# The muon g-2 and lattice QCD hadronic vacuum polarization
may point to new, long-lived neutral hadrons

Glennys R. Farrar  Center for Cosmology and Particle Physics, Department of
Physics, New York University, NY, NY 10003, USA

###### Abstract

The experimental value the muon g-2 is $4.2\sigma$ larger than the Standard
Model prediction when the hadronic vacuum polarization contribution (HVP) is
determined from the measured R-ratio,
$\sigma(e^{+}e^{-}\\!\rightarrow\\!\mathit{hadrons})/\sigma(e^{+}e^{-}\\!\rightarrow\\!\mu^{+}\mu^{-})$;
the HVP calculated in lattice QCD also significantly exceeds the measured
R-ratio value. We show here that both these discrepancies can be explained by
an undetected contribution to $e^{+}e^{-}\\!\rightarrow\\!\mathit{hadrons}$ as
could arise from production of previously unidentified neutral, long-lived
hadrons. We suggest two potential candidates for the new hadrons and propose
several experimental tests.

The magnetic moment of the electron has been of fundamental importance since
Dirac’s prediction that it should be twice the Bohr magneton. The anomalous
magnetic moment g-2 of the muon, the amount it differs from the simple Dirac
prediction, has become a fundamental test of particle physics because it is
sensitive to new particles or forces which are inaccessible to collider
experiments, and it can be measured to extreme accuracy, a few parts in 10
billion. A significant discrepancy, now $4.2\sigma$, has emerged between the
measured muon g-2 [1] and the prediction from the Standard Model of particle
physics [2]. The discrepancy, if persistent, would signal fundamental forces
or particles not yet incorporated in the Standard Model or, as suggested here,
yet-to-be discovered hadrons within the Standard Model.

A crucial component of the Standard Model prediction of the muon g-2 is the
determination of the Hadronic Vacuum Polarization (HVP) contribution, from
experimental measurements of
$e^{+}\\!e^{-}\\!\\!\\!\rightarrow\\!\textit{hadrons}\,$in the “R-ratio”
method.111The HVP contribution to the muon g-2 is given in the R-ratio method
by  $a_{\mu}^{\rm HVP,LO}=\frac{\alpha^{2}}{3\pi^{2}}\int
ds\frac{R(s)K(s)}{s}~{},$ (1) where $s$ is the center-of-mass energy squared,
$R(s)\equiv\sigma(e^{+}\\!e^{-}\\!\rightarrow\\!\textit{hadrons})/\sigma_{\rm
Born}(e^{+}\\!e^{-}\\!\rightarrow\\!\mu^{+}\\!\mu^{-})$ and $\sigma_{\rm
Born}(e^{+}\\!e^{-}\\!\rightarrow\\!\mu^{+}\\!\mu^{-})=4\pi\alpha^{2}/3s$; the
function $K(s)$ is given in [2] and is approximately $m_{\mu}^{2}/3s$.
Intriguingly, if the HVP value calculated in lattice QCD by the BMW
collaboration [3] is used instead of the measured R-ratio value, the muon g-2
agrees with the SM prediction within the uncertainties. Recently, Ref. [4] –
combining the results of multiple lattice QCD groups – sharpened the
discrepancy with the experimental R-ratio to 4.2 $\sigma$.

Occam’s razor suggests that rather than g-2 being anomalous and the lattice
QCD calculations being inconsistent with $e^{+}e^{-}$ data, coincidentally by
the same amount, there may be a discrepancy between the measurement of
$e^{+}e^{-}\rightarrow\mathit{hadrons}$ and its true value. We show here that
this can be due to a class of undetected hadronic final states in $e^{+}e^{-}$
collisions. The lattice QCD calculations sum over all states composed of
quarks and gluons independently of their detectability, so would not be
subject to the same deficit. Recast in this way, the muon g-2 and lattice HVP
discrepancies are replaced by an “R-ratio discrepancy” – a difference between
the R-ratio measured in $e^{+}e^{-}$ experiments and its true value when all
hadronic final states are included.

