Higher order QCD corrections via local subtraction

Transcription

1 Higher order QCD corrections via local subtraction Gábor Somogyi MTA-DE Particle Physics Research Group, Debrecen with U. Aglietti, P. Bolzoni, V. Del Duca, C. Duhr, S.-O. Moch, Z. Trócsányi University of Debrecen, May 3th 04 Gábor Somogyi Higher order QCD corrections via local subtraction page

2 QCD at the LHC Complicated environment, QCD must be understood/modeled as best as feasible parton model - beams of partons radiation off incoming partons primary hard scattering (µ Q Λ QCD ) radiation off outgoing partons (Q > µ > Λ QCD ) hadronization and heavy hadron decay (µ Λ QCD ) multiple parton interactions, underlying event Gábor Somogyi Higher order QCD corrections via local subtraction page

3 The hard process in perturbation theory The scale of the hard scattering is µ Λ QCD, so by asymptotic freedom, it can be treated in perturbation theory, i.e., by expansion in powers of the strong coupling, α S (µ). Consider a generic cross section for producing m jets ( ) σ m = α p S σm LO +α Sσm NLO +α Sσm NNLO +... Representative Feynman-diagrams How many terms to compute? Gábor Somogyi Higher order QCD corrections via local subtraction page 3

4 Why NNLO? LO prediction: order of magnitude estimate, rough shapes of distributions NLO is mandatory for meaningful normalization and shape predictions NNLO may be relevant NLO corrections are large: Higgs production from gluon fusion in hadron collisions for benchmark processes measured with high experimental accuracy: α s measurements form e + e event shapes W, Z production heavy quark hadroproduction reliable error estimate is needed: processes relevant for PDF determination important background processes (Anastasiou, Dixon, Melnikov, Petriello, Phys. Rev. D69 (004) ) Gábor Somogyi Higher order QCD corrections via local subtraction page 4

5 NNLO ingredients A generic m-jet cross section at NNLO involves Tree-level squared matrix elements with m + 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 processes doubly-virtual contribution (VV) Gábor Somogyi Higher order QCD corrections via local subtraction page 5

6 NNLO ingredients A generic m-jet cross section at NNLO involves Tree-level squared matrix elements with m + 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 processes doubly-virtual contribution (VV) Assuming we know the relevant matrix elements, can we use those matrix elements to compute cross sections? Gábor Somogyi Higher order QCD corrections via local subtraction page 5

7 The problem - IR singularities Consider the NNLO correction to a generic m-jet observable σ NNLO = dσ RR J + dσm+ RV J m+ + dσm VV Jm. m+ m Doubly-real dσ RR J Tree MEs with -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 ) poles from PS integration explicit ǫ poles up to O(ǫ ) 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 Higher order QCD corrections via local subtraction page 6

8 The problem - IR singularities Consider the NNLO correction to a generic m-jet observable σ NNLO = dσ RR J + dσm+ RV J m+ + dσm VV Jm. 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 Higher order QCD corrections via local subtraction page 6

9 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 Higher order QCD corrections via local subtraction page 7

10 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, 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 Higher order QCD corrections via local subtraction page 7

11 Approaches - developments Refinement of the sector decomposition algorithm (Anastasiou, Lazopoulos, Herzog 00) uses non-linear mappings to disentangle overlapping singularities the aim is to increase efficiency by reducing the large number of sectors/terms generated during decomposition first application: fully exclusive H b b decay at NNLO Refinement of phase space integration via sector decomposition (Anastasiou, Lazopoulos, Herzog 0) (Czakon 00; Boughezal, Melnikov, Petriello 0) FKS-like approach to double real radiation in t t production sector decomposition used to make singular contributions explicit, guided by known universal IR structure first NNLO computation of pp t t total cross section (Baernreuther, Czakon, Mitov 0) Gábor Somogyi Higher order QCD corrections via local subtraction page 8

12 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 Higher order QCD corrections via local subtraction page 9

