Slides

Overview of recent advances
in calculations of two-photon
exchange effects
Peter Blunden
University of Manitoba
Intense Electron Beams Workshop
June 18, 2015
Outline
• Summary of key results (circa 2003-2008)
Review: Arrington, PGB, Melnitchouk, Prog. Nucl. Part. Phys., (2011)
–impact on form factor measurements
–what is connection to 2nd Born approximation?
–what happens at very low Q2 ?
–how do resonances and partonic description enter as Q2
increases?
• Recent advances
–improved hadronic model parameters (fit to data)
–use of dispersion relations and connection to data
–new experimental results
RRINGTON
Proton GE /GM Ratio
G. 1. $Color online& Ratio of electric to magnetic form factor
xtracted by Rosenbluth measurements $hollow squares& and
the JLab measurements of recoil polarization $solid circles&.
dashed line is the fit to the polarization transfer data.
LT method
ε 2 2
2
2
(Q
)
+
σ
=
G
(Q
)
G
R
M
E
VERVIEW OF FORM FACTOR MEASUREMENTS
τ
We begin with a brief description of the Rosenbluth sepan and recoil polarization
GE fromtechniques,
ε ploton the exslope infocusing
g data and potential problems with the extraction teches.
suppressed at large Q2
PHYSICAL REVIEW C 68, 0343
2
2
#1
(+
(
)
and
(
#
G
/G
Mp
E p ). This enhancement of th
Rosenbluth (Longitudinal-Transverse)
mental uncertainties can become quite large when
Separation
of ( values covered is small or when # (!Q 2 /4M 2p )
This is especially important when one combines hi
from one experiment with low-( data from another
the ( dependence of the cross section. In this case, a
the normalization between the datasets will lead to
Polarization
Q 2 values where the data are com
in G E2 for all Transfer
p
, p G E p !G M p , G E p contributes at most 8.3% $4.3%
total cross section at Q 2 !5(10) GeV2 , so a norm
difference of 1% between a high-( and low-( mea
would change the ratio , p G E p /G M p by 12%
!5 GeV2 and 23%method
at Q 2 !10 GeV2 , more if + ( $
fore, it is vital that!
one properly accounts for the un
in the relative normalization of the data sets when e
P
G
τ
(1
+
ε)
E
T sensitivity t
the form factor
ratios.
The
decreasing
=−
G2Mvalues limits the2εrange PofLapplicability o
large Q
bluth extractions; this was the original motivatio
polarization transfer measurements, whose sensiti
PT,Lasrecoil
proton
not decrease
rapidly with Q 2 .
PT
polarization in �e p → e p�
B. Recoil polarization technique
A. Rosenbluth technique
In polarized elastic electron-proton scattering, p
Two-photon exchange
interference between Born and two-photon exchange amplitudes
X
Mγγ
M0
contribution to cross section:
δ
(2γ)
=
�
�
†
2Re M0 Mγγ
2
|M0 |
standard “soft photon approximation” (used in most data analyses)
approximate integrand in Mγγ by values at γ ∗ poles
neglect nucleon structure (no form factors)
Mo, Tsai (1969)
Assuming OPE
Rosenbluth
Pol. transfer
about 50% TPE + ??
about 80% TPE + ??
"#2$
tio
lope
s
r
e
f
s
n tran
iza
Polar
2
M
!G
"#2$
er slope
f
s
n
a
r
t
n
o
i
t
a
Polariz
G2E
!G2M
G2E
Various Approaches (circa 2003-2008)
!!
#!
#"
"!
"
#"
!"
"
#!
!
!"
• as Q2 increases more and
more parameters,
less and less reliable
!!
!"
!
!"
Low to moderate Q2:
hadronic: N + Δ + N* etc.
"!
