Moments and Structure Functions at Low Q2

1
Moments and Structure Functions at Low Q2
Rolf Ent, DIS2004

• Formalism
• F2 Moments Old Analysis (R Guess)
• E94-110 L/T Separation ? F2, F1, FL Moments
• Low Q2 Intermezzo
• Spin-dependent Moments
• Summary
• (Skipped Extended GDH and Baldin Sum Rules)

2
Inclusive Electron Scattering Formalism
Q2 Four-momentum transfer x Bjorken variable
(Q2/2Mn) n Energy transfer M Nucleon mass W
Final state hadronic mass
U

• Unpolarized structure functions F1(x,Q2) and
  F2(x,Q2), or FT(x,Q2) 2xF1(x,Q2) and FL(x,Q2),
  to separate by measuring R sL/sT
• Polarized structure functions g1(x,Q2) and
  g2(x,Q2)

L
T
3
QCD and the Operator-Product Expansion
1

• Moments of the Structure Function Mn(Q2)
  dx xn-2F(x,Q2)
• Operator Product Expansion
• Mn(Q2) ? (nM02/ Q2)k-1 Bnk(Q2)
• higher
  logarithmic
• twist dependence

0
?
k1
At High Q2 ln(Q2) dependence of
moments one of first
proofs of QCD ? L(QCD) At Low Q2 Unique
regime for JLab to
determine (1/Q2)m ?
Higher Twist effects Lowest moments are
calculable in Lattice QCD (LQCD) least computer
intensive!
4
Moments of F2p _at_ Low Q2
50 of momentum carried by quarks (Momentum Sum
Rule)
Uses old R parameterization
n 2
n 4
n 6
n 8
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Moments of F2p _at_ Low Q2
Proton Charge (Coulomb Sum Rule)
50 of momentum carried by quarks (Momentum Sum
Rule)
n 2
n 4
n 6
n 8
Elastic contribution
(A combination of Hall C CLAS data has been
used to constrain the Constituent Quark radius
Here only Hall C data are shown)
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Moments of F2p _at_ Low Q2
Proton Charge (Coulomb Sum Rule)
_at_ Q2 2 (GeV/c)2 30 of M2 comes from the
resonance region
50 of momentum carried by quarks (Momentum Sum
Rule)
W2 gt 4 GeV2 (DIS)
elastic
total
n 2
n 2
n 4
n 6
D-region
n 8
S11-region
Elastic contribution
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Rosenbluth Separations
Hall C E94-110 a global survey of longitudinal
strength in the resonance region...
sL sT
sT
sL sT
/
(polarization of virtual photon)
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World's L/T Separated Resonance Data
R sL/sT
lt
lt
R sL/sT
(All data for Q2 lt 9 (GeV/c)2)
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World's L/T Separated Resonance Data
R sL/sT
lt

• Now able to study the Q2 dependence of individual
  resonance regions!
• Clear resonant behaviour can be observed!
• Use R to extract
• F2, F1, FL

R sL/sT
(All data for Q2 lt 9 (GeV/c)2)
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E94-110 L-T Separated Structure Functions

• Good agreement between results from (180!)
  direct L/T separations and global fit of all data

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Duality in FT and FL Structure Functions
Duality works well for both FT and FL above Q2
1.5 (GeV/c)2
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Also good agreement with SLAC L/T Data
Alekhin
SLAC
MRST (NNLO) TM
E94-110
MRST (NNLO)
DIS data provide lower x data for moment
extraction
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Low Q2 Intermezzo
F2p _at_ Q2 lt 0.5
F2 Q2
F2 Q2
Resonance region slides to lower x at low Q2,
but does not disappear like Q2 yet
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R _at_ Low Q2 lt 0.5
Low Q2 Intermezzo
Preliminary Still DR 0.1 point-to-point (mainly
due to bin centering assumptions to x 0.1)
(The x lt 0.2, Low Q2 (0.1 GeV2) region pushed
the envelope of the E94-110 (and E99-118)
experiments. Dedicated data for this region were
taken last year.)
R shows surprisingly flat Q2 behaviour down to
very low Q2 (also shown by SLAC R1998
parameterization) ? does not appear to follow
current conservation prescription yet R stays
rather constant
E99118
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n 2 Moments of F2, F1 and FL Mn(Q2) dx
xn-2F(x,Q2)
1
0
Elastic Contributions
DIS SLAC fit to F2 and R Resonances E94-110
(Hall C) fit
F1EL GM2 d(x-1)
F2EL (GE2 tGM2 )d(x-1)
F2
1 t
t Q2/4Mp2
FLEL GE2 d(x-1)
F1
Even flatter Q2 dependence (? smaller higher
twist!) with correct F2 - smaller Q2
dependence in F2 moment than in F1, FL moments
FL
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n 4 Moments of F2, F1 and FL
Neglecting elastics, n 4 Cornwall-Norton
moments have only a small Q2 dependence as well.
F2
Momentum sum rule
ML(n) as(Q2) 4M2(n) 2c?dx xG(x,Q2)
3(n1)
(n1)(n2)
F1
Gluon distributions!
This is only at leading twist and for zero proton
mass ? Must remove non-zero proton mass effects
from data to extract moment of xG(x,Q2) ? Work
in progress
FL
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Nachtmann Moments
n 4
n 2
F2
F2
F1
F1
FL
FL
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Moments of g1p (G1p)
30 of Spin carried by quarks (Ellis-Jaffe Sum
Rule)
CLAS EG1 Data

