrated counterterms). In particular, suitable approximating functions may be obtained by fitting. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 29

58 Example of approximation by fitting Doubly-unresolved soft-collinear master integral I (5) CS (x,y;ǫ) P4(x) + Q3(Y ) + R3(ln(x)) + c P4(x) + Q3(Y ) + R3(ln(x)) + c Y I (5) CS(x, Y ; ǫ,, 3 3ǫ,, 3 3ǫ; 2), O(ǫ 0 ) 0.4 x I (5) CS(x, Y ; ǫ,, 3 3ǫ,, 3 3ǫ; 2), O(ǫ 0 ) 0 4 x Y P4(x) + Q3(Y ) + R3(ln(x)) + c P4(x) + Q3(Y ) + R3(ln(x)) + c < x Y I (5) CS(x, Y ; ǫ,, 3 3ǫ,, 3 3ǫ; 2), O(ǫ 0 ) I (5) CS(x, Y ; ǫ,, 3 3ǫ,, 3 3ǫ; 2), O(ǫ 0 ) 0 < x Y Gábor Somogyi Jet cross sections at NNLO via local subtraction page 30

59 Phase space integrals - methods Several different methods to compute the integrals have been explored use of IBPs to reduce to master integrals + solution of MIs by differential equations use of MB representations to extract pole structure + summation of nested series use of sector decomposition Gábor Somogyi Jet cross sections at NNLO via local subtraction page 3

60 Phase space integrals - methods Method IBP MB SD Analytical M M singly-unresolved integrals bottleneck is the proliferation of denominators iterated singlyunresolved integrals bottleneck is the evaluation of sums easy to automate only in principle, except for lowest order poles Numerical by evaluating the analytic expressions no numbers without full analytical results direct numerical evaluation of MB integrals possible fast and accurate numerical behavior is generally worse than MB method (speed, accuracy) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 3

61 Spinoff - angular integrals in d dimensions Consider the d dimensional angular integral with n denominators (GS, J. Math. Phys. 52 (20) ) Ω j,...,j n = dω d (q) (p q) j (pn q) jn We find (with j = j +...+j n) Ω j,...,j n = 2 2 j 2ǫ π ǫ H[v;(α,A);(β,B);Ls] where H is the so-called H-function of N = n(n+) 2 variables. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 32

62 Spinoff - angular integrals in d dimensions Consider the d dimensional angular integral with n denominators (GS, J. Math. Phys. 52 (20) ) Ω j,...,j n = dω d (q) (p q) j (pn q) jn We find (with j = j +...+j n) Ω j,...,j n = 2 2 j 2ǫ π ǫ H[v;(α,A);(β,B);Ls] where H is the so-called H-function of N = n(n+) 2 variables. We have v = (v,v 2,...,v n,v 22,v 23,...,v n n,v nn), v kl p k p l 2 p 2 k 4 ; k l ; k = l α = (0 N,j,...,j n, j ǫ), β = (j,...,j n,2 j 2ǫ) and Ls = L s...l sn, where L sk is an infinite contour in the complex s k -plane running from i to +i. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 32

63 Spinoff - angular integrals in d dimensions Consider the d dimensional angular integral with n denominators (GS, J. Math. Phys. 52 (20) ) Ω j,...,j n = dω d (q) (p q) j (pn q) jn We find (with j = j +...+j n) Ω j,...,j n = 2 2 j 2ǫ π ǫ H[v;(α,A);(β,B);Ls] where H is the so-called H-function of N = n(n+) 2 variables. We have A = N N M n N, B = [(0) (n+) N ] i.e. B is zero, while the n N dimensional matrix M has the block form: M n N = [ m n n m n (n ) m n ] with m n p = 0 (0) (n p) (p ) 2 0. (p ) (p ) 0 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 32

64 Spinoff - angular integrals in d dimensions Consider the d dimensional angular integral with n denominators (GS, J. Math. Phys. 52 (20) ) Ω j,...,j n = dω d (q) (p q) j (pn q) jn We find (with j = j +...+j n) Ω j,...,j n = 2 2 j 2ǫ π ǫ H[v;(α,A);(β,B);Ls] where H is the so-called H-function of N = n(n+) 2 variables. We have Ω j,...,j n ({v kl };ǫ) = 2 2 j 2ǫ π ǫ n k= Γ(j k)γ(2 j 2ǫ) [ +i n ][ n n ] dz kl i 2πi Γ( z kl) (v kl ) z kl Γ(j k +z k ) Γ( j ǫ z). k= l=k k= where z = n k= l=k n z kl, and z k = k z lk + l= n z kl. l=k Gábor Somogyi Jet cross sections at NNLO via local subtraction page 32

