Theoretical developments
description
Transcript of Theoretical developments
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 f1
q(x) = q(x), g1q(x) = q(x), h1
q(x) = q(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 s/pT
2 in the pT-integration)
Leading order DIS
• In limit of large Q2 the resultof ‘handbag diagram’ survives
• … + contributions from A+ gluonsensuring color gauge invariance
A+ gluons gauge link
Ellis, Furmanski, PetronzioEfremov, 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 f1
q(x) = q(x), g1q(x) = q(x), h1
q(x) = q(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 s/pT
2 in the pT-integration)
Parametrization of lightcone correlator
Jaffe & Ji NP B 375 (1992) 527Jaffe & Ji PRL 71 (1993) 2547
leading part
• M/P+ parts appear as M/Q terms in • T-odd part vanishes for distributions but is important for fragmentation
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 f1
q(x) = q(x), g1q(x) = q(x), h1
q(x) = q(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 s/pT
2 in the pT-integration)
Off-diagonal elements (RL or LR) are chiral-odd functions Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY
Matrix representationfor M = [(x)+]T
Related to thehelicity formalism
Anselmino et al.
Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712
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 f1
q(x) = q(x), g1q(x) = q(x), h1
q(x) = q(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 s/pT
2 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, (x,pT) and (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 resultof ‘handbag diagram’ survives
• Isolating parts encoding soft physics
? ?
Lightfront correlators
no T-constraintT|Ph,X>out = |Ph,X>in
Collins & SoperNP B 194 (1982) 445
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, [](x,pT) and [](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
From AT() m.e.
including the gauge link (in SIDIS)
A+
One needs also AT
G+ = +AT
AT()= AT
(∞) + d G+
Belitsky, Ji, Yuan, hep-ph/0208038Boer, M, Pijlman, hep-ph/0303034
Distribution
A+
A+
including the gauge link (in SIDIS or DY)
SIDIS
SIDIS [-]DY DY [+]
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, [](x,pT) and [](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 (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 quarkdistribution
helicity or chiralitydistribution
transverse spin distr.or transversity
need pT
need pT
need pT
need pT
need pT
T-odd
T-odd
pT-dependent functions
T-odd: g1T g1T – i f1T and h1L
h1L + i h1
Matrix representationfor M = [±](x,pT)+]T
Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712
pT-dependent DF’stwist structure
• For integrated correlator (x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE
• For unintegrated correlators [](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t
• Transverse moments (x,pT) d2pT pT
(x,pT) project out the parts in [](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 [](x) and
[](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
[](x) seems ok, but this seems not the case for subleading functions
in [](x) [so f1T
(1) ok but f(1) problematic: in cos h asymmetry pT f
at order 1/Q gets pQCD correction s f1/|pT|, which shows up as s f1 at order 1]
Difference between [+] and [-] upon integration
integrated quarkdistributions
transverse moments
measured in azimuthal asymmetries
±
Back to the lightcone
pT-dependent DF’stwist structure
• For integrated correlator (x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE
• For unintegrated correlators [](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t
• Transverse moments (x,pT) d2pT pT
(x,pT) project out the parts in [](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 [](x) and
[](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
[](x) seems ok, but this seems not the case for subleading functions
in [](x) [so f1T
(1) ok but f(1) problematic: in cos h asymmetry pT f
at order 1/Q gets pQCD correction s f1/|pT|, which shows up as s f1 at order 1]
Difference between [+] and [-] upon integration
gluonic pole m.e.(T-odd)
In momentum space:
pT-dependent DF’stwist structure
• For integrated correlator (x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE
• For unintegrated correlators [](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t
• Transverse moments (x,pT) d2pT pT
(x,pT) project out the parts in [](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 [](x) and
[](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
[](x) seems ok, but this seems not the case for subleading functions
in [](x) [so f1T
(1) ok but f(1) problematic: in cos h asymmetry pT f
at order 1/Q gets pQCD correction s f1/|pT|, which shows up as s 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 [](x) and
[](x) they have opposite signs
• T-odd parts for fragmentation functions in [](x) and
[](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 X involving three hadrons require a careful analysis
T-odd single spin asymmetry
• with time reversal constraint only even-spin asymmetries• the time reversal constraint cannot be applied in DY or in 1-
particle inclusive DIS or ee
• In those cases single spin asymmetries can be used to select T-odd quantities
*
*
W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)
W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)
W(q;P,S;Ph,Sh) = W(q;P, S;Ph, Sh)
W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)
_
___
_ ____
__ _time
reversal
symmetrystructure
parity
hermiticity
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 [](x) and
[](x) they have opposite signs
• T-odd parts for fragmentation functions in [](x) and
[](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 X involving three hadrons require a careful analysis
Time reversal constraints for distribution functions
Time reversal(x,pT) (x,pT)
G
T-even(real)
T-odd(imaginary)
Conclusion:T-odd effects in SIDIS and DY have opposite signs
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 [](x) and
[](x) they have opposite signs
• T-odd parts for fragmentation functions in [](x) and
[](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 X involving three hadrons require a careful analysis
Time reversal constraints for fragmentation functions
Time reversalout(z,pT)
in(z,pT)
G
T-even(real)
T-odd(imaginary)
Time reversal constraints for fragmentation functions
G out
out
out
out
T-even(real)
T-odd(imaginary)
Time reversalout(z,pT)
in(z,pT)
Conclusion:T-odd effects in SIDIS and ee 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 [](x) and
[](x) they have opposite signs
• T-odd parts for fragmentation functions in [](x) and
[](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 X 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 (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