[a]Fabian Zierler

# Singlet Mesons in Dark Sp(4) Theories

Jong-Wan Lee  Axel Maas  Felix Pressler

###### Abstract

We explore some aspects of $Sp(4)$ gauge theory with two fundamental fermions
in the context of composite Goldstone Dark Matter. We present preliminary
lattice results for the mass of the pseudoscalar iso-singlet meson
$\eta^{\prime}$ using unimproved Wilson fermions and the standard plaquette
action. We find that the $\eta^{\prime}$ is slightly heavier than the
pseudoscalar non-singlets $\pi$ and lighter than the vector mesons $\rho$ for
multiple ensembles with $m_{\pi}/m_{\rho}\geq 0.77$. This pattern is
potentially relevant for Beyond the Standard Model physics model building.
Furthermore, we show that for $N_{f}=1+1$ flavours the disconnected
contributions to the unflavoured pseudoscalar $\pi^{0}$ are small. We
supplement this measurement by a calculation of the $\pi\pi$ scattering length
$a_{0}$ which shows that the ensemble studied might be phenomenologically
relevant for Dark Matter models such as the Strongly Interacting Massive
Particles paradigm.

##  1 Introduction

The structure of symplectic gauge groups $\text{Sp}(2N)$ with fermions has
provided interesting opportunities for physics beyond the Standard Model
(BSM). For a sufficiently small number of fermions they are asymptotically
free, confining and exhibit (spontaneous) chiral symmetry breaking as in QCD
and unitary gauge groups in general. Since the fundamental representation of
Dirac fermions is pseudo-real, however, their global symmetries differ from
those of a unitary gauge group $\text{SU}(N_{c}\geq 3)$. For $N_{f}$ flavors
it leads to the symmetry breaking pattern
$\text{SU}(2N_{f})\rightarrow\text{Sp}(2N_{f})$ [1], which is of considerable
interest for BSM physics. In the case of $N_{f}=2$, this provides a
realization of the $\text{SU}(4)/\text{Sp}(4)$ coset for composite Higgs
models (see e.g. Ref. [2] for a review). Furthermore, the resulting five
(pseudo-)Goldstone bosons play the role of Strongly Interacting Massive
Particles (SIMP) in the context of dark matter (DM), where a large number of
colours appear to be phenomenologically more relevant [3] motivating lattice
investigations beyond $\text{SU}(2)$.

The spectrum of flavour-singlet mesons is particularly interesting for DM
models as they are not protected by flavour symmetry from decays into Standard
Model (SM) particles. Of particular interest is the pseudoscalar singlet
$\eta^{\prime}$ which is associated with the $U_{A}(1)$ axial anomaly. In
$\text{SU}(3)$ gauge theory with two flavours this state is lighter than the
vector mesons [4] and is expected to be degenerate with the other non-singlet
pseudoscalar mesons in the $N_{c}\to\infty$ limit.

On the lattice the non-singlet spectrum of mesons in the $\text{Sp}(4)$ two-
flavour theory has been studied in quenched approximation [5], with dynamical
degenerate fermions [6] and with dynamical non-degenerate fermions [7] (see
also Ref. [8] for a recent review on $\text{Sp}(2N)$ lattice gauge theory).
Similar studies have been performed in $\text{SU}(2)$ gauge theory ( e.g. Ref.
[9]) and first investigations on the spectrum of singlet mesons have been
carried out [10]. In this contribution we present first results on the
$\eta^{\prime}$ meson in $\text{Sp}(4)$ gauge theory with two fundamental
flavours for degenerate and non-degenerate fermions. We also report on the
disconnected contributions to the the unflavoured pseudoscalar $\pi^{0}$ meson
in the non-degenerate case. In a SIMP scenario the fermions need to be massive
to provide a pseudo-Goldstone DM model. However, it is unclear how heavy the
fermions have to be in order for the theory to be a viable DM candidate. We
therefore perform an exploratory investigation of the length of $\pi\pi$
scattering in the isospin $I=2$ channel along the lines of a similar study in
$\text{SU}(2)$ gauge theory [11] and match the results to experimental
constraints.

