Parton physics from Euclidean current-current correlator with a - - PowerPoint PPT Presentation

Parton physics from Euclidean current-current correlator with a valence heavy quark: pion light-cone distribution amplitude as an example

Field Theory Team seminar, RIKEN RCCS, Kobe 18/11/2020 C.-J. David Lin National Chiao Tung University, Taiwan

In collaboration with William Detmold, Anthony Grebe, Issaku Kanamori, Santanu Mondal, Robert Perry and Yong Zhao

Outline

• General HOPE strategy
• HOPE and the pion light-cone wavefunction
• Preliminary numerical result
• Outlook

W.Detmold, A.Grebe, I.Kanamori, CJDL, S.Mondal, R.Perry, Y.Zhao, Contribution to APLAT2020, arXiv:2009.09473. W.Detmold, A.Grebe, I.Kanamori, CJDL, S.Mondal, R.Perry, Y.Zhao, Contribution to APLAT2020, arXiv:2009.09473.

• W. Detmold and CJDL, PRD 73 (2006)

General strategy

Parton distribution from lattice QCD

W µν

S (p, q) =

• d4xeiq·xp, S| [Jµ(x), Jν(0)] |p, S,

The “traditional” approach Hadronic tensor (PDFs from the twist-2 sector)

Parton distribution from lattice QCD

W µν

S (p, q) =

• d4xeiq·xp, S| [Jµ(x), Jν(0)] |p, S,

T µν

S (p, q) =

• d4xeiq·xp, S|T [Jµ(x)Jν(0)] |p, S
• ptical theorem

Imaginary part challenging in Euclidean QCD

The “traditional” approach Hadronic tensor (PDFs from the twist-2 sector)

Parton distribution from lattice QCD

W µν

S (p, q) =

• d4xeiq·xp, S| [Jµ(x), Jν(0)] |p, S,

T µν

S (p, q) =

• d4xeiq·xp, S|T [Jµ(x)Jν(0)] |p, S

T [Jµ(x)Jν(0)] =

• i,n

Ci

• x2, µ2

xµ1 . . . xµnOµνµ1...µn

i

(µ),

• ptical theorem

Imaginary part challenging in Euclidean QCD

local operators, issue of operator mixing leading moments in practice Power divergences arising from Lorentz symmetry breaking

The “traditional” approach Hadronic tensor (PDFs from the twist-2 sector) Light-cone OPE

Parton distribution from lattice QCD

The “new” approach to avoid difficulties in renormalisation

H (p) H (p′) Oµνµ1...µn

i

H (p) H (p′) Onon−local

General idea: Inserting non-local, instead of local, operator

Parton distribution from lattice QCD

The “new” approach to avoid difficulties in renormalisation

H (p) H (p′) Onon−local

Make certain the absence of on-shell states for analytic continuation

H (p) H (p′) Onon−local

• - - - - - - -

Parton distribution from lattice QCD

The “new” approach to avoid difficulties in renormalisation

H (p) H (p′) Onon−local

Typical examples of the non-local operator A space-like Wilson line (quasi-PDF and pseudo-PDF) Two currents separated by space-like distance Two flavour-changing currents with valence heavy quark

• X. Ji, PRL 110 (2013); A. Radyushkin, PRD 96 (2017)
• V. Braun and D. Mueller, EPJC 55 (2018)

Smeared “local” operators

• Z. Davoudi and M. Savage, PRD 86 (2012)
• W. Detmold and CJDL, PRD 73 (2006)

And other proposals

• A. Chambers et al., PRL 118 (2017); Y. Ma and J.-W. Qiu, PRL 120 (2018);……

Introducing the valence heavy quark

Jµ

Ψ,ψ(x) = Ψ(x)γµψ(x) + ψ(x)γµΨ(x)

T µν

Ψ,ψ(p, q) ≡

• S

p, S|tµν

Ψ,ψ(q)|p, S =

• S
• d4x eiq·xp, S|T
• Jµ

Ψ,ψ(x)Jν Ψ,ψ(0)

• |p, S

propagating in both space and time Compton tensor

• W. Detmold and CJDL, PRD 73 (2006)

Valence Not in the action The “heavy quark” is relativistic The current for computing the even moments of the PDF

Strategy for extracting the moments

• Simple renormalisation for quark bilinears.
• Work with the hierarchy of scales
• Extrapolate to the continuum limit first.