65 Integrated approximate cross sections Gábor Somogyi Jet cross sections at NNLO via local subtraction page 33

66 Structure of the results Integrated approximate cross sections After summing over unobserved flavors, all integrated approximate cross sections can be written as products (in color space) of various insertion operators with lower point cross sections. Insertion operators color and flavor structure of all insertion operators known first two leading poles of kinematical functions entering insertion operators known analytically in all cases (except I (0) 2 ) higher order expansion coefficients computed numerically Gábor Somogyi Jet cross sections at NNLO via local subtraction page 34

67 Integrated approximate cross sections - an example MESSAGE Done once and for all (though admittedly lots of tedious work). Gábor Somogyi Jet cross sections at NNLO via local subtraction page 35

68 Integrated approximate cross sections - an example NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σrv m+ +σvv m = σnnlo m+2 +σm+ NNLO +σm NNLO Each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m ( {[ dσm+ RV + { dσm VV + 2 ] [ dσ RR,A m+2 J m+ dσ RV,A [ dσ RR,A 2 m+2 dσ RR,A 2 m+2 ] + m+ + tree-level and one-loop singly-unresolved integrals tree-level iterated singly-unresolved integrals tree-level doubly-unresolved integrals [ dσ RV,A m+ + ) } dσ RR,A A m+2 ]J m ( ) ]} dσ RR,A A m+2 J m Gábor Somogyi Jet cross sections at NNLO via local subtraction page 35

69 Integrated approximate cross sections - an example Iterated singly-unresolved 2 dσ RR,A 2 m+2 = dσ B m I (0) 2 ({p} m;ǫ) structure of insertion operator in color flavor space [ ( I (0) 2 ({p}m;ǫ) = αs µ 2 ) ǫ ] 2 { 2π Sǫ Q 2 + j,l i + i,k,j,l [ C (0) 2,f i T 2 i + k ] C (0) 2,f i f T 2 k k T 2 i [ S (0),(j,l) 2 C A + i } S (0),(i,k)(j,l) 2 {T i T k,t j T l } ] CS (0),(j,l) 2,f T 2 i i T j T l C (0) 2,f i, C (0) 2,f i f k, S (0),(j,l) 2, CS (0),(j,l) 2,f i and S (0),(i,k)(j,l) 2 are kinematical functions with poles up to O(ǫ 4 ) (also depend on PS cut parameters) kinematical dependence through x i = y iq 2pi Q Q 2 and Y ik,q = y ik y iq y kq Gábor Somogyi Jet cross sections at NNLO via local subtraction page 36

70 Integrated approximate cross sections - an example Iterated singly-unresolved example: e + e 3 jets (momentum assignment is q,2 q,3 g) [ ( I (0) 2 (p αs µ 2 ) ǫ ] 2 { 6C 2,p 2,p 3 ;ǫ) = F +2C A C F +C 2π Sǫ A 2 Q 2 ǫ C2 A 3C FT R n f 2 3 ( 4C A C F + 5C2 A 2 3C AT R n f 2 ( 8CF 2 +C AC F 5C2 A 2 [ 2CF 2 + 0C AC F 6 ) lny 2 ) (lny 3 +lny 23 ) (4C 2 F 6C AC F C 2 A )Σ(y 0,D 0 ) ] } (4CF 2 4C AC F )Σ(y 0,D 0 ) ǫ 3 +O(ǫ 2 ) notice x and Y dependence combine to produce just y ik dependence, as expected dependence on PS cut parameters through Σ(z,N) = lnz N ( z) k k= k should vanish once all integrated approximate cross sections are combined Gábor Somogyi Jet cross sections at NNLO via local subtraction page 36

71 Integrated approximate cross sections - an example Iterated singly-unresolved example: e + e 3 jets (momentum assignment is q,2 q,3 g) [ ( I (0) 2 (p αs µ 2 ) ǫ ] 2 { 6C 2,p 2,p 3 ;ǫ) = F +2C A C F +C 2π Sǫ A 2 Q 2 ǫ C2 A 3C FT R n f 2 3 ( 4C A C F + 5C2 A 2 3C AT R n f 2 ( 8CF 2 +C AC F 5C2 A 2 [ 2CF 2 + 0C AC F 6 ) lny 2 ) (lny 3 +lny 23 ) (4C 2 F 6C AC F C 2 A )Σ(y 0,D 0 ) ] } (4CF 2 4C AC F )Σ(y 0,D 0 ) ǫ 3 +O(ǫ 2 ) notice x and Y dependence combine to produce just y ik dependence, as expected dependence on PS cut parameters through Σ(z,N) = lnz N ( z) k k= k should vanish once all integrated approximate cross sections are combined Gábor Somogyi Jet cross sections at NNLO via local subtraction page 36

