NYU-TH-09/06/15

Strongly Coupled Condensate of High Density Matter

Gregory Gabadadze

Center for Cosmology and Particle Physics, Department of Physics,

New York University, New York, NY 10003, USA

###### Abstract

Arguments are summarized, that neutral matter made of helium, carbon, etc.,
should form a quantum liquid at the above-atomic but below-nuclear densities
for which the charged spin-0 nuclei can condense. The resulting substance has
distinctive features, such as a mass gap in the bosonic sector and a gap-less
spectrum of quasifermions, which determine its thermodynamic properties. I
discuss an effective field theory description of this substance, and as an
example, consider its application to calculation of a static potential between
heavy charged impurities. The potential exhibits a long-range oscillatory
behavior in which both the fermionic and bosonic low-energy degree of freedom
contribute. Observational consequences of the condensate for cooling of
helium-core white dwarf stars are briefly discussed.

Based on a talk given at the international workshop
“Crossing the boundaries: Gauge dynamics at strong coupling”
honoring the 60th birthday of M.A. Shifman
Minneapolis, May 14-17, 2009

## Foreword

Like many in the audience, I first met Misha on the pages of journal
publications, before meeting him in person. While working on an undergraduate
thesis at Moscow University, I came across Misha’s review paper “Anomalies and
Low-Energy Theorems of Quantum Chromodynamics” [1]. Impressions of that work
were very distinct – a clear exposition of subtle field theory aspects of the
quantum anomalies, culminating in creative applications to low-energy hadron
phenomenology. The work stood out by its originality, depth, inspiration and
balance of the formalism and applications – the remarkable signatures of
Misha’s enormous contribution to theoretical physics at the forefront of both
field theory and particle phenomenology.

I met Misha in person in Minneapolis in 1998. The discussion with him was very
inspiring. Soon, in Aspen, we started to work on a project. A bit later I
ceased the opportunity to get exposed to two years of a unique FTPI
experience. We continued to work on and off on various projects since then. I
value those works very highly, and feel privileged, as I’m sure many of you do
too, for having such a collaborator.

Happy 60th Birthday Misha!

## Description of charged condensate

Consider a neutral system of a large number of nuclei each having charge $Z$,
and neutralizing electrons. If average inter-particle separations in this
system are much smaller than the atomic scale, $\sim 10^{-8}~{}cm$, while
being much larger than the nuclear scale, $\sim 10^{-13}~{}cm$, neither the
atomic nor nuclear effects will play any significant role. Moreover, the
nuclei can also be treated as point-like particles.

In what follows we focus on spin-0 nuclei with $Z\leq 8$ (helium, carbon,
oxygen), and consider the electron number-density in the interval
$J_{0}\simeq(0.1-5~{}MeV)^{3}$. Then the electron Fermi energy will exceed the
electron-electron and electron-nucleus Coulomb interaction energy. Moreover,
at temperatures below $\sim 10^{7}~{}K$, which are of interest here, the
system of electron represents a degenerate Fermi gas.

Since the nuclei (we also call them ions below) are heavier, temperature at
which they’ll start to exhibit quantum properties will be lower. Let us define
the “critical” temperature $T_{c}$, at which the de Broglie wavelengths of the
ions begin to overlap

$T_{c}\simeq\frac{4\pi^{2}}{3m_{H}d^{2}}\,,~{}~{}~{}~{}~{}d\equiv\left(3Z\over
4\pi J_{0}\right)^{1/3}\,,$ (1)

where, $m_{H}$ denotes the mass of the ion (the subscript
${}^{\prime\prime}H^{\prime\prime}$ stands for heavy), and $d$ denotes the
average separation between the ions111The de Broglie wavelength above is
defined as $\lambda_{dB}=2\pi/|{\bf k}|$, where ${\bf
k}^{2}/2m_{H}=3k_{B}T/2$. We define $T_{c}$ as the temperature at which
$\lambda_{dB}\simeq d$. Note that this differs by a numerical factor of
$\sqrt{2\pi/3}$ from the standard definition of the thermal de Broglie
wavelength, $\Lambda\equiv\sqrt{2\pi/mk_{B}T}$, that appears in the partition
function of an ideal gas of number-density $n$ in the dimensionless
combination $\Lambda^{3}n$..

Somewhat below $T_{c}$ quantum-mechanical uncertainties in the ion positions
become greater than an average inter-ion separation. Hence the latter concept
looses its meaning as a microscopic characteristic of the system; the ions
enter a quantum-mechanical regime of indistinguishability. Then, the many-body
wavefunction of the spin-0 ions should be symmetrized, and this would
unavoidably lead to probabilistic “attraction” of the bosons to condense,
i.e., to occupy one and the same quantum state. We refer the system of
condensed nuclei and electrons as charged condensate.

In the condensate the scalars occupy a quantum state with zero momentum.
Moreover, small fluctuations of the bosonic sector happen to have a mass gap,
$m_{\gamma}=(Ze^{2}J_{0}/m_{H})^{1/2}$, which exceeds $T_{c}$ by more than an
order of magnitude. Therefore, once bosons are in the charged condensate,
their phonons cannot be thermally excited. However, the gap-less fermionic
degrees of freedom are thermally excited, and carry the most of the entropy of
the entire system [2]-[5].

