ROM2F/2008/25

Stückelino Dark Matter in Anomalous $U(1)^{\prime}$ Models

Francesco<EMAIL_ADDRESS>Andrea
<EMAIL_ADDRESS>Andrea
<EMAIL_ADDRESS>Antonio
<EMAIL_ADDRESS>

♮ Dipartimento di Fisica dell’Università di Roma , “Tor Vergata” and

I.N.F.N. - Sezione di Roma “Tor Vergata”

Via della Ricerca Scientifica, 1 - 00133 Roma, ITALY

♭ National Institute of Chemical Physics and Biophysics,

Ravala 10, Tallinn 10143, Estonia

We study a possible dark matter candidate in the framework of a minimal
anomalous $U(1)^{\prime}$ extension of the MSSM. It turns out that in a
suitable decoupling limit the Stückelino, the fermionic degree of freedom of
the Stückelberg multiplet, is the lightest supersymmetric particle (LSP). We
compute the relic density of this particle including coannihilations with the
next to lightest supersymmetric particle (NLSP) and with the next to next to
lightest supersymmetric particle (NNLSP) which are assumed almost degenerate
in mass. This assumption is needed in order to satisfy the stringent limits
that the Wilkinson Microwave Anisotropy Probe (WMAP) puts on the relic
density. We find that the WMAP constraints can be satisifed by different NLSP
and NNLSP configurations as a function of the mass gap with the LSP. These
results hold in the parameter space region where the model remains
perturbative.

##  1 Introduction

A great deal of work has been done recently to embed the standard model of
particle physics (SM) into a brane construction [1, 2, 3, 4]. This research is
part of the effort, initiated in [5], to build a fully realistic four
dimensional vacuum out of string theory. While the original models were
formulated in the framework of the heterotic string, the most recent efforts
were formulated for type II strings in order to take advantage of the recent
work on moduli stabilization using fluxes. Such brane constructions naturally
lead to extra anomalous $U(1)$’s in the four dimensional low-energy theory
and, in turn, to the presence of possible heavy $Z^{\prime}$ particles in the
spectrum. These particles should be among the early findings of LHC and
besides for the above cited models they are also a prediction of many other
theoretical models of the unification of forces (see [6] for a recent review).
It is then of some interest to know if these $Z^{\prime}$ particles contribute
to the cancelation of the gauge anomaly in the way predicted from string
theory or not. In [7] some of the present authors have studied a
supersymmetric (SUSY) extension of the minimal supersymmetric standard model
(MSSM) in which the anomaly is canceled à la Green-Schwarz. The model is only
string-inspired and is not the low-energy sector of some brane construction.
The reason of this choice rests in our curiosity to explore the phenomenology
of these models keeping a high degree of flexibility, while avoiding the
intricacies and uncertainties connected with a string theory construction. For
previous work along these lines we refer to [8]-[15]. In this work we perform
a consistency check of our model [7] by evaluating the thermal relic density
and comparing it against the WMAP data.

The cancelation of the $U(1)^{\prime}$ anomaly in our model requires the
introduction of an extra complex scalar field whose supersymmetric partner is
called the Stückelino. We will see in the following that if the latter is the
lightest supersymmetric particle (LSP) its interactions are such to define it
as an XWIMP (i.e. a weakly interacting particle with couplings at least one
order of magnitude less than the standard weak interactions): in fact it will
also turn out to be a cold relic. If the Stückelino is the lightest
supersymmetric particle (LSP), its relic density turns out to be too high with
respect to the experimental data. This is why, following [16], we favor a next
to lightest supersymmetric particle (NLSP) with a mass close to the LSP. We
show that an interesting scenario arises also for three particle
coannihilation processes. In these two cases, the model is consistent with the
experimental data. Moreover in the three particle case we find configurations
in which the LSP and the NLSP do not need to be nearly degenerate in mass. In
this case the mass gap between the two can be of the order of $20\%$. This is
the plan of the paper: in Section 2 we describe our model. In Section 3 and 4
we find the LSP and study the Stückelino interactions. Finally in Section 5 we
compute the relic density. Section 6 is a summary of our results.

##  2 Model Setup

In this section we briefly discuss our theoretical framework. We assume an
extension of the MSSM with an additional abelian vector multiplet $V^{(0)}$
with arbitrary charges. The anomalies are canceled with the Green-Schwarz (GS)
mechanism and with the Generalized Chern-Simons (GCS) terms. All the details
can be found in [7]. All the MSSM fields are charged under the additional
vector multiplet $V^{(0)}$, with charges which are given in Table 1, where
$Q_{i},L_{i}$ are the left handed quarks and leptons respectively while
$U^{c}_{i},D^{c}_{i},E^{c}_{i}$ are the right handed up and down quarks and
the electrically charged leptons. The superscript $c$ stands for charge
conjugation. The index $i=1,2,3$ denotes the three different families.
$H_{u,d}$ are the two Higgs scalars.

| SU(3)c | SU(2)L | U(1)Y |  U(1)${}^{\prime}~{}$
---|---|---|---|---
$Q_{i}$ | ${\bf 3}$ | ${\bf 2}$ | $1/6$ | $Q_{Q}$
$U^{c}_{i}$ | $\bar{\bf 3}$ | ${\bf 1}$ | $-2/3$ | $Q_{U^{c}}$
$D^{c}_{i}$ | $\bar{\bf 3}$ | ${\bf 1}$ | $1/3$ | $Q_{D^{c}}$
$L_{i}$ | ${\bf 1}$ | ${\bf 2}$ | $-1/2$ | $Q_{L}$
$E^{c}_{i}$ | ${\bf 1}$ | ${\bf 1}$ | $1$ | $Q_{E^{c}}$
$H_{u}$ | ${\bf 1}$ | ${\bf 2}$ | $1/2$ | $Q_{H_{u}}$
$H_{d}$ | ${\bf 1}$ | ${\bf 2}$ | $-1/2$ | $Q_{H_{d}}$
Table 1: Charge assignment.

The key feature of this model is the mechanism of anomaly cancelation. As it
is well known, the MSSM is anomaly free. In our MSSM extension all the
anomalies that involve only the $SU(3)$, $SU(2)$ and $U(1)_{Y}$ factors vanish
identically. However, triangles with $U(1)^{\prime}$ in the external legs in
general are potentially anomalous. These anomalies are555We are working in an
effective field theory framework and we ignore throughout the paper all the
gravitational effects. In particular, we do not consider the gravitational
anomalies which, however, could be canceled by the Green-Schwarz mechanism.

$\displaystyle U(1)^{\prime}-U(1)^{\prime}-U(1)^{\prime}:$ $\displaystyle\
\mathcal{A}^{(0)}$ (1) $\displaystyle U(1)^{\prime}-U(1)_{Y}-U(1)_{Y}:$
$\displaystyle\ \mathcal{A}^{(1)}$ (2) $\displaystyle
U(1)^{\prime}-SU(2)-SU(2):$ $\displaystyle\ \mathcal{A}^{(2)}$ (3)
$\displaystyle U(1)^{\prime}-SU(3)-SU(3):$ $\displaystyle\ \mathcal{A}^{(3)}$
(4) $\displaystyle U(1)^{\prime}-U(1)^{\prime}-U(1)_{Y}:$ $\displaystyle\
\mathcal{A}^{(4)}$ (5)

All the remaining anomalies that involve $U(1)^{\prime}$s vanish identically
due to group theoretical arguments (see Chapter 22 of [17]). Consistency of
the model is achieved by the contribution of a Stückelberg field $S$ and its
appropriate couplings to the anomalous $U(1)^{\prime}$. The Stückelberg
lagrangian written in terms of superfields is [15]

$\mathcal{L}_{S}={\frac{1}{4}}\left.\left(S+S^{\dagger}+4b_{3}V^{(0)}\right)^{2}\right|_{\theta^{2}\bar{\theta}^{2}}\\!\\!-{\frac{1}{2}}\left\\{\left[\sum_{a=0}^{2}b^{(a)}_{2}S\textnormal{Tr}\left(W^{(a)}W^{(a)}\right)+b^{(4)}_{2}SW^{(1)}W^{(0)}\right]_{\theta^{2}}\\!\\!\\!\\!\\!\\!+h.c.\\!\right\\}$
(6)

where the index $a=0,\ldots,3$ runs over the
$U(1)^{\prime},\,U(1)_{Y},\,SU(2)$ and $SU(3)$ gauge groups respectively. The
Stückelberg multiplet is a chiral superfield

$S=s+i\sqrt{2}\theta\psi_{S}+\theta^{2}F_{S}-i\theta\sigma^{\mu}\bar{\theta}\partial_{\mu}s+{\frac{\sqrt{2}}{2}}\theta^{2}\bar{\theta}\bar{\sigma}^{\mu}\partial_{\mu}\psi_{S}-{\frac{1}{4}}\theta^{2}\bar{\theta}^{2}\Box
s$ (7)

The lowest component of $S$ is a complex scalar field $s=\alpha+i\phi$. In our
scenario the scalar $\phi$ is eaten up in the Stückelberg mechanism to give
mass to the gauge field. On the other end, the scalar $\alpha$ is the dilaton
of string theory and its value must be determined somehow. We will not
investigate the way in which this happens, but we will just retain the final
result: $\alpha$ drops out of our effective lagrangian. In the string
literature (see for instance [7]-[16]) the fields $\phi$, $\alpha$ and
$\psi_{S}$ are respectively known as axion, saxion and axino. In this paper,
to avoid confusion with the much better known QCD axion/axino system (see for
instance [18]-[22]), we will adopt the convention of [23] and, from now on, we
will refer to $s$ as the Stückelberg scalar and to $\psi_{S}$ as the
Stückelino.