72 Integrated approximate cross sections - an example Iterated singly-unresolved example: e + e 3 jets (momentum assignment is q,2 q,3 g) higher order expansion coefficients can be computed numerically [ ( I (0) 2 (p αs µ 2 ) ǫ ] 2 0,p 2,p 3 ;ǫ) = 2π Sǫ Q 2 i= 4color kinematical point parametrized by y ij Col ǫ i I (Col,i) 2,3j (p,p 2,p 3 )+O(ǫ ) y 2 = , y 3 = , y 23 = Col O(ǫ 4 ) O(ǫ 3 ) O(ǫ 2 ) O(ǫ ) O(ǫ 0 ) C 2 F C AC F C 2 A C FT Rn f C AT Rn f Gábor Somogyi Jet cross sections at NNLO via local subtraction page 36

73 Integrated approximate cross sections - an example Iterated singly-unresolved example: e + e 3 jets (momentum assignment is q,2 q,3 g) higher order expansion coefficients can be computed numerically [ ( I (0) 2 (p αs µ 2 ) ǫ ] 2 0,p 2,p 3 ;ǫ) = 2π Sǫ Q 2 i= 4color kinematical point parametrized by y ij Col ǫ i I (Col,i) 2,3j (p,p 2,p 3 )+O(ǫ ) y 2 = , y 3 = , y 23 = Col O(ǫ 4 ) O(ǫ 3 ) O(ǫ 2 ) O(ǫ ) O(ǫ 0 ) C 2 F C AC F C 2 A C FT Rn f C AT Rn f Gábor Somogyi Jet cross sections at NNLO via local subtraction page 36

74 Integrated approximate cross sections - an example Iterated singly-unresolved example: e + e 3 jets (momentum assignment is q,2 q,3 g) higher order expansion coefficients can be computed numerically [ ( I (0) 2 (p αs µ 2 ) ǫ ] 2 0,p 2,p 3 ;ǫ) = 2π Sǫ Q 2 i= 4color kinematical point parametrized by y ij Col ǫ i I (Col,i) 2,3j (p,p 2,p 3 )+O(ǫ ) y 2 = , y 3 = , y 23 = Col O(ǫ 4 ) O(ǫ 3 ) O(ǫ 2 ) O(ǫ ) O(ǫ 0 ) C 2 F C AC F C 2 A C FT Rn f C AT Rn f Gábor Somogyi Jet cross sections at NNLO via local subtraction page 36

75 Overview Counterterm Types of integrals M M dσrr,a m+2 tree level singly-unresolved dσrv,a m+ one-loop singly-unresolved ( dσrr,a m+2 ) A tree level iterated singly-unresolved () 2 dσrr,a 2 m+2 tree level iterated singly-unresolved (2) 2 dσrr,a 2 m+2 tree level doubly-unresolved Done M M M M M M M M M M / M Gábor Somogyi Jet cross sections at NNLO via local subtraction page 37

76 Outlook Gábor Somogyi Jet cross sections at NNLO via local subtraction page 38

77 Present status NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σm+ RV +σm VV = σm+2 NNLO +σm+ NNLO +σm NNLO Each integrable in four dimensions σm+2 NNLO = σm+ NNLO = σm NNLO = m+2 m+ m { [ ]} dσm+2 RR J m+2 dσ RR,A 2 m+2 J m dσ RR,A m+2 J m+ dσ RR,A 2 m+2 J m {[ ] [ ( dσm+ RV + dσrr,a m+2 J m+ dσ RV,A ) } m+ + dσrr,a A m+2 ]J m { dσm VV + unintegrated RR counterterms unintegrated RV counterterms 2 ] dσ RR,A 2 m+2 dσ RR,A 2 m+2 + [ ( dσ RV,A ) ]} m+ + dσrr,a A m+2 J m tree-level and one-loop singly-unresolved integrals tree-level iterated singly-unresolved integrals tree-level doubly-unresolved integrals Gábor Somogyi Jet cross sections at NNLO via local subtraction page 39

78 Present status NNLO correction is the sum of three terms σ NNLO = σm+2 RR +σm+ RV +σm VV = σm+2 NNLO +σm+ NNLO +σm NNLO Numerical Monte Carlo integration (single CPU, 50 hours).6.5 C-parameter distribution LO result NLO result NLO+RR+RV.5.4 LO result NLO result NLO+RR+RV Thrust distribution / 0 C d /dc / 0 (-T) d /dt / 0 C d /dc C RV piece -. RR piece / 0 (-T) d /dt T RV piece RR piece Gábor Somogyi Jet cross sections at NNLO via local subtraction page 39

