New Extraction of the Proton Radius from ep-Scattering Data
PSAS 2016
Hebrew University of Jerusalem, Jerusalem, IL
Gabriel Lee
Technion – Israel Institute of Technology
Phys. Rev. D 92, 013013 [arXiv:1505.01489] with J. Arrington, R. Hill
May 26, 2016
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
1 / 34
Outline
1
Background
2
Form Factors and Radiative Corrections
3
Uncorrelated and Constant Systematics
4
Correlated Systematics
5
Results
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
2 / 34
• • • We do not use the following data for averages, fits, limits, etc. • • •
P before 2010
0.879
rE
± 0.005 ± 0.006
0.912 ± 0.009 ± 0.007
0.871 ± 0.009 ± 0.003
0.84184± 0.00036 ± 0.00056
0.8768 ± 0.0069
0.008
0.844 +
− 0.004
0.897 ± 0.018
0.8750 ± 0.0068
0.895 ± 0.010 ± 0.013
0.830 ± 0.040 ± 0.040
0.883 ± 0.014
0.880 ± 0.015
0.847 ± 0.008
0.877 ± 0.024
BERNAUER
BORISYUK
HILL
POHL
MOHR
10
10
10
10
08
BELUSHKIN
07
BLUNDEN
05
MOHR
05
SICK
03
24 ESCHRICH
01
MELNIKOV
00
ROSENFELDR...
00
MERGELL
96
WONG
94
0.865 ± 0.020
MCCORD
91
0.862 ± 0.012
SIMON
80
0.880 ± 0.030
BORKOWSKI 74
0.810 ± 0.020
AKIMOV
72
0.800 ± 0.025
FREREJACQ... 66
0.805 ± 0.011
HAND
63
24 ESCHRICH 01 actually gives !r2 " = (0.69 ± 0.06
Two developments in 2010:
e p → e p form factor
reanalyzes old e p data
z-expansion reanalysis
µ p-atom Lamb shift
RVUE 2006 CODATA value
SPEC
Dispersion analysis
SICK 03 + 2γ correction
RVUE 2002 CODATA value
e p → e p reanalysis
ep → ep
1S Lamb Shift in H
e p + Coul. corrections
e p + disp. relations
reanalysis of Mainz e p
data
ep → ep
ep → ep
ep → ep
ep → ep
e p → e p (CH2 tgt.)
ep → ep
± 0.06) fm2 .
PDG, 2012
p MAGNETIC RADIUS
#!
2 "
I
This is the rms magnetic radius,
r
.
M
High-statistics, precision
ep-scattering experiment
at MAMI by the A1 collaboration.
I
0.777± 0.013measurements
± 0.010
New spectroscopic
in µH at PSI.
BERNAUER
10 SPEC
VALUE (fm)
DOCUMENT ID
TECN
COMMENT
e p → e p form factor
• • • We do not use the following data for averages, fits, limits, etc. • • •
0.876± 0.010 ± 0.016
0.854± 0.005
Gabriel Lee (Technion)
BORISYUK
BELUSHKIN
Extraction of r
p
10
07
reanalyzes old e p → e p data
Dispersion analysis
from ep-Scattering Data
May 26, 2016
3 / 34
!"#$#%&'()"*+&,)-./0&'".0.0&
P
rE
since 2010
™ =/#%.3&(5-"#*+%&>)86&0(.<$&+?4@&)$&!AB&
"!&C&D@1EF1EGH2I&<8& "#$%&'()'*+',%)&'-,+./*'011&'2345236'728389'
"!&C&D@1ED12GKLI&<8& :;+#<;=;=)'*+',%)&'>?=*;?*'44@&'036'728349'
J%4"+3+-+%$+-&&
4"+3.0.#%&
™ 2ɗ&-.03"+4)%35&6+$7++%&8/#%.3&)%-& ™ ,+0/:$0&<"#8&+:+3$"#%.3&)%-&&
)9+")*+&+:+3$"#%.3&8+)0/"+8+%$0&;& &&&&&03)$$+".%*&8+)0/"+8+%$0&)*"++&
1&
&&&&&&
From M. Meziane
We perform a reanalysis of the ep-scattering data and simultaneously fit
the charge and magnetic radii of the proton.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
4 / 34
Dataset Nomenclature
We consider data with maximum momentum transfer Q2 < 1.0 GeV2 . We split the available
elastic ep-scattering data into two datasets:
I
“Mainz”: high-statistics dataset, 1422 data points in the full dataset with Q2max < 1.0 GeV2 .
Bernauer et al. (2014)
I
“world”: compilation of datasets from other experiments, 363 data points plus 43
polarization measurements for Q2max < 1.0 GeV2 .
see e.g. Arrington et al. (2003, 2007), Zhan et al. (2011)
Polarization experiments directly measure the form factor ratio (µp GE )/GM .
