# Optimal Control of the F1-ATPase Molecular Motor

Deepak Gupta  Department of Physics, Simon Fraser University, Burnaby, British
Columbia V5A 1S6, Canada  Steven J. Large  Department of Physics, Simon Fraser
University, Burnaby, British Columbia V5A 1S6, Canada  Shoichi Toyabe
Department of Applied Physics, Tohoku University, Aoba 6-6-05, Sendai,
980-8579, Japan  David A. Sivak  Department of Physics, Simon Fraser
University, Burnaby, British Columbia V5A 1S6, Canada<EMAIL_ADDRESS>
###### Abstract

F1-ATPase is a rotary molecular motor that in vivo is subject to strong
nonequilibrium driving forces. There is great interest in understanding the
operational principles governing its high efficiency of free-energy
transduction. Here we use a near-equilibrium framework to design a non-trivial
control protocol to minimize dissipation in rotating $\mathrm{F}_{1}$ to
synthesize ATP. We find that the designed protocol requires much less work
than a naive (constant-velocity) protocol across a wide range of protocol
durations. Our analysis points to a possible mechanism for energetically
efficient driving of $\mathrm{F}_{1}$  _in vivo_ and provides insight into
free-energy transduction for a broader class of biomolecular and synthetic
machines.

###### keywords:

molecular motors, stochastic fluctuations, nonequilibrium thermodynamics,
free-energy transduction, ATP synthase

revised_SI Institute for Theoretical Physics, Technical University of Berlin,
Hardenbergstr. 36, D-10623 Berlin, Germany Current address: Viewpoint
Investment Partners, Calgary, Alberta, Canada

Nanometer-sized biomolecular machines convert between different forms of free
energy, while remaining in contact with a thermal environment. Consequently,
fluctuations play a crucial role in a machine’s dynamics 1, 2; nevertheless, a
machine achieves (on average) directed motion that is consistent with the
second law of thermodynamics by transducing free energy, often stored in
nonequilibrium chemical concentrations in the surrounding environment 3. It is
of paramount interest to unravel the design principles governing effective
free-energy transduction in biomolecular machines 4, 5, 6.

${\rm F}_{\rm o}{\rm F}_{1}$-ATP synthase has attracted particular attention
7, 8, 9, 10. This molecular motor is responsible for $\sim$95% of the cellular
synthesis of adenosine triphosphate (ATP) 11, 12. ATP production by the
$\mathrm{F}_{1}$ subunit is driven by rotation of its $\gamma$-shaft. _In
vivo_ , the $\gamma$-shaft rotates by utilizing free energy from proton flux
through the membrane-embedded $\mathrm{F}_{\mathrm{o}}$ subunit. The
$\gamma$-shaft can rotate as fast as $\sim$350 revolutions per second 9, while
maintaining high efficiency 13. It is thus of significant interest to measure
and quantitatively understand energy conversion during rapid mechanical
driving of $\mathrm{F}_{1}$ to synthesize ATP 14, 15, 16.

FoF1 can operate in either direction 17: An excess of ATP drives counter-
rotation of the $\gamma$-shaft and transports protons against their
concentration difference. This reversibility is also observed in isolated F1,
which under sufficiently high torque synthesizes ATP 18, 19, but in the
absence of torque hydrolyzes ATP and counter-rotates the $\gamma$-shaft 20,
17. Experiments suggest that this ‘hydrolysis mode’ proceeds via 120∘
rotations of the $\gamma$-shaft 21, composed of two substeps: an $80^{\circ}$
step involving ATP binding, and a $40^{\circ}$ step involving ATP hydrolysis
(catalysis) 22. Overall the $\gamma$-shaft rotates 360∘ by hydrolyzing 3 ATPs.

Rapidly driving the $\gamma$-shaft by applying external torque inevitably
produces dissipation, the difference between the work performed and the free
energy transduced during synthesis or hydrolysis. It remains enigmatic how ATP
synthase achieves highly efficient energetic conversion despite such rapid
operation. In particular, what manner of rapid forced rotation with the
$\gamma$-shaft achieves efficient energy transmission?

Here we theoretically address efficient driving procedures (_protocols_) to
rapidly force rotation of the $\gamma$-shaft. A near-equilibrium framework 23
has proven experimentally useful in designing a protocol for switching between
folded and unfolded conformations of single DNA hairpins 24, and similarly
successful in simulations of barrier crossing 25, 26, rotary motors 27, Ising
models 28, 29, 30, and other model systems 31, 32, 33, 34. In this Letter we
design a protocol that (near equilibrium) minimizes dissipation in
experimentally accessible rotation of the $\gamma$-shaft driving
$\mathrm{F}_{1}$ to synthesize ATP. We find that the designed protocol
outperforms the naive (constant-velocity) protocol for a considerable range of
protocol durations, often far from equilibrium. Such protocols hint at how
$\mathrm{F}_{\mathrm{o}}$ might rotate the $\gamma$-shaft in an efficient
manner.