(PGB et al., Phys. Rev. Lett 91, 142304 (2003))
Moderate to high Q2:
• GPD approach: assumption of hard photon
interaction with 1 active quark
• Embed in nucleon using Generalized
Parton Distributions
• Valid only in certain kinematic
range (|s,t,u| ≫ M²)
• pQCD: recent work indicates two active
quarks dominate
“handbag”
“cat’s ears”
(Afanasev et al., Phys. Rev. D 72, 013008 (2005))
Nucleon (elastic) intermediate state
• positive slope
• vanishes as ε→ 1
• nonlinearity grows with
increasing Q2
• GM dominates in loop
integral
0
2
−
2
δ (ε,Q )
Q =1 GeV
2
−0.02
2
−0.04
3
6
−0.06
0
0.4
0.2
0.6
0.8
1
ε
• changes sign at low Q2
• agrees well with static limit
for point particle (no form
factors in loop and Q²→ 0)
• GE dominates in loop
integral
0.01
−
2
δ (ε,Q )
0.01
0.001
0.1
0
0.5
−0.01
2
Q =1 GeV
2
−0.02
0
0.2
0.4
0.6
ε
0.8
1
Pointlike limit (e.g.
e- µ+)
δγγ
Static limit:
2
δhard (ε, Q )
�
x = (1 + �)/(1 − �)
Q = ∞ (massless limit)
0.03
1
2
Q2 → 0 or M → ∞
απ
δhard =
1+x
0.02
10
0.1
0.01
2
Q =0
Agrees with McKinley & Feshbach (1948)
2nd Born result with x=1/sin(θ/2)
(static limit)
0
Massless limit:
δhard
α
=
π (x2 + 1)
2α
Q2
=−
ln η ln 2 + δhard
π
λ
�
0
Q → ∞ or M → 0
2
ln
�
x+1
x−1
�
0.2
0.4
0.6
0.8
1
ε
�
�
�
�
�
� 2
���
x+1
x−1
x −1
2
2
2
+ x π + ln
+ ln
− ln
2
2
4
Agrees with Nieuwenhiuzen (1971) and Afanasev et al. (2005)
Suggests hard scattering from one active quark per se cannot be
responsible for a reduction in cross section at backward angles.
Fixed E (Novosibirsk kinematics)
_
e -p correction
0.01
1.000 GeV
1.595 GeV
0.00
b
-0.01
-0.02
-0.03
-0.04
-0.05
0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
e
Fixed E (VEPP-3 Novosibirsk kinematics)
_
e -p correction
0.01
1.000 GeV
1.595 GeV
0.00
b
-0.01
-0.02
-0.03
-0.04
-0.05
0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
e
Agrees with 2nd Born expression at small angles
• At forward angles TPE dominated by Coulomb distortion, while at
backward angles exchange of 2 hard photons contributes
Delta intermediate states
5
e! (p3)
e(p1)
γ
γ
k
P (p2 )
P ! (p4)
N, ∆
(a)
N, ∆
(b)
• γNΔ transition well-studied
amplitude for the box diagram in Fig. 1(a) is given as,
inelastic contribution
• Dominant
!
−i
dk
i(p/ + /p − k/)
(−ieγ )u(p )
M
−i
u(p )(−ieγ )
important
• =More
(2π)
(p +as
p −Q²
k) −increases
m + iε
(p − k) + iε
FIG. 1: Two-photon exchange diagrams with ∆ excitation for elastic ep scattering.
(a,∆)
4
1
4
3
µ
1
2
2
2
ν
2
e
1
4
2
3/2
−i(k/ + M∆ )Pαβ (k) νβ
−i
µα
×
ΓγN →∆ (k, k − p2 )u(p2 ),
u(p4 )Γγ∆→N (k, p4 − k)
(k − p2 )2 + iε
k 2 − M∆2 + iε
(4)
where
3/2
Pαβ (k)
= gαβ
γα γβ (k/γαkβ + kα γβ k/)
−
,
−
2
3
3k
(5)
is the spin-3/2 projector. Amplitude for the cross-box diagram Fig. 1(b) M (b,∆) can be
Resonance (Δ) contribution:
γ(qα)
+ Δ(pµ)
→N
qα+
γNΔ vertex
p µ!
