# Nonlinear Regge trajectories and saturation of the Hagedorn spectrum

István Szanyi<EMAIL_ADDRESS>Wigner Research Centre for Physics,
H-1525 Budapest 114, POBox 49, Hungary  Eötvös University, H-1117 Budapest,
Pázmány P. s. 1/A, Hungary  MATE Institute of Technology, Károly Róbert
Campus, H-3200 Gyöngyös, Mátrai út 36, Hungary  Tamás Biró
<EMAIL_ADDRESS>Wigner Research Centre for Physics, H-1525 Budapest 114,
POBox 49, Hungary  László Jenkovszky<EMAIL_ADDRESS>Bogolyubov Institute
for Theoretical Physics (BITP), Ukrainian National Academy of Sciences
14-b, Metrologicheskaya str., Kiev, 03680, Ukraine  Vladyslav Libov
<EMAIL_ADDRESS>Bogolyubov Institute for Theoretical Physics (BITP),
Ukrainian National Academy of Sciences
14-b, Metrologicheskaya str., Kiev, 03680, Ukraine

###### Abstract

We argue that two seemingly different phenomena, namely the well-known
saturation of the Hagedorn exponential distribution and the less familiar
saturation of Regge trajectories at resonance masses $m\approx$ 2–2.5 GeV are
related and have the same origin: quark deconfinement. We show that the slope
of the real part of non-linear Regge trajectories determines the prefactor
$f(m)$ in Hagedorn’s resonance mass density distribution $\rho(m)$. While the
Hagedorn distribution comes from statistics, Regge trajectories contain
dynamics.

###### pacs:

13.75, 13.85.-t

##  I Introduction

The spectrum of hadron resonances is among the central problems of high-energy
physics. The properties of highly excited resonances are intimately connected
with the problem of confinement. We revise the spectrum of hadronic resonances
by relating seemingly two different viewpoints: statistical (Hagedorn) and
dynamical (Regge).

The density of hadron states follows the law [1]

$\rho(m)=f(m)\exp(m/T_{H}),$ (1)

where $f(m)$ is a slowly varying function of mass and $T_{H}$ is the Hagedorn
temperature, originally considered as the limiting temperature but later
interpreted as the temperature of the color deconfinement phase transition
where hadrons ”boil” transforming the matter into a boiling quark-gluon soup.

Over 50 years after the publication of R. Hagedorn’s paper [1] on the spectrum
of resonances many important details still remain open. In spite of many
efforts by different groups of authors [2, 3, 4, 5] involving more new data on
resonances, the calculated value of the Hagedorn temperature shows a
surprisingly widespread from $T_{H}=141$ MeV to $T_{H}=340$ MeV, depending on
the parametrization and the set of data (baryons, mesons) used. The
discrepancies may have different origin, in particular: a) the large
uncertainties in the specification and identification of heavy resonances, b)
the analytical form of the Hagedorn spectrum, in particular, the form of the
function $f(m)$ multiplying Hagedorn’s exponential. In the present paper, we
address both issues.

In our analyses, we rely on the latest results of the Particle Data Group [6].
At the same time, we are aware of recent important developments and progress
in theoretical studies of the Hagedorn spectrum and of the ultimate
temperature based on gauge theories [7], lattice QCD (by which one
distinguishes between two different types of resonances: isolated ones in
vacuum, and resonance at a certain temperature), strings, supergravity [8, 9,
10]. Its possible manifestation in the early universe was discussed in Refs.
[11, 12, 13].

Our approach, however, is limited to the world of observed resonances,
summarized by the Particle Data Group and theoretical methods based on
analyticity, unitarity, and duality.

Crucial and original in our paper is the identification of the function $f(m)$
with the derivative of the relevant Regge trajectory. In the spirit of the
analytic $S$-matrix approach, Regge trajectories encode an essential part of
the strong interaction dynamics, they are building blocks of the theory. There
were many attempts to find analytic forms of the non-linear complex Regge
trajectories, based on mechanical analogs (string), quantum chromodynamics
etc., suggesting different and approximate solutions applicable in a limited
range. In our approach, we rely on duality and constraints based on
analyticity and unitarity, constraining the threshold and asymptotic behavior
of the trajectories. An important constraint, affecting the spectrum near its
critical point is an upper bound on the real part of Regge trajectories’,
coming from dual models with Mandelstam analyticity [14]. Construction of
explicit models of the trajectories satisfying the above constraints is a non-
trivial problem. In the present paper, we propose explicit models of such
trajectories allowing explicit calculations and compatible with the data on
resonances.

Below we argue that while Hagedorn’s exponential rise comes mainly from the
proliferation of spin and isospin degeneracy of states with increasing mass,
that can be counted directly, the prefactor $f(m)$ reflects dynamics, encoded
in Regge trajectories given that $f(m)=\alpha^{\prime}(m)$, where
$\alpha^{\prime}(m)$ is the slope of the trajectory.

The low-mass, $m<1.8$ GeV spectra do not exhibit any surprise by following
Hagedorn’s exponential. The only open questions are the value of the Hagedorn
temperature $T_{H}$ and possible differences between the spectra for various
particles. The spectra beyond $m=1.8$ GeV are different: on the experimental
side, the high-mass resonances tend to gradually disappear, their status
becoming uncertain. An update of the Hagedorn mass spectrum can be found in
[2, 3, 4, 5]. Different Hagedorn temperatures for mesons and baryons from
experimental mass spectra, compound hadrons, and combinatorial saturation were
studied in [2].

In any case, the most important issue is the existence of a “melting point”,
where the resonances are transformed to a continuum of a boiling soup of
quarks and gluons. This critical region/point is studied by various methods:
statistics and thermodynamics, quantum chromodynamics, lattice quantum
chromodynamics (QCD) calculations, the MIT bag models, Regge poles and
relations derived within analytic $S$ matrix theory. The research objectives
are the order of the assumed phase transition or crossover transition,
fluctuations and correlations of conserved charges and the Fourier
coefficients of net-baryon density. For a recent treatment of these issues see
Ref. [15].

In the present paper, we extend the existing panorama by appending the
dynamics arising from the behavior of Regge trajectories. Regge trajectories
are rich objects containing information on the spectrum of resonances.
Although they are usually approximated by infinitely rising linear functions,
predicting an infinite number of resonances, in fact, the analytic theory
confines the rise of their real part limiting the number of resonances in
Nature. Below we show how the nonlinear complex trajectories affect the
Hagedorn spectrum.

More details on the Hagedorn spectrum, Hagedorn temperature, and hot phases of
hadronic matter can be found in the writings of Johann Rafelski [16, 17, 18].

The paper is organized as follows. In Subsec. II.1 we discuss various
approaches to the density of hadron states, including the Hagedorn model. In
Subsec. II.2 we discuss complex, non-linear Regge trajectories and their
relation to Hagedorn’s density of states. Both, the density of states and the
hadron spectrum are finite and interrelated, as shown in Subsec. II.3 by a
relevant explicit example. In Sec. III we discuss the relation between the
spectra and the statistical properties of the nuclear matter with possible
equations of state of hadronic matter.

##  II Melting hadrons

The spectrum of resonances and their statistical properties are interrelated.
In this section we study the relation between the mass density of hadronic
states given by the Hagedorn spectrum and the dynamics emerging from Regge
pole models, inspecting non-linear Regge trajectories.

In spite of the huge number of papers, the subject remains a topical problem
of hadron dynamics with numerous open questions. In this paper we address the
following issues:

  * •

the behavior of the meson mass spectrum in the high-mass region;

  * •

the role of the critical temperature and the prefactor in $\rho(m)$ in the
Hagedorn model of hadronic spectra;

  * •

the finiteness of the Hagedorn spectrum and its consequences.

