Calculations of γZ
corrections
Carl E. Carlson
William and Mary
γZ box(ing) workshop, Dec. 16-17, 2013, JLab
Our relevant papers
•
“Contributions from γZ box diagrams to parity violating elastic
ep scattering,” Rislow & Carlson, Phys.Rev. D83 (2011)
113007
•
“Resonance Region Structure Functions and Parity Violating
Deep Inelastic Scattering,” Carlson & Rislow, Phys.Rev. D85
(2012) 073002
•
“Modification of electromagnetic structure functions for the
γZ-box diagram,” Rislow & Carlson, Phys.Rev. D88 (2013)
013018
2
Weak Charge of the Proton
•
QwP extracted from parity-violating, ep scattering: e
•
AP V =
+
e
AP V
Z
p
•
R
L
L
Lowest order diagrams
and result: g Ae
p
R
GF
p Q2 QpW
=
4⇡↵ 2
g Vp
Lowest order definition of QwP :
Qp,LO
=
W
e p
4gA
gV = 1
3
4 sin2 ✓W
One-loop result
e
e
Z
Z
Z
Z
+
W
W
_
f
f
+
W
Z
W
+
+
W
p
p
J
Z
QpW = (1 +
p
J
Z
⇢+
p
p
J
J
e)
1
W
4 sin2 ✓W +
4
J
0
e
J 3=J Z+2 sin2
W
J
+ ⇤W W + ⇤ZZ + ⇤
Z
Running Weinberg Angle (Czarnecki and Marciano)
Running
Weinberg
Angle
(Czarnecki
Marciano)
Running
Weinberg
Angle
(Czarnecki
andand
Marciano)
Running Weinberg Angle (Czarnecki and Marciano)
+ + + + …
+ + + + …
+ + + + …
Only “Pinched” Part
Only “Pinched” Part
Degrassi and Sirlin, PRD 46,
Only “Pinched” Part
Degrassi
and Sirlin, PRD 46,
3104 (1992)
Degrassi3104
and Sirlin,
(1992)PRD 46,
3104 (1992)
5
5
5
5
Running Weinberg Angle
Running Weinberg Angle
Czarnecki and Marciano Running
Erler and Langacker
Int. J. Mod. Phys. A15, 2365 (2000)
Particle Data Group (2012)
6
6
Q2 dependent uncertainties associated with the predicted
corrections were folded into the systematic error of each
(PRL 99,122003 (2007))
point.
The effects of varying the maximum Q2 or θ of the
data included in the fit, or varying the dipole mass in the
strange form-factors, were studied and found to be small
relative to the fit uncertainty above Q2 ∼0.25 (GeV/c)2 .
0.18
In
O
0.16
0.15
0.14
0.13
p
2
0.3
#! This Experiment Data Rotated to the Forward-Angle Limit
!$$$$HAPPEX
!
"$$$SAMPLE
$$$PVA4
%$$$G0
!
SM (prediction)
0.17
C1u + C1d
0.4
2
A/A 0 ! Q W " Q B(Q , =0)
Weak
Extrapolation
Weak
ChargeCharge
Extrapolation
0.2
0.12
-0.70
!
0.1
#
0.0
0.0
!
"
0.1
0.2
0.3 0.4
Q 2 [GeV] 2
7
0.5
0.6
FIG. 3. The co
constants C1u
The more horiz
7
straint
on the
more vertical (b
isting Q2 < 0.6
:
• Definition:
𝜸Z Box
γZ Box
γZ Box
• Definition:
γZ
Box
• Definition:
Definition:
• Unlike WW and ZZ boxes, the γZ box cannot be analyzed using
pQCD.
•
• Unlike
WWthe
andγZZZ
boxes,
thebe
γZanalyzed
box cannot
be analyzed
• Unlike WW and
ZZ boxes,
box
cannot
using
pQCD.
pQCD.
W and ZZ boxes, the γZ box cannot be analyzed using
•
For WW or ZZ boxes, two heavy propagators enough to
ensure contributing momenta are big. Calculate w/pQCD.
Here, one heavy propagator not enough. Low momenta in11
loop, perturbative calculation unreliable.:
11
8
𝜸Z Box
•
Box
γZγZBox
Gorchtein
andHorowitz
Horowitz
(PRL
102,
091806
(2009))
had
•
Gorchtein
and
(PRL
102,
091806
(2009))
had
insight
• Gorchtein
and
Horowitz
(PRL
102,
091806
(2009))
had
insight
insight
to calculate
the amplitude
dispersively
to
calculate
the
amplitude
dispersively
to calculate the amplitude dispersively
•
OpticalTheorem
theorem,
•
Optical
• Optical Theorem
9
13
13
𝜸Z γZγZBox
equations
Box Equations
Equations
Box
•
Imaginary
Part:
•• Imaginary
Part:
Imaginary
part
•
Dispersion
Relations:
Dispersion
relations,
•• Dispersion
Relations:
14
14
10
𝜸ZγZγZBox
equations
Box
Equations
Box Equations
•• Imaginary
Part:
Imaginary
Part:
Imaginary
part
Equations
•
Structure Functions
must be modeled.
