020008 2010 A. Vindigni A. A. Fedorenko, CNRS-Lab. de Physique, ENS de Lyon,
France. 020008

We numerically study the geometry of a driven elastic string at its sample-
dependent depinning threshold in random-periodic media. We find that the
anisotropic finite-size scaling of the average square width $\overline{w^{2}}$
and of its associated probability distribution are both controlled by the
ratio $k=M/L^{\zeta_{\mathrm{dep}}}$, where $\zeta_{\mathrm{dep}}$ is the
random-manifold depinning roughness exponent, $L$ is the longitudinal size of
the string and $M$ the transverse periodicity of the random medium. The
rescaled average square width $\overline{w^{2}}/L^{2\zeta_{\mathrm{dep}}}$
displays a non-trivial single minimum for a finite value of $k$. We show that
the initial decrease for small $k$ reflects the crossover at $k\sim 1$ from
the random-periodic to the random-manifold roughness. The increase for very
large $k$ implies that the increasingly rare critical configurations,
accompanying the crossover to Gumbel critical-force statistics, display
anomalous roughness properties: a transverse-periodicity scaling in spite that
$\overline{w^{2}}\ll M$, and subleading corrections to the standard random-
manifold longitudinal-size scaling. Our results are relevant to understanding
the dimensional crossover from interface to particle depinning.

# Anisotropic finite-size scaling of an elastic string at the depinning
threshold in a random-periodic medium

S. Bustingorry  [inst1] A. B. Kolton[inst1] E-mail<EMAIL_ADDRESS>mail<EMAIL_ADDRESS>
(20 October 2010; 1 December 2010)

††volume: 2

99 inst1 CONICET, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Río
Negro, Argentina.

##  1 Introduction

The study of the static and dynamic properties of $d$-dimensional elastic
interfaces in $d+1$-dimensional random media is of interest in a wide range of
physical systems. Some concrete experimental examples are magnetic [1, 2, 3,
4] or ferroelectric [5, 6] domain walls, contact lines of liquids [7], fluid
invasion in porous media [8, 9], and fractures [10, 11]. In all these systems,
the basic physics is controlled by the competition between quenched disorder
(induced by the presence of impurities in the host materials) which promotes
the wandering of the elastic object, against the elastic forces which tend to
make the elastic object flat. One of the most dramatic and worth understanding
manifestations of this competition is the response of these systems to an
external drive.

The mean square width or roughness of the interface is one of the most basic
quantities in the study of pinned interfaces. In the absence of an external
drive, the ground state of the system is disordered but well characterized by
a self-affine rough geometry with a diverging typical width $w\sim
L^{\zeta_{\mathrm{eq}}}$, where $L$ is the linear size of the elastic object
and $\zeta_{\mathrm{eq}}$ is the equilibrium roughness exponent. When the
external force is increased from zero, the ground state becomes unstable and
the interface is locked in metastable states. To overcome the barriers
separating them and reach a finite steady-state velocity $v$ it is necessary
to exceed a finite critical force, above which barriers disappear and no
metastable states exist. For directed $d$-dimensional elastic interfaces with
convex elastic energies in a $D=d+1$ dimensional space with disorder, the
critical point is unique, characterized by the critical force $F=F_{c}$ and
its associated critical configuration [12]. This critical configuration is
also rough and self-affine such that $w\sim L^{\zeta_{\mathrm{dep}}}$ with
$\zeta_{\mathrm{dep}}$ the depinning roughness exponent. When approaching the
threshold from above, the steady-state average velocity vanishes like
$v\sim(F-F_{c})^{\beta}$ and the correlation length characterizing the
cooperative avalanche-like motion diverges as $\xi\sim(F-F_{c})^{-\nu}$ for
$F>F_{c}$, where $\beta$ is the velocity exponent and $\nu$ is the depinning
correlation length exponent [13, 14, 15, 16]. At finite temperature and for
$F\ll F_{c}$, the system presents an ultra-slow steady-state creep motion with
universal features [17, 18] directly correlated with its multi-affine geometry
[19, 20]. At very small temperatures the absence of a divergent correlation
length below $F_{c}$ shows that depinning must be regarded as a non-standard
phase transition [20, 21] while exactly at $F=F_{c}$, the transition is
smeared-out with the velocity vanishing as $v\sim T^{\psi}$, with $\psi$, the
so-called thermal rounding exponent [22, 23, 24, 25, 26, 27].

During the last years, numerical simulations have played an important role to
understand the physics behind the depinning transition thanks to the
development of powerful exact algorithms. In particular, the development of an
exact algorithm able to target efficiently the critical configuration and
critical force for a given sample [28, 29] has allowed to study, precisely,
the self-affine rough geometry at depinning [7, 28, 29, 30, 31], the sample-
to-sample critical force distribution [32], the critical exponents of the
depinning transition [26, 27, 33], the renormalized disorder correlator [34],
and the avalanche-size distribution in quasistatic motion [35]. Moreover, the
same algorithm has allowed to study, precisely, the transient universal
dynamics at depinning [36, 37], and an extension of it has allowed to study
low-temperature creep dynamics [20, 21].