13 Basics of subtraction Strategy: rearrange IR singularities between various contributions by subtracting and adding back suitably defined approximate cross sections. subtraction terms match the singularity structure of real emission point wise (in d dimensions) phase space integrals over real radiation rendered convergent integrated forms of subtraction terms have the same pole structure as virtual contribution explicit ǫ-poles cancel point by point The construction of a general (i.e. process- and observable-independent) subtraction algorithm made possible by the universal structure of IR singularities, embodied in so-called IR factorization formulae is not unique, hence several approaches (FKS, dipole, antenna,...) Gábor Somogyi Higher order QCD corrections via local subtraction page 0

14 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 Higher order QCD corrections via local subtraction page

15 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σ RR +σrv m+ +σvv m = σ NNLO +σm+ NNLO +σm NNLO each integrable in four dimensions σ NNLO = σm+ NNLO = σm NNLO = m+ m { [ ]} dσ RR J dσ RR,A J m dσ RR,A J m+ dσ RR,A J m ( {[ dσm+ RV + { dσm VV + ] [ dσ RR,A J m+ dσ RV,A [ dσ RR,A dσ RR,A ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A ]J m ( ) ]} dσ RR,A A J m Gábor Somogyi Higher order QCD corrections via local subtraction page

16 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σ RR +σrv m+ +σvv m = σ NNLO +σm+ NNLO +σm NNLO each integrable in four dimensions σ NNLO = σm+ NNLO = σm NNLO = m+ m { [ ]} dσ RR J dσ RR,A J m dσ RR,A J m+ dσ RR,A J m ( {[ dσm+ RV + { dσm VV + ] [ dσ RR,A J m+ dσ RV,A [ dσ RR,A dσ RR,A ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A ]J m ( ) ]} dσ RR,A A J m. dσ RR,A regularizes the doubly-unresolved limits of dσ RR Gábor Somogyi Higher order QCD corrections via local subtraction page

17 Structure of the NNLO correction Rewrite the NNLO correction as a sum of three terms σ NNLO = σ RR +σrv m+ +σvv m = σ NNLO +σm+ NNLO +σm NNLO each integrable in four dimensions σ NNLO = σm+ NNLO = σm NNLO = m+ m { [ ]} dσ RR J dσ RR,A J m dσ RR,A J m+ dσ RR,A J m ( {[ dσm+ RV + { dσm VV + ] [ dσ RR,A J m+ dσ RV,A [ dσ RR,A dσ RR,A ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A ]J m ( ) ]} dσ RR,A A J m. dσ RR,A regularizes the doubly-unresolved limits of dσ RR. dσ RR,A regularizes the singly-unresolved limits of dσ RR Gábor Somogyi Higher order QCD corrections via local subtraction page

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

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

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

21 Defining a subtraction scheme Two issues must be addressed. What to subtract?. How to add it back? Gábor Somogyi Higher order QCD corrections via local subtraction page 3

22 Defining a subtraction scheme Two issues must be addressed. What to subtract?. How to add it back? Strategy: IR limits are process independent and known. Start by defining subtraction terms based on IR limit formulae the result is trivially general and explicit. Worry about integrating them later, since this is in principle a very narrowly defined problem, given., but in practice turns out to be very cumbersome Gábor Somogyi Higher order QCD corrections via local subtraction page 3

23 IR factorization formulae The structure of IR singularities is universal, i.e., does not depend on the process. The general structure of these formulae is the same in all limits M (0) n+p ({p}n+p) R ( 8πα sµ ǫ) p SingR ({p} p) M (0) n ({p} n) R Gábor Somogyi Higher order QCD corrections via local subtraction page 4

24 IR factorization formulae Collinear and soft currents at NNLO are known Tree level 3-parton splitting functions and double soft gg and q q currents (Campbell, Glover 997; Catani, Grazzini 998; Del Duca, Frizzo, Maltoni 999; Kosower 00) One-loop -parton splitting functions and soft gluon current (Bern, Dixon, Dunbar, Kosower 994; Bern, Del Duca, Kilgore, Schmidt 998-9; Kosower, Uwer 999; Catani, Grazzini 000; Kosower 003) Gábor Somogyi Higher order QCD corrections via local subtraction page 4

