# A multi-reflection time-of-flight mass spectrometer for the offline ion
source of the PUMA experiment

M. Schlaicha Corresponding author<EMAIL_ADDRESS>J. Fischera
P. Fischerb C. Klinka A. Obertellia A. Schmidta
L. Schweikhardb and F. Wienholtza

(aTechnische Universität Darmstadt, Institut für Kernphysik,
Schloßgartenstraße 9, 64289 Darmstadt, Germany
bUniversität Greifswald, Institut für Physik, Felix-Hausdorff-Straße 6, 17489
Greifswald, Germany)

##  1 Introduction

To date, the many-body problem of nuclear physics cannot be solved exactly and
approximations are required [1]. To contribute to the understanding of nuclear
structure, radioactive ion beam facilities provide the investigation of a
broad variety of short-lived nuclei which allow to explore nuclear systems
beyond isospin symmetry. On the neutron rich side of the nuclear landscape,
nuclei form neutron skins, referring to an excess of neutrons on the nuclear
surface. Its size, the neutron skin thickness, can be determined by various
methods [2, 3, 4, 5, 6] and is correlated to the nuclear equation of state
through the slope of the symmetry energy, i.e. the contribution of the proton-
to-neutron asymmetry in nuclear matter [7]. Bridging the gap between the
smallest and largest nucleonic systems known, this correlation is also of
relevance for the structure of neutron stars [8, 9]. Proceeding to even more
asymmetric systems than neutron-skin nuclei, halo nuclei located at the
nuclear drip lines [10] offer the possibility of studying nuclear few-body
systems [11]. These are nuclei in which one or more nucleons have an increased
probability density within classically forbidden regions, far beyond the
nuclear core potential [12, 13].
To contribute to the understanding of phenomena that occur beyond the half-
density radius of nuclei, such as neutron skins and halo nuclei, the
antiProton Unstable Matter Annihilation (PUMA) experiment will investigate the
nucleon composition at the tail of the matter density distribution of stable
as well as radioactive isotopes using low-energy antiprotons [14, 15]. By
trapping both matter and antimatter together and studying the antiproton-
nucleon annihilation residuals, which are mainly pions, the PUMA experiment
aims to determine the proton-to-neutron ratio in the tail of the nuclear
matter density distribution. However, there is to date no facility worldwide
that provides both radioactive nuclei and low-energy antiprotons at the same
location. The PUMA experiment therefore plans to transport the antiprotons
within the CERN site from the Extra Low ENergy Antiproton (ELENA) facility
[16], located at the Antimatter Factory, to the Isotope mass Separator On-Line
DEvice (ISOLDE) [17, 18] in a transportable Penning trap.
In order to start with stable-ion experiments, PUMA requires mass-separated
and cooled ion bunches that can be obtained from a broad mass range of
isotopes. Therefore, a versatile offline ion source system is currently under
development at the Technical University of Darmstadt. Following the ion
source, a multi-reflection time-of-flight mass spectrometer (MR-ToF MS) [19,
20] for beam purification and a radio-frequency quadrupole (RFQ) for particle
accumulation and beam cooling [21, 22, 23, 24] will be used. State-of-the-art
MR-ToF systems provide mass resolving powers beyond $10^{5}$ [25, 26], reached
within milliseconds [27]. As such, they are the tool of choice for fast, high-
precision ion beam purification.

Figure 1: CAD representation of the PUMA offline ion source beamline. The ions
generated in the ion source are mass-separated in the MR-ToF MS and cooled and
bunched in the RFQ. Three movable time-of-flight (ToF) detectors allow a
stepwise characterisation of the ion beam.

##  2 The PUMA offline ion source beamline

The offline ion source beamline can be divided into three main sections which
- in beam direction - are the ion source, the MR-ToF MS and the RFQ (see Fig.
1). They are followed by a fourth section to gradually improve the vacuum
behind a 90-degree bend towards the PUMA beamline. Furthermore, the offline
ion source is complemented with three movable ion detection units, three iris-
type apertures for adjustments of the gas flow and ion optical components for
focusing, steering and energy modification (not labelled in Fig. 1). The
offline setup is to be installed at the PUMA experimental area located at the
ELENA facility.

###  2.1 Overview of the setup

In a commercial electron impact ionisation source (IQE 12/38, SPECS Surface
Nano Analysis GmbH, Berlin, Germany), a gaseous material is ionised and the
ions are extracted continuously with beam currents of a few
$\text{\,}\mathrm{pA}$ up to $800\text{\,}\mathrm{nA}$ (maximum current
specified by manufacturer for argon ion-beam production). The ion source is
equipped with two pairs of deflection plates ($6\text{\,}\mathrm{mm}$ $\times$
$16\text{\,}\mathrm{mm}$), each $5\text{\,}\mathrm{mm}$ spaced, which are
aligned perpendicular to each other. When switching the electric potentials
applied to the plates between an ion blocking and transmission state, chopped
ion packets can be extracted, which is required for the operation of the MR-
ToF MS. Bunches with a full width of half maximum (FWHM) minimum of about
$250\text{\,}\mathrm{ns}$ can be extracted. The ion source is attached to a
quadrupole ion bender, which provides the possibility to add other ion source
types at a later stage. Downstream, the MR-ToF MS allows beam analysis and
mass-selective ion ejection towards the RFQ, in which the ions remain trapped
until up to $10^{5}$ ions are accumulated in one bunch. While ion accumulation
is the main purpose of the RFQ, the ion bunch emittance can simultaneously be
reduced within milliseconds through buffer gas cooling. Subsequently, the ions
are ejected and guided towards the PUMA Penning trap, in which they annihilate
with the trapped antiprotons.

###  2.2 The MR-ToF mass spectrometer

MR-ToF systems, operated in various nuclear laboratories around the world [28,
29, 26, 30, 31], share the operating principle that charged particles are
trapped between two opposing ion-optical mirrors and ejected after up to
thousands of revolutions. In this way, the ion flight path can reach several
kilometres in experimental setups that measure only a few meters. Furthermore,
the reflecting potentials are tuned to retain small temporal bunch widths by
compensating flight-time differences resulting from small discrepancies in the
ions’ energies. Ion species differing in mass-over-charge ratio and
accelerated by the same electric potential can thus be separated with isobaric
precision. The two electrostatic mirrors are usually made of stacks of
cylindrical electrodes for better control of the potential gradients and are
separated by a field-free drift region [32].
The design of the PUMA MR-ToF MS is a further development that started from
the present MR-ToF setup at the University of Greifswald [25, 33, 34, 35],
which includes lens and steering optics in front of the entrance and behind
the exit mirror, in-trap deflector electrodes for in-trap separation of
unwanted ion species [36] and an in-trap lift electrode. The latter can be
used to capture ions at adjustable storing energies as well as to eject a
desired ion species after mass separation [37, 38, 39]. The PUMA offline ion
source is intended to allow for fast accumulation of up to $10^{5}$ mass-
selected ions inside the RFQ before they are sent to the antiprotons in the
PUMA Penning trap to obtain annihilation event rates that can be distinguished
from background annihilations. Therefore, the design aims to maximise the
transversal acceptance of the MR-ToF device to allow for a large ion
throughput while guaranteeing a sufficient mass resolving power of up to 105.

Intuitively, the acceptance is expected to increase with the inner diameter of
the electrodes. To our knowledge, however, a correlation is not known and,
thus, it is investigated in the following with simulations. The maximum
possible electrode size is limited by the vacuum tubes, which are chosen to
have an inner diameter of $155\text{\,}\mathrm{mm}$ in the present design. Due
to limited space of the experimental site at ELENA, it is decided to allow a
maximum length of $1.2\text{\,}\mathrm{m}$ for the MR-ToF MS. In analogy to
the Greifswald design, the MR-ToF system should include injection and ejection
ion optics, the in-trap deflector and lift electrodes. However, it is decided
not to include the support for temperature compensation [33] in the new MR-ToF
system, as it has been found that the stability of the potentials is much more
important than the influence of temperature fluctuations [40]. This allows a
simpler and more modular design. Furthermore, similar to the Greifswald
design, it is decided to use six electrodes per ion-mirror stack, including
one electrode on negative potential which is intended to form lens-like
potentials for radial refocusing and five electrodes on positive potential.
The operation of existing MR-ToF systems has shown that this provides
sufficient degrees of freedom to shape the potential profile to reach mass
resolving powers beyond 100,000.

