Moments of pion light-cone wavefunction using OPE on the lattice

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William Detmold^∗, C.-J. David Lin^∗∗and

Santanu Mondal^∗∗

Moments of pion light-cone wavefunction using OPE on the lattice

William Detmold^∗, C.-J. David Lin^∗∗ and Santanu Mondal^∗∗

Workshop of recent developments in QCD and quantum field theories

10^th November, 2017

*Centre for Theoretical Physics, Massachusetts Institute of Technology

**Institute of Physics, National Chiao-Tung University, Taiwan

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Outline

Pion Light-Cone (LC) wave function and its moments Moments using lattice OPE with a valance heavy quark¹ Lattice correlators

Exploratory numerical results Summary

1proposed byDetmold and Lin (Phys.Rev. D73 (2006)).

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Pion LC wave function/distribution amplitude:

hπ⁺(p)|d (z

2) γ5γ_µ u(−z

2)|0i = −ip_µf_π Z1

0

d ξ e^{i (ξ p}^z²^{−ξ p}^z²⁾φ_π(ξ ) ξ = 1 − ξ

|π⁺(p)i → Ground state of the pseudoscalar π⁺ meson with on shell momentum p²= m²_π.

fraction ξ of pion momentum is carried by u quark.

Moments:

an= Z1

0

d ξ ξⁿφπ(ξ ).

OPE:

hπ⁺(p)|O^µ¹^..µⁿ|0i = f_πan−1[p^µ¹. . . p^µⁿ− Traces]

O^µ¹^..µⁿ = ψ γ^{µ¹γ⁵(iD^µ²). . . (iD^µⁿ^})ψ − Traces

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hπ⁺(p)|ψ(0) γ₅γµ ψ (0)|0i = ifπp_µ

=⇒ a0= 1

In the isospin limit m_u= m_d:

φ_π(ξ ) = φ_π(ξ )

=⇒ Odd moments vanish → lowest non-trivial moment is a2. Lattice calculations of the second moment are available → precision calculation of the higher moments are needed to get the correct shape of the φ_π(ξ ).

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Euclidean OPE with a valance heavy quark

Heavy-light currents:

V_Ψ,ψ^µ = Ψγ^µψ + ψ γ^µΨ A^µ_Ψ,ψ = Ψγ^µγ⁵ψ + ψ γ^µγ⁵Ψ

ψ : light quarks, Ψ: fictitious, relativistic, valance quark which is heavy.

−→Simplify the lattice calculation.

−→Removes the higher twist contributions.

Scale hierarchy required:

ΛQCD<< mΨ∼p q²<<1

a

=⇒ Fine lattices are required.

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p p+q

q

p p p-q p

q q

VV type operator:

T_Ψ,ψ^{µ ν}(p, q) = Z

d⁴x e^iqxhπ⁺(p)|T [V_Ψ,ψ^µ (x )V_Ψ,ψ^ν (0)]|0i OPE:

Z

d⁴x e^iqxT [V_Ψ,ψ^µ (x )V_Ψ,ψ^ν (0)] = ψγ^µ−i (i /D + /q) + mΨ

(iD + q)²+ m²_Ψ γ^νψ + ψ γ^ν

−i (i /D − /q) + mΨ

(iD − q)²+ m²_Ψ γ^µψ Taylor expansion:

−i (i /D + /q) + mΨ

(iD − q)²+ m²_Ψ = −−i (i /D + /q) + mΨ

Q²+ D²− m²_Ψ

∞ n=0∑

−2iq.D Q²+ D²− m²_Ψ

!n

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Higher twist terms → expansion parameter: ^−2iq.D+D2

Q2 −m2Ψ

!

→ Extra powers of p² → small.

Antisymmetric in µ and ν:

Z

d⁴x e^iqx T [V_Ψ,ψ^µ (x )V_Ψ,ψ^ν (0)] = i 2

∞ n=0,even∑

1

( ˜Q²)ⁿ⁺¹ψ γ_λγ5(i /D_µ₁). . . (i /D_µ_n)ψε_{µ ν ρ λ} Q˜²= Q²− M_Ψ²+ α, mΨ= MΨ−¹₂α, MΨ→ mass of heavy-light pseudoscalar meson.