25 IR factorization formulae Collinear and soft currents at NNLO are known Tree level 3-parton splitting functions and double soft gg and q q currents (Campbell, Glover 997; Catani, Grazzini 998; Del Duca, Frizzo, Maltoni 999; Kosower 00) One-loop -parton splitting functions and soft gluon current (Bern, Dixon, Dunbar, Kosower 994; Bern, Del Duca, Kilgore, Schmidt 998-9; Kosower, Uwer 999; Catani, Grazzini 000; Kosower 003) Are these useful for NNLO? Gábor Somogyi Higher order QCD corrections via local subtraction page 4

26 IR factorization formulae Limit formulae cannot be used as subtraction terms as they stand. Consider collinear: p i p r M (0) n+ (p i,p r,...) i r 8πα sµ ǫ s ˆP(0) f i f r (z i,z r,k ;ǫ) M (0) n (p i +p r,...) ir soft: p r 0 Issues M (0) n+ (pr,...) r 0 8πα sµ ǫ i,k s ik M (0) s ir s n,(i,k) ( kr pr,...) the singular function associated to a specific limit, Sing R ({p} p) may be singular in other IR limits as well both Sing R ({p} p) and M (0) n ({p} n) are only well-defined in the strict R limit Gábor Somogyi Higher order QCD corrections via local subtraction page 4

27 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+ = i [ 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} r,s { p} m : dφ ({p} ;Q) = dφ m({ p} m ;Q)[dp,m ] 3. Integration of the counterterms over the phase space of the unresolved parton(s). Gábor Somogyi Higher order QCD corrections via local subtraction page 5

28 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.. 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 Higher order QCD corrections via local subtraction page 6

29 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 Higher order QCD corrections via local subtraction page 7

30 NNLO subtraction terms Can check the ratio of the real emission matrix element and the sum of all subtractions for all IR limits Singe collinear limit e + e q qggg y 45 = 0-6 y 45 = 0-8 y 45 = Double soft limit e + e q qggg y 4Q y 5Q = 0-4 y 4Q y 5Q = 0-6 y 4Q y 5Q = # events 0 5 # events R R Gábor Somogyi Higher order QCD corrections via local subtraction page 8

31 Integrating the subtractions Momentum mappings used to define the counterterms {p} n+p R { p} n dφ n+p({p};q) = dφ n({ p} (R) n ;Q)[dp (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 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] X R ({p} n+p) = ( 8πα sµ ǫ) p SingR (p (R) p ) M (0) n ({ p} (R) n ) Can compute once and for all the integral over unresolved partons X R ({p} n+p) = ( 8πα sµ ǫ) [ ] p Sing R (p p (R) ) p p M (0) n ({ p} (R) n ) Gábor Somogyi Higher order QCD corrections via local subtraction page 9

32 Solving the integrlas Master integrals use algebraic and symmetry relations to reduce to a basic set of integrals note: this is not the usual notion of MIs (no IBPs used) Strategy for computing the master integrals. write phase space in terms of angles and energies. angular integrals in terms of Mellin-Barnes representations 3. resolve the ǫ poles by analytic continuation 4. MB integrals to Euler-type integrals, pole coefficients are finite parametric integrals. choose explicit parametrization of phase space. write the parametric integral representation in chosen variables 3. resolve the ǫ poles by sector decomposition 4. pole coefficients are finite parametric integrals 5. evaluate the parametric integrals in terms of multiple polylogs 6. simplify result (optional) Gábor Somogyi Higher order QCD corrections via local subtraction page 0

