Excited States and Precision Calculations of Hadron Structure

David Richards Jefferson Lab

QCD Evolution Workshop Jefferson Lab

22-26 May, 2017

Systematic Uncertainties• Non-zero lattice spacing and continuum limit a → 0 • Finite volume V →∞ • mπ → mπphys

• Number of quark flavors

• Isolating ground states• Nucleon Charges• Nucleon Charge Radius

Precision Calculations of Nucleon Charges

Hadron StructureM Constantinou, arXiv:1511.00214

• Governs beta-decay rate • Important for proton-proton fusion

rate in solar models • Benchmark for lattice QCD

calculations of hadron structure

e.g. novel interactions probed in ultra-cold neutron decay

d

u

V, A, S, T, P

d u

d u

n

p

gT = ZT gS = ZS

€

Heff ⊃GF εS u d × e (1− γ 5)ν e + εT u σµν d ×e σ µν (1− γ 5)ν e[ ]

€

p u d n

€

p u σµν d n

R Gupta, 2014

Calculation of Physics Observables

Our paradigm: nucleon mass

Noise:

whence

C(t)/

pC�2(t) ' e�(mN�3m⇡/2)t

C(t) =X

~x

hN(~x, t)N(0)i =X

n

| An

|2 e

�Ent

C

�

2(t) =X

~x

hNN(~x, t)NN(0)i �! e

�3m⇡t

Use local nucleon interpolating operators [uC�5(1± �4)d]u

Replace quark field by spatially extended (smeared) quark field

�! (1� �2r2/4N)N

Excited States: Smearing RadiiYoon et al., Phys. Rev. D 93, 114506 (2016)

me↵ = lnC(t)/C(t+ 1) ! E0

Construct matrix of correlators: different smearing radii

Variational Method

Delineate contributions using variational method: solve

Eigenvectors, with metric C(t0), are orthonormal and project onto the respective states

Subleading terms ➙ Excited states

C(t)v(N)(t, t0) = �N (t, t0)C(t0)v(N)(t, t0).

v(N0)†C(t0)v

(N) = �N,N 0

�N (t, t0) ! e�EN (t�t0)(1 +O(e��E(t�t0)))

C

ij

(t) =X

~x

hNi

(~x, t)Nj

(0)i =X

n

A

i

n

A

j†n

e

�Ent

Nucleon Mass - II

Variational method: single-state domination nearer source increase signal-to-noise

me↵ = lnC(t)/C(t+ 1) ! E0

}Variational

Fixed Smearing

Variational Method

Grey Band : tsep ! 1

1.20

1.25

1.30

1.35

1.40

1.45

1.50

-6 -4 -2 0 2 4 6

g A

τ - tsep/2

tsep=10tsep=12tsep=14

tsep=16V357V579

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

-8 -6 -4 -2 0 2 4 6 8

g A

τ - tsep/2

Extraptsep=10tsep=12

tsep=14tsep=16tsep=18

Fixed source smearing

C

3pt� (tsep, ⌧) =

X

~x,~y

hN(~x, tsep)�(~y, ⌧)N(0, 0)i

= | A0|2h0 | � | 0ie�M0tsep +

| A1 |2 h1 | � | 1ie�M1tsep +

A0A⇤1h0 | � | 1ie�M0⌧

e

�M1(tsep�⌧). . .

Variational Method

Variational Comparison - II

Variational - III

Controlling excited states essential for precision calculations!

Renormalized Charges

ID Lattice Theory a fm M⇡(MeV) gu�dA gu�d

S gu�dT gu�d

V

a127m285 2+1 clover-on-clover 0.127(2) 285(6) 1.249(28) 0.89(5) 1.023(21) 1.014(28)a12m310 2+1+1 clover-on-HISQ 0.121(1) 310(3) 1.229(14) 0.84(4) 1.055(36) 0.969(22)a094m280 2+1 clover-on-clover 0.094(1) 278(3) 1.208(33) 0.99(9) 0.973(36) 0.998(26)a09m310 2+1+1 clover-on-HISQ 0.089(1) 313(3) 1.231(33) 0.84(10) 1.024(42) 0.975(33)a091m170 2+1 clover-on-clover 0.091(1) 166(2) 1.210(19) 0.86(9) 0.996(23) 1.012(21)a09m220 2+1+1 clover-on-HISQ 0.087(1) 226(2) 1.249(35) 0.80(12) 1.039(36) 0.969(32)a09m130 2+1+1 clover-on-HISQ 0.087(1) 138(1) 1.230(29) 0.90(11) 0.975(38) 0.971(32)

Consistency between different actions Matrix Elements of 1st excited state?

