Physics Letters B 759 (2016) 126–130
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Improved empirical parametrizations of the γ
∗
N → N(1535)
transition amplitudes and the Siegert’s theorem
G. Ramalho
International Institute of Physics, Federal University of Rio Grande do Norte, Campus Lagoa Nova – Anel Viário da UFRN, Lagoa Nova, Natal-RN, 59070-405, Brazil
a r t i c l e i n f o a b s t r a c t
Article history:
Received
18 February 2016
Received
in revised form 9 May 2016
Accepted
20 May 2016
Available
online 25 May 2016
Editor:
J.-P. Blaizot
Some empirical parametrizations of the γ
∗
N → N(1535) transition amplitudes violate the Siegert’s
theorem, that relates the longitudinal and the transverse amplitudes in the pseudo-threshold limit
(nucleon and resonance at rest). In the case of the electromagnetic transition from the nucleon
(mass M) to the resonance N(1525) (mass M
R
), the Siegert’s theorem is sometimes expressed by the
relation |q|A
1/2
= λS
1/2
in the pseudo-threshold limit, when the photon momentum |q| vanishes, and
λ =
√
2(M
R
− M). In this article, we argue that the Siegert’s theorem should be expressed by the
relation A
1/2
= λS
1/2
/|q|, in the limit |q| → 0. This result is a consequence of the relation S
1/2
∝|q|,
when |q| → 0, as suggested by the analysis of the transition form factors and by the orthogonality
between the nucleon and N(1535) states. We propose then new empirical parametrizations for the
γ
∗
N → N(1535) helicity amplitudes, that are consistent with the data and the Siegert’s theorem. The
proposed parametrizations follow closely the MAID2007 parametrization, except for a small deviation in
the amplitudes A
1/2
and S
1/2
when Q
2
< 1.5GeV
2
.
© 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The information relative to the structure of the electromag-
netic
transitions between the nucleon and the nucleon excita-
tions
(γ
∗
N → N
∗
) has been parametrized using different forms [1,
2].
The representations in terms of helicity amplitudes, longitudi-
nal
and transverse, can be defined independently of the propri-
eties
of the resonances. Alternatively, one can use a representation
in terms of structure form factors, that emphasize precisely the
symmetries associated with the nucleon resonances. The helicity
amplitudes and the structure form factors are functions of the
transition four-momentum transfer (q) squared, q
2
, but are often
represented in terms of Q
2
=−q
2
, particularly in nucleon elec-
troexcitation
reactions ( Q
2
> 0). In general the different helicity
amplitudes are independent functions, except in some specific lim-
its.
The same holds for the form factors.
Taking
the case of the nucleon as example: the electric and
the magnetic form factors, G
E
and G
M
, are independent func-
tions,
except in the threshold limit, Q
2
=−4M
2
, where G
E
= G
M
(threshold of the γ
∗
→ N
¯
N reaction). In the case of the γ
∗
N → N
∗
transitions, there are constraints between helicity amplitudes, or
E-mail address: gilberto.ramalho@iip.ufrn.br.
between form factors, at the pseudo-threshold limit. The pseudo-
threshold
limit is the limit where the photon momentum |q|
vanishes, and both particles, the nucleon (N) and the resonance,
labeled here in general as R, are at rest. In the pseudo-threshold
Q
2
= Q
2
PS
=−(M
R
− M)
2
[3,4].
The
condition that expresses the relation between different am-
plitudes
(or form factors) at the pseudo-threshold is usually re-
ferred as
the Siegert’s theorem. The Siegert’s theorem was intro-
duced
first in studies related with nuclear physics [3,5] and was
later used in pion electroproduction reactions [6–9].
In this work, we study in particular the constraints of the
Siegert’s theorem in the γ
∗
N →N(1535) transition, where N(1535)
is a spin
1
2
state with negative parity ( J
P
=
1
2
−
). We will show
in particular that some parametrizations of the γ
∗
N → N(1535)
transition amplitudes, like the MAID2007 parametrization [8–10],
are not consistent with the Siegert’s theorem. In order to grant
that the Siegert’s theorem is valid, one needs to ensure that
S
1/2
∝|q|, near |q| = 0. In the present article, we propose then
new parametrizations for the amplitudes A
1/2
and S
1/2
, that are
consistent with both, the empirical data and the Siegert’s theorem.
The consequences of the Siegert’s theorem for the γ
∗
N →
(
1232) and γ
∗
N → N(1520) helicity amplitudes are discussed
in a separate article [11].
http://dx.doi.org/10.1016/j.physletb.2016.05.060
0370-2693/
© 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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