79 Conclusions Want to bias you towards differential NNLO is a relevant subject subtraction is the method of choice for general, efficient calculations Local subtraction at NNLO general, explicit, local subtraction scheme for computing NNLO jet cross sections, for processes with no colored particles in the initial state investigated various methods to compute the integrated counterterms: IBP s, MB, SD integration of all singly-unresolved and iterated singly-unresolved counterterms finished integration of doubly-unresolved counterterms underway Immediate next steps finish doubly-unresolved integrals: only triply-collinear and double soft counterterms left consolidate numerics of integrated approximate cross sections Gábor Somogyi Jet cross sections at NNLO via local subtraction page 40

80 Backup slides Gábor Somogyi Jet cross sections at NNLO via local subtraction page 4

81 Doubly real counterterms I - IR limits The doubly real cross section Singly-unresolved IR limits dσ RR m+2 = dφ m+2(p,...,p m+2) M (0) m+2 (p,...,pm+2) 2 Collinear: C ir is a symbolic operator that takes the p i p r limit C ir M (0) m+2 (p i,p r,...) 2 = 8πα sµ 2ǫ s ir M (0) m+ (p ir,...) ˆP fi f r (z i,z r,k ;ǫ) M (0) m+ (p ir,...) Soft: S r is a symbolic operator that takes the p r 0 limit S r M (0) m+2 (pr,...) 2 = 8πα sµ 2ǫ i,k 2 S ik(r) M (0) m+ ( pr,...) T it k M (0) m+ ( pr,...) NOTICE overlap of limits when p i p r and p r 0 (matching) momenta in factorized matrix elements on r.h.s. conserve momentum only in the strict limit (extension) arguments of ˆP fi f r (z i, z r and k ) only defined in strict limit (extension) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 42

82 Doubly real counterterms I - matching Structure of overlaps C ir S r Gábor Somogyi Jet cross sections at NNLO via local subtraction page 43

83 Doubly real counterterms I - matching Structure of overlaps C ir S r Candidate subtraction term A M (0) m+2 2 = i i r 2 C ir M (0) m+2 2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 43

84 Doubly real counterterms I - matching Structure of overlaps C ir C ir S r or S rc ir S r Candidate subtraction term A M (0) m+2 2 = i [ i r 2 C ir +S r ] M (0) m+2 2 has correct singularity structure in all limits BUT performs double subtraction when limits overlap (soft-collinear) Gábor Somogyi Jet cross sections at NNLO via local subtraction page 43

85 Doubly real counterterms I - matching Structure of overlaps C ir C ir S r S r Candidate subtraction term A M (0) m+2 2 = i [ 2 C ir +S r i r i r C ir S r ] M (0) m+2 2 has correct singularity structure in all limits is free of double subtraction BUT is still only well-defined in the strict limits Gábor Somogyi Jet cross sections at NNLO via local subtraction page 43

86 Doubly real counterterms I - matching Structure of overlaps C ir C ir S r S r Candidate subtraction term A M (0) m+2 2 = i [ 2 C ir +S r i r i r C ir S r ] M (0) m+2 2 has correct singularity structure in all limits is free of double subtraction BUT is still only well-defined in the strict limits Matching is nontrivial collinear and soft limits do not commute: C irs r S rc ir. Hence it is not obvious a priori which limit to remove. Gábor Somogyi Jet cross sections at NNLO via local subtraction page 43

87 Doubly real counterterms I - extension Extension over full phase space requires momentum mappings {p} m+2 { p} m+ implement exact momentum conservation respect the structure of cancellations lead to exact phase space factorization Momentum mappings separate collinear and soft momentum mappings both distribute recoil momentum democratically can be generalized to any number of unresolved momenta Gábor Somogyi Jet cross sections at NNLO via local subtraction page 44

88 Doubly real counterterms I - extension Collinear mapping p µ ir = (p µ i +p r µ α irq µ ), p µ n = p n µ, n i,r α ir α ir α ir = [ ] y (ir)q y 2(ir)Q 2 4yir momentum conservation phase space factorization p µ ir + n i,r p µ = p µ i +p µ r + n i,r dφ m+2({p};q) = dφ m+({ p} (ir) ;Q)[dp (ir),m+ (pr, pir;q)] [dp (ir),m+ (pr, pir;q)] = dα( α)2m( ǫ) s ĩr Q 2π dφ2(pi,pr;p (ir)) m + 2 i r m + 2 ĩr p µ i r Gábor Somogyi Jet cross sections at NNLO via local subtraction page 44