Aside: χ2 fitting uses the optimize.leastsq in S CI P Y.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
5 / 34
Outline
1
Background
2
Form Factors and Radiative Corrections
3
Uncorrelated and Constant Systematics
4
Correlated Systematics
5
Results
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
6 / 34
rE and ep Scattering
I
Mott cross-section for scattering of a relativistic electron off a recoiling point-like proton is
dσ α2
θ E0
=
.
cos2
dΩ M
2 E
4E 2 sin4 θ2
I
The Rosenbluth formula generalizes the above,
i
dσ dσ −q 2
1 h 2
τ
1
=
GE + G2M , τ =
, =
dΩ R
dΩ M 1 + τ
4M 2
1 + 2(1 + τ ) tan2
I
.
The Sachs form factors GE (q 2 ), GM (q 2 ) account for the finite size of the proton. In terms
of the standard Dirac (F1 ) and Pauli (F2 ) form factors,
q
p
q
p!
= =Γµ (q 2p) =
GE + τ GM µ
γ +
1p! + τ
|
{z
}
F1 (q 2 )
I
θ
2
i
2M
σ µν qν
GM − GE
.
1+τ
| {z }
F2 (q 2 )
The radii are defined by
Figure 1: Diagrammatic representation of tree-level matching for the one-photon
amplitude in the full theory and in
∂G The black dot pin the dia6 NRQED.
gram on the right-hand
hr2side
i ≡represents insertions
, Gone-photon
of NRQED
E (0) =
2 q 2 =0
G(0)
∂q
operators.
1, GpM (0) = µp .
1
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
7 / 34
Earlier Ansäntze for GE , GM
dσ dΩ
R
=
i
τ
1 h 2
GE + G2M
dΩ M 1 + τ
dσ Earlier analyses used simple functional forms for GE , GM :
Gpoly (q 2 ) =
k
max
X
ak (q 2 )k ,
polynomials/Taylor expansions,
k=0
Ginvpoly (q 2 ) = Pk
max
1
2 k
k=0 ak (q )
Gcf (q 2 ) =
1
a0 + a1
q2
,
inverse polynomials,
,
continued fractions.
2
q
1+a2 1+...
The expansions are truncated at some kmax , with a finite number of coefficients.
The above functional forms exhibit pathological behaviour with increasing kmax .
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
Hill & Paz (2010)
May 26, 2016
8 / 34
ts and different functional forms of GE (q ) that were fit to the data over a period of 50
TheofBounded
z Expansion
traction
the proton charge
radius from scattering data is complicated by the unknown
onal behavior of the form factor. We are faced with the tradeoff between introducing
any parameters (which limits predictive power) and too few parameters 2(which biases
I QCD constrains the form factors to be analytic in t = q outside of a time-like cut beginning
s). Here
we describe a procedure that provides model-independent constraints on the
2
onal behavior
of =
the4m
form
factor.
The constraints
make
use of the
knownthis
analytic
at tcut
two-pion
production
threshold.
Clearly
presents an issue with
π , the
2
ties of theconvergence
form factor, viewed
as a functioninofthe
thevariable
complex qvariable
t = q 2 = −Q2 .
for expansions
.
Hill & Paz (2010)
t
z
z(t; tcut , t0 ) =
−Q2max
I
4m2π
√
√
t −t− tcut −t0
√ cut
√
tcut −t+ tcut −t0
By a conformal map, we obtain a true small-expansion variable z for the physical region.
Figure 1: Conformal mapping of the cut plane to the unit circle.
GE =
k
max
X
ak [z(q 2 )]k ,
GM =
k
max
X
bk [z(q 2 )]k .
illustrated in figure 1, the form factor is k=0
analytic outside of a cut at timelike
values
k=0
6] beginning2 at the two-pion production threshold, t ≥ 4m2π .2 In a restricted region
I Qmax is the maximum momentum
2
transfer
in
a
given
set
of
data.
sical kinematics accessed experimentally, −Qmax ≤ t ≤ 0, the distance to singularities
s the existence
of a point
small that
expansion
parameter.
We) begin
by performing a conformal
I t0 is the
is mapped
to z(t
0 = 0. We have used the simple choice t0
is defined in
Section
3.1.
have
checked
that the results do not vary significantly for the choice t0 .
re and throughout, mπ = 140 MeV denotes the charged pion mass, and mN = 940 MeV is the nucleon
I By including other data, such as from ππ → N N̄ or eN scattering, it is
= 0, but
possible to move
the tcut to larger values, improving the convergence of the expansion.