In practice, the algorithm for targeting the critical configuration [28, 29]
has been numerically applied to directed interfaces of linear size $L$
displacing in a disordered potential of transverse dimension $M$, applying
periodic boundary conditions in both directions in order to avoid border
effects. This is thus equivalent to an elastic string displacing in a
disordered cylinder. The aspect ratio between longitudinal $L$ and transverse
$M$ periodicities must be carefully chosen, in order to have the desired
thermodynamic limit corresponding to a given experimental realization. In Ref.
[32] it was indeed shown that the critical force distribution $P(F_{c})$
displays three regimes associated with $M$: (i) At very small $M$ compared
with the typical width $L^{\zeta_{\mathrm{dep}}}$ of the interface, the
interface wraps the computational box several times in the transverse
direction, as shown schematically in Fig. 1(b), and therefore the periodicity
of the random medium is relevant and $P(F_{c})$ is Gaussian; (ii) At very
large $M$ compared with $L^{\zeta_{\mathrm{dep}}}$, as shown schematically in
Fig. 1(c), periodicity effects are absent but then the critical force, being
the maximum among many independent sub-critical forces, obeys extreme value
statistics and $P(F_{c})$ becomes a Gumbel distribution; (iii) In the
intermediate regime, where $M\approx L^{\zeta_{\mathrm{dep}}}$ and periodicity
effects are still irrelevant, as shown schematically in Fig. 1(a), the
distribution function is in between the Gaussian and the Gumbel distribution.
It has been argued that only the last case, where $M\approx
L^{\zeta_{\mathrm{dep}}}$, corresponds to the random-manifold depinning
universality class (periodicity effects absent) with a finite critical force
in the thermodynamic limit $L,M\to\infty$. This criterion does not give,
however, the optimal value of the proportionality factor between $M$ and
$L^{\zeta_{\mathrm{dep}}}$, and must be modified at finite velocity since the
crossover to the random-periodic universality class at large length-scales
depends also on the velocity [38]. To avoid this problem, it has been
therefore proposed to define the critical scaling in the fixed center of mass
ensemble [39]. The crossover from the random-manifold to the random-periodic
universality class is, however, physically interesting, as it can occur in
periodic elastic systems such as elastic chains. Remarkably, although the
mapping from a periodic elastic system (with given lattice parameter) in a
random potential to a non-periodic elastic system (such as an interface) in a
random potential with periodic boundary conditions is not exact, it was
recently shown that the lattice parameter does play the role of $M$ for
elastic interfaces with regard to the geometrical or roughness properties
[38]. Since the periodicity can often be experimentally tuned in such periodic
systems it is thus worth studying in detail the geometry of critical
interfaces of size $L$ as a function of $M$ with periodic boundary conditions,
and thus complement the study of the critical force in such systems [32].

In this paper, we study in detail, using numerical simulations, the
geometrical properties of the one-dimensional interface or elastic string
critical configuration in a random-periodic pinning potential as a function of
the aspect ratio parameter $k$, conveniently defined as
$k=M/L^{\zeta_{\mathrm{dep}}}$. We show that $k$ is indeed the only parameter
controlling the finite-size scaling (i.e. the dependence of observables with
the dimensions $L$ and $M$) of the average square width and its sample-to-
sample probability distribution. The scaled average square width
$\overline{w^{2}}L^{-2{\zeta_{\mathrm{dep}}}}$ is described by a universal
function of $k$ displaying a non-trivial single minimum at a finite value of
$k$. We show that while for small $k$ this reflects the crossover at $k\sim 1$
from the random-periodic to the random-manifold depinning universality class,
for large $k$ it implies that in the regime where the depinning threshold is
controlled by extreme value (Gumbel) statistics, critical configurations also
become rougher, and display an anomalous roughness scaling.

Figure 1:  (a) Elastic string driven by a force $F$ in a random-periodic
medium with periodic boundary conditions. It is described by a displacement
field $u(z)$ and has a mean width $w$. The anisotropic finite-size scaling of
width fluctuations are controlled by the aspect-ratio parameter
$k=M/L^{\zeta_{\mathrm{dep}}}$, with ${\zeta_{\mathrm{dep}}}$ the random-
manifold roughness exponent at depinning. In the case $k\ll 1$ (b) periodicity
effects are important, while when $k\gg 1$ (c) they are not important but the
roughness scaling of the critical configuration is anomalous.

##  2 Method

The model we consider here is an elastic string in $(1+1)$ dimensions
described by a single valued function $u(z,t)$, which gives the transverse
displacement $u$ as a function of the longitudinal direction $z$ and the time
$t$ [see Fig. 1(a)]. The zero-temperature dynamics of the model is given by

$\gamma\,\partial_{t}u(z,t)=c\,\partial^{2}_{z}u(z,t)+F_{p}(u,z)+F,$ (1)

where $\gamma$ is the friction coefficient and $c$ the elastic constant. The
first term in the right hand side derives from an harmonic elastic energy. The
effects of a random-bond type disorder is given by the pinning force
$F_{p}(u,z)=-\partial_{u}U(u,z)$. The disorder potential $U(u,z)$ has zero
average and sample-to-sample fluctuations given by

$\overline{\left[U(u,z)-U(u^{\prime},z^{\prime})\right]^{2}}=\delta(z-z^{\prime})\,R^{2}(u-u^{\prime}),$
(2)

where the overline indicates average over disorder realizations and $R(u)$
stands for a correlator of finite range $r_{f}$ [18]. Finally, $F$ represents
the uniform external drive acting on the string. Physically, this model can
phenomenologically describe, for instance, a magnetic domain wall in a thin
film ferromagnetic material with weak and randomly located imperfections [1],
being $F$ proportional to an applied external magnetic field pushing the wall
in the energetically favorable direction.

In order to numerically solve Eq. (1), the system is discretized in the
$z$-direction in $L$ segments of size $\delta z=1$, i.e. $z\to j=0,...,L-1$,
while keeping $u_{j}(t)$ as a continuous variable. To model the continuous
random potential, a cubic spline is used, which passes through $M$ regularly
spaced uncorrelated Gaussian number points [30]. For the numerical simulations
performed here we have used, without loss of generality, $\gamma=1$, $c=1$ and
$r_{f}=1$ and a disorder intensity $R(0)=1$. In both spatial dimensions we
have used periodic boundary conditions, thus defining a $L\times M$ system.