•
•• Dispersion
Relations:
Dispersion
Relations:
Dispersion
relations,
Structure Functions
must be modeled.
Labels proton current.
Labels proton current.
1614
11
Vector Boxes
The Vector
Box plots
Plots
Vector
box
Hall et al.
Carlson and Rislow
Gorchtein et al.
PRD 88, 013011 (2013)
PRD 83, 113007 (2011)
PRC 84, 015502 (2011)
•
• Differences come from the treatment of the
19
Differences
come
from the treatment of the structure functions
structure
functions.
13
Us
Carlson and Rislow Analysis
divided up the energy regions and modified the
•• WeWesplit
the energy regions and modified the structure
structure functions.
functions.
W (GeV)
Capella et al.
CTEQ
(Phys. Lett. B 337,
358 (1994))
2.5
(PRD 82, 074024 (2010))
(modified)
Christy and Bosted
(PRC 81, 055213 (2010))
(modified)
8
14
Q2 (GeV)2
20
Evaluation in scaling region
Evaluation of Scaling Region
•
Calculateddirectly
directlyusing
usingPDFs
PDFs
• Calculated
• We use CTEQ
•
•
We use CTEQ
• Alternative
–Alternative
Hall et al. use ABM11 (PRD 86, 054009 (2012))
•
Hall et al. use ABM11 (PRD 86, 054009 (2012))
22
15
Evaluation in resonance region
•
Evaluation
of Resonance
Region
Evaluation
of
Resonance
Region
All later calculations modify Christy-Bosted
electromagnetic
fits.modify
(May Christy
also useBosted
MAID.)
•
All
later
calculations
electromagnetic
• All later calculations modify Christy Bosted electromagnetic
fits. fits.
has
7 resonances
a smooth
background
CBfit7fit
have
7 resonances
and
a smooth
background
• CB• fitCB
has
resonances
and aand
smooth
background
•
•
• Resonances
modified
bybycorrective
Resonances
modified
corrective
ratio:
• Resonances
modified
by corrective
ratio:ratio:
;
16
;
23
23
Rislow and
Carlson Vector
Cres
Vector
C
res
Rislow and Carlson Vector Cres
• Definition of structure functions:
• Definition of structure functions:
Definition of structure functions:
•
•
• Cres inCres
terms
of helicity
amplitudes:
in terms
of helicity
amplitudes:
• Cres in terms of helicity amplitudes:
24
24
17
Rislow
and
Carlson
C
Rislow
and Carlson
Vector
res Cres
Vector
CVector
res
• We•constructed
helicityhelicity
amplitudes
using SU(6)
We constructed
amplitudes
using wave
SU(6) wave
functions:
functions:
We constructed helicity amplitudes using SU(6) wave functions:
•
=
=
Phenomenological constraints used to fit A and B:
•• Phenomenological
constraints
used to
fit Atoand
• Phenomenological
constraints
used
fit AB:and B:
=
-1 for Q
= 0Q2 = 0
-12 for
=+1 for +1
2
high
forQhigh
Q2
25
18
25
Cres for D13(1520)
forDD13(1520)
(1520)
CCresresfor
13
•
SU(6)
wave
function
forfor
(1520):
SU(6)
wave
function
proton:
• • SU(6)
wave
function
for
DD1313
(1520):
Spatial
Spatial
Flavor
Flavor
Spin
Spin
•
Helicity
Amplitudes:
Helicity
amplitudes
• • Helicity
Amplitudes:
28
28
19
Cres for D13(1520)
Cres
forfor
D13D(1520)
Cres
13(1520)
•
• •A and
B relations
forfor
Dfor
(1520):
13D
AAand
BBrelations
and
relations
D(1520):
13(1520):
13
•
Λ found
by comparing
amplitudes
to MAID:
• •Λ found
by by
comparing
amplitudes
to MAID:
Λ found
comparing
amplitudes
to MAID:
29
20
29
Our
vector
C
res
Rislow and Carlson Vector C
res
30
21
R
&
C
Background
Correction
Rislow
and
Carlson
Background
Correction
Rislow
and
Carlson
Background
Correction
Rislow
and
Carlson
Background
Correction
•
In
limit
where
all
light
quarks
(u,
d,d,s)
s)s)are
are
equally
likely:
•In• • limit
Inlimit
limit
