'
Domain wall valence on staggered sea:
Hadron Structure
$
George T. Fleming
c
&
Yale University
[Lattice Hadron Physics Collaboration]
• Summary of LHPC hadron structure proposal.
• Pion and nucleon electromagnetic form factors.
• Nucleon magnetic moments, gA, . . .
%
LHPC Hadron Structure project on USQCD resources
Dru Renner
University of Arizona
Richard Brower, James Osborn
Boston University
Rebecca Irwin, Michael Ramsey-Musolf
California Institute of Technology
Robert Edwards, David Richards
Thomas Jefferson National Accelerator Facility
Bojan Bistrovic, Jonathan Bratt, Patrick Dreher, Oliver Jahn,
John Negele, Kostas Orginos, Andrew Pochinsky, Dmitry Sigaev
Massachusetts Institute of Technology
Matthias Burkardt, Michael Engelhardt
New Mexico State University
George Fleming
Yale University
Constantia Alexandrou, Antonios Tsapalis
Philippe de Forcrand
University of Cyprus
ETH-Zürich and CERN
Wolfram Schroers
Philipp Hägler
DESY, Zeuthen
Vrije Universiteit, Amsterdam
LHPC Hadron Structure project on USQCD resources
• This years plan for valence DWF propagator computation:
– a = 0.13 fm, mπ = 250 – 350 MeV
– a = 0.09 fm, mπ = 250 – 775 MeV
– a = 0.06 fm, mπ = 350 – 500 MeV
• Available USQCD Resources:
– 65% of time on gigE mesh clusters at JLab
– 9% of time on USQCD’s QCDOC at BNL
• Planned observable calculations include:
Form factors at high Q2 : F1 , F2 , GA , GP , Fπ , . . .
Transition form factors: ∆ → N , ρ → π, π ∗ → π, π ∗ → ρ
Moments of structure functions for nucleon and pion: hxn iq , hxn i∆q , hxn iδq
Generalized form factors which give access to transverse structure, quark spin ∆Σ and orbital
angular momentum Lq of nucleon.
– Non-perturbative renormalization of hxi moments in RI/MOM scheme.
– Starting to incorporate disconnected contributions in all of the above . . .
–
–
–
–
2)
The
Pion
Electromagnetic
Form
Factor
F
(Q
π
• Widely considered a good observable for studying the interplay between perturbative
and non-perturbative descriptions of QCD
– Asymptotic normalization fixed by precise measurement of pion decay
8παs(Q2)fπ2
Fπ (Q ) =
Q2
2
as Q2 → ∞
– For small Q2, vector meson dominance (VMD) gives an accurate description, with
normalization Fπ (0) = 1 given by charge conservation
Fπ (Q2) ≈
1
2
1 + Q m2ρ
for Q2 m2ρ
• Not too hard to calculate on the lattice since there are no disconnected diagrams.
• Experimental results are coming for Q2 ≥ 1 GeV2
Pion Form Factor: A QCD Laboratory?
Presented by G. M. Huber at 2004 JLab User’s Meeting.
The Experimental Situation
• Existing data fit VMD monopole formulae too well. Where’s perturbative QCD?
• Results from Lattice QCD simulations will have a huge impact on the debate.
Experimental Methods (I)
±
Elastic Scattering of 300 GeV pions on atomic electrons
−
• Elastic π +e scattering provides direct
measurement of Fπ , (One Photon Exchange).
2
0.2
2
Q Fπ (GeV)
• Kinematics of pion backscatter in CM
frame limits maximum Q2.
CERN NA7: Amendolia et al., Nucl. Phys. B 277, 168 (1986)
• CERN NA7: Even with 300 GeV pions
from SPS incident on liquid hydrogen
target, Q2max . 0.288 GeV2.
0.1
0
0
0.1
0.2
2
2
Q (GeV)
• Hypothetically, for 1 TeV pions from Tevatron, Q2max . 1 GeV2.
• For 7 TeV pions from LHC, Q2max . 7.1 GeV2.