89 Doubly real counterterms I - extension Soft mapping p µ n = Λ µ ν[q,(q p r)/λ r](p ν n/λ r), n r, λ r = y rq, Λ µ ν[k, K ] = g ν µ 2(K + K ) µ (K + K ) ν (K + K + 2Kµ K ν ) 2 K 2 momentum conservation p µ = p r µ + n r phase space factorization n r dφ m+2({p};q) = dφ m+({ p} (r) ;Q)[dp (r),m+ (pr;q)] [dp (r),m+ (pr;q)] = dy( y)m( ǫ) Q 2 2π dφ2(pr,k;q) m + 2 r m + 2 p µ r K Gábor Somogyi Jet cross sections at NNLO via local subtraction page 44

90 Doubly real counterterms I - extension The collinear and soft momentum mappings define extensions of the limit formulae over the full phase space C ir M (0) m+2 2 C (0,0) ir S r M (0) m+2 2 S r (0,0) C ir S r M (0) m+2 2 C ir S r (0,0) On the r.h.s. C (0,0) ir, S r (0,0) and C ir S r (0,0) are functions of the original momenta that inherit the notation of the operators, but have nothing to do with taking limits. Precise definitions of momentum fractions z i, z r and transverse momentum k that appear in the AP functions are available, but not exhibited. The true subtraction term A M (0) m+2 2 A M (0) m+2 2 = i [ i r 2 C(0,0) ir +S r (0,0) i r ] C ir S r (0,0) The approximate cross section dσ RR,A m+2 = dφ m+ [dp ]A M (0) m+2 2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 44

91 Doubly real counterterms II - IR limits Doubly-unresolved IR limits Triple collinear: C irs is the operator that takes the p i p r p s limit C irs M (0) m+2 (p i,p r,p s...) 2 = (8πα sµ 2ǫ ) 2 ˆP sirs 2 fi f rf s M (0) m (p irs,...) 2 Double collinear: C ir;js is the operator that takes the p i p r and p j p s limit C ir;js M (0) m+2 (p i,p r,p j,p s...) 2 = (8πα sµ 2ǫ ) 2 s ir s js ˆPfi f r ˆPfj f s M (0) m (p ir,p js,...) 2 Soft-collinear: CS ir;s is the operator that takes the p i p r and p s 0 limit CS ir;s M (0) m+2 (p i,p r,p s...) 2 = (8πα sµ 2ǫ ) 2 j,k Double soft: S rs is the operator that takes the p r,p s 0 limit S r M (0) m+2 (pr,ps...) 2 = (8πα sµ 2ǫ ) 2 [ i,k,j,l 4 C A i,k 2 S jk(s) ˆP fi f s r M (0) m,(j,k) (pir, ir ps,...) 2 8 S ik(r)s jl (r) M (0) m,(i,k),(j,l) ( pr, ps,...) 2 S ik (r,s) M (0) m,(i,k) ( pr, ps,...) 2 ] Gábor Somogyi Jet cross sections at NNLO via local subtraction page 45

92 Doubly real counterterms II - matching Structure of overlaps CS ir;s C ir;js C irs S rs Gábor Somogyi Jet cross sections at NNLO via local subtraction page 46

93 Doubly real counterterms II - matching Structure of overlaps CS ir;s C ir;js C irs S rs Candidate subtraction term A 2 M (0) m+2 2 = { [ 6 C irs r s r i r,s ] } M (0) m+2 2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 46

94 Doubly real counterterms II - matching Structure of overlaps CS ir;s C ir;js C irs S rs Candidate subtraction term A 2 M (0) m+2 2 = { r s r i r,s [ 6 C irs + j i,r,s 8 C ir;js ] } M (0) m+2 2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 46

95 Doubly real counterterms II - matching Structure of overlaps CS ir;s C ir;js C irs S rs Candidate subtraction term A 2 M (0) m+2 2 = { r s r i r,s [ 6 C irs + j i,r,s 8 C ir;js + ] 2 CS ir;s } M (0) m+2 2 Gábor Somogyi Jet cross sections at NNLO via local subtraction page 46

96 Doubly real counterterms II - matching Structure of overlaps CS ir;s C ir;js C irs S rs Candidate subtraction term A 2 M (0) m+2 2 = { r s r i r,s [ 6 C irs + j i,r,s 8 C ir;js + ] 2 CS ir;s + 2 Srs } M (0) m+2 2 has correct singularity structure in all limits BUT performs double and triple subtraction when limits overlap Gábor Somogyi Jet cross sections at NNLO via local subtraction page 46