Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space

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1 Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space Petra Kovačíková (DESY, Zeuthen) Corfu Summer School 4 th September 21 Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 1/3

2 Outline Introduction Drell-Yan mechanism The Drell-Yan hadronic Cross Section NLO α s corrections Parton distribution functions Mass fatorization The Calculation of Drell-Yan cross section using Mellin transform Results and Checks Conclusions and Outlook Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 2/3

3 The Drell-Yan Process Massive lepton pair production in hadron-hadron collision, M 2 l 1 l 2 >> 1 GeV 2 P 1 p 1 P 2 p 2 ˆσ γ/z, W +/ X l 1 (p 3 ) l 2 (p 4 ) M l1 l 2 = Q 2 = (p 2 + p 3 ) 2 CM energy of hadrons s = (P 1 + P 2 ) 2 p i = x i P i x i (, 1) CM energy of partons ŝ = (p 1 + p 2 ) 2 = sx 1 x 2 Neutral Current pp γ l lx M l l M Z M Z 91.2 GeV pp Z l lx Charged Current M l l M Z pp W ± lνx M l l M W M W 8.4 GeV [Drell,Yan 197] Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 3/3

4 Drell-Yan at the Tevatron and the LHC σ (nb) proton - (anti)proton cross sections σ tot σ b σ jet (E T jet > s/2) σ W σ Z σ jet (E T jet > 1 GeV) σ t σ jet (E T jet > s/4) Tevatron LHC events/sec for L = 1 33 cm -2 s -1 Large total cross sections for W and Z production Number of events N σl d t LHC W: L d t 3 nb 1, N 2 Z: L d t 23 nb 1, N 15 [Gianotti,ATLAS collaboration ICHEP, July 21] Clear signal Backrgound for new phyiscs measurements σ Higgs (M H = 15 GeV) σ Higgs (M H = 5 GeV) Detector calibration, luminosity monitoring, constraints on PDFs s (TeV) Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 4/3

5 The Drell-Yan Process Higher order NLO QCD corrections [ Altarelli, Ellis, Martinelli 1979] NNLO [ Hamberg, Matsuura, van Neerven 199; Haarlander, Kilgore 22] NNLO double differential cross section [Anastasiou, Dixon, Melnikov, Petriello 24; Catani, Ferrera, Grazzini 21] new constraints on PDFs [Alekhin, Melnikov, Petriello 26] Electroweak corrections up to NLO [Hollik, Wackeroth 1996; Vicini et.al. 29] Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 5/3

6 The Hadronic Cross Section Tevatron LHC (1 TeV) NLO/LO = NNLO/NLO = ( ) y W = 1 2 ln x 1 x 2 NLO/LO = NNLO/NLO = [Catani, Ferrera, Grazzini 21] Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 6/3

7 The Drell-Yan Process Hadronic cross section d σ V pp l 1 l 2 d Q 2 = a,b 1 d x 1 1 d ˆσ ab V d x 2 f a (x 1 )f b (x 2 ) (x 1, x 2, Q 2, α s (µ 2 r)) }{{} d Q }{{ 2 } PDFs Hard Cross section V = γ, Z, W ± a, b = q, q, g x 1,2 (, 1) Partonic cross section up to NNLO d ˆσ V ab l l d Q 2 = N V }{{} Norm. factor C V }{{} EW couplings ab (x) }{{} Coefficient functions ab (x) = () ab (x) + α ( s 4π (1) ab (x) + αs ) 2 (2) ab 4π (x), x = Q2 ŝ Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 7/3

8 The Partonic Cross Section Leading order () qq (x) = δ(1 x) Next-to-leading order Virtual contribution ( (1) qq,virt(x) δ(1 x) Real contribution ˆσ NLO n 2 ǫ 2 3 ǫ (1) qq,real(x) δ(1 x) 2 ǫ 2 2 ǫ + finite part = n d ˆσ virtual + n+1 d ˆσ real ) + finite part 1 + x 2 (1 x) + [ ln(1 x) ] (1 x) + }{{} Plus distribution Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 8/3

9 The Partonic Cross Section The plus distributions Singular functions that have origin in phase space integration Are defined by their integrals with a smooth function (such as parton distribution functions) 1 f(x)h + (x)d x = Have to be treated carefuly The sum of real and virtual part 1 h(x)[f(x) f(1)]d x (1) qq (x) 2 [ 3 ǫ C f 2 δ(1 x) x2 (1 x) + ] } {{ } Splitting function P qq () (x) +finite part Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 9/3

