Table Of ContentSpin observables in pion photoproduction
from a unitary and causal effective field theory
1
1 A.M. Gasparyan1,2, M.F.M. Lutz1
0
2 1GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH,
n Planckstrasse 1, 64291 Darmstadt, Germany
a
J 2SSC RF ITEP, Bolshaya Cheremushkinskaya 25,
4
117218 Moscow, Russia
]
h
p January 5, 2011
-
p
e
h
[ Abstract
2 PionphotoproductionisanalyzedwiththechiralLagrangian. Partial-
v
wave amplitudes are obtained by an analytic extrapolation of sub-
8
4 threshold reaction amplitudes computed in chiral perturbation the-
9
ory, where the constraints set by electromagnetic-gauge invariance,
5
causality and unitarity are used to stabilize the extrapolation. The
2. √
experimental data set is reproduced up to energies s (cid:39) 1300 MeV in
1
0 terms of the parameters relevant at order Q3. We present and discuss
1 predictions for various spin observables.
:
v
i
X
1 Introduction
r
a
Chiralperturbationtheoryisasystematictoolforstudyinglow-energyhadron
dynamics. Particularly pion-nucleon scattering and pion photoproduction
were considered in [1, 2, 3, 4]. The application of χPT is however limited
to the near threshold region. A method to extrapolate χPT results beyond
the threshold region using analyticity and unitarity constraints was proposed
recently in [5]. We focus on results obtained for pion photoproduction. The
predictions for spin observables as currently being measured at MAMI are
confronted with previous theoretical predictions.
1
2 Chiral symmetry, causality and unitarity
Our approach is based on the chiral Lagrangian involving pion, nucleon and
photon fields [4, 2]. The terms relevant at the order Q3 for pion elastic
scattering and pion photoproduction are listed below1
1 g
L = − N¯ γµ(cid:0)(cid:126)τ ·(cid:0)(cid:126)π ×(∂ (cid:126)π)(cid:1)(cid:1)N + A N¯ γ γµ(cid:0)(cid:126)τ ·(∂ (cid:126)π)(cid:1)N
int 4f2 µ 2f 5 µ
(cid:110) 1+τ g (cid:111)
− e (cid:0)(cid:126)π ×(∂ (cid:126)π)(cid:1) +N¯ γ 3 N − A N¯ γ γ (cid:0)(cid:126)τ ×(cid:126)π(cid:1) N Aµ
µ 3 µ 2 2f 5 µ 3
e κ +κ τ e2
− N¯ σ s v 3 N Fµν + (cid:15)µναβπ F F
4m µν 2 32π2f 3 µν αβ
N
2c c (cid:110) (cid:111)
− 1 m2 N¯ ((cid:126)π ·(cid:126)π)N − 2 N¯ (∂ (cid:126)π)·(∂ (cid:126)π)(∂µ∂νN)+h.c.
f2 π 2f2m2 µ ν
N
c c
+ 3 N¯ (∂ (cid:126)π)·(∂µ(cid:126)π)N − 4 N¯ σµν (cid:0)(cid:126)τ ·(cid:0)(∂ (cid:126)π)×(∂ (cid:126)π)(cid:1)(cid:1)N
f2 µ 2f2 µ ν
d +d
− i 1 2 N¯ (cid:0)(cid:126)τ ·(cid:0)(∂ (cid:126)π)×(∂ ∂ (cid:126)π)(cid:1)(cid:1)(∂νN)+h.c.
f2m µ ν µ
N
id
+ 3 N¯ (cid:0)(cid:126)τ ·(cid:0)(∂ (cid:126)π)×(∂ ∂ (cid:126)π)(cid:1)(cid:1)(∂ν∂µ∂λN)+h.c.
f2m3 µ ν λ
N
m2 d
− 2i π 5 N¯ (cid:0)(cid:126)τ ·(cid:0)(cid:126)π ×(∂ (cid:126)π)(cid:1)(cid:1)(∂µN)+h.c.
f2m µ
N
ie
− (cid:15)µναβN¯ (cid:0)d (∂ π )+d (cid:0)(cid:126)τ ·(∂ (cid:126)π)(cid:1)(cid:1)(∂ N)F +h.c.