• Lorentz covariant form
• Spin ½ decoupled
• Obeys gauge symmetries
3 coupling constants g1, g2, and g3
At Δ pole:
g1 Magnetic (dominant contribution)
g2-g1 Electric
g3 Coulomb
Take dipole form factor FΔ(q2) = 1/(1-q2/Λ2)2
with Λ = 0.75 GeV (softer than nucleon form factors, with Λ = 0.84 GeV)
Zero width approximation (okay for Re part of δ)
Other resonances (Kondratyuk & PGB, PRC 2007)


N (P11), Δ (P33) + D13, D33, P11, S11, S31
Parameters from dressed K-matrix model
Results
• contribution of heavier resonances
much smaller than N and Δ
• D13 next most important (consistent
with second resonance shape of
Compton scattering cross section)
• partial cancellation between spin 1/2
and spin 3/2
• leads to better agreement, especially
at high Q2
Fit to SuperRosenbluth (JLAB) data
Effect on ratio µp GE/GM
Raw results
Corrected with TPE
Recent Advances
Experiment
• Qweak parity-violation experiment, and the γZ box diagram contribution
• Discrepancy between proton charge radius as measured in atomic H, muonic H,
and electron scattering
• TPE effect on ratio of e+p to e-p cross sections
Theory
• Use improved γNΔ form factors based on most recent data
• Use dispersion integrals to relate Real and Imaginary parts. Imaginary parts fixed by
cross section data
• Valid at forward angles: must use models to extrapolate
• Incomplete: not all data is available (e.g. axial hadron coupling and isospin
dependence in γZ diagrams
• Model-independent analysis of corrections in forward kinematics in dispersive formalism
(sum rule based on total photoabsorption cross section)
TPE effect on ratio of e+p to e-p cross sections
CLAS collaboration (2015)
week ending
13 FEBRUARY 2015
Q²=1.45 GeV²
VIEW LETTERS
1.08
Zhou and Yang (N only)
Blunden et al. (N only)
(e+p)
(e p)
1.06
VEPP-3 Novosibirsk (2015)
PRL 114, 062005 (2015)
PHYSICAL REVIEW
World data
1.04
Zhou and Yang (N+ )
1.02
CLAS TPE
1
0.98
Point-like proton
0.96
0.94
0
0.1
0.2
0.3
0.4
0.6
0.7
World data
0.8
0.9
PHYSICAL REVIEW LETTERS
PRL 114, 062005 (2015)
1.04
Zhou and Yang (N only)
1.03
(e+p)
(e p)
0.5
FIG. 2 (color online). Experimental data (points) and some predictions (
right panels correspond, respectively, to run I and run II. Data points: ope
[25], closed triangle [27], and closed circle—this experiment. Error bar
uncertainties; the shaded bands show the total systematic uncertainty and t
(cyan dash-dotted line), [38] (red thin solid line), [39] (blue thick solid lin
line), and [42] (black dotted line).
Blunden et al. (N only)
Zhou and Yang (N+ )
1.02
CLAS TPE
1.01
1
Point-like proton
0.99
0.98
week ending
13 FEBRUARY 2015
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Q2 (GeV/c)2
FIG. 3 (color online).
ε =0.88
Ratio
of eþ p=e− p cross sections cor2
2
Table II provides the experimental results: the values of
R2γ with the total statistical and systematic uncertainties.
These results are obtained assuming that R2γ is equal to
unity at the normalization points (RLNP
2γ ¼ 1). Also listed
are the kinematic parameters of the measurement, the Δϕ,
chiizin
The
norm
corr
TPE using dispersion relations
(Borisyuk & Kobushkin, Phys. Rev. C 78, 2008)
2 Im
�
dk
�
��
• Imaginary part determined by
unitarity
• Only on-shell form factors
• Real part determined from
dispersion relations
• For elastic (N) intermediate state,
numerical differences between
one loop (solid) and dispersion
(dashed) analyses are tiny (all due
to (F₂ × F₂) term in box vertices
h
x
PGB
B&K
See also recent work by Tomalak & Vanderhaeghen, Eur. Phys. J. A. (2015) 51: 24
∆2γ
0.00
Graczyk, Phys. Rev. C 88, 065205 (2013)
-0.01
1.TPE extracted from data
using Bayesian analysis
2.Use model fit that includes
N and Δ
-0.02
-0.03
Q2=1, N
Q2=1, N+P33
-0.04
Q2=3, N
Q2=3, N+P33
-0.05
0
0.2
0.4
ε
0.6
0.8
KRZYSZTOF M. GRACZYK
1
0.015
PHYSICAL REVIEW C 88, 065205 (2013)
9
BNN
0.85 ± 0.01
fit I
fit II
0.898 ± 0.001
0.867 ± 0.002
∆2γ
FIG. 7. (Color online) #2γ [Eq. (14)] computed for either !(N )
"
)
TPE
contributions.
