[PPT] - Disconnected beyond 2+2 Thomas Blum Norman Christ Ma sashi Hayakawa PowerPoint Presentation

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Disconnected beyond 2+2

Thomas Blum UConn/RBRC Norman Christ Columbia Masashi Hayakawa Nagoya Taku Izubuchi BNL/RBRC Luchang Jin UConn/RBRC Chulwoo Jung BNL Christoph Lehner BNL Cheng Tu UConn

and the RBC/UKQCD collaborations Jun 18, 2018 Helmholtz-Institut Mainz Second Plenary Workshop of the Muon g-2 Theory Initiative

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HLbL Connected diagrams 2/29

xsrc xsnk y′, σ′ z′, κ′ x′, ρ′ xop, ν z, κ y, σ x, ρ xsrc xsnk y′, σ′ z′, κ′ x′, ρ′ xop, ν z, κ y, σ x, ρ xsrc xsnk y′, σ′ z′, κ′ x′, ρ′ xop, ν z, κ y, σ x, ρ

Permutations of the three internal photons are not shown.
There should be gluons exchange between and within the quark loops, but are not drawn.

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Disconnected diagrams 3/29

One diagram (the biggest diagram below, referred to as 2 + 2) do not vanish even in the

SU(3) limit.

We extend the method and computed this leading disconnected diagram as well.

xsrc xsnk z′, κ′ y′, σ′ x′, ρ′ xop, ν z, κ y, σ x, ρ

xsrc xsnk y′, σ′ x′, ρ′ z′, κ′ xop, ν z, κ y, σ x, ρ xsrc xsnk y′, σ′ z′, κ′ x′, ρ′ xop, ν z, κ y, σ x, ρ xsrc xsnk z′, κ′ y′, σ′ x′, ρ′ xop, ν z, κ y, σ x, ρ xsrc xsnk y′, σ′ x′, ρ′ z′, κ′ xop, ν z, κ y, σ x, ρ

xsrc xsnk z′, κ′ y′, σ′ x′, ρ′ xop, ν z, κ y, σ x, ρ

Permutations of the three internal photons are not shown.
There should be gluons exchange between and within the quark loops, but are not drawn.
We need to make sure that the loops are connected by gluons by “vacuum” subtraction.

So the diagrams are 1-particle irreducible.

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Disconnected diagram beyond 2+2 4/29

xsrc xsnk y′, σ′ x′, ρ′ z′, κ′ xop, ν z, κ y, σ x, ρ

xsrc xsnk y′, σ′ z′, κ′ x′, ρ′ xop, ν z, κ y, σ x, ρ

St

il lw

r

k in g in pr

g

r ess.

The right loop has been calculated by Christoph Lehner (can also be used to calculated

disconnected HVP) and saved to disk.

The left loop can be evaluated by two point source propagators at x and y. We can then

randomly sample x and y, similar to the way we evaluted the connected diagrams.

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5/29

Pion Transition Form Factor (TFF) on Lattice: RBC results

Thomas Blum UConn/RBRC Norman Christ Columbia Masashi Hayakawa Nagoya Taku Izubuchi BNL/RBRC Luchang Jin UConn/RBRC Chulwoo Jung BNL Christoph Lehner BNL Cheng Tu UConn

and the RBC/UKQCD collaborations Jun 18, 2018 Helmholtz-Institut Mainz Second Plenary Workshop of the Muon g-2 Theory Initiative

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Outline 6/29

We will be working in Euclidean space by default.

Pion TF

F f

r

mu l a t ion

Model and Lattice results
Contribution to HLbL with pion TFF

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Pion TFF formulation 7/29

0|T i Jµ(u) i Jν(v)|π0(p ) (1) Momentum space TFF F(q1

2, q2 2) (X.D. Ji, C. Jung, [hep-lat/0101014]):

d4u e−iq1·u−iq2·v 0|T i Jµ(u) i Jν(v)|π0(p

) = i 4 π2 Fπ ǫµ,ν,ρ,σ q1,ρ q2,σ F(q1

2, q2 2).