##  2 Lattice setup

We study four-dimensional $\text{Sp}(4)$ gauge theory with two fundamental
fermions on the Euclidean lattice using the Wilson plaquette action and
unimproved Wilson fermions. All our calculations have been performed with the
HiRep code [12] extended to support symplectic gauge theories [13, 14]. To
generate gauge configurations we use the Hybrid Monte Carlo (HMC) algorithm
for degenerate fermions and the rational HMC (RHMC) algorithm for non-
degenerate ones. In the latter case we monitored the lowest eigenvalue of the
Dirac operator as its determinant is not guaranteed to be positive definite.
For the masses considered in this work we do not observe any hints of a sign
problem [7], where the details of the ensembles are found in Table 1.

###  2.1 Pseudoscalar Mesons

For $N_{f}=2$ with degenerate fermions mesons appear either in $10$-plets,
$5$-plets or as singlets under the global $\text{Sp}(4)_{F}$ flavour symmetry
which arises after the symmetry breaking through fermion condensates and/or
fermion masses [1, 15]. The theory has a $5$-plet of pseudoscalar mesons which
are the (pseudo-)Goldstones of global symmetry breaking, and an additional
flavour singlet $\eta^{\prime}$ which is connected to the axial anomaly. When
introducing strong isospin breaking by setting $m_{u}\neq m_{d}$, the $5$-plet
decomposes into a $4$-plet of flavoured mesons and a flavour singlet $\pi^{0}$
under the remaining $\text{SU}(2)_{u}\times\text{SU}(2)_{d}$ global flavour
symmetry. Following the standard procedure we extract their masses from the
meson propagator $\langle O(n)\bar{O}(m)\rangle$. Here, $n,m$ are labels for
lattice sites. We are interested in operators of the form

$\displaystyle O_{1}(n)$ $\displaystyle=\bar{u}\Gamma d,$ (1) $\displaystyle
O_{2}(n)$ $\displaystyle=\left(\bar{u}\Gamma u-\bar{d}\Gamma
d\right)/\sqrt{2},$ (2) $\displaystyle O_{3}(n)$
$\displaystyle=\left(\bar{u}\Gamma u+\bar{d}\Gamma d\right)/\sqrt{2},$ (3)

where $u$ and $d$ denote two distinct Dirac flavours and $\Gamma$ is an
appropriate gamma structure. For $\Gamma=\gamma_{5}$ these operators probe the
pseudoscalar mesons. The four flavoured pseudoscalars $\pi^{\pm}$ are probed
by $O_{1}$, $\pi^{0}$ by $O_{2}$, and the flavour singlet $\eta^{\prime}$ by
$O_{3}$. The difference between the $\pi^{0}$ and $\eta^{\prime}$ arises in
the form of disconnected diagrams. Physically, they correspond to pure gluonic
propagation. In the isospin-symmetric limit these contributions cancel for
$\pi^{0}$ and its non-singlet nature for $m_{u}=m_{d}$ becomes manifest.

Other singlet states exist for any $\Gamma$ where the non-singlets for
degenerate fermions appear in a $5$-plet. However, no singlet states appear
for other gamma structures, such as $\Gamma=\gamma^{\mu}$ interpolating the
vector mesons $\rho$. States involving more fermions and/or additional gluons
are still possible. The state that would source the $\phi/\omega$ meson in QCD
is part of the $\rho$ $10$-plet and under isospin breaking the $10$-plet
decomposes into a flavoured $4$-plet and an unflavoured $6$-plet under the
remaining $\text{SU}(2)_{u}\times\text{SU}(2)_{d}$ symmetry. Another singlet
of interest is the scalar flavour-singlet which corresponds to the
$\sigma/f_{0}$ in QCD. This meson has the same quantum numbers as the vacuum.
Therefore, its correlator contains an additional constant several orders of
magnitude larger than the exponentially decaying signal with the ground state
energy, the mass of the scalar meson, at late times. As an adequate
subtraction of this constant is needed, we defer the study of the scalar
flavour-singlet to future work.