For further discussions it is useful to rewrite the expression for $T_{c}$ in
terms of the mass density $\rho\equiv m_{H}J_{0}$ measured in $g/cm^{3}$:

$\displaystyle T_{c}=\rho^{2/3}\,\left({3.5\cdot 10^{2}\over
Z^{5/3}}\right)~{}K\,,$ (2)

where the baryon number of an ion was assumed to equal twice the number of
protons, $A=2Z$ (true for helium, carbon, oxygen…). Thus, for
$\rho=10^{6}~{}g/cm^{3}$ and helium-4 nuclei we get $T_{c}\simeq 10^{6}~{}K$,
while for the carbon nuclei with the same mass density $T_{c}\simeq 2\cdot
10^{5}~{}K$.

Temperature at which the condensation phase transition takes place,
$T_{condens}$, need not coincide with $T_{c}$. Moreover, we would expect
$T_{condens}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}T_{c}$. Calculation of $T_{condens}$
from the fundamental principles of this theory is hard. However, we can obtain
an interval in which $T_{condens}$ should fit. For this we introduce the
following parametrization:

$\displaystyle T_{condens}=\zeta\,T_{c}\,,$ (3)

where $\zeta$ is an unknown dimensionless parameter that should depend on
density more mildly than $T_{c}$ does. Numerically, however, this parameter
should vary in the interval $0.1\ll\zeta\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1$: The point
$\zeta=0.1$ would corresponds to the temperature of the Bose-Einstein (BE)
condensation of a free gas for which, $T^{BE}_{condens}\simeq 1.3/m_{H}d^{2}$,
is known from the fundamental principles. The condensation temperature in our
system should be higher than $T^{BE}_{condens}$ since the repulsion makes
easier for the condensation to take place [6]. In our case, repulsive
interactions between the bosons are strong – the Coulomb energy is at least an
order of magnitude greater that any other energy scale in the system. Hence,
we should expect $\zeta\gg 0.1$. On the other hand, given the definition of
$T_{c}$, the parameter $\zeta$ cannot be greater than unity. In what follows
we will retain $\zeta$ in our expressions, but use $\zeta\simeq 1$ when it
comes to numerical estimates.

The condensation will take place after gradual cooling, only if $T_{condens}$
is greater than the temperature at which the substance could crystallize. A
classical plasma crystallizes when the Coulomb energy becomes about $\sim 180$
times greater than the average thermal energy per particle [7, 8, 9]. This
gives the following crystallization temperature222The presented formula for
the crystallization temperature is entirely classical. The temperature scale
that determines the classical versus quantum nature of the crystallization
transition is the Debye temperature $\theta_{D}\simeq 4\cdot
10^{3}\rho^{1/2}~{}K$. Often, $\theta_{D}$ may significantly exceed
$T_{\text{cryst}}$ [10]. In such cases, quantum zero-point oscillations should
be taken into account. This seems to delay the formation of quantum crystal,
lowering $T_{\text{cryst}}$ from its classical value at most by about $\sim
10\%$ [11]. Since this is a small change, we will ignore it in our estimates.

$T_{\text{cryst}}\simeq\rho^{1/3}\left(0.8\cdot 10^{3}Z^{5/3}\right)~{}K\,.$
(4)

Note that the density dependence of $T_{c}$ is different from that of
$T_{\text{cryst}}$ – for higher densities $T_{c}$ grows faster, making
condensation more and more favorable! One can define the “equality” density
for which $T_{condens}=T_{\text{cryst}}$:

$\displaystyle\rho_{\rm
eq}=\left({2.3\over\zeta}\right)^{3}Z^{10}\,g/cm^{3}\,.$ (5)

For helium, $Z=2$, and $\rho_{\rm eq}\simeq 10^{4}~{}g/cm^{3}$, while for
carbon, $Z=6$, and $\rho_{\rm eq}\simeq 10^{9}~{}g/cm^{3}$ (as mentioned
above, we use $\zeta\simeq 1$). These results are very sensitive to the value
of $\zeta$; for instance, $\rho_{\rm eq}$ could be an order of magnitude
higher if $\zeta\simeq 0.5$. Irrespective of this uncertainty, however, the
obtained densities are in the right ballpark of average densities present in
helium-core white dwarfs $\sim 10^{6}~{}g/cm$, (for carbon dwarfs, they’re
closer to those expected in high density regions only [5].)

Is the charged condensate a ground state of the system at hand? For the higher
values of the density interval considered, the crystal would not exist due to
strong zero-point oscillations. At lower densities, the crystalline state has
lower free energy (at least near zero temperature) due to more favorable
Coulomb binding. Hence, the condensate can only be a metastable state. The
question arises whether after condensation at $\sim T_{condens}$ the system
could transition at lower temperatures $\sim T_{\rm cryst}$ to the crystal
state, as soon as the latter becomes available.

In the condensate, the boson positions are entirely uncertain while their
momenta equal to zero. In order for such a system to crystallize later on,
each of the bosons should acquire energy of the zero-point oscillations of
crystal ions. As long as this energy, $\sim(Ze^{2}J_{0}/m_{H})^{1/2}$, is much
greater than $T_{\rm cryst}$, no thermal fluctuations can excite the condensed
bosons to transition to the crystalline state. The latter condition is well-
satisfied for all the densities considered in this work. There could, however,
exist a spontaneous transition of a region of size $R_{c}$ to the crystallized
state via tunneling. The value of $R_{c}$, and the rate of this transition,
will be determined, among other things, by tension of the interface between
the condensate and crystal state, which is hard to evaluate. However, for
estimates the following qualitative arguments should suffice: the height of
the barrier for each particle is $(Ze^{2}J_{0}/m_{H})^{1/2}=m_{\gamma}$, while
the number of bosons in the $R_{c}$ region $\sim R_{c}^{3}J_{0}/Z$. Hence, the
transition rate should scale as ${\rm exp}(-m_{\gamma}J_{0}R^{4}_{c}/Z)$.
Since we expect that $R_{c}>1/m_{\gamma}$, the rate is strongly suppressed for
the parameters at hand.