In this paper we show that certain types of hadronic final states in
$e^{+}e^{-}$ collisions would have been systematically missed because of the
stringent trigger, event-selection and analysis requirements imposed to
suppress the severe backgrounds from beam-gas interactions and Bhabha
scattering. For instance, traditional triggers required at least one charged
particle from the interaction region, or an energy deposit pattern in the
detector consistent with photons from one or more $\pi^{0}$’s from the
interaction point. Such experiments exclude events with, for example, pair
production of neutral long-lived hadrons, $X\bar{X}$, which carry a large
fraction of the energy. We further identify a process which naturally fills
the gap between the R-ratio prediction and the value derived from the observed
muon g-2 and lattice HVP. In the following we 1) demonstrate that an R-ratio
discrepancy due to missed hadronic states can naturally be at the root of the
apparent g-2 and lattice anomalies, 2) suggest possible candidates for the
required new particles and 3) propose experiments to test this scenario.

The R-ratio method applied to current $e^{+}e^{-}\rightarrow\mathit{hadrons}$
data gives an HVP contribution $a_{\mu}\equiv\rm{(g-2)_{\rm had}}\times
10^{10}=693.1\pm 4.0$ [2]. This is $25.1\pm 5.9$ less than the value required
for the Standard Model prediction to agree perfectly with the measured g-2,
and $13.7\pm 5.5$ less than the BMW lattice QCD calculation. Let us call this
difference the HVP deficit, $\delta a_{\mu}\approx 20$. About 75% of the HVP
calculated from the R-ratio is due to the $\rho$ meson and 6% to the $\phi$
meson, so the HVP deficit, $\delta a_{\mu}/a_{\mu}\approx 3\%$, is about half
the size of the $\phi$ meson contribution. A natural solution to the HVP
deficit would thus be production of another vector meson, here designated $V$,
which decays almost exclusively to undetected final states $X\bar{X}$. We give
below several examples which fill the HVP gap. Then we review the event
selection criteria of $e^{+}\\!e^{-}\\!\\!\\!\rightarrow\\!\textit{hadrons}\,$
experiments, establishing that $X\bar{X}$ production would have been missed.
The $V\rightarrow X\bar{X}$ scenario is natural and its consistency with
experiments can be verified without a dedicated detector simulation, but
associated production,
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X+\\!\textit{hadrons}$, can also
explain the R-ratio deficit and also evades event selection requirements of
current experiments, as long as a sufficiently small fraction of the CM energy
goes into ordinary particles; detailed study of these more complex final
states is left to future work.

In order to explain both the muon g-2 and the lattice HVP anomalies, the
missed events must contain particles present in the lattice gauge calculation,
i.e., they must be made of $u,d,s$ or $c$ quarks, antiquarks and gluons. The
lattice HVP calculation can correctly incorporate the contribution of hadronic
states without those states having yet been explicitly identified via lattice
QCD calculations. This is because the lattice HVP calculation sums over
intermediate states and is therefore inherently more robust than lattice
predictions for individual states, much as perturbative QCD can accurately
describe Deep Inelastic Scattering while individual production cross-sections
are still not calculable. Thus an HVP excess would naturally be the first
indication of undiscovered neutral long-lived states in the hadron
spectrum.222Neutral, because charged hadrons could not have escaped detection,
and long-lived in order not to have been discovered by displaced vertex
searches; see below.  Future, more accurate lattice HVP calculations will
identify the mass range predominantly responsible for the excess with respect
to the observed R-ratio contribution [6, 7, 8] and hence narrow the mass
region to be searched – both experimentally and on the lattice – for the new
states. We can anticipate that the new states will have mass $\lesssim$ few
GeV, since non-perturbative QCD effects are most prominent at low energy and
moreover the contribution of a state to the HVP decreases with increasing
mass.