The Stückelberg multiplet $S$ transforms under $U(1)^{\prime}$ as

$\displaystyle V^{(0)}$ $\displaystyle\to$ $\displaystyle
V^{(0)}+i\left(\Lambda-\Lambda^{\dagger}\right)$ $\displaystyle S$
$\displaystyle\to$ $\displaystyle S-4i~{}b_{3}~{}\Lambda$ (8)

where $b_{3}$ is a constant related to the $Z^{\prime}$ mass. In our model
there are two mechanisms that give mass to the gauge bosons: (i) the
Stückelberg mechanism and (ii) the Higgs mechanism. In this extension of the
MSSM, the mass terms for the gauge fields for
${Q_{H_{u}}}=-{Q_{H_{d}}}=0$666We impose this condition to simplify our
computations and to give analytical expressions of limited dimensions. There
are no obstructions to set ${Q_{H_{u}}}=-{Q_{H_{d}}}\neq 0$. are given by

$\mathcal{L}_{M}=\frac{1}{2}\left(V^{(0)}_{\mu}\ V^{(1)}_{\mu}\
V^{(2)}_{3\mu}\right)M^{2}\left(\begin{array}[]{c}V^{(0)\mu}\\\ V^{(1)\mu}\\\
V^{(2)\mu}_{3}\end{array}\right)$ (9)

with $M^{2}$ being the gauge boson mass matrix

$M^{2}=\left(\begin{array}[]{ccc}M_{V^{(0)}}&~{}~{}~{}0&~{}~{}~{}0\\\
...&g_{1}^{2}\frac{v^{2}}{4}&-g_{1}g_{2}\frac{v^{2}}{4}\\\
...&...&g_{2}^{2}\frac{v^{2}}{4}\\\ \end{array}\right)$ (10)

where $M_{V^{(0)}}=4b_{3}g_{0}$ is the mass parameter for the anomalous $U(1)$
and it is assumed to be in the TeV range. The lower dots denote the obvious
terms under symmetrization. After diagonalization, we obtain the eigenstates

$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{2}V^{(1)}_{\mu}+g_{1}V^{(2)}_{3\mu}}{\sqrt{g_{1}^{2}+g_{2}^{2}}}$
(11) $\displaystyle Z_{0\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{2}V^{(2)}_{3\mu}-g_{1}V^{(1)}_{\mu}}{\sqrt{g_{1}^{2}+g_{2}^{2}}}$
(12) $\displaystyle Z^{\prime}_{\mu}$ $\displaystyle=$ $\displaystyle
V^{(0)}_{\mu}$ (13)

and the corresponding masses

$\displaystyle M^{2}_{\gamma}$ $\displaystyle=$ $\displaystyle 0$ (14)
$\displaystyle M^{2}_{Z_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(g_{1}^{2}+g_{2}^{2}\right)v^{2}$ (15)
$\displaystyle M^{2}_{Z^{\prime}}$ $\displaystyle=$ $\displaystyle
M_{V^{(0)}}^{2}$ (16)

Finally the rotation matrix from the hypercharge to the photon basis is

$\displaystyle\left(\begin{array}[]{c}Z^{\prime}_{\mu}\\\ Z_{0\mu}\\\
A_{\mu}\end{array}\right)$ $\displaystyle=$ $\displaystyle
O_{ij}\left(\begin{array}[]{c}V^{(0)}_{\mu}\\\ V^{(1)}_{\mu}\\\
V^{(2)}_{3\mu}\end{array}\right)=\left(\begin{array}[]{ccc}1&0{}{}&0\\\
0&-\sin\theta_{W}&\cos\theta_{W}\\\ 0&\cos\theta_{W}&\sin\theta_{W}\\\
\end{array}\right)\left(\begin{array}[]{c}V^{(0)}_{\mu}\\\ V^{(1)}_{\mu}\\\
V^{(2)}_{3\mu}\end{array}\right)$ (29)

where $i,j=0,1,2$.

We now give the expansion of the lagrangian piece $\mathcal{L}_{S}$ defined in
(6) in component fields only for the part that is needed in the following
sections. Using the Wess-Zumino gauge we get

$\displaystyle\mathcal{L}_{\text{St\"{u}ckelino}}$ $\displaystyle=$
$\displaystyle{\frac{i}{4}}\psi_{S}\sigma^{\mu}\partial_{\mu}{\bar{\psi}}_{S}-\sqrt{2}b_{3}\psi_{S}\lambda^{(0)}-{\frac{i}{2\sqrt{2}}}\sum_{a=0}^{2}b^{(a)}_{2}\textnormal{Tr}\left(\lambda^{(a)}\sigma^{\mu}\bar{\sigma}^{\nu}F_{\mu\nu}^{(a)}\right)\psi_{S}$
(30)
$\displaystyle-{\frac{i}{2\sqrt{2}}}b^{(4)}_{2}\left[{\frac{1}{2}}\lambda^{(1)}\sigma^{\mu}\bar{\sigma}^{\nu}F_{\mu\nu}^{(0)}\psi_{S}+(0\leftrightarrow
1)\right]+h.c.$

As it was pointed out in [8], the Stückelberg mechanism is not enough to
cancel all the anomalies. Mixed anomalies between anomalous and non-anomalous
factors require an additional mechanism to ensure consistency of the model:
non-gauge invariant GCS terms must be added. In our case, the GCS terms have
the form [10]

$\displaystyle\mathcal{L}_{GCS}$ $\displaystyle=$ $\displaystyle-
d_{4}\left[\left(V^{(1)}D^{\alpha}V^{(0)}-V^{(0)}D^{\alpha}V^{(1)}\right)W^{(0)}_{\alpha}+h.c.\right]_{\theta^{2}\bar{\theta}^{2}}+$
(31)
$\displaystyle+d_{5}\left[\left(V^{(1)}D^{\alpha}V^{(0)}-V^{(0)}D^{\alpha}V^{(1)}\right)W^{(1)}_{\alpha}+h.c.\right]_{\theta^{2}\bar{\theta}^{2}}+$
$\displaystyle+d_{6}\textnormal{Tr}\bigg{[}\left(V^{(2)}D^{\alpha}V^{(0)}-V^{(0)}D^{\alpha}V^{(2)}\right)W^{(2)}_{\alpha}+n.a.c+h.c.\bigg{]}_{\theta^{2}\bar{\theta}^{2}}$

where $n.a.c.$ refers to non-abelian completion terms. The $b$ constants in
(6) and the $d$ constants in (31) are fixed by the anomaly cancelation
procedure (for details see [7]).

For a symmetric distribution of the anomaly, we have

$\displaystyle b^{(0)}_{2}b_{3}=-\frac{\mathcal{A}^{(0)}}{384\pi^{2}}~{}\qquad
b^{(1)}_{2}b_{3}=-\frac{\mathcal{A}^{(1)}}{128\pi^{2}}\qquad
b^{(2)}_{2}b_{3}=-\frac{\mathcal{A}^{(2)}}{64\pi^{2}}~{}\qquad
b^{(4)}_{2}b_{3}=-\frac{\mathcal{A}^{(4)}}{128\pi^{2}}$
$\displaystyle~{}~{}~{}~{}d_{4}=-\frac{\mathcal{A}^{(4)}}{384\pi^{2}}~{}~{}~{}~{}~{}\qquad
d_{5}=\frac{\mathcal{A}^{(1)}}{192\pi^{2}}~{}~{}~{}~{}~{}~{}\qquad
d_{6}=\frac{\mathcal{A}^{(2)}}{96\pi^{2}}$ (32)

It is worth noting that the GCS coefficients $d_{4,5,6}$ are fully determined
in terms of the $\mathcal{A}$’s by the gauge invariance, while the
$b_{2}^{(a)}$’s depend only on the free parameter $b_{3}$, which is related to
the mass of the anomalous $U(1)$.

The soft breaking sector of the model is given by

$\mathcal{L}_{soft}=\mathcal{L}_{soft}^{MSSM}-{\frac{1}{2}}\left(M_{0}\lambda^{(0)}\lambda^{(0)}+h.c.\right)-{\frac{1}{2}}\left(\frac{M_{S}}{2}\psi_{S}\psi_{S}+h.c.\right)$
(33)

where $\mathcal{L}_{soft}^{MSSM}$ is the usual soft susy breaking lagrangian
while $\lambda^{(0)}$ is the gaugino of the added $U(1)^{\prime}$ and
$\psi_{S}$ is the Stückelino. The Stückelino soft mass term deserves some
comment: from [24] we know that a fermionic mass term for a chiral multiplet
is not allowed in presence of Yukawa interactions in which this chiral
multiplet is involved. But in the classical Lagrangian the Stückelberg
multiplet cannot contribute to superpotential terms given that the gauge
invariance given from our U(1)’ symmetry (8) requires non-holomorphicity in
the chiral fields. In fact in our model both the Stückelino and the scalar
$\phi$ couple only through GS interactions. It is worth noting that a mass
term for the scalar $\phi$ is instead not allowed since it transforms non
trivially under the anomalous $U(1)^{\prime}$ gauge transformation (8).

##  3 Neutralino Sector

Assuming the conservation of R-parity the LSP is a good weak interacting
massive particle (WIMP) dark matter candidate. As in the MSSM the LSP is given
by a linear combination of fields in the neutralino sector. The general form
of the neutralino mass matrix is given in [7]. Written in the interaction
eigenstate basis $(\psi^{0})^{T}=(\psi_{S},\ \lambda^{(0)},\ \lambda^{(1)},\
\lambda^{(2)}_{3},\ \tilde{h}_{d}^{0},\ \tilde{h}_{u}^{0})$ it is a six-by-six
matrix. From the point of view of the strength of the interactions the two
extra states are not on the same footing with respect to the standard ones.
The Stückelino and the extra gaugino $\lambda^{(0)}$ dubbed primeino are in
fact extremely weak interacting massive particle (XWIMP). Thus we are
interested in situations in which the extremely weak sector is decoupled from
the standard one and the LSP belongs to this sector. This can be achieved at
tree level with the choice

${Q_{H_{u}}}={Q_{H_{d}}}=0$ (34)

The neutralino mass matrix ${\bf M}_{\tilde{N}}$ becomes

${\bf
M}_{\tilde{N}}=\left(\begin{array}[]{cccccc}\frac{M_{S}}{2}&\frac{M_{V^{(0)}}}{\sqrt{2}}&0&0&0&0\\\
\dots&M_{0}&0&0&0&0\\\
\dots&\dots&M_{1}&0&-\frac{g_{1}v_{d}}{2}&\frac{g_{1}v_{u}}{2}\\\
\dots&\dots&\dots&M_{2}&\frac{g_{2}v_{d}}{2}&-\frac{g_{2}v_{u}}{2}\\\
\dots&\dots&\dots&\dots&0&-\mu\\\
\dots&\dots&\dots&\dots&\dots&0\end{array}\right)~{}~{}~{}~{}$ (35)

where $M_{S},~{}M_{0},~{}M_{1},~{}M_{2}$ are the soft masses coming from the
soft breaking terms (33) while $M_{V^{(0)}}$ is given in (10). It is worth
noting that the D terms and kinetic mixing terms can be neglected in the tree-
level computations of the eigenvalues and eigenstates.

Moreover, we make the assumption that

$\ M_{0}\gg M_{S},M_{V^{(0)}}$ (36)

so that the Stückelino is the LSP. This assumption is motivated by the
interaction strengths of the two extra states: the Stückelino interacts via
the vertex shown in Fig. 1b which can easily be seen (from (30)) to be
proportional to the coefficient $b^{(a)}_{2}$ of (32) which, in turn, is
inversely proportional to $b_{3}$ that is the $Z^{\prime}$ mass given that
$M_{Z^{\prime}}=M_{V^{(0)}}=4b_{3}g_{0}$. Given these considerations the
vertex shown in Fig. 1b is then of order $\sim g_{0}g_{a}^{2}/M_{Z^{\prime}}$.
The primeino interacts via the vertex in Fig. 1a which is of order $\sim
g_{a}$ that is the standard strength of weak interactions. Assuming the two
decoupling relations (34) and (36) we will see in the following sections that
the dominant contribution in the (co)annihilation processes is that of the
primeino, which is of the type of a standard gaugino interaction.

Figure 1: (a) Gaugino-fermion-sfermion interaction vertex. (b) Stückelino-
gaugino-vector interaction vertex.

