11institutetext: Institutionen för fysik och astronomi, Uppsala universitet,
Box 516, S-75120 Uppsala, Sweden 22institutetext: National Centre for Nuclear
Research, 02-093 Warsaw, Poland

# Electromagnetic transition form factors
and Dalitz decays of hyperons

Nora Saloneaddr1,addr2  Stefan Leupoldaddr1

(August 29, 2024)

###### Abstract

Dalitz decays of a hyperon resonance to a ground-state hyperon and an
electron-positron pair can give access to some information about the composite
structure of hyperons. We present expressions for the multi-differential decay
rates in terms of general transition form factors for spin-parity combinations
$J^{P}=\frac{1}{2}^{\pm},\frac{3}{2}^{\pm}$ of the hyperon resonance. Even if
the spin of the initial hyperon resonance is not measured, the self-analyzing
weak decay of the “final” ground-state hyperon contains information about the
relative phase between combinations of transition form factors. This relative
phase is non-vanishing because of the unstable nature of the hyperon
resonance. If all form factor combinations in the differential decay formulae
are replaced by their respective values at the photon point, one obtains a QED
type approximation, which might be interpreted as characterizing hypothetical
hyperons with point-like structure. We compare the QED type approximation to a
more realistic form factor scenario for the lowest-lying singly-strange
hyperon resonances. In this way we explore which accuracy in the measurements
of the differential Dalitz decay rates is required in order to distinguish the
composite-structure case from the pointlike case. Based on the QED type
approximation we obtain as a by-product a rough prediction for the ratio
between the Dalitz decay width and the corresponding photon decay width.

##  1 Motivation

Electromagnetic form factors have become an important tool to study the
structure of strongly interacting objects, see e.g. Granados:2017cib ;
Alarcon:2017asr ; Leupold:2017ngs ; Junker:2019vvy ; Husek:2019wmt ;
Landsberg:1986fd ; Czerwinski:2012ry ; Miller:2007uy ; Pacetti:2015iqa ;
Punjabi:2015bba ; Devenish:1975jd ; Korner:1976hv ; Carlson:1985mm ;
Pascalutsa:2006up ; Aznauryan:2011qj ; Tiator:2011pw ; Eichmann:2018ytt ;
Kaxiras:1985zv ; Kubis:2000aa ; Sanchis-Alepuz:2017mir ; Ablikim:2019vaj ;
Ramalho:2019koj ; Ramalho:2020tnn  and references therein. In the near future,
photon and Dalitz decays of hyperons will be measured at GSI/FAIR by
HADES+PANDA Ramstein:2019kaz , $Y^{*}\to Y\gamma$ and $Y^{*}\to
Y\,e^{+}e^{-}$, respectively. Here $Y^{*}$ denotes a singly-strange hyperon
resonance and $Y$ a ground-state hyperon ($\Lambda$ or $\Sigma$).

In the present work we want to explore what it takes to extract more
information from the Dalitz decays than from the photon decays. In other
words: How accurately does one need to measure the Dalitz decay distribution
to determine an energy dependence of the form factors? In turn, the shape of a
form factor is related to the information about the intrinsic structure
Alarcon:2017asr ; Miller:2007uy ; Tiator:2011pw . To this end, we introduce
the most general transition form factors for spin-parity combinations
$J^{P}=\frac{1}{2}^{\pm},\frac{3}{2}^{\pm}$ of $Y^{*}$. This is similar in
spirit to the developments of Korner:1976hv ; Perotti:2018wxm  for
$e^{+}e^{-}\to Y^{*}\bar{Y}$, but for a different kinematical regime. For
general considerations about transition form factors see also Devenish:1975jd
; Carlson:1985mm . In practice, what we focus on are the low-lying hyperon
resonances $\Lambda(1405)$ with $J^{P}=\frac{1}{2}^{-}$, $\Lambda(1520)$ with
$J^{P}=\frac{3}{2}^{-}$ pdg . For completeness we cover also the cases of
$J^{P}=\frac{3}{2}^{+}$ and $J^{P}=\frac{1}{2}^{+}$. Examples for the latter
are the states $\Sigma(1385)$ and $\Sigma^{0}$, respectively. Transitions of
those states have been studied by the group of one of the authors in
Granados:2017cib ; Junker:2019vvy ; Husek:2019wmt ; Nair:2018mwa ;
Holmberg:2018dtv .

The $\Lambda(1405)$ is a very interesting state; see e.g. Mai:2020ltx ;
Dalitz:1967fp ; Siegel:1988rq ; Jido:2003cb ; GarciaRecio:2003ks ; Geng:2007hz
; Sekihara:2008qk ; Hall:2014uca  and references therein. It is the lowest-
lying baryon state with negative parity pdg .
Naively, one would think that the lowest-lying baryon with negative parity
should contain only up and down quarks since those are significantly lighter
than the s-quark. Yet, the $\Lambda(1405)$ as the lightest baryon with
negative parity has a stangeness of $-1$, i.e. must contain (at least) one
strange quark. This triggered many discussions about the nature of the
$\Lambda(1405)$ as a state that might not fit into the quark model, which
describes baryons as three-quark states. As an alternative, a bound state of
nucleon and antikaon (“hadronic molecule”) has been proposed. It has also been
suggested that there might actually be two coupled-channel hadronic-molecule
states Jido:2003cb , one coupling stronger to the nucleon-antikaon structure
and one stronger to $\Sigma$-pion, which constitutes the main decay mode of
the $\Lambda(1405)$. From a more quantitative point of view the proper
question is how much overlap a physical $\Lambda(1405)$ state has e.g. with a
three-quark or with a proton-antikaon field configuration etc. In any case, it
is conceivable that different pictures about the nature of the $\Lambda(1405)$
lead to somewhat different size predictions. More generally, different ideas
about the intrinsic structure can lead to different predictions for the
differential Dalitz decay rates.

The second concrete example for which we will provide quantitative results is
the $\Lambda(1520)$. It is the strange baryon next in mass after covering
$\Sigma$ and $\Sigma(1385)$ previously Granados:2017cib ; Junker:2019vvy ;
Husek:2019wmt ; Nair:2018mwa ; Holmberg:2018dtv  and the $\Lambda(1405)$ in
the present work. Above the $\Lambda(1520)$ our low-energy techniques might
fail to work. To provide a self-contained work, it makes sense to cover also
the $\Lambda(1520)$ in the present paper.

Though the $\Lambda(1520)$ is often regarded as a typical quark-model state,
see e.g. the mini-review in pdg , there are also ideas that suggest the
$\Lambda(1520)$ essentially as a hadron molecule partner to the
$\Lambda(1405)$ when interchanging ground-state baryons from the nucleon octet
by spin-3/2 states from the $\Delta$ decuplet Kolomeitsev:2003kt . Like for
the $\Lambda(1405)$, it can be expected that different pictures about the
structure of the $\Lambda(1520)$ Roca:2006pu ; Roca:2006sz  lead to different
predictions for the differential Dalitz decay rates.

The purpose of the present paper is not about developing particular models for
the structure of the $\Lambda(1405)$ or the $\Lambda(1520)$. Yet we take the
large interest in these states as a motivation to perform a model-independent
analysis of the capability of Dalitz decays to access their respective
intrinsic structure.

The paper is structured in the following way. In the next section we present
the framework to introduce electromagnetic transition form factors in the most
general way for initial states with spin 3/2 or 1/2 and final states with spin
1/2. We will calculate the decay rates for radiative (photon) decays and
Dalitz decays and also include the possibility that the “final” hyperon
emerging from the Dalitz decay performs a further weak decay into nucleon and
pion. In sect. 3 we introduce our parametrization for the transition form
factors that features transition radii. We also specify a “structureless” case
that we can contrast with the case of an extended structure. In sect. 4 we
apply our framework concretely to the two initial states $\Lambda(1405)$ and
$\Lambda(1520)$ (and the final state of a ground state $\Lambda$). Further
discussion and a summary is provided in sect. 5. Appendices are added for
technical purposes, but also to explain some conceptual issues that do not fit
into the main body of the text.

##  2 Constraint-free form factors, helicity amplitudes and differential decay
widths

From a formal point of view we study electromagnetic transitions from a baryon
resonance with spin 1/2 or 3/2 to a baryon with spin 1/2. For this transition,
we disregard parity violating processes, i.e. we focus on transitions mediated
by the strong and electromagnetic interactions. Prominent examples that can be
described by this framework are the decays:
$\Lambda(1520)\to\Lambda\gamma^{(*)},\Sigma^{0}\gamma^{(*)}$;
$\Lambda(1405)\to\Lambda\gamma^{(*)},\Sigma^{0}\gamma^{(*)}$;
$\Sigma(1385)\to\Lambda\gamma^{(*)},\Sigma\gamma^{(*)}$;
$\Sigma^{0}\to\Lambda\gamma^{(*)}$.
Electroweak processes like $\Xi^{0}\to\Lambda\gamma^{(*)}$ would need a
(straightforward) extension of the formalism and are not covered in the
present work. We will present formulae for all parity combinations. When it
comes to concrete applications we will focus on two processes, namely
$\Lambda(1520)\to\Lambda\gamma^{(*)}$ and
$\Lambda(1405)\to\Lambda\gamma^{(*)}$.

Generically we study the process $Y^{*}\to Y\gamma^{(*)}$ where the star for
the hyperon $Y$ denotes an excited hyperon, i.e. a resonance, while
$\gamma^{*}$ refers to a virtual photon $\gamma$.

###  2.1 Transition $\frac{3}{2}^{\mp}\to\frac{1}{2}^{\pm}$

If the initial baryon $Y^{*}$ has spin 3/2 and opposite parity to the final
baryon $Y$, the most general decomposition of the transition respecting
Lorentz invariance, current conservation and parity symmetry can be written as
(cf. also Korner:1976hv )

$\langle
p_{Y},\lambda_{Y}|j^{\mu}(0)|p_{Y^{*}},\lambda_{Y^{*}}\rangle=e\bar{u}(p_{Y},\lambda_{Y})\Gamma^{\mu\nu}_{-}u_{\nu}(p_{Y^{*}},\lambda_{Y^{*}})$

(1)

with

$\begin{split}\qquad\Gamma^{\mu\nu}_{-}=&-iH_{1}(q^{2})\,m_{Y^{*}}\,\left(\gamma^{\mu}q^{\nu}-\not{q}\,g^{\mu\nu}\right)\\\
&+iH_{2}(q^{2})\left(q^{\nu}p_{Y^{*}}^{\mu}-(q\cdot
p_{Y^{*}})\,g^{\mu\nu}\right)\\\
&+iH_{3}(q^{2})\left(q^{\mu}q^{\nu}-q^{2}g^{\mu\nu}\right)\end{split}$ (2)

and $q:=p_{Y^{*}}-p_{Y}$. Here $j_{\mu}$ denotes the electromagnetic current
and $e$ the charge of the proton. The helicity of the initial (final) baryon
is denoted by $\lambda_{Y^{*}}$ ($\lambda_{Y}$). Our conventions for the
spin-3/2 vector-spinor $u_{\nu}$ have been spelled out in Junker:2019vvy .
Note that no $\gamma_{5}$ appears here since either none or both of the
involved baryons have natural parity Devenish:1975jd  (and the electromagnetic
current is a vector current and has natural parity). The three quantities
$H_{i}$, $i=1,2,3$, constitute constraint-free transition form factors in the
sense of a Bardeen-Tung-Tarrach (BTT) construction Bardeen:1969aw ;
Tarrach:1975tu .