###  II.1 Density of states (Hagedorn distribution) and resonance spectra
(Regge trajectories)

The idea of the spectral description of a strongly interacting gas was
suggested by Belenky and Landau [19]. This approach treats resonances on equal
footing with stable hadrons. The expression for pressure in this thermodynamic
approach in the Boltzmann approximation is given by:

$p=\sum_{i}g_{i}p(m_{i})=\int_{M_{1}}^{M_{2}}\limits\\!dm\,\rho(m)\,p(m),$ (2)

with

$p(m)=\frac{T^{2}m^{2}}{2\pi^{2}}K_{2}\bigl{(}\frac{m}{T}\bigr{)},$ (3)

where $M_{1}$ and $M_{2}$ are the masses of the lightest and heaviest hadrons,
respectively, and $g_{i}$-s are particle degenerations.

It was suggested in Refs. [20, 21, 22] that for fixed isospin and hypercharge
a cubic density of states, $\rho(m)\sim m^{3},$ fits the data. Moreover, as
argued in Ref. [20], the cubic spectrum can be related to collinear Regge
trajectories. Indeed, following the arguments of Burakovsky [23], on a linear
trajectory with negative intercept, $\alpha(t)=\alpha^{\prime}t-1$, some
integer values of $\alpha(t)=J$ correspond to states with negative spin,
$J=\alpha(t_{J}),$ with squared masses $m^{2}(J)=t_{J}.$ Since a spin-$J$
state has multiplicity $2J+1$, the total number of states with spin $0\leq
J\leq j$ at $t=m(j)^{2}$ is given by

$N(j)=\sum_{J=0}^{j}(2J+1)=(j+1)^{2}=\alpha^{\prime 2}m^{4}(j).$ (4)

Hence the density of states per unit mass interval is obtained as the
derivative of this cumulative quantity,

$\rho(m)=\frac{dN(m)}{dm}=4\alpha^{\prime 2}m^{3},$ (5)

and it grows as the cubic power of the mass. Consequently for a finite number
of collinear trajectories, $N$, the corresponding mass spectrum is given as

$\rho(m)=4N\alpha^{\prime 2}m^{3}.$ (6)

A different view on the spectra was advocated by Shuryak [21], who suggested
to use a quadratic parametrization, completely different from the conventional
form:

$\rho(m)\sim m^{2}.$

In both the statistical bootstrap model [1] and in the dual resonance model
[24, 25] the resonance spectrum takes the form of Eq. (1). In the dual
resonance model $f(m)\sim\frac{d}{dm}\mathfrak{Re}\alpha(m^{2}).$ In this work
we use non-linear complex Regge trajectories to determine this prefactor as
discussed in the next subsections.

The meson and baryon spectra differ, in particular by their slopes, as shown
e.g. in Refs. [2, 3]. More important is the question of the asymptotic
behavior of $\rho(m)$ for large masses. In theory, Hagedorn’s exponential may
rise indefinitely, however, starting from $m\approx 2.5~{}\textrm{GeV}$
resonances are not observed. The question arises whether it is a ”technical”
issue (the resonances gradually fade becoming too wide to be detected) or
there is a critical point where they melt to a continuum transforming the
hadron matter to a ”boiling soup”. Remarkably this point can be illuminated by
means of Regge trajectories, as we shall demonstrate it in the following.

In the present paper, we concentrate on the meson spectrum, more specifically
that of $\rho$ and its excitations. This familiar trajectory is chosen just as
a representative example. Other trajectories, including baryonic ones as well
as those with heavy ($c$ and $b$) flavors will be studied later. We are
interested in the high-mass behavior, starting from $m\approx 1.8$ GeV. Beyond
this value the exponential behavior of the Hagedorn spectrum is expected to
change drastically. Referring to perfect low- and intermediate mass fits of
[3], we concentrate now on its behavior above $1.8$ GeV.

Note that instead of comparing the density of states $\rho(m)$ to the data it
is customary to accumulate states of masses lower than $m$,

$N_{exp}=\sum_{i}g_{i}\Theta(m-m_{i}),$ (7)

where $g_{i}$ is the degeneracy of the i-th state with mass $m_{i}$ in spin
$J$ and isotopical spin $I$, i.e.,

$g_{i}=\Bigg{\\{}\begin{array}[]{ll}(2J_{i}+1)(2I_{i}+1),&\textrm{for non-
strange mesons}\\\ 4(2J_{i}+1),&\textrm{for strange mesons}\\\
2(2J_{i}+1)(2I_{i}+1),&\textrm{for baryons}\end{array}$

The theoretical equivalent of Eq. (7) is

$N_{theor}(m)=\int_{m_{\pi}}^{m}\rho(m^{\prime})dm^{\prime},$ (8)

where the lower integration limit is given by the mass of the pion. We
identify $f(m)$ in $\rho(m)$ with the slope of the relevant non-linear complex
Regge trajectory $\alpha^{\prime}(m)$. In the next subsection we discuss the
properties of these trajectories following from the analytic $S-$matrix theory
and duality, and present an explicit example of such a trajectory.

###  II.2 Regge trajectories

At low and intermediate masses, light hadrons fit linear Regge trajectories
with a universal slope, $\alpha^{\prime}\approx 0.85$ GeV2. As masses
increase, the spectrum changes: resonances tend to disappear. The origin and
details of this change are disputable.

Termination of resonances, associated with a ”ionization point” was also
studied in a different class of dual models, based on logarithmic trajectories
[26].

Possible links between the Hagedorn spectra and Regge trajectories appear in
the statistical bootstrap and dual models [25], according to which the
prefactor $f(m)$ in Eq. (1) depends on the slope of the relevant Regge
trajectory, $\alpha^{\prime}(m^{2})$, which is a constant for linear Regge
trajectories.

We extend the Hagedorn model by introducing the slope of relevant non-linear
Regge trajectories. Anticipating a detailed quantitative analysis, one may
observe immediately that a flattening of $\mathfrak{Re}\alpha(s=m^{2})$ 111We
use the (here positive) variables $s$ or $t$ interchangeably with crossing-
symmetry in mind., shown in Fig. 1 , results in a decrease of the relevant
slope $\alpha^{\prime}(m)$ and a corresponding change in the Hagedorn
spectrum. Following Eq. (1) we parametrize

$\rho(m)=\left(\frac{d}{dm}\mathfrak{Re}\alpha(m^{2})\right)\times\exp(m/T_{H}).$
(9)

Based on the decreasing factor $\mathfrak{Re}\alpha^{\prime}$ in Eq. (9) the
exponential rise of the density of states slows down near to the melting point
around $m\approx 2-2.5$ GeV. The cumulative spectrum Eqs. (7) and (8),
accordingly tends to a constant value.

Any Regge trajectory should satisfy the followings (for a comprehensive review
see [27]):

  * •

threshold behavior imposed by unitarity;

  * •

asymptotic constraints: the rise of real part of Regge trajectories is
limited, $\Re\alpha(s)\leq\gamma\sqrt{t}\ln t,\ \ s\to\infty$;

  * •

compatibility with the nearly linear behavior in the resonance region (Chew-
Frautschi plot).

The threshold behavior of Regge trajectories is constrained by unitarity. As
shown by Barut and Zwanziger [28], $t$-channel unitarity constrains the Regge
trajectories near the threshold, $t\rightarrow t_{0}$ to the form

$\mathfrak{Im}\alpha(t)\sim(t-t_{0})^{\mathfrak{Re}\alpha(t_{0})+1/2}.$ (10)

Here $t_{0}$ is the lightest threshold, e.g. $4m_{\pi}^{2}$ for the meson
trajectories. Since $\mathfrak{Re}\alpha(4m_{\pi}^{2})$ is small, a square-
root threshold is a reasonable approximation to the above constraint.

In the resonance region below flattening near $m=\sqrt{s}\lesssim 2.5$ MeV the
meson and baryon trajectories are nearly linear (Chew-Frautschi plot). Fixed-
angle scaling behavior of the amplitude constrains the trajectories even more,
down to a logarithmic behavior [29].

The combination of the threshold behavior Eq. (10) and the above asymptotic
behavior suggested the explicit model [30] of trajectories as a sum of square-
root thresholds (see the details in Subsec. II.3).

Figure 1: Typical behavior of the real (a) and imaginary (b) parts of Regge
trajectories in dual models with Mandelstam analyticity [14].