##  4 Stückelino Interactions

The Stückelino interactions can be read off from the interaction lagrangian
(30). The relevant Stückelino-MSSM neutralino interaction term, written in
terms of four components Majorana spinors777The gamma matrices $\gamma^{\mu}$
are in the Weyl representation. We use capital letters for four components
spinors and lower case letters for two components spinors. , is given by:

$\mathcal{L}=i\sqrt{2}g_{1}^{2}b_{2}^{(1)}\bar{\Lambda}^{(1)}\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}V^{(1)}_{\nu})\Psi_{S}+i{\frac{\sqrt{2}}{2}}g_{2}^{2}b_{2}^{(2)}\bar{\Lambda}^{(2)}_{3}\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}V^{(2)}_{3\nu})\Psi_{S}$
(37)

where the $b_{2}^{(a)}$ coefficients are given in (32). The related
interaction vertex Feynman rule is

$C^{(a)}\gamma_{5}[\gamma^{\mu},\gamma^{\nu}]ik_{\mu}$ (38)

where $k_{\mu}$ is the momentum of the outgoing vector and the $C^{(a)}$’s are

$\displaystyle C^{(1)}=\sqrt{2}g_{1}^{2}b_{2}^{(1)}$ $\displaystyle
C^{(2)}={\frac{\sqrt{2}}{2}}g_{2}^{2}b_{2}^{(2)}$ (39)

The interaction Lagrangian (37) expressed in the mass eigenstates basis is

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
i\sum_{j}\tilde{N_{j}}\left(\cos\theta_{W}C^{(1)}N_{(1)j}+\sin\theta_{W}C^{(2)}N_{(2)j}\right)\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}A_{\nu})\Psi_{S}+$
(40) $\displaystyle
i\sum_{j}\tilde{N_{j}}\left(-\sin\theta_{W}C^{(1)}N_{(1)j}+\cos\theta_{W}C^{(2)}N_{(2)j}\right)\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}{Z_{0}}_{\nu})\Psi_{S}$

where $\tilde{N_{j}}$ is a generic neutralino and $N_{ij}$ is the matrix that
diagonalizes (35). We remind that the Stückelino $\Psi_{S}$ is directly a mass
eigenstate because of the decoupling (36). We stress that these are only
interactions between the Stückelino and the MSSM neutralinos. All the
Stuckelino interactions in (30) include also analogous interactions involving
the charged wino or the primeino.

The factors in (39) are of naive dimension higher than four. They then contain
the parameters $b_{2}^{(a)}$ which are inversely proportional to the mass of
the $Z^{\prime}$ (see (32)). Given these interactions, our Stückelino will be
an extremely weak interacting particle, according to the definition we gave in
the introduction. The simplifying assumption (34) has now decoupled our
Stückelino and primeino from the standard MSSM sectors and, at tree level, the
relevant diagrams are given in Figs. 2 and 3. We can now give a rough estimate
of the effective interaction, comparing with the standard Fermi coupling,
$G_{F}$, of weak interactions: given a mass of the $Z^{\prime}$ boson of the
order of the TeV, the effective coupling in Fig. 2 is $G_{F}^{\prime}\approx
10^{-4}G_{F}$ and that in Fig. 3 is $G_{F}^{\prime}\approx 10^{-2}G_{F}$.

Figure 2: Annihilation of two Stückelinos into two gauge vectors via the
exchange of a gaugino.

Let us go back now to Fig. 2, where we denoted with $p_{1}$ and $p_{2}$ the
incoming momenta of the Stückelinos while $k_{1}$ and $k_{2}$ are the two
outcoming momenta of the gauge bosons in the final state. We will concentrate
on the case with two photons in the final state. In this case the result for
the differential cross section is given by

$\
\frac{d\sigma}{d\Omega}=\frac{4M_{S}^{2}\omega_{1}}{16\pi^{2}(\omega_{1}+\omega_{2})^{2}(\sqrt{M_{S}^{2}-E_{2}^{2}})}\sum_{i,j=1}^{2}\mathcal{M}_{i}\mathcal{M}^{*}_{j}$
(41)

where $\omega_{1}$ and $\omega_{2}$ are the energies of the two outcoming
photons. Each amplitude $\mathcal{M}_{i}$ is proportional to the relative
coefficient $(C^{(a)})^{2}$ whose generic form is given in (39). In our
scenario the Stückelino annihilations alone, being the cross section (41)
extremely weak, cannot give a relic density in the WMAP preferred range. Thus,
in the scenario of an XWIMP Stückelino, we are forced to consider
coannihilations between the Stückelino and the NLSP [16]. Several scenarios
can be considered for the NLSP. We can split them into two major classes: one
in which the NLSP is either a pure bino or a pure wino, and thus a
coannihilation with a third MSSM particle is needed in order to recover the
WMAP result, and one in which the NLSP is a generic MSSM neutralino with a
non-negligible bino and/or wino component. In both classes in order to have
effective coannihilations, the NLSP (and eventually the other MSSM particle
involved in the coannihilation process) must be almost degenerate in mass.
Furthermore, in the most common applications, the cross sections for the
annihilations of the the LSP and the NLSP and that of the coannihilations
between the LSP and NSLP are roughly of the same order of magnitude. In our
case this last condition will be stretched and the cross section we are going
to discuss will differ for some orders of magnitude. This situation is not
completely new in literature: already in [25] in which the NLSP is the stop
these differences are of order $10^{-2}$. In [16] differences of order $\leq
10^{-4}$ are considered, while in [26] $10^{-4}\div 10^{-5}$ differences are
found888This can be extracted from Fig.4 of the previous reference after an
appropriate rescaling.. But what assures us that the two species are still in
thermal equilibrium and do not decouple separately? The existence of
interactions of the type $LSP+MSSM1\to NLSP+MSSM2$ is then required to keep
the LSP in equilibrium [27]. $MSSM1,MSSM2$ are two MSSM particles. Furthermore
$MSSM1$ better be relativistic so that its abundance is much larger of any
cold particle to foster the above reaction. The above reaction will then keep
the LSP at equilibrium and the formalism of coannihilation can be safely
employed. As we will see in the next section, all of these requirements are
met in our scenario.

As a first example we then consider a pure bino as the NLSP. The allowed
coannihilation processes with the Stückelino are those which involve an
exchange of a photon or a $Z_{0}$ in the intermediate state and with a SM
fermion-antifermion pair, Higgses and $W$’s in the final state. The diagram
with the fermion-antifermion in the final state is sketched in Fig. 3. The
differential cross section in the center of mass frame has the following
general form

$\frac{d\sigma}{d\Omega}\propto\frac{1}{s}\frac{p_{f}}{p_{i}}|\mathcal{M}|^{2}$
(42)

where $s$ is the usual Mandelstam variable and $p_{f,i}$ is the spatial
momentum of the outgoing (incoming) particles. On dimensional ground
$|\mathcal{M}|^{2}$ has at least a linear dependence on $p_{f}$ and this
implies that the dominant contribution comes from the diagram with the SM
fermion-antifermion pair $f$ and $\bar{f}$ in the final state:

$\Psi_{S}\lambda^{(a)}\rightarrow f\bar{f}$ (43)

Figure 3: Coannihilation of a Stückelino and a bino into a $f\bar{f}$ pair via
the exchange of a photon or a $Z_{0}$.

The resulting differential cross section, computed in the center of mass
frame, is

$\frac{d\sigma}{d\Omega}=\sum_{f}c_{f}\frac{\sqrt{(E_{3}-m_{f})^{2}}}{64\pi^{2}(E_{1}+E_{2})^{2}\sqrt{(E_{1}^{2}-M_{S}^{2})}}(\mathcal{M}_{\gamma}^{2}+\mathcal{M}_{Z_{0}}^{2}+\mathcal{M}^{*}_{\gamma}\mathcal{M}_{Z_{0}}+\mathcal{M}_{\gamma}\mathcal{M}_{Z_{0}}^{*})$
(44)

where the sum is extended to all the SM fermions (with mass $m_{f}$) while
$c_{f}$ is a color factor. Details of the amplitude computation can be found
in Appendix A.

##  5 Stückelino Relic Density

In this section we compute the relic density of the Stückelino. The case of
the Stückelino as a cold dark matter candidate has been studied for the first
time in [22]. As we said in the previous section we study two scenarios: the
first in which the Stückelino coannihilates with only one NLSP degenerate in
mass (a generic MSSM neutralino), the second in which there is an additional
supersymmetric particle (either a chargino or a stau) involved in the
coannihilation process with the Stückelino and the NLSP. In the following we
will be largely following [16]. Since this is a first study and given also the
simplifying choice (34) we will defer a complete analysis to a future work and
will content ourselves with showing that our model can accommodate for WMAP
data. Then, following this philosophy we will not solve the Boltzmann equation
numerically but, in agreement with [16], we will argue that, if the ratio
between the thermally averaged cross sections of the coannihilation of Fig. 3
and that of a typical neutralino annihilation is much less than one, a relic
density satisfying the WMAP requirements can be found.

Just to fix the notation we briefly review the relic density computation for
$N$ interacting species [25, 26, 27]. The requirements discussed in the
previous section are met by our model given the lagrangian (30) and the
condition of (34). In this case all channels are open to interactions and the
Stückelino has an interaction with the photon and the bino of strength
$b_{2}^{(1)}$999There is another interesting possibility if we insist on
imposing (34): taking also $Q_{L}=0$ SUSY terms of the type $\int
d^{2}\theta\int
d^{2}\bar{\theta}\log(S+S^{\dagger}+V^{(0)})/M_{S}(LH_{u}/M_{S}^{2}+H_{u}^{\dagger}L^{\dagger}/M_{S}^{2})$
can be safely added to the lagrangian. This term induces a vertex between the
Stückelino, neutrino and Higgs field which can be a viable candidate to keep
the Stückelino in thermal equilibrium..

The Boltzmann equation for $N$ particle species is given by:

$\frac{dn}{dt}=-3Hn-\sum_{i,j=1}^{N}\langle\sigma_{ij}v_{ij}\rangle(n_{i}n_{j}-n^{eq}_{i}n^{eq}_{j})$
(45)

where $n_{i}$ denotes the number density per unit of comoving volume of the
species $i=1,\ldots,N$ ($i=1$ refers to the LSP, $i=2$ refers to the NLSP, and
so on), $n=\sum_{i}n_{i}$, $H$ is the Hubble constant, $\sigma_{ij}$ is the
annihilation cross section between a species $i$ and a species $j$, $v_{ij}$
is the modulus of the relative velocity while $n_{i}^{eq}$ is the equilibrium
number density of the species $i$ given by:

$\frac{n_{i}^{eq}}{n^{eq}}=\frac{g_{i}\left(1+\Delta_{i}\right)^{3/2}e^{-\Delta_{i}x_{f}}}{\sum_{i}g_{i}\left(1+\Delta_{i}\right)^{3/2}e^{-\Delta_{i}x_{f}}}$
(46)

where $g_{i}$ are the internal degrees of freedom,
$\Delta_{i}=(m_{i}-m_{1})/m_{1}$. $x_{f}=m_{1}/T$ is known once (45) is
solved. A preliminary estimate of $x_{f}$ can be obtained by solving the
equation

$x\simeq\ln\left(x^{1/2}M_{P}\,m_{LSP}\left<\sigma v\right>\right)$ (47)

which can be obtained from the decoupling condition. This can be done using
the estimate101010To check this crude estimate in the case of our Stückelino,
we have also solved numerically $\left<\sigma v\right>$ sweeping the
temperatures range $1\div 100$ GeV. We have then fit the results to recover a
function in good agreement with (48).

