Fits and Related Systematics for the
.
Hadronic Vacuum Polarization on the Lattice
Santi Peris
(U.A. Barcelona)
Based on coll. with Christopher Aubin (Fordham U.), Tom Blum (Connecticut U.),
Maarten Golterman (SFSU) & Kim Maltman (York U. & U. Adelaide)
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.1/16
Lots
of
activity
!
, 052001 (2003)
•
•
•
T. Blum, Phys. Rev. Lett. 91
•
•
•
•
P. Boyle, L. Del Debbio, E. Kerrane and J. Zanotti, Phys. Rev. D 85, 074504 (2012)
•
A. Francis, B. Jaeger, H. B. Meyer and H. Wittig, Phys. Rev. D 88, 054502 (2013)
[arXiv:1306.2532 [hep-lat]].
•
C. Aubin, T. Blum, M. Golterman and S. Peris, Phys. Rev. D 88, 074505 (2013)
[arXiv:1307.4701 [hep-lat]].
•
C. Aubin, T. Blum, M. Golterman and S. Peris, Phys. Rev. D 86, 054509 (2012)
[arXiv:1205.3695 [hep-lat]].
•
M. Golterman, K. Maltman and S. Peris, Phys. Rev. D 88, 114508 (2013)
[arXiv:1309.2153 [hep-lat]].
•
B. Chakraborty, C. T. H. Davies, G. C. Donald, R. J. Dowdall, J. Koponen,
G. P. Lepage and T. Teubner, arXiv:1403.1778 [hep-lat].
•
...
C. Aubin and T. Blum, Phys. Rev. D 75, 114502 (2007)
X. Feng, K. Jansen, M. Petschlies and D. B. Renner, Phys. Rev. Lett. 107, 081802
(2011)
M. Della Morte, B. Jager, A. Juttner and H. Wittig, JHEP 1203, 055 (2012)
G. M. de Divitiis, R. Petronzio and N. Tantalo, Phys. Lett. B 718, 589 (2012)
X. Feng, S. Hashimoto, G. Hotzel, K. Jansen, M. Petschlies and D. B. Renner,
arXiv:1305.5878 [hep-lat].
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.2/16
Introduction
P ∼
(g − 2)HV
µ
Lautrup-de Rafael ’69
Blum ’02
h
i
R∞
2
2
2
f (Q )
Π(Q ) − Π(0)
0 dQ
| {z }
known
P
Π(Q2 ) − Π(0) =⇒ (g − 2)HV
µ
⋆ integrand strongly peaked
at
Q2 ∼ m2µ /4
∼ 0.003 GeV2
(g-2) integrand
0.015
0.010
0.005
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Q2
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.3/16
Introduction
P ∼
(g − 2)HV
µ
Lautrup-de Rafael ’69
Blum ’02
h
i
R∞
2
2
2
f (Q )
Π(Q ) − Π(0)
0 dQ
| {z }
known
P
Π(Q2 ) − Π(0) =⇒ (g − 2)HV
µ
⋆ integrand strongly peaked
at
Q2 ∼ m2µ /4
∼ 0.003 GeV2
if no good data in region of curvature =⇒ possibly wrong results !
(g-2) integrand
(even with good χ2 )
0.015
Need reliable fitting function !
0.010
0.005
how to test this theoretical error?
0.000
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Q2
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.3/16
A τ -based model for I = 1 contributions
0.14
Boito, Cata, Golterman, Jamin, Mahdavi, Maltman, Osborne, SP
0.12
0.10
t ≤ tmin GeV2 → OPAL data.
t ≥ tmin GeV2 :
0.08
0.06
0.04
ImΠ(t) = ρ Pert.Th. (t) + e−δ−γ t sin(α + βt)
0.02
0.00
0.0
’11 + ’12
0.5
1.0
1.5
2.0
2
s0 HGeV L
2.5
3.0
Π(Q2 ) = −Q2
R∞
4m2
π
dt ImΠ(t)
π t(t+Q2 )
We take tmin = 1.5 GeV2 .