(2) Coordinate space TFF Fc(x, z2) (previously presented at UConn by Cheng Tu): 0|T i Jµ(u) i Jν(v)|π0(p ) = i 4 π2 Fπ ǫµ,ν,ρ,σ (−i ∂ρ

u) (−i ∂σ v) F ′(p· (u − v), (u − v)2) eip·v,

(3) Let r = u − v, Fc(x, r2) is the Fourier transformation of F ′(p· r, r2): F ′(p· r, r2) =

−∞

∞

dx Fc(x, r2) eixp·r. (4) Interestingly, we can prove that: Fc(x, r2) = 0 if x < 0 or x > 1. (5)

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Pion TFF formulation 8/29

0|T i Jµ(u) i Jν(v)|π0(p ) = i 4 π2 Fπ ǫµ,ν,ρ,σ(−i ∂ρ

u) (−i ∂σ v)

×

1

dx Fc(x, (u − v)2) π0(xu + (1 − x) v)

π0(p

)

(6)

= i 4 π2 Fπ ǫµ,ν,ρ,σ ×

1

dx [−∂ρ

u Fc(x, (u − v)2)] ∂σπ0(xu + (1 − x) v)

π0(p

)

(7)

The coordinate space form factor Fc(x, r2) can be interpreted this way:

The dependence on x describe the distribution of the pion source along the segment

between the two EM currents.

In the r2 → 0 limit, the function can be factorized into pion d

ist r ibut ion a mpl it u des (PD A ). At tree level, Fc(x, r2) is the same as PDA after normalization. Fc(x, r2) ∼ x(1 − x).

The parameter r = (u − v) is the separation between the two EM currents.

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Pion TFF formulation: proof for 0 x 1 9/29

Define P ˆ

π0 =

d3p

(2π)3 |π0(p ) 1 2 Eπ0,p

π0(p

)|. (8)

π0(x) P

ˆ

π0 π0(y)

= G(x − y) =
d4p

(2π)4 eip·(x−y) p2 + mπ

2

(9)

T [i Jµ(u)i Jν(v)]P

ˆ

π0 π0(w)

=
i

4 π2 Fπ ǫµ,ν,ρ,σ

−∞

∞

dx [−∂ρ

u Fc(x, (u − v)2)] ∂σπ0(xu + (1 − x) v)P

ˆ

π0π0(w)

=

i 4 π2 Fπ ǫµ,ν,ρ,σ

−∞

∞

dx [−∂ρ

u Fc(x, (u − v)2)] ∂σG(x u + (1 − x)v − w).

(10) Let w = xu + (1 − x)v, the above expression should not be singular when x > 1 or x < 0. Therefore Fc(x, (u − v)2) should be zero for x outside of [0, 1].

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Pion TFF formulation 10/29

Let f(|r|) be the function which describes the strength of the π0γ γ coupling:

1

d x [−∂ρ

r Fc(x, r2)] = 2 zρ

2Fπ

2

3 1 (r2)2

f(|r|).

(11)

Based on Chiral anomaly, (π2/2)

∞ (2Fπ 2/3)f(r) 2 r d r = 1, (F(0, 0) = 1).

Based on OPE, in the r → 0 limit, f(|r|) → 1, (F(Q2, Q2) → 8π2Fπ

2/(3 Q2)).

For HLbL, the long distance contribution should be dominated by the π0 exchange process, where the π0 propagator for a relatively long distance, while the two photons created/anni- hilated the pion are fairly close. Therefore, the x dependence of Fc(x, r2) is less important. Instead, we should focus on the total strength f(|r|).

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Outline 11/29

We will be working in Euclidean space by default.

Pion TFF formulation
M
d

ela n d L a t t ic e r esul t s

Contribution to HLbL with pion TFF

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Pion TFF formulation 12/29

Vector Meson Dominance model F VMD(q1

2, q2 2) =

mV

2

q1

2 + mV 2

mV

2

q2

2 + mV 2

(12) Two Ends model F TE(q1

2, q2 2) =

mV

2 /2

q1

2 + mV 2 + mV 2 /2

q2

2 + mV 2

(13) Lowest Meson Dominance model F LMD(q1

2, q2 2) = 8π2Fπ 2

3mV

2 F TE(q1 2, q2 2) +

1 − 8π2Fπ

2

3mV

2

F VMD(q1

2, q2 2)

(14) Relation between Momentum space form and Coordinate space form: F(q1

2, q2 2) =

d4z e−iq1·r
1

dx Fc(x, r2) eixp·r =

1

dx

d4r e−i((1−x)q1−xq2)·r Fc(x, r2)

(15)

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Pion TFF formulation: VMD model 13/29

F VMD(q1

2, q2 2) =

mV

2

q1

2 + mV 2

mV

2

q2

2 + mV 2

=

1

d x mV

4

[(1 − x)(q1

2 + mV 2 ) + x(q2 2 + mV 2 )]2

=

1

d x mV

4

[[(1 − x)q1 − x q2]2 + mV

2 − x(1 − x)mπ 2]2

(16) Recall F(q1

2, q2 2) =

1

dx

d4r e−i((1−x)q1−xq2)·r Fc(x, r2)

(17) Fc

VMD(x, r2) =

d4p

(2π)4 mV

4 eip·r

[p2 + mV

2 − x(1 − x)mπ 2]2

(18) The dependence on x is very weak. Pion is uniformly created/annihilated between the two EM currents.