8 α 3 9 α β µν
f m
N
d −d
+ i 14 15 N¯ σµν (cid:0)(∂ (cid:126)π)·(∂ ∂ (cid:126)π)(cid:1)(∂λN)+h.c.
2f2m ν µ λ
N
m2 d em2 d
− π 18 N¯ γ γµ(cid:0)(cid:126)τ ·(∂ (cid:126)π)(cid:1)N − π 18 N¯ γ γµ(cid:0)(cid:126)τ ×(cid:126)π(cid:1) N A
f 5 µ f 5 3 µ
e(d −2d )
+ 22 21 N¯ γ γµ(cid:0)(cid:126)τ ×∂ν(cid:126)π(cid:1) N F
2f 5 3 µν
ed
+ 20 N¯ γ γµ(cid:0)(cid:126)τ ×(∂ (cid:126)π)(cid:1) (∂ν∂λN)F +h.c.. (1)
2f m2 5 λ 3 µν
N
A strict chiral expansion of the amplitude to the order Q3 includes tree-level
graphs, loop diagrams, and counter terms. Counter terms depend on a few
1Note a typo in Eq. (1) of [5].
2
unknown parameters, which we adjusted to the empirical data on πN elastic
scattering and pion photoproduction. The extrapolation of the amplitudes
obtained within ChPT is performed utilizing constraints imposed by basic
principles of analyticity and unitarity. For each partial wave we solved the
non-linear integral equation
√ √
TJP( s) = UJP( s)
ab ab √
(cid:88)(cid:90) ∞ dw s−µ T∗,JP(w)ρJP(w)TJP(w)
+ M ac c√d db , (2)
π w−µ w− s−i(cid:15)
µ M
c,d thr
√
where the generalized potential, U(JP)( s), is the part of the amplitude
ab
that contains left-hand cuts only. The phase-space matrix ρJP(w) reflects
cd
our particular convention for the partial-wave amplitudes, that are free of
kinematical constraints. The matching scale µ is required as to arrive at
M
approximate crossing symmetric results. For more details we refer to [5].
3 Results
Thelow-energyconstantsrelevantfortheelasticpion-nucleonscatteringwere
determined in [5]. The empirical s- and p-wave phase shifts are well repro-
√
duced up to the energy s ≈ 1300 MeV. Above this energy inelastic channels
become important. The only exception is the P partial wave where the in-
11
fluence of inelastic channels is significantly larger, that results in a slightly
worse description of the phase shift. A convincing convergence pattern when
going from Q1 to Q3 calculation was observed. In addition to the low-energy
constants of the Lagrangian (1) there are CDD pole parameters characteriz-
ing the Delta and Roper resonances [5].
Thepion-photoproductions-andp-wavemultipolesarequiteconstrained.
There are only four additional low-energy constants and four CDD-pole pa-
rameters for the twelve multipoles to be reproduced. Nevertheless a good
agreement to the existing partial wave analyzes was achieved. In order to
avoidtheambiguitiesinthedifferentpartial-waveanalyzeswedeterminedthe
parameters from the experimental data directly, where we excluded the near-
threshold data in the fit. Our results for the differential cross sections, beam
asymmetries, and helicity asymmetry for the reaction channels γp → π0p,
γp → π+n, γn → π−p are in agreement with experimental data from thresh-
√
old up to s = 1300 MeV. Fig. 1 confronts our prediction for the neutral
3
8 50 75
1/2
s =1073.6 MeV
4 25 50
]
r
s
b/ s1/2=1076.6 MeV s1/2=1079.5 MeV
n
[ 0 0 25
M. -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
C.140 200 300
Ω 1/2
s =1082.5 MeV
d
/
σ
d 90 150 200
1/2 1/2
s =1085.5 MeV s =1088.5 MeV
40 100 100
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
cosθ cosθ cosθ
C.M. C.M. C.M.