GOAL: test fitting ansätze accuracy in lattice determinations
• Take typical lattice Q2 values + lattice covariance matrix
(e.g., Aubin et al. ’12, 643 × 144 lattice, a = 0.06 fm, mπ = 220MeV, periodic BCs)
• Generate fake lattice data for Π(Q2 ) − Π(0) and compare with true answer from model.
• You should try this model to check your systematics: it’s very physically motivated !
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.4/16
Fitting functions
Aubin, Blum, Golterman, SP ’12
⋆ Padés, model independent, they enjoy a convergence theorem for N → ∞:
Π(Q2 ) = Π(0) + Q2
Π(0), a′ s and b′ s are fitting parameters.
a0 +
|
!
N
X
ar
2+b
Q
r
r=1
{z
}
Pade
⋆ VMD is not a Pade, since you fix b1 = Mρ2 . ( true Π(Q2 ) has cut starting at 4m2π ...)
We have:
a0 6= 0 =⇒ [N, N ] Pade; a0 = 0 =⇒ [N − 1, N ] Pade.
For instance:
•
•
etc...
a1
Q2 +b1
a0 +
is a [0,1] Pade =⇒
a1
2
Q +b1
Π(Q2 )
is a [1,1] Pade =⇒
= Π(0) +
Π(Q2 )
Q2
= Π(0) +
a1
Q2 +b1
Q2
a0 +
a1
2
Q +b1
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.5/16
Example: Stieltjes. No errors.
Toy Vac. Pol. function (ImΠ ≥ 0), with a cut, −∞ ≤ Q2 ≤ −1 :
Π(0) − Π(Q2 )
Q2
=
Z
∞
1
dt
1
(t + Q2 ) t
2
log 1 + Q
1
Q2
Theorem: As N → ∞ , with ai , bi determined from the function (and derivatives) at
Q2 = 0, or at multiple points,
=
a0 + a1 Q2 + a2 Q4 + ... + aN Q2N
1 + b1 Q2 + ... + bN Q2N
−→
Π(0) − Π(Q2 )
Q2
everywhere in a compact region in complex Q2 , except on the cut.
Relat. Error
1.0
0.020
Poles
@33D
0.5
0.015
0.010
x-14
@44D
0.005
@55D
0.000
2
4
6
8
Q2
10
-12
x-10 -8x -6
x x -4
xxxxx-2
xxxxxx
Q2
-0.5
-1.0
Poles6= Physics
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.6/16
Model Fits
Golterman, Maltman, SP
’13
P|
−7 .
“Exact result”: (g − 2)HV
µ
Q2 ≤1 GeV2 = 1.204 × 10
Fit interval 0 < Q2 ≤ 1 GeV2 , (49 points).
VMD
VMD+
[0, 1]
[1, 1]
[1, 2]
[2, 2]
(g − 2)µ × 107
Pull = (exact - fit) / error
Error×107
χ2 /dof
2189/47 ×
67.4/46
285/46
61.4/45
55.0/44
54.6/43
Pull
VMD “flavors” :
•
VMD:
•
VMD+:
[0,1] Pade with b1 = Mρ2 .
[1,1] Pade with b1 = Mρ2 (i.e. VMD + linear polynomial)
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.7/16
Model Fits
Golterman, Maltman, SP
’13
P|
−7 .
“Exact result”: (g − 2)HV
µ
Q2 ≤1 GeV2 = 1.2059 × 10
Fit interval 0 < Q2 ≤ 1 GeV2 , (49 points).
VMD
VMD+
[0, 1]
[1, 1]
[1, 2]
[2, 2]
(g − 2)µ × 107
1.3201
Pull = (exact - fit) / error
Error×107
0.0052
χ2 /dof
2189/47
67.4/46
285/46
61.4/45
55.0/44
54.6/43
Pull
-
VMD “flavors” :
•
VMD has a bad χ2 and (g − 2).
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.7/16
Model Fits
Golterman, Maltman, SP
’13
P|
−7 .