There are several reasons why the non-linear and complex nature of the Regge
trajectories is often ignored, namely: 1) the observed spectrum of meson and
baryon resonances (Chew-Frautschi plot) seem to confirm their linearity; 2) in
the scattering region, $t<0,$ the differential cross-section,
$d\sigma/dt\sim\exp((2\alpha(t)-2)\ln s)$ is nearly exponential in $t$; 3)
Dual models, e.g. the Veneziano amplitude are valid only in the narrow-width
approximation, corresponding to linear Regge trajectories (hadronic strings).
Deviation from linearity is unavoidable, but its practical realization is not
easy. Attempts are known in the literature, see e.g. [31, 32, 23] and
references therein.

As a final remark, we comment on a typical feature of dual analytic models,
namely the extremely broad resonance approximation, suggested in [14] by which
the unitarization procedure, contrary to the narrow resonance (Veneziano-like)
models Fig. 25 of Ref. [14], move the resonance pole from the real (negative)
axis to the physical region of broad resonances, harmless to its basic
properties, e.g. polynomial boundedness (for more details see [14]). This
property is essential for the parametrization of our Regge trajectory
$\alpha(t)$. Due to our factor $f(m)=(Re\alpha(m))’$, the spectrum is a
“mixture” of narrow and wide resonances, therefore not purely “Hagedornian”
anymore.

###  II.3 The $\rho$-trajectory and the Hagedorn spectrum

Trajectories satisfying the above requirements have been studied extensively
in the past. A particularly simple and transparent non-linear trajectory was
suggested in Refs. [30, 33] and is defined as a sum of square-root thresholds
formed by stable particles, allowed by quantum numbers. While the imaginary
part of such a trajectory starts to be nonzero by exceeding the lightest
threshold and rises indefinitely, its real part terminates at the heaviest
threshold (see Fig. 1).

Specifically, Ref. [33] defines trajectories as

$\alpha(s)=\lambda-\sum_{i}\gamma_{i}\sqrt{s_{i}-s},$ (11)

where all two-particle stable thresholds are included in the sum. For the
$\rho$-meson trajectory two meson-meson and four baryon-antibaryon channels
are taken into account, as listed in Table 1 along with corresponding
thresholds $s_{i}$ and weights $\gamma_{i}$. Note, that for simplicity the
same weight is used for all baryon-antibaryon channels.

Channel | $s_{i}$ | $\gamma_{i}$
---|---|---
$\pi\pi$ | 0.078 | 0.127
$K\overline{K}$ | 0.976 | 0.093
$N\overline{N}$ | 3.52 | 0.761
$\Lambda\overline{\Sigma}$ | 5.31 | 0.761
$\Sigma\overline{\Sigma}$ | 5.66 | 0.761
$\Xi\overline{\Xi}$ | 6.98 | 0.761
Table 1: Parametrisation of the $\rho$-trajectory [33].

The real and imaginary parts of the trajectory described by Eq. (11) with
parameters from Table 1 are shown in Figure 2. The vertical lines indicate
thresholds $s_{i}$. As expected, above the highest threshold
($s_{6}=6.98~{}\textrm{GeV}^{2}$) the real part reduces to a constant,
$\mathfrak{Re}\alpha=\lambda$.

Figure 2: Real and imaginary parts of the $\rho$-trajectory. Figure 3: Mass
density $\rho(m)$ calculated by using the derivative of the smoothed
$\rho$-meson trajectory as a prefactor, Eq. (13). The exponential density
without any prefactor is also shown (dashed line), using the same temperature
and normalization. Above the highest threshold the derivative, and hence the
density vanishes (not seen due to the logarithmic scale).

Putting $s=m^{2}$ and differentiating the real part of the trajectory with
respect to $m$ we obtain

$\frac{d}{dm}\mathfrak{Re}\alpha(m^{2})=\sum_{i}\frac{\gamma_{i}m}{\sqrt{s_{i}-m^{2}}},$
(12)

where for a given $m$ the sum includes only terms satisfying $m^{2}<s_{i}$.
Substitution into (9) yields

$\displaystyle\rho(m)=\frac{d}{dm}\mathfrak{Re}\alpha(m^{2})\exp(m/T)$ (13)
$\displaystyle=\sum_{i\mathrel{\mathop{\mathchar
58\relax}}\,s_{i}>m^{2}}\frac{\gamma_{i}m}{\sqrt{s_{i}-m^{2}}}\exp(m/T).$

As in Eq. (12), the sum includes for a given $m$ only terms with
$s_{i}>m^{2}$; the derivative and thus the mass density vanish above the
highest threshold. At the thresholds $s_{i}$, the derivative and hence the
density diverge. However, the relevant integral, Eq. (8) stays finite. For
simplicity, we smooth the trajectory $\mathfrak{Re}\,\alpha(s)$ in the
vicinity of the thresholds using splines. With this procedure the derivative
becomes finite and continuous, and can be integrated numerically. An example
of the resulting mass density $\rho(m)$ is shown in Figure 3.

The mass density $\rho(m)$ can now be integrated using Eq. (8) in order to
obtain the theoretical prediction for the mass spectrum, $N_{theor}(m)$.
Figure 4 shows the result, together with the experimental cumulative spectrum
$N_{exp}(m)$ using the states listed by the Particle Data Group [6].

Figure 4: Hadron mass spectrum $N_{theor}$ from Eq. (8) using the mass density
of Eq. (13) (blue line) compared to data (black histogram). Additionally,
$N_{theor}$ obtained from a simple exponential mass density is also shown
(green dashed line).

As expected, $N_{theor}(m)$ flattens above the highest threshold
(corresponding to $m\sim 2.65~{}\textrm{GeV}$), since the density $\rho(m)$
defined with Eq. (13) vanishes in that region. This is consistent with the
experiment, since no unflavored mesons have been observed above $m\sim
2.5~{}\textrm{GeV}$ [6]. On the other hand, the integrated spectrum using a
simple form $\rho=e^{m/T}$ is rising indefinitely, thus failing to describe
flattening of the data at high masses.

The temperature and normalization used in Fig. 4 were determined from a least-
square fit to data. The fit was performed separately for two models (a simple
exponent and an exponent with a prefactor). The optimal temperature for the
pure exponent is $T=0.41~{}\textrm{GeV}$, slightly larger but generally
consistent with the previous studies. Inclusion of the prefactor steepens the
curve (see e.g. Fig. 3), thus a larger temperature ($T=1.45~{}\textrm{GeV}$)
is required to fit the data.

Resonances may melt for two reasons: 1) for square-root $\alpha(s)$
trajectories (11), whose real part terminates at some large mass and
consequently the derivative vanishes, or 2) for densities of resonances
modelled by power-like functions rather than exponentials.

The above is only a representative example intended to show the interrelation
between the Hagedorn spectrum and the one based on non-linear complex Regge
trajectories (Chew-Frautchi plot). The present results can be extended in
several directions:

  * •

by including other trajectories - mesonic and/or baryonic;

  * •

by including heavy flavors such as $b$ quarks, both in the Hagedorn and Regge
spectra.

  * •

working with alternative parametrizations of non-linear, complex Regge
trajectories.

##  III Boiling quarks and gluons

In the previous section, we inspected the spectrum of resonances by combining
two different approaches - statistical (Hagedorn) and dynamical (Regge). We
have focused on the region of heaviest resonances, the region where hadrons
may melt transforming in a boiling ”soup” of quarks and gluons. Melting may
happen in different ways, characterized by the details for a phase transition
of colorless hadronic states into a quark-gluon soup, whatever it be [34]. In
terms of hadron strings this process corresponds to breakdown (fragmentation)
of a string. Lacking any theory of confinement providing a quantitative
description of interacting string, we will not pursue this model. Instead we
use thermodynamics adequate in this situation. To complement the previous
section, we present our arguments below related to the possible change of
phase from a different, thermodynamic perspective.