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Pion TFF formulation: TE model 14/29

F TE(q1

2, q2 2) =

mV

2 /2

q1

2 + mV 2 + mV 2 /2

q2

2 + mV 2

=

1

d x δ(x) + δ(x − 1) 2 mV

2

((1 − x)q1 − x q2)2 + mV

2

(19) Recall F(q1

2, q2 2) =

1

dx

d4r e−i((1−x)q1−xq2)·r Fc(x, r2)

(20) Fc

TE(x, r2) = δ(x) + δ(x − 1)

2

d4p

(2π)4 mV

2 eip·r

p2 + mV

2

(21) The value for x is either 0 or 1. Pion is created/annihilated at the two ends of the segment between the two EM currents location.

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Pion TFF formulation: LMD model 15/29

F LMD(q1

2, q2 2) = 8π2Fπ 2

3mV

2 F TE(q1 2, q2 2) +

1 − 8π2Fπ

2

3mV

2

F VMD(q1

2, q2 2)

(22)

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 f(r) r (fm) VMD model TE model LMD model

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Lattice results 16/29

RBC/UKQCD 243 × 64 Iwasaki+DSDR ensemble: mπ = 139 MeV, a−1 = 1.015 GeV. With zt = 0, f(|z|) can be evaluated with (tsep = 10a) 0|T i Jµ(z) i Jν(0)|π0(p = 0) = i 4 π2 Fπ ǫµ,ν,ρ,σ 2 zρ i pσ

2Fπ

2

3 1 (z2)2

f(|z|),

(23) Using 16 configurations and the point source propagators generated by computing the leading disconnected contribution to HLbL, we obtained:

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0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 f(r) r (fm) VMD model TE model LMD model 24D lattice

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Outline 17/29

We will be working in Euclidean space by default.

Pion TFF formulation
Model and Lattice results
C
n

t r ibut ion t

H

L bL w it h pion TF F

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Contribution to HLbL with pion TFF 18/29

xsrc xsnk x′ y′ z′ xop x y z Rmax = max {|x − y|, |y − z|, |z − x|} (24) Rmin = minx{|x − y|, |y − z|, |z − x|} (25) We will evaluate with the following parameter. mV = 770 MeV Fπ = 93 MeV (26) The QED part, we use the weighting function developed in [arXiv:1705.01067] by us.

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Contribution to HLbL with pion TFF: VMD 19/29

Discretization effect: a = 0.223 fm, 0.171 fm, 0.114fm.

−1 1 2 3 4 5 6 7 5 10 15 20 25 30 F2(0) (10−10) Rmax 24nt48-coarse-2-phys sub 24nt48-coarse-2-phys no-sub 32nt64-coarse-phys sub 32nt64-coarse-phys no-sub 48nt96-phys sub 48nt96-phys no-sub

The unit of x-axis is 0.223 fm, the lattice spacing of the coarsest lattice.

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Contribution to HLbL with pion TFF: VMD 20/29

Finite volume effect: a = 0.223fm, L = 5.5 fm, 7.3 fm, 10.9fm.

−1 1 2 3 4 5 6 7 5 10 15 20 25 30 35 40 F2(0) (10−10) Rmax 24nt48-coarse-2-phys sub 24nt48-coarse-2-phys no-sub 32nt64-coarse-2-phys sub 32nt64-coarse-2-phys no-sub 48nt96-coarse-2-phys sub 48nt96-coarse-2-phys no-sub

The unit of x-axis is 0.223 fm, the lattice spacing of the lattices.

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Contribution to HLbL with pion TFF: VMD 21/29

Extrapolations:

Infinite volume: use the largest volume L = 10.9fm as approximation.
Continuum: extrapolate from the two lattice spacing a=0.223fm,0.114fm, O(a2) scaling.