The critical configuration $u_{c}(z)$ and force $F_{c}$ are defined from the
pinned (zero-velocity) configuration with the largest driving force $F$ in the
long time limit dynamics. They are thus the real solutions of

$c\,\partial^{2}_{z}u(z)+F_{p}(u,z)+F=0,$ (3)

such that for $F>F_{c}$ there are no further real solutions (pinned
configurations). Middleton theorems [12] assure that for Eqs. (3) the solution
exists and it is unique for both $u_{c}(z)$ and $F_{c}$, and that above
$F_{c}$ the string trajectory in an $L$ dimensional phase-space is trapped
into a periodic attractor (for a system with periodic boundary conditions as
the one we consider). In other words, the critical configuration is the
marginal fixed point solution or critical state of the dynamics, being $F_{c}$
the critical point control parameter of a Hopf bifurcation. Solving the
$L$-dimensional system of Eqs. (3) for large $L$ directly is a formidable
task, due to the non-linearity of the pinning force $F_{p}$. On the other
hand, solving the long-time dynamics at different driving forces $F$ to
localize $F_{c}$ and $u_{c}$ is very inefficient due to the critical slowing
down. Fortunately, Middleton theorems, and in particular the “non-passing
rule”, can be used again to devise a precise and very efficient algorithm
which allows to obtain the critical force $F_{c}$ and the critical
configuration $u^{c}_{j}$ for each independent disorder realization
iteratively without solving the actual dynamics nor directly inverting the
system of Eqs. (3) [30]. Once the critical force and the critical
configuration are determined with this algorithm, we can compute the different
observables. In particular, the square width or roughness of the string at the
critical point for a given disorder realization is defined as

$w^{2}=\frac{1}{L}\sum_{j=0}^{L-1}\left[u^{c}_{j}-\frac{1}{L}\sum_{k=0}^{L-1}u^{c}_{k}\right]^{2}.$
(4)

Computing $w^{2}$ for different disorder realizations allows us to compute its
disorder average $\overline{w^{2}}$ and the sample-to-sample probability
distribution $P(w^{2})$. In addition, the average structure factor associated
to the critical configuration is

$S_{q}=\frac{1}{L}\overline{\left|\sum_{j=0}^{L-1}u^{c}_{j}\,e^{-iqj}\right|^{2}},$
(5)

where $q=2\pi n/L$, with $n=1,...,L-1$. One can show, using a simple
dimensional analysis, that given a roughness exponent $\zeta$, such that
$\overline{w^{2}}\sim L^{2\zeta}$, the structure factor behaves as $S(q)\sim
q^{-(1+2\zeta)}$ for small $q$, thus yielding an estimate to $\zeta$ without
changing $L$. To compute averages over disorder and sample-to-sample
fluctuations, we consider between $10^{3}$ and $10^{4}$ independent disorder
realizations depending on the size of the system.

##  3 Results

###  3.1 Roughness at the critical point

Figure 2:  The scaling of $\overline{w^{2}}$ for the critical configuration at
different $M$ values as indicated. The curves for $M=64$ and $16384$ are
shifted upwards for clarity. The dashed and dotted lines are guides to the eye
showing the expected slopes corresponding to the different roughness
exponents.

Figure 2 shows the scaling of the square width of the critical configuration
$\overline{w^{2}}$ with the longitudinal size of the system $L$ for
$L=32,64,128,256,512$ and different values of $M$. When $M$ is small, $M=8$,
for all the $L$ values shown we observe $\overline{w^{2}}\sim
L^{2\zeta_{\mathrm{L}}}$ with $\zeta_{\mathrm{L}}=1.5$, corresponding to the
Larkin exponent in $(1+1)$ dimensions. This value is different from the value
$\zeta_{\mathrm{dep}}=1.25$ [33, 40] expected for the random-manifold
universality class, and is thus indicating that the periodicity effects are
important for this joint values of $M$ and $L$. This situation is
schematically represented in Fig. 1(b). This result is a numerical
confirmation of the two-loop functional renormalization group result of Ref.
[16] which shows that the $\zeta=0$ fixed point, leading to a universal
logarithmic growth of displacements at equilibrium is unstable. The
fluctuations are governed, instead, by a coarse-grained generated random-force
as in the Larkin model, yielding a roughness exponent
$\zeta_{\mathrm{L}}=(4-d)/2$ in $d$ dimensions [16], which agrees with our
result for $d=1$. We can thus say that for small enough $M$ (compared to $L$)
the system belongs to the same random-periodic depinning universality class as
charge density wave systems [14, 41], which strictly correspond to $M=1$.

When $M$ is large, on the other hand, $M=16384$ in Fig. 2, for all the $L$
values considered the exponent is consistent with $\zeta_{\mathrm{dep}}$, of
the random-manifold universality class. This situation is schematically
represented in Fig. 1(c), and we will show later that, for this elongated
samples, the effects of extreme value statistics are already visible.

For intermediate values of $M$, such as $M=64$ in Fig. 2, we can observe the
crossover in the scale-dependent roughness exponent
$\zeta(L)\sim\frac{1}{2}\frac{d\log w^{2}}{d\log L}$ changing from
$\zeta_{\mathrm{dep}}$ to $\zeta_{\mathrm{L}}$ as $L$ increases, as indicated
by the dashed and dotted lines. This crossover, from the random-manifold to
the random-periodic depinning geometry, occurs at a characteristic distance
$l^{*}\sim M^{1/\zeta_{\mathrm{dep}}}$, when the width in the random-manifold
regime reaches the transverse dimension or periodicity $M$. At finite
velocity, this crossover length remains constant up to a non-trivial
characteristic velocity and then decreases with increasing velocity [38].