We have introduced the three transition form factors $H_{i}$ such that they
all have the same dimensionality (two inverse mass dimensions). In general,
these transition form factors are complex quantities. Thus the appearance of
the explicit $i$’s in the defining eq. (2) is a pure convention. However,
there is some meaning to this choice. Suppose one calculates contributions to
the transition form factors from an effective Lagrangian that satisfies charge
conjugation symmetry. Then, a tree-level calculation will yield purely real
results for $H_{i}$. In other words, conventions have been chosen such that
only loops create imaginary parts for $H_{i}$. We substantiate this further in
A.

Next, we introduce dimensionless helicity amplitudes: we define

$\displaystyle H_{-}(q^{2})$ $\displaystyle:=$
$\displaystyle-\left(m_{Y^{*}}-m_{Y}\right)m_{Y^{*}}H_{1}(q^{2})$
$\displaystyle{}+\frac{1}{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}+q^{2}\right)H_{2}(q^{2})+q^{2}H_{3}(q^{2})\,,$
$\displaystyle H_{0}(q^{2})$ $\displaystyle:=$ $\displaystyle-\
(m_{Y^{*}}-m_{Y})\,m_{Y^{*}}H_{1}(q^{2})$
$\displaystyle{}+(m_{Y^{*}}-m_{Y})\,m_{Y^{*}}H_{2}(q^{2})$
$\displaystyle{}+\frac{m_{Y^{*}}-m_{Y}}{2m_{Y^{*}}}\left(m_{Y^{*}}^{2}-m_{Y}^{2}+q^{2}\right)H_{3}(q^{2})\,,$
$\displaystyle H_{+}(q^{2})$ $\displaystyle:=$
$\displaystyle-\left(m_{Y^{*}}m_{Y}-m_{Y}^{2}+q^{2}\right)H_{1}(q^{2})$
$\displaystyle{}+\frac{1}{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}+q^{2}\right)H_{2}(q^{2})+q^{2}H_{3}(q^{2})\,.$

In a frame where the baryon momenta are aligned, the helicity flip amplitude
$H_{-}$ is related to the combinations
$(\lambda_{Y^{*}},\lambda_{Y})=(3/2,1/2),(-3/2,-1/2)$. The other helicity flip
amplitude $H_{+}$ is related to the combinations
$(\lambda_{Y^{*}},\lambda_{Y})=(-1/2,1/2),(1/2,-1/2)$. Finally the non-flip
amplitude $H_{0}$ relates to $\lambda_{Y^{*}}=\lambda_{Y}=\pm 1/2$. Note that
the spin111We avoid here the phrase “helicity” since the virtual photon might
be at rest. of the real or virtual photon along the axis defined by the flight
direction of the hyperons is given by
$\lambda_{\gamma}=\lambda_{Y^{*}}-\lambda_{Y}$.

The advantage of the helicity amplitudes over the three transition form
factors $H_{i}$, $i=1,2,3$, is that there are no interference terms when
calculating the decay widths for the reactions $Y^{*}\to Y\gamma$ and
$Y^{*}\to Y\,e^{+}e^{-}$. We will see this explicitly below. A second
advantage is that one can use quark counting rules to determine the high-
energy behavior Carlson:1985mm  of the helicity amplitudes for large values of
space-like $q^{2}<0$. The main topic of this work are Dalitz decays. Here the
photon virtuality $q^{2}$ is time-like and has an upper limit given by
$(m_{Y^{*}}-m_{Y})^{2}$. Therefore high-energy constraints are not so relevant
for the physics discussed here. Nonetheless, for completeness, we collect the
high-energy behavior of all transition form factors and helicity amplitudes in
B.

While the transition form factors $H_{1}$, $H_{2}$ and $H_{3}$ are free from
kinematical constraints, the helicity amplitudes satisfy

$\begin{split}\qquad
H_{+}((m_{Y^{*}}-m_{Y})^{2})&=H_{0}((m_{Y^{*}}-m_{Y})^{2})\\\
&=H_{-}((m_{Y^{*}}-m_{Y})^{2})\end{split}$ (4)

and

$\begin{split}\qquad&\frac{2(m_{Y^{*}}+m_{Y})}{m_{Y^{*}}-m_{Y}}H_{0}((m_{Y^{*}}+m_{Y})^{2})\\\
&=H_{+}((m_{Y^{*}}+m_{Y})^{2})+H_{-}((m_{Y^{*}}+m_{Y})^{2})\,.\end{split}$ (5)

From a technical point of view, these relations are easy to deduce from the
definitions of the helicity amplitudes (2.1). Later, the constraint (4) will
be very important for our model independent low-energy parametrization of the
transitions. Therefore we feel obliged to offer also a physical instead of
purely technical motivation for the kinematical constraints. This physical
explanation is provided in C.

The width for the two-body radiative decay
$Y^{*}\to Y\gamma$ is given by

$\Gamma_{2}=e^{2}\frac{\left(m_{Y^{*}}+m_{Y}\right)^{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}\right)}{96\pi
m_{Y^{*}}^{3}}\\\ \times\left[3|H_{-}(0)|^{2}+|H_{+}(0)|^{2}\right]\,.$ (6)

As already announced there are no interference terms between the helicity
amplitudes. The absence of $H_{0}$ signals nothing but the non-existence of a
longitudinally polarized real photon.

To describe in the one-photon approximation the Dalitz decay $Y^{*}\to
Y\gamma^{*}\to Y\,e^{+}e^{-}$ we choose a frame where the virtual photon (and
therefore the electron-positron pair) is at rest. In this frame, $\theta$
denotes the angle between the hyperon $Y$ and the electron. The double-
differential three-body (Dalitz) decay width is given by

$\begin{split}&\frac{\text{d}\Gamma_{3}}{\text{d}q^{2}\text{d}(\cos{\theta})}=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{3}\,192m_{Y^{*}}^{3}}\frac{(m_{Y^{*}}+m_{Y})^{2}-q^{2}}{q^{2}}\\\
&\hskip
2.84526pt\times\bigg{\\{}\bigg{(}1+\cos^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}\theta\bigg{)}\big{[}3|H_{-}(q^{2})|^{2}+|H_{+}(q^{2})|^{2}\big{]}\\\
&\hskip
5.69054pt+\bigg{(}\sin^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}\theta\bigg{)}\frac{4q^{2}}{(m_{Y^{*}}-m_{Y})^{2}}|H_{0}(q^{2})|^{2}\bigg{\\}}\,.\end{split}$

(7)

Here we have used the velocity of the electron in the rest frame of the
electron-positron pair given by $\beta_{e}=\sqrt{1-4m_{e}^{2}/q^{2}}$. The
momentum of $Y^{*}$ and $Y$ in the rest frame of the virtual photon is given
by

$\qquad\qquad\qquad
p_{z}:=\frac{\lambda^{1/2}(m_{Y^{*}}^{2},m_{Y}^{2},q^{2})}{2\sqrt{q^{2}}}$ (8)

with the Källén function

$\qquad\lambda(a,b,c):=a^{2}+b^{2}+c^{2}-2(ab+bc+ac)\,.$ (9)

To obtain the integrated Dalitz decay width, we note that the lower (upper)
integration limit of the $\cos\theta$ integration is $-1$ ($+1$), i.e. the
$\theta$ integration would go from $\pi$ to 0 and not the other way. This
convention makes the right-hand side of (7) positive.

It is nice to see how all physical constraints are visible in (7). For
$q^{2}=(2m_{e})^{2}$ the leptons are produced at rest. Thus one cannot define
an angle between electron and hyperon momentum. The right-hand side of (7)
should show no angular dependence. This is indeed the case. At the other end
of the phase space, i.e. for $q^{2}=(m_{Y^{*}}-m_{Y})^{2}$ the hyperons are at
rest. Again one cannot define an angle between electron and hyperon momentum.
Using the kinematical constraint (4), one can see that also here the angular
dependence disappears.

Even though the electromagnetic transition that we consider respects parity
symmetry, we can allow for a further decay of the “final” hyperon. Focusing
now on $Y=\Lambda$ (or $\Sigma^{\pm}$) this last decay is mediated by the weak
interaction and does violate the parity symmetry. As a consequence, this decay
populates different partial waves and gives rise to an interference pattern.
In total, we study now the decay sequence $Y^{*}\to Y\,e^{+}e^{-}$ and
$Y\to\pi N$. Thus we have a four-body final state. In principle, this gives
rise to 5 independent kinematical variables. However, the intermediate $Y$
state is so long-living pdg  that its mass is fixed (and in the experimental
analyses one triggers on a displaced vertex Thome:2012bdy ;
IkegamiAndersson:2020rau ). In addition, one can show that the squared matrix
element of the four-body decay is independent of specific combinations of
four-momenta. This feature is discussed in D. As a consequence, the specific
four-body decay depends on three independent variables, one variable more than
the already considered Dalitz decay.