The Hagedorn exponential spectrum of resonances Eq. (1) results in a
singularity in the thermodynamic functions at critical temperature $T=T_{c}$
and in an infinite number of effective degrees of freedom in the hadronic
phase. Also, as shown in Ref. [35], a Hagedorn-like mass spectrum is
incompatible with the existence of the quark-gluon phase. To form a quark
phase from the hadronic phase, the hadron spectrum cannot grow more quickly
than a power. This is possible [21] in case of a simple power parametrization
$\rho\sim m^{k},$ compatible with $k\approx 3,$ for the observable mass
spectrum in the interval 0.2–1.5 GeV. Assuming ideal contributions to
thermodynamical quantities we hence take energy density in the form

$\epsilon\>=\>\int_{0}^{\infty}\limits\rho(m)T^{4}\sigma(m/T)dm\>=\>\lambda_{k}T^{k+5},$
(14)

and obtain the corresponding pressure and sound velocity square as
follows222Note that the definition of entropy density $s$, energy density
$\epsilon$ and velocity of sound $c_{s}$ in case of $\mu=0$:
$s(T)=p^{\prime}(T),\ \epsilon(T)=Ts-p,\ \
c_{s}^{2}=\frac{dp}{d\epsilon}=\frac{p^{\prime}}{Tp^{\prime\prime}}=\frac{s}{Ts^{\prime}}.$
:

$p=\frac{\lambda_{k}}{k+4}T^{k+5},\ \ c_{s}^{2}=1/(k+4).$ (15)

The above empirical power-like behavior has also theoretical background. In
Ref. [36] an asymptotic, $T\gg m$, EOS was derived by using the $S$-matrix
formulation of statistical mechanics [37]. It was shown that the existence of
the forward cone in hadronic interactions with non-decreasing total cross
sections, $i.e$, pomeron dominance, confirmed by numerous experiments at high
energies, results in an asymptotic, $T\gg m$ EOS $p(T)\sim T^{6}$ where $m$ is
a characteristic hadron (e.g. pion) mass. The inclusion of non-asymptotic
(secondary) Regge terms produces a minimum in the $p(T)$ dependence at
negative pressure, with far-reaching observable consequences.

Such an asymptotic formula for the pressure, different from the ”standard”
$p(T)\sim T^{4}$, was derived, on different grounds also in Ref. [21].

The unorthodox $p\sim T^{6}$ asymptotic behavior is orthogonal to the
”canonical” (perturbative QCD) form $\sim T^{4}$. Still, it cannot be rejected
e.g. when assuming a screening of the action of large-distance van der Waals
forces at high temperatures and densities.

In Ref. [38] the asymptotic form $\sim T^{6}$ was extended to lower
temperatures by adding non-asymptotic Regge-pole exchanges. The resulting EOS
is

$p(T)=aT^{4}-bT^{5}+cT^{6},$ (16)

where $a,\ b,\ c$ are parameters connected with Regge-pole fits to high-energy
hadron scattering. The remarkable property of this EOS, apart from the non-
standard asymptotic behavior, $\sim T^{6},$ is the appearance of the non-
asymptotic term $T^{5}$ with negative sign, creating a local minimum with
negative pressure. This metastable state with negative pressure was shown [39]
to produce inflation of the universe.

In the next part, we discuss the phase transition of colorless hadronic states
into quark-gluon soup in the framework of modified quark bag models.

The standard bag equation of state assuming, for simplicity, vanishing
chemical potential, $\mu=0$:

$p_{q}(T)=\frac{\pi^{2}}{90}\nu_{q}T^{4}-B,$ (17)
$p_{h}(T)=\frac{\pi^{2}}{90}\nu_{h}T^{4},$ (18)

where $p_{q}(T)$ and $p_{h}(T)$ are pressure in the quark-gluon plasma (QGP)
and in the hadronic gas phase, respectively, $B$ is the bag constant, and
$\nu_{q(h)}$ is the number of degrees of freedom in the QGP (hadronic gas).
From Eqs. (17) and (18) one finds the characteristic parameters of the phase
transition (see Fig. 5a):

$p_{c}=B\nu_{h}/(\nu_{q}-\nu_{h}),\ \
T_{c}=[90B/\pi^{2}(\nu_{q}-\nu_{h})]^{1/4}.$ (19)

Since $s(T)=dp(T)/dT,$ the relevant formula for the entropy density can be
rewritten as (see Fig. 5b)

$s(T)=(2\pi^{2}T^{3}/45){\nu_{h}[1-\Theta(T-T_{c})+\nu_{q}\Theta(T-T_{c})}.$
(20) $s^{*}(T)=s(T)/T^{3},\ \ s^{*}_{c}=s_{c}/T^{3}_{c},\ \
s_{c}=\frac{\pi^{2}T_{c}^{3}}{45}(\nu_{h}+\nu_{q}).$ (21)

The above simple bag model EOS can also be modified [41, 42] by making the bag
”constant” $T$-dependent, $B(T)=AT$, to produce a metastable QGP state with
negative temperature (see Fig. 5c and Fig. 5d):

$p_{q}(T)=(\pi^{2}/90)\nu_{q}T^{4}-AT,\ \ \
p_{h}(T)=(\pi^{2}/90)\nu_{h}T^{4}.$ (22)

Figure 5: Icons $a$ and $b$ show the standard bag EoS, Eq. (18), with constant
$B$; here $s^{*}(T)=s(T)/T^{3}.$ The dotted line corresponds to the modified
bag EoS (24). Icons $c$ and $d$ feature the Källmann EoS [41], where $B=AT$.
Figure 6: Modified bag EOS Eq. (25), including metastable states. Icon $a$ is
the $p(T)$ dependence while icon $b$ is $s^{*}(T)=s(T)/T^{3}$.

In Ref. [40] EOS Eq. (20) was modified by the replacement

$\Theta(T-T_{c})\rightarrow\frac{1}{2}\left[1+\tanh\left(\frac{T-T_{c}}{\Delta
T}\right)\right],$ (23)

where $\Delta T$ is a parameter related to the smoothness of a crossover
transition. With the above substitution, the EOS can be rewritten as

$\frac{T-T_{c}}{\Delta T}={\rm arctanh}(\Gamma\Delta s^{*}),$ (24)

where $\Gamma=45/[\pi^{2}(\nu_{q}-\nu_{h})],\ \ \Delta s^{*}=s^{*}-s^{*}_{c}$,
behaving as shown by the dotted line in Fig. 5b. This modification, however,
smoothes down first order phase transitions in EOS, thus excluding possible
metastable states. It may serve as a springboard for a further modification
suggested in Ref. [43], namely:

$(T-T_{c})/\Delta T={\rm arctanh}(\Gamma\Delta s^{*})-\gamma\Delta s^{*},$
(25)

where $\gamma$ is the ”metastability parameter”. For $\Gamma-\gamma>0$, EOS
Eq. (25) is of the same type as Eq. (24), while for $\Gamma-\gamma<0$ a loop
emerges in the entropy density (Fig. 6b, similar to the loop in the density to
pressure dependence, described by the Van der Waals equation or the EOS of a
magnet in vicinity of the Curie point, replacing $\Delta s^{*}$ with the order
parameter of the phase transition and substituting $(T-T_{c})/\Delta T$ by the
corresponding conjugate field. For $\Gamma-\gamma<0$ the curve $s^{*}(T)$
contains a non-physical region (interval $CD$) with $ds/dT<0$. The intervals
($AC$) and ($DE$) of $s^{*}(T)$ correspond to metastable states, $C$ and $D$
being the spinodal points where $ds/dT\sim c_{s}^{-2}$, with $c_{s}$ being the
sound speed.

Another important feature of the EOS Eq. (25) is that for $\Gamma-\gamma=0$ it
describes second order phase transitions, with singular behavior of the
thermal capacity at $T=T_{c}.$ Really, in this case, we have
$T-T_{c}\sim(\Delta s^{*})^{3}$ near $T=T_{c}$.

##  IV Summary and conclusions

We related two different approaches to the critical point in the hadron
spectra using statistical (Hagedorn) and dynamical (Regge) models. Our
innovation is in the use of non-linear Regge trajectories, predicting a
limited spectrum of hadron states, ignored in most of the papers on Regge-pole
theory. Although our analyses is limited to the simple case of the $\rho$
meson spectrum, this technique and the results can (and will) be extended to
other hadrons.