## Effective field theory description

We use a low-energy effective field theory description to study the charged
condensate. Even though realistic temperatures in the system may be well above
zero, we focus on the zero-temperature limit. The relevance of this limit is
justified a posteriori and goes as follows: the spin-0 nuclei undergo the
condensation to the zero-momentum state; their phonons cannot be excited since
their gap, $m_{\gamma}$, is greater than $T_{c}$. On the other hand, gap-less
near-the-Fermi-surface quasielectrons will be excited. Therefore, all the
thermal fluctuations will end up being stored in the fermionic quasiparticles.
For the latter, however, the finite temperature effects aren’t significant
since their Fermi energy is so much higher, $T/J_{0}^{1/3}\ll 10^{-2}$. We
note that the finite temperature effects, in a general setup with condensed
bosons, were calculated in Refs. [12, 13].

We begin at scales that are well below the heavy mass scale $m_{H}$, but
somewhat above the scale set by ${\rm max}[\mu_{f},m_{e}]$, where $\mu_{f}$
and $m_{e}$ are the electron chemical potential and mass respectively. Hence
the electrons are described by their Dirac Lagrangian, while for the
description of the nuclei we will use a charged scalar order parameter
$\Phi(x)$. As it was shown in [4], in a non-relativistic approximation for the
nuclei, the effective Lagrangian proposed by Greiter, Wilczek and Witten (GWW)
[14] in a context of superconductivity, is also applicable here, given that an
appropriate reinterpretation of its variables and parameters is made.

The construction of the GWW Lagrangian is based on the following fundamental
principles: it is consistent with the translational, rotational, Galilean and
the global $U(1)$ symmetries, preserves the algebraic relation between the
charged current density and momentum density, gives the Schrödinger equation
for the order parameter in the lowest order, and is gauge invariant [14].
Combined with the electron dynamics the GWW Lagrangian reads (we omit for
simplicity the Maxwell term):

$\displaystyle{\cal L}_{eff}={\cal P}\left({i\over
2}(\Phi^{*}D_{0}\Phi-(D_{0}\Phi)^{*}\Phi)-{|D_{j}\Phi|^{2}\over
2m_{H}}\right)\,+{\bar{\psi}}(i\gamma^{\mu}D^{f}_{\mu}-m_{f})\psi,$ (6)

where we use the standard notations for covariant derivatives with the
appropriate charge assignments: $D_{0}\equiv(\partial_{0}-iZeA_{0})$,
$D_{j}\equiv(\partial_{j}-iZeA_{j})$, $D^{f}_{\mu}=\partial_{\mu}+ieA_{\mu}$,
while ${\cal P}(x)$ stands for a general polynomial function of its argument.
The coefficients of this polynomial, ${\cal
P}(x)=\sum^{\infty}_{n=0}C_{n}x^{n}$, are dimensionful parameters that are
inversely proportional to powers of a short-distance cutoff of the effective
field theory333In general one should also add to the Lagrangian terms
$\mu_{NR}\Phi^{*}\Phi$, $\lambda(\Phi^{*}\Phi)^{2}/m_{H}^{2}$,
$\lambda_{1}(\Phi^{*}\Phi){\bar{\psi}}\psi/(m_{H}J_{0}^{1/3})$, and other
higher dimensional operators that are consistent with all the symmetries and
conditions that lead to (6) (the Yukawa term is not). Here $\mu_{NR}$ denotes
a non-relativistic chemical potential for the scalars. These terms are not
important for the low-temperature spectrum of small perturbations we’re
interested in, as long as $\lambda,\lambda_{1}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1$ and $J_{0}\ll
m^{3}_{H}$. However, near the phase transition point it is the sign of
$\mu_{NR}$ that would distinguish between the broken and symmetric phases, so
these terms should be included for the discussion of the symmetry restoration.
We also note that the scalar part of (6) is somewhat similar to the Ginzburg-
Landau (GL) Lagrangian for superconductivity. However, there are significant
differences between them, one such difference being that the coherence length
in the GL theory is many orders of magnitude greater than the average
interelectron separation, while in the present case, the “size of the scalar”
$\Phi$ is smaller that the average interparticle distance..

Once the basic Lagrangian is fixed, we introduce the electron chemical
potential term $\mu_{f}\psi^{+}\psi\,$, where
$\mu_{f}=\epsilon_{F}=[(3\pi^{2}J_{0})^{2/3}+m_{f}^{2}]^{1/2}$. This is the
only term that at the tree level sets a frame in which the electron total
momentum is zero.

There exists a homogeneous solution of the equations of motion that follow
from the effective Lagrangian (6) [3]:

$\displaystyle Z|\Phi|^{2}=J_{0}\,,~{}~{}~{}A_{\mu}=0,~{}~{}~{}~{}{\cal
P}^{\prime}(0)=1\,.$ (7)

(We use the unitary gauge $\Phi=|\Phi|$). The condition ${\cal
P}^{\prime}(0)=1$ is satisfied by any polynomial functions ${\cal P}(x)$ for
which the first coefficient is normalized to unity

$\displaystyle{\cal P}(x)=x+C_{2}x^{2}+...\,.$ (8)

The above solution describes a neutral system of negatively charged electrons
of charge density $-eJ_{0}$, and positively charged scalar condensate of
charge density $Ze\Phi^{+}\Phi=eJ_{0}$ [4, 5].