33 List of master integrals Int I (k) C,0 I (k) C, I (k) C, I (k) C,3 I (k) C,4 I (k,l) C,5 I (k,l) C,6 I (k) C,7 IC,8 status Int IS,0 IS, IS, I (k) S,3 IS,4 IS,5 IS,6 IS,7 status Int ICS,0 ICS, I (k) CS, ICS,3 ICS,4 status Int I (k,l) C, I (k,l) C, I (k) C,3 I (k,l) C,4 I (k) C,5 I (k) C,6 I (k) C,7 I (k) C,8 I (k) C,9 I (k) C,0 status Int IS, IS, IS,3 IS,4 IS,5 IS,6 IS,7 IS,8 IS,9 IS,0 IS, IS, IS,3 status Int I (k) S, I (k) S, I (k) S,3 I (k) S,4 I (k) S,5 IS,6 IS,7 status Int I (k) CS, ICS, ICS,3 status Int I (j,k,l,m) C, I (j,k,l,m) C, I (j,k,l,m) C,3 I (j,k,l,m) C,4 I (j,k,l,m) C,5 I (k,l) C,6 status Int I (k) CS, I (k) CS, I (k) CS,3 I (k) CS,4 I (k) CS,5 status IS,4 IS,5 IS,6 IS,7 IS,8 IS,9 IS,0 IS, IS, IS,3 IS,8 IS,9 IS,0 IS, IS, IS,3 Gábor Somogyi Higher order QCD corrections via local subtraction page

34 Phase space integrals - an example Abelian double soft counterterm: among many others, in dσ RR,A we find ( ) S rs (0,0) ab = (8παsµ ǫ ) s ik s jl M (0) 4 s ir s kr s js s m,(i,k)(j,l) ({ p}) ls i,k,j,l ( y rq y sq +y rs) d 0 m( ǫ) Θ(y 0 y rq y sq +y rs) The set of m momenta, { p}, is obtained by a momentum mapping which leads to an exact factorization of phase space S {p} rs { p} : dφ ({p};q) = dφ m({ p};q)[dp (rs),m ] r s r s K m + m + Gábor Somogyi Higher order QCD corrections via local subtraction page

35 Phase space integrals - an example Abelian double soft counterterm: among many others, in dσ RR,A we find ( ) S rs (0,0) ab = (8παsµ ǫ ) s ik s jl M (0) 4 s ir s kr s js s m,(i,k)(j,l) ({ p}) ls i,k,j,l ( y rq y sq +y rs) d 0 m( ǫ) Θ(y 0 y rq y sq +y rs) The set of m momenta, { p}, is obtained by a momentum mapping which leads to an exact factorization of phase space S {p} rs { p} : dφ ({p};q) = dφ m({ p};q)[dp (rs),m ] Then we must compute ( ) [ ( ab [dp (rs),m ] S rs (0,0) αs µ π Sǫ Q ) ǫ ] i,k,j,l [S (0) rs ] (i,k),(j,l) M (0) m,(i,k)(j,l) ({ p}) where [S (0) rs ] (i,k),(j,l) [S (0) rs ] (i,k),(j,l) (p i,p k,p j,p l,ǫ,y 0,d 0) is a kinematics dependent function. Gábor Somogyi Higher order QCD corrections via local subtraction page

36 Abelian double soft integral For simplicity, consider the terms in the sum where j = i and l = k: [S (0) rs ] (i,k),(i,k). Kinematical dependence is through cosχ ik = (p i,p k ), we set cosχ ik = Y ik,q, i.e., Y ik,q is between zero and one. Using angles and energies in the Q rest frame with some specific orientation to parametrize the factorized phase space measure, [dp (rs),m ], we find that [S (0) rs ] (i,k),(i,k) is proportional to I S, (Y ik,q ;ǫ,y 0,d 0 ) = ( ǫ) B y0 ( ǫ,d 4Γ4 0 ) y0 πγ Y ik,q dy y ǫ ( y) d 0 +ǫ ( ǫ) ǫ 0 d(cosϑ)(sinϑ) ǫ d(cosϕ)(sinϕ) ǫ[ f(ϑ,ϕ;0) ] [ f(ϑ,ϕ;y ik,q ) ] [ Y(y,ϑ,ϕ;Y ik,q ) ] ǫ ( F ǫ, ǫ, ǫ, Y(y,ϑ,ϕ;Yik,Q ) ) where f(ϑ,ϕ;y ik,q ) = Y ik,q ( Y ik,q )sinϑcosϕ ( Y ik,q )χcosϑ Y(y,ϑ,ϕ;χ) = 4( y)y ik,q [( y)+y f(ϑ,ϕ;0)][( y)+y f(ϑ,ϕ;y ik,q )] Gábor Somogyi Higher order QCD corrections via local subtraction page 3