##  3 Simulations

For the optimisation of the electrode geometry, simulations are performed
using SIMION® [41]. This allows to define customised electrode geometries to
which a potential can be assigned. The software then calculates the resulting
electric fields by solving the Laplace equation. Based on the potential array
thus determined, ion trajectories are calculated as a function of the starting
values of the ions. In all simulations described, we use cations. The
application to anions requires a polarity change of the specified potential
values.

###  3.1 Simulation method

Using SIMION®, the ion-optical acceptance of a given electrode geometry is
determined by measuring the emittance of the ions that successfully pass
through the MR-ToF system. However, this strongly depends on the six-
dimensional potential set which is applied to the mirror electrodes. Here, we
assume that the same potential is applied mirror-symmetrically to opposing
electrodes. A potential combination that allows sufficiently isochronous ion
reflection (time dispersion of the ion bunch per revolution in the order of a
few tens of picoseconds), again depends on the electrode geometry and can only
be found with a more time-consuming optimisation algorithm. However, this
approach can result in local minima, which means that comparability between
different geometries is not necessarily given. Therefore, an alternative
measure is used that indirectly reflects the ion acceptance and is comparable
for different designs, regardless of the electrode size and shape: We use the
volume of the subspace formed by all mirror potential sets
$V\in\mathbb{R}^{6}$ that allow a given ion bunch to be trapped without
losses. If the initial conditions for the ions are not changed for the
trajectory simulations of different geometries, the solution volume is
expected to reflect the acceptance depending on the proposed geometry
modifications. Since it is not possible to calculate the described volume
analytically, its size is determined using a Monte Carlo (MC) approach.
The voltages that can be applied to the mirror electrodes are limited by the
maximum operation voltage of $5\text{\,}\mathrm{kV}$ for standard SHV
connectors. Furthermore, we restrict the outer five mirror electrodes to a
repulsive potential, while the innermost electrode is used as lens with a
negative potential applied to it. This provides a finite sample volume
$\Omega_{\text{m}}\subset\mathbb{R}^{6}$ of voltage combinations for the
mirror electrodes. The used mirror potentials
$V_{\text{i}}\in\Omega_{\text{m}}$ are taken randomly from the defined voltage
interval. With the weight function $P(V_{\text{i}})\in\\{0,1\\}$, which is
equal to 1 if the applied potential profile allows to trap all ions on loss-
free trajectories and equal to 0 if this is not the case, the MC simulation
approaches the desired relative solution volume

$V_{\text{rel}}=\frac{\int_{\textbf{V}\in\Omega_{\text{m}}}P(\textbf{V})d\Omega_{\text{m}}}{\int_{\textbf{V}\in\Omega_{\text{m}}}d\Omega_{\text{m}}}$
(1)

by calculating the mean value

$\langle V_{\text{rel}}\rangle=\frac{1}{N}\sum_{i=1}^{N}P(V_{i}).$ (2)

Here, $N$ is the number of voltage configurations tested. An illustration of
the potential volume determined in this kind of MC simulation is provided in
Sec. 3.4. Unless stated otherwise, 100 85Rb+ ions trapped for 20 revolutions
are used in each of the following simulations.

Table 1: Dimensions of the six mirror electrodes for two different mirror concepts to be compared using the MC simulation. The sizes are chosen to sum up to the same total length of the electrode stack. The labels M1 - M6 correspond to the mirror electrodes from inside out (see Fig. 3). The spacing between the electrodes remains unchanged. mirror concept | $l_{\text{M1}}$ | $l_{\text{M2}}$ | $l_{\text{M3}}$ | $l_{\text{M4}}$ | $l_{\text{M5}}$ | $l_{\text{M6}}$ | spacing
---|---|---|---|---|---|---|---
Greifswald MR-ToF analyser | $40\text{\,}\mathrm{mm}$ | $30\text{\,}\mathrm{mm}$ | $20\text{\,}\mathrm{mm}$ | $12\text{\,}\mathrm{mm}$ | $12\text{\,}\mathrm{mm}$ | $12\text{\,}\mathrm{mm}$ | $5\text{\,}\mathrm{mm}$
unified mirror electrodes | $42\text{\,}\mathrm{mm}$ | $18\text{\,}\mathrm{mm}$ | $18\text{\,}\mathrm{mm}$ | $18\text{\,}\mathrm{mm}$ | $18\text{\,}\mathrm{mm}$ | $12\text{\,}\mathrm{mm}$ | $5\text{\,}\mathrm{mm}$
Table 2: Results of the relative solution space $V_{\text{rel}}$ in percent of the simulation varying the mirror electrode length $l_{\text{M}}$ with a fixed inner diameter $d_{\text{M}}=$68\text{\,}\mathrm{mm}$$. $l_{\text{M}}$ | $15\text{\,}\mathrm{mm}$ | $20\text{\,}\mathrm{mm}$ | $25\text{\,}\mathrm{mm}$ | $30\text{\,}\mathrm{mm}$ | $35\text{\,}\mathrm{mm}$
---|---|---|---|---|---
$V_{\text{rel}}$ | $5.1(2)\text{\,}\mathrm{\char 37\relax}$ | $7.9(3)\text{\,}\mathrm{\char 37\relax}$ | $8.8(3)\text{\,}\mathrm{\char 37\relax}$ | $10.8(3)\text{\,}\mathrm{\char 37\relax}$ | $10.4(3)\text{\,}\mathrm{\char 37\relax}$

###  3.2 Determination of the mirror electrode size

To converge to a final electrode design, the MC simulation approach is applied
to MR-ToF systems with varying electrode dimensions. The design currently in
use at the University of Greifswald has mirror electrodes of different
lengths. As a first step, it is investigated whether the unification of the
mirror electrode length has an effect on the solution volume. Identical mirror
electrodes provide the advantages to make use of mass production. Therefore,
the MC simulation is applied to two different mirror concepts with electrode
lengths as specified in Tab. 1. The length of the unified electrodes is chosen
such that the total length of the electrostatic mirror equals that of the
Greifswald MR-ToF analyser while the length of the negative mirror electrode
(M1) is on the same order of magnitude.
The ion bunch used for the simulation has a transverse 2$\sigma$ emittance of
3.7$\pi$ $\text{\,}\mathrm{m}\mathrm{m}~{}\mathrm{m}\mathrm{rad}$ and a
Gaussian energy distribution around $2000\text{\,}\mathrm{eV}$ with a standard
deviation of $50\text{\,}\mathrm{eV}$. According to the MC simulation, this
leads to the solution volumes of
$V_{\text{rel,u}}=$1.20(3)\text{\,}\mathrm{\char 37\relax}$$ for unified
electrodes and $V_{\text{rel,G}}=$1.26(3)\text{\,}\mathrm{\char 37\relax}$$
for the Greifswald design. Considering the uncertainties, the results are
compatible with each other. Therefore, the advantages of uniform electrodes
are considered to be more important and it is decided to use mirror electrodes
of equal length, which has yet to be determined.
As stated above, it is to be expected that the acceptance of the MR-ToF
analyser is increased by increasing the inner diameter of the system, i.e. the
diameter of the in-trap lift $d_{\text{lift}}$ and that of the mirror
electrodes $d_{\text{M}}$. Also, their ratio $d_{\text{lift}}$/$d_{\text{M}}$
as well as the ratio $l_{\text{M}}$/$d_{\text{M}}$, where $l_{\text{M}}$ is
the length of a mirror electrode, may affect the acceptance. To investigate
the effect of the mirror electrode size, MC simulations are conducted for
fifteen different mirror designs with varying electrode lengths
($8\text{\,}\mathrm{mm}$, $11\text{\,}\mathrm{mm}$, $14\text{\,}\mathrm{mm}$,
$17\text{\,}\mathrm{mm}$ and $20\text{\,}\mathrm{mm}$) and inner diameters
($18\text{\,}\mathrm{mm}$, $30\text{\,}\mathrm{mm}$ and
$50\text{\,}\mathrm{mm}$). The values are chosen to meet the mechanical and
experimental length restrictions and to feature reasonable dimensional
differences in order to see a trend of $V_{\text{rel}}$ as a function of the
electrodes’ dimension. Further, the diameter value of $18\text{\,}\mathrm{mm}$
was chosen to match the mirror diameter to that of the in-trap lift used in
this simulation.

Figure 2: Relative solution space found in the MC simulations as a function of
the mirror electrode inner diameter and length. For more details see the text.