Production of $X\bar{X}$

One of the ways that $X\bar{X}$ pairs can be produced in $e^{+}e^{-}$
collisions is through a vector meson. Such contributions are well-described as
Breit-Wigner resonances characterized by their mass, width and branching
fractions into the initial and final states. We shall use the $\rho^{0}$ and
$\phi$ and their charmonium and bottomonium analogs, $\psi(3770)$ and
$\Upsilon(4S)$, for guidance in modeling $X\bar{X}$ production via
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$. All these are
$J^{PC}=1^{--}$ resonances (as required to match the virtual photon quantum
numbers) which decay predominantly to two spin-0 particles. We also explore
the possibility that $X$ has spin-1/2.

Following the analysis of the SND collaboration at VEPP for
$e^{+}e^{-}\rightarrow\phi\rightarrow K_{L}K_{S}$ [9], we write

$\sigma_{e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}}(s)\\!=\\!12\pi\\!\left(\frac{q_{X}(s)}{q_{X}(m_{V}^{2})}\right)^{2L+1}\\!\\!\frac{\Gamma_{X}\,\Gamma_{e}\,(m_{V}/\sqrt{s})^{5}}{[(m_{V}^{2}-s)^{2}+\Gamma_{V}^{2}\,s]}~{},$
(2)

where $q_{X}(s)\\!\equiv\\!\sqrt{s/4-m_{X}^{2}}$ is the magnitude of the
center-of-mass 3-momenta of $X$ and $\bar{X}$ at $s=E_{\rm CM}^{2}$ and $L$ is
the orbital angular momentum of the final $X\bar{X}$. $L=1$ is required to
conserve angular momentum when $X$ is a scalar or pseudoscalar, while $L=0$
for $X$ having spin-1/2. The parameters of the Breit-Wigner are $m_{V}$,
$m_{X}$, the total width $\Gamma_{V}$, and the partial widths $\Gamma_{X}$ and
$\Gamma_{e}$, for $V$ decay to $X\bar{X}$ and $e^{+}e^{-}$. The $V$ cannot
have a significant probability of decaying to visible hadrons or we would
already know of its existence, so to good approximation
$\Gamma_{X}=\Gamma_{V}$ since $\Gamma_{e}<<\Gamma_{V}$. Allowing for multiple
resonances is a straightforward extension.

Figure 1: The undetected hadronic contribution to the R-ratio from $V$ decay
to $X\bar{X}$, for four illustrative models taking $m_{X}=2$ GeV for
definiteness. Details of the models are given in Table 1 and the scaling with
mass and other parameters is discussed in the text. Cases a-c are for spin-0
and d (shown in red) is for spin-1/2.

Figure 1 shows $R_{V}(s)$, the increment in the R-ratio due to $X\bar{X}$
production, for four choices of mass and width of the resonance $V$ selected
for making certain points in the discussion of experimental tests below. For
each example, $\delta a_{\mu}=20.2$ and the combined fit to the muon g-2 and
the BMW HVP has $\chi_{\rm dof}^{2}=0.8$; Table 1 lists the details of each
case. The widths adopted are representative of values for the vector mesons
$\\{\rho^{0},\,\phi,\,\psi(3770),\,\Upsilon(4S)\\}$, for which
$\Gamma_{V}=\\{147,4,27,20\\}$ MeV, $\Gamma_{e}=\\{7,1.3,0.3,0.3\\}$ keV, and
$\int\\!\sigma_{V}dE_{cm}=\\{278,31,0.75,0.25\\}$ nb-GeV. This shows that a
vector meson with properties like those we know, decaying predominantly to a
previously unobserved long-lived neutral, naturally gives rise to the observed
increment in the R-ratio. There could also be multiple $V_{i},X_{i}$ systems
making individually smaller contributions.

