Quasi-PDFs and Pseudo-PDFs

Transcription

1 s and s A.V. Radyushkin Physics Department, Old Dominion University & Theory Center, Jefferson Lab 2017 May 23, 2017

2 Densities and Experimentally, one works with hadrons Theoretically, we work with quarks Hadron to Transition p ẓ 0. M( (pz), z 2 ) Can be described in momentum or coordinate space Concept of PDFs does not rely on spin complications p φ(0)φ(z) p =M( (pz), z 2 ) p Lorentz: p φ(0)φ(z) p depends on z through (pz) and z 2

3 ẓ 0. M( (pz), z 2 ) p p P(x, z 2 ): Fourier transform with respect to (pz) M( (pz), z 2 ) = dx e ix(pz) P(x, z 2 ) Should be valid in general for a very wide class of functions Non-trivial: limits of integration over x Support region is dictated by properties of Feynman diagrams Is determined by denominators of propagators Not affected by numerators present in non-scalar theories

4 distribution ẓ 0. p M( (pz), z 2 ) = M( (pz), z 2 ) p dx e ix(pz) P(x, z 2 ) (pz) is Ioffe time [(pz) = Mz 0 in rest frame p = {M, 0, 0, 0}] M(, z 2 ) is distribution Inverse transformation M(, z 2 ) = P(x, z 2 ) = 1 2π dx e ix P(x, z 2 ) d e ix M(, z 2 ) We do not need here z 2 = 0 or p 2 = 0

5 Collinear Distribution ẓ 0. p M( (pz), z 2 ) Take light-like z = z : collinear parton distribution f(x) = P(x, 0) M( p +z, 0) = p dx f(x) e ixp +z Usual interpretation: parton carries fraction x of hadron p + z 2 0 nontrivial in, since M(, z 2 ) has ln z 2 singularities Reflect perturbative evolution of parton densities Within OPE, ln z 2 singularities are subtracted, e.g., by dimensional renormalization ln(1/z 2 ) ln µ 2 Resulting PDFs depend on renormalization scale µ, f(x) f(x, µ 2 ) In P(x, z 2 ) pseudo-pdfs, 1/z 2 serves as cut-off scale

6 s and quasi-pdfs Treat target momentum p as longitudinal p = (E, 0, P ) Take z with z and z = {z 1, z 2} components (z + = 0), then (pz) = p +z ; define M(, z 2 ) = dx e ix d 2 k e i(k z ) F(x, k ) 2 carries xp + and has transverse momentum k Rotational invariance in z plane: this depends on k 2 only Take z = {0, 0, 0, z 3}, define p φ(0)φ(z 3) p M(P z }{{} 3, z3 2 ) = }{{} 2 /P 2 Inverse Fourier transformation Q(y, P ) = 1 2π dy e iyp z 3 Q(y, P ) d e iy M(, 2 /P 2 ) Q(y, P ) tends to f(y) in P limit, as far as M(, 2 /P 2 ) M(, 0).

7 s vs s z 3 P 1 P 2 s Q(y, P ): integration of M(, z3) 2 over z 3 = /P lines Becomes horizontal z 3 = 0 line in P limit PDF Q(y, P ) has perturbative evolution wrt P for large P Support region < y < Psedo-PDFs: integration of M(, z3) 2 over z 3 =const lines Always has x 1 support P 3 P(x, z 2 3) has perturbative evolution wrt 1/z 3 for small z 3 PDF f(x, C 2 /z 2 3) for small z 3 C = matching coefficient, C MS = 2e γ E 1.12

8 between quasi-pdfs and s Write definition of quasi-pdf Q(y, P ) = 1 2π Write definition of s M(, z z 2 2) = d e iy M(, 2 /P 2 ) dx e ix dk 1dk 2 e ik 1z 1 ik 2 z 2 F(x, k1 2 + k2) 2 Take z 1 = 0, z 2 = /P and combine expressions to get Q(y, P )/P = dx Introduce momentum in k 3 yp R(k 3, P ) = Q(k 3/P, P )/P = dk 1F(x, k (y x) 2 P 2 ) R(x, k 3) is F(x, κ 2 ) integrated over k 1 R(x, k 3) dx R(x, k 3 xp ) dk 1F(x, k k 2 3)

9 Convolution nature of quasi-pdfs R(k 3, P ) = Q(k 3/P, P )/P = Take hadron at rest, p = {M, 0, 0, 0} R(k 3, P = 0) r(k 3) = 1D distribution obtained from density M(0, z3) 2 = p φ(0)φ(z 3) p p=0 M(0, z 2 3) = dx R(x, k 3 xp ) dx R(x, k 3) dk 3 r(k 3) e ik 3z 3 r(k 3) = primordial distribution of k 3 in rest frame In moving hadron, parton momentum k 3 = xp + (k 3 xp ) comes a) from motion of hadron as a whole (part xp ) governed by x-dependence of F(x, κ 2 ) b) remaining part k 3 xp governed by κ 2 dependence of reflecting primordial rest-frame distribution