−1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 F2(0) (10−10) Rmax (fm) sub no-sub sub Rmin ≤ 1.0 fm no-sub Rmin ≤ 1.0 fm sub Rmin ≤ 0.5 fm no-sub Rmin ≤ 0.5 fm

Errorbars are statistical only.

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Contribution to HLbL with pion TFF: TE 22/29

Discretization effect: a = 0.223 fm, 0.171 fm, 0.114fm.

−2 2 4 6 8 10 12 14 5 10 15 20 25 30 F2(0) (10−10) Rmax 24nt48-coarse-2-phys sub 24nt48-coarse-2-phys no-sub 32nt64-coarse-phys sub 32nt64-coarse-phys no-sub 48nt96-phys sub 48nt96-phys no-sub

The unit of x-axis is 0.223 fm, the lattice spacing of the coarsest lattice.

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Contribution to HLbL with pion TFF: TE 23/29

Finite volume effect: a = 0.223fm, L = 5.5 fm, 7.3 fm, 10.9fm.

−2 2 4 6 8 10 12 5 10 15 20 25 30 35 40 F2(0) (10−10) Rmax 24nt48-coarse-2-phys sub 24nt48-coarse-2-phys no-sub 32nt64-coarse-2-phys sub 32nt64-coarse-2-phys no-sub 48nt96-coarse-2-phys sub 48nt96-coarse-2-phys no-sub

The unit of x-axis is 0.223 fm, the lattice spacing of the lattices.

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Contribution to HLbL with pion TFF: TE 24/29

Extrapolations:

Infinite volume: use the largest volume L = 10.9fm as approximation.
Continuum: extrapolate from the two lattice spacing a=0.223fm,0.114fm, O(a2) scaling.

−2 2 4 6 8 10 12 14 1 2 3 4 5 6 7 8 F2(0) (10−10) Rmax (fm) sub no-sub sub Rmin ≤ 1.0 fm no-sub Rmin ≤ 1.0 fm sub Rmin ≤ 0.5 fm no-sub Rmin ≤ 0.5 fm

Errorbars are statistical only.

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Contribution to HLbL with pion TFF: LMD 25/29

Discretization effect: a = 0.223 fm, 0.171 fm, 0.114fm.

−1 1 2 3 4 5 6 7 8 9 5 10 15 20 25 30 F2(0) (10−10) Rmax 24nt48-coarse-2-phys sub 24nt48-coarse-2-phys no-sub 32nt64-coarse-phys sub 32nt64-coarse-phys no-sub 48nt96-phys sub 48nt96-phys no-sub

The unit of x-axis is 0.223 fm, the lattice spacing of the coarsest lattice.

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Contribution to HLbL with pion TFF: LMD 26/29

Finite volume effect: a = 0.223fm, L = 5.5 fm, 7.3 fm, 10.9fm.

−1 1 2 3 4 5 6 7 8 9 5 10 15 20 25 30 35 40 F2(0) (10−10) Rmax 24nt48-coarse-2-phys sub 24nt48-coarse-2-phys no-sub 32nt64-coarse-2-phys sub 32nt64-coarse-2-phys no-sub 48nt96-coarse-2-phys sub 48nt96-coarse-2-phys no-sub

The unit of x-axis is 0.223 fm, the lattice spacing of the lattices.

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Contribution to HLbL with pion TFF: LMD 27/29

Extrapolations:

Infinite volume: use the largest volume L = 10.9fm as approximation.
Continuum: extrapolate from the two lattice spacing a=0.223fm,0.114fm, O(a2) scaling.

−1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 F2(0) (10−10) Rmax (fm) sub no-sub sub Rmin ≤ 1.0 fm no-sub Rmin ≤ 1.0 fm sub Rmin ≤ 0.5 fm no-sub Rmin ≤ 0.5 fm

Errorbars are statistical only.

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Conclusion 28/29

We developed coordinate space formulation of the pion transition form factor (TFF). It

turned out to be pion distribution amplitude at short distance.

We defined f(r) to describe the r (relative coordinate between the two EM currents)

dependence of the coordinate space TFF.

We calculated f(r) for three models (VMD, TE, LMD) and using lattice data (24D).
LMD model is designed to satisfy both the Chiral anomaly constraint and the OPE
constraint. Among the three models, it also agrees the lattice data the most.
We computed the π0 contribution to HLbL using the three models using lattice.

− The total results agree with the momentum space more analytical evalution. − We also obtained the models results in the long distance region, which can be used to correct the QCD finite volume errors of the lattice calculation of HLbL.

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