Theoretical developments Transversity workshop Trento, 17/06/2004 Theoretical developments piet mulders pjg.mulders@few.vu.nl
Content Spin structure & transversity Transverse momenta & azimuthal asymmetries T-odd phenomena & single spin asymmetries
Collinear hard processes, e.g. DIS Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators DF’s are quark densities that are directly linked to lightcone wave functions squared Perturbative QCD evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)
Leading order DIS A+ In limit of large Q2 the result of ‘handbag diagram’ survives … + contributions from A+ gluons ensuring color gauge invariance A+ gluons gauge link Ellis, Furmanski, Petronzio Efremov, Radyushkin A+
Collinear hard processes, e.g. DIS Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators DF’s are quark densities that are directly linked to lightcone wave functions squared Perturbative QCD evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)
Parametrization of lightcone correlator M/P+ parts appear as M/Q terms in s T-odd part vanishes for distributions but is important for fragmentation Jaffe & Ji NP B 375 (1992) 527 Jaffe & Ji PRL 71 (1993) 2547 leading part
Collinear hard processes, e.g. DIS Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators DF’s are quark densities that are directly linked to lightcone wave functions squared Perturbative QCD evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)
Matrix representation for M = [F(x)g+]T Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712 Related to the helicity formalism Anselmino et al. Off-diagonal elements (RL or LR) are chiral-odd functions Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY
Collinear hard processes, e.g. DIS Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators DF’s are quark densities that are directly linked to lightcone wave functions squared Perturbative QCD evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)
Non-collinear processes, e.g. SIDIS Relevant in electroweak processes with two hadrons (SIDIS, DY) Beyond just extending DIS by tagging quarks … Transverse momenta of partons become relevant, appearing in azimuthal asymmetries DF’s and FF’s depend on two variables, F(x,pT) and D(z,kT) Gauge link structure is process dependent ( []) pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance This allows T-odd functions h1^ and f1T^ (H1^ and D1T^) appearing in single spin asymmetries
? ? Leading order SIDIS In limit of large Q2 only result of ‘handbag diagram’ survives Isolating parts encoding soft physics ? ?
Lightfront correlators Collins & Soper NP B 194 (1982) 445 no T-constraint T|Ph,X>out = |Ph,X>in Jaffe & Ji, PRL 71 (1993) 2547; PRD 57 (1998) 3057
Non-collinear processes, e.g. SIDIS Relevant in electroweak processes with two hadrons (SIDIS, DY) Beyond just extending DIS by tagging quarks … Transverse momenta of partons become relevant, appearing in azimuthal asymmetries DF’s and FF’s depend on two variables, F[](x,pT) and D[](z,kT) Gauge link structure is process dependent ( []) pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance This allows T-odd functions h1^ and f1T^ (H1^ and D1T^) appearing in single spin asymmetries
including the gauge link (in SIDIS) Distribution including the gauge link (in SIDIS) A+ One needs also AT G+a = +ATa ATa(x)= ATa(∞) + dh G+a Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M, Pijlman, hep-ph/0303034 From <y(0)AT()y(x)> m.e.