10 Splitting Functions P ab (z) describes probability of a parton with momentum p radiates a soft or collinear parton zp zp zp zp p p p p Perturbative quantities calculated up to NNLO [Moch, Vermaseren, Vogt 23] P Nm LO ab (x, µ 2 ) = m k= α k+1 s (µ 2 ) P k 4π ab(x) Main ingredient of the evolution equation of PDFs Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 1/3

11 The Hadronic cross section When the bare parton distribution functions in the cross section are expressed in terms of renormalized ones, in MS scheme f q, (x) = f q (x, µ 2 f) α { s 1 ( µ 2 )} 2π ǫ ln 4π + γ f E + ln µ 2 { 1 d ξ ( x ) 1 ξ P qq () d ξ ( x ) } f (ξ) + ξ ξ P qg () f g, (ξ), ξ x the convolution with the divergent partonic cross section [ (1) qq (x) = 2P qq () (x) 1 ( µ 2 )] ǫ ln4π + γ E + ln Q 2 x + finite part leads to finite hadronic cross section as a result of factorization theorem. ( µ 2 ) (1) qq (x, µ 2 f) finite part 2P qq () f (x) ln Q 2 Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 11/3

12 Parton Distribution Functions if a (x, µ 2 f) a = q, q, g x (, 1) Distribution of momentum fraction x of the parton a in the proton Non-perturbative objects Extraction from experiment (DIS) at low scale Q 2 few GeV 2 fitting Scale dependence described by evolution equations (DGLAP) f a (x, µ 2 f ) lnµ 2 f = b 1 x d ξ ξ P ab (x/ξ, µ 2 }{{ r) f b (ξ, µ 2 } f) splitting functions Solution - system of integro differential equations Universal - process independent quantities Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 12/3

13 Kinematics of PDFs LHC at 7 TeV (at 14 TeV) x 1,2 = (M/7 TeV) exp(±y) Q=M Q 2 (GeV 2 ) M=1 GeV M=1 TeV y= Rapidity of the vector boson in CM frame of partons ( ) y = 1 2 ln x 1 x 2 x 1,2 = M s e ±y M=1 GeV DIS (HERA, fixed target) 1 S.M x [S.Moch] Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 13/3

14 The Hadronic Cross Section d σ pp V (Q 2 ) d Q 2 = a,b 1 d x 1 1 d x 2 f a (x 1, Q 2 )f b (x 2, Q 2 ) d ˆσV ab (x 1, x 2, Q 2 ) d Q 2 = N V τ a,b C V (f a f b V ab)(τ); Q = µ f = µ r d ˆσ d Q 2 = N V C V ab (x) Evolution of PDFs 1 ( d xδ x τ ) x 1 x 2 = 1 τ = Q2 s f a (x, Q 2 ) lnq 2 = b 1 x d ξ ξ P ab(x/ξ, Q 2 )f b (ξ, Q 2 ) f a(x, Q 2 ) ln Q 2 = b (P ab f b )(x) Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 14/3

15 Convolution Definition (f 1 f 2 f k )(x) = 1 1 d x 1 d x d x k δ(x x 1 x 2...x k )f 1 (x 1 )f 2 (x 2 )...f(x k ) Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 15/3

16 Convolution and The Mellin Transform The Mellin transform M[f(x)] = 1 N = c + ρe iϕ, ρ, c R, d xx N 1 f(x) = f(n) 1 d xx c 1 f(x) abs. convergent Mellin transform of the convolution is a product of functions in N space M[(f 1 f 2 f k )(x)](n) m k=1 M[f k ](N) The cross section (f a f b V ab)(τ) f a (N)f b (N) V ab(n) W(N) Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 16/3

17 The Inverse Mellin Transform Inverse Mellin Transform f(x) = 1 2πi c+i c i d N x N f(n) Im(N) The original function W(τ) can be recovered W(τ) = 1 2πi c+i c i d N x N W(N) N i c Re(N) Only one integration Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 17/3

18 The Mellin Transform - Example f(x) = x 2 f(n) = f(x) = = 1 1 2πi d xx 2 x N 1 f(x) = 1 N + 2 c+i c i d N x N 1 N (N + 2) 2πi 2πix N (N + 2) N= 2 = x 2 Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 18/3

19 The Inverse Mellin transform z Im(N) Numerical evaluation [Vogt 24] 1 2πi = 1 π c+i c i d N x N f(n = c + ρe iϕ ) d ρ Im[e iϕ x c ρeiϕ f(n)] N i c ϕ Re(N) Integration only up to ρ max due to the factor exp[ρ ln(1/x) cosϕ] Parameters ϕ, c and number of moments N i can be tuned to get better accuracy Mellin transform of coefficient functions and parton distribution functions with an analytic dependence on N C must exist Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 19/3