Calculation of the spectrum of small perturbations is straightforward. The
Lagrangian density for the fluctuations in the quadratic approximation reads
[2]

$\displaystyle{\cal L}_{2}=-{1\over 4}F_{\mu\nu}^{2}+{1\over
2}m_{0}^{2}A_{0}^{2}-{1\over 2}m_{\gamma}^{2}A_{j}^{2}+{1\over
2}\,A_{0}{(2m_{H}m_{\gamma})^{2}\over-\Delta}A_{0}\,,$ (9)

where $\Delta$ denotes the Laplacian, and the last term emerged due to mixing
of $A_{0}$ with the fluctuation of the $|\Phi|$, which we integrated out. As
before,

$\displaystyle m_{\gamma}^{2}\equiv{Ze^{2}J_{0}\over m_{H}}\,,$ (10)

and $m_{0}^{2}=m_{\gamma}^{2}+C_{2}e^{2}J^{2}_{0}$. At this stage we retained
the fermionic fluctuations only in the Thomas-Fermi approximation [3]; an
important refinement of this approximation, discussed in [4], will be included
below.

That there are no pathologies in (9), such as ghost and/or tachyons, can be
seen by calculating the Hamiltonian density:

$\displaystyle{\cal H}={\pi_{j}^{2}\over 2}+{F^{2}_{ij}\over 4}+{1\over
2}(\partial_{j}\pi_{j})\left(m_{0}^{2}+{4M^{4}\over-\Delta}\right)^{-1}(\partial_{j}\pi_{j})+{1\over
2}m_{\gamma}^{2}A_{j}^{2}\,.$ (11)

Here, $M^{2}\equiv m_{H}m_{\gamma}$ and $\pi_{j}\equiv-F_{0j}$. The
Hamiltonian is positive semi-definite. Moreover, the spectrum has a mass gap
determined by $m_{\gamma}$ (10). There are two transverse polarizations of a
massive photon, as well as the longitudinal mode, the phonon, with the same
mass $m_{\gamma}$ [2].

The massive bosonic collective excitations give rise to exponentially
suppressed contributions to the value of specific heat of the charged
condensate since typically $m_{\gamma}\gg T_{c}$. The suppression scales as
${\rm exp}(-m_{\gamma}/T)$, where $T\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}T_{c}$. This is in
contrast with the crystal, where the dominant contribution to the specific
heat comes from a gap-less phonon, and scales with temperature as $T^{3}$.

As to the electrons, their behavior is similar in both crystal and condensate
cases. At temperatures of interest they form a degenerate Fermi gas with gap-
less excitations near the Fermi surface. Their contribution to the specific
heat scales linearly with temperature. In the case of crystallized substance
this is sub-dominant to the specific heat due to the crystal phonon. For the
charged condensate, however, the (quasi)electron fluctuations are the dominant
contributors to the specific heat.

To study the effects of collective bosonic and fermionic modes, as an
interesting example, we look at a potential between two impurity nuclei (say
hydrogen, or helium-3) of charge $Q_{1}$ and $Q_{2}$. The calculation of the
propagator that involves the light collective modes (for relativistic
fermions) gives the following result [4]:

$\displaystyle V_{stat}=\alpha_{\rm em}{Q_{1}Q_{2}}\left({e^{-Mr}\over\,r}{\rm
cos}(Mr)\,+{4\alpha_{\rm em}\over\pi}{k_{F}^{5}{\rm sin}(2k_{F}r)\over
M^{8}r^{4}}\right)\,.$ (12)

The first, exponentially suppressed term modulated by a periodic function, is
due to cancellation between the screened Coulomb potential and that of a
phonon [4]. The fact of such a cancellation, and that it could give rise to
the oscillatory behavior of the exponentially screened potential was pointed
out before in Ref. [15] in the context of superconductivity444I’d like to
thank Ki-Myeong Lee who recently brought the paper [15] to my attention..

Most important, however, is the second term in (12) that has a long-range [4].
It dominates over the exponentially suppressed term in (12) for scales of
physical interest, and exhibits the power-like behavior modulated by a
periodic function.

The potential (12) is a generalization of the Friedel potential to the case
when in addition to the fermionic excitations there are also collective modes
due to the charged condensate. The long-range oscillating term in (12) is also
a result of a subtraction between the conventional Friedel term and the long-
range oscillating term due to a phonon. As a result, its magnitude is
suppressed compared to what it would have been in a theory without the
condensed charged bosons [4] (see, [16] for the discussion of the conventional
Friedel potential, and Ref. [13] for its recent detailed study in the presence
of the charged condensate at finite temperature.)555Note that for spin-
dependent interactions the same effects of the charged condensate would give a
generalization of the Ruderman-Kittel-Kasuya-Yosida (RKKY) potential [17].

The potential (12) is not sign-definite. In particular, it can give rise to
attraction between like charges; this attraction is due to collective
excitations of both fermionic and bosonic degrees of freedom. This represents
a generalization of the Kohn-Luttinger [18] effect to the case where on top of
the fermionic excitations the collective modes of the charged condensate are
also contributing666In the charged condensate Cooper pairs of electrons can
also be formed, however, the corresponding transition temperature, and the
magnitude of the gap, are suppressed by a factor ${\rm exp}(-1/e_{eff}^{2})$,
where $e_{eff}^{2}$ is proportional to the value of the inter-electron
potential that contains both screened Coulomb and phonon exchange. The fact
that this potential has attractive domain, but is very small, is suggested by
the static potential found in [4] (see also eq. (12) above); the latter is
suppressed by a power of a large scale $M$. Furthermore, taking into account
the frequency dependence of the propagator in the Eliashberg equation does not
seem to change qualitatively the conclusion on a strong suppression of the
Green’s function and pairing temperature. Hence, even though the bosonic
sector (condensed nuclei) is superconducting at reasonably high temperatures
$\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{6}~{}K$, interactions with gap-less fermions could
dissipate the superconducting currents. Only at extremely low temperatures,
exponentially close to the absolute zero, the electrons could also form a gap
leading to superconductivity of the whole system. In the present work we
consider temperatures at which electrons are not condensed into Cooper pairs,
and ignore the finite temperature effects..