In general, the disconnected contributions suffer more strongly from noise and
thus higher statistics is needed. Additionally, they require all-to-all
propagators. We use noisy [16], diluted [17] sources and find that spin
dilution and even-odd dilution on $Z_{2}\times Z_{2}$ noise sources with
$n_{\text{hit}}=128$ are sufficient to obtain a signal for the pseudoscalar
disconnected diagrams for the ensembles listed in Table 1. From this we build
an unbiased estimator for the disconnected pieces as outlined in Ref. [10].

The disconnected contributions still suffer from substantial noise and we only
find a signal for a few timeslices at small and intermediate $t$. However, it
has been noted that the they receive relatively small corrections from the
excited state contamination compared to the connected diagrams. Hence, we
first remove the excited state contributions present in the connected pieces.
This is done by fitting the connected pieces – which have a good signal over
all timeslices – at large $t$ by a single exponential of the form

$\displaystyle C_{\pi,\text{conn}}^{\text{1exp}}(t)$
$\displaystyle=A_{0}\left(e^{-m_{\text{conn}}t}+e^{-m_{\text{conn}}(T-t)}\right),$
(4)

and use this to build an improved correlator for the $\eta^{\prime}$ and the
$\pi^{0}$ [18]

$\displaystyle C_{\eta^{\prime}}^{\text{1exp}}(t)$
$\displaystyle=C_{\pi,\text{conn}}^{\text{1exp}}(t)+C_{\eta^{\prime},\text{disc.}}(t),$
(5) $\displaystyle C_{\pi^{0}}^{\text{1exp}}(t)$
$\displaystyle=C_{\pi,\text{conn}}^{\text{1exp}}(t)+C_{\pi^{0},\text{disc.}}(t).$
(6)

With the excited, connected contributions removed we obtain plateaus in the
effective masses at much earlier time $t$. In the left panel of Fig. 1 we
depict the effective masses of the singlet mesons with and without excited
state subtraction in the connected pieces for an exemplified case of
degenerate fermions. From the plateaus in the effective mass we determine the
range of $t$ for which we perform a single-exponential fit to the improved
correlators. The masses extracted using this method are found in Table 1.
Similarly, we apply the subtraction method to the non-degenerate case for both
pseudoscalar singlets $\pi^{0}$ and $\eta^{\prime}$ as shown in the right
panel of Fig. 1.

Figure 1: (left) Effective masses of the $\eta^{\prime}$ correlator before
subtraction of excited state contributions (orange squares) and after
subtraction (blue circles). We only show data points for which we can extract
a signal. The subtracted effective mass shows a clear plateau over several
timeslices. (right) Effective masses for the pseudoscalar singlets $\pi^{0}$
and $\eta^{\prime}$ in the $N_{f}=1+1$ theory after excited state subtraction.
Additionally, we give the masses for the unflavoured vector mesons and the
$\pi^{0}$ in the connected-only approximation denoted by $m(\pi^{0}_{c})$.

$\beta$ | $m_{0}$ | $L$ | $T$ | $n_{\text{conf}}$ | $m_{\pi}$ | $m_{\rho}$ | $m_{\eta^{\prime}}$
---|---|---|---|---|---|---|---
6.9 | -0.87 | 12 | 24 | 309 | 0.743(3) | 0.852(4) | 0.780(13)
6.9 | -0.89 | 12 | 24 | 575 | 0.631(3) | 0.753(5) | 0.669(14)
6.9 | -0.90 | 12 | 24 | 2767 | 0.5664(15) | 0.703(3) | 0.615(9)
6.9 | -0.90 | 14 | 24 | 932 | 0.5642(19) | 0.698(4) | 0.614(12)
6.9 | -0.91 | 14 | 24 | 334 | 0.484(6) | 0.624(12) | 0.57(3)