Besides the invariant mass $\sqrt{q^{2}}$ of the dilepton pair (the photon
virtuality $\gamma^{*}$) and the angle $\theta$ between the electron and the
hyperons in the rest frame of $\gamma^{*}$, one could use a second relative
angle. It is convenient to define this angle in the frame where the decaying
$Y$ hyperon is at rest. In this frame the electron-positron pair defines a
plane. One could use the angle between the plane’s normal and the direction of
the nucleon. Since it is a relative angle, its definition does not depend on
the choice of a coordinate system, only on the choice of a proper frame of
reference. Yet, to connect to the formalism developed in Perotti:2018wxm ;
Faldt:2013gka ; Faldt:2016qee ; Faldt:2017kgy ; Faldt:2017yqt ; Faldt:2019zdl
we introduce a fixed coordinate system and find for the four-body decay
$Y^{*}\to Y\gamma^{*}\to\pi N\,e^{+}e^{-}$ the differential decay width

$\begin{split}&\frac{\text{d}\Gamma_{4}}{\text{d}q^{2}\text{d}(\cos{\theta})\text{d}\Omega_{N}}\\\
&=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{4}\,384m_{Y^{*}}^{3}}\,\text{Br}_{Y\to\pi
N}\,\frac{(m_{Y^{*}}+m_{Y})^{2}-q^{2}}{q^{2}}\\\ &\hskip
2.84526pt\times\bigg{\\{}\bigg{(}1+\cos^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}\theta\bigg{)}\big{[}|H_{+}(q^{2})|^{2}+3|H_{-}(q^{2})|^{2}\big{]}\\\
&\hskip
14.22636pt+\bigg{(}\sin^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}\theta\bigg{)}\frac{4q^{2}}{\left(m_{Y^{*}}-m_{Y}\right)^{2}}|H_{0}(q^{2})|^{2}\\\
&\hskip
14.22636pt-\frac{4\sqrt{q^{2}}\beta_{e}^{2}}{m_{Y^{*}}-m_{Y}}\,\alpha_{Y}\,\text{Im}[H_{0}(q^{2})H_{+}^{*}(q^{2})]\\\
&\hskip
25.60747pt\times\sin\theta\cos\theta\sin\theta_{N}\sin\phi_{N}\bigg{\\}}\,.\end{split}$

(10)

We recall that $q^{2}$ is the square of the dilepton mass or photon virtuality
and $\theta$ denotes the angle between one of the hyperons and the electron in
the rest frame of the dilepton (rest frame of the virtual photon). In
addition, $\theta_{N}$ and $\phi_{N}$ are the angles of the nucleon three-
momentum measured in the rest frame of $Y$. The coordinate system in this
frame is defined by $\vec{q}$ pointing in the negative $z$-direction (i.e. in
the rest frame of the virtual photon the $Y$ direction defined the positive
$z$-axis) and the electron moves in the $x$-$z$ plane with positive momentum
projection on the $x$-axis. In this frame, $\theta_{N}$ is the angle of the
nucleon momentum relative to the $z$-axis and $\phi_{N}$ is the angle between
the $x$-axis and the projection of the nucleon momentum on the $x$-$y$ plane,
i.e.

$\displaystyle\qquad\textbf{p}_{N}=p_{\rm f}\,(\sin\theta_{N}$
$\displaystyle\cos\phi_{N},\sin\theta_{N}\sin\phi_{N},\cos\theta_{N})\,,$ q
$\displaystyle=|\textbf{q}|\,(0,0,-1)\,,$ (11)
$\displaystyle\textbf{p}_{e^{-}}\cdot\textbf{e}_{y}=0\,,\quad$
$\displaystyle\textbf{p}_{e^{-}}\cdot\textbf{e}_{x}>0\,,\quad\textbf{e}_{y}=\frac{\textbf{p}_{e^{-}}\times\textbf{q}}{|\textbf{p}_{e^{-}}\times\textbf{q}\,|}\,,$

with the momentum $p_{\rm f}$ of the nucleon in the rest frame of the decaying
$Y$ hyperon. We provide the momentum and also the corresponding energy:

$\qquad\qquad\qquad p_{\rm
f}=\frac{\lambda^{1/2}(m_{Y}^{2},m_{N}^{2},m_{\pi}^{2})}{2m_{Y}}$ (12)

and

$\qquad\qquad\qquad E_{N}=\frac{m_{Y}^{2}+m_{N}^{2}-m_{\pi}^{2}}{2m_{Y}}\,,$
(13)

with the Källén function defined in (9). Note the subtlety that $\theta$ is
measured in the rest frame of the virtual photon while $\Omega_{N}$ denotes
angles in the rest frame of the $Y$ hyperon. In terms of Lorentz invariant
quantities the angles are related to

$\displaystyle p_{Y}\cdot k_{e}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\lambda^{1/2}(m_{Y^{*}}^{2},m_{Y}^{2},q^{2})\,\beta_{e}\,\cos\theta\,,$
$\displaystyle\epsilon_{\mu\nu\alpha\beta}\,k_{e}^{\mu}\,p_{Y}^{\nu}\,p_{N}^{\alpha}\,q^{\beta}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\,\sqrt{q^{2}}\,\lambda^{1/2}(m_{Y^{*}}^{2},m_{Y}^{2},q^{2})\,p_{\rm
f}\,\beta_{e}$ (14) $\displaystyle\times\sin\theta\sin\theta_{N}\sin\phi_{N}$

with $k_{e}:=p_{e^{-}}-p_{e^{+}}$, $q=p_{e^{-}}+p_{e^{+}}=p_{Y^{*}}-p_{Y}$ and
the convention pesschr  for the Levi-Civita symbol: $\epsilon_{0123}=-1$.
Other aspects of the four-body decay are discussed in D.

The final weak decay of a spin-1/2 hyperon to a nucleon and a pion is driven
by the matrix element pdg

$\qquad{\cal M}_{\rm
weak}=G_{F}\,m_{\pi}^{2}\,\bar{u}_{N}(p_{N})\left(A-B\gamma_{5}\right)u_{Y}(p_{Y})\,.$
(15)

It is useful to introduce the asymmetry parameter

$\qquad\qquad\qquad\alpha_{Y}:=\frac{2{\rm
Re}(T^{*}_{s}T_{p})}{|T_{s}|^{2}+|T_{p}|^{2}}$ (16)

with the s-wave amplitude $T_{s}:=A$, the p-wave amplitude $T_{p}:=p_{\rm
f}\,B/(E_{N}+m_{N})$ and mass $m_{N}$, energy $E_{N}$ and momentum $p_{\rm f}$
of the nucleon in the rest frame of the decaying hyperon $Y$; see also
appendix A of Faldt:2013gka  for further details that are useful for practical
calculations.

For stable particles, e.g. nucleons, the electromagnetic form factors are
complex for positive transferred momentum $q^{2}\geq 4m_{N}^{2}$ i.e. for the
reaction $e^{+}e^{-}\to N\bar{N}$ in the time-like region of $q^{2}$. On the
other hand, the form factors are real for the space-like region $q^{2}<0$,
i.e. for the scattering process $e^{-}N\to e^{-}N$. However, for resonances,
e.g. $Y^{*}$, the TFFs are complex for all values of $q^{2}$ Junker:2019vvy .
Therefore the interference terms in (10) can in principle be measured. They
contain information that is complementary to the moduli that are accessible by
the Dalitz decay parametrized by (7). A calculation of such interference terms
is beyond the scope of this work. Yet, the results of Junker:2019vvy  suggest
that such interference terms are relatively small. Note that the spin-parity
combination considered in Junker:2019vvy  refers strictly speaking to
subsection 2.3 below, but semi-quantitatively we expect a similar pattern.
High experimental accuracy will be required to access these interference terms
in the Dalitz decay region. Yet, it might be worth to extract this additional
structure information. It would also be interesting to see how different
models for the structure of hyperon resonances differ in their predictions for
such interference terms. In this context we stress once more that in practice
these interference terms are driven by the fact that resonances are unstable
with respect to the strong interaction. Quasi-stable states (which decay only
because of the electromagnetic or weak interaction) would have tiny imaginary
parts of form factors. For the same reason, models that treat resonances as
stable are not capable to provide predictions for such interference terms. For
measurements of the latter in the production region of hyperon-antihyperon
pairs see Ablikim:2019vaj . Note also that one needs an additional (weak)
decay (more generally a decay that populates more than one partial wave) to
make the interference term visible. If the asymmetry parameter $\alpha_{Y}$
vanished, one would not see an interference effect in (10).

###  2.2 Transition $\frac{1}{2}^{\mp}\to\frac{1}{2}^{\pm}$

The structure of this subsection follows closely the previous one. If the
initial baryon $Y^{*}$ has spin 1/2 and opposite parity to the final baryon
$Y$, the most general decomposition of the transition respecting Lorentz
invariance, current conservation and parity symmetry can be written as (cf.
also Korner:1976hv )

$\langle
p_{Y},\lambda_{Y}|j^{\mu}(0)|p_{Y^{*}},\lambda_{Y^{*}}\rangle=e\bar{u}(p_{Y},\lambda_{Y})\,\Gamma^{\mu}_{-}\,u(p_{Y^{*}},\lambda_{Y^{*}})$

(17)

with

$\Gamma^{\mu}_{-}=\tilde{F}_{2}(q^{2})\,m_{Y^{*}}\,\sigma^{\mu\beta}q_{\beta}\gamma_{5}+i\tilde{F}_{3}(q^{2})\left(q^{2}\gamma^{\mu}-\not{q}q^{\mu}\right)\gamma_{5}\,.$

(18)

Note the appearance of a $\gamma_{5}$ since one of the baryons has unnatural
parity. The two quantities $\tilde{F}_{2}$ and $\tilde{F}_{3}$ constitute
constraint-free transition form factors in the sense of a BTT construction
Bardeen:1969aw ; Tarrach:1975tu . The labeling is motivated in A.

We introduce dimensionless helicity amplitudes:

$\displaystyle\qquad\tilde{F}_{0}(q^{2}):={}$
$\displaystyle(m_{Y^{*}}-m_{Y})^{2}\tilde{F}_{3}(q^{2})$
$\displaystyle{}-(m_{Y^{*}}-m_{Y})\,m_{Y^{*}}\,\tilde{F}_{2}(q^{2})\,,$ (19)
$\displaystyle\qquad\tilde{F}_{+}(q^{2}):={}$ $\displaystyle
q^{2}\tilde{F}_{3}(q^{2})-(m_{Y^{*}}-m_{Y})\,m_{Y^{*}}\,\tilde{F}_{2}(q^{2})\,.$

In a frame where the baryon momenta are aligned, the helicity flip amplitude
$\tilde{F}_{+}$ is related to the combinations
$(\lambda_{Y^{*}},\lambda_{Y})=(-1/2,1/2),(1/2,-1/2)$. The non-flip amplitude
$\tilde{F}_{0}$ relates to $\lambda_{Y^{*}}=\lambda_{Y}=\pm 1/2$.

While the transition form factors $\tilde{F}_{2}$ and $\tilde{F}_{3}$ are free
from kinematical constraints, the helicity amplitudes satisfy

$\qquad\tilde{F}_{+}((m_{Y^{*}}-m_{Y})^{2})=\tilde{F}_{0}((m_{Y^{*}}-m_{Y})^{2})\,.$
(20)

We will use this later for a model independent low-energy parametrization of
the transitions.
The constraint (20) can be easily deduced from the definitions (19).
Physically it follows from the fact that one partial wave is dominant over the
other at the end of the phase space of the Dalitz decay $Y^{*}\to
Y\,e^{+}e^{-}$; see the related discussion in C.