The
form
from
r !(N +
P
0.01
33 IV. Values of the proton radius !rE2factors
TABLE
" obtained
fromRef.
the [7] are
BNN
andinelastic
HM fits inTPE
femtometers.
pplied.
The
correction, given by !(P33 ), is computed
ithin the P33 (full) model. The values of Q2 are in units of GeV2 .
Q2=3
2
Q =2
0.005
0
our attention was not particularly focused on the Q2 → 0 limit.
Certainly, accurate calculations of the proton radius require
more careful discussion, as is reported in Refs. [54,64–68].
Above Q2 = 1 GeV2 the BNN FF ratio µp GE /GM ,
2
Q =1
P33(Full)
P33(SU(6))
-0.005
0
0.2
0.4
0.6
ε
0.8
1
Rev. D 91, 014023 (2015)
PHYSICAL REVIEW D 91, 014023 (2015)
0.012
Nucleon Form Factor source:
Sachs-monopoles
Sachs-dipoles
dispersion relation approach
Dirac,Pauli-dipoles
calc. by Kondratyuk et al.
0.02
0.015
Nucleon Form Factor source:
Sachs-monopoles
Sachs-dipoles
dispersion relation approach
0.01
0.008
δ2γ,∆
δ2γ,∆
Lorenz
et al., Phys.
HEORETICAL CONSTRAINTS AND
SYSTEMATIC
…
0.01
0.006
0.004
0.005
0.002
0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
ε
0.6
0.8
1
ε
• Used γNΔ form factors fit to recent data
IG. 4 (color online). Dependence of the TPE with Δ intermediate state on the nucleon form factors at Q2 ¼ 3 GeV2 . Left panel: NΔγ
ertex as given by Kondratyuk. Right panel: NΔγ vertex directly matched to helicity amplitudes from electroproduction of nucleon
esonances.
calculation (red, þ) and
thePGB
nucleon þ Δ-TPE calculation
&
• Find smaller results than Kondratyuk
(black, x). Here, we omit the error bars to show the
urther calculations for different kinematics and assumpons are not directly comparable [36–39].
corrections
more clearly.
remains
1 GeV for
with softer form factor
Λ=0.75
GeV Qthan
forbelow
nucleon)
•3.(consistent
the shown MAMI data. Besides the last MAMI data se
Application to cross sections
2
We apply our calculated corrections to the electronroton scattering data with the highest quoted precision
t low Q2. The corresponding measurements have been
arried out at the Mainz Microtron (MAMI) for six
ifferent energies of the incoming electron beam with
2
with the highest precision, we partly include in the
following analysis former world data on electron-proton
scattering. First, this serves as a consistency check, and
second, for an evaluation of the proton structure dependence of the third Zemach moment (see below), a larger
• Claim substantial effect on the determination of the
proton charge radius from scattering data
e! (p3)
e(p1)
γ
γ
k
P (p2 )
N, ∆
(a)
P ! (p4)
1
N, ∆
(b)
Zhou and Yang, arXiv: 1407.2711 (2015)
FIG. 1: Two-photon exchange diagrams with ∆ excitation for elastic ep scattering.
amplitude for the box diagram in Fig. 1(a) is given as,
!