$\left<\sigma v\right>\simeq G^{2}\,m_{LSP}^{2}x^{-5/2}$ (48)

where the effective coupling $G$ can be the $G_{F},G_{F}^{\prime}$ introduced
in Section 4. In the mass range $m_{LSP}=10\div 1000$ GeV, by plugging in (48)
$G=G_{F}$ we would get $x_{f}=25\div 30$ (if we would take our Stückelino as a
separate species, that is we would use $G=G_{F}^{\prime}$, we would get those
values divided by half).

Eq. (45) can be rewritten in a useful way by defining the thermal average of
the effective cross section

$\langle\sigma_{eff}v\rangle\equiv\sum_{i,j=1}^{N}\langle\sigma_{ij}v_{ij}\rangle\frac{n_{i}^{eq}}{n^{eq}}\frac{n_{j}^{eq}}{n^{eq}}$
(49)

obtaining

$\frac{dn}{dt}=-3Hn-\langle\sigma_{eff}v\rangle(n^{2}-(n^{eq})^{2})$ (50)

where $n^{eq}=\sum_{i}n_{i}^{eq}$. It is sensible to use these approximations
when the LSP is kept in equilibrium by a relativistic particle in the thermal
background as discussed in the previous section. Given the typical values of
$x_{f}$ discussed above, the ratio between the number density per comoving
volume of this relativistic species and that of a cold relic is
$n^{eq}_{rel}/n^{eq}_{cold}=10^{4}\div 10^{6}$, which is sufficient to keep
the Stückelino coupled until the end of coannihilations.

As a rule of thumb [28] a first order estimate of the relic density is given
by

$\Omega_{\chi}h^{2}\simeq\frac{10^{-27}\,{\rm
cm^{3}\,s^{-1}}}{\langle\sigma_{eff}v\rangle}$ (51)

To give a rough idea of the role played by coannihilations we plotted in Fig.
4 the relic density estimate (51) induced by an electro-weak cross section
((48) with $G=G_{F}$), a Stückelino cross section ((48) with
$G=G^{\prime}_{F}$) and two coannihilations cross sections estimations. We can
see that the Stückelino annihilations cannot give a relic density in the WMAP
data range, while coannihilations can do it. Moreover we can see that the
effect of coannihilations on the MSSM is to decrease the efficiency of the
MSSM annihilations (while they increase the Stückelino one) and to increase
the LSP mass value (according to an increasing mass gap) in order to agree
with WMAP data. We stress that Fig. 4 does not take into account several MSSM
parameters such as the sfermion masses, neutralino composition etc., so it is
just a rough estimate that, however, clarifies the role played by the
coannihilations.

Figure 4: Relic density estimation for a pure Stückelino cross section
(black), a pure electro-weak cross section (red), a coannihilating cross
section with a Stückelino-neutralino mass gap $\Delta=1\%$ (blue) and a
coannihilating cross section with $\Delta=5\%$ (dashed blue). The WMAP data
range is the gray strip. The Figure on the right is a zoom of that on the left
around the region of interest.

In the following we will now give a better estimation of
$\langle\sigma_{eff}v\rangle$ in the two cases $N=2$ and $N=3$ using what we
have learnt in our scenario and the coannihilation cross section of Fig. 3
presented in detail in Appendix A.

  * •

$N=2$ case. Assuming that the relative velocities are all equal $v_{ij}\equiv
v$ we get:

$\langle\sigma_{eff}^{(2)}v\rangle=\langle\sigma_{22}v\rangle\frac{\langle\sigma_{11}v\rangle/\langle\sigma_{22}v\rangle+2\langle\sigma_{12}v\rangle/\langle\sigma_{22}v\rangle
Q+Q^{2}}{(1+Q)^{2}}$ (52)

where $Q=n_{2}^{eq}/n_{1}^{eq}$. The first term in the numerator can be
neglected because the Stückelino annihilation cross section is suppressed by a
factor $(C^{(a)})^{4}$ with respect to the MSSM neutralino annihilations (see
the previous section) and thus
$\langle\sigma_{11}v\rangle\ll\langle\sigma_{22}v\rangle$. The second term
involves the coannihilation cross section. Let us consider the case in which
the NLSP is a generic MSSM neutralino (a linear combination of
$\lambda^{(1)},\ \lambda^{(2)}_{3},\ \tilde{h}_{d}^{0},\ \tilde{h}_{u}^{0}$)
with a non-vanishing bino or wino components. As we saw in the previous
section each amplitude is generically proportional to $C^{(a)}g_{i}$ with
$i=1,2$. Without loss of generality we consider the diagram which involves the
bino component $\Psi_{S}\lambda^{(1)}\rightarrow f\bar{f}$ and a photon
exchange in the intermediate channel, i.e the $\mathcal{M}_{\gamma}^{2}$
amplitude in (44). We get

$C^{2}_{\gamma}=(C^{(1)}\cos\theta_{W})^{2}=2(b_{2}^{(1)})^{2}g_{1}^{4}\cos^{2}\theta_{W}$
(53)

From the expression of the mixed $U(1)^{\prime}-U(1)_{Y}-U(1)_{Y}$ anomaly
(see [7]) and from the (32) we have the following relation

$b_{2}^{(1)}=\frac{3(3Q_{Q}+Q_{L})}{256\pi^{2}b_{3}}$ (54)

where $b_{3}=M_{Z^{\prime}}/4g_{0}$. With the assumption $M_{Z^{\prime}}=1$
TeV as in [7] we finally get

$\frac{C^{2}_{\gamma}}{e^{2}}\simeq 5.76\times
10^{-12}(3g_{0}Q_{Q}+g_{0}Q_{L})^{2}\,{\rm GeV}^{-2}$ (55)

where $e$ is the electric charge. We get similar expressions for the other
three terms in (44). This result has to be compared to the typical weak cross
section $\langle\sigma_{22}v\rangle\simeq 10^{-9}\,{\rm GeV}^{-2}$. As long as
the charges and the coupling constant of the extra $U(1)$ satisfy the
perturbative requirement

$g_{0}^{2}\cdot(3Q_{Q}+Q_{L})^{2}<16$ (56)

the following upper bound is satisfied:

$\frac{\langle\sigma_{12}v\rangle}{\langle\sigma_{22}v\rangle}\lesssim
10^{-6}$ (57)

in the case of a pure bino, while

$\frac{\langle\sigma_{12}v\rangle}{\langle\sigma_{22}v\rangle}\lesssim
10^{-5}$ (58)

in the case of a pure wino. Accordingly to eqs. (51), (52), (57) and (58) the
relic density gets rescaled as [16]

$\left(\Omega
h^{2}\right)^{(2)}\simeq\left[\frac{1+Q}{Q}\right]^{2}\left(\Omega
h^{2}\right)^{(1)}$ (59)

We performed a random sampling of MSSM models in which the NLSP is a pure bino
or a mixed bino-higgsino (the case of a pure wino falls back into the $N=3$
case due to the wino-chargino mass degeneracy) and we computed the relic
density in presence of coannihilations using the DarkSUSY package [29]. These
two situations are easily realized in some corners of the mSUGRA parameter
space. Thus in our scan we assumed this scenario in order to fix the pattern
of the supersymmetry breaking parameters at weak scale. We emphasize here that
this choice is completely arbitrary, and it is assumed only for simplicity,
since in our model [7] the supersymmetry breaking mechanism is not specified.
In the former case there is no model which satisfies the WMAP constraints
[30]:

$0.0913\leq\Omega h^{2}\leq 0.1285$ (60)

since the annihilation cross section of a pure bino is too low and the
rescaling (59) is not enough to get the right relic density. In the latter
case the higgsino component tends to increase the annihilation cross section
and thus we find models which satisfy the WMAP constraints. The results are
summarized in Fig. 5 for $\Delta_{2}=1\%$ and $\Delta_{2}=5\%$.

Figure 5: Stückelino relic density in the case in which the NLSP is a linear
combination bino-higgsino. Red (darker) points denote models which satisfy
WMAP data. Left panel: $\Delta_{2}=1\%$. Right panel: $\Delta_{2}=5\%$.

In order to fulfill the WMAP data (red (darker) points in the plot (5)) the
Stückelino mass must be in the range $50\;\text{GeV}\lesssim M_{S}\lesssim
700\;\text{GeV}$ in the limit $\Delta_{2}\to 0$, where the lowest bound is
given by the current experimental constraints [31].

  * •

$N=3$ case. This is the case in which there is a third MSSM particle almost
degenerate in mass with the LSP and the NLSP. Typical situations of this kind
arise when the NLSP and the next to next to lightest supersymmetric particle
(NNLSP) are respectively the bino and the stau or the wino and the lightest
chargino. Expanding in an explicit way all the terms in the sum (49) we get:

$\displaystyle\langle\sigma_{\text{eff}}^{(3)}v\rangle$ $\displaystyle=$
$\displaystyle\langle\sigma_{11}v\rangle\gamma_{1}^{2}+\langle\sigma_{12}v\rangle\gamma_{1}\gamma_{2}+\langle\sigma_{13}v\rangle\gamma_{1}\gamma_{3}+$
(61)
$\displaystyle\langle\sigma_{21}v\rangle\gamma_{2}\gamma_{1}+\langle\sigma_{22}v\rangle\gamma_{2}^{2}+\langle\sigma_{23}v\rangle\gamma_{2}\gamma_{3}+$
$\displaystyle\langle\sigma_{31}v\rangle\gamma_{3}\gamma_{1}+\langle\sigma_{32}v\rangle\gamma_{3}\gamma_{2}+\langle\sigma_{33}v\rangle\gamma_{3}^{2}$
$\displaystyle=$
$\displaystyle\Big{[}\langle\sigma_{11}v\rangle(n^{\text{eq}}_{1})^{2}+\langle\sigma_{12}v\rangle
n^{\text{eq}}_{1}n^{\text{eq}}_{2}+\langle\sigma_{13}v\rangle
n^{\text{eq}}_{1}n^{\text{eq}}_{3}+$ $\displaystyle\
\langle\sigma_{21}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{1}+\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+$
$\displaystyle\phantom{\Big{[}}\langle\sigma_{31}v\rangle
n^{\text{eq}}_{3}n^{\text{eq}}_{1}+\langle\sigma_{32}v\rangle
n^{\text{eq}}_{3}n^{\text{eq}}_{2}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}\Big{]}\frac{1}{(n^{\text{eq}})^{2}}$
$\displaystyle\simeq$
$\displaystyle\frac{\big{[}\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}\big{]}}{(n^{\text{eq}})^{2}}$

where in the last line we have neglected the terms
$\langle\sigma_{11}v\rangle$, $\langle\sigma_{12}v\rangle$ and
$\langle\sigma_{13}v\rangle$ since these are the thermal averaged cross
sections which involve the Stückelino. By introducing a new set of variables
defined by