The respective decay widths for the two-body radiative decay $Y^{*}\to
Y\gamma$, the three-body Dalitz decay $Y^{*}\to Y\gamma^{*}\to Y\,e^{+}e^{-}$,
and the four-body decay $Y^{*}\to Y\gamma^{*}\to\pi N\,e^{+}e^{-}$ are given
by

$\quad\Gamma_{2}=\frac{e^{2}|\tilde{F}_{+}(0)|^{2}\left(m_{Y^{*}}+m_{Y}\right)^{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}\right)}{8\pi
m_{Y^{*}}^{3}}\,,$ (21)
$\displaystyle\frac{\text{d}\Gamma_{3}}{\text{d}q^{2}\text{d}(\cos{\theta})}=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{3}\,16m_{Y^{*}}^{3}}\frac{(m_{Y^{*}}+m_{Y})^{2}-q^{2}}{q^{2}}$
$\displaystyle\quad\times\bigg{\\{}\bigg{(}1+\cos^{2}{\theta}+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}{\theta}\bigg{)}\,|\tilde{F}_{+}(q^{2})|^{2}$
(22)
$\displaystyle\qquad+\bigg{(}\sin^{2}{\theta}+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}{\theta}\bigg{)}\frac{q^{2}}{(m_{Y^{*}}-m_{Y})^{2}}|\tilde{F}_{0}(q^{2})|^{2}\bigg{\\}}\,,$

and

$\begin{split}&\frac{\text{d}\Gamma_{4}}{\text{d}q^{2}\text{d}(\cos{\theta})\text{d}\Omega_{N}}\\\
&=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{4}\,32m_{Y^{*}}^{3}}\text{Br}_{Y\to\pi
N}\,\frac{(m_{Y^{*}}+m_{Y})^{2}-q^{2}}{q^{2}}\\\ &\hskip
5.69054pt\times\bigg{\\{}\bigg{(}1+\cos^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}\theta\bigg{)}|\tilde{F}_{+}(q^{2})|^{2}\\\
&\hskip
8.53581pt+\bigg{(}\sin^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}\theta\bigg{)}\frac{q^{2}}{(m_{Y^{*}}-m_{Y})^{2}}|\tilde{F}_{0}(q^{2})|^{2}\\\
&\hskip
8.53581pt+\frac{2\sqrt{q^{2}}\beta_{e}^{2}}{m_{Y^{*}}-m_{Y}}\,\alpha_{Y}\,\text{Im}[\tilde{F}_{0}(q^{2})\tilde{F}_{+}^{*}(q^{2})]\\\
&\qquad\times\sin\theta\cos\theta\sin\theta_{N}\sin\phi_{N}\bigg{\\}}\,.\end{split}$

(23)

Again, it is nice to see how all angular dependence disappears for the cases
where specific three-vectors vanish and therefore do not allow to define
relative angles. For $q^{2}=(2m_{e})^{2}$ one can see directly how all angular
dependence vanishes in the curly brackets of (22) and (23). For
$q^{2}=(m_{Y^{*}}-m_{Y})^{2}$ this is achieved by the kinematical constraint
(20).

###  2.3 Transition $\frac{3}{2}^{\pm}\to\frac{1}{2}^{\pm}$

The previous two subsections were devoted to cases that we will study in more
detail later. For completeness, we also add two more combinations in the
present and the succeeding subsection.

If the initial baryon $Y^{*}$ has spin 3/2 and the same parity as the final
baryon $Y$, the corresponding decomposition is given by

$\langle
p_{Y},\lambda_{Y}|j^{\mu}(0)|p_{Y^{*}},\lambda_{Y^{*}}\rangle=e\bar{u}(p_{Y},\lambda_{Y})\Gamma^{\mu\nu}_{+}u_{\nu}(p_{Y^{*}},\lambda_{Y^{*}})$

(24)
$\begin{split}\qquad\Gamma^{\mu\nu}_{+}={}&\tilde{H}_{1}(q^{2})\,m_{Y^{*}}\left(\gamma^{\mu}q^{\nu}-\not{q}g^{\mu\nu}\right)\gamma_{5}\\\
&+\tilde{H}_{2}(q^{2})\left(q^{\nu}p_{Y^{*}}^{\mu}-(q\cdot
p_{Y^{*}})g^{\mu\nu}\right)\gamma_{5}\\\
&+\tilde{H}_{3}(q^{2})\left(q^{\mu}q^{\nu}-q^{2}g^{\mu\nu}\right)\gamma_{5}\,.\end{split}$
(25)

We see the appearance of a $\gamma_{5}$ since one of the baryons has unnatural
parity. The three quantities $\tilde{H}_{i}$, $i=1,2,3$ constitute constraint-
free transition form factors in the sense of a BTT construction Bardeen:1969aw
; Tarrach:1975tu .

We note that formally $\Gamma^{\mu\nu}_{+}$ is obtained from (2) by
multiplying with $-i\gamma_{5}$ from the left Korner:1976hv  and changing
$H\to\tilde{H}$. One can rephrase this by stating that one obtains (24), (25)
from (1), (2) by replacing $\bar{u}$ by $-i\bar{u}\gamma_{5}$ for the spinor
of the Y hyperon. The interesting aspect is that $-i\bar{u}\gamma_{5}$
satisfies the same equation of motion as $\bar{u}$ but with the mass replaced
by its negative. As a consequence, most of the relations that we present now
for the transition $3/2^{\pm}\to 1/2^{\pm}$ can be obtained from the
corresponding relations for the transition $3/2^{\mp}\to 1/2^{\pm}$ from
subsection 2.1 by just replacing $m_{Y}\to-m_{Y}$ (and changing
$H\to\tilde{H}$).

Again we introduce dimensionless helicity amplitudes:

$\displaystyle\tilde{H}_{-}(q^{2})$ $\displaystyle:=$
$\displaystyle-\left(m_{Y^{*}}+m_{Y}\right)m_{Y^{*}}\,\tilde{H}_{1}(q^{2})$
$\displaystyle{}+\frac{1}{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}+q^{2}\right)\tilde{H}_{2}(q^{2})+q^{2}\tilde{H}_{3}(q^{2})\,,$
$\displaystyle\tilde{H}_{0}(q^{2})$ $\displaystyle:=$
$\displaystyle-\left(m_{Y^{*}}+m_{Y}\right)m_{Y^{*}}\,\tilde{H}_{1}(q^{2})$
$\displaystyle{}+\left(m_{Y^{*}}+m_{Y}\right)m_{Y^{*}}\,\tilde{H}_{2}(q^{2})$
$\displaystyle{}+\frac{m_{Y^{*}}+m_{Y}}{2m_{Y^{*}}}\left(m_{Y^{*}}^{2}-m_{Y}^{2}+q^{2}\right)\,\tilde{H}_{3}(q^{2})\,,$
$\displaystyle\tilde{H}_{+}(q^{2})$ $\displaystyle:=$
$\displaystyle-\left(q^{2}-m_{Y^{*}}m_{Y}-m_{Y}^{2}\right)\tilde{H}_{1}(q^{2})$
$\displaystyle{}+\frac{1}{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}+q^{2}\right)\tilde{H}_{2}(q^{2})+q^{2}\tilde{H}_{3}(q^{2})\,.$

Note that the conventions for the helicity amplitudes are in line with
Carlson:1985mm , but opposite to Junker:2019vvy .

The kinematical constraints obtain the form

$\begin{split}\qquad\tilde{H}_{+}((m_{Y^{*}}+m_{Y})^{2})&=\tilde{H}_{0}((m_{Y^{*}}+m_{Y})^{2})\\\
&=\tilde{H}_{-}((m_{Y^{*}}+m_{Y})^{2})\end{split}$ (27)

and

$\begin{split}\qquad&\frac{2(m_{Y^{*}}-m_{Y})}{m_{Y^{*}}+m_{Y}}\tilde{H}_{0}((m_{Y^{*}}-m_{Y})^{2})\\\
&=\tilde{H}_{+}((m_{Y^{*}}-m_{Y})^{2})+\tilde{H}_{-}((m_{Y^{*}}-m_{Y})^{2})\,.\end{split}$
(28)

The width for the two-body radiative decay $Y^{*}\to Y\gamma$ is given by

$\Gamma_{2}=e^{2}\left[3|\tilde{H}_{-}(0)|^{2}+|\tilde{H}_{+}(0)|^{2}\right]\\\
\times\frac{\left(m_{Y^{*}}-m_{Y}\right)^{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}\right)}{96\pi
m_{Y^{*}}^{3}}\,.$ (29)

The differential decay width for the Dalitz decay $Y^{*}\to Y\gamma^{*}\to
Y\,e^{+}e^{-}$ can be expressed as

$\begin{split}&\frac{\text{d}\Gamma_{3}}{\text{d}q^{2}\text{d}(\cos{\theta})}=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{3}\,192m_{Y^{*}}^{3}}\frac{(m_{Y^{*}}-m_{Y})^{2}-q^{2}}{q^{2}}\\\
&\
\times\bigg{\\{}\bigg{(}1+\cos^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}\theta\bigg{)}\big{[}3|\tilde{H}_{-}(q^{2})|^{2}+|\tilde{H}_{+}(q^{2})|^{2}\big{]}\\\
&\quad+\bigg{(}\sin^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}\theta\bigg{)}\frac{4q^{2}}{(m_{Y^{*}}+m_{Y})^{2}}|\tilde{H}_{0}(q^{2})|^{2}\bigg{\\}}\,.\end{split}$

(30)

The four-body decay $Y^{*}\to Y\gamma^{*}\to\pi N\,e^{+}e^{-}$ has the
following differential decay width:

$\begin{split}&\frac{\text{d}\Gamma_{4}}{\text{d}q^{2}\text{d}(\cos{\theta})\text{d}\Omega_{N}}\\\
&=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{4}\,384m_{Y^{*}}^{3}}\,\text{Br}_{Y\to\pi
N}\,\frac{(m_{Y^{*}}-m_{Y})^{2}-q^{2}}{q^{2}}\\\ &\hskip
2.84526pt\times\bigg{\\{}\bigg{(}1+\cos^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}\theta\bigg{)}\big{[}|\tilde{H}_{+}(q^{2})|^{2}+3|\tilde{H}_{-}(q^{2})|^{2}\big{]}\\\
&\hskip
8.53581pt+\bigg{(}\sin^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}\theta\bigg{)}\frac{4q^{2}}{\left(m_{Y^{*}}+m_{Y}\right)^{2}}|\tilde{H}_{0}(q^{2})|^{2}\\\
&\hskip
8.53581pt+\frac{4\sqrt{q^{2}}\beta_{e}^{2}}{m_{Y^{*}}+m_{Y}}\,\alpha_{Y}\,\text{Im}[\tilde{H}_{0}(q^{2})\tilde{H}_{+}^{*}(q^{2})]\\\
&\qquad\times\sin\theta\cos\theta\sin\theta_{N}\sin\phi_{N}\bigg{\\}}\,.\end{split}$

(31)

In the last equation we have a notable exception to the “rule” that the
formulae of the present subsection are obtained from the corresponding ones in
subsection 2.1 by the replacements $m_{Y}\to-m_{Y}$ and $H\to\tilde{H}$. The
interference term has the opposite sign to the one in (10).