The results are given in Fig. 2. Confirming the expectations, we see that
$V_{\text{rel}}$ grows with increasing inner diameter, which is interpreted as
an increased acceptance. Moreover, it turns out that there is an optimum
electrode length as a function of its diameter, which becomes more apparent
for larger diameters. Based on these results, we decide to use mirror
electrodes with the largest diameter compatible with the current design
constraints, which is $d_{\text{M}}=$68\text{\,}\mathrm{mm}$$ and is simulated
below.
With the diameter determined, the mirror electrode length is varied, keeping
the in-trap lift electrode dimensions at
$l_{\text{lift}}=$600\text{\,}\mathrm{mm}$$ and
$d_{\text{lift}}=$18\text{\,}\mathrm{mm}$$. As can be seen in Tab. 2, a
maximum $V_{\text{rel}}$ is found for
$l_{\text{M}}\geq$30\text{\,}\mathrm{mm}$$. Larger mirror electrodes are not
considered because long electrodes come at the expense of a shorter in-trap
lift electrode and thus reduced longitudinal acceptance due to the given limit
of the MR-ToF MS length (see Sec. 2.2). The negative mirror electrode (M1) is
optimised using the same algorithm leading to a length of
$40\text{\,}\mathrm{mm}$.

Figure 3: Cross-sectional view of the final MR-ToF MS geometry implementation
in SIMION® including all relevant dimensions. The distance between each
electrode is $5\text{\,}\mathrm{mm}$. Due to mirror symmetry, only one half of
the device is shown. For design reasons, the inner diameter of the mirror
electrodes later had to be reduced from $68\text{\,}\mathrm{mm}$ to
$62\text{\,}\mathrm{mm}$. Figure 4: Acceptance plot of the Greifswald (left)
and Darmstadt (right) MR-ToF MS geometries. The respective applied potential
profiles are selected based on the largest mass resolving power found in the
final MC simulation. The second orthogonal part of the phase space, the z-z’
plane (beam axis along the x-axis), is redundant due to the rotational
symmetry of the electrodes.

###  3.3 Final electrode dimensions

Finalising the geometry of the MR-ToF system, the MC algorithm is used to
define the in-trap lift diameter and that of the aperture on the injection and
ejection side of the mirror. $30\text{\,}\mathrm{mm}$ is determined for both
dimensions. Furthermore, the length of the in-trap lift,
$480\text{\,}\mathrm{mm}$, is chosen to ensure that the MR-ToF system uses the
entire length of the vacuum chamber. A variation of the aperture of mirror
electrode M6, which forms the transition between the mirror electrode stack
and the lens electrode assembly, shows no relevant influence on the solution
volume. It is adjusted to the lift and lens diameter. Due to minor technical
difficulties in the design of the mirror electrode assembly, the inner
diameter had to be reduced from $68\text{\,}\mathrm{mm}$ to
$62\text{\,}\mathrm{mm}$. An overview of the final geometry and relevant
dimensions is given in Fig. 3. It should be noted that all the investigations
shown are multi-dimensional problems and the result of each simulation is
affected by all electrode dimensions. However, the simulation effort for an
absolute optimisation of the MR-ToF MS geometry in dependence of all relevant
dimensions on each other is not feasible in terms of simulation cost.
After specifying the electrode geometry, final MC simulations are performed to
compare the solution spaces obtained from the Greifswald and the new Darmstadt
design. A solution volume of $1.19(3)\text{\,}\mathrm{\char 37\relax}$ is
found for the Greifswald MR-ToF device and $6.54(7)\text{\,}\mathrm{\char
37\relax}$ for the new PUMA MR-ToF system. The solution space is increased by
a factor of about six for the considered ion distribution. This result is
interpreted as an increased acceptance. To investigate the acceptance change
directly, the potential profiles leading to the largest mass resolving power
$R=t/\left(2\Delta t\right)$ are selected for each solution space, where $t$
is the mean ToF and $\Delta t$ the FWHM ToF of the simulated ions. Based on
these two mirror potential sets and geometries, the starting values of the
ions that can be successfully trapped for 100 revolutions and ejected are
measured. To include the capture and ejection process, the ions are generated
$10\text{\,}\mathrm{mm}$ in front of the injection lens and stopped
$10\text{\,}\mathrm{mm}$ behind the ejection lens. The result is plotted in
Fig. 4 together with the 2$\sigma$ ellipse containing
$91\text{\,}\mathrm{\char 37\relax}$ of the measured data points. Compared to
the Greifswald design the area of the 2$\sigma$ ellipse increases from
$9.75\pi\text{\,}\mathrm{m}\mathrm{m}~{}\mathrm{m}\mathrm{rad}$ to
$75.38\pi\text{\,}\mathrm{m}\mathrm{m}~{}\mathrm{m}\mathrm{rad}$, directly
showing the increase in acceptance by a factor of about eight.

Table 3: Set of mirror and in-trap lift potentials optimised for a $3000\text{\,}\mathrm{eV}$ 85Rb+ beam. electrode | M1 | M2 | M3 | M4 | M5 | M6 | in-trap lift
---|---|---|---|---|---|---|---
potential in $\text{\,}\mathrm{V}$ | $-3710\text{\,}\mathrm{V}$ | $745\text{\,}\mathrm{V}$ | $893\text{\,}\mathrm{V}$ | $1393\text{\,}\mathrm{V}$ | $2160\text{\,}\mathrm{V}$ | $2681\text{\,}\mathrm{V}$ | $1003\text{\,}\mathrm{V}$
Figure 5: Potential space obtained from the MC simulation described in Sec.
3.1 for the final MR-ToF MS geometry. The spot size indicates the final bunch
width of simulated ion bunch and the colour its penetration depth into the
electrostatic mirror volume, which has a total length of
$199\text{\,}\mathrm{mm}$.

###  3.4 Potential profile determination

The MC simulation approach described is not only used to improve the system’s
acceptance, but also to determine suitable potential profiles. Figure 5
illustrates the potential configurations obtained from the MC simulation
performed for the final electrode design. While the spot size reflects the
value of the final bunch width, the larger the plotted circle the larger the
width of the ToF distribution, the colour indicates the penetration depth of
the ions into the $199\text{\,}\mathrm{mm}$ long electrostatic mirror.
Highlighted by clusters of points of the same colour, subspaces of the
potential volume can be identified that allow ion storage at different
penetration depths. Stable trajectories are found that have reversal points
over almost the entire span of the mirror volume. A penetration depth of
$60\text{\,}\mathrm{mm}$ corresponds to the first $10\text{\,}\mathrm{mm}$ of
the mirror electrode M2, while a depths of $199\text{\,}\mathrm{mm}$ forms the
end of the mirror electrode volume. In order to include all mirror electrodes
to contribute to the reflecting mirror potential, voltage configurations that
allow a penetration depth of about 160 - $200\text{\,}\mathrm{mm}$ are
considered. In this way, the available six degrees of freedom are exploited,
which is expected to be beneficial for the compensation of the time dispersion
of the trapped ion bunches. By additionally separating those that lead to the
smallest possible bunch widths, suitable mirror potentials can be determined.
For the further improvement of the mirror potential shape, a simplex
optimisation routine (Nelder-Mead downhill type) is used, included in a module
provided by SIMION®. The algorithm varies the potentials applied to the six
mirror electrodes while locally minimising a user-specific metric. This allows
to reduce the final bunch width further and thus increase the mass resolving
power after a given number of revolutions. Based on this approach, the mirror
potential set to be used for commissioning measurements is determined for a
$3000\text{\,}\mathrm{eV}$ 85Rb+ beam captured with a trapping energy of about
$2000\text{\,}\mathrm{eV}$ inside the MR-ToF MS. The relevant potentials are
provided in Tab. 3. While the injection and ejection lens are set to
$U_{\text{lens}}=$2800\text{\,}\mathrm{V}$$, all steering and in-trap
deflector segments are kept at $0\text{\,}\mathrm{V}$. With this specific
potential configuration, the mass resolving power achievable is investigated
varying the initial ion distribution. Keeping the transverse 2$\sigma$
emittance constant at 8$\pi$
$\text{\,}\mathrm{m}\mathrm{m}~{}\mathrm{m}\mathrm{rad}$, initial bunch widths
$\Delta t_{\text{i}}$ of $10\text{\,}\mathrm{ns}$, $50\text{\,}\mathrm{ns}$
and $100\text{\,}\mathrm{ns}$ (FWHM) are combined with energy distributions
$\Delta E_{\text{i}}$ of $1\text{\,}\mathrm{eV}$, $2\text{\,}\mathrm{eV}$ and
$3\text{\,}\mathrm{eV}$ (FWHM). In the ideal case of $\Delta
t_{\text{i}}=$10\text{\,}\mathrm{ns}$$ and $\Delta
E_{\text{i}}=$1\text{\,}\mathrm{eV}$$, the mass resolving power reaches a
value of about 630,000 after 2000 revolutions. In contrast, at $\Delta
t_{\text{i}}=$100\text{\,}\mathrm{ns}$$ and $\Delta
E_{\text{i}}=$3\text{\,}\mathrm{eV}$$ the mass resolving power reaches a value
of about 230,000 for similar storing times. Although the resolving power for
ion selection is generally smaller, the values obtained are sufficient to
achieve the objectives of the offline ion source to provide isotopically pure
stable ion bunches.