1
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
9 / 34
Bounded z Expansion Fit to Mainz Data
1.00
0.99
0.98
0.97
0.96
0.94
rE [fm]
0.92
0.90
0.99
0.98
0.97
0.96
0.88
1.02
1.00
0.98
σ/σdip
0.86
1.2
rM [fm]
1.1
1.06
1.04
1.02
1.00
0.98
0.96
0.94
1.10
1.05
1.0
1.00
0.95
0.9
1.20
1.15
1.10
1.05
1.00
0.95
0.90
0.8
0.7
0.0
0.96
0.1
0.2
0.3
0.4 0.5 0.6
Q2max [GeV2]
0.7
0.8
0.9
1.0
0
20
40
60
80
100
120
140
Scattering angle [deg]
t0 = 0, kmax = 12, |ak |max = 5, |bk |max = 5µp ∼ 2.8 × 5. Gaussian bound on ak , bk in χ2 .
R: spectrometer B, A, C
2
For Qmax = 1.0 GeV2 (statistics-only errors),
rE = 0.920(9) fm,rM = 0.743(25) fm.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
10 / 34
kmax Dependence
1750
χ2
1700
1650
1600
0.94
I
We can also test the
dependence of the fit results
on the choice of kmax .
I
The fit has converged for
kmax = 10.
I
We use a default of kmax = 12
in fits.
rE [fm]
0.92
0.90
0.88
0.86
rM [fm]
0.80
0.75
0.70
0.65
4
Gabriel Lee (Technion)
6
8
kmax
10
Extraction of r
12
p
from ep-Scattering Data
May 26, 2016
11 / 34
Unbounded z Expansion Fits
Fits using unbounded z expansion performed by Lorenz et al.
1700
I
Eur. Phys. J. A48, 151; Phys. Lett. B737, 57
Sum rules such as (t0 = 0)
χ2
1650
GE (q 2 = 0) =
1600
tell us ak → 0 as the k becomes
large.
1500
1.0
rE [fm]
0.9
I
The Sachs form factors are also
known to fall off as Q4 up to logs
for large Q2 (dipole-like
behaviour at large Q2 ).
I
To test enlarging the bound, we
took |ak |max = |bk |max /µp = 10,
and found rE = 0.916(11) fm,
rM = 0.752(34) fm.
I
However, as |ak |max → ∞, |ak |
for large k takes on unreasonably
large values, in conflict with QCD.
0.8
0.7
0.6
0.5
3.0
2.5
rM [fm]
ak = 1
k=0
1550
2.0
1.5
1.0
0.5
5
k
max
X
6
Gabriel Lee (Technion)
7
kmax
8
9
Extraction of r
10
p
from ep-Scattering Data
May 26, 2016
12 / 34
One-Loop O(α) Radiative Corrections
k0
k
(a)
(b)
must
subtract
offFirst
radiative
corrections
that
are part
ofelectron
the experimental
The
proton
form 2:
factors
are
defined
the matrix
element
of
exchange. on
A prot
Figure
order
realfrom
radiative
corrections
forone-photon
scattering
measurement:
consistent
definition
of
the
form
factors
is
required
to
compare
extracted
radii.
0
p
p
possible attachments
of the
radiated photon.
I
(a)
k
k
k0
k0
p
p
p0
p0
4.1
(a)
I
(b)
(c)
k0
Single photon exchange
p0
Let us rigorously
define
the(c)charge
radius
of a composite
fermion(b)such(b)as the
(b)
(c)
(a)
(b)
(b)
(c) (d)
(a)
(in particular,
IR
finite)
quantity
in
the
presence
of
radiative
corrections.
Figure 1: Virtual radiative correctionsFigure
through
first order
in
for pointcorrections
particle (top
particle
line)
2:
First
real↵.radiative
for
electron
scattering
on proto
Figure
2: First order
realorder
radiative
corrections
for the
electron
scattering
on proton.
In (
We know how to compute
results for the
electron
vertex
correction
and
leptonic
scattering on a composite particle (bottom
particle
line). Wavefunction
renormalization contributions
possible
attachments
of the radiated
amplitude
for
one
exchanged
photon,
possible
attachments
of the radiated
photon. photon.
Through one-loop
order,
only essential
difficulty
is with Two-Photon
are not
shown
explicitly.
contributions
to the vacuum
polarization
in perturbation
theory.
(a)
(a)
Exchange:
present
technology
to compute
from
firstthe
principles.
4⇡↵
From
previousbeyond
dispersive
analyses
of e+ e− 1→
data,
we
expect
correction from 0
(e) photon
0 (e)µ
4.1 hadrons
Single
M1 B=
ūtheorem
(k
) exchange
(k 0 , k)u(e) (k)ū(p) (p0 ) (p)
4.1
Single
photon
exchange
2 for which the
µ (p ,
A2i ! A2i +
simple
convexity
following from (14) no longer applies. It may
2
i ,to
2
ˆ
hadronic vacuum polarization
be
smaller
than
current
achieved
precision
in probability
scattering
q
(d)
Let
us)rigorously
definetheorems
the charge
radius of
a composite
fermion such as the
1
⇧(q
to
pursue
more
general
“physical
convexity”
involving
multiple
(d) be interesting
Let us rigorously define the charge radius of a composite fermion such as the proton
(d)
(in
particular,
IR
finite)
quantity
in
the
presence
of
radiative
corrections.
sums
and
correlated
errors.redone
selected
fitsfinite)
with
aquantity
global search
thatFriar
acorrections.
true
experiments.