###  2.4 Transition $\frac{1}{2}^{\pm}\to\frac{1}{2}^{\pm}$

Finally we study the case of a transition between two hyperons with the same
spin and parity assignments. One such case, the transition from $\Sigma^{0}$
to $\Lambda$ has been studied in detail in Granados:2017cib ; Husek:2019wmt .

The decomposition is given by

$\langle
p_{Y},\lambda_{Y}|j^{\mu}(0)|p_{Y^{*}},\lambda_{Y^{*}}\rangle=e\bar{u}(p_{Y},\lambda_{Y})\,\Gamma^{\mu}_{+}\,u(p_{Y^{*}},\lambda_{Y^{*}})$

(32)

with

$\Gamma^{\mu}_{+}=iF_{2}(q^{2})\,m_{Y^{*}}\,\sigma^{\mu\beta}q_{\beta}+F_{3}(q^{2})\left(q^{2}\gamma^{\mu}-\not{q}q^{\mu}\right)\,.$
(33)

We introduce the dimensionless helicity amplitudes by

$\begin{split}\qquad F_{0}(q^{2}):=&(m_{Y^{*}}+m_{Y})^{2}F_{3}(q^{2})\\\
&-(m_{Y^{*}}+m_{Y})\,m_{Y^{*}}\,F_{2}(q^{2})\,,\\\
F_{+}(q^{2}):=&q^{2}F_{3}(q^{2})-(m_{Y^{*}}+m_{Y})\,m_{Y^{*}}\,F_{2}(q^{2})\,.\end{split}$
(34)

These helicity amplitudes satisfy the kinematical constraint

$\qquad F_{+}((m_{Y^{*}}+m_{Y})^{2})=F_{0}((m_{Y^{*}}+m_{Y})^{2})\,.$ (35)

The respective decay widths for the two-body radiative decay $Y^{*}\to
Y\gamma$, the three-body Dalitz decay $Y^{*}\to Y\gamma^{*}\to Y\,e^{+}e^{-}$,
and the four-body decay $Y^{*}\to Y\gamma^{*}\to\pi N\,e^{+}e^{-}$ are given
by

$\quad\Gamma_{2}=\frac{e^{2}|F_{+}(0)|^{2}\left(m_{Y^{*}}-m_{Y}\right)^{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}\right)}{8\pi
m_{Y^{*}}^{3}}\,,$ (36)

$\begin{split}&\frac{\text{d}\Gamma_{3}}{\text{d}q^{2}\text{d}(\cos{\theta})}=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{3}\,16m_{Y^{*}}^{3}}\frac{(m_{Y^{*}}-m_{Y})^{2}-q^{2}}{q^{2}}\\\
&\hskip
8.53581pt\times\bigg{\\{}\bigg{(}1+\cos^{2}{\theta}+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}{\theta}\bigg{)}\,|F_{+}(q^{2})|^{2}\\\
&\hskip
11.38109pt+\bigg{(}\sin^{2}{\theta}+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}{\theta}\bigg{)}\frac{q^{2}}{(m_{Y^{*}}+m_{Y})^{2}}|F_{0}(q^{2})|^{2}\bigg{\\}}\,,\end{split}$

(37)

and

$\begin{split}&\frac{\text{d}\Gamma_{4}}{\text{d}q^{2}\text{d}(\cos{\theta})\text{d}\Omega_{N}}\\\
&=\frac{e^{4}p_{z}\sqrt{q^{2}}\beta_{e}}{(2\pi)^{4}\,32m_{Y^{*}}^{3}}\text{Br}_{Y\to\pi
N}\,\frac{(m_{Y^{*}}-m_{Y})^{2}-q^{2}}{q^{2}}\\\
&\qquad\times\bigg{\\{}\bigg{(}1+\cos^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}\theta\bigg{)}|F_{+}(q^{2})|^{2}\\\
&\qquad\quad+\bigg{(}\sin^{2}\theta+\frac{4m_{e}^{2}}{q^{2}}\cos^{2}\theta\bigg{)}\frac{q^{2}}{(m_{Y^{*}}+m_{Y})^{2}}|F_{0}(q^{2})|^{2}\\\
&\qquad\quad-\frac{2\sqrt{q^{2}}\beta_{e}^{2}}{m_{Y^{*}}+m_{Y}}\,\alpha_{Y}\,\text{Im}[F_{0}(q^{2})F_{+}^{*}(q^{2})]\\\
&\qquad\qquad\times\sin\theta\cos\theta\sin\theta_{N}\sin\phi_{N}\bigg{\\}}\,.\end{split}$

(38)

##  3 QED type case and form factor parametrization

In this section we will give a quantitative meaning to the phrases
“structureless” and “extended structure”. We will call the former “QED type”.
For the latter we introduce a radius.

A real photon does not resolve the intrinsic structure of the composite
hyperons that take part in the electromagnetic transitions. It is the photon
virtuality that relates to the resolution; see e.g. the discussion in
Alarcon:2017asr ; Miller:2007uy . In this spirit, we define a QED type case
Junker:2019vvy  by modifying the Dalitz decay formula (7) such that it fits to
the radiative decay formula (6). To this end we replace e.g. in (7)

$\displaystyle\big{[}3|H_{-}(q^{2})|^{2}+|H_{+}(q^{2})|^{2}\big{]}$
$\displaystyle\to\big{[}3|H_{-}(0)|^{2}+|H_{+}(0)|^{2}\big{]}\,,$
$\displaystyle\frac{4q^{2}}{(m_{Y^{*}}-m_{Y})^{2}}\,$
$\displaystyle|H_{0}(q^{2})|^{2}\to 0\,,$ (39)

i.e. we replace all virtualities $q^{2}$ by $0$ for these building blocks. We
can do this for all spin-parity combinations. In this way we obtain a Dalitz
decay formula for “structureless” fermions. If we divide out the corresponding
radiative decay width we get

$\displaystyle\frac{1}{\Gamma_{2}}\frac{\text{d}\Gamma_{\text{QED
type}}}{\text{d}q^{2}\text{d}(\cos{\theta})}=\frac{(m_{Y^{*}}\pm
m_{Y})^{2}-q^{2}}{q^{2}\left(m_{Y^{*}}\pm
m_{Y}\right)^{2}\left(m_{Y^{*}}^{2}-m_{Y}^{2}\right)}$
$\displaystyle\phantom{mmm}\times\frac{e^{2}p_{z}\sqrt{q^{2}}\beta_{e}}{(4\pi)^{2}}\bigg{(}1+\cos^{2}{\theta}+\frac{4m_{e}^{2}}{q^{2}}\sin^{2}{\theta}\bigg{)}\,.\phantom{m}$
(40)

Here the upper (lower) sign refers to the cases where the parities of initial
and final fermion differ (are the same).

Obviously, this formula (40) is independent of any transition form factors or
helicity amplitudes. If an experiment cannot resolve the difference between
nature and (40), it cannot reveal if hyperons have an intrinsic structure or
not. This does not mean that the determination of radiative decay widths does
not contain any interesting information Kaxiras:1985zv , but no visible
deviation from (40) means that the such measured Dalitz decays do not contain
more information than radiative decays, i.e. decays with real photons.

The QED type case (40) defines our baseline to which we want to compare the
case where hyperons do have an intrinsic structure. From now on we focus on
decays of the two negative-parity resonances $\Lambda(1520)$ and
$\Lambda(1405)$. The mass difference between the considered resonances and the
ground-state hyperons is not very large. Consequently the energy range
$\sqrt{q^{2}}$ (dilepton invariant mass) that is explored by the Dalitz decays
is rather limited, it ranges from two times the electron mass up to the mass
difference $m_{Y^{*}}-m_{Y}$. For a rough estimate of the importance of the
$q^{2}$ dependence, we approximate any helicity amplitude $G(q^{2})$ in the
following way:

$\qquad\qquad G(q^{2})\approx G(0)\left(1+\frac{1}{6}q^{2}\langle
r^{2}\rangle\right)$ (41)

where we have introduced the radius via

$\qquad\qquad\langle
r^{2}\rangle:=\frac{6}{G(0)}\left.\frac{\text{d}G(q^{2})}{\text{d}q^{2}}\right|_{q^{2}=0}\,.$
(42)

In view of the typical size of hadrons of about $1\,$fm, we assume

$\qquad\qquad\quad 0\leq\langle r^{2}\rangle\leq 1\,\text{fm}^{2}\approx
25\,\text{GeV}^{-2}\,.$ (43)

In practice, this will provide us with a rough upper limit of the deviation of
a differential decay rate from the corresponding QED type case.

We would like to stress that (43) is not entirely accurate. As already pointed
out, the transition form factors and helicity amplitudes for resonances are
not real-valued in the Dalitz decay region.222They are not even real-valued in
the space-like region of electron-hadron scattering. Strictly speaking, this
carries over to the squared radii. The physical reason for being complex is
the inelastic two-step process of strong decay, $Y^{*}\to\pi Y^{\prime}$, and
rescattering, $\pi Y^{\prime}\to\gamma^{*}Y$. Here $Y^{\prime}$ denotes
another hyperon. However, the imaginary part of a transition form factor or
helicity amplitude will not be very large if the total decay width of the
resonance is sufficiently small. This is the case for the resonances that we
consider. For the electromagnetic transitions $\Sigma(1385)\to\Lambda$ the
imaginary parts of squared radii have been explicitly calculated in
Junker:2019vvy . There, the real parts were in the ballpark of (43) and the
imaginary parts were much smaller. Therefore we assume that the real part of
any $\langle r^{2}\rangle$ is dominant over the imaginary part.

What remains to be shown to justify (43) as a reasonable approximation? We
still have to argue why the real part should have positive sign. Indeed, there
is a well-known case where the squared radius of a hadron seems to be
negative: the electric charge radius of the neutron pdg . Yet, this is a
somewhat misleading case. For the electric form factor of the neutron, the
radius cannot be defined via (42), since the charge of the neutron, $G(0)$,
vanishes. As a remedy one drops $G(0)$ in the definition of the charge radius
of the neutron. However, one can take a somewhat different route to the same
physical information. Instead of electric charge, one can look at the isospin.
The isoscalar and isovector form factors of the nucleon have all non-vanishing
“charges”. One can define an isoscalar and an isovector radius based on (42),
see, e.g., Leupold:2017ngs ; Kubis:2000aa . Those two radii are positive and
in the order of 1 fm, in full agreement with (43). Also the results of
Junker:2019vvy  support this estimate. Finally we note that even the radius of
the pion transition form factor Hoferichter:2014vra  fits into this picture.
For all these cases, the radii are less than 1 fm. Thus we believe that (43)
defines a reasonable conservative range and we expect that for all hadrons,
reality is closer to 1 fm than to 0.

##  4 Concrete results for negative-parity resonances

The parametrization (41) seems to indicate that we have two free parameters
per helicity amplitude. However, this is not the case because we have to obey
the kinematical constraints (4) or (20), respectively. This will help us to
express all relevant quantities solely in terms of radii.