##  4 Technical design

A sectional view of the system is shown in Fig. 6. It includes the
electrostatic mirrors, lens and steering electrodes in front of the entrance
mirror and behind exit mirror as well as in-trap deflector electrodes. The
central drift tube serves as an in-trap potential lift.

Figure 6: CAD representation in half-section view of the MR-ToF MS. Ions are
trapped between two electrostatic mirrors (indicated in red), which are
separated by a drift tube. Lens and steering electrodes as well as in-trap
deflectors are included.

To maintain high flexibility, all in-vacuum components are combined into
individual subassemblies that are placed on separate base plates, see Sec. 4.1
and 4.2. All modules are in turn supported by a base plate ranging over the
full length, which is referred to as the optical table in the following. Apart
from electrically nonconducting materials and electrode supporting base plates
made of aluminium and titanium, all in-vacuum components are made of stainless
steel (material number 1.4404). For electrical insulating components, alumina
ceramic and Macor® glass ceramic are chosen, which allow bake-out temperatures
up to $800\text{\,}\mathrm{\SIUnitSymbolCelsius}$. With the materials used, it
is expected to reach a vacuum below ${10}^{-9}\text{\,}\mathrm{mbar}$ inside
the MR-ToF MS during operation, reducing ion losses, especially for long
storage times.

###  4.1 Electrode assemblies

Separated by the drift tube, the symmetrical arms of the MR-ToF mass
spectrometer are formed by three subassemblies, which each serve individual
purposes. This modular character offers the advantage of flexible assembly and
easy modification for adaptations. Each subassembly is mounted onto its own
base plate before it is fixed to the $1200\text{\,}\mathrm{mm}$ long,
$90\text{\,}\mathrm{mm}$ wide and $4\text{\,}\mathrm{mm}$ thick optical table.
Thus, it can be dismounted, modified and reintegrated without changing the
remaining system. In the following, the three electrode modules are described
in more detail, from the inside out.

Figure 7: CAD representation of the individual electrode modules in cross-
section view. The innermost electrode stack (a) includes a support for the
drift tube and an in-trap deflector electrode for mass-selective ion
deflection. Formed by a shielding electrode and six mirror electrodes, the
electrostatic mirror (b) creates the trapping potential. At the injection and
ejection sides, respectively, an einzel lens and an ion steerer allow for ion-
beam adjustments (c). Both arms of the MR-ToF mass spectrometer are arranged
with mirror symmetry.

The innermost electrode assembly includes a support electrode for the drift
tube as well as an in-trap deflector electrode. The central drift tube can be
plugged onto the support and is fixed to it by integrated screws. The in-trap
deflector, which enables mass-selective ion deflection during the ion trapping
phase, is formed by a quartered ring electrode which connects to the drift
section. When applying an attractive or a repulsive potential to a deflector
segment (or opposite potentials to opposing segments), ions can be kicked off
their trajectory and are thus removed from the beam [36]. Both components, the
drift tube support and the in-trap deflector, are supported by two holder
rings from which they are insulated by ceramic spacers. In order to define the
axial orientation, all components are mounted on four threaded rods. Their
axial distance is specified and fixed by nuts on these rods (see Fig. 7a).
Following the innermost module outwards, the mirror-electrode stack forms the
core of the MR-ToF system. A shielding electrode and six mirror electrodes are
stacked with a distance of $5\text{\,}\mathrm{mm}$, electrically insulated by
MACOR® spacer plates and mounted on a common base plate. To keep the
electrodes axially aligned, insulated holder bars fix the electrode
orientation at two additional positions (see Fig. 7b).
At both ends of the MR-ToF MS, a lens and ion steering module completes the
assembly. Similar to the in-trap module, presented above, a lens electrode and
a quartered steering electrode, identical in construction to the in-trap
deflector, are supported by four threaded rods and two mounting rings. One of
the latter, together with the steering electrode, serves as grounded lens
segment in front of and behind the lens electrode (see Fig. 7c). To protect
the ceramics from mechanical stress during temperature fluctuations, the
assemblies, which are secured by threaded rods, are spring-loaded. It allows
movement of the electrodes and decouples the electrodes from the thermal
expansion of the threaded rods. All electrodes are connected to standard SHV
feedthroughs via in-vacuum Kapton cables and beryllium-copper connectors,
which are plugged onto set screws integrated in the electrodes.

###  4.2 In-vacuum support system

The optical table that supports the complete MR-ToF system is made of
titanium. It is reinforced by two additional $1200\text{\,}\mathrm{mm}$ long,
$30\text{\,}\mathrm{mm}$ wide and $3\text{\,}\mathrm{mm}$ thick titanium
plates to form a U-shaped profile. Originally, only one plate was planned but
it was found that the plate bends too much under the weight of the electrodes.
The material has been chosen because of the low thermal expansion coefficient,
$8.35\text{\times}{10}^{-6}\text{\,}\mathrm{K}^{-1}$ for temperatures between
$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$, compared to
$16.5\text{\times}{10}^{-6}\text{\,}\mathrm{K}^{-1}$ for 1.4404-type stainless
steel and $23.03\text{\times}{10}^{-6}\text{\,}\mathrm{K}^{-1}$ for aluminium
[42]. The lowest possible thermal expansion is in the interest of a stable
potential profile and thus an improved mass resolving power for elongated
measurement timescales. Consequently, it is advantageous to limit temperature
fluctuations of the in-vacuum components and electrodes. They may most likely
arise from variations of the temperature in the laboratory environment.
Therefore, it is preferable to decouple the in-vacuum system as much as
possible from the surrounding vacuum chamber. Hence, the titanium plate and,
thus, the entire MR-ToF electrode system is designed to rest only on the
conical tips of four set screws while its position is fixed by two additional
screws. The set screws that define the support points are height-adjustable
through threaded holes integrated in two dedicated mounting plates attached to
the vacuum chamber.

##  5 Current status

The PUMA offline ion source setup as shown in Fig. 1 has been assembled from
the ion source to the Paul trap. The electron impact ionisation source has
been commissioned with different gases, such as air, hydrogen, argon, neon and
xenon. With an electron energy of about $100\text{\,}\mathrm{eV}$, multiply-
charged ions can be generated. For the commissioning of the MR-ToF MS, using
argon as target material, a continuous ion beam of approximately
$10\text{\,}\mathrm{pA}$ is extracted and chopped by pulsing a pair of
deflector electrodes included in the ion source with a
$200\text{\,}\mathrm{ns}$ gate from a deflection to a transmission state. This
results in ion packets with a FWHM of the ToF distribution of about
$250\text{\,}\mathrm{ns}$, measured with a ToF detector behind the MR-ToF MS.
Due to the largest abundance, 40Ar+ was selected to be used for first test
measurements. Starting from the potential configuration determined in Sec. 3.4
and compensating possible misalignments by manually tuning the mirror
electrodes and the in-trap lift voltage, the ToF focus could be set on the
detector behind the device after trapping the ions for about 100 - 150
revolutions. For longer storing times and the given voltage applied to the in-
trap lift electrode, the ToF signal is defocused again. This is shown in Fig.
8 together with a plot of the mass resolving power as a function of the number
of revolutions that the 40Ar+ ions are trapped in the MR-ToF MS for. With a
signal width of $\Delta t=$34\text{\,}\mathrm{ns}$$ (FWHM), the highest mass
resolving power of $R=m/\Delta m=t/2\Delta t\sim 50,000$ is reached after 150
revolutions.