These examples take $m_{X}=2$ GeV but they can be rescaled to other $m_{X}$
values: for a fixed $\Gamma_{e}$, $\delta a_{\mu}$ scales approximately as
$(m_{X}\,\rm{or}\,m_{V})^{-2.55}$. The contribution of
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$ to $a_{\mu}$ depends
weakly on the separation between resonance mass and threshold, $m_{V}-2m_{X}$,
unless this becomes larger than the width of $V$, and it increases slower than
linearly with increasing $\Gamma_{V}$. The distinctly higher peak in $R_{V}$
seen in Fig. 1 for the spin-1/2 case is because the Breit-Wigner damps more
rapidly at high energy for $L=0$ than for $L=1$, but inclusion of form-factor
effects for the $L=1$ case would reduce the difference. As the last two
columns of Table 1 show, the integrated contribution is similar in all cases.

case | L | $m_{V}$ | $m_{X}$ | $\Gamma_{V}$ | $\Gamma_{e}$ | $\int R_{V}dE_{cm}$ | $\int\\!\sigma_{V}dE_{cm}$
---|---|---|---|---|---|---|---
a | 1 | 4.005 GeV | 2 GeV | 10 MeV | 5 keV | 12.1 GeV | 56.8 nb-GeV
b | 1 | 4.01 GeV | 2 GeV | 30 MeV | 5.1 keV | 12.4 GeV | 57.1 nb-GeV
c | 1 | 4.003 GeV | 2 GeV | 100 MeV | 0.4 keV | 13.2 GeV | 57.7 nb-GeV
d | 0 | 4.01 GeV | 2 GeV | 30 MeV | 38 keV | 10.3 GeV | 55.4 nb-GeV
Table 1:  The parameters of the 4 illustrative models shown in Fig. 1. The
last column shows the integrated cross section
$\int\sigma_{e^{+}e^{-}\\!\rightarrow V}dE_{cm}$.

Experiments

Here we review the most important $e^{+}e^{-}$ experiments, historically and
for the R-ratio determination, to ascertain whether $X\bar{X}$ final states
would be included. We do not include experiments measuring a specific final
state such as $K^{+}K^{-}$, which are not relevant for this discussion.
Reference [2] gives an overview of all of the experiments, as well as the
methodology used in the R-ratio determination of the HVP.
$\bullet$ Mark I at SPEAR, the experiment that discovered the $\psi$, reported
measurements of the total cross section for hadron production in $e^{+}e^{-}$
collisions between 2.6 and 7.8 GeV [10]. They only recorded events with at
least two charged particles, and further required reconstructed tracks to
intersect in the collision region.
$\bullet$ The Crystal Ball collaboration measured the total hadronic and D*
cross sections in the range $3.87\,{\rm GeV}<E_{\rm CM}<4.5$ GeV [11]. The
heart of the Crystal Ball was a hermetic (93% of $4\pi$) array of NaI crystals
in which photons and electrons deposit virtually all of their energy. An inner
tracking system tagged charged particles. In order to qualify as a hadronic
candidate, an event without charged tracks had to i) have a total energy
deposit of at least 35% of $E_{\rm cm}$ and ii) pass a “maximum asymmetry cut”
of 0.85, such that along no axis is the difference in energy between
hemispheres greater than 15% of the total observed energy. These event
selection criteria mean that
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$ events would generally
not be detected. Condition i) is not met except for possible instances of
$\bar{X}$ annihilation, because the total energy deposit is limited by the
kinetic energy of the $X\bar{X}$: $m_{V}-2m_{X}\lesssim\mathcal{O}(100\,{\rm
MeV})<<0.35\,E_{\rm CM}\approx 1.3$ GeV. And in case $\bar{X}$ carries baryon
number -1 or lower and annihilates in the detector, condition ii) would
generally cause the event to be rejected on grounds of too-large energy
asymmetry: in the direction of the $\bar{X}$ annihilation at least
$m_{X}+m_{N}$ less the mass of the final state products would be released,
while in the opposite hemisphere the maximum KE is
$m_{V}/2-m_{X},~{}\mathcal{O}(10^{\prime}s\,{\rm MeV})$.
$\bullet$ BES, the Beijing Spectrometer at the Beijing Electron Positron
Collider (BEPC), is the most powerful instrument now measuring $e^{+}e^{-}$
collisions below a CM energy of around 5 GeV. BEPC is a symmetric $e^{+}e^{-}$
collider producing a very high luminosity. The BESII collaboration took data
at 85 CM energies between 2-5 GeV (34 energies between 3700 and 4200 MeV with
roughly 10 MeV spacing, closer near known structures) [12]; results from the
BESIII energy scan have not yet been published. Event selection criteria for
BESII were mainly as described in Ref. [13]. They require an event to have at
least two charged particles and a total deposited energy in the EM calorimeter
of at least 0.28 $E_{\rm beam}$, so they would exclude a $V\rightarrow
X\bar{X}$ process. An additional requirement is that events with just two
charged particles must have at least two additional neutral clusters of more
than 100 MeV in the EM calorimeter, not too co-linear with the charged tracks.
(The neutral clusters could be produced by a photon or neutral particle such
as $K_{L}$ or neutron).

