Lattice Operator Product
Expansion
and the Structure Functions
C.-J. David Lin
CTS-NTU
NTHU, Hsinchu
21/09/06
Based on
W. Detmold and C-JDL, Phys Rev D73, 014501 (2006).
Outline
• Hadronic tensor and the OPE.
• The OPE on the lattice.
• Extracting moments from lattice data.
• Application to the pion distribution amplitude.
• Conclusion.
Deeply Inelastic
Lepton-Hadron Scattering, I
A good review: Aneesh Manohar, hep-ph/9204208.
k', E'
k, E
q = k - k'
X
p
Dimensionless hadronic tensor WSµν (p, q)
→ decomposed into the structure functions F1,2 and g1,2 .
p, λ
ν
µ
q
q
p, λ'
Dimensionless hadronic tensor TSµν (p, q)
→ decomposed into the structure functions F̃1,2 and g̃1,2 .
• Optical theorem relates Imag of F̃ , g̃ to F , g.
• The structure functions are functions of
x = −q 2 /2p · q and −q 2/p2 .
Deeply Inelastic
Lepton-Hadron Scattering, II
• DIS is the study of the regime
−q 2/p2 → ∞ at fixed x = −q 2/2p · q.
→ Asymptotic freedom of QCD.
→ The Bjorken scaling.
• Physical region is 0 ≤ x ≤ 1.
• The DIS regime can be shown to be dominated by
the structure of the hadron along the light-cone.
→ Difficult for field theory formulated in Euclidean
space.
• Can perform an operator product expansion.
→ A short distance expansion, works at x → ∞.
→ Extract information, i.e., moments of the
structure functions, in this unphysical region.
Hadronic tensor and the OPE
WSµν (p, q)
=
|
Z
d4 z eiq·z hp, S| [J µ (z), J ν (0)] |p, Si
{z
}
Imaginary part of
TSµν (p, q) =
|
Z
d4z eiq·z hp, S|T [J µ (z)J ν (0)] |p, Si
{z
}
T [J µ (z)J ν (0)] =
the OPE
P
Ci (z 2, µ2)zµ1 . . . zµn Oiµνµ1...µn (µ)
twist = dimension - spin
Powers of 1/x in the matrix elements.
Lattice calculations
• Analytic continuation
→ Difficult to obtain TSµν directly.
• Operator mixing and renormalisation
→ Difficult for high-spin operators.
The OPE on the lattice
General features
µ
ν
hp, S|T [J (z)J (0)] |p, Si =
|
{z
Simulation
}
X
|
Ci (z 2, µ2 ) zµ1 . . . zµn hp, S|Oiµνµ1 ...µn (µ)|p, Si
} |
{z
Analytical calculation
{z
Fits
• First investigated in kaon physics.
C. Dawson et al., 1998.
Our proposal
• Simulation of
P
S
TSµν → Continuum limit.
→ No power divergence.
• Perform the OPE in Euclidean space
• Fit the matrix elements.
→ No need for analytic continuation.
→ No need for operator matching.
• Not obtaining TSµν in Minkowski space directly.
}
The OPE on the lattice
Specific features
• A fictitious “valence” heavy quark Ψ and current
µ
(z) = Ψ̄(z)γ µ ψ(z) + ψ̄(z)γ µ Ψ(z).
JΨ,ψ
• Study the Euclidean Compton scattering tensor
Z
h
E
i
D
X
µ
µν
ν
4
iq·z
d z e
TΨ,ψ =
p, S T JΨ,ψ (z)JΨ,ψ (0) p, S
S
• Sum the target-mass effects.
• Compute the twist-two matrix elements.
Why a “valence” heavy quark?
• Two large scales, q 2 and mΨ .
p
ΛQCD ≪ mΨ ∼ q 2 ≪
1
a
• Remove many higher-twist contributions.
• No all-to-all propagator in the simulation.
• The Fourier transform is practical → z4 ∼ 1/mΨ .
The OPE on the lattice
Some details
i(iD
/+q/)+mΨ
(iD+q)2 +m2Ψ
/+q/)+mΨ
= − i(iD
Q2 +D 2 −m2
Q2 = −q 2,
Ψ
P∞
n=0
Higher-twist
components
n
−2i q·D
Q2 +D 2 −m2Ψ
MΨ = mψ + α/2
2 + αM + β
Q2 + D2 − m2Ψ = Q̃2 = Q2 − MΨ
Ψ
µν
µν
.
− TΨ,d
No contribution in TΨ,u
Removed because Ψ is non-dynamical.
Λ2QCD
q 2 +m2Ψ
Extrating moments from data
The Euclidean Compton tensor
X
{µ
p, S ψ̄γ 1 (iDµ2 ) . . .
S
X
S
{µν}
p, S ψ̄
TΨ,ψ (p, q)
=
iD
i
{µ1
∞
X
n=2
even
iD
µn }
. . . iD
µn }
− tr p, S
− tr p, S
=
Anψ (µ2 ) [pµ1 . . . pµn − tr]
=
Ânψ (µ2 ) [pµ1 . . . pµn − tr]
(2)
(3)
Anψ (µ2 ) ζ n F Cn(1) (η), Cn−1
(η), Cn−2
(η), n, q 2 , Q̃2 , µ2
∞
M (mΨ − m) µν X
Cˆn Ânψ (µ2 ) ζ n Cn(1) (η)
δ
−2i
2
Q̃
n=0
even
• The Gegenbauer polynomial: target-mass effects.
η = √p·q2
p q2
2 is the large scale for the OPE.
• Q̃2 ∼ −q 2 − MΨ
√22
pq
ζ = Q̃2
• Remove Ânψ by choosing µ 6= ν.
Extracting moments from data
Plots
{34}
TΨ,ψ (p, q)
∞
X
=
Anψ (µ2 )f (n)
n=2,even
p = (0, 0, 0, iM )
q = (0, 0,
,
p
q02 − Q2 , iq0 )
M = 1.2 GeV
0
fHnL
-0.1
-0.2
Α = 0.4 GeV
Α = 1.0 GeV
Α = 1.2 GeV
-0.3
-0.4
-0.5
MQ = 3.54 GeV, q0 = 2.76 GeV, Q2 = 1.5 GeV2
0
5
10
n
15
20
M = 1.2 GeV
0
fHnL
-0.1
-0.2
-0.3
-0.4
MQ = 2.1 GeV
-0.5
q0 = 1.98 GeV, Q2 = -3.85 GeV2
0
5
10
n
15
20
Extracting moments from data
Correlators
µν
µν
No contribution for TΨ,u
− TΨ,d
Additional contractions if light-light current is used.
µν
µν
− TΨ,d
Contributes to TΨ,u
Application to the Pion
Distribution Amplitude
¯
Related to the matrix element hπ|T d(z)γµ γ5u(0) |0i.
• A crucial input in B → ππ decays via QCD factorisation.
• The OPE leads to the need of the matrix elements
µ
µ
{µ
n} 2
1
0 .
π(p) ψ̄γ γ5 (iD ) . . . iD
• These can be obtained by applying the OPE to the
“unphysical” matrix element
h
i E
D
µ
ν
π(p) T VΨ,ψ (z)AΨ,ψ (0) 0 .
Conclusion
• Extract moments via the OPE on the lattice.
• Can be applied to other nucleon structure functions.
• Can be applied to the pion distribution amplitude
• Numerical work is being carried out.