Figure 3:  Structure factor of the critical configuration for $L=256$ and
different $M$ values, as indicated. The curves for $M=64$ and $16384$ are
shifted upwards for clarity. The dashed and dotted lines are guides to the eye
showing the expected slopes corresponding to the different roughness
exponents.

The above mentioned geometrical crossover can be studied in more details
through the analysis of the structure factor $S(q)$, for a line of fixed size
$L$. In Fig. 3 we show $S(q)$ for $L=256$ and $M=8,64,16384$. For the
intermediate value $M=64$ a crossover between the two regimes is visible, and
can be described by

$S_{q}\sim\left\\{\begin{array}[]{ll}q^{-(1+2\zeta_{\mathrm{L}})}&q\ll
q^{*},\\\ q^{-(1+2\zeta_{\mathrm{dep}})}&q\gg q^{*}.\\\ \end{array}\right.$
(6)

with $q^{*}$ expected to scale as $q^{*}\sim l^{*-1}\sim
M^{-1/\zeta_{\mathrm{dep}}}$. Therefore, the structure factor should scale as
$S_{q}M^{-(2+1/\zeta_{\mathrm{dep}})}=H(x)$, where the scaled variable is
$x=q\,M^{1/\zeta_{\mathrm{dep}}}\sim q/q^{*}$ and the scaling function behaves
as

$H(x)\sim\left\\{\begin{array}[]{ll}x^{-(1+2\zeta_{\mathrm{L}})}&x\ll 1,\\\
x^{-(1+2\zeta_{\mathrm{dep}})}&x\gg 1.\\\ \end{array}\right.$ (7)

The collapse of Fig. 4 for $L=256$ and different values of $M=2^{p}$ with
$p=3,4,...,14$ shows that this scaling form is a very good approximation.
However, as we show below, small corrections can be expected fully in the
random-manifold regime in the large $ML^{-\zeta_{\mathrm{dep}}}$ limit of very
elongated samples.

Figure 4:  Scaling of the structure factor of the critical configuration for
$L=256$ and different values of the transverse size $M=2^{p}$ with
$p=3,4,...,14$ $M$. Although the values of the two exponents are very close,
the change in the slope of the scaling function against the scaling variable
$x=q\,M^{1/\zeta_{\mathrm{dep}}}$ is clearly observed. Figure 5:  (a) Squared
width of the critical configuration as a function of $M$ for different system
sizes $L$ as indicated. (b) Scaling of the width in (a), showing that the
relevant control parameter is $M/L^{\zeta_{\mathrm{dep}}}$. The dashed line in
(a) and (b) corresponds to $\overline{w^{2}}=M^{2}$, which is always to the
left of the minimum of $\overline{w^{2}}$ occurring at
$k^{*}=m^{*}L^{-\zeta_{\mathrm{dep}}}$. The solid line indicates
$k^{2(1-\zeta_{\mathrm{L}}/\zeta_{\mathrm{dep}})}$ which is the behavior
expected purely from the random-periodic to random-manifold crossover at the
characteristic distance $l^{*}\sim M^{1/\zeta_{\mathrm{dep}}}$.

In Fig. 5(a), we show $\overline{w^{2}}$ as a function of the transverse
periodicity $M$ for different values of the longitudinal periodicity $L$.
Remarkably, $\overline{w^{2}}$ is a non-monotonic function of $M$. For small
$M$ it decreases towards an $L$ dependent minimum $m^{*}$, and then increases
with increasing $M$, in the regime where the extreme value statistics starts
to affect the distribution of the critical force [32]. Since the only typical
transverse scale in Fig. 5(a) is set by the minimum $m^{*}$, we can expect
$\overline{w^{2}}\sim m^{*2}G(M/m^{*})$ with $G(x)$ some universal function.
On the other hand, since the only relevant characteristic length-scale of the
problem is set by the crossover between the random-periodic regime and the
random-manifold regime, we can simply write $m^{*}\sim
L^{\zeta_{\mathrm{dep}}}$ and therefore

$\overline{w^{2}}\,L^{-2\zeta_{\mathrm{dep}}}\sim
G(M\,L^{-\zeta_{\mathrm{dep}}}).$ (8)

This scaling form is confirmed in Fig. 5(b) and shows that the aspect-ratio
parameter $k=ML^{-\zeta_{\mathrm{dep}}}$ fully controls the anisotropic
finite-size scaling of the problem. It is worth, however, noting some
interesting consequences of the result of Fig. 5(b), as we describe below.

Since at very small $k$ the interface is in the random-periodic regime, Eq.
(8) should led to $\overline{w^{2}}\sim L^{2\zeta_{\mathrm{L}}}$ and therefore
one deduces that,

$G(k)\sim k^{2(1-{\zeta_{\mathrm{L}}}/{\zeta_{\mathrm{dep}}})},\;\;\;k\ll
k^{*},$ (9)

where $k^{*}=m^{*}L^{-\zeta_{\mathrm{dep}}}$. The fact that the random-
periodic roughness exponent ${\zeta_{\mathrm{L}}}=3/2$ is larger than the
random-manifold one ${\zeta_{\mathrm{dep}}}\approx 5/4$ consequently implies
an initial decrease of $G(k)$ as $G(k)\sim k^{-2/5}$, as shown in Fig. 5(b) by
the solid line. Periodicity effects, or the crossover from random-periodic to
random-manifold, thus explain the initial decrease of $G(k)$ seen in Fig.
5(b), or the initial decrease of $\overline{w^{2}}$ against $M$ for fixed $L$,
seen in Fig. 5(a). At this respect, it is then worth noting that the line
$\overline{w^{2}}=M^{2}$, shown by a dashed line, lies completely in the
regime $k<k^{*}$ implying that the naive criterion $\overline{w^{2}}<M^{2}$ is
not enough to avoid periodicity effects, and to have the system fully in the
random-manifold regime. As we show later, this is related with the shape of
the probability distribution of $P(w^{2})$ which displays sample-to-sample
fluctuations of the order of the average $\overline{w^{2}}$.