where
all
light
quarks
(u,
are
equally
likely:
In
where
all
light
quarks
(u,
d,
equally
likely:
where all light quarks (u, d, s) are equally likely:
•
Invalence
valence
quark
limit
(dand
and
u’s):
• • valence
In
quark
limit
(d
222u’s):
In
valence
quark
limit
(d
and
u’s):
• In
quark
limit
(d and
2 u’s):
•
Wetook
took
their
average
thecorrection:
correction:
took
their
average
asasthe
the
correction:
• • We
We
their
average
as
• We
took
their
average
as the
correction:
31
22
3131
Alternative resonance analysis
Christy
andand
Bosted
Modification
Christy
Bosted
Modification
•
Isospin
rotate
neutron
amplitudes:
• Isospin
rotate
neutron
amplitudes:
• Isospin
rotate
neutron
amplitudes:
•
Rewrite
neutral
current
• Rewrite
neutral
current:
• Rewrite
neutral
current:
23
•
• C calculated using PDG photoproduction data
Cres calculated using PDG photoproduction data
• GHRM used PDG data at Q = 0, dropped relative Q
res
2
2
dependence.
•
2
Can also use MAID to obtain neutron amplitudes, at all Q .
24
CresCresEffect
onγZ γZ
Effect on
Box Box
•
•
•
•Solid:
Solid:
Constituent
Quark
Model
Carlsonand
andRislow
Rislow
Constituent
Quark
Model
ofof
Carlson
• Blue: PDG (used by Gorchtein et al. and Hall et al.)
Blue: PDG (used by Gorchtein et al. and Hall et al.) • Red: MAID (Eur.Phys.J.ST 198, 141 (2011))
Red:– MAID
198,Resonance
141 (2011)) Green: (Eur.Phys.J.ST
MAID without Roper
39
– Green: MAID without Roper Resonance
25
Axial Box Analyses
Axial Box Calculations
Axial Box Calculations
Blunden et al.
Carlson and Rislow
PRL 107, 081801 (2011)
PRD 88, 013018 (2013)
27
46
Comments on
A
𝜸
☐Z
•
The ☐𝜸ZV ≈ 0.005 just discussed compares to ☐𝜸Z ≈
0.0051 quoted on “old days”. Pure coincidence. This was
just for ☐𝜸ZA. •
☐𝜸ZA can be calculated anew. With the DR treatment there
are no logs to guess arguments of.
A
Z ( E)
•
=
2
E dE
E >0
E
2
E2
Im
A
Z (E
)
Notes
•
•
•
Not zero at threshold
The E’ makes high energies important
𝜸Z
Most of result comes from scaling region for F3 , where
it can be obtained reliably
28
Current Axial Box Results
Axial Box Calculations
Blunden et al.
Carlson and Rislow
PRL 107, 081801 (2011)
PRD 88, 013018 (2013)
46
29
Sameand
split
ofAnalysis
regions
Carlson
Rislow
• We split the energy regions and modified the structure
functions.
W (GeV)
2
CTEQ
(modified)
CTEQ
(PRD 82, 074024 (2010))
Christy and Bosted
(PRC 81, 055213 (2010))
(modified)
8
30
Q2 (GeV)2
47
•
Evaluationof
of Axial
Axial Structure
Function
Evaluation
Structure
Function
Scaling
Region:
region
• Scaling
•
W, low Q ,
• High
High
W, Low Q :
2
2
CTEQ
48
31
•
•
Resonance Region:
Blunden et al. use Lalakulich et al. (PRD74,014009) for F3
•
Fewer resonances than the Christy Bosted fit:
fits for D13(1520), P11(1440), P33(1232), and S11(1535)
• Resonance Region:
•
– Blunden et al. use Lalakulich et al. (PRD 74, 014009)
We continue modifying the Christy Bosted fit.
– We continue modifying the Christy Bosted fit.
32
Rislow & Carlson Axial Cres
Rislow and Carlson Axial Cres
50
33
Rislow and Carlson Background Correction
R & C Background Correction
• Smooth Background.
Smooth background
•
••
Average
Average5/3.
5/3
51
34
Axial Box Contributions
Axial Box Contributions
Blunden et al.
Carlson and Rislow
PRL 107, 081801 (2011)
PRD 88, 013018 (2013)
(Overall results already given)
35
53
Summary
•
•
•
•
•
The world is saved (almost), regarding the 𝜸Z corr. to QWeak.