37 Abelian double soft integral This integral is equal to (y 0 =, d 0 = 3 3ǫ) I S, (Y;ǫ,,3 3ǫ) = = ǫ 4 [ ǫ 3 ln(y) 3 ]+ ǫ [ ( Li ( Y)+ln (Y) π Y ( Y) + 9 ) ] ln(y)+ ( Y) +6 + [ ( 5 8Li3 ( Y) + 6Li 3(Y) ǫ Li ( Y)ln(Y) 5 5 ln3 (Y)+ 3 5 ln( Y)ln (Y)+π ln(y) 78ζ(3) ) 5 ( 3 + Y 3 4( Y) + 5 ) ( ( Li ( Y)+ln (Y)) 6π 7 4 ( Y) 3 4( Y) ) ln(y)+ Note the Y limit is finite 9 4( Y) + 63 ] +O(ǫ 0 ) lim I S,(Y;ǫ,,3 3ǫ) = 3 ( ) 7 Y ǫ 4+ ǫ 3+ ǫ 4 π + ( ) 393 ǫ 4 6π 4ζ(3) +O(ǫ 0 ) Gábor Somogyi Higher order QCD corrections via local subtraction page 3

38 Abelian double soft integral Finite term is computed numerically (y 0 =, d 0 = 3 3ǫ) 80 Finite part I S, (Y) / Y Gábor Somogyi Higher order QCD corrections via local subtraction page 3

39 Analytic vs. numeric As a matter of principle A rigorous proof of cancellation of IR poles requires the poles of integrated counterterms in analytic form. However An actual implementation needs numbers for the finite parts of the integrated counterterms. These finite parts are smooth functions of kinematic variables. Hence Numerical forms of the finite parts are sufficient for practical purposes. The final results can be conveniently given by interpolating tables or approximating functions computed once and for all. In particular, suitable approximating functions may be obtained by fitting. Gábor Somogyi Higher order QCD corrections via local subtraction page 4

40 Example of approximation by fitting Doubly-unresolved double-collinear master integral I C,6(x ir,x js;ǫ,,3 3ǫ,k,l) I C,6 (x ir,x js ;ǫ,α 0,d 0 ;k,l) = x ir x js dαdβ dv duα ǫ β ǫ ( α β) d 0 ( ǫ) 0 0 [α+( α β)x ir ] ǫ [β +( α β)x js ] ǫ v ǫ ( v) ǫ u ǫ ( u) ǫ ( ) α+( α β)xir v k ( ) β +( α β)xjs u l Θ(α 0 α β) α+( α β)x ir β +( α β)x js Gábor Somogyi Higher order QCD corrections via local subtraction page 5

41 Example of approximation by fitting Doubly-unresolved double-collinear master integral I C,6(x ir,x js;ǫ,,3 3ǫ,k,l) poles (up to O(ǫ 4 )) extracted via sector decomposition numerical values of pole coefficients computed for a 7 7 grid with precision of 0 7 define three regions (note: result is symmetric in x ir,x js) asymptotic: x ir,x js < 0 4 non-asymptotic: x ir,x js > 0 border: x ir < 0 or x js < 0 in each region, fit with ansatz F(x ir,x js) = p i,l i C m;p,p ;l,l (x p ir xp js )(logl (x ir)log l (x js)) where p +p m with m a free parameter, while l +l n and n is predicted Gábor Somogyi Higher order QCD corrections via local subtraction page 5

42 Example of approximation by fitting Doubly-unresolved double-collinear master integral I C,6(x ir,x js;ǫ,,3 3ǫ,k,l) Gábor Somogyi Higher order QCD corrections via local subtraction page 5

43 Methods of integration - angular integrals Consider the d dimensional angular integral with n denominators (GS 0) Ω j,...,j n = dω d (q) (p q) j (pn q) jn This admits the following Mellin-Barnes representation (j = j +...+j n) Ω j,...,j n ({v kl };ǫ) = j ǫ π ǫ n k= Γ(j k)γ( j ǫ) [ +i n ][ n n ] dz kl i πi Γ( z kl) (v kl ) z kl Γ(j k +z k ) Γ( j ǫ z). k= l=k k= where v kl = p k p l for k l and v kk = p k 4 while n n k n z = z kl and z k = z lk + z kl. k= l=k l= l=k Gábor Somogyi Higher order QCD corrections via local subtraction page 6

