Deep inelastic scattering and the OPE in lattice QCD

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1 Deep inelastic scattering and the OPE in lattice QCD William Detmold [WD & CJD Lin PRD 73, (2006)]

2 DIS structure of hadrons Deep-inelastic scattering process critical to development of QCD k, E k, E q X q2 x = 2 p q p Momentum fraction carried by parton Hadron structure parameterised through structre functions

3 Deep inelastic scattering Cross-section: dσ dωde = e 4 16π 2 Q 4 l µνw µν (p, q) Lepton tensor l µν well known Hadron tensor related to forward Compton scattering by optical theorem W µν = 1 2π Im[T µν ] Contains information on target structure

4 DIS on spin-1/2 targets W µν = F 1 g µν + F 2 p µ p ν ν + i ν g 1ɛ µνλσ q λ s σ + i ν 2 g 2ɛ µνλσ q λ (p qs σ s qp σ ) Four structure functions:, F 1,2 g 1,2 Depend on Bjorken x and Q 2 g 1,2 require polarised beam and target Terms q µ dropped since q µ l µν = q ν l µν = 0

5 Parton distributions Factorisation theorem: F (x, Q 2 ) = i ( ) x dz C i z, Q2 µ 2, α s(µ 2 ) f i (z, α s (µ 2 )) Wilson coefficeint (perturbative) Parton distributions: Parton distribution (non-perturbative) u(x), s(x), δd(x), g(x) Universal: DIS, Drell-Yan, W +/- production Can be fitted from experimental data (MRST, GRV, CTEQ,...) Lattice: insight, information on unmeasured PDFs

6 Operator product expansion Wilsonian (light-cone) operator product expansion: O i Compton tensor expanded in matrix elements of local operators Classify by twist = dimension -spin Near Bjorken limit, twist-2 operators dominate Higher twists supressed by powers of Q 2 i

7 Hadron structure in Lattice QCD Lattice techniques can investigate light-cone hadron structure from QCD Two difficulties: Euclidean space Reduced symmetry: O(4) H(4) Focused on calculating matrix elements of twist-2 operators Mellin moments of PDFs x n q = 1 1 dx xn q(x)

8 Local matrix element Matrix elements of twist-two operators p qγ {µ1 D µ2... D µn }q p = x n q p µ1... p µn Lattice calculations via ratio of correlators x n q

9 Local matrix element Matrix elements of twist-two operators p qγ {µ1 D µ2... D µn }q p = x n q p µ1... p µn Lattice calculations via ratio of correlators Limited to low moments by power-divergent operator mixing Choose safe irreps of H(4) only for n 4 Renormalisation becomes extremely complex for large n (perturbative and non-perturbative)

10 Summary of lattice results QCDSF, LHP and RBCK,... collaborations Matrix elements in proton, pion, rho Lowest three moments of q(x), Δq(x), δq(x) Also calculate moments of GPDs Chiral and infinite volume extrapolations Only isovector moments accessible - isosinglet MEs have disconnected contractions: all-to-all propagators or external fields [hep-lat/ ]

11 Example: axial charge g A = x 0 u d simplest twist-two operator hep-lat/ Unquenched data

12 Moments PDFs Well defined problem: inverse Mellin transform Rigourously requires all integer moments! How useful are just 3 moments? Fit parametric form for PDF But: standard PDF parameterisations have 6 or more parameters A different approach is needed

13 OPE in Euclidean space OPE of Compton tensor can be studied directly in Euclidean space 1. Calculate current-current commutator 2. Extrapolate to continuum - restore O(4) 3. Match to Euclidean OPE to extract matrix elements of local operators Determines the same moments as in Minkowski space. Analytic continuation is in Wilson coefficients Still a very difficult calculation [Schierholz 99]

14 Fictitious heavy quarks Consider Compton tensor for currents that couple light quarks to heavy fictitious quark No disconnected contractions practical Heavy quark mass acts like photon virtuality to suppress higher twists Heavy quark integrated out after OPE Determine same PDF moments Effects in perturbative Wilson coefficients Ψ