Figure 8: Ion intensity of 40Ar+ ions measured as a function of the ToF after
the ejection from the MR-ToF MS $\tau$ and the number of revolutions $N$ for
which the ions are trapped in the MR-ToF MS for (top). Evolution of the mass
resolving power as a function of $N$ (bottom). Figure 9: Ion intensity of Xe+
ions measured as a function of the ToF after the ejection from the MR-ToF MS
$\tau$ and the number of revolutions $N$ for which the ions are trapped in the
MR-ToF MS for. All three plots show the same measurement, where once the
deflector is switched off (left), once the deflector is switched on and
synchronised to A=132 (centre) and once synchronised to A=131 (right). In case
of the right plot, the capture time is shifted by
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to keep the 132Xe+ ions in the lift
for a larger number of revolutions.

In order to demonstrate the capability for isotopic purification, the in-trap
deflector electrodes are commissioned using xenon as target gas, which has
nine natural isotopes. A fast high voltage switch (Stahl-Electronics, HS-2000)
is used to switch the potential applied to two opposite segments of the
quadruple segmented deflector electrode between $0\text{\,}\mathrm{V}$ and
$\pm 100\text{\,}\mathrm{V}$, respectively. Thus, the in-trap deflector can be
switched between a deflecting ($200\text{\,}\mathrm{V}$ potential difference)
and a transmission state ($0\text{\,}\mathrm{V}$ potential difference). The
rising and falling times are typically below $500\text{\,}\mathrm{ns}$ and
therefore enable undisturbed deflection of ion species separated by a few
$\text{\,}\mathrm{\SIUnitSymbolMicro s}$. By synchronising the switch to the
revolution period of a desired ion mass-over-charge ratio, the deflector can
be used for a mass-selective transversal ion ejection [36]. The left-hand plot
in Fig. 9 shows the measured ion intensity of singly-charged xenon isotopes as
a function of the time-of-flight $\tau$ after the ejection from the MR-ToF MS
(abscissa) and the number of revolutions the ions are trapped in the MR-ToF MS
for (ordinate). Here, the deflector is not active. Below
$\tau=$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ (above
$\tau=$18\text{\,}\mathrm{\SIUnitSymbolMicro s}$$) ions with a smaller
(larger) mass-to-charge ratio are not located in the in-trap lift electrode
during switching, resulting in them not being ejected from the MR-ToF MS. The
centre plot shows the same measurement but with the deflector activated using
a duty cycle of 67%. The latter refers to the ratio between the length of the
deflection state of $27\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and the
revolution period of $40.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$ for A/z=132
ions, where A is the mass number and z the charge state. In this case, the
transmission window of the deflector is synchronised and centred on mass
A/z=132. In the right-hand plot, the measurement is repeated with the
transmission window synchronised and centred to mass A/z=131, allowing to
isolate 131Xe ions. To increase the number of revolutions for which 131Xe ions
can still be ejected, the capture time was decreased by
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$, which is the time between the ion
extraction from the source and the switching of the in-trap lift. This results
in a shift of $\tau$ by $2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to a longer
ToF.

##  6 Conclusion and outlook

In the framework of the PUMA project, a new MR-ToF-MS has been developed for
the PUMA offline ion source setup, allowing for fast isotopic ion beam
purification. To increase the transmission of the device and, thus, decreasing
the accumulation time of $10^{5}$ ions in an RFQ ion trap, which is mandatory
for PUMA, SIMION® simulations have been performed modifying the electrode
geometry of an MR-ToF MS design from the University of Greifswald. This has
predominantly led to an increase in the inner diameters and the corresponding
adjustment of the individual electrode lengths. As a result, the transmission
could be increased by a factor of about eight, as shown by benchmarking
simulations with realistic mirror potential profiles. To facilitate the
assembly and allow for later modifications, the design is based on several
electrode sub-assemblies. The setup can be used to capture the ions of
interest and separate them from unwanted ion species. Furthermore, focus and
steering electrodes are included in the design on both the injection and the
ejection sides of the device.
The MR-ToF MS has been commissioned, using a chopped argon ion beam, extracted
from a commercial electron impact ionisation source. First mirror tuning
attempts have led so far to a mass resolving power of 50,000, trapping Ar+
ions for 150 revolutions. Furthermore, the in-trap deflector has been
commissioned using Xe+ ions. For the examples 131Xe+ and 132Xe+, it could be
demonstrated that the device is able to successfully isolate isotopes of
interest from a stable ion spectrum.
Already during the simulation phase, an increasing number of institutes
expressed their interest in an MR-ToF device. As a result, the Darmstadt’s MR-
ToF (DA’s MR-ToF) Collaboration has been formed to distribute the
manufacturing and acquisition of the different parts among seven institutions.
These collaboration partners are the Technical University of Darmstadt, the
University of Greifswald, the University of Groningen, the University of
Manchester, the Massachusetts Institute of Technology, the Johannes Gutenberg
University Mainz and the University of Innsbruck.

## Acknowledgements

We gratefully acknowledge the discussions with the staff of the mechanical
workshop at TU Darmstadt and thank the contributing institutes and workshops
for their support. The PUMA project is funded by the European Research Council
through the ERC grant PUMA-726276. J.F., C.K., A.O., A.S. and M.S. appreciate
the support from the Alexander von Humboldt Foundation.

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# Electromagnetic triangle anomaly and neutral pion condensation in QCD vacuum

Gaoqing Cao  Physics Department and Center for Particle Physics and Field
Theory, Fudan University, Shanghai 200433, China.  Xu-Guang Huang  Physics
Department and Center for Particle Physics and Field Theory, Fudan University,
Shanghai 200433, China.

###### Abstract

We study the QCD vacuum structure under the influence of an electromagnetic
field with a nonzero second Lorentz invariant $I_{2}={\bm{E}}\cdot{\bm{B}}$.
We show that the presence of $I_{2}$ can induce neutral pion ($\pi^{0}$)
condensation in the QCD vacuum through the electromagnetic triangle anomaly.
Within the frameworks of chiral perturbation theory at leading small-momenta
expansion as well as the Nambu–Jona-Lasinio model at leading $1/N_{c}$
expansion, we quantify the dependence of the $\pi^{0}$ condensate on $I_{2}$.
The stability of the $\pi^{0}$-condensed vacuum against the Schwinger charged
pair production due to electric field is also discussed.

###### pacs:

12.38.Aw, 12.39.Fe, 11.30.Rd

##  I Introduction

The vacuum structure of Quantum Chromodynamics (QCD) has been intensively
studied over many years and a variety of interesting properties have been
extracted. For example, it is well-known that the normal QCD vacuum possesses
the spontaneous (approximate) chiral symmetry breaking (CSB) which can be
characterized by the scalar quark-antiquark condensate
$\langle{\bar{\psi}}\psi\rangle\neq 0$; while the pseudoscalar condensates
like $\langle{\bar{\psi}}i\gamma_{5}\tau_{3}\psi\rangle$ with $\tau_{3}$ the
third Pauli matrix are not permitted — an assertion known as the Vafa-Witten
theorem Vafa:1984xg .

Under proper conditions, the QCD vacuum structure can be altered, usually
through phase transitions. Examples include the pion superfluid phase at high
isospin chemical potential Son:2000xc ; Kogut:2002zg ; He:2005nk  and the
color superconducting phase at high baryon chemical potential Alford:1997zt ;
Rapp:1997zu ; Alford:2007xm ; in the first case the isospin symmetry is broken
while the chiral symmetry is restored and in the second case the color
symmetry is broken while the chiral symmetry may or may not be broken.

Strong electromagnetic (EM) fields provide another way to modify the QCD
vacuum. Strong magnetic fields may exist in compact stars Duncan:1992hi ;
Thompson:1993hn ; Olausen:2013bpa , heavy-ion collisions Skokov:2009qp ;
Deng:2012pc ; Huang:2015oca , and early universe Vachaspati:1991nm ;
Baym:1995fk ; Grasso:2000wj . Strong electric fields may also be accessible in
heavy-ion collisions Hirono:2012rt ; Deng:2014uja ; Voronyuk:2014rna . The
presence of the magnetic field is known to enhance the scalar
$\langle{\bar{\psi}}\psi\rangle$ condensate at zero quark chemical potentials
which is called the magnetic catalysis of the CSB Gusynin:1994re ;
Gusynin:1995nb ; Miransky:2015ava ; Klimenko:1991he , while the effect of the
magnetic field on the CSB at finite temperature or chemical potentials shows
novel features which are still not fully understood Preis:2010cq ; Bali:2011qj
; Bali:2012zg ; Fukushima:2012kc ; Kojo:2012js ; Chao:2013qpa . It was also
proposed that when $|e{\bm{B}}|\gtrsim m_{\rho}^{2}$ with $m_{\rho}$ the mass
of the rho meson, the QCD vacuum will become a superconductor due to charged
rho condensation Chernodub:2011mc . The effect of electric field on the CSB
was also studied Klevansky:1989vi ; Babansky:1997zh ; Cohen:2007bt  and it was
found that the electric field always tends to break the scalar quark-antiquark
pairs and thus weaken the CSB.