44 Methods of integration - MB to parametric integrals Basic idea is to express products of gamma functions as real integrals I = = +i i +i i dz dz πi πi Γ[a+z +z]γ[b z z] vz dz dz πi πi Γ[a+b] 0 vz dtt a +z +z ( t) b z z v z vz if R(a+z +z ) > 0 and R(b z z ) > 0 so the t integral converges Eliminate enough gamma functions to be able to perform the MB integrals can eliminate all gamma functions for real integrals, then use +i i dz πi vz = δ( v), v > 0 For multidimensional MB integrals, sometimes it is more useful to eliminate just the gamma functions that couple the MB integrations. This turns the multidimensional MB integral into products of d MB integrals. After solving the remaining MB integrals, we get the desired parametric representation. Gábor Somogyi Higher order QCD corrections via local subtraction page 7

45 Methods of integration - symbolic integration Assume P and Q are polynomials and the following integral converges I(x) = dt dt dt 3... P(x,t,t,t 3,...) 0 Q(x,t,t,t 3,...) The t integration assuming the denominator is a product of factors all linear in t, after partial fractioning, we will need to compute dt 0 t n n = is non-trivial dt = lnt, t e.g., we have 0, 0 dt [t a(x,t,...)] n, dt t a(x,t,...) = ln[t a(x,t,...)] [ ] dt t a(x,t,...) = ln a(x,t,...) this is elementary, although there is some fine print for definite integrals Gábor Somogyi Higher order QCD corrections via local subtraction page 8

46 Methods of integration - symbolic integration Assume P and Q are polynomials and the following integral converges I(x) = dt dt dt 3... P(x,t,t,t 3,...) 0 Q(x,t,t,t 3,...) The t integration assuming the new denominator is a product of factors all linear in t, after partial fractioning aside from the integrals we already encountered we will have to compute 0 dt t n [ ] ln, a(x,t,...) 0 [ ] dt [t b(x,t 3,...)] n ln, a(x,t,...) if a(x,t,...) is also linear in t, we can use the functional identities for the logarithm [ln(ab) = lna+lnb, ln(/a) = lna] to write [ ] ln = ln[a (x,t 3,...) t ] ln[a (x,t 3,...) t ] a(x,t,...) again, n = is non-trivial dt t lnt = ln (t ), dt t ln( t ) = Li (t ) Gábor Somogyi Higher order QCD corrections via local subtraction page 8

47 Methods of integration - symbolic integration Assume P and Q are polynomials and the following integral converges I(x) = dt dt dt 3... P(x,t,t,t 3,...) 0 Q(x,t,t,t 3,...) The t integration (cont.) e.g., we have 0 [ ] dt t b(x,t 3,...) ln(t ) = Li b(x,t 3,...) Gábor Somogyi Higher order QCD corrections via local subtraction page 8

48 Methods of integration - symbolic integration Assume P and Q are polynomials and the following integral converges I(x) = dt dt dt 3... P(x,t,t,t 3,...) 0 Q(x,t,t,t 3,...) Before going to the t 3 integration, notice. at each step, we needed to introduce a new transcendental function, ln, Li. we needed to know the functional identities for ln to proceed Gábor Somogyi Higher order QCD corrections via local subtraction page 8

49 Methods of integration - symbolic integration Assume P and Q are polynomials and the following integral converges I(x) = dt dt dt 3... P(x,t,t,t 3,...) 0 Q(x,t,t,t 3,...) The t 3 integration assuming the new denominator is a product of factors all linear in t 3, after partial fractioning aside from the integrals we already encountered we will have to compute 0 dt 3 t n 3 [ ] p(x,t3,...) Li, q(x,t 3,...) 0 [ ] dt 3 [t 3 c(x,t 4,...)] n Li p(x,t3,...), q(x,t 3,...) will need to introduce new transcendental functions multiple polylogs will need to use the functional identities for Li to reduce to some standard form symbols, coporoducts, Hopf algebra of multiple polylogs Gábor Somogyi Higher order QCD corrections via local subtraction page 8