The totally asymmetric allosteric model (TASAM) 35 of $\mathrm{F}_{1}$
describes the evolution of a rotational degree of freedom $\theta\in[0,2\pi]$
obeying periodic boundary conditions, corresponding to a bead attached to the
$\gamma$-shaft (Fig. 1a provides a schematic of the modeled experiment). The
TASAM is constructed to recover the steady-state and kinetic behavior of F1
hydrolyzing ATP during single-molecule experiments 36, 37. The
$\ell=40^{\circ}$ step is modeled by the system switching between two harmonic
potentials of the same spring constant $k\approx 20~{}k_{\rm B}T/{\rm
rad}^{2}$, and with minima offset by the free-energy difference
$\widetilde{\Delta\mu}=5.2~{}k_{\rm B}T$ between the catalytic-dwell and
binding-dwell states. Coarse-graining over the fast 40∘ step gives the
effective potential

$\displaystyle\beta U_{n}(\theta)\equiv-\ln[e^{-\frac{1}{2}\beta
k(\theta+\ell-n\xi)^{2}-\beta\widetilde{\Delta\mu}}+e^{-\frac{1}{2}\beta
k(\theta-n\xi)^{2}}\big{]}\ ,$ (1)

for $n=0,\pm 1,\pm 2,\dots$. $\beta\equiv 1/(k_{\rm B}T)$ is the inverse
temperature. The first and second terms in square brackets respectively
indicate the catalytic-dwell and binding-dwell states. The $80^{\circ}$ step
is modeled by a hop between adjacent effective potentials $U_{n}(\theta)$ and
$U_{{n}\pm 1}(\theta)$ angularly separated by $\xi=2\pi/3~{}{\rm rad}$ [see
$U_{1,2}(\theta)$ in Fig. 1b].

Figure 1:  a) Schematic of modeled experiment. b) Effective potential
$U_{1,2}(\theta)$ (dotted), trap potential $U_{\rm trap}(\theta|\lambda)$
(black dashed), and total potentials $U^{\rm tot}_{1,2}(\theta|\lambda)\equiv
U_{1,2}(\theta)+U_{\rm trap}(\theta|\lambda)$ (solid). c) Potential of mean
force $U_{\rm PMF}(\theta)$ (dotted), trap potential $U_{\rm
trap}(\theta|\lambda)$ (black dashed), and total potential $U^{\rm tot}_{\rm
PMF}(\theta|\lambda)\equiv U_{\rm PMF}(\theta)+U_{\rm trap}(\theta|\lambda)$
(solid). Each potential is plotted as a function of $\gamma$-shaft angle
$\theta$, for fixed trap minimum $\lambda=0.41$ (vertical line), chemical
drive $\Delta\mu_{\rm ATP}=18~{}k_{\rm B}T$, and trap strength
$E^{\ddagger}=30~{}k_{\rm B}T/{\rm rad}^{2}$.

Thus, the overall hydrolysis/synthesis of one ATP molecule is modeled by
switching the effective potential, $U_{n}(\theta)\to U_{{n}\pm 1}(\theta)$,
where the ‘+’ (‘-’) sign is for hydrolysis (synthesis). The corresponding
transition rates obey the local detailed-balance condition 35

$\displaystyle\dfrac{R^{+}_{n}(\theta)}{R^{-}_{n+1}(\theta)}=e^{\beta[U_{n}(\theta)-U_{n+1}(\theta)+\Delta\mu_{\rm
ATP}]}\ ,$ (2)

for chemical-potential difference (hereafter _chemical drive_) $\Delta\mu_{\rm
ATP}\geq 0$ due to synthesis of one ATP, favoring ATP hydrolysis and
$\gamma$-shaft counter-rotation. The potential switches with respective
forward and backward transition rates

$\displaystyle R_{n}^{+}(\theta)$ $\displaystyle=\Gamma,$ (3a) $\displaystyle
R_{n+1}^{-}(\theta)$ $\displaystyle=\Gamma
e^{-\beta[U_{n}(\theta)-U_{n+1}(\theta)+\Delta\mu_{\rm ATP}]}\ .$ (3b)

$\Gamma$ is a rate constant characterizing the chemical reaction (Supporting
Information 38 and Fig. S1 relate $\Gamma$ and [ATP] at fixed $\Delta\mu_{\rm
ATP}$). Among the range of possible splittings of angular dependence between
the forward and backward transition rates that satisfy local detailed-balance
(2), the experimental kinetics in Refs. 37, 36 are best fit by this splitting
35.