The experiments described above used the “scan method” of stepping through the
energy range with steps small compared to structure in the cross section.
Another approach is the Initial State Radiation (ISR) method, in which the CM
collision energy $\sqrt{s}$ is typically fixed to a resonance such as an
$\Upsilon$, but the CM energy of the $e^{+}e^{-}$ annihilation process takes a
continuous value over a large range due to emission of a hard ISR photon. The
effective $E_{\rm cm}$ of the $e^{+}e^{-}$ annihilation in the ISR method is
completely determined by the accelerator collision energy, $\sqrt{s}$, and the
ISR photon energy in the CM, $E_{\gamma}^{*}$: $E_{\rm
cm}^{2}=s-2E^{*}_{\gamma}\sqrt{s}.$ Thus the ISR method has the advantage of
not missing a narrow resonance between steps in the scan, although limitations
in measuring $E_{\gamma}^{*}$ limit the resolution in $E_{\rm cm}$. The price
paid with the ISR method is less well-determined collision kinematics and
smaller cross section. In principle – with sufficiently good ISR photon energy
resolution – the ISR method provides an ideal approach to performing a
completely inclusive search which can discover an otherwise invisible
contribution to $\sigma(e^{+}e^{-}\rightarrow$ everything). However up to now
the only use of the ISR method, in the context of measuring $R(s)$, has been
for exclusive final states.

$\bullet$ BaBar, at the SLAC asymmetric B-factory, was sensitive to
$V\rightarrow X\bar{X}$ via direct production for $m_{V}$ around 10 GeV, and
through ISR for lower masses [14]. An in-depth study of the inclusive ISR
capabilities with BaBar is given in an unpublished thesis [15]. Although BaBar
has not yet reported an inclusive total $e^{+}e^{-}$ cross section, they have
used the ISR technique to place limits on the kinetic mixing parameter between
the QED photon and a possible massive “dark photon” [16]. That analysis fits
the observed spectrum of ISR photons to a zero-width resonance plus a smooth
background of unspecified shape. Their event selection aimed to identify
events with exactly one ISR photon and no other activity, hence demanded that
the ISR photon not be accompanied by any charged particles or any neutral
particle energy deposit. The BaBar dark photon search can be directly applied
to constrain the $e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$
scenario if the $V$ is narrow and the $X$ and $\bar{X}$ do not interact.
However neither of these conditions is generic to the missing hadrons
scenario. As a first step, removing the requirement that the $X$ or $\bar{X}$
not interact in the detector could test the examples given in Fig. 1 when the
width of $V$ is small compared to the detector resolution.

To summarize, events in which the energy is dominantly carried by new neutral
hadrons would not have been identified as an
$e^{+}e^{-}\rightarrow\textit{hadrons}$ event by experiments to date, unless
the $X\bar{X}$ were accompanied by sufficiently energetic charged particles or
deposit a significant fraction of their own energy in the detector. Other
types of events can also escape detection, but this must be established on a
case-by-case basis.

How could an $X$ particle not be already known?