Jegerlehner
(1996),
et al. (1999)
(inparticle
particular,
IR
in thestrategy
presencetoofverify
radiative
To beg
Figure 1: Virtual
radiative corrections through
first
order
in ↵. for
point
(top particle
amplitude
forline)
one exchanged photon,
I
Virtual
radiative
corrections
through
first order
in ↵.has
forbeen
point
(top
particle
line)
minimum
found
by
the
inductive
search
assumes
Figure
Virtual
radiative
corrections
through
order
in ↵.
for
point
particle
(topthat
particle
line)convexity.
3particle
scattering
on 1:
a composite
particle (bottom
particle
line). first
Wavefunction
renormalization
amplitude
for contributions
one
exchanged
photon,
g on a composite
particle
(bottomparticle
particle(bottom
line). Wavefunction
contributions contributions
scattering
on
a composite
particle
line).renormalization
Wavefunction
Ishown
4⇡↵ are
1
are not
explicitly.
For
soft
bremsstrahlung
and
two-photon
exchange
(TPE),
there
two
conventions for
14renormalization
0
M1 = 1 2
ū(e) (k 0 ) (e)µ (k 0 , k)u(e) (k)ū(p) (p0 ) (p)
,p
hown explicitly.
4⇡↵
µ (p
are not shown explicitly.
(p)
3.4 Deficiencies in other parameterizations
ˆ(k 0 2)) (e)µ (k 0 , k)u(e) (k)ū(p) (p0 ) (p) (p0 , p)u
q 1ū(e)⇧(q
M1 =
(p
µ (2000)
ˆ 2)
subtraction of infrared divergences.
Tsai (1961), Maximon & Tjon
q 2 1 ⇧(q
We remark that several parameterizations of the proton form3 factors in common use rely on flawed
where ↵ = 7.297 ⇥ 210 is the fine structure constant. Applying onshell ren
2
A2i ! A
Bi2 , for which the simple convexity
theorem following
from A
(14)
no longer
applies.
It may
3 in
theoretical
assumptions.
simple
expansion
q contribution
[11] isstructure
valid onlyconstant.
for
momentum
transfers
Ii +At
where
↵Taylor
=we7.297
⇥ 10
is the
Applying
onshell(e)
renormaliz
present,
we
cannot
calculate
the
remainder
of
the
TPE
from
(e)
write
(e)
0 qapplies.
µmay
2Itfine
µ⌫ first
0 principles.
2
2simplewhich
+be
Bi2interesting
, for
the
convexity
theorem
following
from
(14)
no
longer
2(14)
pursue
general
“physical
convexity”
theorems
involving
multiple
A2i which
! A2i to
+
B
the simple
convexity
following
from
no2⇡Itlonger
may
below
piontheorem
production
threshold
 4m
⇡probability
0.08applies.
GeV2 . Convergence
of a sequence of Padé approxi , for more
we
write
⌫ 2
1
sting
pursue
more to
general
“physical
convexity”
theorems
involving
multiple
probability
i
sumstobe
and
correlated
errors.redone
selected
fits
with
a
global
search
strategy
to
verify
that
a
true
(e)
(e) 2
interesting
pursue
more
general
“physical
convexity”
theorems
involving
multiple
probability
imants, implemented either directly as a ratio of polynomials
or= asµ aFe continued
fractionµ⌫[17],
(e) [16],
0
2
0
(k
,
k)
(q
,
)
+
(k
k)
F
(q
, )
⌫
1
2
correlated
errors.redone
fits with
a global
search
to verify
that ato
true
i µ⌫ 2m0factors,
(e)
minimum
been
found selected
byerrors.redone
the inductive
search
that
assumes
convexity.
(e)that
µ (e) 2
requires
positivity
of
the spectral
function
in
the dispersive
representations
sumshas
and
correlated
selected
fits
with astrategy
global
search
strategy
verify
a=
true
(k 0 , k)
F1 (q , ) +of the form
(k e k)⌫ Fa2 (q 2 , ) ,
6
has been
found by
inductive
search
that
assumes
convexity.