• Elastic not included in
• Moment as shown ?
• With Elastic included
• no zero crossing, and
• Q2 dependence far
• smoother

k of proton (GDH Sum Rule)
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Moments of g1p (G1p)
30 of Spin carried by quarks (Ellis-Jaffe Sum
Rule)
CLAS EG1 Data

• Elastic not included in
• Moment as shown ?
• With Elastic included
• no zero crossing, and
• Q2 dependence far
• smoother

Zero crossing mainly due to cancellation of D
(negative) and S11 Resonances
k of proton (GDH Sum Rule)
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Moments of g1p (G1p)
30 of Spin carried by quarks (Ellis-Jaffe Sum
Rule)
CLAS EG1 Data

• Elastic not included in
• Moment as shown ?
• With Elastic included
• no zero crossing, and
• Q2 dependence far
• smoother

s1/2 s3/2

• SU(6) unbroken Mp MD
• D ground state of s3/2

Zero crossing mainly due to cancellation of D
(negative) and S11 Resonances

• In this scenario the D is
• a d function and all higher
• twist ? it plays the same
• role as the elastic in F2

k of proton (GDH Sum Rule)
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Moments of g1n and g1p-g1n

• Hall A 3He(e,e) to extract
• g1n and its moment G1n (E94-010)
• Similar ideas as with proton
• Here, whole region negative

G1n
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Moments of g1n and g1p-g1n
Combine Hall A g1n with Hall B g1p data

• Hall A 3He(e,e) to extract
• g1n and its moment G1n
• Similar ideas as with proton
• Here, whole region negative

G1n
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Moments of g1n and g1p-g1n
Combine Hall A g1n with Hall B g1p data

• Hall A 3He(e,e) to extract
• g1n and its moment G1n
• Similar ideas as with proton
• Here, whole region negative

Bjorken Sum Rule (Verification of QCD)
Chiral Perturbation Theory (cPT)
G1n
G1p - G1n
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Moments of g2n (G2n)
Burkhardt-Cottingham Sum Rule
Hall A 3He(e,e) (E94-010)
Dispersion relation for forward spin-flip Compton
amplitude (similar assumption as GDH). Doesnt
follow from OPE and valid at all Q2 Many
scenarios of g2 low x behavior would invalidate
the sum rule. Here, the part of the integral
that comes from the nucleon resonance region is
shown. Its 0, mostly from a cancellation of the
large elastic and D contributions
Contribution beyond resonance region
as calculated from g1
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Summary Moments at Low Q2
Resonances are an integral part of the Structure
Function Moments at Low Q2 (Note QCD deals with
the Moments and does not care what contributes
to the moments!) 2xF1, FL, FT Structure
Functions well determined in the Resonance Region
(for Q2 lt 5 (GeV/c)2) The Structure Function
Moments have a smooth behaviour as a function of
Q2, and in fact pick up almost uniquely
the quark-quark interactions in the SU(6) ground
states. This seems to indicate that the
parton-hadron transition occurs in a local and
small region, with only few resonances ?
Quark-Hadron Duality Revised Nachtmann
Moments (using measured R) at low Q2 show very
limited Q2 dependence. F2 Moments have less Q2
dependence than F1 and FL moments.
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Why is IGDH(Q2) interesting?
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Extended Baldin Sum Rule
Q2 0, photoproduction
Baldin Sum Rule
GDH Sum Rule
Q2 gt 0, electroproduction
Extended GDH Sum Rule
Extended Baldin Sum Rule
Where ? anomalous magnetic moment of the
nucleon. a, ß electric and magnetic
polarizabilities respectively ?0
pion photoproduction threshold
Need L/T separated data!
D. Drechsel, B. Pasquini, M. Vanderhaeghen
hep-ph/0212124 Dec 2002
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Measurement of the extended Baldin Sum Rule for
the proton
(PDG)
(D. Babusci et al. 1998)

• Extended Baldin Integral goes smoothly from Q2
  gt 5 to real photon point (Q2 0).

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Multiply with Q4/2M to emphasize transition of
Baldin Sum Rule to Perturbative DIS Region
2xF1 Moment