$Q_{i}=\frac{n^{\text{eq}}_{i}}{n^{\text{eq}}_{1}}=\frac{g_{i}}{g_{1}}(1+\Delta_{i})^{3/2}e^{-x_{f}\Delta_{i}}\qquad\text{for}\
i=2,3$ (62)

where $g_{i}$ are the internal degrees of freedom of the particle species,
$x_{f}=m_{1}/T$ and $\Delta_{i}=(m_{i}-m_{1})/m_{1}$, we obtain

$\displaystyle\langle\sigma_{\text{eff}}^{(3)}v\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{\langle\sigma_{22}v\rangle
Q_{2}^{2}+2\langle\sigma_{23}v\rangle Q_{2}Q_{3}+\langle\sigma_{33}v\rangle
Q_{3}^{2}}{\left(1+Q_{2}+Q_{3}\right)^{2}}$ (63)

Under the assumption $(m_{3}-m_{2})/m_{1}\ll 1/x_{f}$, $Q_{3}/Q_{2}\simeq
g_{3}/g_{2}$ we finally get

$\langle\sigma_{\text{eff}}^{(3)}v\rangle\simeq\frac{Q_{2}^{2}}{\left[1+\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}\right]^{2}}\langle\sigma_{\text{MSSM}}v\rangle$
(64)

where

$\langle\sigma_{\text{MSSM}}v\rangle=\langle\sigma_{22}v\rangle+2\frac{g_{3}}{g_{2}}\langle\sigma_{23}v\rangle+\left(\frac{g_{3}}{g_{2}}\right)^{2}\langle\sigma_{33}v\rangle$
(65)

In order to compute the rescaling factor between the relic density of our
model and the MSSM relic density we have to express $\sigma_{\text{MSSM}}$ in
terms of a two coannihilating species effective cross section. This is given
by

$\displaystyle\langle\sigma_{\text{eff}}^{(2)}v\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}}{(n^{\text{eq}})^{2}}$
(66) $\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}}{(n^{\text{eq}}_{2}+n^{\text{eq}}_{3})^{2}}$
$\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}}{(n^{\text{eq}}_{2})^{2}(1+n^{\text{eq}}_{3}/n^{\text{eq}}_{2})^{2}}$
$\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle+2\langle\sigma_{23}v\rangle
Q_{23}+\langle\sigma_{33}v\rangle Q_{23}^{2}}{(1+Q_{23})^{2}}$

where

$\displaystyle Q_{23}$ $\displaystyle=$ $\displaystyle
n^{\text{eq}}_{3}/n^{\text{eq}}_{2}=\frac{g_{3}}{g_{2}}\left(1+\frac{m_{3}-m_{2}}{m_{2}}\right)^{3/2}e^{-x_{f}\frac{m_{3}-m_{2}}{m_{2}}}$
(67) $\displaystyle\simeq$ $\displaystyle\frac{g_{3}}{g_{2}}$

since $(m_{3}-m_{2})/m_{1}\ll 1/x_{f}$ and $m_{2}>m_{1}$ then
$(m_{3}-m_{2})/m_{2}\ll 1/x_{f}$. We remind the reader that the values of
$n^{\text{eq}}_{2}$, $n^{\text{eq}}_{3}$ and $n^{\text{eq}}$ are different
with respect to those in the former case since now there are only two species
in the thermal bath. We then find

$\displaystyle\langle\sigma_{\text{eff}}^{(2)}v\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{\langle\sigma_{22}v\rangle+2\frac{g_{3}}{g_{2}}\langle\sigma_{23}v\rangle+\left(\frac{g_{3}}{g_{2}}\right)^{2}\langle\sigma_{33}v\rangle}{\left(1+\frac{g_{3}}{g_{2}}\right)^{2}}$
(68) $\displaystyle\simeq$
$\displaystyle\frac{\langle\sigma_{\text{MSSM}}v\rangle}{\left(1+\frac{g_{3}}{g_{2}}\right)^{2}}$

and inserting back this relation into (64) we obtain

$\langle\sigma_{\text{eff}}^{(3)}v\rangle\simeq\left[\frac{\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}{1+\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}\right]^{2}\langle\sigma_{\text{eff}}^{(2)}v\rangle$
(69)

The rescaling factor between the three and two particle species relic density
is given by the following relation

$\left(\Omega
h^{2}\right)^{(3)}\simeq\left[\frac{1+\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}{\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}\right]^{2}\left(\Omega
h^{2}\right)^{(2)}$ (70)

We performed a random sampling of MSSM models with bino-stau and wino-chargino
coannihilations. The first situation is realized in some corners of the mSUGRA
parameter space111111Or in the so called Constrained MSSM (CMSSM). while the
second situation is naturally realized in anomaly mediated supersymmetry
breaking scenarios. For each model we computed the relic density $\left(\Omega
h^{2}\right)^{(2)}$ for the two coannihilating species with the DarkSUSY
package [29]. We finally computed $\left(\Omega h^{2}\right)^{(3)}$ using
(70). The bino-stau models which satisfy the WMAP constraints have a
Stückelino mass in the range $100\;\text{GeV}\lesssim M_{S}\lesssim
350\;\text{GeV}$ in the limit $\Delta_{2}\to 0$. As the mass gap increases the
number of allowed models drastically decreases and eventually vanishes for
$\Delta\simeq 5\%$. In the wino-chargino case, models which satisfy the WMAP
constraints are shown in Fig. 6 for four reference values of $\Delta_{2}$. The
space of parameters with $\Delta_{2}\lesssim 5\%$ and a Stückelino mass
$M_{S}\gtrsim 700$ GeV is favored while as the mass gap increases lower
Stückelino masses become favored, e.g. $100\;\text{GeV}\lesssim M_{S}<200$ GeV
($\Delta_{2}\simeq 20\%$).

Figure 6: Stückelino relic density in the case in which the NLSP is a wino
while the NNLSP is the lightest chargino. Red (darker) points denote models
which satisfy WMAP data. Upper left panel: $\Delta_{2}=20\%$. Upper right
panel: $\Delta_{2}=10\%$. Lower left panel: $\Delta_{2}=5\%$. Lower right
panel: $\Delta_{2}=1\%$.

##  6 Conclusions

We studied a possible dark matter candidate in the framework of our minimal
anomalous $U(1)^{\prime}$ extension of the MSSM [7]. In the decoupling limit
(34) and under the assumption $M_{0}\gg M_{S},M_{V^{(0)}}$ the Stückelino
turns out to be the LSP. Being an XWIMP the Stückelino annihilation cross
section is suppressed with respect to the typical weak interaction cross
sections. This implies that in order to satisfy the WMAP constraints on the
relic density we must have at least a NLSP almost degenerate in mass with the
Stückelino. We considered the case with two and three coannihilating particles
and we found some configuration which satisfies the WMAP constraints. The
results depend on the mass gap between the Stückelino and the NLSP. In the
exact degeneracy limit $\Delta_{2}\to 0$ the allowed models have a Stückelino
mass in the range $50\;\text{GeV}\lesssim M_{S}\lesssim 700\;\text{GeV}$ for
the bino-higgsino coannihilation case while $900\;\text{GeV}\lesssim
M_{S}\lesssim 2\;\text{TeV}$ for the wino-chargino coannihilation case. When
the mass gap is $\Delta_{2}\simeq 20\%$ the allowed models are those with
wino-chargino coannihilations and a Stückelino mass of
$100\;\text{GeV}\lesssim M_{S}<200$ GeV. Finally let us comment on the
differences between our scenario and that studied in the work [16]. In our
framework the $U(1)^{\prime}$ does not arise from a hidden sector and thus all
the MSSM fields can be charged under this extra abelian gauge group. This is
the most relevant feature which could also be detected experimentally (see for
example [6]). Moreover, in our scenario the Stückelino interactions are
suppressed with respect to the weak interactions due to the GS couplings while
in [16] the mechanism to suppress the couplings and give an XWIMP is provided
by the kinetic mixing between the $U(1)^{\prime}$ and $U(1)_{Y}$.

##  Appendix A Amplitude for $\lambda_{1}+\psi_{S}\to f\bar{f}$

In this Appendix we give some details of the amplitude computation for the
process $\lambda_{1}+\psi_{S}\to f\bar{f}$,

$\mathcal{M}=-ik^{\mu}\bar{v}_{S}\gamma_{5}\left[\gamma_{\mu},\gamma_{\nu}\right]u_{1}\left[eq_{f}C_{\gamma}\frac{\eta^{\nu\rho}}{k^{2}}\bar{u}_{f}\gamma_{\rho}v_{f}+\frac{g_{Z_{0}}}{2}C_{Z_{0}}\frac{\eta^{\nu\rho}}{k^{2}-M_{Z_{0}}^{2}}\bar{u}_{f}\gamma_{\rho}(v_{f}^{Z_{0}}-a_{f}^{Z_{0}}\gamma_{5})v_{f}\right]$
(71)

where $q_{f}$ denote the electric charges, $v_{f}^{Z_{0}}$ and $a_{f}^{Z_{0}}$
are the vectorial and axial couplings with $Z_{0}$,
$C_{\gamma}=C^{(1)}\cos\theta_{W}$, $C_{Z_{0}}=-C^{(1)}\sin\theta_{W}$ while
$k^{2}=s$ is the momentum of the intermediate gauge boson. The corresponding
square modulus is