20 Mellin Transforms Coefficient functions MT s up to next-to-next-to leading order known [Vermaseren 1998; Blümlein, Ravindran 25] Harmonic polylogarithms harmonic sums [Vermaseren - harmpol.h (2), summer.h (1999)] Analytic continuations of harmonic sums in terms of polygamma functions PDFs Choose a PDF set and parametrize it at the initial scale µ 2 few GeV 2 xf(x, µ 2 ) x a (1 x) b β(n + a, b + 1) }{{} Euler beta function Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 2/3

21 The Calculation - Toy PDFs "N-space" "x-space" Mellin Inverse (c++) Cross section QCD-Pegasus(FORTRAN) Coefficients(FORM) PDFs QCD-Pegasus [Vogt 24] Coefficient functions - transformation of x space results using using harmpol.h, summer.h [ Hamberg, Matsuura, van Neerven 199; Vermaseren 1998] - direct extraction of N space coefficients [Blümlein, Ravindran 25] N space integration using 32 point Gaussian quadrature Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 21/3

22 QCD Pegasus FORTRAN package performing evolution of polarized and unpolarized evolution of PDFs up to NNLO using Mellin space technique [Vogt 24] Output in both N space and x space Two possible input parametrizations xf i (x, µ 2 ) = N i p i,1 x p i,2 (1 x) p i,3 (1 + p i,5 x p i,4 + p i,6 x) xf i (x, µ 2 ) = N i p i,1 x p i,2 (1 x) p i,3 (1 + p i,4 x.5 + p i,5 x + p i,6 x 1.5 ) Defalut values of p i correspond roughly to the cteq5m PDF set Choice of µ f µ r VFNS vs FFNS avogt/pegasus.html Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 22/3

23 The Calculation - Checks (Toy PDFs) "x-space" Integrate (CUBA) Cross section QCD-Pegasus(FORTRAN) Coefficients(FORTRAN) PDFs QCD-Pegasus (FORTRAN) [Vogt 24] Coefficient functions in x space [ Hamberg, Matsuura, van Neerven 199] x space integration - CUHRE algorithm using CUBA library [Hahn 25] Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 23/3

24 Comparison N space vs x space, toy PDFs 1.5 xspace vs nspace, toy pdfs lo nlo [%] -.5 [%, smaller int.error] lo nlo x Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 24/3

25 Calculation - Checks, LHAPDF "x-space" Integrate (CUBA) Cross section LHAPDF Coefficients(FORTRAN) PDFs LHAPDF [HepForge] Comparison - MCFM [Campbell, Ellis] - DYNNLO [Catani, Cieri, Ferrera, de Florian, Grazzini 29; Catani, Grazzini 27] Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 25/3

26 Comparison of x space with MCFM 5e+7 4.5e+7 xspace mcfm 4e+7 W + (nlo, cteq5m), zero width approx 3.5e+7 3e+7 σ [fb] 2.5e+7 2e+7 1.5e+7 1e+7 5e+6 [%].8.6 RelErr w.r.t. MCFM e x Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 26/3

27 Discussion Speed (no optimization of either code so far) Ratio of standard x-space calculation (σ(x)) vs Mellin space approach (σ N (x)) using LHAPDF grid for x-space, required relative error on CUBA integration 1 3. x = 1 2 σ(x)/σ N (x) 1 2 x = 1 4 σ(x)/σ N (x) 1 3 Accuracy relative error of nspace and xspace in the error on x space integration No need of special care of plus distributions in N space Coefficient functions in N space for other processes (DIS, Higgs production) are availiable PDFs need improvement Restricted by the functional form Using different PDF sets requires user to do own parametrization, functions with more parameters may be necessary Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 27/3

28 Summary and Outlook Higher order corrections for Drell-Yan process are necessary for precise predictions Calculation in Mellin space is fast and accurate NNLO DY, possible parametrization Deep Inelastic Scattering Release the code Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 28/3

29 Backup The plus distribution (1 x) 1 2ǫ = δ(1 x) 2ǫ + 1 [ ln(1 x) ] 2ǫ (1 x) + 1 x + + O(ǫ2 ) Analytic continuation of a Harmonic Sum S k (N) = ( 1)k+1 (k 1)! ψk 1 (N + 1) + ζ(k) (1) Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 29/3

30 The Partonic Cross Section Leading order W exchange d ˆσ W q k q l l l d Q 2 = Γ W l ν πα M W sinθ 2 W N c (Q 2 MW 2 )2 + MW 2 V Γ2 qk q l 2 δ(1 x) W }{{}}{{}}{{} C W () N W qq (x) () qq (x) = δ(1 x) Higher Order Corrections to the Drell-Yan Cross Section in the Mellin Space p. 3/3