## Applications to White Dwarfs

The above described system of electrons and nuclei constitutes cores of white
dwarf stars. Up to a factor of a few, these are roughly Earth’s size solar-
mass objects; their mass density may range over
$\sim(10^{6}-10^{10})~{}g/cm^{3}$, most of them being near the lower edge of
this interval. Since the dwarf stars exhausted thermonuclear fuel in their
cores already, they evolve by cooling [19]; the ones that we consider in this
work cool from $\sim 10^{7}~{}K$ down to lower temperatures.

As a typical dwarf star cools down, the Coulomb interaction energy in a
classical plasma of charged nuclei will significantly exceed their classical
thermal energy, and the nuclei, in order to minimize energy, would organize
themselves into a crystal lattice [20]. In most of these cases quantum effects
of the nuclei should be negligible; for instance, the Debye temperature should
be less than the temperature at which crystallization takes place, and the de
Broglie wavelengths of the nuclei should be much smaller than the average
internuclear separations. This indeed is the case in majority of white dwarf
stars, the cores of which are composed of carbon and/or oxygen nuclei and span
the interval of mass densities around $\sim(10^{6}-10^{8})~{}g/cm^{3}$.

However, there exists a class of dwarf stars in which the nuclei enter the
quantum regime before the classical crystallization process sets in [10, 11].
Among these, furthermore, there is a relatively small subclass of the dwarf
stars for which the temperature $T_{c}$, is higher than the would-be
crystallization temperature $T_{cryst}$ [5]. In such dwarf cores the charged
condensation should be expected to take place.

White dwarfs composed of helium constitute a smaller sub-class of dwarf stars
(see, [21, 22] are references therein); they exhibit best conditions for the
charged condensation. Most of helium dwarfs are believed to be formed in
binary systems, where the removal of the envelope off the dwarf progenitor red
giant by its binary companion happened before helium ignition, producing a
remnant that evolves to a white dwarf with a helium core. Helium dwarf masses
range from $\sim 0.5~{}M_{\odot}$ down to as low as $(0.18-0.19)~{}M_{\odot}$,
while their envelopes are mainly composed of hydrogen.

Using the approach of [23], and following [5] we will consider an over-
simplified model of a reference helium star of mass $M=0.5~{}M_{\odot}$ with
the atmospheric mass fractions of the hydrogen, and heavy elements
(metallicity) respectively equal to

$\displaystyle X\simeq 0.99,\quad\quad Z_{m}\simeq(0.0002-0.002)~{}.$ (13)

The lower value of the metallicity $Z_{m}\simeq 0.0002$ is appropriate for the
recently discovered 24 He WDs in NGC 6397 [22], but for completeness, we
consider a wider range for this parameter.

It is straightforward to find the following expression for the cooling time of
a star in the classical regime [23]

$\displaystyle
t_{He}=\frac{k_{B}}{CAm_{u}}\left[\frac{3}{5}(T_{f}^{-\frac{5}{2}}-T_{0}^{-\frac{5}{2}})+Z\frac{\pi^{2}}{3}\frac{k_{B}}{E_{F}}(T_{f}^{-\frac{3}{2}}-T_{0}^{-\frac{3}{2}})\right],$
(14)

where $T_{f}$ and $T_{0}$ denote the final and initial core temperatures. The
first term in the bracket on the right hand side corresponds to cooling due to
classical gas of the ions and the second term corresponds to the contribution
coming from the Fermi sea. The latter is sub-dominant in the range of final
temperatures we are interested in (the factor Z in front of this term is due
to $Z$ electrons per ion). Since $T_{f}\ll T_{0}$, the age of a dwarf star
typically doesn’t depend on the initial temperature. Neglecting the fermion
contribution, we find time that is needed to cool down to critical temperature
$T_{f}=T_{c}$

$\displaystyle
t_{He}=\frac{3}{5}\frac{k_{B}T_{c}M}{Am_{u}L(T_{c})}\simeq(0.76-7.6)~{}\text{Gyr}\,.$
(15)

Where an order of magnitude interval in (15) is due to the interval in the
envelope metallicity composition given in (13). We also find the corresponding
luminosities

$\displaystyle
L(T_{c})\simeq(10^{8}~{}erg/s)\frac{M}{M_{\odot}}\left(\frac{T_{c}}{\text{K}}\right)^{{7/2}}\simeq
1.5\cdot(10^{-4}-10^{-5})L_{\odot}\,,$ (16)

which are in the range of observable luminosities ($L_{\odot}\simeq 3.84\cdot
10^{33}~{}erg/s$).

After the condensation, specific heat of the system dramatically drops as the
collective excitations of the condensed nuclei become massive and “get
extinct”. A contribution from the Fermi sea, which is strongly suppressed by
the value of Fermi energy, becomes the dominant one. The phase transition
itself would take some time to complete, and the drop-off in specific heat
will not be instantaneous.