“Exact result”: (g − 2)HV
µ
Q2 ≤1 GeV2 = 1.2059 × 10
Fit interval 0 < Q2 ≤ 1 GeV2 , (49 points).
VMD
VMD+
[0, 1]
[1, 1]
[1, 2]
[2, 2]
•
•
(g − 2)µ × 107
1.3201
1.0658
Pull = (exact - fit) / error
Error×107
0.0052
0.0076
χ2 /dof
2189/47
67.4/46
285/46
61.4/45
55.0/44
54.6/43
Pull
18
VMD has a bad χ2 and (g − 2).
VMD+ also gets it wrong although the χ2 is good =⇒ DANGER !
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.7/16
Model Fits
Golterman, Maltman, SP
’13
P|
−7 .
“Exact result”: (g − 2)HV
µ
Q2 ≤1 GeV2 = 1.2059 × 10
Fit interval 0 < Q2 ≤ 1 GeV2 , (49 points).
VMD
VMD+
[0, 1]
[1, 1]
[1, 2]
[2, 2]
•
•
•
(g − 2)µ × 107
1.3201
1.0658
0.8703
1.116
1.182
1.177
Pull = (exact - fit) / error
Error×107
0.0052
0.0076
0.0095
0.022
0.043
0.058
χ2 /dof
2189/47
67.4/46
285/46
61.4/45
55.0/44
54.6/43
Pull
18
4
0.5
0.5
VMD has a bad χ2 and (g − 2).
VMD+ also gets it wrong although the χ2 is good =⇒ DANGER !
Pades [1,2] and [2,2] get it right, but the error is ∼ 4%.
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.7/16
Moral
Take, e.g., the VMD+ case
You may think this is a good fit for an accurate (g − 2)µ :
solid line=VMD+ Fit
Π(Q2 ) − Π(0)
0.08
0.06
0.04
0.02
0.00
0.0
0.5
1.0
1.5
2.0
2.5
Q2
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.8/16
Moral
while, in fact, this is what you should be looking at:
dashed blue=Truth
solid line=VMD+ Fit
(g-2) Integrand
0.015
0.010
0.005
0.000
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Q2
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.8/16
Moral
while, in fact, this is what you should be looking at:
dashed blue=Truth
solid line=[2,2] Pade Fit
(g-2) Integrand
0.015
0.010
0.005
0.000
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Q2
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.8/16
Other Pade strategies are possible...
Chakraborty et al. arXiv:1403.1778
Aubin et al. (unpublished)
Instead of fitting Pades to lattice data in a Q2 interval, determine Taylor coeff’s around
Q2 = 0 and then build Pades to compute avac.pol.
.
µ
2
2
Π(Q ) − Π(0) = Q
with Πj =
(−1)j+1
(2j+2)!
R
2
4
6
Π1 + Π2 Q + Π3 Q + Π4 Q + ...
d4 x t2j+2 hJ1 (x)J1 (0)i.
In our model Π1 = −0.178565 etc... and construct [0, 1], [1, 1], [1, 2], .. to compute
avac.pol.
(0 ≤ Q2 ≤ 1 GeV2 ).
µ
ÈPade-ExactȐExact
0.012
@0,1D
Lattice Errors: Chakraborty et al. use s and c quarks
0.010
0.008
@1,1D
and find
0.006
0.004
0.002
then
@1,2D
Pade
δaµ
aµ
∼ǫ
δΠj
Πj
∼jǫ
(both w/o corr’s or w 100% corr’s)
and u, d quarks ??
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.9/16
Conclusions
•
VMD-type fits turn out to be unreliable for an accuracy in (g − 2)µ of few per cent.
•
Benchmark your fitting method: Use a (reasonable) model to generate fake data
with your lattice Q2 values and cov. matrix to get a good check on your systematic
error. (If you are interested we could try to help).
•
Do not only rely on the χ2 of your fit for assessing the accuracy in (g − 2)µ .
•
Blow up the region of the integrand around Q2 ∼ m2µ . You need good data there.
(Twisted BC’s (Della Morte et al. ’12, Aubin et al. ’13), analytic continuation (Feng
et al. ’13), ...)