In this work, we showed that the observed saturation of the number of hadrons
as a function of mass can be easily explained in the Regge theory, by using
non-linear trajectories and attributing the prefactor in Hagedorn’s density to
the slope of the corresponding Regge trajectory. Due to flattening of the
latter, the slope vanishes, and the number of states flattens. In this way, we
enter the most intriguing of the strong interaction, namely the expected
transition of excited colorless hadrons into the hypothetical boiling soup of
quarks and gluons.

Flattening of the exponential density of states and of the linear rise of
Regge trajectories point to the same phenomenon, namely quark deconfinement
(“melting” of hadrons). The two phenomena are correlated but they are not
identical. Their combined study and further fits to the data may reduce the
available freedom of the relevant parametrizations and tell us more about the
unset of deconfinement.

The maximum mass of existing resonances depends on the parameters fitted to
the observed resonances. Resonances tend to disappear (fade) beyond some mass.
Their non-observability may have two reasons: either they are not visible
because of their large width (decay time) or they melt losing their individual
features. These phenomena, especially the second, ”boiling” phase can be
studied also in the framework of thermodynamics, e.g., by studying the
relevant equation of state (EoS).

## Acknowledgements

We thank Wojciech Broniowski for useful correspondence on the subject of this
paper.

LJ was supported by the National Academy of Sciences of Ukraine’s Grant N
1230/22-1 ”Fundamental properties of matter” and by the NKFIH Grant no.
K133046; TB acknowledges support from the Hungarian National Office for
Research, Development and Innovation, NKFIH under the project number K123815;
ISz was supported the by the NKFIH Grant no. K133046 and by the ÚNKP-22-3 New
National Excellence Program of the Ministry for Innovation and Technology from
the source of the National Research, Development and Innovation Fund.

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#  Identifying Disappearance of a White Dwarf Binary with LISA

Naoki Seto
Department of Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

We discuss the prospect of identifying a white dwarf binary merger by
monitoring disappearance of its nearly monochromatic gravitational wave. For a
ten-year operation of the laser interferometer space antenna (LISA), the
chance probability of observing such an event is roughly estimated to be 20%.
By simply using short-term coherent signal integrations, we might determine
the merger time with an accuracy of $\sim 3$-10 days. Also considering its
expected sky localizability $\sim 0.1$-$0.01{\rm deg^{2}}$, LISA might make an
interesting contribution to the multi-messenger study on a merger event.

###### keywords:

gravitational waves — binaries: close

##  1 introduction

LISA is expected to detect $\sim 10^{4}$ white dwarf binaries (WDBs) in our
Galaxy as nearly monochromatic gravitational wave (GW) sources (Hils, Bender,
& Webbink, 1990; Nelemans et al., 2001; Nissanke et al., 2012; Breivik et al.,
2018; Lamberts et al., 2019; Amaro-Seoane et al., 2022). Their GW frequencies
slightly change due to the radiation reaction, and the majority of them are
considered to merge eventually. Given the estimated Galactic merger rate
$R\rm\sim 0.02yr^{-1}$ (see e.g., Nelemans et al. 2001), we might actually
have a merger event with a chance probability of $\sim 0.2(T/{\rm 10yr})$,
during LISA’s operation period $T$.

A WDB merger in our Galaxy will be an intriguing observational target for
multi-messenger astronomy (see e.g., Schwab 2018 and also Shen et al. 2012;
Dan et al. 2014) and can help our understanding of white dwarfs (potentially
including their explosion process). As an omni-directional GW detector free
from interstellar extinction, LISA can provide us with critical information
(e.g. sky locations, orbital phases) for follow-up electromagnetic (EM)
searches (e.g., Korol et al. 2017). Therefore, in this paper, we discuss
identification of a WDB merger in LISA’s data.

From various estimations (Amaro-Seoane et al. 2022 and references therein),
the majority of merging WDBs are likely to have total masses less than
$1M_{\odot}$. In the connection to type Ia supernovae, one might be interested
in WDBs with the total masses larger than the Chandrasekhar mass. However,
their merger rate is estimated to be $\sim 5-7$ times smaller than the total
rate quoted above (Nelemans et al. 2001, see also Maoz, Hallakoun, & Badenes
2018).

According to recent numerical simulations with a careful treatment for the
initial conditions (Dan et al. 2011 see also Benz et al. 1990; Rasio & Shapiro
1992; D’Souza et al. 2006; Lorén-Aguilar, Isern, & García-Berro 2009; Raskin
et al. 2012; Pakmor et al. 2013), a typical merging WDB will continue to emit
nearly monochromatic GW, until just before the donor star is tidally disrupted
and the system virtually turns off GW emission. On the basis of this overall
GW emission pattern, we examine a forceful approach for the identification of
a WDB merger by repeatedly checking the resultant disappearance of its nearly
monochromatic GW. We pay an attention to the estimation of the merger time,
which will be important for EM observations.

This paper is organized as follows. In §2, we briefly describe the evolution
of a merging WDB and the associated GW emission. In §3, we discuss LISA’s
observation for the GW signal, including estimation of the merger time with
matched filtering analysis. In §4, we discuss related topics such as the cases
with other space interferometers. §5 is devoted to a short summary.

##  2 Evolution of a merging WDB

###  2.1 Nearly Monochromatic Waves

As an approximation to a detached WDB emitting a nearly monochromatic GW, we
briefly discuss a circular binary composed by two point masses $m_{1}$ and
$m_{2}$ ($m_{1}\leq m_{2}$). Using the Kepler’s law, we can write down the
wave frequency

$\displaystyle f=\pi^{-1}G^{1/2}(m_{1}+m_{2})^{1/2}a^{-3/2}$ (1)

with the orbital separation $a$ and the gravitational constant $G$. At the
quadrupole order, the angular averaged strain amplitude is written as

$\displaystyle h=\frac{8G^{5/3}{\cal M}^{5/3}\pi^{2/3}f^{2/3}}{5^{1/2}c^{4}d}$
(2)

with the distance $d$ to the binary (below fixed at the representative
distance $d=10$kpc) and the speed of light $c$ (Robson, Cornish, & Liu, 2019).
Here we put the chirp mass ${\cal
M}\equiv(m_{1}m_{2})^{3/5}(m_{1}+m_{2})^{-1/5}$.

The frequency evolution due to the radiation reaction is written as

$\displaystyle\frac{df}{dt}$ $\displaystyle=$
$\displaystyle\frac{96G^{5/3}\pi^{8/3}{\cal M}^{5/3}f^{11/3}}{5c^{5}}$
$\displaystyle=$ $\displaystyle 2.5\times 10^{-6}\left(\frac{f}{\rm
20.8mHz}\right)^{11/3}\left(\frac{{\cal M}}{0.38M_{\odot}}\right)^{5/3}{\rm
Hz\,yr^{-1}}.$

The tidal interaction between the two star will modify our expressions
(1)-(2.1) to some extent (e.g., $\sim 10\%$ for Eq. (2.1) as in Wolz et al.
2021). However, in this paper, we are mainly interested in the identification
of a WDB merger event from LISA’s data, by just using the disappearance of its
GW (not trying to accurately estimate e.g., its chirp mass). Later, we use Eq.
(2) in this context, and it will work reasonably well, up to very close to the
final merger (Dan et al., 2011).

Given the merger rate (or more appropriately the merger flux) $R$ of Galactic
WDBs, we can apply the continuity equation in the frequency space and
approximately evaluate their frequency distribution $dN/df$ as

$\displaystyle\frac{dN}{df}$ $\displaystyle\sim$ $\displaystyle R(df/dt)^{-1}$
(4) $\displaystyle\sim$ $\displaystyle 3.4\times 10^{6}\left(\frac{R}{\rm
0.02yr^{-1}}\right)\left(\frac{f}{\rm 4mHz}\right)^{-11/3}{\rm Hz^{-1}}.$ (5)

This expression will be valid in the frequency regime $f\lesssim 10$mHz where
the merger flux $R$ is expected to be nearly constant (Seto, 2022). Here, for
simplicity, we ignored the chirp mass distribution and put ${\cal
M}=0.38M_{\odot}$. From an integral of Eq. (5), the number of Galactic WDBs
above $\sim 4$mHz is roughly estimated to be $N(>{\rm 4mHz})\sim 5\times
10^{3}$.