Some clarification is in order here. Of course, the ansatz (41) is an
approximation that holds only for sufficiently low values of $q^{2}$. We want
to use this approximation for the whole Dalitz decay region for cases where
the mass difference between the decaying and the final hyperon is sufficiently
small. The kinematical constraints (4) or (20) lie in this low-energy region
because they lie at the end of the kinematically allowed Dalitz decay region.
Therefore we can use the kinematical constraints to reduce the number of free
parameters and focus on the impact of the radii on the results. Note that this
line of reasoning would not work for positive-parity resonances. In
particular, the constraint (35) does not lie in the low-energy region where
the ansatz (41) would make sense. Albeit not of completely general use, the
kinematical constraints are absolutely suited for the cases that we consider
further, namely for the hyperon resonances $\Lambda(1520)$ and $\Lambda(1405)$
with spin-parity assignment $3/2^{-}$ and $1/2^{-}$, respectively pdg .

###  4.1 Transition radii and the decay $\Lambda(1405)\to\Lambda\,e^{+}e^{-}$

Using (41) for the helicity amplitudes $\tilde{F}_{+}$ and $\tilde{F}_{0}$
allows us to rewrite the kinematical constraint (20) into

$\qquad\tilde{F}_{0}(0)\approx\tilde{F}_{+}(0)\,\frac{1+(m_{Y^{*}}-m_{Y})^{2}\langle
r^{2}\rangle_{+}/6}{1+(m_{Y^{*}}-m_{Y})^{2}\langle r^{2}\rangle_{0}/6}\,.$
(44)

The remaining dependence on $\tilde{F}_{+}(0)$ cancels out in the ratio of the
Dalitz decay width (22) and the radiative decay width (21). Thus the such
normalized decay width
$\text{d}\Gamma_{3}/(\text{d}q^{2}\text{d}(\cos\theta))/\Gamma_{2}$ depends
only on the (squared) transition radii $\langle r^{2}\rangle_{+}$ and $\langle
r^{2}\rangle_{0}$.

Figure 1: Comparison between radius structure and QED-type approximation for
the $\frac{1}{2}^{-}\to\frac{1}{2}^{+}$ transition. Top panel: both radii at
maximal value (43); middle panels: one radius at max, one zero; bottom panel:
both radii put to zero.

In the following, we will focus on the once-integrated normalized decay widths

$\qquad\qquad\qquad\frac{1}{\Gamma_{2}}\frac{\text{d}\Gamma_{3}}{\text{d}q^{2}}\quad\mbox{and}\quad\frac{1}{\Gamma_{2}}\frac{\text{d}\Gamma_{3}}{\text{d}(\cos\theta)}\,,$
(45)

to study the dependence on the transition radii and compare to the QED type
case.

The $q^{2}$ dependence is depicted in fig. 1. We observe that the radius
related to the helicity flip amplitude $\tilde{F}_{+}$ causes a significant
deviation from the QED type case. The effect from the non-flip amplitude is
minor. This is related to the additional $q^{2}$ factor that multiplies
$\tilde{F}_{0}$ in (22) and to the larger weight of $1+\cos^{2}\theta$
relative to $\sin^{2}\theta=1-\cos^{2}\theta$ when integrating over
$\cos\theta$. Note that the QED type case is not defined by putting all radii
to zero. Therefore, there is a small difference between the two curves in the
bottom panel of fig. 1. We could have defined the QED type case in a different
way. But instead we use the occasion to point out that there is in principle
an ambiguity in defining the structureless QED case. In practice, however,
this ambiguity is small.

Depending on the composition of the $\Lambda(1405)$ as a dominantly three-
quark or dominantly hadron molecule state, its helicity flip transition radius
can be expected to be somewhat different. Still we think that 1 fm is a
reasonable estimate in any case. We regard the plots of fig. 1 as interesting
for experimentalists who aim to reveal the intrinsic structure of the
$\Lambda(1405)$ using Dalitz decays. For this endeavor one needs to achieve an
experimental accuracy that can discriminate at least between the two lines in
the top panel of fig. 1. To study differences between different scenarios for
the structure of the $\Lambda(1405)$ requires an even better accuracy.

Next we turn to the angular distribution. If one integrates (22) or (40) over
$q^{2}$ (in the range $(2m_{e})^{2}\leq q^{2}\leq(m_{Y^{*}}-m_{Y})^{2}$), one
obtains a constant term and one linear in $\cos^{2}\theta$. Thus the general
structure is

$\qquad\qquad\qquad\frac{1}{\Gamma_{2}}\frac{\text{d}\Gamma_{3}}{\text{d}(\cos\theta)}=A\,(\cos^{2}\theta+C)\,.$
(46)

If one integrates finally over the angle, one obtains the total decay width
for the process $Y^{*}\to Y\,e^{+}e^{-}$, normalized to the width for the
process $Y^{*}\to Y\gamma$, i.e.

$\qquad\qquad\qquad\qquad\frac{\Gamma_{3}}{\Gamma_{2}}=\frac{2}{3}\,A+2AC\,.$
(47)

In table 1 we provide $A$, $C$ and the width ratio of (47) for the QED type
case and different radius combinations.

$\big{(}\langle r^{2}\rangle_{0};\langle r^{2}\rangle_{+}\big{)}$ GeV-2 | $A$ | $C$ | $\frac{2}{3}A+2AC$
---|---|---|---
(0 ; 25) | $0.00252$ | 1.43 | 0.00889
(25 ; 25) | $0.00262$ | 1.34 | 0.00876
(0 ; 0) | $0.00253$ | 1.30 | 0.00830
(25 ; 0) | $0.00259$ | 1.26 | 0.00823
QED type | $0.00273$ | 1.14 | 0.00804
Table 1: Parameters $A$ and $C$, and decay width ratio for the
$J^{P}=\frac{1}{2}^{-}$ initial state in QED approximation and with a radius
structure.

Concerning the overall scaling factor $A$, we observe that different radii
lead to modifications on a 10% level. However, for this parameter the
ambiguity how to define a structureless case is in the same order.

For the parameter $C$, larger helicity flip transition radii lead to larger
values. In total, we observe variations up to about 20%. Interestingly, a
radius $\langle r^{2}\rangle_{0}$ in the non-flip amplitude counteracts the
effect of a radius $\langle r^{2}\rangle_{+}$ in the helicity flip amplitude.

We predict that the Dalitz decay width, normalized to the radiative decay
width, is about 0.8 to 0.9%. This fits to the rule of thumb that an additional
QED vertex provides a suppression factor of about $\alpha=e^{2}/(4\pi)\approx
10^{-2}$.

###  4.2 Transition radii and the decay $\Lambda(1520)\to\Lambda\,e^{+}e^{-}$

We perform the same procedure for the decays of the spin-$3/2^{-}$ hyperon
resonance $Y^{*}=\Lambda(1520)$.
For the numerical results we focus on $Y=\Lambda$, the lightest strange
baryon. The approximation (41) is utilized for the helicity amplitudes
$H_{+}$, $H_{0}$, $H_{-}$. It is supposed to be valid in the whole Dalitz
decay region, $(2m_{e})^{2}\leq q^{2}\leq(m_{Y^{*}}-m_{Y})^{2}$. The
kinematical constraints (4) are used to eliminate the dependence on
$H_{...}(0)$ for the ratio of Dalitz decay width and radiative width. What
remains is the dependence on the three squared radii, $\langle
r^{2}\rangle_{+}$, $\langle r^{2}\rangle_{0}$, $\langle r^{2}\rangle_{-}$.
Figs. 2 and 3 illustrate this dependence as a function of $q^{2}$ for the
normalized singly-differential Dalitz decay width.

Figure 2: Same as fig. 1 but for the $\frac{3}{2}^{-}\to\frac{1}{2}^{+}$
transition. Top panel: all three radii at maximal value (43); other panels:
two radii at max, third at zero.

Figure 3: Same as fig. 2 but with less many radii put at maximum value (43).
Bottom panel: all radii put to zero; other panels: one radius at max, rest at
zero.

What the plots show, first of all, is the fact that $\langle r^{2}\rangle_{-}$
matters most. Whenever it is large, there is a significant deviation from the
QED type case. Whenever it is small, the results with radii are close to the
structureless QED case. Essentially this can be traced back to the explicit
factor of 3 in (7) that boosts the importance of $H_{-}$ relative to the other
two helicity amplitudes. In addition, the importance of $|H_{0}|^{2}$ is
demoted by the explicit $q^{2}$ factor in front of it and the different weight
in the angular average, an effect already observed for the spin-1/2 case of
the previous subsection. Again, we believe that the plots of figs. 2 and 3 are
important for experimentalists to judge how accurate their results must be to
disentangle extended from structureless objects.

We turn to the angular dependence. Formulae (46) and (47) can again be used
and the dependence of the parameters $A$ and $C$ on the radii is collected in
table 2.

$\big{(}\langle r^{2}\rangle_{0};\langle r^{2}\rangle_{+};\langle r^{2}\rangle_{-}\big{)}$ | $A$ | $C$ | $\frac{2}{3}A+2AC$
---|---|---|---
GeV-2
(0 ; 25 ; 25) | $0.00267$ | 1.55 | 0.0101
(0 ; 0 ; 25) | $0.00270$ | 1.42 | 0.00945
(25 ; 25 ; 25) | $0.00290$ | 1.35 | 0.00976
(0 ; 25 ; 0) | $0.00272$ | 1.31 | 0.00894
(25 ; 0 ; 25) | $0.00285$ | 1.29 | 0.00924
(0 ; 0 ; 0) | $0.00273$ | 1.28 | 0.00881
(25 ; 25 ; 0) | $0.00282$ | 1.23 | 0.00881
(25 ; 0 ; 0) | $0.00281$ | 1.22 | 0.00870
QED type | $0.00292$ | 1.13 | 0.00855
Table 2: Same as table 1 but for the $J^{P}=\frac{3}{2}^{-}$ initial state.

Similar to the spin-1/2 case of the previous subsection we observe a variation
in $A$ of about 10%, but no clear tendency of the impact of finite radii.

Concerning the parameter $C$ we observe variations of up to 30%. The value of
$C$ is increased for larger values of $\langle r^{2}\rangle_{-}$ or $\langle
r^{2}\rangle_{+}$ and decreased for a larger value of $\langle
r^{2}\rangle_{0}$. The impact of $\langle r^{2}\rangle_{+}$ is less pronounced
than the impact of $\langle r^{2}\rangle_{-}$, which we explain again by the
explicit factor of 3 in (7).

For the integrated Dalitz decay width, normalized to the photon decay width,
we predict 0.9 to 1%, slightly larger than our prediction for the
$\Lambda(1405)$ of the previous subsection. We attribute this to the larger
phase space available for the heavier $\Lambda(1520)$.