−
Experimental Methods (II)
e + p → e− + π + + n
(One Photon Exchange)
• Differential cross section depends on Q2,
s and t
p
d2 σ
= σL + σT + 2( + 1)σLT cos φπ
dΩ
+σTT cos 2φπ
!−1
• Interference terms σLT and σTT extracted from fit and subtracted from
dσ/dΩ.
2
15
10
5
2
0
2
Q = 1.0 (GeV/c)
2
-t = 0.09 - 0.13 (GeV/c)
o
o
θπq = 1.1 - 6.8
2
=
2 |q|2
θe
1+
tan2
2
Q
2
Phys. Rev. Lett. 86, 1713 (2001)
2π d σ/(dt dφπ) (µb/(GeV/c) )
• Experiments hold Q2, s and t fixed
and measure dσ/dΩ vs. φπ and virtual
photon polarization .
0
π/2
π
3π/2
2π
φπ
Experimental Methods (III)
L/T (or Rosenbluth) separation
• Subtracted dσ/dΩ is σL + σT. Separation of σL and σT by dependence.
• Near the unphysical pion pole
:
(t → Mπ2 )
−tQ2
2
2
2
gπN
σL ∼
N (t)Fπ (Q , t)
2
2
(t − Mπ )
• Experiments access t < 0. Extrapolation introduces model dependence.
dσ/dt (µb/(GeV/c)2)
Phys. Rev. Lett. 86, 1713 (2001)
40
2
Q =0.60
25
2
Q =0.75
20
30
15
20
10
10
0
5
0
0.1
0.2
0
0
0.1
0.2
10
20
2
Q2=1.60
Q =1.00
10
0
0
0.1
0.2
0
0
0.1
0.2
-t (GeV/c)2
• Figure shows fits of VGL Regge model to JLab data where Fπ (Q2) is a free parameter.
Model fits σL reasonably well.
2)
Lattice
techniques
for
extracting
F
(Q
π
• Form factor definition.
π(pf ) Vµ(pf − pi, t) π(pi)
t,q
cont
= ZV π(pf ) Vµ(pf − pi, t) π(pi)
lat
t i ,pi t ,p
f f
2
= Fπ (Q )(pf + pi)µ
• We use the sequential source method.
• Main disadvantage: Momentum pf and pion quantum numbers are fixed at the sink.
• Main advantages: Variety of meson form factors, e.g. ρ → γπ computable from
same set of propagators.
• Largest Q2 available in Breit frame (pf = −pi). Sequential propagators for several
pf .
• Fπ (Q2) extracted either from simultaneous fit to several two and three-point functions
or by constructing ratios.
Three point functions, smearings, and ratios
• The three point correlator with local (L) or smeared (S) source A and sink B:
ΓAB (t , t, t , p , p ) = e−i(xf −x)·pf π B (x , t ) |V (x, t)| π A(x , t ) e−i(x−xi)·pi
π4π
i
f
i
f
f
f
4
i
i
• After inserting complete set of states and taking large Euclidean times ti t tf :
ΓAB
π4π
ZB (pf )ZA∗ (pi) −(tf −t)E(p ) −(t−ti)E(p )
f e
i
→
e
π(pf ) V4(pf − pi, t) π(pi)
4E(pf )E(pi)
• For the two point correlator:
ΓAB
ππ (ti , tf , p)
ZB (p)ZA∗ (p) −(tf −ti)E(p)
→
e
2E(p)
ti tf
• Crucial point: ZL(p) = ZL. Also ZS (p) = ZS (“|p|”) and E(p) = E(“|p|”).
• Ratio gives Fπ (Q2, t) independent of Z’s and exponentials:
AB
CL
Γ
(t
,
t,
t
,
p
,
p
)
Γ
(t
,
t,
p
)
2
Z
E(|p
|)
f
V
i
f
f
f
π4π i
ππ i
Fπ (Q2, t) =
CB
ΓAL
E(|pi|) + E(|pf )|
ππ (ti , t, pi ) Γππ (ti , tf , pf )
Strategy for data analysis
• When operators are smeared, a gauge-invariant Gaussian smearing function is used.