2me
property
which
is not
satisfied.
minimum
hasthe
been
found by
the inductive
search
that
assumes
convexity.
i
(p)
(p)
2
µ⌫ 0
(p)
(p) 0
µ (p) 2 (p) (p0 , p) = µ F1 (qµ⌫
22 , )
, ) +0
(p p)
⌫ F2 (q
3.4 Deficiencies in other parameterizations
(p) 2
1 (p) (p0 , p) = µ F1(p) (q2 , ) + i µ⌫2m
(p0 p p)⌫⌫F2 2
(q , ) ,
eficiencies
other parameterizations
3.4 in
Deficiencies
in other parameterizations
We remark
that
several parameterizations
of4the Radiative
proton form factors
in common
use rely on flawed
p 2mp2
corrections
p
Gabriel LeeA(Technion)
Extraction
r use
from
ep
-Scattering
Data factors are normalized as (at q = 0 we
Maymay
26, 2016
34
where
thetransfers
form
take the 13
IR/ finite
assumptions.
simpleofTaylor
expansion
q 2 [11]
is common
valid of
only
for
momentum
rktheoretical
that several
parameterizations
the proton
forminfactors
in
rely
on
flawed
where ↵ = 7.297 ⇥ 10
we write
is the fine structure constant. Applying onshell ren
(k , k) =
F
(q , ) +
i
2m
(k
k) F
(q ,
(p , p) =
F
(q , ) +
i
2m
(p
p) F
(q ,
Finite TPE Corrections
p
L
I
I
L
p!
k−L+p
k!
k
k + L − p!
k!
Figure 3: Feynman diagrams for two-photon exchange with momentum labels.
The standard procedure for modelling the finite part of the TPE is by “Sticking in Form
Factors” (SIFF). Treat the proton as a propagating Dirac particle and insert Γµ at each of
the vertices, using simple form factor ansätze for F1 , F2 .
Blunden et al. (2003, 2005)
2
We investigated the model dependence of this
calculation:
F1 = F2 /(µp − 1) = (1 − q 2 /Λ2 )−1 ,
F1 = F2 /(µp − 1) = (1 − q 2 /Λ2 )−2 ,
Fi =
3
X
j=1
I
p
L − p!
L−p
k
p!
aij
,
bij − q 2
3
X
j=1
aij
= Fi (0) ,
bij
monopole, Λ2 = 0.71 GeV2 ,
dipole, Λ2 = 0.71 GeV2 ,
Blunden et al. sum of monopoles (2005).
The A1 collaboration instead applies the Feshbach correction
δF
McKinley & Feshbach (1948)
sin(θ/2)(1 − sin(θ/2))
= απ
> 0,,
cos2 (θ/2)
which is the Q2 = 0 limit of the Coulomb distortion computed by Rosenfelder. It can also
be understood as Coulomb exchange between e and p in the Mp → ∞ limit. Rosenfelder (1999)
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
14 / 34
Effect of TPE on Fit to Mainz Dataset
0.94
0.92
rE [fm]
0.90
0.88
0.86
0.84
0.82
1.10
0.1
0.2
0.3
0.4 0.5 0.6
Qmax [GeV2 ]
0.7
0.8
0.9
1.0
I No finite TPE correction.
1.0
I Feshbach: used by default in A1 collaboration’s analysis of Mainz dataset.
rM [fm]
I SIFF dipole
0.9
I SIFF Blunden: used in previous analyses of world dataset.
We use the Blunden convention for the remainder of the fits.
0.8
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
15 / 34
Outline
1
Background
2
Form Factors and Radiative Corrections
3
Uncorrelated and Constant Systematics
4
Correlated Systematics
5
Results
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
16 / 34
ated
errors
Thesystematic
A1 Approach
tions around initial fit
rescaling factor
ll
(data-fit)/stat.error
I
The A1 analysis groups the Mainz dataset into 18
subsets: 3 spectrometers × 6 beam energies.
I
For each subset, the differences between the fit and
measured cross sections, scaled by the uncertainties,
are fit to a Gaussian.
I
The width of the Gaussian is used as the scaling factor
κ for the statistical uncertainties in the subset.
Concerns:
maticsI drive
In the A1 analysis, the χ2
red
for the fit to the entire dataset with scaled errors is ≈ 1.15.
In our bounded z expansion fit, we find χ2red per subset similar to the A1 Gaussian widths.
I Expressing the total A1 uncertainties as quadrature sums of statistical and uncorrelated
uncertainties,
q
kinematics
2
2
dσi,A1 = κi dσi,stat = dσi,
stat + dσi,syst ,
I
ndent of
statistics
dσsyst
is as low as 0.05% for some points. This seems unreasonably small.
is) I Multiple data points at the same kinematic settings drive the “effective systematic
details: backup slide
uncertainties” even lower.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
17 / 34
Rebinning
Certain systematic uncertainties are experimentally difficult to constrain below 0.1%, such as:
I
time-dependent efficiencies,
I
rate-dependent variations,
I
beam-energy uncertainties,
I
spectrometer angle offsets.