Yoon et al., Phys. Rev. D 95, 074508 (2017)

arXiv:1704.01114, Berkowitz et al

Feynman-Hellman Method proceeds through looking at variation of a spectral function w.r.t external current.

@En@� = hn | H� | ni where H = H0 + �H�

See talk by K. Orginos

Nucleon Charge Radius

Bouchard, Chang, Orginos, Richards, PoS LATTICE2016 (2016) 170

Proton EM form factors• Nucleon Pauli and Dirac Form Factors described in

terms of matrix element of vector current

hN | Vµ | Ni(~q) = u(~pf )

Fq(q

2)�µ + �µ⌫q⌫F2(q2)

2mN

�u(~pi)

• Alternatively, Sach’s form factors determined in experiment GE(Q

2) = F1(Q2)� Q2

4M2F2(Q

2)

GM (Q2) = F1(Q2) + F2(Q

2)

Charge radius is slope at Q2 = 0@GE(Q2)

@Q2

����Q2=0

= �1

6hr2i = @F1(Q2)

@Q2

����Q2=0

� F2(0)

4M2

EM Form factors

15

Approved expt E12-07-109PRAD: E12-11-106

Research Management Plan March 3, 2014 p. 5

2.2 Proton Form-Factor Ratio Measurements up to Q2= 12 GeV2 using Recoil Polarization

Introduction The experiment GEp (E12-07-109) was approved by PAC32 in August of 2007 and wasthe experiment that provided the original motivation for the Super Bigbite Spectrometer. It will measurethe Sachs Form Factors ratio Gp

E/GpM of the proton using the polarization-transfer method in the reaction

p(�e, e��p). The polarization of the recoil proton will be measured using a large-acceptance spectrometer,based on the Super Bigbite magnet, that will incorporate a double polarimeter instrumented with GEMtrackers and a highly-segmented hadron calorimeter.

The electron will be detected in coincidence by a electromagnetic calorimeter that is sometimes referredto as “BigCal”. PAC35 allocated 45 days of beam time for the proposed measurement and recommended amaximum value of Q2 = 12 GeV2.

These parameters were used to readjust the original plan of measurements which will be made at threevalues of Q2 : 5, 8, and 12 GeV2 , while achieving an error in the ratio Gp

E/GpM of 0.07. The projected results

are shown in Fig 3, in which we show results from earlier GpEmeasurements, and the anticipated errors for the

present GEp experiment. The excellent precision that GEp will obtain even at 12 GeV2 is clearly evident.Additional measurements at even higher values of Q2 will be evaluated after SBS commissioning.

Figure 3: GpE/Gp

M existing measurements and expected statistical accuracy for the GEp experiment. Theprojected errors for the measurements made with the Super Bigbite Spectrometer are indicated by the filledblue squares, corresponding to 45-day run with the recommended highest value of momentum transfer 12GeV2.

Equipment A schematic representation of the experiment is shown in Fig. 4.

Q2 . 4.1 GeV2Q2 . 8.2 GeV2

LHPC, Syritsyn, Gambhir, Orginos et al, Lattice 2016

UKQCD, Lellouch, Richards et al., NPB444 (1995) 401

Nucleon Charge Radius

Direct calculation of charge radius through coordinate-space moments

Boosted interpolating operatorsBali et al., Phys. Rev. D 93, 094515 (2016)Green et al, arXiv:1404.40

Differing treatment of Excited states

Isgur-Wise Function and CKM matrix

Lattice

Extract Vcb if know intercept at zero recoil

UKQCD, L. Lellouch et al., Nucl. Phys. B444, 401 (1995), hep-lat/9410013

Calculate slope at zero recoil..