Experimentally, $\mathrm{F}_{1}$ is driven by confining the magnetic bead
attached to the $\gamma$-shaft in a magnetic trap 19, 18, 39, 40, whose
minimum is dynamically rotated. Modeling such a magnetic trap, here we
consider a sinusoidal trap potential (see Fig. 1),

$\displaystyle U_{\rm trap}(\theta|\lambda)\equiv-\tfrac{1}{2}E^{\ddagger}\cos
2(\theta-2\pi\lambda)\ ,$ (4)

with time-dependent control parameter $\lambda\in[0,1]$ determining the angle
of the two minima differing by 180∘ and separated by barriers of height
$E^{\ddagger}$ (parametrizing trap strength).

Thus, subject to this external trap, the $\gamma$-shaft angle $\theta$
dynamically evolves according to

$\displaystyle\dot{\theta}=-\beta D\,\partial_{\theta}U_{n}^{\rm
tot}(\theta|\lambda)+\sqrt{2D}~{}\eta(t)\ ,$ (5)

where the dot indicates a time derivative, $U_{n}^{\rm
tot}(\theta|\lambda)\equiv U_{n}(\theta)+U_{\rm trap}(\theta|\lambda)$ is the
total potential, $D=13.7~{}{\rm rad}^{2}/{\rm s}$ is the rotational diffusion
constant 36, and $\eta(t)$ is Gaussian white noise with zero mean and unit
variance. For $\Delta\mu_{\rm ATP}\neq 0$, the system eventually reaches a
nonequilibrium steady state 41. Chemical transition rates (3) and mechanical
dynamics (5), subject to potentials (1) and (4), constitute the TASAM.

In the limit of fast switching between effective potentials when chemistry is
fast compared to mechanics ($\Gamma\to\infty$, e.g., at high [ATP]) 35, the
dynamics (5) reduce to

$\displaystyle\dot{\theta}=-\beta D\,\partial_{\theta}U^{\rm tot}_{\rm
PMF}(\theta|\lambda)+\sqrt{2D}\,\eta(t)\ ,$ (6)

for total potential energy $U^{\rm tot}_{\rm PMF}(\theta|\lambda)\equiv U_{\rm
PMF}(\theta)+U_{\rm trap}(\theta|\lambda)$ and potential of mean force,
averaging over all effective potentials (see Fig. 1c):

$\displaystyle\beta U_{\rm
PMF}(\theta)\equiv-\ln\sum_{n=-\infty}^{+\infty}e^{-\beta[U_{n}(\theta)-n\Delta\mu_{\rm
ATP}]}\ .$ (7)

We seek a driving protocol that minimizes dissipation while rotating the
$\gamma$-shaft (for theoretical developments when control is much finer-
grained, see Refs. 42, 43). For a harmonically confined Brownian particle,
minimum-dissipation protocols have been analytically solved for arbitrary
protocol duration 44, 45. For more complicated scenarios such as this model,
no analytical solution is known; nevertheless, linear-response theory gives an
approximately dissipation-minimizing protocol 23. Up to the linear-response
approximation, the instantaneous excess power (that exceeding the quasistatic
power) during dynamic variation of control parameter $\lambda$ is

$\displaystyle P_{\rm
ex}(t)\approx\zeta(\lambda)\bigg{(}\dfrac{\mathrm{d}\lambda}{\mathrm{d}t}\bigg{)}^{2}\
.$ (8)

Its time integral over protocol duration $t_{\rm prot}$ gives the excess work
$W_{\rm ex}\equiv W-\Delta F=\int_{0}^{t_{\rm prot}}~{}{\mathrm{d}}t~{}P_{\rm
ex}(t)$, for protocol work $W$ and equilibrium free-energy change $\Delta F$
from initial to final control-parameter values. $\zeta(\lambda)$ is a
generalized friction coefficient obtained here by integrating the equilibrium
torque autocovariance:

$\displaystyle\zeta(\lambda)\equiv\beta\int_{0}^{\infty}~{}{\mathrm{d}}t~{}\langle\delta\tau(0)~{}\delta\tau(t)\rangle_{\lambda}\
.$ (9)