We would expect these uncertainties to be identical for the repeated measurements. Simply
adding a fixed systematic to all points in the dataset would underestimate the systematic error for
these repeated data points. We therefore combine these before adding a fixed systematic to the
statistical uncertainty in quadrature.
We perform the following:
I
Remove one set of points at Ebeam = 315 MeV, θ = 30.01◦ with inconsistent scatter.
I
Identify 407 kinematic settings with multiple data points.
I
“Rebin” these to obtain a dataset of 657 points.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
18 / 34
Constant Systematics
After rebinning, we investigate the effect of adding a 0.25% and a 0.3% fixed systematic, e.g. for
a data point with cross section σi ,
q
2
2
dσi = dσi,
stat + (0.003σi ) .
Fitting the rebinned dataset after these two modifications, we find
rE = 0.908(13) fm, rM = 0.766(33) fm.
spec.
A
B
C
beam
180
315
450
585
720
855
180
315
450
585
720
855
180
315
450
585
720
855
Gabriel Lee (Technion)
Nσ
χ2red
29
23
25
28
29
21
61
46
68
60
57
66
24
24
25
18
32
21
0.59
0.54
1.52
1.54
1.05
0.92
0.85
1.05
0.90
0.61
1.29
1.88
0.88
1.16
1.53
0.83
1.11
0.79
χ2red
CL (%)
96.1
96.4
4.8
3.4
39.9
56.8
79.8
38.5
71.7
99.2
6.9
0.002
63.3
27.2
4.3
66.3
30.2
73.7
Extraction of r
0.46
0.44
1.00
1.03
0.87
0.77
0.65
0.76
0.67
0.50
0.97
1.15
0.68
0.78
1.08
0.65
0.90
0.62
p
CL (%)
99.4
99.1
46.7
42.8
66.4
76.0
98.3
88.5
98.2
99.96
53.7
19.6
88.0
76.8
35.9
86.4
62.3
90.5
from ep-Scattering Data
Cols. 4 and 5 (6 and 7)
give the results after the
inclusion of a uniform
0.25% (final 0.3–0.4%)
uncorrelated systematic.
The 0.4% applies to
Ebeam = 855 MeV, spec B.
May 26, 2016
19 / 34
Outline
1
Background
2
Form Factors and Radiative Corrections
3
Uncorrelated and Constant Systematics
4
Correlated Systematics
5
Results
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
20 / 34
The A1 Approach (Again)
In the Mainz dataset, each data point includes three additional quantities:
I
two cross sections corresponding to variations of the energy cut on bremsstrahlung of the
electron,
I
kinematic-dependent factor, linear in the scattering angle θ , which accounts for efficiency
changes, normalization drifts, variations in spectrometer acceptance, and background
misestimations.
The entire dataset is refit either:
I
using the minimum or maximum cross sections from variations on the energy cut,
dividing or multiplying central values of the cross sections by the linear factor.
In each case, the largest difference of the resulting fit from the central values is taken as the
difference, and
q
I
∆rsyst =
(∆rEcut )2 + (∆rcorr )2 .
We find the bremsstrahlung energy cut has little impact on the radius central values: translates to
an uncertainty in rE of 0.003 fm and in rM of 0.009 fm.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
21 / 34
Our Approach
The linear factor is written as
1 + δcorr = 1 + a
In the A1 analysis:
x − xmin
.
xmax − xmin
I
x = θ,
I
18 values of θmax , θmin for each spectrometer-Ebeam subset,
I
a ≈ 0.2%, same sign for all subsets.
We choose:
I
x = θ, 1/θ, Q2 , 1/Q2 , E 0 , 1/E 0 , 1/ sin4 (θ/2),
I
Three groupings: by spec (3), spec-Ebeam (18), and normalization (34),
I
a = 0.5%, and same sign.
Different variables modify the functional form of the correction within each subset;
however, the endpoints are always fixed to have a correction of 0 and 0.5%.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
22 / 34
Our Findings
x
Q2
1/Q2
θ
1/θ
Q2max [GeV2 ]
0.05
0.5
1
0.05
0.5
1
0.05
0.5
1
0.05
0.5
1
∆rE [fm]
∓0.017
∓0.016
∓0.015
±0.041
±0.025
±0.023
∓0.022
∓0.018
∓0.017
±0.036
±0.024
±0.021
∆rM [fm]
±0.021
∓0.022
∓0.026
∓0.046
±0.016
±0.021
±0.027
∓0.021
∓0.025
∓0.039
±0.018
±0.022
Multiplication (top sign) or division (bottom sign), spectrometer-Ebeam (18)
I
A factor of 2.5 bigger than the A1 analysis, mainly due to increase in a.
I
Different variable choices yield similar results, largest effect from 1/Q2 .
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
23 / 34
Our Findings (cont.)
I
Norm. grouping (34) yielded uncertainties that were typically 20–30% larger for rE
compared to the spec-Ebeam (18), with smaller increases for the uncertainty on rM .