##  5 Further discussion, summary and outlook

We have provided a comprehensive framework that first considers the most
general electromagnetic transition form factors, free of kinematical
constraints. Those form factors are perfectly suited for a dispersive
representation Junker:2019vvy , a perspective that we might explore in the
future. We have related these form factors to helicity amplitudes and shown
how they make their appearance in the decay rates of the processes $Y^{*}\to
Y\gamma$ and $Y^{*}\to Y\,e^{-}e^{+}$. We covered the cases where the initial
hyperon $Y^{*}$ can have spin 1/2 or 3/2. The final hyperon $Y$ was assumed to
have spin 1/2. All parity combinations have been covered.

Of course, these relations are in principle not new and easy to deduce, e.g.,
from Korner:1976hv . Yet, we have decided to spell out all definitions and
conventions in detail to keep the work self-contained and to facilitate
comparisons between different works. We have further extended the framework to
include also a final weak decay of the ground-state hyperon $Y$, very much in
spirit of Perotti:2018wxm , though with different kinematics. We have stressed
that the emerging interference term between a helicity flip and the non-flip
amplitude exists also in the Dalitz decay region.

In a second step, we have focused on the specific transitions of
$\Lambda(1405)$ and $\Lambda(1520)$ to the ground-state $\Lambda$. In view of
the not too large kinematical range that is covered by the Dalitz decays we
have parametrized the helicity amplitudes in terms of transition radii. Such
radii are commonly used for the study of form factors at low energies, e.g.
for the electric or magnetic form factors of proton and neutron, i.e. for
their helicity non-flip or flip amplitudes. As a function of such transition
radii, we have investigated how much accuracy is needed for an experiment to
discriminate between hypothetical structureless hyperons and a more realistic
case. Indeed, we regard the scenario where all radii are non-vanishing as the
most realistic one.

Concerning the dependence on the invariant mass of the electron-positron pair,
our setup shows a significant deviation from the structureless QED type case,
in particular in the $q^{2}$ range above $0.01\,$GeV2 for the transition from
$\Lambda(1405)$ to $\Lambda$ and above $0.02\,$GeV2 for the transition from
$\Lambda(1520)$ to $\Lambda$. We see also significant differences for the
parameter $C$ that parametrizes the deviation of the angular dependence from a
pure $\cos^{2}$ distribution.

Of course, it is completely straightforward to extend this radius analysis
also to the electromagnetic transitions $\Lambda(1405)\to\Sigma^{0}$,
$\Lambda(1405)\to\Sigma^{0}(1385)$, $\Lambda(1520)\to\Sigma^{0}$, and with a
little generalization also to $\Lambda(1520)\to\Sigma^{0}(1385)$. But the
smaller the mass difference between initial and final hyperon, the less
additional information is contained in the Dalitz decays relative to the real-
photon case. Therefore we have focused on the cases where the final hyperon is
as light as possible.

More generally, the framework provided here can also be applied to double- or
triple-strange hyperon resonances and to baryon resonances where strange
quarks are replaced by heavier quarks.

###### Acknowledgements.

SL thanks T. Galatyuk, A. Kupść, B. Ramstein, P. Salabura and K. Schönning for
many valuable discussions that inspired this investigation. This work has been
supported by the Swedish Research Council (Vetenskapsrådet) (grant number
2019-04303). Partial support has been provided by the Polish National Science
Centre through the grant 2019/35/O/ST2/02907.

##  Appendix A Toy-model Lagrangians

The following Lagrangians are hermitian and invariant with respect to charge
conjugation symmetry. They are also parity symmetric, but it is the first two
properties that make the coupling constants real-valued. Note that we do not
make use of these Lagrangians in our actual calculations. In this sense, these
are toy-model Lagrangians. But we use them to motivate the appearance or
absence of $i$’s for our definitions of the transition form factors $H_{j}$,
$\tilde{H}_{j}$, $F_{j}$, and $\tilde{F}_{j}$.

A real-valued contribution to $H_{1}$ as introduced in (2) comes from a tree-
level calculation based on

$\displaystyle\qquad\qquad{\cal
L}_{H1}=a_{1}\left(\bar{\Psi}\gamma^{\mu}F_{\mu\nu}\Psi^{\nu}+\bar{\Psi}^{\nu}\gamma^{\mu}F_{\mu\nu}\Psi\right)\,,$
(48)

where $a_{1}\in\mathds{R}$. In the same way one finds a contribution from

$\displaystyle\qquad\qquad{\cal
L}_{H2}=ia_{2}\left(\bar{\Psi}F_{\mu\nu}\partial^{\mu}\Psi^{\nu}-\partial^{\mu}\bar{\Psi}^{\nu}F_{\mu\nu}\Psi\right)$
(49)

for $H_{2}$ and from

$\displaystyle\qquad\qquad{\cal
L}_{H3}=ia_{3}\left(\bar{\Psi}\partial^{\mu}F_{\mu\nu}\Psi^{\nu}-\bar{\Psi}^{\nu}\partial^{\mu}F_{\mu\nu}\Psi\right)$
(50)

for $H_{3}$. Also here the coupling constants must be real,
$a_{2},a_{3}\in\mathds{R}$.

Similarly, the Lagrangians

$\qquad\tilde{\cal
L}_{H1}=i\tilde{a}_{1}\left(\bar{\Psi}\gamma^{\mu}\gamma_{5}F_{\mu\nu}\Psi^{\nu}-\bar{\Psi}^{\nu}\gamma^{\mu}\gamma_{5}F_{\mu\nu}\Psi\right)\,,$
(51) $\qquad\tilde{\cal
L}_{H2}=\tilde{a}_{2}\left(\bar{\Psi}\gamma_{5}F_{\mu\nu}\partial^{\mu}\Psi^{\nu}-\partial^{\mu}\bar{\Psi}^{\nu}\gamma_{5}F_{\mu\nu}\Psi\right)\,,$
(52) $\qquad\tilde{\cal
L}_{H3}=\tilde{a}_{3}\left(\bar{\Psi}\gamma_{5}\partial^{\mu}F_{\mu\nu}\Psi^{\nu}-\bar{\Psi}^{\nu}\gamma_{5}\partial^{\mu}F_{\mu\nu}\Psi\right)$
(53)

contribute to $\tilde{H}_{1}$, $\tilde{H}_{2}$, $\tilde{H}_{3}$, respectively.
We recall that the latter transition form factors have been introduced in
(25). The Lagrangians are only hermitian and invariant with respect to charge
conjugation, if the coupling constants are real, i.e.
$\tilde{a}_{1},\tilde{a}_{2},\tilde{a}_{3}\in\mathds{R}$.

Turning to spin-1/2, we have the Lagrangians

$\qquad{\cal
L}_{F2}=b_{2}\left(\bar{\Psi}_{Y}\sigma_{\mu\nu}F^{\mu\nu}\Psi_{Y^{*}}+\bar{\Psi}_{Y^{*}}\sigma_{\mu\nu}F^{\mu\nu}\Psi_{Y}\right)$
(54)

and

$\quad{\cal
L}_{F3}=b_{3}\left(\bar{\Psi}_{Y}\gamma_{\mu}\partial_{\nu}F^{\mu\nu}\Psi_{Y^{*}}+\bar{\Psi}_{Y^{*}}\gamma_{\mu}\partial_{\nu}F^{\mu\nu}\Psi_{Y}\right)$
(55)

contributing with real-valued results to $F_{2}$ and $F_{3}$, respectively.
Finally

$\tilde{\cal
L}_{F2}=i\tilde{b}_{2}\left(\bar{\Psi}_{Y}\sigma_{\mu\nu}\gamma_{5}F^{\mu\nu}\Psi_{Y^{*}}+\bar{\Psi}_{Y^{*}}\sigma_{\mu\nu}\gamma_{5}F^{\mu\nu}\Psi_{Y}\right)$

(56)

and

$\tilde{\cal
L}_{F3}=i\tilde{b}_{3}\left(\bar{\Psi}_{Y}\gamma_{\mu}\gamma_{5}\partial_{\nu}F^{\mu\nu}\Psi_{Y^{*}}-\bar{\Psi}_{Y^{*}}\gamma_{\mu}\gamma_{5}\partial_{\nu}F^{\mu\nu}\Psi_{Y}\right)$

(57)

contribute to $\tilde{F}_{2}$ and $\tilde{F}_{3}$, respectively. Hermiticity
and charge conjugation symmetry demand
$b_{2},b_{3},\tilde{b}_{2},\tilde{b}_{3}\in\mathds{R}$.

An explanation is in order concerning the labels 2 and 3 for the transition
form factors $F_{i}$ and $\tilde{F}_{i}$. In a (here formal) low-energy
counting scheme where the hyperons’ three-momenta and the electromagnetic
potentials and their four-momenta are counted as soft and of same size, the
Lagrangians (54) and (56) are of second order, while (55) and (57) are of
third order.

In addition, the construction in (54) resembles the Pauli term that describes
the anomalous magnetic moment of a spin-1/2 state, here generalized to
transitions. For the nucleon, the Pauli form factor is often called $F_{2}$;
see e.g. Kubis:2000aa  and references therein.

For the transitions involving spin-3/2 initial states this formal power
counting leads to a mismatch between the two parity sectors, cf. e.g.
Holmberg:2019ltw ; Junker:2019vvy . Therefore we have just enumerated the
transition form factors $H_{i}$ and $\tilde{H}_{i}$ from 1 to 3.

##  Appendix B High-energy behavior

In this appendix we use the methods of Carlson:1985mm  to present the high-
energy behavior of all transition form factors. This procedure uses the
implicit assumption that the considered hyperons have a minimal quark content
of three quarks. For the validity of the framework, it does not matter how
large the overlap of the physical state with a three-quark configuration is
Aznauryan:2011qj , as long as it is non-zero.