4
−i
d
k
i(p/1 + /p2 − k/)
(a,∆)
(−ieγν )u(p1 )
M
= −i
u(p3 )(−ieγµ )
(2π)4
(p1 + p2 − k)2 − m2e + iε
(p4 − k)2 + iε
3/2
−i(k/ + M∆ )Pαβ (k) νβ
−i
µα
×
ΓγN →∆ (k, k − p2 )u(p2 ),
u(p4 )Γγ∆→N (k, p4 − k)
2
2
2
(k − p2 ) + iε
k − M∆ + iε
(4)
where
3/2
Pαβ (k) = gαβ −
γα γβ (k/γαkβ + kα γβ k/)
−
,
2
3
3k
(5)
is the spin-3/2 projector. Amplitude for the cross-box diagram Fig. 1(b) M (b,∆) can be
Include
all
3
multipoles,
with
form
factors
fit
to
recent
written down in similar manner. The amplitude in Eq. (4) is IR finite because when
2 = 3 GeV2 . The vertical (blue) dashed lines correspond to a value o
IG. 3:CLAS
δ∆ vs.
"
at
Q
data
the four-momentum of the photon approaches zero, the γN∆ vertex functions Γ s also
"
approaches
we do of
notour
havehadronic
to include an
infinitesimal
mass in the See tex
= 0.904 above
whichzero.
theTherefore
predictions
model
could photon
be questionable.
Q2 �3 GeV
0.000
Plot vs. energy instead of ε
Re�∆� �
Im�∆� �
• Imaginary part well-behaved
• Dispersive integral also wellbehaved
(e.g. vanishes at ε→ 0)
�0.001
�0.002
Magnetic + Electric + Coulomb
�0.003
�0.004
Magnetic only
5
Q2 �3 GeV
Magnetic only
0.000
�0.005
Magnetic + Electric + Coulomb
�0.010
�0.015
�0.020
5
10
15
20
25
Ee �GeV�
15
20
25
Ee �GeV�
30
• Real part from loop
0.010
0.005
10
30
35
40
35
40
calculation diverges linearly
with energy (violation of
Froissart bound)
• Problem due to momentumdependent vertices,
uncontrained by on-shell
condition
Resonance (Δ) contribution:
γ(qα)
+ Δ(pµ)
→N
qα+
p µ!
3 coupling constants g1, g2, and g3
At Δ pole:
g1 Magnetic (dominant contribution)
g2-g1 Electric
g3 Coulomb
γNΔ vertex
Dispersion method
k₁
#!
S † = 1 − iM†
!!
†
SS = 1
†
�
#!
#"
"
�
!
†
−i
M
−
M
=
2�m
M
=
M
M
Unitarity →
�
�
1
∗
�m �f |M|i� =
dρ
�f |M |n��n|M|i�
2
n
!"
!
S = 1 + iM
on shell
"!
"
dispersion relation
imaginary part given by
• For dipole form factors, 2D integral can be done
analytically; expressible in terms of elementary functions.
• Can also be done numerically for more general form
factor parametrizations
νth extends into the unphysical region (ε<0)
0.015
0.015
Q2=3
Q2 � 3 GeV2
0.010
0.01
2
Q =2
∆2γ
∆�
KRZYSZTOF M. GRACZYK
0.005
Q2 � 1 GeV2
0.005
0
2
Q =1
0.000
Q2 � 0.1 GeV2
�0.005
0.0
0.2
P33(Full)
P33(SU(6))
-0.005
0.4
0.6
Ε
PGB: dispersive calculation
0.8
1.0
0
0.2
0.4
0.6
0.8
1
ε
FIG. 8. (Color online) !2γ [Eq. (14)] given by the resonance
Graczyk,
Phys. Rev.
88,for065205
(2013)and
only. Calculations
areC
done
the P33 [SU(6)]
P33 contribution
loop
calculation
The values
of Q2 are in units of GeV2 .
P33 (full) models.
Both techniques agree reasonably
atwith predictions of the Borisyuk
also a qualitativewell
agreement
and Kobushkin model [13]. In both approaches the TPE !(P )
low ε (small E), but only the
dispersive
correction
is positive in themethod
low and intermediate Q range and
it reduces the total TPE correction.
gives a vanishing contribution
as εto→
1.), the function ! (P ) depends on
In contrast
! (N
2
2γ
2γ
33
33
the details of the hadronic model. Indeed, there are small but
noticeable differences between the predictions of KBMT05
and the P33 (full) model. To illustrate the model dependence
Why? Isn’t this contrary to Cutkowsky rules?