15 Lattice correlators Quark contractions in Compton correlator Greatly simplified with heavy quark: no disconnected contributions in isovector case

16 Heavy quark Compton tensor Leading twist contributions Higher twist contributions Significantly reduced by heavy quark

17 Heavy quark DIS Compton tensor with heavy quark T µν (p, q) = S d 4 x e iq x p, S T [ ] J µ Ψ,ψ (x)j Ψ,ψ(0) ν p, S Heavy-light vector current J µ Ψ,ψ = ψγµ Ψ + ΨΓ µ ψ ψ i Operator product expansion: heavy propagator (id + q/ ) + m Ψ (i D + q) 2 + m 2 Ψ i (id ) + q/ + m Ψ ψ = ψ Q 2 + D 2 m 2 Ψ n=0 ( 2i q D Q 2 + D 2 m 2 Ψ ) n ψ Denominator Q 2 = Q 2 M 2 Ψ + αm Ψ + β n Higher twists? Heavy-light meson mass Higher twists ~Λ 2 Binding energy ~Λ

18 Higher twists in OPE Derivative squared generates towers of higher twist operators n=0 ψ ( 2i q D + D 2 Q 2 m 2 Ψ Corrections should be ) n ψ... q µ1... q µ3ψ D µ 1 D 4 D µ 2 D µ 3 D 2 ψ... O [ (Λ 2 QCD /Q 2) j ] Depend on n since non-abelian Can be include in lattice analysis Possible to use a free fermion field: no HT

19 Heavy quark DIS Usual twist-two operators Twist-3 scalar operators also contribute: e(x) Matrix elements S S O µ 1...µ n ψ = ψγ {µ 1 Ô µ 1...µ n ψ p, S O µ 1...µ n ψ p, S Ôµ 1...µ n ψ = ψ ( i D µ 2 ( i D µ 1 ) ) ( i D µ n} ) ψ traces ( i D µ n ) ψ traces p, S = A n ψ(µ 2 ) [p µ 1... p µ n traces] p, S = i M Ân ψ(µ 2 ) [p µ 1... p µ n traces]

20 T {µν} Ψ,ψ = i +4 p{µ q ν} p q PDF moments [ + qµ q ν q 2 Leads to n=2,even [ Heavy quark DIS { [ A n ψ(µ)ζ n δ µν Q2 nc n (1) (η) 2ηC (2) C n q 2 n(n 1) Q2 (n 1)ηC (2) n 1 C (η) 4η2 C (3) n q 2 n(n 1) C n Q2 q 2 n(n 2)C (1) 2i M(m Ψ m) Q 2 δ µν n=0,even Wilson coefficients n 2 (η) ] C n η n C(2) n 1 (η) n (η) 2η(2n 3)C (2) n 1 (η) + 8η2 C (3) n(n 1) Ĉ n  n ψ(µ)ζ n C n (1) (η) n 1 (η) Gegenbauer polynomials + C nc n (1) (η) ] n 2 (η) [ + pµ p ν Q2 8η 2 (p q) 2 C C (3) n 2 (η) n n(n 1) 2C n ( C (1) n ] (η) 2 η ) ] } n C(2) n 1 (η) where ζ = p2 q 2 Q 2 η = p q p2 q 2 Gegenbauer polynomials from target mass [ corrections: powers of p 2 /q 2] j Not small

21 Heavy quark DIS... or more comprehendibly (target rest frame) T {34} Ψ,ψ (p, q) = where f(n) is completely known: { f(n) = q0 2 Q2 ζ n 2 q 0 [ n=2,even A n ψ(µ)f(n) Q2 (n 1)ηC (2) n 1 C (η) 4η2 C (3) n Q 2 n(n 1) n 2 (η) Kinematic variables & Wilson coefficients +... Parameters α, β known perturbatively Fits to calculations of LHS A n ψ(µ), α, β

22 Lattice details Scale hierarchy: Λ QCD Q, m Ψ a 1 Need fine lattice spacings: a 1 5 GeV Heavy quark quenched: very cheap Use variety of masses Different q µ also easy Correlator analysis is straightforward Lattice renormalisation and matching is simple