$\beta$ | $m_{0}^{1}$ | $m_{0}^{2}$ | $L$ | $T$ | $n_{\text{conf}}$ | $m_{\pi^{0}_{c}}$ | $m_{\pi^{0}}$ | $m_{\pi^{\pm}}$ | $m_{\rho^{0}}$ | $m_{\rho^{\pm}}$ | $m_{\eta^{\prime}}$
---|---|---|---|---|---|---|---|---|---|---|---
6.9 | -0.90 | -0.75 | 14 | 24 | 599 | 0.804(2) | 0.812(4) | 0.918(2) | 0.928(2) | 1.003(2) | 0.821(7)
6.9 | -0.90 | -0.80 | 14 | 24 | 599 | 0.760(3) | 0.765(5) | 0.828(2) | 0.884(2) | 0.922(2) | 0.778(9)
6.9 | -0.90 | -0.85 | 14 | 24 | 291 | 0.691(2) | 0.691(3) | 0.714(2) | 0.811(2) | 0.822(3) | 0.719(6)
6.9 | -0.90 | -0.89 | 14 | 24 | 255 | 0.597(3) | 0.600(5) | 0.600(4) | 0.724(5) | 0.728(7) | 0.63(1)

Table 1: Lattice ensembles and meson masses in lattice units for the $N_{f}=2$
theory (upper) and for strong isospin breaking $N_{f}=1+1$ (lower). We denote
the bare fermion masses by $m_{0}^{(1,2)}$ and the inverse gauge coupling by
$\beta$. All simulations were performed on lattices of the size $L^{3}\times
T$. For reference we also give the mass of the unflavoured pseudoscalar in the
connected-only approximation in the $N_{f}=1+1$ theory denoted by
$m_{\pi^{0}_{c}}$.

Figure 2: Masses of the pseudoscalar and vector mesons in $N_{f}=2$ (left) and
in $N_{f}=1+1$ (right) in units of the lightest pseudoscalar meson mass. Note,
that in the latter case $\pi^{\pm}$ and $\rho^{\pm}$ denote the $4$-plet of
flavoured mesons. Similarly, $\rho^{0}$ denotes the $6$-plet of unflavoured
vector mesons.

We plot the spectrum of pseudoscalar and vector mesons for both the degenerate
and non-degenerate case in Fig. 2. We see that for the large fermion masses
the mass of $\eta^{\prime}$ is slightly larger than that of $\pi$.
Furthermore, we observe that for non-degenerate fermions the disconnected
contributions to the $\pi^{0}$ meson is small and the connected-only
approximation of Ref. [7] appears to be justified in this mass-regime.
However, it is a priori unknown if the parameter space studied here is able to
reproduce the Dark Matter self-scattering rates and Dark Matter masses needed
in a SIMP scenario [7]. In the next section, we address this by performing an
exploratory calculation of the $\pi\pi$ scattering lengths and match it to
existing experimental constraints.

##  3 Dark Matter self-scattering: $\pi\pi$ scattering lengths at isospin
$I=2$

Dark Matter self-scattering might address some issues such as the core-cusp
problem and strongly interacting Dark Matter models naturally provide self-
interactions. Together with the current experimental constraints on the self-
interaction of DM we perform a preliminary test on the viability of our
parameter choices by computing an estimate of the $\pi\pi$ scattering length
$a_{0}$ in the isospin $I=2$ channel.

A similar, more in-depth study has already been performed in $\text{SU}(2)$
[11]. We follow their approach. Since we are (currently) only interested in a
order of magnitude estimation, we defer a full Lüscher analysis for future
work. We study the ensemble with the highest available statistics given by the
bare gauge coupling $\beta=6.9$ and a degenerate, bare fermion mass
$m_{0}=-0.9$ on a $24\times 12^{3}$ lattice. We extract the energy shift
$\delta E_{\pi\pi}$ from [11]

$\displaystyle
R(t)=\frac{C_{\pi\pi}(t)-C_{\pi\pi}(t+1)}{C_{\pi}^{2}(t)-C_{\pi}^{2}(t+1)},$
(7)

by fitting it to its large time behaviour of

$\displaystyle R(t\to\infty)=A\left[\cosh\left(\delta
E_{\pi\pi}(t-T/2)\right)+\sinh\left(\delta
E_{\pi\pi}(t-T/2)\right)\coth\left(m_{\pi}(t-T/2)\right)\right].$ (8)

Here, $C_{\pi}$ and $C_{\pi\pi}$ denote the one- and two-pseudoscalar
correlation functions, respectively. From this we estimate the scattering
length according to the relation