Then match to the short-distance OPE results. Extract the moments without power divergence.

T µν

Ψ,ψ(p, q) ≡

• S

p, S|tµν

Ψ,ψ(q)|p, S =

• S
• d4x eiq·xp, S|T
• Jµ

Ψ,ψ(x)Jν Ψ,ψ(0)

• |p, S

Jµ

Ψ,ψ(x) = Ψ(x)γµψ(x) + ψ(x)γµΨ(x)

T µν

Ψ,ψ(p, q)

ΛQCD << p q2 ≤ mΨ << 1

a

Heavy scales for short-distance OPE. Avoid branch point in Minkowski space

p (q + p)2 ∼ (mN + mΨ)2

at

Short-distance OPE & valence heavy quark

T µν

Ψ,v = T µν Ψ,u − T µν Ψ,d.

leading twist, absent in higher twist, absent leading and higher twist These are the leading-twist contributions that we are after. ambiguity in heavy quark mass

HOPE and pion light-cone distribution amplitude

W.Detmold, A.Grebe, I.Kanamori, CJDL, S.Mondal, R.Perry, Y.Zhao, Contribution to APLAT2020, arXiv:2009.09473.

Pion light-cone wavefunction

Mellin moments

h0|ψ(z2n)/ nγ5W[z2n, z1n]ψ(z1n)

• π+(p)

↵ = ifπ(p · n) Z 1 dx e−i(z1x+z2(1−x))p·nφπ(x, µ2)

h⇠niµ2 = Z 1

−1

d⇠ ⇠n(⇠, µ2)

⟨0|Oμ1...μn

ψ

|π(p)⟩ = fπ ⟨ξn−1⟩ [pμ1 . . . pμn − traces] Oμ1...μn

ψ

= ¯ ψγ5γ{μ1(iDμ2) . . . (iDμn})ψ − traces

OPE

π

Phenomenological relevance

Pion form factor in QCD exclusive processes Important input for flavour physics

T II

i

Φ

M1

Φ

M2

Φ

B

B M1 M2

Hadronic tensor for computing pion LCDA

+

• q

µ

T µν(p, q) = Z d4z eiq·z h0| T [Jµ

A(z/2)Jν A(z/2)] |⇡(p)i

U µν(p, q) = 1 2 ✓ T µν(p, q) T νµ(p, q) ◆ Z

Jµ

A = ¯

Ψµ5 + ¯ µ5Ψ,

is the valence, relativistic heavy quark

Ψ

π(p)

Jμ

A(z/2)

Jν

A(−z/2)

the cross term

OPE for the hadronic tensor: Euclidean result

• ⌘ =

p · q p p2q2 p p ⇣ = p p2q2 ˜ Q2

C2

n(⌘)

, higher-twist : target-mass effect tree-level OPE

• ne-loop

fit lattice data

˜ Q2 = q2 + m2

Ψ

U µν(p, q) = 2✏µναβqαpβ ˜ Q2

∞

X

n even

⇣nC2

n(⌘)

2n(n + 1)C(n)

W ( ˜

Q2)fπh⇠ni + O(1/ ˜ Q3)

GeV in this talk

μ = 2

OPE for : issue in fitting higher moments

Uμν

U µν(p, q) ⇠

∞

X

n=0

hξni ωn ω = 2p · q ˜ Q2 = 2p · q q2

4 + q2 + m2 Q

+ 2iEπq4 q2

4 + q2 + m2 Q

ω Re{ω} Im{ω} −1 1

, suppressing higher-moments need large p to make ω → 1 allowed region for imaginary q4

OPE for : issue in fitting higher moments

Uμν

p =(0,0,0), q =(0,0,1) p =(1,0,0), q =( 1

2,0,1)

p =(4,0,0), q =(2,0,1)

1 2 3 4 5 6 10-7 10-5 0.001 0.100 n Re[W(n)]

U µν(p, q) = 2✏µναβqαpβ ˜ Q2

∞

X

n even

⇣nC2

n(⌘)