23 Compton correlator G µν (4) (p, q, t, τ; Γ) = x,z = x,z y N,N e ip x e iq y Γ βα 0 χ α (x, t))j µ Ψ,ψ(y + z, τ + τ 2 )J ν Ψ,ψ(z, τ 2 )χ β(0, 0) 0 y s,s e i(p pn) x e i(pn pn ) z e iq y e (EN +EN ) τ 2 Γβα 0 χ α (0) E N, p N, s E N, p N, s J µ Ψ,ψ(y, τ)j ν Ψ,ψ(0) E N, p N, s E N, p N, s χ β (0) 0 t e E 0t y e iq y s,s Γ βα 0 χ α (0) E 0, p, s E 0, p, s J µ Ψ,ψ(y, τ)j ν Ψ,ψ(0) E 0, p, s E 0, p, s χ β (0) 0 G (2) (p, t; Γ) = x e ip x Γ βα 0 χ α (0)χ β (x, t) 0 T {µν} Ψ,ψ (p, q) = 4M a τ e iq 4τ {µν} G t lim (4) (p, q, t, τ; Γ 4 ) G (2) (p, t; Γ 4 )

24 Moment extraction Want to extract 6-8 moments Two scenarios for n dependence 0.6 Moments constant Moments falling f n 0 f n n M Ψ = 3.5 GeV, Q 2 = 1.5 GeV 2, q 0 = 2.8 GeV n M Ψ = 2.1 GeV, Q 2 = 3.9 GeV 2, q 0 = 2.0 GeV α = 0.4, 1.2 GeV, β = 0, M N = 1.2 GeV

25 Other observables Can consider correlator of vector/axial-vector currents (neutrino scattering): odd moments Can use unphysical currents Eg: scalar/vector correlator moments of transversity distribution Moments of GPDs also accessible Moments of meson distribution amplitudes

26 Pion distribution amplitude Distribution amplitudes (light-cone WFs) important for QCDF/SCET in hard processes Lattice can determine moments of meson distribution amplitudes: e.g. φ π (x) ξ n π = 1 0 ξn φ π (ξ)dξ Local ME method limited to one moment! OPE method can determine higher moments Not related to a physical process

27 Pion distribution amplitude Lattice calculations of the tensor S µν Ψ,ψ (p, q) = H-L vector current d 4 x e i q x π + (p) T [V µ Ψ,ψ (x)aν Ψ,ψ(0)] 0 After OPE determine same matrix elements as for distribution amplitude: H-L axial current π + (p) ψγ {µ 1 γ 5 (i D) µ 2... (i D) µ n} ψ 0 = f π ξ n 1 π [p µ 1... p µ n traces] Numerical work under construction Computationally similar to pion FF

28 OPE on the lattice Improve understanding of hadron structure from Euclidean lattice calculations Moments of parton distributions Moments of generalised parton distributions Moments of distribution amplitudes Use of OPE without OPE made practical for isovector distributions by heavy quark Exploratory studies feasible today

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30 Supplementary Slides

31 Gegenbauer polynomials C (λ) n (z) = n 2 k=0 ( 1) k Γ(n k + λ) k!(n 2k)!Γ(λ) (2z) n 2k z C(λ) n (z) = 2λC (λ+1) n 1 (z) C (λ) n (z) = 2(λ + n + 1)z 2λ + n C (λ) n+1 (z) n + 2 2λ + n C(λ) n+2 (z) Back

32 Target mass corrections Result from deviation of bound-state from light cone OPE basis constructed from operators of definite spin trace subtractions Eg: p µ 1 p µ 2 tr = p µ 1 p µ 2 p 2 g µ 1µ 2 p µ 1... p µ n tr = n 2 j=0 ( 1) j (n j)! 2 j n! M 2j g {µ 1µ 2... g µ 2j 1µ 2j p µ 2j+1... p µ n} Contraction with q μ s leads to Gegenbauers Back