Discussion on Higher Twist Questions and some answers

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Discussion on Higher TwistQuestions and (some)
answers
HIGHER TWIST ISSUES IN PVDIS
June 2009

• Piet Mulders

mulders_at_few.vu.nl
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Basic questions
INTRODUCTION

• At JLab energies consideration of higher twist is
  essential. What assumptions are reasonable?
• How interesting are higher twist contributions in
  itself?
• Issues
• Q-dependence
• Description at operator level
• Models

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General features
DATA

• Structure functions have x and Q-dependence
• Flavor dependence of twist 2 and twist 4 is in
  general different. This is calculable in
  multi-parton treatment.
• x-dependence of twist 2 and twist 4 is expected
  to differ. Semi-rigorous counting rules provide
  (1-x)-dependence.
• Data have 1/Q2 dependence and ln Q2 dependence.
  Forget the latter, probably 1/Q4 gives more
  problems.
• Magnitude (scale LHT) not constrained by
  fundamental principles. Expected to be small from
  QCD scaling arguments/models.

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From partons to OPE
OPERATORS

• Hard scattering process involves quarks and
  gluons. Link to external particles
• Confinement leads to hadrons as sources for
  quarks
• and source for quarks gluons
• and .

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From partons to OPE
QUARK-QUARK CORRELATORS

• Thus, the theoretical description/calculation
  for hard processes involves instead of
  ui(p)uj(p) forward matrix elements of the form

_
quark momentum
Operator combinations
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(non-)collinearity of parton correlators
(NON-)COLLINEARITY

• The cross section can be expressed in hard
  squared QCD-amplitudes and distribution and
  fragmentation functions entering in forward
  matrix elements of nonlocal combinations of quark
  and gluon field operators (f ? y or G). These are
  the (hopefully universal) objects we are after,
  useful in parametrizations and modelling.

Distribution functions
lightfront x 0
TMD
collinear
lightcone
local
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Spin and twist expansion
(NON-)COLLINEARITY
Spin n (Pm1Pmn traces) Twist t dimension -
spin

• Local matrix elements in F
• Operators can be classified via their
• canonical dimensions and spin (OPE)
• Nonlocal matrix elements in F(x)
• Parametrized in terms of (collinear)
  distribution functions f(x) that involve
  operators of different spin but with one specific
  twist t that determines the power of (M/Q)t-2 in
  observables (cross sections and asymmetries).
• Moments give local operators.
• Nonlocal matrix elements in F(x,pT)
• Parametrized in terms of TMD distribution
  functions f(x,pT2) that involve operators of
  different spin and different twist. The lowest
  twist determines the operational twist t of the
  TMD functions and determines the power of
  (M/Q)t-2 in observables.
• Transverse moments give
• collinear functions.

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Collinear parametrizations
(NON-)COLLINEARITY

• Gauge invariant correlators ? distribution
  functions
• Collinear quark correlators (leading part, no
  n-dependence)
• i.e. massless fermions with momentum distribution
  f1q(x) q(x), chiral distribution g1q(x) Dq(x)
  and transverse spin polarization h1q(x) dq(x)
  in a spin ½ hadron
• Collinear gluon correlators (leading part)
• i.e. massless gauge bosons with momentum
  distribution f1g(x) g(x) and polarized
  distribution g1g(x) Dg(x)

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The operators
HIGHER TWIST OPERATOR BASIS
n is direction conjugate to P, P.n
1 Transverse (a) is orthogonal to P and n

• Twist-2 operator combinations
• Twist-3 operator combinations (including
  transverse gluon field), relevant in TMDs and in
  situations with transverse pol.
• Twist-4 operator combinations

One can find appropriate basis sets of
operators within matrix elements one can use
equations of motion
four-quark operator
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Level 1 (rigorous)
QUESTIONS AND ANSWERS

• Type of multi-parton operators provides flavor
  dependence. An analysis in terms of twist-4 PDFs
  would be useful (along lines of work done by
  Boer, Jakob, Mulders).
• It could reveal relations between twist-4
  coefficients aHT and e.g. d2, which probably do
  not exist.

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Level 2 (semi-rigorous)
QUESTIONS AND ANSWERS

• Arguments for x-dependence depend on mild
  assumptions like correlators F(x,pT) falling with
  pT and/or MX
• Arguments for scale LHT
• Bag model

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Level 3 (non-rigorous)
QUESTIONS AND ANSWERS

diquark

• Diquarks describe nonperturbative aspects of
  nucleon structure, usually model-dependent, which
  may explain specific spin/flavor correlations.
• Large uncertainties!

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Fundamental importance
AN OWN BULLET IN THE PROPOSAL FOR HIGHER TWIST?

• In principle yes, in practice
• Unpolarized twist-2
• Longitudinally polarized twist-2
• Transversely polarized or TMD (for DIS twist-3)
• Unpolarized twist-4 in inclusive DIS

HARDER TO MEASURE
THEORETICALLY HARDER