###  2.2 Transition around the Merger

Due to the radiation reaction, the orbital separation of a detached WDB
decreases gradually and the less massive donor $m_{1}$ will eventually fill
its Roche-lobe, initiating mass transfer to the accreter $m_{2}$ (Paczyński,
1967; Marsh, Nelemans, & Steeghs, 2004). The time evolution during this mass
transfer phase has been examined by numerical simulations (Dan et al., 2011).
A typical merging WDB (like those analyzed below) will continue to emit nearly
monochromatic GW (roughly described in the previous subsection), until the
donor loses a small fraction of its mass due to an unstable mass transfer (see
e.g., Figs 7 and 17 in Dan et al. 2011).

Then, in a short transitional period $\Delta t_{\rm t}$ (corresponding to
$\lesssim 10$ wave cycles), the donor is tidally disrupted, and the system
will soon settle down into a nearly axisymmetric configuration, virtually
stopping GW emission (Dan et al. 2011, see also Yoshida 2021; Moran-Fraile et
al. 2023). We define the merger time $t_{\rm m}$ as the beginning of the
transition period $\Delta t_{\rm t}$. One of our primary objectives below is
to clarify how well we can observationally determine the merger time $t_{\rm
m}$ only with LISA.

We also define the merger frequency $f_{\rm m}$ as the GW frequency at end of
the nearly monochromatic emission (just before the transition period).
Similarly, we put the associated characteristic strain amplitude $h_{\rm m}$
which is obtained by plugging in the final frequency $f=f_{\rm m}$ in Eq. (2).

Next, following Marsh, Nelemans, & Steeghs (2004), let us approximately
estimate the merger frequency $f_{\rm m}$, by using the condition of the
Roche-lobe filling of the less massive (donor) star $m_{1}$. We apply the mass
radius relation $r(m_{1})$ for completely degenerate helium given in Verbunt &
Rappaport (1988), ignoring (potentially existing) diffuse outer envelope.
Meanwhile the Roche-lobe radius is approximately given by Paczyński (1967) as

$\displaystyle R_{L}=2\cdot 3^{-4/3}m_{1}^{1/3}(m_{1}+m_{2})^{-1/3}a.$ (6)

Then we put $R_{L}=r(m_{1})$ for the donor $m_{1}$ at the onset of the mass
overflow. From Eqs. (1) and (6), the merger frequency is written as

$\displaystyle f_{\rm m}(m_{1})=\frac{2^{3/2}G^{1/2}m_{1}^{1/2}}{9\pi
r(m_{1})^{3/2}},$ (7)

depending only on the mass $m_{1}$ of the donor.

In this paper, considering the mass distribution of WDBs, we examine the three
WDB models presented in Table 1. We set the model A as the base model for our
study with the lower-mass comparative model B. The reference model C has the
total mass of $1.5M_{\odot}$. In Table 1, we present their merger frequencies
(7) . From a comparison with a more detailed numerical study in Dan et al.
(2011), our rough estimation is expected to have accuracy within $\sim 15\%$
for the presented donor masses $m_{1}$.

At the frequency $f_{\rm m}=20.8$mHz, under the point particle approximation
the base model A has the radiation reaction time scale $f/{\dot{f}}\sim
10^{4}$yr. Therefore, even $\sim 10$yr before the merger, we approximately
have its frequency and amplitude as $f\sim f_{\rm m}$ and $h\sim h_{\rm m}$.

Table 1: Nearly monochromatic GWs from merging WDBs (A: our base model, B: the lower mass model, C: the higher mass model). We put the donor mass $m_{1}$ and the accreter mass $m_{2}$. The binary merges around the frequency $f_{\rm m}$. We show the signal-to-noise ratios $\rho$ during the final two years before the merger with LISA (at the distance of $d=10$kpc). We present the short-term integration period $T_{\rm s}$ to get signal-to-noise ratio of $\rho_{\rm th}=7.5$. The corresponding rotation cycles $f_{\rm m}\cdot T_{\rm s}$ are also presented.  model | A (base) | B | C
---|---|---|---
$m_{1}$ | 0.4 $M_{\odot}$ | 0.3 $M_{\odot}$ | 0.6 $M_{\odot}$
$m_{2}/m_{1}$ | 1.2 | 1.2 | 1.5
$f_{\rm m}$ | 20.8 mHz | 15.0 mHz | 35.1 mHz
$\rho$ ($T=$2yr) | $85$ | 47 | $210$
$T_{\rm s}$ ($\rho_{\rm th}=$7.5) | 5.7 day | 18.6 day | 0.93 day
$f_{\rm m}\cdot T_{\rm s}$ | 10000 | 24000 | 2800

###  2.3 Evolutionary Stages

On the basis of the previous two subsections, we next divide the time
evolution of a merging WDB into the four stages (i)-(iv), as a preparation for
the matched filtering analysis mainly discussed in the next section.

Figure 1: Schematic diagram for observation of a GW signal from a merging WDB.
We set the time origin $t=0$ at the beginning of LISA’s operation and put the
merger time by $t=t_{\rm m}$. We divide the time sequence into the four stages
(i)-(iv) with the duration $\Delta t_{\rm p}$ and $\Delta t_{\rm t}$
respectively for (ii) and (iii). For the identification of the merger time
$t_{\rm m}$, we introduce the time segmentation with an appropriately selected
duration $T_{\rm s}$. The merger time $t_{\rm m}$ is in the segment $k=k_{\rm
m}$.

For coherent signal integration with the matched filtering analysis, the phase
of the target GW should be accurately characterized by a limited number of
fitting parameters (e.g., the polynomial expansion with the derivative
coefficients $f^{(n)}$ at some epoch). Unlike a well separated WDBs, even in
the time region of nearly monochromatic GW emission, a WDB close to the merger
might have complicated GW phase modulations which should be handled with a
short-term phenomenological phase modeling. As a precaution for such a
potential difficulty, we conservatively divide the time region of nearly
monochromatic emission into two stages (i) and (ii). We put $\Delta t_{\rm p}$
as the total duration of the late monochromatic stage (ii). Sorting out the
temporal sequence, around the merger time $t_{\rm m}$, we now have the
following four stages.

(i) Early monochromatic stage at $t<t_{\rm m}-\Delta t_{\rm p}$. The binary
emits nearly monochromatic GW at the frequency $\sim f_{\rm m}$ with the
amplitude $\sim h_{\rm m}$. Its intrinsic GW phase can be coherently described
by a simple expression.

(ii) Late monochromatic stage at $t_{\rm m}-\Delta t_{\rm p}<t<t_{\rm m}$. The
binary continues to emit nearly monochromatic wave at $\sim f_{\rm m}$ and
$\sim h_{\rm m}$. We need phenomenological phase model with a short
integration period.

(iii) Transition stage at $t_{\rm m}<t<t_{\rm m}+\Delta t_{\rm t}$. The
disruption of the donor proceeds rapidly and the emitted GW no longer has
regular monochromatic pattern, due to hydrodymanical effects.

(iv) Post-merger stage at $t_{\rm m}+\Delta t_{\rm t}<t$. Virtually no GW is
emitted, as the merged system settles down to a nearly stationary and
axisymmetric configuration.

Here we can practically assume $\Delta t_{\rm p}\gg\Delta t_{\rm t}$
(otherwise we do no need to introduce the late stage (ii)). In contrast to the
transition duration $\Delta t_{\rm t}$ deduced from existing numerical
simulations, the magnitude of $\Delta t_{\rm p}$ is currently unclear. We need
to accurately follow the longer-term dynamical evolution at high mass
resolution. Numerical approach would be highly demanding, and analytical study
might be more advantageous. Observational studies on known short-period WDBs
might be also useful (see e.g. Strohmayer 2021; Munday et al. 2023 for RX
J0806.3+1527). Below, we leave the duration $\Delta t_{\rm p}$ as a free
parameter, just assuming $\Delta t_{\rm p}\ll 1$yr. We should also comment
that the boundary between (i) and (ii) is not mathematically distinct. It is
rather a practical one and depends on the complexity of the phase modeling
allowed for the stage (i).