Figure 1: Near threshold differential cross section for the reaction γp → π0p
with data taken from [6, 7]. Shown are results from our coupled-channel
theoryincludingisospinbreakingeffectsasareimpliedbytheuseofempirical
pion and nucleon masses. The solid lines correspond to our calculation with
only s- and p-wave multipoles included. The effect of higher partial waves is
shown by the dashed lines.
pion-photo production with the near-threshold MAMI data [6, 7]. This data
allows one to extract the electric s-wave multipole E . Its energy depen-
0+
dence reveals a prominent cusp effect at the opening of the π+n channel. As
shown Fig. 2 this structure is reproduced by our calculation, which discrim-
inates the channels with neutral and charged pions.
Further information about the low-energy photoproduction dynamics is
encoded in the p-wave threshold multipoles. In order to disentangle the
three independent p-wave amplitudes it is insufficient to measure the differ-
ential cross section only. The near-threshold beam asymmetry in γp → π0p
was measured by MAMI [7]. Our results are in striking disagreement with
that measurement. We predict the beam asymmetry to change sign close to
threshold, in a similar manner as predicted before in the dynamical model
of Kamalov et al. [8]. In Fig. 3 we compare the two different results on the
beam asymmetry. Though there is qualitative agreement, important quanti-
tative differences remain. It is interesting to observe that d-wave multipoles
appear to play an important role in the near-threshold region. This was
4
Schmidt et al., 2001
1 Re
Im
]
+
π
m
/
3
-0 0
1
[
+
0
E
-1
145 150 155 160 165
E [MeV]
lab
Figure 2: Energy dependence of the E multipole close to the pion produc-
0+
tion threshold. The data are from [7].
discussed also in [9]. Thus the beam asymmetry may not be the optimal
quantity to extract the p-wave threshold amplitudes. Currently a new data
set on the beam asymmetry is being analyzed at MAMI.
We conclude that it is important to take further data on spin observables
other than the beam asymmetry. Two cases are currently been studied at
MAMI close to threshold. The target asymmetry T and the double polar-
ization observable F. Since there are different phase conventions used in the
literature the reader may appreciate that we detail the relevant expression
in the convention used in [5]. It holds
dσ p¯
= cm (cid:0)|H |2 +|H |2 +|H |2 +|H |2(cid:1) ,
N SA SP D
dΩ 2p
cm
dσ p¯
Σ = cm (cid:60)(H H∗ −H H∗) ,
dΩ p SP SA N D
cm
dσ p¯
T = cm (cid:61)(H H∗ +H H∗ ) ,
dΩ p SP N D SA
cm
dσ p¯
F = cm (cid:60)(H H∗ +H H∗ ) , (3)
dΩ p SP N D SA
cm
5
0.3
s and p waves
full θ=90
S.S.Kamalov et al. (1999)
θ=120
0.2
p
0
π
>
-
p
γ 0.1
Σ
θ=60
θ=120
0
160 180 200 220
E [MeV]
lab
Figure 3: Energy dependence of the beam asymmetry for three particular
angles.
with x = cosθ and
H = m√N cos θ (cid:88)(cid:0)tJ −tJ (cid:1)(cid:16)P(cid:48) (x)−P(cid:48) (x)(cid:17) ,
N 4π s 2 +,1 −,1 J+1 J−1
2 2
J
H = − m√N sin θ (cid:88)(cid:0)tJ +tJ (cid:1)(cid:16)P(cid:48) (x)+P(cid:48) (x)(cid:17) ,
SA 4π s 2 +,1 −,1 J+1 J−1
2 2
J
H = m√N −sinθcos 2θ (cid:88)(cid:0)tJ −tJ (cid:1)(cid:16)P(cid:48)(cid:48) (x)−P(cid:48)(cid:48) (x)(cid:17) ,
SP 4π s (cid:113) +,2 −,2 J+1 J−1
(J − 1)(J + 3) 2 2
J
2 2
H = m√N sinθsin 2θ (cid:88)(cid:0)tJ +tJ (cid:1)(cid:16)P(cid:48)(cid:48) (x)+P(cid:48)(cid:48) (x)(cid:17) .
D 4π s (cid:113) +,2 −,2 J+1 J−1
(J − 1)(J + 3) 2 2
J
2 2
In Fig. 4 an Fig. 5 we show our predictions for the energy dependence
of the target asymmetry and the F observable in comparison with the pre-
dictions of the dynamical model of Ref. [8] calculated at different angles.