Showing plots of Π(Q2 ) for large Q2 is very misleading.
•
Model independence is necessary: Pades are a useful example (convergent and
efficient at low orders).
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.10/16
BACK-UP SLIDES
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.11/16
“IN GOD WE TRUST,
ALL THE OTHERS MUST BRING DATA.”
(W. Edwards Deming, American Statistician, 1900-1993.)
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.12/16
Science Fiction
P|
−7 .)
(Recall exact value: (g − 2)HV
µ
Q2 ≤1 GeV2 = 1.2059 × 10
Reduce the previous covariance matrix by 104 :
VMD
VMD+
[0, 1]
[1, 1]
[1, 2]
[2, 2]
[2, 3]
(g − 2)µ × 107
1.3210
1.0732
0.89774
1.1011
1.1644
1.1884
1.1987
Error×107
0.00005
0.00008
0.00010
0.0002
0.0004
0.0015
0.0028
χ2 /dof
2 × 107 /47
7 × 104 /46
2 × 107 /46
5 × 104 /45
1340/44
76.4/43 (?)
42.0/42
Pull
12
2.6
•
•
VMD-type fits are very bad.
•
•
[2, 3] reaches error comparable to present e+ e− and τ -data based determination.
Pade fits are eventually better, but need good data around the peak for χ2 errors to
be reliable.
To reduce “Pull”, need twisting or larger volumes to have very good data in the
region of curvature of the integrand. See also the strategies in de Divitiis et al. ’13
and Feng et al. ’13.
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.13/16
Duality Violations(I)
• OPE valid in euclidean, but not in minkowski. We know that spectrum 6= OPE
Im PHtL
• We expect
t
( @ large t ) :
ImΠDV ∼ Im(Π − ΠOP E ) ∼
κ
e−γt
| {z }
OPE asympt.
• ΠDV (s) → 0 as |s| → ∞. Then:
−∞
−s0
sin(α + βt)
|
{z
}
Regge
(Cata-Golterman-S.P. ’05)
Re q 2
1
− 2πi
H
|z|=s0
dz w(z) ΠDV (z) =
−
Z
∞
ds
s0
w(s)
1
ImΠDV
π
(s)
| {z }
extrapolation!
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.14/16
D. Violations(II)
Blok-Shifman-Zhang ’98; Cata-Golterman-SP ’05’08; Jamin ’11
Explicit realization only models, no theory. Take ΛQCD = 1; F ∼ 0.1, decay constant.
• 1 resonance (M → M + iΓ/2):
F2
F2
−→ 2
√
q2 − n
q −n−i nΓ
• Regge-like tower: n = 1, 2, 3, ...
Π(q 2 )
∼
∼
∞
X
F2
z+n
n
ψ(z) =
,
P
• For q 2 > 0 −→ ψ(z) = ψ(−z) −
ζ ≃ 1 − O(
1
)
Nc
1
z
cn
zn
− π cot(πz)
X cn
−q 2
2
ImΠ(q ) ∼ Im(log z)+Im
+ F e Nc
n
z
2
,
cut, q 2 >0
d log Γ(z)
dz
• For q 2 < 0 −→ Π(q 2 ) ∼ log z +
z = (−q 2 )ζ
| {z }
,
sin(α + β q 2 )
F ∼ 0.1
;
α, β ∼ 1
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.15/16
Uncorrelated Fits
P|
−7 .)
(Recall exact value: (g − 2)HV
µ
Q2 ≤1 GeV2 = 1.2059 × 10
VMD
VMD+
[0, 1]
[1, 1]
[1, 2]
•
(g − 2)µ × 107
1.2145
1.085
0.999
1.17
1.30
Error×107
0.0082
0.017
0.023
0.074
0.32
Q2 /dof
75.2/47
15.0/46
20.1/46
13.8/45
13.6/44
Pull
-1.1
7
9
0.4
-0.3
“Error" column is the result of linear propagation
Fits and Related Systematics for the.Hadronic Vacuum Polarization on the Lattice – p.16/16