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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.

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theoretical developments

Transversity workshop

Trento, 17/06/2004

Theoretical developments

piet mulders

pjg.mulders@few.vu.nl

content
Content
  • Spin structure & transversity
  • Transverse momenta & azimuthal asymmetries
  • T-odd phenomena & single spin asymmetries
collinear hard processes e g dis
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+

Ellis, Furmanski, Petronzio

Efremov, Radyushkin

A+ gluons

 gauge link

Leadingorder DIS
  • In limit of large Q2 the result

of ‘handbag diagram’ survives

  • … + contributions from A+ gluons

ensuring color gauge invariance

collinear hard processes e g dis1
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

leading part

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

collinear hard processes e g dis2
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

Matrix representationfor M = [F(x)g+]T

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 dis3
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
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
Leading order SIDIS
  • In limit of large Q2 only result

of ‘handbag diagram’ survives

  • Isolating parts encoding soft physics

?

?

lightfront correlators
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 sidis1
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
distribution
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.

distribution1
Distribution

including the gauge link (in SIDIS or DY)

A+

SIDIS

A+

DY

SIDIS F[-]

DY F[+]

non collinear processes e g sidis2
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 p t
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
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 p t g t

pT-dependent

functions

Matrix representationfor M = [F[±](x,pT)g+]T

T-odd: g1T g1T – i f1T^ and h1L^  h1L^ + i h1^

Bacchetta, Boglione, Henneman & Mulders

PRL 85 (2000) 712

p t dependent df s twist structure
pT-dependent DF’stwist 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 momentsFa(x,pT)   d2pTpTa 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 pTaf 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
Difference between F[+] and F[-] upon integration

Back to the lightcone

integrated quark

distributions

transverse

moments

measured in azimuthal asymmetries

±

p t dependent df s twist structure1
pT-dependent DF’stwist 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 momentsFa(x,pT)   d2pTpTa 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 pTaf 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 integration1
Difference between F[+] and F[-] upon integration

In momentum space:

gluonic pole m.e.

(T-odd)

p t dependent df s twist structure2
pT-dependent DF’stwist 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 momentsFa(x,pT)   d2pTpTa 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 pTaf 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
  • 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

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)

_

_

_

_

_

_

_

_

_

_

_

_

T-oddsingle spin asymmetry

symmetry

structure

hermiticity

*

parity

  • 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

time

reversal

*

t odd phenomena1
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
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 phenomena2
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
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 functions1
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 phenomena3
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
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 processes1
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