50 Multiple polylogarithms The appropriate generalization of log and classical polylogs (Goncharov 998, 00) z dt G(a,...,a n;z) = G(a,...,a n;t) with G(z) = 0 t a G(0,...,0;z) = }{{} n! lnn (z) n Logarithms and classical polylogs are special cases, e.g., G(a,...,a;z) = ( }{{} n! lnn z ) ( ) z, G(0,...,0,a;z) = Li n a }{{} a n n Functional relations among G s Problem: after the (n )-st step of integration, the n-th variable can appear in the a i dt n t n b G(a (t n,...),...,a n (t n,...);z(t n,...)) Must reduce to canonical form, where t n is only in the last entry. Unfortunately the functional equations among G s that would be needed to do this are often unknown and need to be derived. Gábor Somogyi Higher order QCD corrections via local subtraction page 9

51 Symbols, coproducts Symbols are a tool for obtaining functional equations among G s (Goncharov 009; Goncharov, Spradlin, Vergu, Volovich 00; Duhr, Gangl, Rhodes 0) The symbol is a way of associating to a multiple polylog a tensor in a certain tensor space. n ( ) ai a i+ S(G(a n,...,a ;a n)) = S(G(a n,...,a i,a i+,...,a ;a n)) a i= i a i e.g., ( ) S n! lnn (z) = z... z } {{ } n times, S(Li n(z)) = ( z) z... z }{{} (n ) times Functional equations between multiple polylogs become algebraic equations between tensors. The idea of symbols can be refined based on the Hofp algebra structure of multiple polylogs coproduct (Duhr 0) With these refinements one can build algorithms to reduce multiple polylogs to canonical form. Gábor Somogyi Higher order QCD corrections via local subtraction page 30

52 Integrated approximate cross sections Recall the NNLO correction is a sum of three terms σ NNLO = σ RR +σrv m+ +σvv m = σnnlo +σm+ NNLO +σm NNLO each integrable in four dimensions σ NNLO = σm+ NNLO = σm NNLO = m+ m { [ ]} dσ RR J dσ RR,A J m dσ RR,A J m+ dσ RR,A J m ( {[ dσm+ RV + { dσm VV + ] [ dσ RR,A J m+ dσ RV,A [ dσ RR,A dσ RR,A ] + m+ + [ dσ RV,A m+ + ) } dσ RR,A A ]J m ( ) ]} dσ RR,A A J m Gábor Somogyi Higher order QCD corrections via local subtraction page 3

53 Integrated approximate cross sections Recall the NNLO correction is a sum of three terms σ NNLO = σ RR +σrv m+ +σvv m = σnnlo +σm+ NNLO +σm NNLO each integrable in four dimensions σ NNLO = σm+ NNLO = σm NNLO = m+ m { [ ]} dσ RR J dσ RR,A J m dσ RR,A J m+ dσ RR,A J m {[ ] [ ( dσm+ RV + dσrr,a J m+ dσ RV,A m+ + { dσ VV m + Integrated approximate cross sections [ ] dσ RR,A dσ RR,A + [ dσ RV,A m+ + ) } dσ RR,A A ]J m ( ) ]} dσrr,a A J m 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. Can be computed once and for all (though admittedly lots of tedious work). Gábor Somogyi Higher order QCD corrections via local subtraction page 3

54 Integrated approximate cross sections - an example Doubly unresolved dσ RR,A = dσ B m I (0) ({p} m;ǫ) structure of insertion operator in color flavor space [ ( I (0) ({p}m;ǫ) = αs µ ) ǫ ] { π Sǫ Q + j,l i + i,k,j,l [ C (0),f i T i + k C (0),f i f k T k ] T i [ S (0),(j,l) C A + i } S (0),(i,k)(j,l) {T i T k,t j T l } ] CS (0),(j,l),f T i i T j T l C (0),f i, C (0),f i f k, S (0),(j,l), CS (0),(j,l),f i and S (0),(i,k)(j,l) are kinematical functions with poles up to O(ǫ 4 ) kinematical dependence through x i = y iq p i Q Q and Y ik,q = y ik y iq y kq Gábor Somogyi Higher order QCD corrections via local subtraction page 3