Dispersive
""
#!
"! "!
"
#"
!"
#" #"
!"
"!
!!
!"
"
##! !
!
!"
=
#"
!"
#!
!! !!
!"
Im
!!
!!
!"
!
!"
Loop
"!
!
!
""
"! "!
"
#!
#"
q₂
!"
+! "
#
#! !
q₁
!"
!"
#" #"
# "!!
!"
"!
##! !
!
!"
"
=∫
#"
!! !!
!"
#!
!!
!"
Re
!!
!"
!
contact term
" Im part = 0
"!
#
"
Borisyuk & Kobushkin, arXiv:1506.02682 (2015)
0.15
2
2
0.4
0.6
Q = 1.0 GeV
S11
S31
P11
P31
P13
P33
D13
D33
%&/&, %
0.1
0.05
0
• Include other spin 1/2 and 3/2
0
0.2
0.8
1
!
1.5
2
2
0.4
0.6
Q = 5.0 GeV
S11
S31
P11
P31
P13
P33
D13
D33
1
0.5
%&/&, %
resonances using MAID
helicity amplitudes
• Include a finite width
• Contributions tend to cancel,
in qualitative agreement with
Kondratyuk & Blunden result
-0.05
0
-0.5
-1
-1.5
0
0.2
!
0.8
1
2
values of the electron beam energy relevant for the Ma
A1 experiment with the proton [1] and the deuteron [
target, the latter being currently under analysis. It
compared to the experimental sensitivity of the Ma
corrections
in
forward
kinematics
experiments that is obtained as
�
2 µν
µ ν
+ fGorchtein,
(P
·
q
)
g
+
(q
·
q
)P
P
2
1 arXiv: 1406.1612
1
2
(2014)
�
−(P · q1 )(P µ q1ν + P ν q2µ ) .
Model-independent analysis of
e of(forward
the relation angles, low Q²) using dispersive analysis
R , exp.
u(k) =
·k1 )
t (P
(P ·K)
−
q12
−
4(P · K)
q22
ū(k � )P/ u(k),
δσR E
(29)
2
/|t| = −RE
/3 (1 ± δRE /RE ) .
(
For the proton, the experimental result of RE = 0.879
fm translates
Brown, into
Phys Rev D 1, 1432 (1970)
also
q1 , t, q12 , q22 )
hout this calculation, Im f2 (P ·
ken at t = q12 = q22 = 0. In these kinematics,
heorem relates this imaginary part to the
otoabsorption cross section σT as
P · q1 , 0, 0, 0) = −2σT /[(P · q1 )e2 ].
(31)
definition of Eq. (14) and identifying the
ntegrals in Eq. (30) with the previously inin Eq. (27), one can express the leading
δσʀ(E,t)/|t| (GeV⁻²)
TPE
amplitude
Φ(E): neglectSee
ensor
contraction
and consistently
2
q1,2
and ∼ q12 q22 in the numerator, the imagRE , exp.
δσR
/|t| = −6.61(12) GeV−2 .
(
the TPE amplitude can be cast in the form
�
This experimental sensitivity is compared in Fig 2 to
d3�k1
1
4
�
= e ū(k )P/ u(k)
(30)
numerical evaluation of Eq. (34) in the kinematics of
2
2
3
(2π) 2E1 q1 q2
A1 Mainz experiment [1] The energy dependence (diff
�
�
2
2
) + (P · k1 )
(P · k1 )
t
− q12 − q22 Imf2 ,
2(P · K)
(P · K)
E = 180 MeV
photoabsorption 0.16
E = 315 MeV
plitude f1 does not contribute atTotal
the leading
E = 450 MeV
cross section
curacy. According to the power counting
0.14
Exp. Sensitivity
0.12
0.1
0.08
0.06
0
0.01
0.02
0.03
|t| (GeV²)
0.04
Summary
• Lots of interesting new theoretical work motivated
by new experimental results
• Dispersive method promising approach with
connection to data in forward angle limit
–A similar approach is essential for the γZ box in Qweak
parity-violation kinematics