Angle brackets $\langle\dots\rangle_{\lambda}$ indicate a steady-state average
at fixed $\lambda$. $\delta\tau(t)\equiv\tau(t)-\langle\tau\rangle_{\lambda}$
is the deviation of the conjugate torque $\tau\equiv-\partial_{\lambda}U_{\rm
trap}(\theta|\lambda)=2\pi E^{\ddagger}\sin 2(\theta-2\pi\lambda)$ from its
equilibrium average. The generalized friction coefficient can be decomposed as

$\displaystyle\zeta(\lambda)=\beta\langle(\delta\tau)^{2}\rangle_{\lambda}~{}t_{\rm
relax}(\lambda)\ ,$ (10)

the product of the torque variance $\langle(\delta\tau)^{2}\rangle_{\lambda}$
and the torque relaxation time

$\displaystyle t_{\rm
relax}(\lambda)\equiv\int_{0}^{\infty}~{}{\mathrm{d}}t~{}\dfrac{\langle\delta\tau(0)~{}\delta\tau(t)\rangle_{\lambda}}{\langle(\delta\tau)^{2}\rangle_{\lambda}}\
.$ (11)

In the linear-response regime, the minimum-dissipation protocol proceeds with
velocity inversely proportional to the square root of the generalized friction
coefficient 23:

$\displaystyle\dfrac{{\mathrm{d}}\lambda^{\rm
des}}{{\mathrm{d}}t}\propto[\zeta(\lambda)]^{-1/2}\ .$ (12)

Dissipation is reduced by driving slower where system resistance is greatest
(due to large fluctuations and/or slow relaxation) and compensating by driving
faster where system resistance is least. Imposing boundary conditions
$\lambda(0)=\lambda_{\rm i}$ and $\lambda(t_{\rm prot})=\lambda_{\rm f}$ fixes
the proportionality constant. Such a designed protocol $\lambda^{\rm des}(t)$
gives (up to linear response) constant excess power.

Figure 2a shows $U^{\rm tot}_{\rm PMF}(\theta|\lambda)$ for different trap
minima $\lambda$ and fast switching. For some $\lambda$’s, the total potential
has two metastable states, with a small ($<0.5~{}k_{\rm B}T$) barrier.

Figure 2: (a) Total potential energy as a function of $\gamma$-shaft angle
$\theta$. (b) Torque autocovariance as a function of time. (c) Torque
variance, (d) torque relaxation time, (e) friction coefficient, and (f)
protocol velocity, each as a function of trap minimum $\lambda$. (g) Protocol
as a function of time. (f,g) Red lines: naive (constant-velocity) protocols;
points/curves: designed protocols. Potential switching rate $\Gamma=\infty$
(a,b), and $10/100/1000/\infty\,{\rm s}^{-1}$ (c-g). Throughout, trap strength
$E^{\ddagger}=30~{}k_{\rm B}T/{\rm rad}^{2}$ and chemical drive
$\Delta\mu_{\rm ATP}=18~{}k_{\rm B}T$. In all figures, error bars show one
standard error of the mean.

The friction coefficient (9) is obtained through measurement of the
equilibrium torque autocovariance for a stationary trap (i.e., fixed
$\lambda$) and hence fixed total potential-energy landscape (Fig. 2a). Figure
2b shows this torque autocovariance function. For those trap minima $\lambda$
giving two metastable states (Fig. 2a), the torque autocovariance relaxes
particularly slowly.

Figure 3: Steady-state excess power while driving ATP synthesis, as a function
of trap minimum $\lambda$, for protocol durations $t_{\rm prot}=0.125,~{}0.5$,
and 2 s, for naive (red) and designed (blue) protocols (Fig. 2g) in the full
model (points) and the linear-response approximation (8) (curves). Throughout,
trap strength $E^{\ddagger}=30~{}k_{\rm B}T/{\rm rad}^{2}$, chemical drive
$\Delta\mu_{\rm ATP}=18~{}k_{\rm B}T$, and potential switching rate
$\Gamma\to\infty$. Gray curves at bottom depict the total potential at the
corresponding trap minimum.

Figure 2c shows the torque variance computed from the equilibrium torque
fluctuations. The torque variance is independent of $\Gamma$, as expected
since varying $\Gamma$ modifies system relaxation between landscapes but not
the landscapes themselves. Similarly, Fig. 2d shows the torque relaxation time
(11), $t_{\rm relax}(\lambda)$. The product of the torque variance and torque
relaxation time gives the generalized friction coefficient (Fig. 2e). The peak
relaxation time (generally for trap minimum $\lambda$ such that the system has
significant probability in multiple potentials) is higher for smaller
$\Gamma$, which slows system relaxation between distinct potentials. The
torque variance, torque relaxation time, and friction coefficient each have a
local maximum for the trap minimum near the shoulder of $U_{n}(\theta)$ (Fig.
1b) and $U_{\rm PMF}(\theta)$ (Fig. 1c). As expected, these local maxima in
friction coefficient coincide with the experimental local minima of F1’s
rotational velocity during hydrolysis 15.