Moment Methods

• Introduce three-momentum projected three-point function

• Now take derivative w.r.t. k2

C3pt(t, t0) =X

~x,~x

0

DNa

t,~x

�t

0,~x

0Nb

0,~0

Ee�ikx

0z

C

03pt(t, t

0) =X

~x,~x

0

�x

0z

2ksin (kx0

z

)DN

a

t,~x

�t

0,~x

0N

b

0,~0

E

limk

2!0C

03pt(t, t

0) =X

~x,~x

0

�x

02z

2

DN

a

t,~x

�t

0,~x

0N

b

0,~0

E.

whence

Odd moments vanish by symmetry

Moment Methods - II

• Analogous expressions for two-point functions:

C2pt(t) =X

~x

DN b

t,~x

Nb

0,~0

Ee�ikxz

⇒C

02pt(t) =

X

~x

�x

z

2ksin (kx

z

)DN

b

t,~x

N

b

0,~0

E

⇒limk

2!0C

02pt(t) =

X

~x

�x

2z

2

DN

b

t,~x

N

b

0,~0

E.

Lowest coordinate-space moment ⇔ slope at zero momentum

Moment Methods - II

• Analogous expressions for two-point functions:

C2pt(t) =X

~x

DN b

t,~x

Nb

0,~0

Ee�ikxz

⇒C

02pt(t) =

X

~x

�x

z

2ksin (kx

z

)DN

b

t,~x

N

b

0,~0

E

⇒limk

2!0C

02pt(t) =

X

~x

�x

2z

2

DN

b

t,~x

N

b

0,~0

E.

Lowest coordinate-space moment ⇔ slope at zero momentum

Lattice Details• Two degenerate light-quark flavors, and strange quark

set to its physical valuea ' 0.12 fm

m⇡ ' 400 MeV

Lattice Size : 243 ⇥ 64

• To gain control over finite-volume effects, replicate in z direction: 24⇥ 24⇥ 48⇥ 64

Two-point correlatorln [C2pt(t, xz)]

Any polynomial moment in xz converges

lnC2pt(t, xz)/C2pt(t, xz + 1)

“Effective mass”

Three-point correlatorln [C3pt(t

0, x

0z)]

“Effective mass”

• Spatial moments push the peak of the correlator away from origin

• Larger finite volume corrections compared to regular correlators

Fitting the data…

C3pt(t, t0) =X

n,m

Z†an (0)�nm(k2)Zb

m(k2)

4Mn(0)Em(k2)e�Mn(0)(t�t0)e�Em(k2)t0

where Z†an (0) ⌘ h⌦|Na|n, pi = (0, 0, 0)i

Zbm(k2) ⌘ hm, pi = (0, 0, k)|N b |⌦i

�nm(k2) ⌘ hn, pi = (0, 0, 0) |�|m, pi = (0, 0, k)i

C2pt(t) =X

m

Zb†m (k2)Zb

m(k2)

2Em(k2)e�Em(k2)t

Allow for multi-state contributions in the fit

Fitting - II• Now look at the functional form of derivatives:

C 02pt(t) =

X

m

C2ptm (t)

✓2Zb0

m(k2)

Zbm(k2)

� 1

2[Em(k2)]2� t

2Em(k2)

◆

C 03pt(t, t

0) =X

n,m

C3ptnm(t, t0)

⇢�0nm(k2)

�nm(k2)+

Zb0m(k2)

Zbm(k2)

� 1

2[Em(k2)]2� t0

2Em(k2)

�

spatially extended sources

Second distance scale

Fitting - III

In practice we use multi-exponential, Bayesian fits

F1 Form Factor

Outlook• Controlling the contribution from excited states in study of hadron

structure is a crucial for precise and accurate calculations • The approach of the variational method is a powerful way of

addressing systematic uncertainties due to excited state • Current basis of operators based on quasi-local sources. Exploring

basis that admits non-zero orbital structure

27

!D

m=�1 = ip2

⇣ !D

x

� i !D

y

⌘

!D

m=0 = i !D

z

!D

m=+1 = � ip2

⇣ !D

x

+ i !D

y

⌘.

Starting point

Introduce circular basis:

Dudek, Edwards, arXiv:1201.2349R.G.Edwards et al., arXiv:1104.5152

• Structure of Excited States - Raul Briceno