$\displaystyle|\mathcal{M}|^{2}$ $\displaystyle=$
$\displaystyle-64\left[T_{a}\left(\frac{a_{f}C_{Z_{0}}g_{Z_{0}}}{k^{2}-M_{Z_{0}}^{2}}\right)^{2}+T_{v}\left(\frac{2C_{\gamma}eq_{f}}{k^{2}}+\frac{C_{Z_{0}}g_{Z_{0}}v_{f}}{k^{2}-M_{Z_{0}}^{2}}\right)^{2}\right]$
(72)

with

$\displaystyle T_{v}$ $\displaystyle=$ $\displaystyle
m_{f}^{4}(p_{\lambda_{1}}p_{S})+M_{1}M_{S}\Big{[}2m_{f}^{4}+3(p_{f}p_{\bar{f}})m_{f}^{2}+(p_{f}p_{\bar{f}})^{2}\Big{]}+$
$\displaystyle-(p_{f}p_{\bar{f}})\Big{[}(p_{\lambda_{1}}p_{f})(p_{f}p_{S})+(p_{\lambda_{1}}p_{\bar{f}})(p_{\bar{f}}p_{S})\Big{]}+m_{f}^{2}\Big{[}(p_{\lambda_{1}}p_{S})(p_{f}p_{\bar{f}})+$
$\displaystyle-2(p_{\lambda_{1}}p_{f})(p_{f}p_{S})-(p_{\lambda_{1}}p_{\bar{f}})(p_{f}p_{S})-(p_{\lambda_{1}}p_{f})(p_{\bar{f}}p_{S})-2(p_{\lambda_{1}}p_{\bar{f}})(p_{\bar{f}}p_{S})\Big{]}$
$\displaystyle T_{a}$ $\displaystyle=$
$\displaystyle\Big{[}(p_{\lambda_{1}}p_{\bar{f}})(p_{f}p_{S})+(p_{\lambda_{1}}p_{f})(p_{\bar{f}}p_{S})\Big{]}m_{f}^{2}-M_{1}M_{S}\left[m_{f}^{4}-(p_{f}p_{\bar{f}})^{2}\right]+$
(73)
$\displaystyle-(p_{f}p_{\bar{f}})\Big{[}(p_{\lambda_{1}}p_{f})(p_{f}p_{S})+(p_{\lambda_{1}}p_{\bar{f}})(p_{\bar{f}}p_{S})\Big{]}$

where $p_{\lambda_{1}}$, $p_{S}$, $p_{f}$ and $p_{\bar{f}}$ are the bino,
Stückelino and SM fermions 4-momenta respectively. Writing all the momenta in
function of $s$ and integrating over the solid angle we get

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\sigma=c_{f}\left(g_{1}^{2}b_{2}^{(1)}\right)^{2}\sqrt{s-4m_{f}^{2}}\times$
(74)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\times\frac{\Big{[}-2M_{1}^{4}+\left(4M_{S}^{2}+s\right)M_{1}^{2}-6M_{S}sM_{1}-2M_{S}^{4}+s^{2}+M_{S}^{2}s\Big{]}}{12\pi\left(M_{Z_{0}}^{2}-s\right)^{2}s^{5/2}\sqrt{M_{1}^{4}-2\left(M_{S}^{2}+s\right)M_{1}^{2}+\left(M_{S}^{2}-s\right)^{2}}}\times$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\times\Bigg{[}\left(2m_{f}^{2}+s\right)\Big{(}2\cos\theta_{W}eq_{f}\left(M_{Z_{0}}^{2}-s\right)+\sin\theta_{W}g_{Z_{0}}v_{f}s\Big{)}^{2}+\left(\sin\theta_{W}g_{Z_{0}}a_{f}\right)^{2}s^{2}\left(s-4m_{f}^{2}\right)\Bigg{]}$

Acknowledgements

A. R. would like to thank Prof. Michael Green and DAMTP for hospitality and
the Marie Curie Research Training Network “Superstring Theory”, contract MRTN-
CT-2004-512194 and contract ESF JD164 for financial support during the
completion of this paper. F.Fucito wants to thank N.Fornengo and P.Ullio for
discussions.

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# Dialogue-Contextualized Re-ranking for Medical History-Taking

Jian Zhu
University of British Columbia
<EMAIL_ADDRESS>
Ilya Valmianski
AuxHealth
<EMAIL_ADDRESS>
&Anitha Kannan
Curai Health
<EMAIL_ADDRESS>
Work done during an internship at Curai HealthWork done while at Curai Health

###### Abstract

AI-driven medical history-taking is an important component in symptom
checking, automated patient intake, triage, and other AI virtual care
applications. As history-taking is extremely varied, machine learning models
require a significant amount of data to train. To overcome this challenge,
existing systems are developed using indirect data or expert knowledge. This
leads to a training-inference gap as models are trained on different kinds of
data than what they observe at inference time. In this work, we present a two-
stage re-ranking approach that helps close the training-inference gap by re-
ranking the first-stage question candidates using a dialogue-contextualized
model. For this, we propose a new model, global re-ranker, which cross-encodes
the dialogue with all questions simultaneously, and compare it with several
existing neural baselines. We test both transformer and S4-based language
model backbones. We find that relative to the expert system, the best
performance is achieved by our proposed global re-ranker with a transformer
backbone, resulting in a 30% higher normalized discount cumulative gain (nDCG)
and a 77% higher mean average precision (mAP). As part of this work, we also
release pre-trained checkpoints for bi-directional and autoregressive S4
models trained on Wikipedia and PubMed data.

##  1 Introduction

History taking is a critical component of a medical encounter [15]. It
involves collecting relevant patient-reported information such as presenting
symptoms, patient concerns as well as the past medical, psychological and
social history. This information forms the basis of subsequent patient triage,
diagnosis, and care planning. While history taking is an important component
of the medical encounter, it is also one of the most time-consuming components
[37, 6] and when done incompletely can lead to triage, diagnostic, and
treatment errors [15].

Creating tools for automating portions of history taking has been an on-going
effort for more than five decades [45]. The simplest of such tools are static
pre-visit questionnaires that are now used widely in US healthcare. However,
static questionnaires tend to be long, ask not very relevant questions, and
are not customized to patients’ needs. More recently, there has been work on
building intelligent systems that can adjust questions based on patient
responses (see [41, 9] and citations therein). However, developing the medical
reasoning necessary for these systems is difficult. Existing approaches
include using reinforcement learning with simulated patients [38, 19],
supervised learning on clinical notes [41], and expert systems [9].

In all of the previous works, the medical reasoning system was built on top of
data that was a proxy for real doctor-patient interactions. This is because,
on the one hand, there is little available data consisting of doctor-patient
history-taking dialogue, on the other hand, the space of possible questions
asked during history-taking is very large. Thus, training a history-taking
model requires significant amounts of labeled interaction data. However, this
data is more readily available from indirect sources such as medical notes,
expert knowledge, or simulations. This creates a training-inference gap: the
data that is used to train the model is not fully representative of the data
that the model sees at inference time.

This training-inference gap has a significant impact on the quality of history
taking, especially since only a few questions can be asked in a given
encounter. This calls for an approach that reconciles the difficulty of
supporting a large set of potential questions to a small set of pertinent
questions attuned to the patient’s health issue, with only a small amount of
direct training data.

In this paper, we start with an expert system and show how to use a relatively
small amount of real doctor-patient dialogue data to close this training-
inference gap. We take inspiration from the information retrieval literature
where “retrieve and re-rank” is a popular paradigm for computationally
efficient retrieval of documents from a large corpus [28, 22]. In our case,
the “retrieve” part is performed by the expert systems which retrieves a list
of possible questions to ask the patient, and a dialogue-trained re-ranker
then “re-ranks” the possible questions. Because the re-ranking model takes the
original expert system’s candidate questions, it does not need to predict over
the space of all possible questions. Instead, it only needs to re-rank from a
much smaller subset, which greatly simplifies the machine-learning task. Our
model takes both the previous dialogue and the possible questions as free text
entries, which means that the system can operate even if the underlying expert
system is replaced with something else.

Our contributions are as follows:

  1. 1.

We propose a two-step approach to history-taking question selection where we
use an expert system to retrieve a list of candidate questions and then use a
machine-learned re-ranker to get the top question to ask.

  2. 2.

We propose a novel “global re-ranker” which embeds both the preceding dialogue
and candidate questions into a single long string. We then train long context
language models to predict the relevance of each question simultaneously.

  3. 3.

We perform a careful study of other re-rankers for this task. This includes
different architectures such as bi-encoder, cross-encoder, and autoregressive
re-rankers. We examine different long context models including S4 (bi-
directional and autoregressive), Nystromformer (bi-directional, variants with
Nystrom attention and with full attention), and LongT5 (autoregressive). We
examine the effect of different loss functions from the pointwise, pairwise,
and listwise families. Finally, we perform some ablation studies on the
context length and the initial retrieval ordering.

  4. 4.

We release checkpoints for S4 pre-trained both bidirectionally and
autoregressively on the English subset of
Wikipedia111https://huggingface.co/datasets/wikipedia and Pubmed PMC Open
Access Subset222https://www.ncbi.nlm.nih.gov/pmc/tools/textmining/ datasets.

  5. 5.

We find that our global re-ranker approach performs better than other more
traditional approaches. Furthermore, all re-rankers significantly improve the
original expert system performance. We also find that S4-based, while worse
than full-attention transformers, is competitive with the Nystrom-attention
transformer.

##  2 Generalizable Insights about Machine Learning in the Context of
Healthcare

One of the main challenges in using deep learning for healthcare is the lack
of large annotated datasets. Obtaining large amounts of annotated data is
costly and time-consuming because annotations need to be provided by trained
healthcare professionals. Recent works have successfully leveraged the
progress in the development of large language models that are trained on web-
scale data. In many tasks, including medical history taking discussed in this
paper, this approach introduces a training-inference gap: the data used to
train the model does not fully represent the data that the model sees at
inference time. In this context, our approach of retrieving a candidate set of
answers (we use an expert system as the base model to provide candidates, but
this can come from a large language model, too) and then using a learned
reranker based on small amounts of labeled data is a promising alternative. As
we show in this paper, such a reranker can be trained from public data sources
and then fined tuned to the task.

##  3 Related work

We study re-ranking history-taking recommendations based on doctor-patient
dialogue. These dialogues tend to be long and exceed the typical token-length
limits of transformer models. As such, there are two bodies of literature
relevant to this work. § 3.1 discusses work on modern neural long-range
language models that are able to encode the entire doctor-patient
conversation. § 3.2 discusses work on re-ranking algorithms that can take the
encoded dialogue and use it to re-rank history-taking questions.

###  3.1 Long-range transformers

Transformers [42] have become the mainstream architecture for natural language
processing. With the self-attention mechanism, transformers can attend to all
tokens in a sequence simultaneously, thereby being more powerful than
classical architectures like convolutional [21] or long short-term memory
(LSTM) [16] networks. However, due to its $O(n^{2})$ complexity, the original
transformer cannot process long sequences efficiently. Popular pre-trained
language models such as BERT [10] and RoBERTa [24] can only process up to 512
tokens. Efforts have been made to reduce the computational complexity of
transformers, as a variety of efficient transformers that can process long
text sequences have been proposed, such as Reformer [20], Linformer [44],
Longformer [2], BigBird [49], Performer [8], Nystromformer [48], LongT5 [14],
etc. In addition to transformers, alternative approaches have also been shown
to be promising for processing very long sequences, notably state-space models
[12, 13, 26]. The Structured State Space Sequence model (S4) [12] has
significantly outperformed many long-range transformers in the Long Range
Arena benchmark [39]. In this paper we utilize the Nystromformer[48] and the
S4 model[12] as the possible non-autoregressive backbones and LongT5 as an
autoregressive backbone.

###  3.2 Re-ranking

Modern information retrieval (IR) or question answering (QA) systems are
usually divided into two stages [28, 22]. In the first stage, given a query, a
large number of documents are retrieved using an efficient method. In the
second stage, a computationally intensive but more accurate method is used to
re-rank documents retrieved in the first stage. This is analogous to our
problem statement where we use an expert system to retrieve a set of relevant
history-taking questions, and then use a re-ranking algorithm as the second
stage. Relevant related work addresses both the architectural choices that can
be made for the re-rankers, as well as the loss functions used to train them.