$\displaystyle\frac{\delta E_{\pi\pi}}{m_{\pi}}=\frac{4\pi
m_{\pi}a_{0}}{(m_{\pi}L)^{3}}\left(1+c_{1}\frac{m_{\pi}a_{0}}{m_{\pi}L}+c_{2}\left(\frac{m_{\pi}a_{0}}{m_{\pi}L}\right)^{2}\right)$
(9)

with coefficients $c_{1}=-2.837$ and $c_{2}=6.375$ [19]. We obtained values of
$\delta E_{\pi\pi}=0.010(4)$ and $a_{0}=1.0(5)$. Using $\sigma=\pi a_{0}^{2}$
we compare our result with experimental constraints on
$\sigma/m_{\pi}^{phys.}$. Note that we have not yet fixed the physical lattice
spacing $a$. From the experimental constraints
$\sigma/m_{\pi}^{phys.}<0.19~{}\text{cm}^{2}g^{-1}$ [20] and
$\sigma/m_{\pi}^{phys.}<0.13~{}\text{cm}^{2}g^{-1}$ [21] we estimate a lower
limit for the physical dark pion mass and obtain $m_{\pi}>100\text{MeV}$. This
is consistent with the constraints of the Dark Matter relic density [22] and
thus provides a hint that we are looking at phenomenologically interesting
points in parameter space.

Still, we note that this is a rough estimate and a full scattering analysis is
needed. Furthermore, it has been obtained only on a fixed volume for one
lattice spacing and one (degenerate) fermion mass. It was pointed out in Ref.
[11] that finite volume effects are expected to be large in this system. We
also note that the current constraints [21, 20] are so strong that the core-
vs-cusp problem might not be resolved by a velocity-independent cross-section
but that rather an explicit velocity dependence $\sigma(v)$ is needed. This
further motivates going beyond a study of the scattering length to fully
velocity-dependent $\pi\pi$ scattering.

##  4 Summary

We provide preliminary results on the spectrum of pseudoscalar and vector
mesons, including the flavour-singlet, in Sp($4$) gauge theory with $N_{f}=2$
and $N_{f}=1+1$ Dirac fermions. We find that for ensembles with
$m_{\pi}/m_{\rho}>0.77$ the pseudoscalar singlet $\eta^{\prime}$ is
parametrically close to the non-singlet pseudoscalars $\pi$. In the non-
degenerate case we find that for the large quark masses considered here, the
connected-only approximation for the $\pi^{0}$ appears to be justified. We
supplement these spectroscopic investigations by a first estimate of the
$\pi\pi$ scattering length. Combined with current experimental constraints on
Dark Matter self-interaction we find that the ensembles studied in this work
warrant further investigations.

## Acknowledgments

FZ is supported by the Austrian Science Fund research teams grant STRONG-DM
(FG1). The work of J. W. L is supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MSIT)
(NRF-2018R1C1B3001379). The computations have been performed on the Vienna
Scientific Cluster (VSC4). We thank E. Bennett, H. Hsiao, S. Kulkarni, B.
Lucini, S. Mee and M. Piai for helpful discussions.