The quark counting rules are derived for large space-like photon virtuality.
Therefore it is convenient to introduce the large positive quantity
$Q^{2}:=-q^{2}$ and $Q:=\sqrt{Q^{2}}$. One obtains

$\begin{split}H_{+}(-Q^{2})\sim\frac{1}{Q^{4}}\,,&\;H_{0}(-Q^{2})\sim\frac{1}{Q^{6}}\,,\;H_{-}(-Q^{2})\sim\frac{1}{Q^{6}}\,;\\\\[10.00002pt]
\tilde{H}_{+}(-Q^{2})\sim\frac{1}{Q^{4}}\,,&\;\tilde{H}_{0}(-Q^{2})\sim\frac{1}{Q^{6}}\,,\;\tilde{H}_{-}(-Q^{2})\sim\frac{1}{Q^{6}}\,;\\\\[10.00002pt]
F_{+}(-Q^{2})\sim\frac{1}{Q^{4}}\,,&\;F_{0}(-Q^{2})\sim\frac{1}{Q^{6}}\,;\\\\[10.00002pt]
\tilde{F}_{+}(-Q^{2})\sim\frac{1}{Q^{4}}\,,&\;\tilde{F}_{0}(-Q^{2})\sim\frac{1}{Q^{6}}\,,\end{split}$
(58)

which leads to

$\begin{split}H_{1}(-Q^{2})\sim\frac{1}{Q^{6}}\,,&\;H_{2}(-Q^{2})\sim\frac{1}{Q^{8}}\,,\;H_{3}(-Q^{2})\sim\frac{1}{Q^{8}}\,;\\\\[10.00002pt]
\tilde{H}_{1}(-Q^{2})\sim\frac{1}{Q^{6}}\,,&\;\tilde{H}_{2}(-Q^{2})\sim\frac{1}{Q^{8}}\,,\;\tilde{H}_{3}(-Q^{2})\sim\frac{1}{Q^{8}}\,;\\\\[10.00002pt]
F_{2}(-Q^{2})\sim\frac{1}{Q^{6}}\,,&\;F_{3}(-Q^{2})\sim\frac{1}{Q^{6}}\,;\\\\[10.00002pt]
\tilde{F}_{2}(-Q^{2})\sim\frac{1}{Q^{6}}\,,&\;\tilde{F}_{3}(-Q^{2})\sim\frac{1}{Q^{6}}\,.\end{split}$
(59)

##  Appendix C Kinematical constraints and partial waves

In the main part of this paper we use helicity amplitudes, i.e. amplitudes
characterized by the helicities of the initial and final states and by total
angular momentum Jacob:1959at . Instead of helicities, one could also use spin
and orbital angular momentum. This characterization scheme is commonly used in
non-relativistic physics (where the spin-orbit coupling is suppressed). While
we find the helicity amplitudes in general more practical for our purpose, the
classification using orbital angular momentum $l$ is actually helpful where
the considered system becomes non-relativistic. For the Dalitz decays
$Y^{*}\to Y\,e^{+}e^{-}$, this happens at the end of the phase space, i.e. for
$q^{2}\approx(m_{Y^{*}}-m_{Y})^{2}$. Here initial and final hyperon are at
rest and their mass difference goes entirely into the back-to-back motion of
the electron-positron pair. Note that we consider the frame where the virtual
photon is at rest and that we denote the hyperons’ momentum by $p_{z}$, cf.
(8).

In subsection 2.1 we discuss the case of opposite parity of initial spin-3/2
and final spin-1/2 hyperon. In this case, the orbital angular momentum $l$
must be even. In total we find three partial waves: $l=0$ and $s=3/2$; $l=2$
and $s=1/2$; $l=2$ and $s=3/2$. Here $s$ denotes the total spin built from the
spin 1 of the photon and the spin 1/2 of the final hyperon. Since the
amplitude for orbital angular momentum $l$ scales with $p_{z}^{l}$, one
amplitude (the s-wave) becomes dominant at the end of the phase space. This
must find its expression in a relation between the helicity amplitudes.
Indeed, this is what the kinematical constraint (4) means physically.

To complete this discussion, we note that due to crossing symmetry the same
amplitude that describes $Y^{*}\to Y\,e^{+}e^{-}$ can also be used to describe
$e^{+}e^{-}\to Y\bar{Y}^{*}$. A non-relativistic situation emerges at the
production threshold $q^{2}\approx(m_{Y^{*}}+m_{Y})^{2}$. Since the
antifermion $\bar{Y}^{*}$ has opposite parity to $Y^{*}$, the allowed orbital
angular momentum $l$ must now be odd. The three possible partial waves are
$l=1$ and $s=1$; $l=1$ and $s=2$; $l=3$ and $s=2$, where now $s$ denotes the
total spin built from the spin 3/2 of the antihyperon and the spin 1/2 of the
hyperon. At threshold, the two p-waves dominate over the f-wave. Thus one of
the three helicity amplitudes must be related to the other two. This is the
physics content of the kinematical constraint (5).

Analogous considerations can be used for the other spin-parity combinations
that we discuss in this paper. For every case, one can understand where and
how many kinematical constraints emerge for the helicity amplitudes.

##  Appendix D General structure of the squared matrix element for the four-
body decay

We consider the decay $Y^{*}\to Y\,e^{+}e^{-}\to N\pi\,e^{+}e^{-}$. The spin
summed/averaged squared matrix element $\langle|{\cal M}_{4}|^{2}\rangle$ can
be expressed in terms of the four four-vectors $p_{Y}$, $p_{N}$,
$q=p_{e^{-}}+p_{e^{+}}$ and $k_{e}=p_{e^{-}}-p_{e^{+}}$. The Lorentz invariant
combinations that cannot be expressed solely by masses are $q^{2}$,
$k_{e}\cdot p_{Y}$, $q\cdot p_{N}$, $k_{e}\cdot p_{N}$ and
$\epsilon_{\mu\nu\alpha\beta}\,k_{e}^{\mu}\,p_{Y}^{\nu}\,p_{N}^{\alpha}\,q^{\beta}$.

In the following we will show that the general structure is

$\displaystyle\langle|{\cal M}_{4}|^{2}\rangle$ $\displaystyle=$
$\displaystyle J_{1}(q^{2})+J_{2}(q^{2})\,(k_{e}\cdot p_{Y})^{2}$ (60)
$\displaystyle{}+J_{3}(q^{2})\,k_{e}\cdot
p_{Y}\;\epsilon_{\mu\nu\alpha\beta}\,k_{e}^{\mu}\,p_{Y}^{\nu}\,p_{N}^{\alpha}\,q^{\beta}\,.$

In particular, we will show that $\langle|{\cal M}_{4}|^{2}\rangle$ does not
depend on $q\cdot p_{N}$ or $k_{e}\cdot p_{N}$.

An evaluation of the lepton trace shows that $k_{e}$ can only appear pairwise.
It is also easy to see that $p_{N}$ can appear at most linearly. This defines
already the structure with the Levi-Civita tensor. No terms $\sim q\cdot
p_{N},k_{e}\cdot p_{N}$ could show up there, because there is already one
$p_{N}$ contracted with the Levi-Civita tensor. It remains to be shown that
any term without a Levi-Civita tensor does not depend on $q\cdot p_{N}$ or
$k_{e}\cdot p_{N}$.

The Levi-Civita structure emerges from an odd
number of $\gamma_{5}$ matrices while any other structure stems from terms
with an even number of $\gamma_{5}$ matrices. Consequently, the weak coupling
constants $A$ and $B$ from (15) appear each linearly in $J_{3}$, while the
contributions to any other structure are $\sim|A|^{2}$ or $\sim|B|^{2}$. Thus
one has to deal there either with the $A$-type interaction or with the
$B$-type interaction, but not with both at the same time. Both interactions
are formally parity invariant if one considers the nucleon as having opposite
or the same parity as the $Y$. Thus one can use parity arguments to show that
any term that does not contain the Levi-Civita tensor will be independent of
$q\cdot p_{N}$ and $k_{e}\cdot p_{N}$.

Now we consider the rest frame of the $Y$ and a spin quantization axis along
the motion of the nucleon. In this frame, the spin of the $Y$ must be
identical to the spin of the nucleon. Let ${\cal M}_{w}(s_{N},s_{Y})$ be the
decay matrix element caused by the $A$\- or $B$-interaction (exclusive “or”).
Here $s_{N/Y}$ is the spin of the nucleon or $Y$, respectively. We find

$\begin{split}\qquad\qquad\quad{\cal
M}_{w}(s_{N},s_{Y})\sim{}&\delta_{s_{N},s_{Y}}\,,\\\ |{\cal
M}_{w}(+1/2,+1/2)|={}&|{\cal M}_{w}(-1/2,-1/2)|\,.\end{split}$ (61)

Let ${\cal M}_{s}(\ldots,s_{Y})$ denote the decay matrix element for the decay
$Y^{*}\to Y\,e^{+}e^{-}$. The dots denote the spins of the other states. We do
not need to specify them further. In total we find

$\displaystyle\langle|{\cal M}_{4}|^{2}\rangle_{A\;{\rm or}\,B}$
$\displaystyle\sim$
$\displaystyle\sum\limits_{\ldots}\sum\limits_{s_{N},s_{Y},s^{\prime}_{Y}}{\cal
M}_{w}(s_{N},s_{Y})\,{\cal M}_{s}(\ldots,s_{Y})$
$\displaystyle\phantom{mmmmmm}\times{\cal
M}_{s}^{*}(\ldots,s^{\prime}_{Y})\,{\cal M}^{*}_{w}(s_{N},s^{\prime}_{Y})$
$\displaystyle=$ $\displaystyle\sum\limits_{\ldots}\sum\limits_{s_{N}}{\cal
M}_{w}(s_{N},s_{N})\,{\cal M}^{*}_{w}(s_{N},s_{N})$
$\displaystyle\phantom{mmmm}\times{\cal M}_{s}(\ldots,s_{N})\,{\cal
M}_{s}^{*}(\ldots,s_{N})$ $\displaystyle=$
$\displaystyle\sum\limits_{\ldots}\sum\limits_{s_{N}}|{\cal
M}_{w}(s_{N},s_{N})|^{2}\,|{\cal M}_{s}(\ldots,s_{N})|^{2}$ $\displaystyle=$
$\displaystyle\sum\limits_{\ldots}\left(|{\cal M}_{w}(+1/2,+1/2)|^{2}\,|{\cal
M}_{s}(\ldots,+1/2)|^{2}\right.$ $\displaystyle\phantom{mm}\left.+|{\cal
M}_{w}(-1/2,-1/2)|^{2}\,|{\cal M}_{s}(\ldots,-1/2)|^{2}\right)$
$\displaystyle=$ $\displaystyle|{\cal M}_{w}(+1/2,+1/2)|^{2}$
$\displaystyle\times\sum\limits_{\ldots}\left(|{\cal
M}_{s}(\ldots,+1/2)|^{2}+|{\cal M}_{s}(\ldots,-1/2)|^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sum\limits_{s_{N},s_{Y}}|{\cal
M}_{w}(s_{N},s_{Y})|^{2}\right)\left(\sum\limits_{s^{\prime}_{Y},\ldots}|{\cal
M}_{s}(\ldots,s^{\prime}_{Y})|^{2}\right)\,.$

Now we have reached a product of two spin-summed squares of matrix elements.
Both are Lorentz invariant and can be evaluated in any frame. The first one
does not depend on $q$ or $k_{e}$. The second does not depend on $p_{N}$.
Therefore, the products $q\cdot p_{N}$ and $k_{e}\cdot p_{N}$ do not appear.

We note again that this line of reasoning only works if one considers solely
the $A$-type interaction or solely the $B$-type interaction. It does not work
for interference terms $\sim AB^{*},A^{*}B$. But such interference terms are
accompanied by an odd number of $\gamma_{5}$ matrices and therefore give rise
to a Levi-Civita tensor.

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