##  3 Observation with LISA

Next we discuss matched filtering analysis with LISA. We set the origin of the
time coordinate $t=0$ at the beginning of LISA’s operation. Given its planned
operation period 4-10 yr, we below set $t_{\rm m}=2$yr as a conservative
average. For the purpose of concise illustration, we are assumed to make two
kinds of data analysis: the long-term search and the segmented short-term
search.

The purpose of the long-term search is to confidently detect a WDB and
determine its basic parameters, using the data at the early monochromatic
stage (i). This search would be similar to the signal analysis for a standard
(non-merging) WDB. Using the results of the long-term search, we subsequently
perform the segmented short-term searches for identifying the disappearance of
its nearly monochromatic GW at the merger. In reality, we can chronologically
mix these two searches.

###  3.1 The Long-Term Search

Throughout the early monochromatic stage (i), the GW signal would be
coherently searched with manageable numerical costs. The total signal-to-noise
ratio $\rho$ can be estimate as

$\displaystyle\rho=\frac{h_{\rm m}\sqrt{T}}{S_{n}(f_{\rm m})^{1/2}}$ (8)

for the integration time $T$. Here, for simplicity, we ignore the angular
dependence of the GW strain (Cutler, 1998). It is straightforward to include
it in our analysis below.

As presented in Table 1 for $T=t_{\rm m}=2$yr, the expected signal-to-noise
ratio in the stage (i) is $\rho=O(100)$ and much larger than those for typical
Galactic LISA sources existing at lower frequencies. For the base model A (up
to the end of the stage (ii)), the signal-to-noise ratio squared $\rho^{2}$
increases unity in every $\sim 180$ wave cycles, corresponding to $\sim$0.1
day ($\sim 430$ cycles for B and $\sim 50$ cycles for C). Therefore, even
though the catastrophic stage (iii) is interesting, this stage is likely to
have a negligible contribution to the signal-to-noise ratio, given its
rotation cycles of $\lesssim 10$ (Dan et al., 2011). For our signal analysis
scheme, we can omit the stage (iii) and effectively regard that the nearly
monochromatic stage (ii) is directly followed by the quiet stage (iv) at the
merger time $t_{\rm m}$.

Using the high quality signal in the stage (i), we can securely confirm the
existence of the binary. We can also estimate its nearly monochromatic
frequency $f_{\rm m}$ and other basic parameters including the amplitude
$h_{\rm m}$. For instance, the sky position of the binary can be determined
with the aid of the Doppler modulation (Cutler, 1998). For $T\gtrsim 2$yr, we
can apply an asymptotic scaling relation for the area of the error ellipsoid
(Takahashi & Seto, 2002)

$\displaystyle\Delta\Omega\sim
0.01\left(\frac{\rho}{85}\right)^{-2}\left(\frac{f}{\rm
20.8mHz}\right)^{-2}{\rm deg^{2}}.$ (9)

The area becomes $\rm\sim 0.06deg^{2}$ for the model B. The high sky
localization would be beneficial for associated EM observations.

###  3.2 Segmented Short-Term Search

Let us suppose that we have already done the long-term search for a WDB, in
the middle of its early monochromatic stage (i). Now, we divide the upcoming
data streams into segments of a duration $T_{\rm s}$ as shown in Fig. 1. We
assign the label $k$ and put $\rho_{k}$ for the corresponding signal-to-noise
ratio obtained after the short-term coherent matched filtering. The total
number of the segments increases, as LISA’s operation period becomes longer.

Our immediate goal is to observationally determine the specific segment
$k_{m}$ which contains the merger time $t_{\rm m}$ (see Fig. 1). To this end,
we appropriately set the threshold $\rho_{\rm th}$ and the duration $T_{\rm
s}$, so that we can reliably have $\rho_{k}\geq\rho_{\rm th}$ for earlier
segments $k<k_{m}$ and $\rho_{k}<\rho_{\rm th}$ for later segments $k>k_{m}$,
due to the existence of the nearly monochromatic wave before the merger time
$t_{\rm m}$.

Here, in terms of statistical tests, we want to suppress the false dismissal
rate $P_{\rm FDR}$ for $k<k_{m}$ and the false alarm rate $P_{\rm FAR}$ for
$k>k_{m}$, both caused by the flukes of the noise realizations (Buonanno,
Chen, & Vallisneri, 2003). We should notice that, for the target WDB,
presupposing the high-quality information from the already analyzed data (in
the stage (i)), the implication of these two rates are somewhat different from
the blind short-lived binary search with the current LVK network (The LIGO
Scientific Collaboration et al., 2021).

For a segment at $k>k_{m}$, assuming Gaussian noise, the false alarm rate is
estimated to be

$\displaystyle P_{\rm FAR}=\frac{\cal N}{\sqrt{2\pi}}\int_{\rho_{\rm
th}}^{\infty}{\exp\left[-\frac{\rho^{2}}{2}\right]}d\rho=\frac{\cal N}{2}{{\rm
erfc}\left[\frac{\rho_{\rm th}}{2^{1/2}}\right]}$ (10)

with the effective number of the templates $\cal N$ and the complementary
error function $\rm erfc(\cdot)$. Note that the function ${{\rm
erfc}\left[{\rho_{\rm th}}/{2^{1/2}}\right]}/2$ depends very strongly on
$\rho_{\rm th}$ and we have $10^{-2},10^{-3},10^{-5}$ and $10^{-7}$
respectively for $\rho_{\rm th}=2.3$, 3.1, 4.3 and 5.2. The effective number
of templates $\cal N$ depends on the possible phase models (and the
integration time $T_{\rm s}$). After all, in our pilot study, we shifted the
uncertainty of the stage (ii) to that of the template number $\cal N$. In the
actual data analysis, we can use the information of the earlier segments, for
empirically extending the phase models. At present, on the basis of the strong
dependence on $\rho_{\rm th}$, we adopt the reference value $\rho_{\rm th}=5$
for $P_{\rm FAR}=10^{-2}$. If we actually have $\Delta t_{\rm p}\ll T_{\rm
s}$, our setting $\rho_{\rm th}=5$ will result in a very conservative choice.

Next we discuss the false dismissal rate for an earlier segment $k<k_{m}$.
Using the results (e.g., $h_{\rm m}$ and $f_{\rm m}$) from the preceding long-
term search, we can estimate the expectation value

$\displaystyle\left\langle\rho_{k}\right\rangle=\frac{h_{\rm m}\sqrt{T_{\rm
s}}}{S_{n}(f_{\rm m})^{1/2}}$ (11)

as a function of the segment duration $T_{\rm s}$. Then, by adjusting the
duration $T_{\rm s}$, we can evaluate the false dismissal rate as

$\displaystyle P_{\rm FDR}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int^{\rho_{\rm
th}}_{-\infty}{\exp\left[-\frac{(\rho-\left\langle\rho_{k}\right\rangle)^{2}}{2}\right]}d\rho$
(12) $\displaystyle=$ $\displaystyle\frac{1}{2}{{\rm
erfc}\left[\frac{\left\langle\rho_{k}\right\rangle-\rho_{\rm
th}}{2^{1/2}}\right]}.$ (13)

For $P_{\rm FDR}\sim 10^{-2}$, we have

$\displaystyle\left\langle\rho_{k}\right\rangle\simeq\rho_{\rm th}+2.5=7.5.$
(14)

We can inversely obtain the segment duration $T_{\rm s}$ by

$\displaystyle T_{\rm s}=\left\langle\rho_{k}\right\rangle^{2}S_{n}(f_{\rm
m})h_{\rm m}^{-2}.$ (15)

In Table 1, we present the numerical results for $T_{\rm s}$, together with
the corresponding rotation cycles $f_{\rm m}\cdot T_{\rm s}$. For the base
model, we have $T_{\rm s}=5.7$ day and 10000 cycles.