10 Factorized Two sources of k 3 look independent Try factorized model R fact (x, k 3 xp ) = f(x)r(k 3 xp ) (x integral of f(x) is normalized to 1) For original M(, z 2 ) function, this Ansatz corresponds to M fact (, z 2 ) = M(, 0)M(0, z 2 ) Popular idea: Gaussian dependence of on k. Gives r G(k 3) = 1 πλ e k2 3 /Λ2, density : r G(z 3 3) = e z2 3 Λ2 /4 Factorized Gaussian model for momentum distribution R fact G (k 3, P ) = 1 Λ π dx f(x) e (k 3 xp ) 2 /Λ 2

11 for Gaussian model Take simple PDF f(x) = 4(1 x) 3 θ(0 x 1) resembling valence quark R(k, P) P = Λ P = 10Λ P = 50Λ k/λ Changes from a Gaussian shape (for small P ) to a shape resembling stretched PDF (for large P )

12 Small Momenta P Small-P approximation ( x = average x, in our model x = 0.2) R(k 3, P ) = R(k, P) P =0 P = dx f(x) r(k 3 xp ) r(k 3 xp ) P =2 P =5 P = k/

13 Large Momenta P For large P r G(k 3 xp ) = 1 e (k 3 xp ) 2 /Λ 2 1 δ(x k3/p ) πλ P Combination P R(k 3, P ) in large P limit converts into f(k 3/P ) = f(y) For finite P, we have P R(k 3, P ) = Q(y = k 3/P, P ) Q(y, P) P = 50Λ P = 10Λ P = Λ y

14 case in M α (z, p) p ψ(0) γ α Ê(0, z; A)ψ(z) p with standard 0 z straight-line gauge link Ê(0, z; A) Decompose into p α and z α parts M α (z, p) = 2p α M p( (zp), z 2 ) + z α M z( (zp), z 2 ) For : take z = (z, z ) and α = + z α -part drops out F(x, k 2 ) is related to M p(, z 2 ) by scalar formula For quasi- PDF: take time component of M α (z = z 3, p) and define M 0 (z 3, p) = 2p 0 dy Q(y, P ) e iyp z 3 Q(y, P ) is related to F(x, k 2 ) by scalar formula

15 s and z 3 P 1 P 2 s Q(y, P ): integration of M(, z 2 3) over z 3 = /P lines Have x-convolution structure even if M(, z 2 3) factorizes, i.e., M(, z 2 3) = M(, 0)M(0, z 2 3) Fit M(P z 3, z 2 3) by M(, z 2 3) (K. Orginos) and take reduced function P 3 M(, z 2 3) M(, z2 3) M(0, z 2 3 ) In factorized case, gives M(, 0) take its Fourier transform to get PDF f(x) Bonus: z 2 3-dependence due to self-energy of gauge link cancels in ratio (K. Orginos)

16 of LO equation (Braun et al. 1994) for distribution d M(, z3) 2 = αs d ln z3 2 2π CF du B(u)M(u, z3) 2 0 Nonsinglet evolution kernel [ ] 1 + u 2 B(u) = 1 u Valence f(x) = 4(1 x) 3 corresponds to M(, 0) = 12 [ 2 4 sin 2 (/2) ] / M(, 0) B M( ) -1.0 No perturbative evolution for M(0, z 2 3) [vector current is conserved]

17 , evolution In reality: M(, z 2 3) will have residual z 2 3-dependence from a) perturbative evolution visible as ln(1/z 2 3Λ 2 ) spike for small z 2 3 Take for illustration P soft (x, z 2 3) = f(x)e z2 3 Λ2 /4 (corresponding to F soft (x, k 2 ) = f(x)e k2 /Λ2 /πλ 2 ) and α s/π = 0.1 for hard part Γ[0, z 2 3Λ 2 /4] M(,z 2 3) =1 =5 = 10 = z 3 If Λ = 300MeV we have z 3Λ = 1.5 for z 3 = 1 fm

18 , soft M(, z 2 3) may also have residual z 2 3-dependence from b) violation of factorization for soft part Take for illustration P soft (x, z 2 3) = f(x)e x(1 x)z2 3 Λ 2 /4 F soft (x, k 2 ) = f(x)e k2 /x(1 x) Λ 2 /[πx(1 x) Λ 2 ]) To have the same k 2 we need Λ 2 = 15 2 Λ2 M(,z 2 3) =1 =5 z 2 3Λ 2 = 2.25 for z 3 = 1 fm = 10 = z 2 3 2

19 s are hybrids of PDFs and primordial rest-frame momentum Complicated convolution nature of quasi-pdfs necessitates p 3 3 GeV to wipe out primordial effects Alternative approach is to use pseudo-pdfs P(x, z 2 3) related by Fourier transform to M(, z 2 3) s have same x 1 support as PDFs Their z 2 3-dependence for small z 2 3 is governed by a usual evolution equation Using ratio M(, z 2 3)/M(0, z 2 3) of one divides out z 2 3-dependence of primordial rest-frame distribution Ratio excludes z 2 3-dependence coming from gauge link self-energy corrections

20 Primordial s operator O α (0, z; A) involves straight-line link Our differs from stapled-link s used in Drell-Yan and SIDIS processes Stapled links reflect initial or final state interactions inherent in these processes The straight-link s describe structure of a hadron in non-disturbed or primordial state Unlikely that such a can be measured in a scattering experiment Still, it is a well-defined quantum field theory object Hopefully can be measured on the lattice through its connection to pseudo-pdfs