In the zeroth approximation, we can regard the transition to be very fast, and
retain only the fermion contribution to specific heat below $T_{c}$. Then, the
expression for the age of the star for $T_{f}<T_{c}$, reads as follows

$\displaystyle
t_{He}^{\prime}=\frac{k_{B}}{CAm_{u}}\left[\frac{3}{5}(T_{c}^{-\frac{5}{2}}-T_{0}^{-\frac{5}{2}})+Z\frac{\pi^{2}}{3}\frac{k_{B}}{E_{F}}(T_{f}^{-\frac{3}{2}}-T_{0}^{-\frac{3}{2}})\right].$
(17)

Notice the difference of (17) from (14) – in the former $T_{f}<T_{c}$ and it
is $T_{f}$ that enters as final temperature in the fermionic part, while
$T_{c}$ should be taken as the final temperature in the bosonic part.

From the ratio of ages, $\eta={t_{He}/t_{He}^{\prime}}$, for two identical
helium dwarf stars, with and without the interior condensation, we deduce that
the charged condensation substantially increases the rate of cooling– the age
could be twenty times less than it would have been without the condensation
phase [5].

The condensation of the core would induce significant deviations from the
classical curve for helium white dwarfs. What is independent of the
uncertainties involved in these discussions, is the fact that the luminosity
function (LF) will experience a significant drop-off after the charged
condensation phase transition is complete. This is due to the “extinction” of
the bosonic quasiparticles below the phase transition point. In fact, the LF
will drop by a factor of $\sim 200$. This may be relevant for an explanation
of the observed termination of a sequence of the 24 He WD’s found in [22]. See
Ref. [5] for more details.

Finally, the magnetic properties of the charged condensate, which are similar
to those of type II superconductor, and in particular admit the presence of
Abrikosov’s vortices, were studied in Ref. [24]. As was shown there, only very
strong magnetic fields, $\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10^{7}~{}Gauss$, will be able to
penetrate the dwarf cores in the vortices, while weaker fields will be
entirely expelled from it.

Acknowledgments

The above-reported results constitute a part of the work done in collaboration
with Rachel A. Rosen and David Pirtskhalava [2]-[5], [24]. I’d like to thank
Paul Chaikin, Daniel Eisenstein, Leonid Glazman, Andrei Gruzinov, Stefan
Hofmann, Andrew MacFadyen, Juan Maldacena, Aditi Mitra, Slava Mukhanov, Hector
Rubinstein, Malvin Ruderman and Arkady Vainshtein for useful discussions and
correspondence on these topics. The work was supported by the NSF grant
PHY-0758032.

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11institutetext: ShareChat

# Variance Reduction in Ratio Metrics for Efficient Online Experiments

Shubham Baweja  Neeti Pokharna  Aleksei Ustimenko and Olivier Jeunen

###### Abstract

Online controlled experiments, such as A/B-tests, are commonly used by modern
tech companies to enable continuous system improvements. Despite their
paramount importance, A/B-tests are expensive: by their very definition, a
percentage of traffic is assigned an inferior system variant. To ensure
statistical significance on top-level metrics, online experiments typically
run for several weeks. Even then, a considerable amount of experiments will
lead to _inconclusive_ results (i.e. false negatives, or type-II error). The
main culprit for this inefficiency is the variance of the online metrics.
Variance reduction techniques have been proposed in the literature, but their
direct applicability to commonly used ratio metrics (e.g. click-through rate
or user retention) is limited.

In this work, we successfully apply variance reduction techniques to ratio
metrics on a large-scale short-video platform: ShareChat. Our empirical
results show that we can either improve A/B-test confidence in 77% of cases,
or can retain the same level of confidence with 30% fewer data points.
Importantly, we show that the common approach of including as many covariates
as possible in regression is counter-productive, highlighting that control
variates based on Gradient-Boosted Decision Tree predictors are most
effective. We discuss the practicalities of implementing these methods at
scale and showcase the cost reduction they beget.

##  1 Introduction & Motivation

Online controlled experiments (colloquially known as A/B-tests) are a crucial
tool for online businesses to enable data-driven decision making [8, 9].
Nevertheless, they are inherently costly and statistical nuances pose
challenges when implementing them at scale and interpreting their results
correctly [5, 7].

The motivation behind this research is to enhance the confidence and
efficiency of A/B-testing, particularly when dealing with commonly used
_ratio_ metrics such as click-through rates and user retention. These metrics
play a crucial role in evaluating user engagement and platform performance but
inhibit the use of classical statistical tools. Our goal is to increase the
_sensitivity_ of such ratio metrics. This can either lead to  (i) more
conclusive results at constant sample sizes (i.e. reduced type-II error), or
(ii) a reduction of the required sample size to obtain constant $p$-values
(i.e. saving time and experiment cost).

Variance reduction techniques have been proposed in the literature [1, 3, 4,
11], but their direct applicability to ratio metrics is not always
straightforward. Furthermore, a myopic focus on _variance_ can make us blind
to _bias_ [10], implying that _variance_ reduction does not unequivocally lead
to _sensitivity_ improvements.

We discuss the practical application of a synthesis of existing variance
reduction techniques to improve sensitivity (or, analogously, decrease
experimentation cost) on the ShareChat platform. Our empirical observations,
based on a log of historical real-world A/B-experiments, underline our
theoretical insights. We find that the highest variance reduction does not map
to the highest sensitivity increase. Our best-performing method can increase
statistical confidence in 77% of A/B-experiments, or achieve on-par confidence
with 30% fewer samples, directly impacting the cost per experiment for the
business.