2n(n + 1)C(n)

W ( ˜

Q2)fπh⇠ni + O(1/ ˜ Q3) = 2✏µναβqαpβ ˜ Q2

∞

X

n even

W(n)C(n)

W ( ˜

Q2)fπh⇠ni + O(1/ ˜ Q3)

max{ }

In general, need large p to access non-leading moments

Strategy for fitting at low pion momentum

⟨ξ2⟩

choose p q 0 while =0, and being real

⋅ ≠ p3 q3 ≠ 0 q4

imaginary real complex

U 12(p, q) = 2✏12αβqαpβ ˜ Q2

∞

X

n even

⇣nC2

n(⌘)

2n(n + 1)C(n)

W ( ˜

Q2)fπh⇠ni + O(1/ ˜ Q3) = 2(q3p4 q4p3) ˜ Q2  C(0)

W ( ˜

Q2)fπ + 6(p · q)2 p2q2 6( ˜ Q2)2 C(2)

W ( ˜

Q2)fπh⇠2i + . . .

• + O(1/ ˜

Q3) 

• 
• = 2iq3Eπ

˜ Q2  C(0)

W ( ˜

Q2)fπ + 6(p · q)2 p2q2 6( ˜ Q2)2 C(2)

W ( ˜

Q2)fπh⇠2i + . . .

• + O(1/ ˜

Q3)

The largest contribution to Re[ ] is from

U12 ⟨ξ2⟩

Correlators for lattice calculation

Cµν

3 (τe, τm; pe, pm) =

Z d3xe d3xm eipe·xeeipm·xm h0|T ⇥ Jµ

A(xe, τe)Jν A(xm, τm)O† π(0, 0)

⇤ |0i

C2(⌧π, p) = Z d3x eip·xh0|Oπ(x, ⌧π)O†

π(0, 0)|0i

Excited states?

and the Fourier transform for

Rμν Uμν

Rµν(τ; p, q)= Z d3z eiq·zh0|T h Jµ ⇣z 2 ⌘ Jν ⇣ z 2 ⌘i |π(p)i

z = xe xm,

• p = pe + pm, q = 1

2(pm pe)

U µν(p, q) ⌘ Z dτ eiq4τR[µν](τ; p, q)

From and , one can construct

Cμν

3

C2

Then the hadronic tensor can be obtained via

Exploratory quenched calculation @ MeV

Mπ ≈ 560

p = (1,0,0) q = (1/2,0,1) in units of GeV

2π/L ∼ 0.64

is improved without improving the axial current

Uμν O(a)

Wilson plaquette and non-perturbatively improved clover actions

bare mΨ fitted mΨ 1.0 GeV 2.0 GeV 1.6 GeV 2.6 GeV 2.5 GeV 3.3 GeV a (fm) ˆ L3 ⇥ ˆ T Nconfig Nsrc 0.081 243 ⇥ 48 650 2 0.060 323 ⇥ 64 450 3 0.048 403 ⇥ 80 250 3 0.041 483 ⇥ 96 341 3

Excited-state contamination

e = 0.24 fm e = 0.36 fm e = 0.48 fm e = 0.6 fm e = 0.72 fm e = 0.84 fm

0.0 0.1 0.2 0.3 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (fm)

• 1[Im(U)] (GeV2)

0.3 0.4 0.5 0.6 0.7 0.8 0.024 0.025 0.026 0.027 0.028 0.029 e (fm)

• 1[Im(U)] (GeV2)

fm

a = 0.060

Continuum extrapolation of U12

0.0 0.1 a2 (GeV−2) −0.00008 −0.00006 Re[U12(p, q)] (GeV) q4 = −1.2 GeV mQ = 2.0 GeV 0.0 0.1 a2 (GeV−2) 0.026 0.028 0.030 0.032 Im[U12(p, q)] (GeV) q4 = −1.2 GeV mQ = 2.0 GeV

Preliminary Preliminary

Continuum extrapolation of U12

0.0 0.1 a2 (GeV−2) −0.00006 −0.00005 −0.00004 −0.00003 Re[U12(p, q)] (GeV) q4 = −2.0 GeV mQ = 2.0 GeV 0.0 0.1 a2 (GeV−2) 0.018 0.020 0.022 Im[U12(p, q)] (GeV) q4 = −2.0 GeV mQ = 2.0 GeV