I
Spec-only grouping yielded somewhat smaller uncertainties for rE compared to the
spec-Ebeam , with larger increases for the uncertainty on rM .
I
Systematic effects could differ for the different spectrometers, and the combined effect
might be enhanced or suppressed by the assumption of identical corrections (always
multiplying or dividing, same sign).
I
For rM , we found some cases with cancellations between spectrometers when the linear
correction was applied to all spectrometers vs. each spectrometer individually.
For final results, take uncertainties using x = θ in each spectrometer-beam energy subset as a
representative correlated systematic, and use a ≈ 0.4%, dividing the above corrections by 4/5.
For the full dataset, we obtain
rE = 0.908(13)(3)(14) fm, rM = 0.766(33)(9)(20) fm.
We have expanded the A1 analysis of the correlated systematics, but have not made any drastic
changes to the framework. A larger systematic shift to reconcile the values would require:
I
a range of corrections larger than 0.4%,
I
an extreme functional form,
I
a “tuned” cancellation between subsets to reduce the overall systematic.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
24 / 34
Outline
1
Background
2
Form Factors and Radiative Corrections
3
Uncorrelated and Constant Systematics
4
Correlated Systematics
5
Results
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
25 / 34
rE , rM vs. Q2max for Final Fits
0.94
0.92
rE [fm]
rE [fm]
0.90
0.88
0.86
0.84
0.82
1.3
0.96
0.94
0.92
0.90
0.88
0.86
0.84
0.82
0.80
1.1
1.1
rM [fm]
rM [fm]
1.2
1.0
0.9
1.0
0.9
0.8
0.7
0.0
0.1
0.2
0.3
0.4 0.5 0.6
Q2max [GeV2]
0.7
0.8
0.9
0.8
0.0
1.0
0.1
0.2
0.3
0.4 0.5 0.6
Q2max [GeV2]
0.7
0.8
0.9
1.0
L: final fit to Mainz rebin dataset with 0.3/0.4% fixed systematic
R: final fit to world+pol dataset
t0 = 0, kmax = 12, |ak |max = |bk |max /µp = 5
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
26 / 34
Sensitivity of Statistical Uncertainties to High-Q2 Data
r [fm]
Scattering data at low-Q2 determine radius, from its definition as the slope of the FF at q 2 = 0.
I
Filled: Mainz
I
Hollow: world+pol
I
Squares: rE
I
Circles: rM
Q2max [GeV2 ]
Want to maximize sensitivity, but minimize effect of possible high-Q2 systematics.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
27 / 34
A1 analysis (spline fit)
z expansion
Hydrogen (CODATA)
μ-Hydrogen (CREMA)
Final Results for rE Proton charge radius
+ hadronic TPE
rebin, + 0.3% uncorr. syst.
+ 0.4% corr. syst.
Mainz final (Q2max=0.5 GeV2)
world data (Q2max=0.6 GeV2)
4.2σ
Mainz + world average
0.85
0.9
0.95
rE [fm]
from R. Hill
Mainz = 0.895(14)(14)
rEMainz
world
rE = 0.895(14)(14) fm, rE
=simple
0.916(24)
fm
average:
avg
world = 0.918(24)
rrEE
= 0.904(15) fm
rEavg. = 0.904(15)
27
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
28 / 34
Final Results for rM Proton magnetic radius
A1 analysis (spline fit)
z expansion
+ hadronic TPE
rebin, + 0.3% uncorr. syst.
+ 0.4% corr. syst.
Mainz final (Q2max=0.5 GeV2)
2.5σ
2.7σ
world data (Q2max=0.6 GeV2)
Mainz + world average
0.7
0.8
rM [fm]
0.9
1
from R. Hill
rMMainz = 0.777(34)(17)
Mainz
world
rM
= 0.776(34)(17) fm, rE
= 0.914(35)
fm
simple average:
rEavg. = 0.847(27)
avg world
= 0.913(37)
rMrE = 0.851(26)
fm
28
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
29 / 34
A Possible Resolution: Large Logs
We have included scattering data with momentum transfers as large as Q2 ∼ 1 GeV2 .
I
I
In this regime, QED perturbation theory breaks down due to large logarithms from electron
radiative corrections
Q2 α
log2 2 ≈ 0.5 .
π
me Q2 ∼1 GeV2
Recall the sum of the first-order vacuum polarization and electron vertex and real
bremsstrahlung corrections:
Q2
(η∆E)2
13
Q2
α
log 2 − 1 log
+
log
+
.
.
.
.
δ=
π
me
EE 0
6
m2e
where ∆E is the detector energy resolution.
I
When Q ∼ E ∼ E 0 and me ∼ ∆E , the leading series of logarithms αn log2n (Q2 /m2e )
are resummed by making the replacement,
Yennie, Frautschi, Yuura (1961)
1 + δ → exp(δ) .