including the gauge link (in SIDIS or DY) Distribution including the gauge link (in SIDIS or DY) A+ SIDIS A+ DY SIDIS F[-] DY F[+]
Non-collinear processes, e.g. SIDIS Relevant in electroweak processes with two hadrons (SIDIS, DY) Beyond just extending DIS by tagging quarks … Transverse momenta of partons become relevant, appearing in azimuthal asymmetries DF’s and FF’s depend on two variables, F[](x,pT) and D[](z,kT) Gauge link structure is process dependent ( []) pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance This allows T-odd functions h1^ and f1T^ (H1^ and D1T^) appearing in single spin asymmetries
Parametrization of F(x,pT) Link dependence allows also T-odd distribution functions since T U[0,] T = U[0,-] Functions h1^ and f1T^ (Sivers) nonzero! These functions (of course) exist as fragmentation functions (no T-symmetry) H1^ (Collins) and D1T^
Interpretation unpolarized quark distribution need pT T-odd helicity or chirality distribution need pT T-odd need pT transverse spin distr. or transversity need pT need pT
Matrix representation for M = [F[±](x,pT)g+]T pT-dependent functions T-odd: g1T g1T – i f1T^ and h1L^ h1L^ + i h1^ Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712
pT-dependent DF’s twist structure For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t Transverse moments Fa(x,pT) d2pT pTa F(x,pT) project out the parts in F[](x,pT) proportional to pT . They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon) Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link Factorization of explicit pT-dependent functions requires ‘soft factors’ Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTa f at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]
Difference between F[+] and F[-] upon integration Back to the lightcone integrated quark distributions transverse moments measured in azimuthal asymmetries ±
pT-dependent DF’s twist structure For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t Transverse moments Fa(x,pT) d2pT pTa F(x,pT) project out the parts in F[](x,pT) proportional to pT . They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon) Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link Factorization of explicit pT-dependent functions requires ‘soft factors’ Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTa f at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]
Difference between F[+] and F[-] upon integration In momentum space: gluonic pole m.e. (T-odd)
pT-dependent DF’s twist structure For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t Transverse moments Fa(x,pT) d2pT pTa F(x,pT) project out the parts in F[](x,pT) proportional to pT . They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon) Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link Factorization of explicit pT-dependent functions requires ‘soft factors’ Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTa f at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]
T-odd phenomena T-odd phenomena appear in single spin asymmetries T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links Contributions in other hard processes, such as pp pX involving three hadrons require a careful analysis
T-odd single spin asymmetry structure Wmn(q;P,S;Ph,Sh) = -Wnm(-q;P,S;Ph,Sh) Wmn(q;P,S;Ph,Sh) = Wnm(q;P,S;Ph,Sh) Wmn(q;P,S;Ph,Sh) = Wmn(q;P, -S;Ph, -Sh) Wmn(q;P,S;Ph,Sh) = Wmn(q;P,S;Ph,Sh) _ hermiticity * parity time reversal * with time reversal constraint only even-spin asymmetries the time reversal constraint cannot be applied in DY or in 1-particle inclusive DIS or e+e- In those cases single spin asymmetries can be used to select T-odd quantities
T-odd phenomena T-odd phenomena appear in single spin asymmetries T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links Contributions in other hard processes, such as pp pX involving three hadrons require a careful analysis
Time reversal constraints for distribution functions T-odd (imaginary) Time reversal: F[+](x,pT) F[-](x,pT) pFG F[+] F T-even (real) Conclusion: T-odd effects in SIDIS and DY have opposite signs F[-]
T-odd phenomena T-odd phenomena appear in single spin asymmetries T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links Contributions in other hard processes, such as pp pX involving three hadrons require a careful analysis
Time reversal constraints for fragmentation functions T-odd (imaginary) Time reversal: D[+]out(z,pT) D[-]in(z,pT) pDG D[+] D T-even (real) D[-]
Time reversal constraints for fragmentation functions T-odd (imaginary) Time reversal: D[+]out(z,pT) D[-]in(z,pT) D[+]out pDG out D out T-even (real) D[-]out Conclusion: T-odd effects in SIDIS and e+e- are not related
T-odd phenomena T-odd phenomena appear in single spin asymmetries T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links Contributions in other hard processes, such as pp pX involving three hadrons require a careful analysis
other hard processes qq-scattering as hard subprocess insertions of gluons collinear with parton 1 are possible at many places this leads for ‘external’ parton fields to gauge link to lightcone infinity
other hard processes qq-scattering as hard subprocess insertions of gluons collinear with parton 1 are possible at many places this leads for ‘external’ parton fields to gauge link to lightcone infinity The correlator F(x,pT) enters for each contributing term in squared amplitude with specific link The link may enhance the effect of the gluonic pole contribution involving also specific color factors