• Smearing optimal for ground state overlap of nucleon operators at rest.
• Simultaneous fit of smeared-smeared (SS) and smeared-local (SL) hP P i and
hA4A4i correlators with constraints ZL(p) = ZL, ZS (p) = ZS (|p|) and E(p) =
E(|p|).
• For each Q2, compute Fπ (Q2, t) ratios using extracted energies E(|p|) and choose
plateau ranges [tmin, tmax].
• For each Q2, simultaneously fit two and three point functions to extract constant
Fπ (Q2) over range [tmin, tmax].
• Need small pi, pf to minimize statistical errors but need large Q2 to address physics.
– Signal to noise ∝ exp {− [Eπ (p) − Eπ (0)] t}.
• Staying in Breit frame is an important part of overall program.
DWF Results: Ratio Method
The Nucleon Electromagnetic Form Factors
• Brodsky-Lepage predicted F1 ∼ Q−4
and F2 ∼ Q−6 as Q2 → ∞.
• One Photon Exchange approximation
may not be justified.
• Softer scaling also possible
(hep-ph/0212351)
Q2
F2(Q2)
∼ const
log2(Q2/Λ2) F1(Q2)
2
1.5
(a)
CQM
SU(6)
Soliton
SU(6) + CQ ff
PFSA
VMD
1
2
Q F2p/F1p
• New polarization transfer experiments do
not see expected scaling.
Phys. Rev. Lett. 88, 092301 (2002)
0.5
0
(b)
0.75
QF2p/F1p
• Older L/T separation experiments observed predicted behaviour.
0.5
Jones [11]
This work
0.25
0
0
1
2
3
4
2
2
Q (GeV )
5
6
Nucleon F2/F1 on the Lattice (I)
• But, . . . We don’t control asqtad lattice generation.
2
(GeV )
(I=1)
0.5
Q F2
(I=1)
1.5
2
/ κF1
2
(GeV)
0
mπ = 775 MeV
mπ = 498 MeV
mπ = 359 MeV
1.5
n
expt: Q F2p / F1p
(I=1)
• Exploring sink momenta up to
pf =(2,0,0). May reach Q2 ∼ 4 GeV2
with sufficient statistics.
2.5
1
/ κF1
• Errors shown do not account for data
covariance, are not jackknifed and are
likely overestimated.
3
(I=1)
• Our normalization is F2(Q2) → κ as
Q2 → 0.
VERY PRELIMINARY!!!
Q F2
• Only I = 1 form factors computed so far
to avoid disconnected diagrams. F1I=1 =
F1p − F1n but F1n, F2n not known accurately for Q2 & 1 GeV2.
1
0.5
0
0
1
2
3
4
2
2
Q (GeV )
5
6
Nucleon F2/F1 on the Lattice (II)
VERY PRELIMINARY!!!
4
• F2I=1/F1I=1 → κp −κn as Q2 → 0.
• PDG: κp = 1.792847351(28)
(I=1)
/ F1
(I=1)
• So, comparison of I = 1 with p −
n could be OK with proper chiral
extrapolation.
2
F2
• PDG: κn = −0.91304273(45)
3
1
0
0
mπ = 775 MeV
mπ = 498 MeV
mπ = 359 MeV
Expt: κp - κn
0.5
1
2
1.5
2
2
Q (GeV )
2.5
3
Nucleon axial charge
PRELIMINARY
Philipp Hägler (Vrije Universiteit)
Summary and outlook
• Lots of propagator and observable calculations already completed. Data analysis is
proceeding rapidly. Expect published results soon.
• Hashimoto’s talk has inspired me to look at pion scalar form factor, K`3, etc. Data
is on disk . . .
• Plan to measure mixed asqtad-DWF pion mass discussed in Baer’s talk.
• Reaching higher Q2 is high priority.
• Looking forward to dynamical chiral fermion lattices. . .