The presence of a minimum at $k^{*}$ in the function $G(k)$ and in particular
its slower than power-law increase for $k>k^{*}$ is non-trivial and
constitutes one of the main results of the present work. This result shows
that corrections to the standard scaling $\overline{w^{2}}\sim
L^{\zeta_{\mathrm{dep}}}$ may arise from the aspect-ratio dependence of the
prefactor $G(k)$. On the one hand, $\overline{w^{2}}$ now grows with $M$ for
$L$ fixed, in spite that $\overline{w^{2}}\ll M^{2}$, i.e. transverse-
size/periodicity scaling is present. On the other hand, the scaling of
$\overline{w^{2}}$ with $L$ is slower in this regime, due to subleading
scaling corrections coming from $G(k)$. The precise origin of these
interesting leading and subleading corrections in the finite-size anisotropic
scaling are highly non-trivial. Since the critical configurations in this
regime have the constant roughness exponent $\zeta_{\mathrm{dep}}$ of the
random-manifold universality class, the slow increase of $G(k)$ cannot be
attributed to a geometrical crossover effect, as for the case $k<k^{*}$.
However, we might relate this effect to the crossover in the critical force
statistics, from Gaussian to Gumbel, in the $k\gg k^{*}$ limit [32]. In the
Gumbel regime, the average critical force is expected to increase as
$F_{c}\sim\log(M/L^{\zeta}_{\mathrm{dep}})\equiv\log k$ [39], since the sample
critical force can be roughly regarded as the maximum among
$M/L^{\zeta}_{\mathrm{dep}}$ independent sub-critical forces and
configurations [32]. The increase in the critical force might be therefore
correlated with the slow increase of roughness. The physical connection
between the two is subtle though, since a large critical force in a very
elongated sample could be achieved both by profiting very rare correlated
pinning forces such as accidental columnar defects, or by profiting very rare
non-correlated strong pinning forces. Since in the first case the critical
configuration would be more correlated and in general less rough than for less
elongated samples (smaller $k$), contrary to our numerical data of Fig. 5(b),
we think that the second cause is more plausible. We can thus think that in
the $k\gg k^{*}$ limit of extreme value statistics of $F_{c}$, the effective
disorder strength on the critical configuration increases with $k$. This might
be translated into the universal function $G(k)$, such that
$\overline{w^{2}}\approx L^{2\zeta_{\mathrm{dep}}}G(k)$ can increase for
increasing values of $k$ at fixed $L$ in such regime. A quantitative
description of these scaling corrections remains an open challenge.

###  3.2 Distribution function

Figure 6:  Scaling function $\Phi(x)$ for $L=256$ and different values of
$M=8,128,2048,16384$, which shows the change with the transverse size $M$.

We now analyze sample-to-sample fluctuations of the square width $w^{2}$ by
computing its probability distribution $P(w^{2})$. This property is relevant
as $w^{2}$ fluctuates even in the thermodynamic limit for critical interfaces
with a positive roughness exponent [42]. It has been computed for models with
dynamical disorder such as random-walk [43] or Edwards–Wilkinson interfaces
[44, 45], the Mullins Herrings model [46] and for non-Markovian Gaussian
signals in general [47, 48]. It has also been calculated for non-linear models
such as the one-dimensional Kardar–Parisi–Zhang model [49, 50] and for the
quenched Edwards–Wilkinson model at equilibrium [51].

In particular, the probability distribution $P(w^{2})$ of critical interfaces
at the depinning transition was studied analytically [52], numerically [31]
and also experimentally for contact lines in partial wetting [7]. Remarkably,
non-Gaussian effects in depinning models are found to be smaller than $0.1\%$
[31, 52], thus showing that $P(w^{2})$ is strongly determined by the self-
affine (critical) geometry itself, rather than by the particular mechanism
producing it. As in all the above mentioned systems the width distribution
$P(w^{2})$ at different universality classes of the depinning transition was
found to scale as

$\overline{w^{2}}P(w^{2})\approx\Phi_{\zeta}\left(\frac{w^{2}}{\overline{w^{2}}}\right).$
(10)

with $\Phi_{\zeta}$ an universal function, which only depends on the roughness
exponent $\zeta$ and on boundary conditions when the global width is
considered [47, 48]. In this way, $\overline{w^{2}}$ is the only
characteristic length-scale of the system, absorbing the system longitudinal
size $L$, and all the non-universal parameters of the model such as the
elastic constant of the interface, the strength of the disorder and/or the
temperature. Since $\Phi_{\zeta}$ can be easily generated using non-Markovian
Gaussian signals [53], the quantity $\overline{w^{2}}P(w^{2})$ is a good
observable to extract the roughness exponent of a critical interface from
experimental data.

Figure 7:  Scaling function $\Phi(x)$ for different values of
$L=32,64,128,256$ while keeping (a) $k=M/L^{\zeta_{\mathrm{dep}}}\approx 1$
and (b) $k=M/L^{\zeta_{\mathrm{dep}}}\approx 0.025$. The dotted line
corresponds to the scaling function of the non-disordered Edwards–Wilkinson
equation [43], while the continuous and dashed lines correspond to the scaling
functions of Gaussian signals with $\zeta=1.25$ and $\zeta=1.5$, respectively
[31, 53].