Figure 1: The electromagnetic triangle anomaly.

In this Letter, we study the combined effect of the electric and magnetic
fields on QCD vacuum. We focus on the EM-field configuration where the second
Lorentz invariant $I_{2}={\bm{E}}\cdot{\bm{B}}$ is nonzero. We show that the
presence of $I_{2}$ provides a parity-odd environment in which the otherwise-
forbidden (by Vafa-Witten theorem) neutral pion condensation can occur via the
EM triangle anomaly (see Fig. 1). We calculate the $\pi^{0}$ condensate within
both the chiral perturbation theory (ChPT) and the Nambu–Jona-Lasinio (NJL)
model frameworks and identical result is obtained,

$\displaystyle\frac{\pi^{0}}{m^{*}}=\left\\{\begin{array}[]{ll}{\displaystyle\frac{N_{c}}{4\pi^{2}f^{2}_{\pi}m_{\pi}^{2}}(q_{u}^{2}-q_{d}^{2}){\bm{E}}\cdot{\bm{B}}}\;\;\;{\rm
for}\,\;|I_{2}|<I_{2}^{c},\\\ {\rm
sgn}(I_{2})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm
for}\;|I_{2}|>I_{2}^{c},\end{array}\right.\ $ (3)

where $m^{*}=\sqrt{(\pi^{0})^{2}+\sigma^{2}}$ with
$\sigma\sim\langle{\bar{\psi}}\psi\rangle$ and
$\pi^{0}\sim\langle{\bar{\psi}}i\gamma_{5}\tau_{3}\psi\rangle$ the scalar and
neutral pion condensates in the presence of the EM field, $q_{u}=2e/3$ and
$q_{d}=-e/3$ are charges of $u$ and $d$ quarks with $e>0$ the proton charge,
and $I_{2}^{c}=4\pi^{2}f^{2}_{\pi}m_{\pi}^{2}/[N_{c}(q_{u}^{2}-q_{d}^{2})]$ is
a critical value for $|I_{2}|$ above which the condensate $m^{*}$ is wholly
contributed by $\pi^{0}$ mode. Equation (3) shows that the $I_{2}$ rotates the
chiral condensate from the $\sigma$-direction to the $\pi^{0}$-direction with
the rotation angle given by $\phi=\sin^{-1}(\pi^{0}/m^{*})$ if
$|I_{2}|<I_{2}^{c}$ and $\phi={\rm sgn}(I_{2})\pi/2$ if $|I_{2}|>I_{2}^{c}$
(see Fig. 2).

Figure 2: Illustration of the chiral rotation due to the neutral pion
condensation induced by EM triangle anomaly.

Throughout this article, we restrict to the zero quark chemical potential and
zero temperature case. Before we proceed, we note that the neutral pion
condensation we will study is different from the disoriented chiral condensate
(DCC) Mohanty:2005mv  proposed to possibly occur in heavy-ion collisions, as
the latter is a far-from-equilibrium, transient, phenomenon while the former
is a nearly equilibrium, (quasi-)stationary, state. We point out that, in the
context of DCC, it was also found that the large EM fields in heavy-ion
collisions could also possibly catalyze the formation of the DCC
Minakata:1995gq ; Asakawa:1998st .

##  II Invalidation of the Vafa-Witten Theorem

The EM-triangle-anomaly induced neutral pion condensation does not contradict
the Vafa-Witten theorem. The validity of the Vafa-Witten theorem relies on the
positivity, $\det{\cal D}>0$, where ${\cal D}$ is the Euclidean Dirac operator
Vafa:1984xg ; Vafa:1983tf . However, this is not the case in the presence of a
real electric field. To see this, we write down $\cal D$ explicitely,

$\displaystyle{\cal D}=\gamma_{\mu}(\partial_{\mu}-ig{\cal
A}_{\mu})+Q\gamma_{4}A_{0}+iQ\gamma_{i}A_{i}+M,$ (4)

where ${\cal A}_{\mu}$ ($\mu=1-4$) is the gluon field, $A_{\alpha}$
($\alpha=0-3$) is the EM field, and $M$ is the mass matrix. We choose the
gauge such that ${\bm{E}}=-{\bm{\nabla}}A_{0}$ and $A_{i}$ is time
independent. For two-flavor case, the charge matrix is
$Q/e={\rm{diag}}(2/3,-1/3)=1/6+\tau_{3}/2$ and the mass matrix is $M=m_{0}$.
The crucial observation is that $Q$ is not traceless and thus its role is
similar with the baryon chemical potential $\mu$. It is well-known that the
presence of $\mu$ destroys the positivity of $\det{\cal D}$, and so does
$A_{0}$. In fact, when $A_{0}\neq 0$ there does not exist a matrix $\Gamma$
such that $\Gamma^{-1}{\cal D}\Gamma={\cal D}^{\dagger}$ to guarantee the
positivity. Choosing another gauge, e.g. $A_{0}=0$, does not change the
conclusion. Because in this case, $A_{i}$ must depend on time, e.g. for
constant ${\bm{E}}$, $A_{i}=-E_{i}t$, which after Wick rotation spoils again
the positivity footnote . Note that if $q_{u}=-q_{d}$, $\det{\cal D}$ is semi-
positive Yamamoto:2012bd .

The invalidation of the Vafa-Witten theorem in the electric field has
important physical contents. Physically, in the presence of ${\bm{E}}$, the
Dirac determinant has to be complex or at most semi-positive because it always
receives contributions from the Schwinger poles which represent the particle-
antiparticle pair creation due to ${\bm{E}}$ Schwinger:1951nm .

##  III Chiral perturbation theory calculation

We start with the two-flavor ChPT described by the Lagrangian

$\displaystyle{\cal L}={\cal L}_{0}+{\cal L}_{\rm WZW},$ (5)

where ${\cal L}_{0}$ is the usual chiral Lagrangian given by (we keep only
$O(p^{2})$ terms)

$\displaystyle{\cal L}_{0}=\frac{f_{\pi}^{2}}{4}{\rm
tr}\left[D_{\mu}U^{\dagger}D^{\mu}U+m_{\pi}^{2}(U+U^{\dagger})\right],$ (6)

and the Wess-Zumino-Witten term ${\cal L}_{\rm WZW}$ is given by Wess:1971yu ;
Witten:1983tw ; Scherer:2012xha ; Son:2007ny ; Fukushima:2012fg

$\displaystyle{\cal L}_{\rm
WZW}=\frac{N_{c}}{48\pi^{2}}A_{\mu}\epsilon^{\mu\nu\alpha\beta}[{\rm
tr}\left(QL_{\nu}L_{\alpha}L_{\beta}+QR_{\nu}R_{\alpha}R_{\beta}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-iF_{\alpha\beta}T_{\nu}],$
(7) $\displaystyle T_{\nu}={\rm
tr}\left[Q^{2}(L_{\nu}+R_{\nu})+\frac{1}{2}\left(QUQU^{\dagger}L_{\nu}+QU^{\dagger}QUR_{\nu}\right)\right],$
$\displaystyle L_{\mu}=U\partial_{\mu}U^{\dagger},$ $\displaystyle
R_{\mu}=\partial_{\mu}U^{\dagger}U.$ (8)

In the above, the covariant derivative is given by

$\displaystyle D_{\mu}U=\partial_{\mu}U+A_{\mu}[Q,U],$ (9)

and $U$ is a $2\times 2$ unitary matrix representing the chiral fields, for
which we choose the Weinberg parametrization,

$\displaystyle U=\frac{1}{f_{\pi}}(s+i{\bm{\tau}}\cdot{\bm{t}}),$ (10)

where the fields $s$ and ${\bm{t}}$ fulfill the constraint
$s^{2}+{\bm{t}}^{2}=f_{\pi}^{2}$.

