Higher moments of PDFs in lattice QCD. William Detmold The College of William and Mary & Thomas Jefferson National Accelerator Facility

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1 Higher moments of PDFs in lattice QCD William Detmold The College of William and Mary & Thomas Jefferson National Accelerator Facility

2 Lattice QCD and hadron structure The problem with higher moments (~ high x) Possible avenues for progress New results

3 Hadron structure in LQCD Lattice techniques are the only (known) way to access non-perturbative hadron structure from QCD LQCD formulated in Euclidean space to use importance sampling for functional integrals: SO(3,1) O(4) Light cone physics is not apparent Example: unpolarised pdf q(x, µ) =p O q γ(x) p = dz 2π eizx p q z 2 n n γ U [ z 2 n, z 2 n] q z 2 n p Wilson operator product expansion to the rescue U [ z 2 n, z 2 n] = Pexp ig z/2 z/2 n A(z n)dz

4 Hadron structure in LQCD OPE for the case in question q(x, µ) =p O q γ(x) p = dz 2π eizx p p qγ {µ 0 D µ 1...D µ n} q p = x n q [p µ 0...p µ n traces ] x n q = 1 1 dx xn q(x) Non-perturbative matrix elements can be calculated Determine Mellin moments of PDFs q z 2 n n γ U [ z 2 n, z 2 n] q z 2 n p

5 Lattice technology Measuring hadronic matrix elements has a long history [Martinelli & Sachrajda 86] Measure two- and three-point correlations on an ensemble of gauge configurations operator operator connected disconnected Ratio C3/C2 proportional to lattice matrix element

6 Lattice technology Connect to experimental measurements: bare matrix element continuum renormalisation scheme O cont i = Z i O lat i + j Z ij O lat j Sum over all operators with right quantum numbers Simple cases: just a multiplicative scaling Operator mixing also possible: e.g. flavour singlet operators mix with gluon operators Renormalisation constants can be calculated perturbatively ( ) or non-perturbatively ( )

7 Overview of lattice results [See Dru Renner s talk yesterday] QCDSF, LHP, ETMC and RBCK,... collaborations Lowest three moments of q(x), Δq(x), δq(x) Also calculate form factors, moments of GPDs, and meson and baryon distribution amplitudes Continuum, chiral and infinite volume extrapolations getting sophisticated Only isovector moments calculated rigourously : flavour singlet moments require disconnected contractions Still some discrepancies

8 The problem at high xbj LQCD necessarily formulated on a discrete geometry, typically 4d hypercube: O(4) H(4) Operators classified by H(4) quantum numbers Finite number of irreducible representations Operator mixing: O cont i = Z i O lat i + j Z ij O lat j Allows (forces) mixing with operators of lower dimension Sum over all operators with right quantum numbers Coefficients scale with inverse powers of lattice cutoff Taking the continuum limit is difficult

9 Hypercubic group H(4) (aka W4, D4,...): finite group of rotations by π/2 and reflections H(4) = {(a, π) a Z 4 2, π S 4 } 20 irreducible representations

10 Nice Example Continuum operator O µν = qγ {µ D ν} q belongs to 1 2, , 1 2 =(0, 0) [(1, 0) (0, 1)] (1, 1) Hypercubic decomposition = Lattice operators (symmetric traceless): O 14 + O 41, O (O 11 + O 22 + O 33 ) Have same continuum limit (63 requires p 0) No operators of lower dimension ( )

11 Nasty example Continuum operator 1 2, , , 1 2 O {µνρ} = qγ {µ D ν D ρ} q lives in =4 1 2, , , , 2 3 Hypercubic decomposition = Lattice operators: O 111, O {123}, O {441} 1 2 (O {221} + O {331} ) Same continuum limit but O 111 mixes with qγ 1 q 4 1 and the coefficient absorbs the missing dimensions ( ) Always the case for all n > 4 operators

12 Moments PDFs Well defined problem: inverse Mellin transform Requires all integer moments How useful are just 3 moments? Fit parametric form for PDF Standard parameterisations have ~5 parameters for a given PDF ( ) NB: 3 moments is 2 of qv and 1 of q+q A different approach is needed

13 Options Possible methods to bypass this problem Alternate discretisations Brute force OPE without OPE Hamiltonian/transverse lattice approaches (PDFs directly) [Dalley & Burkardt; Grünewald, Ilgenfritz & Pirner; Vary et al.]??