Z

d⁴x e^iqx hπ⁺(p)|T [V_Ψ,ψ^µ (x )V_Ψ,ψ^ν (0)]|0i =

∞ n=0, even∑

anf (n) f_π

f (n) = i 2

ξⁿ⁺¹ n + 1

h2ηC_n²(η)(q^ρp^λ) p.q

i ε_{µ ν ρ λ}

ξ = pp²q²

Q˜² , η = p.q pp²q²

For simplicity, Wilson coefficients are set to one.

Identical result for the AA type correlator.

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0 2 4 6 8 10

n

1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1

Im[f(n)]

p=(0.5, 0.5, 0.5, 0.9 i) , m_π=0.3, q=(0.5, 0.0, 0.5, 1.13 i), m_Ψ=2, in unit of GeV

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Lattice correlators

π⁺(0, 0) j^eµ(xe, τe)

pe pm

j^mν(xm, τm)

τe> τm>> 0

pm pe

τm> τe>> 0 π⁺(0, 0)

jµ^e(xe, τe)

jν^m(xm, τm)

Euclidean time

C₃^{µ ν}(τm, τe;~pm,~pe) = Z

d³xm Z

d³xe

eî~p^m^·~x^me^−i~pê^·~xêh0|T[jm^µ(~xm, τm)j_e^ν(~xe, τe)Oπ^†(~0, 0)]|0i

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C_3;τ^{µ ν}_m_>τ_e(τm, τe;~pm,~pe) ≈ 1

2E_πδ³(~pπ− (~pm−~pe)) × e^−E^π^τ^eD π (~pπ)

O

† π(~0, 0)

0E Z

d³x e^i~p^m^·~xD 0

jm^µ(~x, τm− τe) j_e^ν(~0, 0) π(~pπ)E

.

The two point function:

Cπ(τπ;~pπ) = Z

d³x e^i~p^π^·~x D

0

Oπ(~x, τ)Oπ^†(~0, 0) 0

E

τπ→∞

−−−→

D

π (~pπ) O

† π(~0, 0)

0

E

2

2E_π × e^−E^π^τ^π. We can take the ratio

R_3;τ^{µ ν}_m_>τ_e(τm− τe;~pm,~p_π) = C_3;τ^{µ ν}

m>τe(τm, τe;~pm,~pm−~pπ) C_π(τe;~pπ) ×D

π (~p_π) O

† π(~0, 0)

0E

= Z

d³x e^i~p^m^·~xD 0

jm^µ(~x, τm− τe) j_e^ν(~0, 0) π(~pπ)E

τm>τe

.

Perform the Fourier transform:^Rdτ e^iq⁴^τ, τ = τm− τe.

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exploratory numerical result

Quenched calculation

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VA channel:

a⁻¹∼ 2 GeV, L³× T = 24³× 48, naive Wilson action at valance, mπ∼ 370 MeV, m_Ψ∼ 1.1 GeV, sample size= 24

0 2 4 6 8 10 12 14

τ_m

1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1

R11 3;τe>τm(τ=τe-τm,pe=111,pm=000)

τe=8 τe=9 τe=10 τe=11 τe=12 τe=13 τe=14 τe=15

0 2 4 6 8

τ

1e-08 1e-06 0.0001 0.01

3 4 5 6 7 8 9 10 11

τ_m

0.4 0.5 0.6 0.7 0.8 0.9 1

Meff π(τm,pπ=111)

S(τ)=∑_{ι=0,..,τ} R(i)

τ = 5

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20 24 28 32 36 40

τ_m

-0.0015 -0.001 -0.0005 0

Im( R[34] )

AA channel, sample size =43

a^-1= 4 GeV, m_π = 290 MeV,

m_Ψ = 2 GeV, p_e = 111, p_m = 100, τ_e = 30, Clover action at valance

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Summary

I have described a new method to calculate the moments of pion distribution amplitude by studying the Euclidean OPE on lattice.

A valance heavy quark is used to make lattice calculation simpler and to give more flexibility.

I have shown our preliminary numerical data and demonstrate our strategy to analyze those.

The method described here can also be used to study parton distribution function of the pion and nucleon.

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Santanu

Mondal^∗∗

Moments of pion light-cone wavefunction using OPE on the lattice

Moments of pion light-cone wavefunction using OPE on the lattice

Outline

Euclidean OPE with a valance heavy quark

Lattice correlators

exploratory numerical result

VA channel:

Summary

Thanks for your attention!