Now we consider constant EM field and make the approximation such that we
treat all the chiral fields in (5) as uniform condensates. We note that the
$\sigma$ and $\pi^{0}$ in Eq. (3) are proportional to $s$ and $t_{3}$,
respectively. In the absence of the EM field, it is well-known that the CSB
takes place along the $\sigma$-direction so that $s=f_{\pi}$ and
${\bm{t}}={\bm{0}}$. The EM field with a configuration where
$I_{2}={\bm{E}}\cdot{\bm{B}}\neq 0$ is coupled to the $\pi^{0}$ field via the
triangle anomaly and raises the possibility of $\pi^{0}$ condensation. If this
happens, the CSB will be rotated from the $\sigma$-direction to
$\pi^{0}$-direction. To explore this possibility, we make the following ansatz
for the condensates,

$\displaystyle s$ $\displaystyle=$ $\displaystyle
f_{\pi}\cos\phi,\;\;t_{3}=f_{\pi}\sin\phi,$ (11) $\displaystyle t_{1}$
$\displaystyle=$ $\displaystyle t_{2}=0,$ (12)

where without loss of generality, we choose $\phi\in[-\pi/2,\pi/2]$ so that
$s\geq 0$. The Lagrangian ${\cal L}$ then becomes

$\displaystyle{\cal L}(\phi)$ $\displaystyle=$ $\displaystyle
f^{2}_{\pi}m_{\pi}^{2}\cos\phi+\frac{N_{c}I_{2}}{4\pi^{2}}{\rm
tr}(Q^{2}\tau_{3})\phi,$ (13)

where we have omitted a total derivative term. As obviously, the second term
is responsible for the triangle anomaly effects such as $\pi^{0}\rightarrow
2\gamma$. It is easy to minimize $-{\cal L}(\phi)$ and obtain

$\displaystyle\sin\phi$ $\displaystyle=$
$\displaystyle\frac{N_{c}I_{2}}{4\pi^{2}f_{\pi}^{2}m_{\pi}^{2}}{\rm
tr}(Q^{2}\tau_{3})$ (14) $\displaystyle=$
$\displaystyle\frac{N_{c}I_{2}}{4\pi^{2}f_{\pi}^{2}m_{\pi}^{2}}(q_{u}^{2}-q_{d}^{2}),\;\;\;\;{\rm
for}\;\;|I_{2}|<I_{2}^{c},$ $\displaystyle\phi$ $\displaystyle=$
$\displaystyle{\rm sgn}\left(I_{2}\right)\frac{\pi}{2},\;\;{\rm
for}\;\;|I_{2}|>I_{2}^{c},$ (15)

where $I_{2}^{c}=4\pi^{2}f_{\pi}^{2}m_{\pi}^{2}/[N_{c}(q_{u}^{2}-q_{d}^{2})]$.
This is equivalent to Eq. (3).

Several comments are in order. (1) In the chiral limit $m_{\pi}\rightarrow 0$,
thus $I_{2}^{c}\rightarrow 0$, an infinitesimal $I_{2}$ will drive the maximum
$\pi^{0}$ condensation, namely, the CSB is completely driven to
$\pi^{0}$-direction once $I_{2}$ is applied. (2) The neutral pion condensation
induced by $I_{2}$ is merely a chiral rotation; it is not a phase transition
as no symmetry is broken along this process. (3) As a low-energy effective
theory of QCD, the ChPT is reliable only when all the physical parameters,
$m_{\pi},p,\sqrt{|e{\bm{B}}|},\sqrt{|e{\bm{E}}|}$, are much smaller than the
typical hadronic scale $\Lambda_{\chi}\sim 1$ GeV. We can estimate that

$\displaystyle\frac{e^{2}I_{2}^{c}}{\Lambda_{\chi}^{4}}\sim\frac{m_{\pi}^{2}}{\Lambda_{\chi}^{2}}\ll
1,$ (16)

which justifies the validity of our result for $I_{2}$ not too larger than
$I_{2}^{c}$. (4) As well-known, when $|e{\bm{E}}|\gg m_{\pi}^{2}$ (but not
larger than $\Lambda_{\chi}^{2}$), the vacuum will be unstable because of the
Schwinger pair production of $\pi^{+}\pi^{-}$. We will discuss this issue in
another section.

##  IV Nambu–Jona-Lasinio Model calculation

The ChPT is built on hadronic degrees of freedom, we now use the NJL model
which is based on quark degrees of freedom to test the consequences of the
ChPT. The Lagrangian of the two-flavor NJL model is given by Klevansky:1992qe

$\displaystyle{\cal L}_{\rm
NJL}={\bar{\psi}}(i\not{D}-m_{0})\psi+G[({\bar{\psi}}\psi)^{2}+({\bar{\psi}}i\gamma_{5}{\bm{\tau}}\psi)^{2}],$
(17)

where $\psi=(u,d)^{T}$ represents the two-flavor quark fields, $m_{0}$ is the
current mass of quarks, $G$ is the four-fermion coupling constant, and

$\displaystyle D_{\mu}=\partial_{\mu}+iQA_{\mu},$ (18)

is the covariant derivative. Note that, in addition to the coupling constant
$G$, the NJL model contains implicitly another parameter, the ultraviolet
cutoff $\Lambda$, which specifies the applicable region of the model. The
values of $G$ and $\Lambda$ are determined by fitting NJL predictions with the
physical hadronic observables like $f_{\pi}$ and
$\langle{\bar{\psi}}\psi\rangle$ in the normal vacuum Klevansky:1992qe .

The NJL model was widely used to study a number of non-perturbative properties
of QCD, especially those related to the chiral symmetry. In fact, the NJL
model shares the same global symmetries with QCD. At the mean-field level
where only the leading-order terms in the $1/N_{c}$ expansion are kept,

$\displaystyle{\cal L}_{\rm
NJL}={\bar{\psi}}(i\not{D}-m-i\pi^{0}\gamma_{5}\tau_{3})\psi-\frac{\sigma^{2}+(\pi^{0})^{2}}{4G},$
(19)

where $m=m_{0}+\sigma$ is the constitute mass of quarks and
$\sigma=-2G\langle{\bar{\psi}}\psi\rangle$ and
$\pi^{0}=-2G\langle{\bar{\psi}}i\gamma_{5}\tau_{3}\psi\rangle$ represent the
sigma and neutral pion condensates. The quark fields can then be integrated
out analytically, which is most conveniently done in Euclidean path integral
and the resultant thermodynamic potential reads

$\displaystyle\Omega=\frac{(m-m_{0})^{2}+(\pi^{0})^{2}}{4G}-\frac{1}{V_{4}}{\rm
Tr}\ln S^{-1},$ (20)

where $V_{4}$ is the Euclidean spacetime volume, ${\rm Tr}$ acts on all
physical spaces, and the quark propagator is

$\displaystyle
S(x,x^{\prime})=-(i\not{D}-m-i\pi^{0}\gamma_{5}\tau_{3})^{-1}\delta^{(4)}(x-x^{\prime}).$
(21)

It is diagonal in flavor space which allows us to calculate it analytically by
adopting the Schwinger proper-time formalism Schwinger:1951nm . The result is

$\displaystyle{\cal S}_{f}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle{-i\over(4\pi)^{2}}\int_{0}^{\infty}{ds\over
s^{2}}\;e^{-iq_{f}\int_{x^{\prime}}^{x}A\cdot dx}\big{[}-{1\over
2}\gamma\big{(}q_{f}F\coth(q_{f}Fs)-q_{f}F\big{)}(x-x^{\prime})+m-{\rm
sgn}(q_{f})i\gamma^{5}\pi^{0}\big{]}$
$\displaystyle\times\exp\Big{\\{}-i\big{[}m^{2}+(\pi^{0})^{2}\big{]}s+{i\over
4}(x-x^{\prime})q_{f}F\coth(q_{f}Fs)(x-x^{\prime})+{i\over 2}q_{f}\sigma
Fs\Big{\\}}{-(q_{f}s)^{2}I_{2}\over\text{Im}\cosh\big{(}iq_{f}s(I_{1}+2iI_{2})^{1/2}\big{)}},$

where $f=u,d$, $I_{1}={\bm{B}}^{2}-{\bm{E}}^{2}$ is the first Lorentz
invariant, and we have suppressed some of the Lorentz scripts, e.g., $\sigma
F$ should be understood as $\sigma^{\mu\nu}F_{\nu\mu}$ with
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. The true vacuum
corresponds to the global minimum of $\Omega$. In the case that $I_{1}=0$ and
$I_{2}>0$, we have numerically minimized $\Omega$ and obtained the
corresponding $m$ and $\pi^{0}$, see Fig. 3. Once $I_{2}$ is turned on, finite
$\pi^{0}$ condensate arises and sigma condensate is suppressed — a feature
representing a chiral rotation from $\sigma$-direction to $\pi^{0}$-direction;
when $I_{2}^{1/4}\gtrsim 0.28$ GeV, the CSB is driven by $\pi^{0}$ condensate
solely. In addition, the total condensate $m^{*}=\sqrt{m^{2}+(\pi^{0})^{2}}$
is weakened and finally killed by $I_{2}$ due to the presence of the electric
field Babansky:1997zh ; Cohen:2007bt .