The discussion above and the
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$ model demonstrate that
certain contributions to the R-ratio, sufficient to remove the discrepancy
with g-2 and lattice QCD, would not have been detected. Events in which most
of the energy is carried by long-lived neutral hadrons do not generally
satisfy event selection requirements. However since the hadron spectrum is
very well studied, it seems surprising that there could be unknown long-lived
hadrons. The minimal conditions for a state $X$ to have been missed are that
it is neutral, not part of a multiplet with charged members which have been
detected, and long-enough lived not to have been noticed through its decays. A
lifetime $\tau\gtrsim 10^{-7}$s is probably sufficient for this, but a
comprehensive examination of experiments is warranted.

If $m_{X}$ is less than a few GeV, and the product of the $X$ production cross
section in hadron collisions and its scattering cross section on nuclei is
small compared to that product for neutrons, it would escape notice in high
energy hadron scattering and collider experiments: a background of vastly more
neutrons is a potent camouflage! Indeed, the neutral particle search of Ref.
[17] explicitly excluded masses below 2 GeV.

As a hadron, the $X$ is presumably mildly or moderately strongly-interacting,
so it loses energy in matter and would not be penetrating enough to be seen in
searches such as FASER [18] or MilliQan [19]. Indeed, discovering a mild-to-
moderately interacting long-lived neutral particle in the GeV-range requires a
dedicated effort. A strategy and detector to find a subdominant population of
long-interaction-length-neutral particles is outlined in Sec. VII.E of Ref.
[5].

What could the $X$ particle or particles be?

The biggest impediment to the existence of a suitable candidate for $X$ is the
requirement that it be long-enough-lived. A long lifetime requires some
mechanism for stabilization.

Phenomena such as instantons, sphalerons and Skyrmions in particle physics,
and topological order and vortices in condensed matter systems, provide
motivation to consider whether QCD may support a not-yet-identified
topologically-stabilized state. If such a state exists, it might be resemble
an exotic, conceivably fermionic glueball-$\eta^{\prime}$ hybrid, suggesting a
mass in the 1-2 GeV range. If it were actually spin-1/2, the $X$ would have to
be pair produced along with an $\bar{X}$, as already envisaged in the
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$ model. A concrete
model would enable evaluation of whether production of such a particle could
explain the HVP deficit.

A less speculative mechanism for a hadron to have a long lifetime is if its
constituent quarks carry a quantum number that is conserved. A candidate of
this type appears in the stable sexaquark scenario [5], where $X$ would be the
scalar, flavor-singlet di-baryon $S$, composed of
$u_{\uparrow}u_{\downarrow}d_{\uparrow}d_{\downarrow}s_{\uparrow}s_{\downarrow}$.
Here, the conservation of baryon number, and of strangeness by the strong
interactions, leads to a very long lifetime if the $S$ is light enough. For
consistency with the stability of the deuteron but otherwise being minimally
restrictive, the mass range of the $S$ is 1850 MeV $<m_{S}\lesssim 2230$
MeV.333For $m_{S}\lesssim 2050$ MeV, the lifetime exceeds the age of the
Universe [20]. See Ref. [5] for more information about such a state and a
discussion of why it would have escaped detection, as well as suggestions for
experimental approaches to discovering it. The
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$ mechanism can
potentially be realized with $X$ being the sexaquark. The exchange of the
flavor-singlet combination of $\omega$ and $\phi$ vector mesons produces an
attractive Yukawa interaction between $S$ and $\bar{S}$. Explicit numerical
solution of the Schroedinger equation shows that $1^{--}$ bound states are
formed for reasonable coupling, with width and mass offsets compatible with
the examples of Table I.444X. Xu, private communication.

However $e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow S\bar{S}$ may not be the
dominant production channel. According to the Vector Meson Dominance model,
$\Gamma_{e}$ is proportional to the square of the mean charge at the origin.
Although the $S$ would have a non-zero charge radius since the strange quark
is significantly heavier than the light quarks, angular momentum conservation
requires the $S\bar{S}$ in the $V$ to have $L=1$, suppressing the charge at
the origin. Thus associated production, e.g.,
$e^{+}e^{-}\rightarrow\bar{S}\Lambda\Lambda$ or $S\bar{\Lambda}\bar{\Lambda}$,
could be more important than
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!S\bar{S}$, especially in view of
the large production cross section
$\sigma(e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda})$ near threshold [21].
Associated production might have evaded detection because the BESII event
selection requires an event with two charged tracks to have accompanying
neutral clusters, and the Crystal Ball anti-asymmetry cut would exclude such
events since the rest mass of the $S$ would not be detected. Dedicated study
with an event generator is needed to delimit this possibility.