In Fig. 6, we show how the scaled distribution function
$\Phi(x)\equiv\overline{w^{2}}\,P(x\;\overline{w^{2}})$ looks like for the
depinning transition in a random-periodic medium for a fixed value $L=256$ and
different values of $M$. We see that $\Phi(x)$ depends on $M$ for small $M$
but converges to a fixed shape for large $M$. We also note that for all $M$
$\Phi(x)$ extends appreciably beyond $x=1$ explaining why the criterion
$\overline{w^{2}}\lesssim M^{2}$ is not enough to be fully in the random-
manifold regime, as noted in Fig. 5.

In Fig. 7, we show the scaling function $\Phi(x)$ for different values of $L$
and $M$ but fixing the aspect-ratio parameter $k=M/L^{\zeta_{\mathrm{dep}}}$,
$k\approx 1>k^{*}$ in Fig. 7(a) and $k\approx 0.025\ll k^{*}$ in Fig. 7(b),
with $k^{*}$ the minimum of $\overline{w^{2}}$. Since data for the same $k$
practically collapses into the same curve, we can write for our case:

$\overline{w^{2}}P(w^{2})=\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right).$
(11)

Therefore, the anisotropic scaling of the probability distribution is fully
controlled by $k$, as it was found for $\overline{w^{2}}$.

In Figs. 7(a) and (b), we also show the universal functions $\Phi_{\zeta_{L}}$
and $\Phi_{\zeta_{\mathrm{dep}}}$ generated using non-Markovian Gaussian
signals [31, 53], and for comparison we also show $\Phi_{1/2}$ corresponding
to the Markovian periodic Gaussian signal or the Edwards–Wilkinson equation
[43]. Comparing this with the collapsed data for depinning, we see that the
function $\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right)$ respects the
limits

$\displaystyle\Phi\left(x,k\to 0\right)$ $\displaystyle=$
$\displaystyle\Phi_{\zeta_{\mathrm{L}}}(x),$
$\displaystyle\Phi\left(x,k\gtrsim k^{*}\right)$ $\displaystyle\approx$
$\displaystyle\Phi_{\zeta_{\mathrm{dep}}}(x),$ (12)

as expected from the existence of the geometric crossover between the
roughness exponents ${\zeta_{\mathrm{L}}}$ for $k\to 0$ and
${\zeta_{\mathrm{dep}}}$ for $k>k^{*}$. For intermediate values $k<k^{*}$,
however, $\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right)$ does not
necessarily coincide with the one of a Gaussian signal function $\Phi_{\zeta}$
for a given $\zeta$, since the critical configuration includes a crossover
length $l^{*}\lesssim L$. Whether multi-affine or effective exponent self-
affine non-Markovian Gaussian signals can be used to describe satisfactorily
these intermediate cases is an interesting open issue.

##  4 Conclusions

We have numerically studied the anisotropic finite-size scaling of the
roughness of a driven elastic string at its sample-dependent depinning
threshold in a random medium with periodic boundary conditions in both the
longitudinal and transverse directions. The average square width
$\overline{w^{2}}$ and its probability distribution are both controlled by the
parameter $k=M/L^{\zeta_{\mathrm{dep}}}$. A non-trivial single minimum for a
finite value of $k$ was found in $\overline{w^{2}}/L^{2\zeta_{\mathrm{dep}}}$.
For small $k$, the initial decrease of $\overline{w^{2}}$ reflects the
crossover from the random-periodic to the random-manifold roughness. For very
large $k$, the growth with $k$ implies that the crossover to Gumbel statistics
in the critical forces induces corrections to $G(k)$, that grow with $k$, to
the string roughness scaling $\overline{w^{2}}\approx
G(k)L^{2\zeta_{\mathrm{dep}}}$. These increasingly rare critical
configurations thus have an anomalous roughness scaling: they have a
transverse-size/periodicity scaling in spite that its width is
$\overline{w^{2}}\ll M^{2}$, and subleading (negative) corrections to the
standard random-manifold longitudinal-size scaling.

Our results could be useful for understanding roughness fluctuations and
scaling in finite experimental systems. The crossover from random-periodic to
random-manifold roughness could be studied in periodic elastic systems with
variable periodicity, such as confined vortex rows [54] and single-files of
macroscopically charged particles [55] or colloids [56], with additional
quenched disorder. The rare-event dominated scaling corrections to the
interface roughness scaling could be studied in systems with a large
transverse dimension, such as domain walls in ferromagnetic nanowires [57].
For the later case, it would be interesting to have a quantitative theory,
making the connection between the extreme value statistics of the depinning
threshold and the anomalous scaling corrections to the roughness of such rare
critical configurations. This would allow to understand the dimensional
crossover, from interface to particle depinning.

###### Acknowledgements.

This work was supported by CNEA, CONICET under Grant No. PIP11220090100051͒,
and ANPCYT under Grant No. PICT2007886. A. B. K. acknowledges Universidad de
Barcelona, Ministerio de Ciencia e Innovación (Spain) and Generalitat de
Catalunya for partial funding through I3 program.

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# The Mu2e Experiment — Searching for Charged Lepton Flavor Violation

M. T. Hedges  on behalf of the Mu2e Collaboration  Department of Physics and
Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN
47907, USA