#### Architectures for second-stage re-ranking.

There are three main type of architectures (1) bi-encoders [22, 29, 36, 40]
(2) cross-encoders [22, 29, 17] and (3) autoregressive re-rankers [30, 33,
27]. In the bi-encoder architecture, the query and the candidate document are
encoded into vector representations by two separate encoders (though these two
encoders can share the same weights [36]). The relevance score is calculated
as the distance between the two vector representations. In a cross-encoder, a
query and a candidate document are concatenated together and fed into the
cross-encoder in a single pass. In most use cases, cross-encoders outperform
bi-encoders in document retrieval and ranking [40, 17]; however, bi-encoders
are usually more efficient than cross-encoders, as all documents in a given
corpus can be pre-computed and stored as dense embeddings for retrieval,
thereby avoiding repeated computations [36]. Recent sequence-to-sequence
models such as T5 [35] have also been applied to autoregressive re-ranking. In
this approach, the query and the documents are usually encoded by the encoder
and the decoder either predicts whether the document is relevant [30] or
directly generates the retrieved text in response to the query [27].

#### Scoring functions for ranking.

There are also several scoring paradigms possible for ranking: (1) “pointwise”
scores where the relevance of a query is computed on a per-document basis
(similar to cross-encoders) [29], (2) “pairwise” scores where documents are
ranked relative to each other in pairs [29], and (3) “listwise” scores where a
list of candidates are ranked simultaneously [23]. Prior studies show that
pairwise and listwise approaches tend to outperform pointwise approaches [29,
33, 27, 50, 7]. In this study, we compare all three (‘pointwise’, ‘pairwise’,
and ‘listwise’) approaches for re-ranking history-taking questions but mainly
focus on listwise approaches.

Figure 1: (a) Bi-encoder, (b) the cross-encoder and (c) the proposed global
reranker.

#### Connection to our proposed global re-ranker.

In this paper we propose a novel re-ranker approach we call “global re-
ranker.” In this approach, all candidate documents are concatenated into a
single input that is then processed by a long context language model. For
schematic comparison between bi-encoder, cross-encoder, and global re-ranker
please see Figure 1. For a more detailed description of the method see § 5.
Concatenating pairs of documents into a single string has been previously done
both in bi-directional [29] and autoregressive [33] paradigms. These models
only investigate pairwise scoring due to length constraints imposed by the
pre-trained transformer. For listwise ranking with more than two documents,
previous approaches focused on ranking only the extracted embeddings [7, 1],
which doesn’t model the deep semantic relationships between candidate
documents.

##  4 Closing the train-inference gap with re-ranking

Figure 2: Overview of the proposed two-stage history-taking workflow. An
expert system suggests candidate questions based on relevant entities
extracted from the dialogue. A machine-learned deep neural network re-ranker
then re-ranks the candidate questions based on the dialogue text.

An overview of our approach to closing the train-inference gap in an existing
history-taking system can be seen in Figure 2. We first use an expert system
to suggest relevant history-taking questions and then use a deep neural
network contextualized by the entire doctor-patient dialogue to re-rank expert
system suggestions.

The goal of re-ranking is, given the prior dialogue context $\boldsymbol{d}$
and a list of $n$ candidate history-taking questions
$Q=[\boldsymbol{q}_{1},\boldsymbol{q}_{2},\dots,\boldsymbol{q}_{n}]$, to
generate a new list $Q^{\prime}$ which consists of (possibly reordered)
questions from $Q$ such that the higher relevance questions appear earlier in
the sequence. In our case, the candidate questions are generated using an in-
house Expert System, and the ground truth labels
$\boldsymbol{y}=[y_{1},y_{2},\dots,y_{n}],y_{i}\in\\{0,1\\}$ represent whether
a doctor asked a given recommended question ($1$ if the question was asked,
$0$ if the question was not asked). A doctor may ask multiple questions at the
same time, thus multiple elements of $\boldsymbol{y}$ can have a value of $1$,
see § 6.1 for more details on how the ground truth is produced. Finally, in
all of the models studied in this work, the re-ranking is achieved by
assigning scores $\boldsymbol{s}=[s_{1},s_{2},\dots,s_{n}]$ to each question
in $Q$, and then constructing $Q^{\prime}$ by reordering using scores in
$\boldsymbol{s}$.

##  5 Global re-ranker

We propose the global re-ranker, an accurate and efficient listwise re-ranking
method. In this approach (see Figure 1(c) for a schematic), the history-taking
dialogue and all candidate history-taking questions are concatenated into a
single text input, using which the model then assigns the ranking scores to
all questions simultaneously. The global re-ranker directly encodes all texts
through the language model, thereby ensuring deep semantic interactions not
only between the dialogue and the candidate questions but also between all
candidate questions.

The input text to the global re-ranker is the concatenation of both the
dialogue context and all the candidate questions: [CLS] $\boldsymbol{d}$ [SEP]
$\boldsymbol{q}_{1}$ [MASK1][SEP] $\boldsymbol{q}_{2}$ [MASK2][SEP]
…$\boldsymbol{q}_{n}$ [MASKn][SEP], where the [SEP] token is used to mark the
boundaries of candidate questions. The [MASKi] token is the pooling token for
the preceding question $\boldsymbol{q}_{i}$. For each pooling token [MASKi],
the global reranker predicts a score $s_{i}$, which represents the relevance
for $\boldsymbol{q}_{i}$. We also added type embeddings to every input tokens
to indicate whether it belongs to the dialogue or the candidate questions. The
actual number of candidate questions provided by the expert system ranged from
3 to 40.

While self-attention itself does not assume any inherent order of the input
sequence, pretrained transformer models usually encode the text sequentially
due to the presence of positional embeddings. In the current task, it is
expected that a language model learns the sequential relations between words
within $\boldsymbol{d}$ and $\boldsymbol{q}_{i}$. From our ablation
experiments (see § 7.2), we found that the best performance is achieved when
the model is agnostic to the order of input questions
$[\boldsymbol{q}_{1},\boldsymbol{q}_{2},\dots,\boldsymbol{q}_{n}]$. In order
to remove the positional bias, we reset the positional embedding when each new
question starts.

We selected three different neural architectures to implement the global
ranker, all of which can process long textual sequences. The first two
approaches are based on the Nystromformer [48], which was originally proposed
to be an efficient transformer. We experiment with Nystromformer with both
Nystrom attention turned on and turned off (in which case it uses full
attention). We use Nystromformer as the base of our “full attention”
transformer because this enables us to leverage the pretrained Nystromformer
checkpoints that had been trained on long texts and retain the good
performance of full attention. We learned from pilot experiments that other
efficient transformers such as Longformer [2] failed to converge. The third
neural architecture is a state-space model, S4, which has been shown to
process long sequences more effectively than many transformers [12].

To train the global re-ranker, we compared a variety of loss functions across
point-wise, pair-wise and listwise approaches in the learning-to-rank
framework [23]. The point-wise baseline was trained with binary cross-entropy.
For pairwise loss functions, we tested the RankNet [3] and LambdaRank [4]. The
listwise loss functions we used were ListNet [5], ListMLE [47], ApproxNDCG
[34] and NeuralNDCG [32], the latter two of which directly optimized the
Normalized Discounted Cumulative Gain (NDCG) metrics.

##  6 Experiments

###  6.1 Data

The medical dialogue data was collected from a portion of real doctor-patient
interactions collected on our text-based medical service platform. In a
typical interaction, the physician asks a series of history-taking questions
that can be entered either as free text or selected from a list of
recommendations. These recommendations are made using the Expert System that
forms the first stage in our proposed workflow. At each dialogue turn where
recommended questions are asked, the doctor selected questions are marked as
relevant and the not-selected questions are marked as irrelevant. This forms a
natural dataset of doctor annotated selections on which we train our re-
rankers.

The dataset consists of 13071 encounters. We filtered non-history-taking
dialogue turns using in-house dialogue segmentation model, similar to [43].
The detailed statistics of our data are displayed in Table 1.

|  Train |  Dev |  Test
---|---|---|---
Num. Encounters | 12105 | 311 | 655
Num. Samples | 26106 | 626 | 1361
Avg. Length of Dialog. | 287.2 | 374.7 | 288.8
Num. Selected Questions | 4.0 | 3.8 | 4.1
Num. Candidate Questions | 27.9 | 26.6 | 28.1
Avg. Length of Questions | 8.0 | 8.1 | 8.0
Table 1: Statistics of different data splits. Text lengths were calculated
based on words.

###  6.2 Metrics

For evaluation, we adopted two common ranking metrics, normalized discounted
cumulative gain (nDCG)[18] and mean average precision (mAP) [22]. The mAP
assumes binary relevance whereas nDCG can work with both binary and continuous
relevance. Specifically for global re-rankers, the average metrics over 5
repeated runs of evaluations were reported. In each run, the order of
candidate questions fed to the global re-ranker was randomly reshuffled to
mitigate positional biases.

###  6.3 Baseline approaches

In addition to the global ranker, we also implement three widely adopted
baseline ranking approaches: bi-encoder, cross-encoder, and autoregressive re-
ranker.

#### Bi-encoder.

In the bi-encoder architecture (see Figure 1(a)), the dialogue query and the
candidate questions are encoded by two separate encoders $f_{D}$ and $f_{Q}$,
and the relevance score between the two resulting vector representations are
computed with cosine similarity. The bi-encoder learns an embedding space
where the dialogue representation is close to the most relevant questions
while being distant from less relevant questions. The training objective is to
minimize the InfoNCE loss function [31] through contrastive learning with 7
negatives randomly sampled from the list of recommended candidate questions by
the Expert System. The temperature parameter of the InfoNCE loss was set to
0.05 throughout the training [11].

#### Cross-encoder.

In the cross-encoder architecture (see Figure 1(b)) the prior dialogue is
concatenated with a candidate question. The cross-encoder $f_{C}$ assigns a
relevance score to this candidate question using a classification head on top
of the contextual representation of the dialogue and the query. We consider
transformers and S4-based models. For transformers, the [CLS] token is treated
as the contextual representation. For the bi-directional S4 re-rankers, we use
average pooling of the last layer to obtain the contextual representations.
All cross-encoder variants are trained with the binary cross-entropy loss.

#### Autoregressive re-ranker.

We also consider autoregressive re-rankers [30, 33]. For a transformer
baseline, we use a pre-trained LongT5 [14]. The query and the document are
concatenated together to form the input sequence: Query: $d$ Document: $q_{i}$
Relevant:, which is fed into the encoder. The decoder then predicts true for
relevant documents or false for irrelevant documents. During inference, a
softmax function is applied to the logits of the true and the false tokens to
normalize the results across multiple queries.

For autoregressive S4, when we followed the Long-T5 method, we found it to
highly unstable and failed to converge, similar to what was found in the
literature [30] on its dependency certain keywords e.g., true/false.
Therefore, we followed the same setting as in the cross-encoder, except that
the underlying model is autoregressive rather than bi-directional. Here, the
concatenated dialogue and a candidate question are fed into the S4 re-ranker
and the average pooling of the last layer is classified as either relevant or
irrelevant through a classification head.