I
In practice, ∆E me , which can introduce another scale into the problem. As a check,
we can instead multiply the cross sections by
±1
Q2
α
Q2
α
exp(δ) → 1 ± δ + log2 2
× exp − log2 2 ,
π
me
π
me
I
This has the same 1-loop corrections, and also resums the leading-logs when there is only
one large ratio of scales, Q2 /m2e .
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
30 / 34
Large Logs (cont.)
1.00
rE [fm]
0.95
−0.2
0.90
−0.21
0.85
0.80
−0.22
δ
0.75
1.3
−0.23
1.2
rM [fm]
1.1
−0.24
1.0
0.9
0.8
−0.25
0.7
0.6
0.0
0.1
0.2
0.3
0.4 0.5 0.6
Q2max [GeV2]
0.7
0.8
0.9
0
1.0
0.2
0.4
0.6
0.8
1
Q2 (GeV2)
Black: systematic two-loop analysis in 1605.02613.
Naive exponentiation of TPE correction with
Black: fit to rebinned Mainz data with 0.3/0.4%
µ2 = M 2 , µ2 = Q2 .
fixed systematics, statistical uncertainties
shown only. of complete next to leading order resummed correction
FIG. 7: Comparison
Blue: ∆E = 10 MeV, upper/lower blue curves
exponentiations using di↵erent factorization scales for the two photon
to the (1 ± δ + . .naive
.)±1 factor.
µ2 = M 2 (dotted red line) and µ2 = Q2 (dashed blue line). See text for deta
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
31 / 34
EFT Analysis of Large Logs
A systematic analysis of the radiative corrections using effective field theory is performed by
R. Hill in 1605.02613, identifying the sources of all large logarithms in the limit Q2 m2 ;
e.g., there are implicit conventions of µ2 = M 2 for vertex corrections vs. µ2 = Q2 for
Maximon-Tjon TPE corrections.
Heavy particle: ∆E E ∼ Q ∼ M . Neglected: α2 log2 (M 2 /(∆E)2 ) small.
Relativistic particle: m, ∆E E, Q M . Neglected: α2 log3 (Q2 /m2 ) ∼ O(α1/2 ).
0.5–1% discrepancies between the NLO resummed EFT prediction and the
phenomenological analysis, which is greater than the assumed < 0.5% systematic error of
the A1 analysis.
I
I
I
−0.15
−0.2
I Leading log resummation.
I Next-to-leading log resummation.
δ
−0.25
I Black: complete next-to-leading order resummation.
I Bands from varying low and high renormalization
2
scales µ2
L , µH between 1/2 ∗ min and
−0.3
−0.35
∆E 2 , m2 and 2 ∗ max of Q2 , E 2 .
0
0.2
0.4
0.6
0.8
1
Q2 (GeV2)
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
32 / 34
Ongoing Work
Combined fit of Mainz+world+pol datasets for determination of form factors GE , GM with:
I
correlated systematic parameters for the Mainz data floating in the fit,
I
implementation of sum rules enforcing dipole-like behaviour of GE , GM at high-Q2 ,
I
inclusion of neutron data.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
May 26, 2016
33 / 34
Conclusion
I
We presented the most comprehensive analysis of existing ep-scattering data:
I
I
I
I
using form factors constrained by QCD,
performing careful studies of existing radiative correction models,
examining the uncorrelated systematics and rebinning the Mainz high-statistics dataset,
reconsidering systematic uncertainties.
I
The Mainz and world values for rE are consistent, but the simple combination of the Mainz
and world values remains 4σ away from the µH spectroscopic value.
I
We find a 2.7σ difference in the Mainz and world values for rM .
I
Experiments have reached a level of precision that demands a more systematic treatment
of the radiative corrections: in particular, the standard treatment of the resummation of
one-loop large logs is inadequate.
I
Stay tuned for future experiments.
I Low-Q2 (10−4 − 10−2 GeV2 ) ep scattering.
I µp scattering at PSI.
I Further measurements of H spectroscopy.
I Further measurements of µH spectroscopy.
I
I
Next-generation lattice QCD.
PRad at JLAB, A1
MUSE
Vutha et al. (2012), Beyer et al. (2013), Peters et al. (2013)
Pohl group at MPI Quantenoptik
Alexandrou et al. (2013), Bhattacharya et al. (2013), Green et al. (2014)
New physics?
I
I
I
New general flavour-conserving nonuniversal interactions.
Parity-violating muonic forces.
MeV-scale force carriers between protons and muons.
Gabriel Lee (Technion)
Extraction of r
p
from ep-Scattering Data
Barger et al. (2011), Carlson & Rislow (2012)
Batell et al. (2011)
Tucker-Smith & Yavin (2011), Izaguirre et al. (2015)
May 26, 2016
34 / 34