6
0.4
θ=120
θ=90
p
0
π
>
- θ=60
p
γ 0.2
T
s and p waves
full
S.S.Kamalov et al. (1999)
0
160 180 200 220
E [MeV]
lab
Figure 4: Energy dependence of the target asymmetry for three particular
angles.
One can see that the target and beam asymmetry are most sensitive to the
details of the dynamics, whereas the F observable is quite similar in both
approaches.
In order to unravel the dynamics close to threshold we detail the contri-
butions of s- and p-waves to the differential cross section and the Σ, T and
F observables. It holds
dσ p¯ (cid:104) 1 1
= cm |E |2 + |P |2 + |P |2 +2(cid:60)(E P∗) cosθ
dΩ p 0+ 2 2 2 3 0+ 1
cm
(cid:16) 1 1 (cid:17) (cid:105)
+ |P |2 − |P |2 − |P |2 cos2θ ,
1 2 3
2 2
dσ p¯
Σ = cm (cid:2)|P |2 −|P |2(cid:3) sin2θ,
3 2
dΩ 2p
cm
dσ p¯
T = − cm (cid:61)(cid:2)(E +P cosθ)(P −P )∗(cid:3)sinθ,
0+ 1 2 3
dΩ p
cm
dσ p¯
F = cm (cid:60)(cid:2)(E +P cosθ)(P −P )∗(cid:3)sinθ, (4)
0+ 1 2 3
dΩ p
cm
7
1
θ=120
0.5
p
0
π
> 0
p - θ=90
γ
F s and p waves
full
S.S.Kamalov et al. (1999)
-0.5
θ=60
-1
150 180 210
E [MeV]
lab
Figure 5: Energy dependence of the double polarization observable F for
three particular angles.
with the linear combinations of p-wave multipoles P = 3E +M −M ,
1 1+ 1+ 1−
P = 3E −M +M , P = 2M +M .
2 1+ 1+ 1− 3 1+ 1−
The expression for the target asymmetry T (Eq. (4)) depends on the
imaginary parts of the multipoles. That is why close to the π0p threshold T
is very small (see Fig. 4). Slightly above the π+n threshold it holds approx-
imately
dσ p¯
T (θ = 90◦) ≈ − cm (cid:61)(E )(P −P ), (5)
0+ 2 3
dΩ p
cm
since the imaginary parts of p-wave multipoles are small. The imaginary part
of E is in turn dominated by the intermediate π+n state. This allows one
0+
to access the difference P −P in the vicinity of the π+n threshold.
2 3
In the beam asymmetry Σ the p-waves multipoles enter only quadrati-
cally and therefore terms containing an interference of the E and d-wave
0+
amplitudes are not suppressed by powers of the πN momentum. Moreover
the magnitudes of |P | and |P | are similar [7] which further diminishes the
2 3
8
relative importance of the p-wave contribution to Σ. In contrast, the quan-
tity F(θ = 90◦) contains the term (cid:60)(E (P −P )∗) but no other competing
0+ 2 3
contribution. Thus measuring F provides a reliable determination of P −P .
2 3
Note that this difference is not small because P and P have opposite signs
2 3
[7].
4 Summary
√
We studied pion photoproduction from threshold up to s = 1300 MeV
with a novel approach developed in [5] based on an analytic extrapolation
of subthreshold amplitudes calculated in ChPT. The free parameters were
adjusted to the pion-nucleon and photoproduction empirical data excluding
the threshold region. Nevertheless the near-threshold MAMI data on the
reaction γp → π0p are described well. The energy dependence of the s-wave
electric E multipole close to threshold including its prominent cusp struc-
0+
ture is also well reproduced. We presented predictions for spin observables
that are planned to be measured or being analyzed at MAMI. Our predic-
tions are compared with results of the dynamical model of Kamalov et al.
[8]. The importance of d-waves for the beam asymmetry close to the πN
threshold was emphasized. This effect indicates that the beam asymmetry
may be not best suited to disentangle the various p-wave threshold multi-
poles as has been anticipated before. We argued that the measurement of
the double polarization observable F suits this purpose much better.
Acknowledgments
We thank M. Ostrick and L. Tiator for stimulating discussions.
9
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