###  6.4 Implementation

#### S4 model pretraining.

The S4 model was based on the original implementation of S4 language
model[12], in which the S4 layers were used as a drop-in replacement for the
self-attention layers in a typical transformer. We implemented a 12-layer
bidirectional and autoregressive S4 models. We set the hidden dimensions to
768 in order to match the parameter count of mainstream pretrained
transformers (such as BERT-base [10]), and the number of state-space machines
(SSM) to 128 with 64 states for each SSM.

Both bidirectional S4 and autoregressive S4 models were pretrained on large-
scale texts. The autoregressive S4 was pretrained with the casual language
modeling task on the whole English subset of Wikipedia. The second iteration
of pretraining, initialized with the pretrained Wikipedia checkpoint, was on
the whole Pubmed PMC Open Access Subset. The bidirectional S4 models were
pretrained on the same datasets but with the mask language modeling task using
the same masking settings as in BERT [10]. The maximum sequence length for
pretraining was set to 8192 and the effective batch size was 256.

All models were optimized with AdamW optimizer with a learning rate of 1e-4
and the learning rate was dynamically adjusted using the Cosine Scheduler with
a warm-up step of 1000. The pretraining took place on 8$\times$RTX 3090 GPU
with 24GB of memory. The training was stopped when the evaluation loss stopped
to decrease ($\sim$12k steps for all models). The autoregressive and bi-
direction checkpoints pre-trained on these datasets will be released together
with this paper.

#### Transformer implementation.

Transformer models were all implemented through the Transformers package [46]
with default dimensions. The autoregressive model was LongT5 [14] initialized
by the long-t5-tglobal-base checkpoint. Other transformers were based on the
Nystromformer [48] with initialization from the public checkpoint uw-
madison/nystromformer-4096.

#### Re-ranker training.

For global re-rankers, the maximum input length was set to 4096 with an
effective batch size of 32. For other models, the effective batch size was 64
and the maximum length was 2048, as this length was enough to cover almost all
of the data samples. Models were trained with a maximum of 5 epochs and only
the model with the best validation performance was kept. All models were
trained using the AdamW optimizer [25] with a learning rate of 5e-5. We used a
cosine scheduler with a warm-up step of 1000 to automatically adjust the
learning rate during training. All ranking models were trained on a single
V100 GPU with 16GB of memory.

##  7 Results

###  7.1 Main results

Model | Dev | Test
---|---|---
nDCG | mAP | nDCG | mAP
Expert System (Baseline) | 0.592 | 0.383 | 0.570 | 0.349
Bi-encoder (Transformer) | 0.690 | 0.548 | 0.677 | 0.531
Cross-encoder |  |  |  |
Transformer | 0.718 | 0.584 | 0.706 | 0.566
Nystromformer | 0.653 | 0.496 | 0.654 | 0.497
Bidirectional S4 (Wiki Pretraining) | 0.648 | 0.490 | 0.641 | 0.481
Bidirectional S4 (Pubmed Pretraining) | 0.643 | 0.483 | 0.630 | 0.464
Autoregressive Re-ranker |  |  |  |
LongT5-base | 0.690 | 0.546 | 0.678 | 0.529
Autoregressive S4 (Wiki Pretraining) | 0.658 | 0.502 | 0.648 | 0.490
Autoregressive S4 (Pubmed Pretraining) | 0.654 | 0.498 | 0.642 | 0.484
Global Re-ranker
Transformer |  |  |  |
\+ Pointwise loss: BCE | 0.744 | 0.618 | 0.743 | 0.618
\+ Pairwise loss: RankNet | 0.739 | 0.612 | 0.735 | 0.603
\+ Pairwise loss: LambdaLoss | 0.739 | 0 616 | 0.739 | 0.612
\+ Listwise loss: ListNet | 0.737 | 0.609 | 0.740 | 0.610
\+ Listwise loss: ListMLE | 0.727 | 0.597 | 0.721 | 0.587
\+ Listwise loss: ApproxNDCG | 0.701 | 0.555 | 0.697 | 0.550
\+ Listwise loss: NeuralNDCG | 0.742 | 0.617 | 0.741 | 0.612
Nystromformer |  |  |  |
\+ Pointwise loss: BCE | 0.684 | 0.537 | 0.678 | 0.530
Bidirectional S4 (Wiki Pretraining) |  |  |  |
\+ Pointwise loss: BCE | 0.667 | 0.516 | 0.663 | 0.510
Bidirectional S4 (PubMed Pretraining) |  |  |  |
\+ Pointwise loss: BCE | 0.697 | 0.556 | 0.670 | 0.518
Table 2: Results of reranking experiments.

Our main results are summarized in Table 2. All neural re-ranking models
outperform the baseline Expert System in both metrics, suggesting that re-
ranking does up-rank the more relevant history-taking questions. Among the
neural baselines, the transformer-based cross-encoder outperforms the bi-
encoder, which is consistent with previous findings [29]. Surprisingly, the
LongT5 autoregressive re-ranker, despite having more parameters (220M
parameters), also performs worse than the cross-encoder ($\sim$110M
parameters).

The best performance is achieved by the global re-ranker for both transformer
and S4 architectures, regardless of the loss functions chosen. Among the
various loss functions, the pointwise binary cross-entropy (BCE) performs the
best. Our hypothesis is that since our ground truth relevance scores are
binary rather than continuous, the current task does not make full use of the
listwise loss functions.

The effectiveness of the global re-ranker lies in the fact that it attends to
the semantic interactions not only between the dialogue and the candidate
questions but also between the candidate questions themselves. This allows the
model to exploit the dependencies between history-taking questions, such as
co-occurrence statistics, to improve ranking outcomes.

It is also worth noting that, despite its outstanding performance in some long
sequence processing benchmark [12], S4 still lags behind transformers in the
current task. One reason could be that the S4 model here has only been pre-
trained on a comparatively small amount of texts, while transformers have been
pre-trained on huge amounts of texts. Furthermore, the text sequences in our
task range from a few hundred to about three thousand words, which might not
be long enough for S4 to reveal its full potential.

###  7.2 Ablation analysis

We conducted ablation analyses on the global re-ranker to assess the impact of
dialogue context length, the effect of type embeddings, and the effect of
shuffling candidate question order. The results are displayed in Table 3.

#### Context length ablations.

When ablating on context length, only the last $N$ tokens of the dialogue were
considered (full model is 4096 tokens, ablation are 3072, 2048, and 1024
tokens). While most of the text sequences were shorter than 1000 tokens,
truncating texts still decreases test performance on some text sequences that
are particularly long (longer than 1024), as some important information could
be removed. In general, the global re-ranker benefits from getting more
dialogue contexts, though this benefit seems to diminish after expanding to
more than 2048 tokens.

#### Effect of position and type embeddings.

We find that the removal of type embeddings (which are learned embeddings that
differentiate whether the token is from dialogue or a candidate question) has
almost no impact on the test performance. We reset the positional embeddings
for each candidate questions in the input sequence, as this might help the
model learn to be agnostic to the order of questions. We trained a model that
used sequential positional embeddings for the input sequence. It turned out
that positional embeddings played a minor role in training the global re-
ranker.

#### Effect of shuffling.

We tested the importance of permutation invariance with regard to the order of
input candidate questions. The list of candidate questions
$[\boldsymbol{q}_{1},\boldsymbol{q}_{2},\dots,\boldsymbol{q}_{n}]$ were
concatenated with the prior dialogue as an input to the model. We found that
while the expert system should produce questions in order or relevance,
performance was significantly higher when the model was trained with shuffled
order. We believe that this forces the model to learn to re-rank the questions
without falling back to the original order of the candidate questions.

Ablation | Dev | Test
---|---|---
nDCG | mAP | nDCG | mAP
\- Maximum length: 4096 (Full) | 0.744 | 0.618 | 0.743 | 0.618
\- Maximum length: 3072 | 0.741 | 0.614 | 0.739 | 0.611
\- Maximum length: 2048 | 0.746 | 0.622 | 0.747 | 0.622
\- Maximum length: 1024 | 0.747 | 0.623 | 0.737 | 0.609
\- No type embedding | 0.733 | 0.610 | 0.732 | 0.607
\- Sequential position embedding | 0.749 | 0.625 | 0.741 | 0.613
\- No random shuffling of questions | 0.515 | 0.313 | 0.523 | 0.319
Table 3: Results of ablation studies on global re-ranker

##  8 Discussion

In this work, we address an important problem of closing the training-
inference gap for automated medical history-taking. Our approach, inspired by
modern neural information retrieval systems, has two stages: (1) we use an
expert system to suggest a list of candidate questions (out of possible
thousands), (2) we train a machine-learned re-ranking model to re-rank expert
system-suggested questions based on the free text of the doctor-patient
dialogue.

To perform re-ranking (stage 2), we introduce a new approach which we call
“global re-ranker”, and compare it to existing neural baselines. We also
explore several language model back-bones including various transformers and
structure-state-space (S4) models333As part of this publication, we release
bi-directional and autoregressive S4 checkpoints pre-trained on the English
Wikipedia and Pubmed PMC Open Access Subset.. We find that while all neural
re-ranking models outperform the original expert system, the global re-ranker
with a full-attention transformer backbone performs the best with a 30%
increase in nDCG and 77% increase in mAP over the first-stage recommendations.

While our results directly show the effectiveness of training a re-ranking
model on top of an expert system for history taking, we believe this approach
can also be applied to other decision support systems. The conditions under
which this approach is beneficial are the following: (1) There exists a
scoring system that has a training-inference gap (2) The space of possible
predictions is very large, and as such would require a lot of data to machine-
learn from scratch. One example beyond history-taking where we believe these
conditions are satisfied is medical diagnosis prediction. There are many
expert-system-derived diagnosis models, and training a diagnosis model from
scratch can be difficult as the space of possible diagnosis is very large. Re-
ranking could be used to close the gap between an off-the-shelf diagnostic
expert system and the practice’s actual patient population outcomes.

#### Limitations

This work is still limited in several ways. While our proposed global re-
ranker had exhibited best overall performance over other ranking models, it is
still computationally inefficient as the full attention transformers have
quadractic computational complexity in processing long sequence. This will
become a more serious bottleneck as the dialogue gets longer or the number of
candidate questions increases. Secondly, the global re-ranker only learns the
association between history-taking questions and the dialogue contexts from
languages but it does not have the underlying medical knowledge. It will be
paramount to augment such models with real medical knowledge such that it
makes more informed decisions and does not biased against low frequency long-
tail history-taking questions. In the future, we plan to investigate more
effective approaches to encode long textual contexts and inject knowledge into
language mdoels.

###  8.1 Ethics

This work was done as part of a quality improvement activity as defined in
45CFR §46.104 (d)(4)(iii) – secondary research for which consent is not
required for the purposes of “health care operations.”

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