55 Integrated approximate cross sections - an example Doubly unresolved dσ RR,A = dσ B m I (0) ({p} m;ǫ) e.g., e + e 3 jets (momentum assignment is q, q,3 g) [ ( I (0) (p αs µ ) ǫ ] { 8C,p,p 3 ;ǫ) = F +0C A C F +3C π Sǫ A Q 4ǫ C A 7C FT R n f 4 3 ( C A C F + 3C A C AT R n f ) (lny 3 +lny 3 ) ( 4CF +C AC F 3C A ] } ǫ 3 +O(ǫ ) [ 6CF + 09C AC F ) lny notice x and Y dependence combine to produce just y ik dependence, as expected Gábor Somogyi Higher order QCD corrections via local subtraction page 3

56 Integrated approximate cross sections - an example Doubly unresolved dσ RR,A = dσ B m I (0) ({p} m;ǫ) e.g., e + e 3 jets (momentum assignment is q, q,3 g) [ ( I (0) (p αs µ ) ǫ ] { 8C,p,p 3 ;ǫ) = F +0C A C F +3C π Sǫ A Q 4ǫ C A 7C FT R n f 4 3 ( C A C F + 3C A C AT R n f ) (lny 3 +lny 3 ) ( 4CF +C AC F 3C A ] } ǫ 3 +O(ǫ ) [ 6CF + 09C AC F ) lny notice x and Y dependence combine to produce just y ik dependence, as expected Gábor Somogyi Higher order QCD corrections via local subtraction page 3

57 Insertion operators Color and flavor structure of all insertion operators known Analytic computation of poles almost complete all MIs known analytically up to and including O(ǫ ) /ǫ parts of 0 of O(300) integrals to be finished Finite parts computed numerically final results in the form of approximating functions obtained by fitting Gábor Somogyi Higher order QCD corrections via local subtraction page 33

58 Present status NNLO correction is the sum of three terms σ NNLO = σ RR +σm+ RV +σm VV = σ NNLO +σm+ NNLO +σm NNLO Each integrable in four dimensions σ NNLO = σ NNLO m+ = σ NNLO m = m+ m { dσ RR J dσ RR,A J m {[ ] [ dσm+ RV + dσrr,a J m+ dσ RV,A m+ + { dσm VV + [ ]} dσ RR,A J m+ dσ RR,A J m ( ) } dσrr,a A ]J m ] dσ RR,A dσ RR,A + [ ( dσ RV,A ) ]} m+ + dσrr,a A J m unintegrated doubly real counterterms (for final state singularities) unintegrated real virtual counterterms (for final state singularities) tree-level and one-loop singly unresolved integrals [to O(ǫ )] tree-level iterated singly unresolved integrals [O(ǫ ) in progress] tree-level doubly unresolved integrals [O(ǫ ) in progress] Gábor Somogyi Higher order QCD corrections via local subtraction page 34

59 Extension to hadronic initial states Essential elements of the construction unchanged write the IR factorization formulae for all limits including initial state splittings in such a form that their overlap structure can be disentangled extend the formulae relevant for initial state singularities over full phase space requires new momentum mappings subtraction terms for final state splittings fine as they stand integrate new subtraction terms analytically and numerically Basic steps clear, but admittedly lots of tedious work Gábor Somogyi Higher order QCD corrections via local subtraction page 35

60 Conclusions NNLO is the new precision frontier Two bottlenecks. can we compute the relevant (-loop) amplitudes?. if yes, can we use those to compute cross sections? Subtraction is the traditional solution to. We have set up general, explicit, local subtraction scheme for computing NNLO corrections in QCD construction of subtraction terms based on IR limit formulae analytic integration of subtraction terms is feasible with modern integration techniques and is almost finished the scheme is worked out in full detail for processes with no colored particles in the initial state Extension to hadron initiated processes conceptually clear, work in progress Gábor Somogyi Higher order QCD corrections via local subtraction page 36

61 Acknowledgments This research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP A/ National Excellence Program. Gábor Somogyi Higher order QCD corrections via local subtraction page 37