An intriguing anomaly reported in [15] is a potential hint of $\bar{S}$
production: BaBar sees an excess of high energy anti-protons relative to
simulation, with no such discrepancy for protons, as shown in Fig. 9.34 of
[15]. This could be a signature of anti-sexaquark annihilation producing
$\bar{\Lambda}$, whose decay produces the observed $\bar{p}$ excess. (The
generic suppression of reactions involving $S$ interconversion with baryons at
low momentum transfer [5] is mitigated by BaBar’s asymmetric collider
configuration causing possible $\bar{S}$’s produced at rest in the CM to have
$\beta\approx 0.5$ relative to the detector.)

Predictions and Tests

Several experimental approaches can shed light on the possibility of an unseen
contribution to $e^{+}e^{-}$ final states:
$\bullet$ Measurement of the $e^{+}e^{-}$ total cross section using ISR with
existing BaBar data, or potentially at Belle II, is an ideal approach if event
selection can be clean enough and background processes adequately modeled. The
condition of no neutral energy deposit should be removed to avoid suppressing
the signal of interacting neutrals, since hadrons generically will interact.
Good resolution on $E_{cm}$ of the $e^{+}e^{-}$annihilation reaction would be
valuable to trace the structure of the cross section and expose differences in
shape relative to the measured (“visible”) $R(s)$, enabling the mass to be
constrained. A corollary of the missed-events scenario is that excess in the
ISR inclusive measurement should be associated with a corresponding population
of events that would not have passed selection cuts for the standard R-ratio
measurements. Seeing a sufficient excess in inclusive ISR over expectations
from the visible R-ratio measurements would validate the proposed mechanism of
an “invisible” contribution to $e+e-$ final states as the solution to the g-2
anomaly, but demonstrating the existence of such an invisible component is not
alone sufficient to discriminate between the production of unseen hadronic
states or Beyond the Standard Model particles. In the latter case, the lattice
HVP anomaly would not be explained.

$\bullet$ Precision measurement of $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$,
using the ratio
$R_{\mu\mu}\equiv\sigma_{e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}}/(4\pi\alpha^{2}/3s)$
to probe the vacuum polarization as proposed in Ref. [22] and done in Ref.
[23]. The present statistical precision on $R_{\mu\mu}$ is somewhat better
than 1% in the scan data and 0.2% in high luminosity data, but the systematic
uncertainty is 2.9%; the latter is expected to improve significantly, possibly
to better than 1%.

Figure 2: The predicted ratio $R_{\mu\mu}$ for the four illustrative cases of
Fig. 1, taking $R_{\rm had}$ without $R_{V}$ to give $R_{\mu\mu}=1.05$. Cases
requiring large $\Gamma_{e}$ to fit the HVP should be testable via
$R_{\mu\mu}$. The inset is a blow-up for cases b and c.

To illustrate the discriminating power of a precision measurement of
$R_{\mu\mu}(s)$, Fig. 2 shows the impact of
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$ on $R_{\mu\mu}$ in the
four models introduced previously to explore solution space (Fig. 1 and Table
1), approximating the visible $R(s)$ in this region as giving a constant
$R_{\mu\mu}=1.05$ in the absence of a $V$:555I thank C. Yuan for this
suggestion, as well as valuable discussions about BESIII experimental
capabilities.