Preliminary Preliminary

Results of U12

−2.5 0.0 2.5 q4 (GeV) −0.00010 −0.00005 0.00000 0.00005 0.00010 Re[U12(p, q)] (GeV)

Continuum 0.041 fm 0.048 fm 0.060 fm 0.081 fm

−4 −2 2 4 q4 (GeV) 0.01 0.02 0.03 0.04 Im[U12(p, q)] (GeV)

Continuum 0.041 fm 0.048 fm 0.060 fm 0.081 fm

Preliminary Preliminary

OPE fits in momentum and position spaces

U 12(p, q) = 2iq3Eπ ˜ Q2  C(0)

W ( ˜

Q2)fπ + 6(p · q)2 p2q2 6( ˜ Q2)2 C(2)

W ( ˜

Q2)fπh⇠2i + . . .

• + O(1/ ˜

Q3)

˜ U µν(p, q, ⌧) = Z d⌧ e−iq4τU µν(p, q)

Momentum space: fit the continuum-limit to

U12

Position space: Fourier transform Allows for determining at finite lattice spacing

⟨ξ2⟩

Offers a different analysis procedure Less sensitive to and

ZA bA

Continuum extrapolation for ⟨ξ2⟩

m = 2.0 GeV m = 2.6 GeV m = 3.3 GeV 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.00 0.05 0.10 0.15 0.20 0.25 0.30 a2 (fm2) 2

Preliminary

Continuum extrapolation for ⟨ξ2⟩

m = 2.0 GeV m = 2.6 GeV m = 3.3 GeV 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.00 0.05 0.10 0.15 0.20 0.25 0.30 a2 (fm2) 2

Preliminary

Continuum extrapolation for ⟨ξ2⟩

m = 2.0 GeV m = 2.6 GeV m = 3.3 GeV 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.00 0.05 0.10 0.15 0.20 0.25 0.30 a2 (fm2) 2

Preliminary

Continuum extrapolation for from

fπ ⟨ξ0⟩

m = 2.0 GeV m = 2.6 GeV m = 3.3 GeV

0.000 0.002 0.004 0.006 0.008 50 100 150 a2 (fm2) f (MeV)

Preliminary

Comparing with other calculations

0.2 0.3 hξ2i Braun et al. (2006) Arthur et al, (2011) Braun et al. (2015) Bali et al. (2019) HOPE (2020)

Preliminary

Conclusion and outlook

The HOPE method is completely worked out for ϕπ(x, μ) In general, need large p for accessing non-leading moments A strategy is found for computing at low p

⟨ξ2⟩

Numerical result shows the validity of the HOPE method Future: higher and other partonic quantities

⟨ξn⟩

Backup slides

Enhancing the signal: the need

Leading contribution in Re[ ] is ~

U12 ⟨ξ2⟩ω2

Leading contribution in Im[ ] is ~

U12 ⟨ξ0⟩

We work with |ω| =

2p ⋅ q ˜ Q

2

< 1

Much noisier compared to Im[ ]

U12

Enhancing the signal: the idea

Re[U µν(p, q)] = Re Z ∞

−∞

dτ Rµν(τ; p, q)e−iq4τ

• /

Z ∞ dτ [Rµν(τ; p, q) Rµν(τ; p, q)] sin(q4τ)

We work with where Minkowskian is imaginary.

|ω| < 1 Uμν

From ;p,q).

Uμν

Minkowski(p, q) = ∫ ∞ −∞

dτ e−q0τ Rμν(τ

• ) = Rµν(τ; p, q)+Rµν(τ; p, q)

(42)

More correlated reduced error is imaginary.

Rμν

Back to Euclidean space:

hermiticity

γ5

Enhancing the signal: the result

R(;p ,q )+R(;-p ,q ) R(;p ,q )-R(-;p ,q )

0.0 0.1 0.2 0.3 0.4 0.5 0.6 20 40 60 80 (fm) 1

• 1[Re(U)] (MeV2)

2 sources on 450 configs

fm

a = 0.060