From the arguments so far, we can easily see that, for the critical segement
$k=k_{m}$, we have $\rho_{k}\geq\rho_{\rm th}$ or $\rho_{k}<\rho_{\rm th}$,
depending on the location of the merger time $t_{\rm m}$ in the segment.

###  3.3 Estimation of the Merger Time

Next we discuss the estimation of the merger time $t_{\rm m}$ from the growing
list $\\{\rho_{k}\\}$ obtained after repeating the short-term matched
filtering. To proceed in a dualistic manner, we map the signal-to-noise ratios
$\rho_{k}$ by

$\displaystyle F_{k}\equiv\theta(\rho_{k}-\rho_{\rm th})$ (16)

with the step function $\theta(\cdot)$. We will have $F_{k}=1$ for $k<k_{m}$
and $F_{k}=0$ for $k>k_{m}$ ($F_{k}=1$ or 0 for $k=k_{m}$). Then, the list
$\\{F_{k}\\}$ should have the following form

$\displaystyle\\{F_{k}\\}=\\{1,1\cdots,1,\underline{1,0},0,0,0,\cdots\\}.$
(17)

The merger time $t_{\rm m}$ should contained in the two specific segments
marked with the underline. Therefore, the half width $\delta t_{\rm m}$ of the
estimation error becomes

$\displaystyle\delta t_{\rm m}=2^{-1}(2T_{\rm s})=T_{\rm s}.$ (18)

So far, we have fix the segmentation. By relaxing this setting, we can further
analyze the two candidate segments above. More specifically, we put a slidable
segment characterized by the starting time $x$ (keeping the duration at
$T_{\rm s}$) as

$\displaystyle k(x)\equiv[x,x+T_{\rm s}].$ (19)

We define the associated signal-to-noise ratio $\rho(x)$ for this segment. We
can identify the maximum value $x=x_{\max}$ which satisfies $F[\rho{(x)}]=1$.
Similarly, we read the minimum value $x=x_{\min}$ for $F([\rho{(x)}]=0$ with
$x_{\max}\leq x_{\min}$. Here, the inequality can be caused by an intricate
pattern of the function $F[\rho{(x)}]$ due the noise. The merger time $t_{\rm
m}$ should be contained simultaneously in the two segments $k(x_{\max})$ and
$k(x_{\min})$. Taking their overlapped part, we can estimate the merger time
with a half width of

$\displaystyle\delta t_{\rm m}\leq T_{\rm s}/2,$ (20)

corresponding to $\lesssim 3$ days for our base model.

The estimated magnitude $\delta t_{\rm m}$ is much larger than the transition
period $\Delta t_{\rm t}$ around the merger. Therefore, as mentioned earlier,
we practically do not need to too rigidly define the merger time $t_{\rm m}$
(e.g., before or after the transition stage (iii)).

##  4 Discussion

So far, we have discussed matched filtering analysis for a merging Galactic
WDB with LISA. Here, from a broader perspective, we mention related topics and
potential extension of this work.

###  4.1 GW data analysis

We have assumed $\Delta t_{\rm p}\ll$1yr, for the duration $\Delta t_{\rm p}$
of the late monochromatic stage (ii). Currently, it is not straightforward to
solidly estimate the duration $\Delta t_{\rm p}$ introduced for precautionary
purpose. As already commented, we can rather evaluate the potential phase
evolution, by empirically using the existing earlier data of the binary
(smoothly from the stage (i)). Similarly, the number of effective templates
$\cal N$ will be estimated reasonably well at the actual data analysis in the
stage (ii).

In a very pessimistic scenario with $\Delta t_{\rm p}\gtrsim$2yr, we might not
have the simple stage (i), which has played a key role in our data analysis
procedure. A semi-coherent search could be a powerful option, as currently
applied to unknown pulsar search with the LVK network (Abbott et al., 2022).
GW from a WDB could be observed at much higher signal-to-noise ratio with much
smaller rotation cycles. Therefore, depending on the phase irregularity, our
task could be much easier than the unknown pulsar search with the LVK network.

At estimating the merger time $t_{\rm m}$, we have used a matched filtering
analysis just for checking the disappearance of a nearly monochromatic GW
signal (without decoding its detailed structure). In reality, the wave phase
might be relatively simple with a precursor signature for the forthcoming
merger. Then, we might predict the merger time $t_{\rm m}$ by carefully
performing matched filtering analysis. If this is the case, our estimation
$\delta t_{\rm m}$ should be regarded as a very conservative bound. On another
front, while we have introduced the mapping (16) for our simple dualistic
argument, the original function $\rho(x)$ can be directly analyzed for the
estimation of the merger time $t_{\rm m}$.

Note that, if the operation period of LISA is longer than $\sim 1$yr, the
Galactic confusion noise is negligible in the frequency regime $f\gtrsim 5$mHz
(Robson, Cornish, & Liu, 2019; Littenberg et al., 2020). Therefore, for a
donor mass $m_{1}\gtrsim 0.3M_{\odot}$ shown in Table 1, the interference with
other binary signals is likely to be insignificant. However, we have $f_{\rm
m}\sim 5$mHz, for a smaller donor mass $m_{2}\sim 0.1M_{\odot}$. For such a
binary, we might need to securely suppress the interference effects at the
LISA multi source-analysis. Consequently, the required segment duration
$T_{\rm s}$ could be longer than that estimated from Eq. (11).

We can try other efficient methods (e.g., the chi-square analysis),
particularly in the context of the low latency analysis. The non-Gaussianity
of detector noises might be also worth studying.

###  4.2 Other Space Interferometers

Here we briefly comment on space interferometers other than LISA. Since a WDB
mainly emits a nearly monochromatic GW, our results scale simply with the
instrumental noise spectrum $S_{n}(f)$, as shown in Eqs. (8) and (15).

Around 20mHz, the sensitivities of Taiji (Ruan et al., 2018) and TianQin (Luo
et al., 2016) are quite similar and approximately two times better than that
of LISA. Meanwhile, at $\sim$20mHz, B-DECIGO (Kawamura et al., 2021) is
designed to have a sensitivity similar to LISA, but its optimal band is in the
higher frequency regime around 100mHz (Isoyama, Nakano, & Nakamura, 2018).
B-DECIGO and its follow-on mission DECIGO (Seto, Kawamura, & Nakamura, 2001)
might thereby enable us to detect extra-Galactic merging WDBs with total
masses higher than the Chandrasekhar mass (possibly relevant for SNIa)
(Maselli, Marassi, & Branchesi, 2020; Kinugawa et al., 2022).

###  4.3 EM Observation

When we try to identify an EM counterpart to a WDB, its binary parameters
(e.g., those related to orbital phase, sky location and inclination) will be
critically useful. Once identified, we can continue to frequently observe the
binary with EM telescopes, even after the operation period of LISA. Moreover,
we might monitor the transitional merger stage (iii) with the telescopes. Such
an observation will provide us with valuable information on white dwarfs and,
possibly, its explosion mechanism. The archived data much before LISA’s launch
might be also useful for studying a WDB in a long time span (Digman & Hirata,
2022).

##  5 summary

The merger rate of Galactic WDBs is roughly estimated to be $\rm\sim
0.02yr^{-1}$. In this paper, we discuss identification of a merger event in
the data streams of LISA. Until very close its final merger, a typical WDB
will emit nearly monochromatic GW around 20mHz. After a short transition
period, it will virtually stop emitting GW. LISA has potential to detect its
nearly monochromatic GW with a signal-to-noise ratio of $O(10^{2})$ and
determine its sky location within $\sim 0.01$-$0.1{\rm deg^{2}}$. By
repeatedly checking the disappearance of its nearly monochromatic GW, we could
robustly estimate its merger time with accuracy of $\sim 3$-10 days.

## Acknowledgements

This work is supported by JSPS Kakenhi Grant-in-Aid for Scientific Research
(Nos. 17H06358, 19K03870 and 23K03385).

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