$R_{\mu\mu}\\!=\\!\rm{Abs}\\!\left[1.025+\frac{3s}{\alpha}\\!\left(\frac{q_{S}(s)}{q_{S}(m_{V}^{2})}\right)^{L}\\!\frac{k_{ee}\,\Gamma_{e}/m_{V}}{(s-m_{V}^{2}+im_{V}\Gamma_{V})}\right]^{2}.$
(3)

Figure 2 shows that some resonance properties for
$e^{+}e^{-}\\!\rightarrow\\!V\\!\rightarrow\\!X\bar{X}$ which solve the HVP
discrepancies should already be excludable by BESIII data – particularly those
requiring large $\Gamma_{e}$ like models a and d. But model c underlines the
fact that an invisible contribution to the R-ratio would be difficult to see
in $R_{\mu\mu}$, if $\Gamma_{e}$ is small.

$\bullet$ Search for $X\bar{X}$ production through interactions in the
detector, analogous to the BESIII measurement of $n\bar{n}$ production in the
2-3.08 GeV range [24]. The $X\bar{X}$ search would be more challenging because
the $n\bar{n}$ study exploited the subset of events in which the $\bar{n}$
annihilates, producing a characteristic asymmetry in the energy deposit in
opposite directions. However annihilation cannot be relied on in either of the
long-lived $X$ scenarios discussed above: a topologically stabilized state
could not annihilate with a nucleon because of baryon number conservation, and
annihilation is much suppressed relative to scattering for an $\bar{S}$, at
least at low energy [5]. So the energy deposit on opposite sides of the
detector would be on average symmetric, and limited by $(E_{\rm
CM}-2m_{X})/2$. Running at the highest beam energy to maximize the energy the
$X\bar{X}$ can deposit, would produce the most visible signal.

Summary

The experimentally measured value of the anomalous magnetic moment of the muon
[1] is 4.2$\sigma$ above the prediction of the Standard Model, if the Hadronic
Vacuum Polarization contribution is inferred from precision measurements of
cross sections for $e^{+}e^{-}\rightarrow\mathit{hadrons}$ using the R-ratio
method [2]. The latest lattice QCD calculations of the HVP [3, 4] also exceed
the R-ratio value at a similar level. We have proposed here that both of these
discrepancies may have a common resolution via an undetected contribution to
$e^{+}e^{-}$$\rightarrow$ hadrons. We showed that, given the event selection
and trigger requirements of the $e^{+}e^{-}$ experiments to date, production
of final states consisting only of new neutral particles $X\bar{X}$, or such
neutrals with a sufficiently discreet accompaniment of ordinary hadrons, would
have gone undetected. Hypothesizing the existence of such a hadron $X$, we
showed as a concrete “proof-of-principle” that production of $X\bar{X}$ pairs
through a vector meson $V$ naturally reconciles the R-ratio, lattice QCD and
muon g-2 results.

To provide a viable explanation for the muon g-2 and lattice HVP puzzles, the
new state $X$ must be composed of $u,d,s,c$ quarks and gluons. Furthermore it
must be sufficiently long-lived not to have already been discovered. We
identified two potential candidates for the required long-lived neutral
particle $X$: a conjectural topologically-stabilized hadron and the stable
sexaquark $uuddss$ introduced elsewhere and detailed in [5]. The natural
$E_{\rm CM}$ range for $X\bar{X}$ production in these cases is $\approx 2-4.5$
GeV.

There are currently no other evident candidates to reconcile both the lattice
HVP and muon g-2 values with the R-ratio, because to contribute to the lattice
HVP, the state must be made of quarks and gluons. As-yet-unknown colored
particles such as color sextet quarks would not do the job because they are
not present in the lattice calculation, moreover the contribution to the HVP
goes roughly as $R(s)/s^{2}$ so it is difficult for high mass particles to
have sufficient impact. The proposed resolution of the discrepancy between
muon g-2, lattice QCD, and direct experimental measurement of the R-ratio,
will impact electroweak precision sector fits and likely make tensions among
those observables more significant [25, 26, 27].

Acknowledgements:
I have benefitted from valuable input from Alex Bondar, Marek Karliner, Yury
Kolomensky, Arkady Vainshtein, Zihui Wang, Xingchen Xu and Changzheng Yuan.
This research was supported by the National Science Foundation Grant NSF-
PHY-2013199 and by the Simons Foundation.

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