###### Abstract

The Mu2e experiment will search for a Standard Model violating rate of
neutrinoless conversion of a muon into an electron in the presence of an
aluminum nucleus. Observation of this charged lepton flavor violating process
would be an unambiguous sign of new physics. Mu2e will improve upon previous
searches for this process by four orders of magnitude. This requires the
world’s highest-intensity muon beam, a detector system capable of efficiently
reconstructing the 105 MeV/c conversion electron signal, and minimizing
sensitivity to background events. A pulsed 8 GeV proton beam strikes a target,
producing pions that decay into muons. Beam outside the pulse must be
suppressed to $<10^{-10}$ to reduce beam-related backgrounds. The muon beam is
guided from the production target along the transport system and onto the
aluminum stopping target. Conversion electrons leave the stopping target and
propagate inside a solenoidal magnetic field to the tracker and
electromagnetic calorimeter. The tracker is a system of straw tube panels
filled with Ar/CO2 at 1 atm that tracks particles inside of a solenoidal
B-field and measures their momenta with $\sim 100$ keV/$c$ resolution to
resolve signal events from decay-in-orbit backgrounds. The CsI calorimeter
provides $E/p$ and is used to seed the track reconstruction algorithm with
$\sigma_{E}/E\sim{}10\%$ and $\sigma_{t}<500~{}\rm{ps}$. Additionally, a novel
cosmic ray veto with greater than 99.99% efficiency brings the expected number
of background events to fewer than one over three years of running. To
normalize the experiment, the stopping target monitor measures the rate of
capture photons from muons incident on the stopping target by using a system
of high-purity germanium and lanthanum bromide scintillators.

##  1 Introduction

The Standard Model of particle physics classifies fundamental fermions into
six quark flavors and six lepton flavors. Of these fermions, only the charged
leptons have never experimentally shown evidence of flavor violation. However,
the discovery of massive neutrinos provides a mechanism for charged lepton
flavor violation (CLFV) at loop level, but is highly suppressed to
unobservably small levels. Thus, any observation of CLFV would unambiguously
indicate physics beyond the Standard Model.

One particularly promising experimental search for CLFV focuses on detection
of direct conversion of a muon to an electron in the Coulomb field of a
nucleus, or $\mu^{-}N-e^{-}N$ conversion. This process produces a
monoenergetic _conversion-electron_ (CE) with energy given by:

$E_{CE}=m_{\mu}{}c^{2}-E_{b}-E_{\text{recoil}}$

where $E_{b}$ is the binding energy of the muon in the $1S$ orbit and
$E_{\text{recoil}}$ is the energy of the recoiling nucleus.

##  2 The Mu2e experiment

The Mu2e experiment [1] will search for $\mu^{-}N-e^{-}N$ by measuring $R_{\mu
e}$, defined as the ratio of the rate of $\mu^{-}N-e^{-}N$ conversion to the
rate of muon capture using an aluminum target nucleus. Specifically, Mu2e will
measure $R_{\mu e}$ on an Al target given by:

$R_{\mu e}=\frac{\mu^{-}+A(Z,N)\to
e^{-}+A(Z,N)}{\mu^{-}+A(Z,N)\to\nu_{\mu}+A(Z-1,N)}$

with a 5$\sigma$ discovery potential of $R_{\mu e}>2\times 10^{-16}$ or a
corresponding upper limit of $R_{\mu e}<8\times 10^{-17}$ (90% CL). This
sensitivity is four orders of magnitude beyond the current bounds set by
SINDRUM II which measured $R_{\mu e}<7\times 10^{-13}$ on Au [2].

For $\mu^{-}N-e^{-}N$ in aluminum, the CE signal is a monoenergetic electron
of $E_{CE}=104.9~{}\rm{MeV}/c$. Mu2e must detect this signal in the presence
of both _intrinsic_ backgrounds such as muon Decay-In-Orbit (DIO) and cosmic-
ray events, and _beam-induced_ backgrounds such as antiproton annihilation and
radiative pion capture (RPC) in the muon stopping target. These two types of
backgrounds drive the design of the Mu2e experiment, which is shown in Fig. 1.

Figure 1: The Mu2e experiment.

DIO events are the primary intrinsic background of concern. Muons incident on
the aluminum stopping target can decay while in atomic orbit to an electron
and two neutrinos, just as a free muon. While the DIO electron momentum
spectrum is similar to that of free muon decay, the presence of the aluminum
nucleus can cause a recoil and results in the kinematic endpoint of DIO
electrons to be equivalent to the momentum of the CE signal. This is
demonstrated graphically in Fig. 2. A full calculation of the DIO spectrum in
aluminum can be found in Ref. [3].

Figure 2: Qualitative representation of electron momentum spectrum due to free
muon decay (blue), muon decay-in-orbit (red), and conversion electrons
(magenta).

Electrons emitted from the stopping target traverse a solenoidal magnetic
field into the Mu2e tracker. Resolving the CE signal from the DIO background
requires the tracker to have excellent momentum resolution and track
reconstruction efficiency. This is obtained by utilizing a low-mass straw-tube
tracking system consisting of 21,600 mylar straws of 15
$\mathrm{\SIUnitSymbolMicro m}$ thickness arranged into 36 panels. The straws
are filled with 1 atm of 80:20 Ar:CO2 with an applied voltage of 1450 V. The
tracker is expected to provide a momentum resolution of
$\sim{}100~{}\rm{keV}/c$ at the CE momentum of $\sim{}105~{}\rm{MeV}/c$. To
maximize track reconstruction efficiency and to minimize effects from the
large currents produced from low-momentum DIO events, the tracker utilizes an
annular design wherein an inner radius of 38 cm remains uninstrumented. This
design, shown in Fig. 3, drastically reduces acceptance to lower momentum
electrons while maintaining acceptance to conversion electrons.

Figure 3: (top) The Mu2e straw tube tracker and (bottom) the acceptance of the
annular design to the radii of 105 MeV/$c$ conversion electrons (green) and
lower momentum backgrounds (black) emitted from the stopping target.

The calorimeter, shown in Fig. 4, consists of two disks of 674 pure CsI
crystals arranged in the same annular design as the tracker. Each CsI crystal
measures 34×34×200 mm3 and is read out by 2 SiPMs. The calorimeter provides
$E/p$ for particle identification with expected $\sigma_{E}/E$ of order 10%
for conversion electrons, as well as timing resolution of $\sigma$t < 500 ps,
which seeds the Mu2e track reconstruction algorithm. Prototype modules
consisting of an array of $3\times 3$ crystals have been tested utilizing a
variable energy electron beam (22 – 120 MeV) have demonstrated an energy
resolution of $\sim$7% and a time resolution of better than $\sim$230 ps [4].