To gain more insights, we write down the gap equations which is valid when
$|I_{2}|<I_{2}^{c}$ (see below):

$\displaystyle{m-m_{0}\over 2G}$ $\displaystyle=$
$\displaystyle\frac{mN_{c}}{4\pi^{2}}\sum_{f=u,d}\int_{0}^{\infty}dse^{-\big{[}m^{2}+(\pi^{0})^{2}\big{]}s}\;{q_{f}^{2}I_{2}{\rm
Re}\cosh[q_{f}s\sqrt{I_{1}+2iI_{2}}]\over{\rm
Im}\cosh[q_{f}s\sqrt{I_{1}+2iI_{2}}]}-{N_{c}\over
4\pi^{2}}{\pi^{0}\over{m^{2}+(\pi^{0})^{2}}}(q_{u}^{2}-q_{d}^{2})I_{2},$ (22)
$\displaystyle{\pi^{0}\over 2G}$ $\displaystyle=$
$\displaystyle\frac{\pi^{0}N_{c}}{4\pi^{2}}\sum_{f=u,d}\int_{0}^{\infty}dse^{-\big{[}m^{2}+(\pi^{0})^{2}\big{]}s}\;{q_{f}^{2}I_{2}{\rm
Re}\cosh[q_{f}s\sqrt{I_{1}+2iI_{2}}]\over{\rm
Im}\cosh[q_{f}s\sqrt{I_{1}+2iI_{2}}]}+{N_{c}\over
4\pi^{2}}{m\over{m^{2}+(\pi^{0})^{2}}}(q_{u}^{2}-q_{d}^{2})I_{2}.$ (23)

By eliminating $m$, we analytically obtain

$\displaystyle\pi_{0}=\frac{N_{c}}{4\pi^{2}}\frac{2G}{m_{0}}(q^{2}_{u}-q^{2}_{d})I_{2}.$
(24)

By assuming the NJL version of the Gell-Mann-Oakes-Renner relation,
$m^{2}_{\pi}f_{\pi}^{2}=m_{0}m^{*}(2G)^{-1}$ Klevansky:1992qe ; GOR , Eq. (24)
immediately reduces to Eq. (3) for $|I_{2}|<I_{2}^{c}$ which is independent of
the model parameters $G$ and $\Lambda$.

Two comments: (1) If $m_{0}=0$ or $m_{\pi}=0$, the above formula is divergent.
This corresponds to the chiral limit case where once $I_{2}$ is turned on the
CSB is immediately rotated to the $\pi^{0}$-direction. (2) When
$|I_{2}|>I_{2}^{c}$, one of the gap equations is not applicable because $m$ is
already zero. This situation is consistent with the ChPT result. In both NJL
and ChPT frameworks, when $|I_{2}|>I_{2}^{c}$, the CSB is found to be totally
driven by $\pi^{0}$ condensation.

Figure 3: The constitute quark mass $m$ and $\pi^{0}$ condensate as functions
of $I_{2}^{1/4}$ obtained by minimizing the thermodynamic potential $\Omega$
at $I_{1}=0$ in NJL model. The presence of $I_{2}$ always tends to diminish
$\sigma$ condensate while drive $\pi^{0}$ condensation.

##  V Stability of the $\pi^{0}$-condensed vacuum

The neutral pion condensation found above requires a nonzero electric field.
As we know, strong electric field can efficiently generate particle-
antiparticle pairs (in the confined phase, mostly $\pi^{+}$ and $\pi^{-}$; in
the deconfined phase, quarks and antiquarks) through the Schwinger mechanism
and drive the vacuum unstable. So it is necessary to analyze that under which
condition the $\pi^{0}$-condensed vacuum is stable. For this purpose, we need
to calculate the pair creation rate. This can be achieved within the ChPT
framework by using the Schwinger proper-time method, see, e.g., Refs.
Kim:2003qp ; Cohen:2007bt . We will not repeat their calculations here, but
just quote their results relevant to our discussion.

In case that $I_{2}\neq 0$, we can always find a frame in which ${\bm{E}}$ and
${\bm{B}}$ are parallel. We will work in such a frame. The imaginary part of
the leading order effective Lagrangian, which is responsible for the
$\pi^{+}\pi^{-}$ production, reads

$\displaystyle{\rm{Im}}{\cal
L}\\!=\\!\frac{e^{2}|EB|}{16\pi^{2}}\\!\\!\\!\sum_{n,l=1}^{\infty}\\!\\!\\!\frac{(\\!-\\!1)^{n+1}}{n}\exp\\!\\!\left[\\!-n\pi\frac{m_{\pi}^{2}\\!+\\!|eB|(2l\\!-\\!1)}{|eE|}\right],$

where $n$ runs over all Schwinger poles and $l$ runs over all Landau levels.
The first term in the summation over $n$ defines the $\pi^{+}\pi^{-}$
production rate per unit volume which reads,

$\displaystyle
w_{\pi^{+}\pi^{-}}=\frac{e^{2}|EB|}{16\pi^{2}}\sum_{l=1}^{\infty}\exp\left[-\pi\frac{m_{\pi}^{2}+|eB|(2l-1)}{|eE|}\right].$
(26)

It is clear that the presence of the magnetic field effectively enhances the
mass of charged pions,

$\displaystyle m^{2}_{\pi^{\pm}}=m^{2}_{\pi}+|eB|,$ (27)

and therefore as long as

$\displaystyle|eE|\ll m^{2}_{\pi}+|eB|,$ (28)

the rate of $\pi^{+}\pi^{-}$ production is strongly suppressed. In this case,
the pair-producing instability takes place over a very long time $t\sim
1/(w_{\pi^{+}\pi^{-}}V)$ ($V$ the volume) and it is meaningful to consider the
“static” property of the vacuum. The “equilibrium state” neutral pion
condensate we have studied so far is sensible when the condition (28) is
satisfied.

##  VI Summary

In this article, we propose a mean to tune the QCD vacuum by applying parallel
electric and magnetic fields. The underlying mechanism is the electromagnetic
triangle anomaly, through which the second Lorentz invariant
$I_{2}={\bm{E}}\cdot{\bm{B}}$ of the EM field can induce a neutral pion
condensate in QCD vacuum and lead to a chiral rotation from the isosinglet
scalar $\sigma$-direction toward the isotriplet pseudoscalar
$\pi^{0}$-direction. By adopting the chiral perturbation theory at leading
order plus the Wess-Zumino-Witten term and the Nambu–Jona-Lasinio model at the
mean-field approximation (leading order in $1/N_{c}$ expansion), we show that
the sine of the rotation angle is universally given by Eq. (3).For weak
electric field, our results may be testable in lattice QCD by adopting similar
strategies as the ones used in lattice simulations at finite baryon chemical
potential DElia:2012zw .

Our finding may have experimental implications in heavy-ion collisions. In
ultra-peripheral heavy-ion collisions the hot quark-gluon matter may not form
but quite large $I_{2}$ with opposite signs above and below the reaction plane
may exist Deng:2012pc  which can induce transient $\pi^{0}$ condensation. The
diphoton spectrum may be used to distinguish the $\pi^{0}$-condensed vacuum
from the normal vacuum, because in the $\pi^{0}$-condensed vacuum the $\sigma$
and $\pi^{0}$ excitations are mixed which will change the diphoton emission
rate and spectrum. We will study these experimental signatures in near future.

_Acknowledgments_ — We thank T. Brauner for very useful comments. GC and XGH
are supported by Shanghai Natural Science Foundation (Grant No. 14ZR1403000),
the Key Laboratory of Quark and Lepton Physics (MOE) of CCNU (Grant No.
QLPL20122), the Young 1000 Talents Program of China, and Scientific Research
Foundation of State Education Ministry for Returned Scholars. GC is also
supported by China Postdoctoral Science Foundation (Grant No. KLH1512072).

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