##  2 Problem Statement, Methodology & Contribution

###  2.1 Statistical Hypothesis Testing

Consider an online controlled experiment with deployed variants $A$ and $B$,
and a metric $\mathcal{M}$ to evaluate. That is, $\mathcal{M}$ is a random
variable whose expectation we wish to estimate under $A$ and $B$. Typically, a
$z$-test is done to ascertain whether the difference in observed metric values
is statistically significant. Denote by $\mu(\mathcal{M})$ the mean of the
observed values, and by $\sigma(\mathcal{M})$ its standard deviation. A
subscript $\mu_{V}$ indicates the mean over observed values for units assigned
to $V$.

The $z$-statistic is then given by Eq. 1, which can be transformed to a two-
tailed $p$-value for the null hypothesis $A\simeq B$ following Eq. 2, where
$\Phi(\cdot)$ represents the cumulative distribution function (CDF) for a
standard Gaussian.  $z_{A\succ
B}(\mathcal{M})=\frac{\mu_{A}(\mathcal{M})-\mu_{B}(\mathcal{M})}{\sqrt{\frac{\sigma_{A}(\mathcal{M})^{2}}{N_{A}}+\frac{\sigma_{B}(\mathcal{M})^{2}}{{N_{B}}}}}$
(1) $\quad p_{A\succ B}(\mathcal{M})=2\Phi\left(-\big{|}z_{A\succ
B}(\mathcal{M})\big{|}\right)$ (2)

For a confidence level of $\alpha$ (typically $\approx 0.05$), we can reject
the null hypothesis when $p_{A\succ B}(\mathcal{M})<\alpha$, and claim a
statistically significant impact on metric $\mathcal{M}$. Many standard
metrics (e.g. daily active users, event counters, et cetera) can be expressed
as Bernoulli-distributed random variables. In these cases, $\mu(\mathcal{M})$
and $\sigma(\mathcal{M})$ can be straightforwardly computed using standard
formulas. Aside from these metrics, modern platforms on the web typically also
care about _ratio_ metrics (e.g. click-through rate, retained users per active
users, et cetera). In these cases, we need to apply the Delta method to
estimate the variance in the denominator of Eq. 1. For a metric
$\mathcal{M}\equiv\frac{\mathcal{M}_{N}}{\mathcal{M}_{D}}$, this yields:

$\sigma(\mathcal{M})^{2}\approx\frac{\mu(\mathcal{M}_{N})^{2}}{\mu(\mathcal{M}_{D})^{2}}\left(\frac{\sigma(\mathcal{M}_{N})^{2}}{\mu(\mathcal{M}_{N})^{2}}+\frac{\sigma(\mathcal{M}_{D})^{2}}{\mu(\mathcal{M}_{D})^{2}}-2\frac{\mathrm{cov}(\mathcal{M}_{N},\mathcal{M}_{D})}{\mu(\mathcal{M}_{N})\mu(\mathcal{M}_{D})}.\right)$
(3)

Alternatively, one can adopt a linearisation approach to obtain a new metric
$L(\mathcal{M})=\mathcal{M}_{N}-c\mathcal{M}_{D}$ that preserves
directionality and statistical power [1]. Here,
$c=\frac{\mu_{C}(\mathcal{M}_{N})}{\mu_{C}(\mathcal{M}_{D})}$, where $C$
corresponds to the _control_ variant.

###  2.2 Variance Reduction in Online Controlled Experiments

Lower $p$-values (i.e. higher $z$-scores) indicate higher confidence in
rejecting the null hypothesis. From Eq. 1, a clear way to increase the
$z$-score is to increase the sample size $N$. As this corresponds to running
the experiment for a longer period or on a larger portion of traffic, this is
costly. An alternative route is to instead decrease the variance of the metric
$\sigma(\mathcal{M})^{2}$. This can be done by leveraging _control variates_.
Suppose we have access to a random variable $\mathcal{M}_{\rm CV}$ that is
correlated with the metric value $\mathcal{M}$, but _independent_ of the
treatment. Then, we can obtain a variance-reduced metric simply by subtracting
it:

$\mathcal{M}_{\rm VR}=\mathcal{M}-\mathcal{M}_{\rm CV}.$ (4)

The CUPED approach proposes to assign pre-experiment values of the same metric
$\mathcal{M}_{\rm pre}$ as the control variate in a linear regression model
[3]:

$\mathcal{M}_{\rm CV}=\theta(\mathcal{M}_{\rm pre}-\overline{\mathcal{M}_{\rm
pre}}),\rm{~{}with~{}}\theta=\frac{{\rm cov}(\mathcal{M},\mathcal{M}_{\rm
pre})}{\sigma(\mathcal{M}_{\rm
pre})^{2}}\rm{~{}estimated~{}on~{}pooled~{}data}.$ (5)

As a natural extension, multiple covariates can be used in a regression model
to obtain a final estimate for the outcome metric. The variance reduction this
approach yields is directly proportional to the correlation between the
original metric $\mathcal{M}$ and the control variate $\mathcal{M}_{\rm CV}$.
Minasayan et al. provide a characterisation of the approximation bias on the
Average Treatment Effect (ATE) that is incurred by this approach [10], which
is multiplicative of the form $1-\frac{k}{N}$ with $k$ the number of
regression covariates. This implies that for fixed $N$, adding more covariates
always leads to a _decrease_ in the ATE estimate. This, in turn, leads to a
_decrease_ in the _numerator_ of Eq. 1, which might (partially) offset
variance reduction: lower variance does not unequivocally imply higher
sensitivity.

Sidestepping the problems of approximation with linear models, Poyarkov et al.
propose to use Gradient-Boosted Decision Trees (GBDTs) to estimate
$\mathcal{M}_{\rm CV}$ directly, reporting significant improvements over
linear models and CUPED [11].