Figure 4: The Mu2e CsI calorimeter.

The remaining significant intrinsic background arises from cosmic rays, which
can enter the experimental apparatus and produce CE-like events. Simulations
have indicated a base rate of order 1 CE-like event per day of running. To
reject these events, Mu2e uses a Cosmic Ray Veto (CRV) which surrounds the
Detector Solenoid that houses the muon stopping target, the tracker, and the
calorimeter. The CRV consists of over 5000 extruded polystyrene scintillators
arranged in four overlapping layers. Scintillation light is collected by
wavelength shifting fibers and detected by SiPMs. This design, shown in Fig.
5, is expected to reject cosmic ray events with over 99.99% efficiency using
triple coincidence of the scintillator layers.

Figure 5: The Mu2e Cosmic Ray Veto.

Beam-induced backgrounds can also limit sensitivity to CE signal. The muon
beam for Mu2e originates from a pulsed proton beam incident on the tungsten
pion Production Target. A single pulse consists of $\sim$4 × 107 protons at 8
GeV, with a pulse duration of $\sim$250 ns and period of 1695 ns—longer than
the muonic-Al lifetime of 864 ns. Pions emerge from the production target and
decay to muons, which are then transferred down the Transport Solenoid and
onto the Al muon Stopping Target inside of the Detector Solenoid. The pulsed
beam allows the stopped muons inside the stopping target to decay during the
time between sequential pulses. This allows blinding the experiment until
after prompt beam backgrounds decay to appropriate levels. This is shown in
Fig. 6, which shows the intensity versus time for various sources of
detectable electrons. As can be seen, using a delayed signal window
drastically reduces sensitivity to beam-induced backgrounds.

Achieving this level of reduction in beam-induced backgrounds requires a
minimal level of beam particles to arrive at the muon stopping target within
the signal window. We require this inter-pulse _extinction_ —defined as the
ratio of out-of-time beam intensity to in-time beam intensity—to be less than
10-10. The Extinction Monitor pixel telescope, shown in Fig. 7, consisting of
eight sequential planes with 20 FE-I4B pixel chips, measures the extinction by
tracking protons scattered downstream off of the Production Target both in-
and out-of-time with the proton pulses.

Figure 6: Timing distributions for the CE signal and various backgrounds, with
the signal window shown at times after the reduction of prompt backgrounds and
before the arrival of backgrounds from the next pulse. Figure 7: The Mu2e
Extinction Monitor, shown relative to the production target.

Finally, the Stopping Target Monitor provides the absolute normalization by
measuring the gamma spectrum from atomic capture of muons on the Al stopping
target using both LaBr3 and high-purity Ge detectors. A mockup of the LaBr3
and HPGe sensors is shown in Fig. 8.

Figure 8: The LaBr3 (left) and HPGe (right) detectors used in the Mu2e
Stopping Target Monitor.

##  3 Mu2e Run 1

The Mu2e experiment is poised to begin Run 1 of physics data with muon beam
during the middle of this decade. Through a dedicated simulation campaign
utilizing expected detector and accelerator performance during the expected
duration of Run 1, we expect a discovery potential of $R_{\mu e}>1\times
10^{-15}$, with a standalone, dedicated publication anticipated this year. The
simulated momentum spectrum for signal and background components is shown in
Fig. 9, and the yields for expected background contributions are shown in
Table 1. In the absence of signal in Run 1, the corresponding limit will be
improved to $R_{\mu e}<6\times 10^{-16}$ (90% CL), corresponding to 1000 times
better than the current world limit. Run 2, with an anticipated start by the
end of the decade, will provide an additional factor of 10 in improvement
beyond Run 1, resulting in an overall $10^{4}$ improvement in sensitivity.

Figure 9: Expected momentum spectra from conversion electrons in the presence of backgrounds during Mu2e Run I. Table 1: Expected yields of backgrounds in Mu2e Run 1 with statistical and systematic uncertainties. | Expected yield
---|---
Cosmics | $0.048\pm 0.01\pm 0.010$
DIO | $0.038\pm 0.002~{}^{+0.026}_{-0.016}$
RPC | $0.011\pm 0.002~{}^{+0.001}_{-0.002}$
Antiprotons | $0.010\pm 0.003~{}^{+0.010}_{-0.004}$
Total |  $0.107\pm 0.032$ (stat $\oplus$ syst)

##  4 Summary

The Mu2e experiment will significantly improve the existing bounds in
searching for a CLFV signal in $\mu^{-}N-e^{-}N$ conversion. Mu2e is currently
poised to begin Run 1 by the middle of the decade, with an expected 1000 times
improvement on the existing world limit. Run 2, providing another factor of 10
improvement, is expected to begin by the end of the decade.

##  5 Acknowledgements

We are grateful for the vital contributions of the Fermilab staff and the
technical staff of the participating institutions. This work was supported by
the US Department of Energy (grant number 140000369); the Istituto Nazionale
di Fisica Nucleare, Italy; the Science and Technology Facilities Council, UK;
the Ministry of Education and Science, Russian Federation; the National
Science Foundation, USA; the Thousand Talents Plan, China; the Helmholtz
Association, Germany; and the EU Horizon 2020 Research and Innovation Program
under the Marie Sklodowska-Curie Grant Agreement Nos. 101006726, 734303,
822185, and 858199. This document was prepared by members of the Mu2e
Collaboration using the resources of the Fermi National Accelerator Laboratory
(Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility.
Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under
Contract No. DE-AC02-07CH11359.

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