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
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
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
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
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
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