###  2.3 Variance Reduction for Ratio Metrics

We train a multiple linear regression model to estimate the ATE on a ratio
metric $\mathcal{M}$ from past online experiments, based on multiple
covariates. Following the classical CUPED method [3] extended for ratio
metrics, the covariates include: the numerator from the pre-experiment period
$\mathcal{M}_{N,{\rm pre}}$; the denominator from the pre-experiment period
$\mathcal{M}_{D,{\rm pre}}$; and the linearised metric for the pre-experiment
period $L(\mathcal{M}_{\rm pre})$ (following [1]). Alternatively, we consider
GBDT predictions for the numerator, denominator, and linearisation
$\widehat{\mathcal{M}_{N}},\widehat{\mathcal{M}_{D}},\widehat{L(\mathcal{M})}$
(following [11]). The output of the regression model is then the control
variate that we plug into Eq. 4 to obtain a variance-reduced metric, on which
we perform a $z$-test following Eqs. 1 and 2.

##  3 Experimental Validation of Improved Sensitivity

Table 1: Empirical insights into variance-reduced metrics on a dataset of past A/B experiments. We consider using pre-experiment metrics $\mathcal{M}_{\rm pre}$, GBDT-based predictions $\widehat{\mathcal{M}}$, or both. Lower variance does not unequivocally imply lower $p$-values, and GBDT-based predictions lead to significantly improved sensitivity with good type-I errors. Covariates | Var. Red. | $\mathbb{P}(p\rm{-value}\downarrow)$ | med. rel. $z$ | Type-I Error
---|---|---|---|---
$\mathcal{M}_{\rm pre}$ | -72.47 % | 15.38 % | 0.78 | 4.3%
$\widehat{\mathcal{M}}$ | -45.66 % |  76.92 % | 1.19 | 4.8%
$\\{\mathcal{M}_{\rm pre}\\}\cup\\{\widehat{\mathcal{M}}\\}$ | -72.62 % | 30.77 % | 0.81 | 5.2%

We consider past online controlled experiments that ran on the ShareChat
application, with known (conclusive) outcomes. That is, for deployed system
variants $A$ and $B$, we know a preference ordering to obtain $A\succ B$. All
experiments ran for either 7 or 14 days, and all variants were assigned
between 1.6 and 3.5 million users. In this preliminary study, we consider 13
known A/B pairs. The ratio metric we consider is _one-day retention_ : the
number of users who were retained from day $D_{0}$ to $D_{1}$, over the number
of active users on day $D_{0}$, aggregated over the experiment period. For
this metric and the variance reduction approaches introduced earlier, we
measure several attributes:  (i) the reduction in variance compared to the
original ratio metric $\mathcal{M}$,  (ii) the fraction of online experiments
that have a lower $p$-value under the variance-reduced metric,  (iii) the
median relative $z$-score, which gives an indication of the sample size
reduction that would be gained at constant sensitivity, and  (iv) the type-I
error measured over 220 two-week A/A-pairs on 3.5 million users per variant.
Table 1 presents these results. Even though pre-experiment metric values
contribute significantly to variance reduction, they also significantly impact
the ATE estimate, which leads to disappointing performance when considering
the percentage of experiments for which we can reject the null hypothesis with
higher confidence. Indeed, when considering solely GBDT-based predictions
$\widehat{\mathcal{M}}$, we obtain lower $p$-values for 77% of experiments.
For the classical CUPED approach using pre-experiment values, this number
deteriorates to 15%. Combining all covariates does not provide solace,
stagnating at 31%. This is a direct result of the bias we discuss in Section
2.2. Furthermore, a median relative $z$-score of $1.19$ implies that we need
$1.19^{2}\approx 1.42$ times fewer data points to achieve the same level of
confidence as the original ratio metric ($1-\frac{1}{1.42}\approx 30\%$; see
e.g. [2, 6]). This directly impacts the speed at which experiments conclude,
and the number of experiments that can run concurrently, greatly improving
experimental velocity. For $\alpha=0.05$, we expect the type-I errors to
converge at 5% as well. We observe that all are very close, ensuring that our
encouraging results are not the result of overfitting.

##  4 Conclusions & Outlook

In this work, we have explored variance reduction techniques specifically
tailored for ratio metrics in the context of A/B-testing. One notable takeaway
from our study is a re-evaluation of the conventional CUPED method [3]. This
approach, which often incorporates multiple covariates (pre-experiment
metrics) can inadvertently lead to overfitting. Instead, we propose using GBDT
predictors which are _unbiased_ estimators and account for complex
relationships among pre-experiment metrics [11]. Leveraging predicted metrics
as covariates in the CUPED framework yields lower variance reduction but
better performance in terms of sensitivity without inflating type-I errors,
providing empirical support for the inclusion of unbiased estimators to allow
for more efficient A/B-testing.

## About the presenter

Aleksei Ustimenko is a staff applied scientist at ShareChat, focusing on A/B
experimentation, statistics and recommender systems. He additionally enjoys
work as a theoretical mathematician, focusing on stochastic calculus.

## About the company

ShareChat (Mohalla Tech Pvt Ltd) is India’s largest homegrown social media
company, with 325+ million Monthly Active Users (MAUs) across all its
platforms, and social media brands such as ShareChat App and Moj under its
portfolio. Today, ShareChat App is India’s leading social media platform with
over 180 million MAUs spread across the country, and Moj is India’s #1 short-
video app with the highest MAU base of nearly 160 million.

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