## References

  * [1] D.A. Kosower, _SYMMETRY BREAKING PATTERNS IN PSEUDOREAL AND REAL GAUGE THEORIES_ , _Phys. Lett. B_ 144 (1984) 215.
  * [2] G. Cacciapaglia, C. Pica and F. Sannino, _Fundamental Composite Dynamics: A Review_ , _Phys. Rept._ 877 (2020) 1 [2002.04914].
  * [3] Y. Hochberg, E. Kuflik, H. Murayama, T. Volansky and J.G. Wacker, _Model for Thermal Relic Dark Matter of Strongly Interacting Massive Particles_ , _Phys. Rev. Lett._ 115 (2015) 021301 [1411.3727].
  * [4] P. Dimopoulos et al., _Topological susceptibility and $\eta^{\prime}$ meson mass from $N_{f}=2$ lattice QCD at the physical point_, _Phys. Rev. D_ 99 (2019) 034511 [1812.08787].
  * [5] E. Bennett, D.K. Hong, J.-W. Lee, C.-J.D. Lin, B. Lucini, M. Mesiti et al., _$Sp(4)$ gauge theories on the lattice: quenched fundamental and antisymmetric fermions_, _Phys. Rev. D_ 101 (2020) 074516 [1912.06505].
  * [6] E. Bennett, D.K. Hong, J.-W. Lee, C.J.D. Lin, B. Lucini, M. Piai et al., _Sp(4) gauge theories on the lattice: $N_{f}=2$ dynamical fundamental fermions_, _JHEP_ 12 (2019) 053 [1909.12662].
  * [7] S. Kulkarni, A. Maas, S. Mee, M. Nikolic, J. Pradler and F. Zierler, _Low-energy effective description of dark $Sp(4)$ theories_, 2202.05191.
  * [8] E. Bennett, J. Holligan, D.K. Hong, H. Hsiao, J.-W. Lee, C.J.D. Lin et al., _Progress in $Sp(2N)$ lattice gauge theories_, _PoS_ LATTICE2021 (2022) 308 [2111.14544].
  * [9] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica and F. Sannino, _SU(2) gauge theory with two fundamental flavors: A minimal template for model building_ , _Phys. Rev. D_ 94 (2016) 094507 [1602.06559].
  * [10] R. Arthur, V. Drach, A. Hietanen, C. Pica and F. Sannino, _$SU(2)$ Gauge Theory with Two Fundamental Flavours: Scalar and Pseudoscalar Spectrum_, 1607.06654.
  * [11] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica and F. Sannino, _Scattering lengths in SU(2) gauge theory with two fundamental fermions_ , _PoS_ LATTICE2014 (2014) 271 [1412.4771].
  * [12] L. Del Debbio, A. Patella and C. Pica, _Higher representations on the lattice: Numerical simulations. SU(2) with adjoint fermions_ , _Phys. Rev. D_ 81 (2010) 094503 [0805.2058].
  * [13] E. Bennett, D.K. Hong, J.-W. Lee, C.J.D. Lin, B. Lucini, M. Piai et al., _Sp(4) gauge theory on the lattice: towards SU(4)/Sp(4) composite Higgs (and beyond)_ , _JHEP_ 03 (2018) 185 [1712.04220].
  * [14] “GitHub - sa2c/HiRep: HiRep repository — github.com.” https://github.com/sa2c/HiRep.
  * [15] V. Drach, T. Janowski and C. Pica, _Update on SU(2) gauge theory with NF = 2 fundamental flavours_ , _EPJ Web Conf._ 175 (2018) 08020 [1710.07218].
  * [16] K. Bitar, A.D. Kennedy, R. Horsley, S. Meyer and P. Rossi, _The QCD Finite Temperature Transition and Hybrid Monte Carlo_ , _Nucl. Phys. B_ 313 (1989) 348.
  * [17] J. Foley, K. Jimmy Juge, A. O’Cais, M. Peardon, S.M. Ryan and J.-I. Skullerud, _Practical all-to-all propagators for lattice QCD_ , _Comput. Phys. Commun._ 172 (2005) 145 [hep-lat/0505023].
  * [18] H. Neff, N. Eicker, T. Lippert, J.W. Negele and K. Schilling, _On the low fermionic eigenmode dominance in QCD on the lattice_ , _Phys. Rev. D_ 64 (2001) 114509 [hep-lat/0106016].
  * [19] M. Lüscher, _Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States_ , _Commun. Math. Phys._ 105 (1986) 153.
  * [20] D. Eckert, S. Ettori, A. Robertson, R. Massey, E. Pointecouteau, D. Harvey et al., _Constraints on dark matter self-interaction from the internal density profiles of X-COP galaxy clusters_ , 2205.01123.
  * [21] K.E. Andrade, J. Fuson, S. Gad-Nasr, D. Kong, Q. Minor, M.G. Roberts et al., _A stringent upper limit on dark matter self-interaction cross-section from cluster strong lensing_ , _Mon. Not. Roy. Astron. Soc._ 510 (2021) 54 [2012.06611].
  * [22] Y. Hochberg, E. Kuflik, T. Volansky and J.G. Wacker, _Mechanism for Thermal Relic Dark Matter of Strongly Interacting Massive Particles_ , _Phys. Rev. Lett._ 113 (2014) 171301 [1402.5143].