•
For goal of 1% or better measurement of QWeak (P2 at Mesa),
energy is about 1/6 of JLab experiment, and corrections and error
in ☐𝜸ZV scale with energy. •
Would like to improve ☐𝜸ZA but believe this is very manageable
I.e., ☐𝜸ZV now calculated.
About (8.1±1.4)% of QWp at Eelec=1.165 GeV. Proportional to Eelec. ☐𝜸ZA also now calculated w/o guesswork on logs
About (6.3±0.6%) of QWp at Eelec threshold. Small dependence on
Eelec.
36
are
nofofthe
thebox
boxdiadia- tree
tree level,
level, when
when higher-order
higher-order radiative
radiative corrections
corrections are
heleading-order
leading-order neglected.
neglected.
ver, additional
Interference
exchange
wever,
additional
Interference between
between photon
photon and
and Z-boson
Z-bosonPV
edtotocalculate
calculatea a produces
produces an
an experimentally
experimentally observable
observable parity-violating
parity-violating
kprocesses.
processes.AA asymmetry
asymmetryAAPVPV ininthe
the scattering
scattering of
of longitudinally
longitudinally polarduced bybyMo
roduced
Mo ized
izedelectrons
electrons on
on an
an unpolarized
unpolarized target
target [14].
[14]. It can be
one-photon
ve corrections
expressed
ininterms
ofofthe
ative
corrections With
expressed
termsexchange,
theabove
above quantities
quantities as
as
ephoton
photoninindiadia22 Born
Born
Born
Born
Born
#
"
#
G
Q
A
%
A
%
A
#
"
#
G
Q
A
%
A
R
L
F
R
L
F
E
M
E
M
A
Born !
Born
approximation,
pproximation,
p
!!!
p
!!!
!
"
AA
!
!
"
!! 22
! 2 ; (6)
PV
PV
'*#GEp
%&#G
&#GMp$ (
##RR%%##LL
4() 22'*#G
4()
dependencetoto
Ep$$ %
dependence
ould
notethat
thatifif where
ld note
where
where
nthethecombined
combined
!! ;
Born ! "2g
Born ! "2gee &GZ G!
Born
e e*G
ZZ G
Born
wouldfactor
factor
*G
G
A
)]d)]
would
!
"2g
;
A
AA
Ep Ep
M ! "2gA
Mp Mp
Ep
A
EE
M
Mp
Mp
AA Ep
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
asymmetries.
qq!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ymmetries. ToTo
Born
e
2 ZZ !!
Born
e
2
ymmetries,
one
&#1
%
&$#1
"
*
A
!
2g
GAAG
GMp
mmetries, one
AA A ! 2gVV &#1 % &$#1 " * $$G
Mp
mation
two%%"1
$$
ation forfortwo"1
%
e
%
2
e
2#1%%&$tan
&$tan2
;
(7)
e presentLetter.
Letter.
**!! 11%%2#1
;
(7)
present
2
2
electromaghetheelectromagolating
asymme- and
"#1"
"4sin
4sin22%%W$=2,
$=2, ggeeA !
! "1=2
"1=2 are
are the
andwhere
wheregge eV!!"#1
ating asymmethe
W
V
A
isdescribed
describedinina a electron
electroncoupling
couplingconstants
constantsfrom
from Eq.
Eq. (1).
(1).
omagneticform
form
Let usus now
now consider
consider the
the case
case when
when electron-proton
electron-proton
magnetic
Let
[Fig.1(b)]
1(b)]isis scattering
scatteringisiscaused
caused by
by the
the exchange
exchange of
of more
more than
than one
non[Fig.
one
photon.Neglecting
Neglectingthe
theelectron
electronmass,
mass, the
the electron-nucleon
electron-nucleon
photon.
scattering amplitude
amplitude can
can be
be presented
presented in
in the
the form
form descattering
described by three complex scalar
invariants; see also [1,3]:
(1) scribed
37
by three complex scalar
invariants;
see also [1,3]:
(1)
Post summary: A
•
change [Fig. 1(c)]. This mechanism of 2! ! Z interference
soft by experimental cuts.
0 same 2order
!2
!
contributes
to sthe observed asymmetry atG
the
2 fixed Q
nal
correction
at
E #*; Q $ !