14 Alternate discretisations Large-scale LQCD universally formulated on a hypercubic geometry 4D fcc geometry (F4) has larger symmetry Some higher moments would be accessible Entirely new simulations (and algorithms) needed Problems with fermions

15 Brute force approach The generic lattice mixing problem O cont i = Z i O lat i + j Z ij O lat j where Zij ~ a -m (m>0) may not be insurmountable Precise calculations of large set of operators at a range of a could work Examples: kaon weak matrix elements, d2 [QCDSF] Gets worse as n increases and hopeless for GPDs

16 OPE in Euclidean space [WD, CJD Lin, Phys.Rev. D73 (2006) ] 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] [see also Braun & Mueller Eur.Phys.J. C55 (2008) 349]

17 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 Ψ Effects in perturbative Wilson coefficients

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

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

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

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

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

23 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 Operator product expansion: heavy propagator i (id ) + q/ + m Ψ i (id ) + q/ + m Ψ ψ (id ψ = ψ + q) 2 + m 2 Ψ Q 2 + D 2 m 2 Ψ Denominator n=0 ( 2i q D Q 2 + D 2 m 2 Ψ Q 2 = Q 2 M 2 Ψ + αm Ψ + β ) n ψ Higher twists? Heavy-light meson mass Binding energy ~Λ Higher twists ~Λ 2

24 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 Operator product expansion: heavy propagator i (id ) + q/ + m Ψ i (id ) + q/ + m Ψ ψ (id ψ = ψ + q) 2 + m 2 Ψ Q 2 + D 2 m 2 Ψ Denominator n=0 ( 2i q D Q 2 + D 2 m 2 Ψ Q 2 = Q 2 M 2 Ψ + αm Ψ + β n ) n ψ Higher twists? Heavy-light meson mass Binding energy ~Λ Higher twists ~Λ 2

25 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 = imân ψ(µ 2 )[p µ 1...p µ n traces]

26 Heavy quark DIS T {µν} Ψ,ψ = i +4 p{µ q ν} p q PDF moments [ + qµ q ν q 2 Leads to n=2,even { [ 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 where n=0,even Wilson coefficients ζ = 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) + C nc n (1) (η) n 2 (η) p2 q 2 Q 2 η = p q p2 q 2 Gegenbauer polynomials 2C n Gegenbauer polynomials [ from target mass corrections: powers of p 2 /q 2] j ] [ + pµ p ν Q2 8η 2 (p q) 2 C C (3) n 2 (η) n n(n 1) ( C (1) n ] (η) 2 η ) ]} n C(2) n 1 (η)

27 Heavy quark DIS... in a more comprehensible form (target rest frame) T {34} Ψ,ψ (p, q) = n=2,even A n ψ(µ)f(n) where f(n) is completely known: { f(n) = q0 2 Q2 ζ n 2 q 0 [ 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

28 Lattice details Scale hierarchy: Λ QCD Q, m Ψ a 1 Naively 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

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

30 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

31 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 [see also Braun & Mueller Eur.Phys.J. C55 (2008) 349]

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

33 Summary Information about high x region is difficult to obtain from LQCD but not impossible Some possible avenues for progress All are significantly more computationally expensive than the current methods for low moments ( )

34 [FIN]

35 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 Depend on n since non-abelian Can be include in lattice analysis ) n ψ...q µ1...q µ3ψ D µ 1 D 4 D µ 2 D µ 3 D 2 ψ... [ (Λ 2 O QCD /Q 2) ] j Possible to use a free fermion field: no HT

36 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

37 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) =4Ma τ e iq 4τ {µν} G t lim (4) (p, q,t,τ;γ 4 ) G (2) (p,t;γ 4 )

38 F4 lattice