! #1 " 4sin %W $F1;2p "O""#
F1;2n
"
F
;
(4)
1;2
as the !Z box discussed in Ref. [10]; it comes in
ring angle, reaching a few
G0A #*; Q2 $ !
us to take a closer look at
the weak
(Weinberg) angle and the
urements
thatmixing
require high
G!E #Q2 $ % +G0E #*; Q2 $
+G #*; Q $;
May also have 2-photon exchange
0
A
2
denote contributions to the neutral weak
implications
of two-photon
strange
(and
heavier) quarks. Other combinalectron scattering. The posfactors
used, namely, GE ! F1 "
ion
of this are
type also
has been
!
F1Ref.
% F[9]),
& ! Q2 =#4M2 $.
, e.g.,
but neither
2 , where
alysis factor
has beenisdone so far.
orm
where G!M #Q2 $ and G!E #Q2 $ are the usual
electric form factors that parametrize matri
nucleon electromagnetic current, and the
order O#)$ from multiphoton exchange a
by the quantities +G0M , +G0E , and G0A .
essions for parity-violating
EquippedPwith
H Ythe
S I corrected
C A L Rexpression
EVIEW
eralized
GZA #Q2form
$ ! factors,
"&3 GAand
#Q2 $ %
"s;
(5)
PRL
94,
212301
(2005)
s using the formalism of
electromagnetic amplitudes, it is straightforw
the two-photon-exchange
the formula
for the parity-violating asymme
%1#"1$
for protons (neutrons) and "s stands
!
matic correction to parityelectromagnetic
form
factors
GE;M includes
with realtheparts
of their box cont
ge-quark
contribution.
The
isovector
axial
properly
two-photon
ring asymmetry at a level of
0
0
2
generalized
versions
Re!G
".
Two
additional
terms,
A
E;M
M
is
normalized
Q ! 0 to the neutron
asymmetry takes the form
kinematics,
which
isat
comwhence,
2
•
0
perimental constraints of the
and A
from interference
between
FIG.
(a),(b) Diagrams
of the Born approximation,
(c) two-the axial-like
A ,1. arise
2
0 % Ain0
his asymmetry.
exchange,
andtwo-photon-exchange
(d) A
theE !Z
scattering
GF Qphoton
%box
AMfor%elastic
AA e-p
%A
terms
in
the
and ;
M amplitude
A
—At tree level,
parity
viola- p
a !!!standard model of electroweak interactions.
Corresponding
APV
!"
p!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
0diagrams
2
0
2
Z-boson
exchange,
used by interference of the
are
implied.
4() cross-box
2 *jG
j % &jG j % 2 &#1 % &$#1 " *2 $G Re#G0 $
1(4)$23.00
rmulas
Ep
Mp
Mp
Ap
© 2005
American Physical
Society for ABorn of Eq. (7)
where
for
AE;M;A 212301-1
are obtained from
the The
corresponding
formulas
E;M;A
A0A # 2geV !1 $212301-2
""GZA Re!G0Ap "
q!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
A0M # %2geA "!1 $ ""!1 % #2 "GZM Re!G0Ap ":
38
(11)
The two-photon-exchange correction reduces the magnitude of both the denominator and the numerator of Eq. (10)
by several percent, leading to partial cancellation of the
effect for the asymmetry and increasing the magnitude of
APV compared to its Born value.
The correction is both Q2 and # dependent. It is increastoward
scattering to
angles
(small #),
there
arebackward
effectiveelectron
modifications
electromagnetic
sm of Fig. 1(c), a I.e.,ing
above 1%. Quantitatively, a 1% increase in the
ich the two pho- andreaching
G
,
plus
a new
e.m.toform
factor,exchange
here called
GAin’
M
magnitude
of
A
due
two-photon
results
PV
uark. The photon
about the same percentage decrease in the magnitude of the
integral is sepa- Calculated
(estimated) mid-last-decade,
using the gauge1.03
t is treated sepa2
2
Q = 2 GeV
eorem, while the
2
2
Q = 5 GeV
GPDs to describe
2
2
Q
=
9
GeV
by a nucleon. The
1.02
Ds as in Ref. [3],
fraction) and Q2
n elastic nucleon
1.01
ions of inclusive
s not extended to
n its present form
1
hich corresponds
0
0.2
0.4
0.8
1
0.6
ε
2
%u < M . The
o restricts fourFIG. 3 (color
online). and
Q2 me,
evolution
of94,
two-photon-exchange
Reference:
Afanasev
PRL
212301 (2005).
2
M .
correction to parity-violating asymmetry for several values of
2
n in Figs. 2 and 3
(dashed curve), and 9 GeV2
Q2 # 2 GeV2 (solid curve), 5 GeV
39
two-photon exusing a partonic
ized form factors
Refs. [3,7], and
ader of the main
plot
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APV(1γ+2γ)/APV(1γ)
•
•
GE