According to (12), excess work is reduced by slowing down where friction is
high, giving more time for thermal fluctuations to overcome the energy barrier
separating metastable states (Fig. 2a) and thereby reducing necessary work.
Figure 2f compares the control-parameter velocities of designed protocols and
naive ones (i.e., with constant velocity). Integrating the control-parameter
velocity gives the protocol as a function of time (Fig. 2g).

To compute excess power and work, we numerically simulate (without assuming
linear response) the full model of the system, Eqs. 5 and 6, driven by naive
and designed protocols across a full rotation driving ATP synthesis
(decreasing $\lambda$).

Figure 3 shows the steady-state excess power of the full model for fast
potential switching rate ($\Gamma\to\infty$), with the linear-response
approximation (8) as a reference. The linear-response approximation is quite
accurate in the long-duration limit since the system remains sufficiently
close to equilibrium. For the naive protocol, excess power is strongly peaked
where the $\gamma$-shaft experiences high resistance (Fig. 2e). The designed
protocol compensates by driving slower there (Fig. 2d), thereby flattening the
excess power across the protocol.

Figure 4: Excess work while driving ATP synthesis. a) Naive (red) and designed
(blue) excess works. b) Ratio of naive and designed excess works. Points show
the full model and curves the linear-response approximation. Color intensity
decreases with switching rate $\Gamma$. Throughout, trap strength
$E^{\ddagger}=30~{}k_{\rm B}T/{\rm rad}^{2}$ and chemical drive
$\Delta\mu_{\rm ATP}=18~{}k_{\rm B}T$.

Finally, Figure 4a shows excess work during naive and designed protocols
driving ATP synthesis, for different switching rates $\Gamma$. As expected,
the work decreases with protocol duration and with $\Gamma$. For longer
durations, the linear-response approximation is quite accurate.

Figure 4b shows the ratio of excess work during naive and designed protocols.
For the slowest protocols and slowest switching rate, designed protocols
reduce excess work $\sim 2.5\times$ compared to naive. At longer protocol
duration, the excess-work ratio for the slowest switching rate is almost twice
that of the fastest switching rate ($\Gamma\to\infty$). A slower switching
rate produces greater variation in the friction coefficient (Fig. 2e), leading
to designed protocols that differ more from naive ones (Fig. 2g) and thus
offering greater advantage to using designed protocols.

Analogous driving of ATP hydrolysis (instead of synthesis) and different trap
strengths and chemical drives produce qualitatively similar generalized
friction coefficients and designed protocols (Fig. S2), excess power along the
protocol (Fig. S3), and excess work (Figs. S4 and S5).

We have studied nonequilibrium driving of an experimentally motivated model
35, 36 of the F1-ATPase molecular motor. We found that the designed protocol
(that near equilibrium minimizes work) indeed significantly decreases work for
a wide range of protocol durations, including some driving F1 far from
equilibrium and dissipating as much as several $k_{\rm B}T$ per ATP
synthesized. This demonstrates the utility of this linear-response framework
significantly beyond its regime of strict validity. This protocol allocates
more time to regions where fluctuations are large and/or relaxation is slow
(and where $\mathrm{F}_{1}$ rotates slowest when hydrolyzing ATP 15), and
hence where greater time for system relaxation can make the most difference.
In this model this is where the total potential is bimodal, due to
$\mathrm{F}_{1}$ fluctuating between distinct chemical states with
correspondingly different mechanics.

Recent work has derived optimal protocols in the opposite limit of very fast
driving 46; these rapid protocols, and interpolations between them and near-
equilibrium ones, would allow systematic characterization of efficiency across
driving speeds. Our simple near-equilibrium theory may also be helpful in
physically constraining machine-learning algorithms 47 or in more direct
numerical calculation 48 of optimal protocols far from equilibrium. Moreover,
there is promise in exploring near-optimal protocols 49, analysis of which
significantly broadens the class of successful control strategies.

Our research opens avenues for future experimental investigation with an
analogous setup to Refs. 39, 40. Such an experiment can be used to design and
implement a rotational driving protocol employing our methodology. Our study
predicts experimental conditions where such designs will produce substantial
energy savings, informing experiments to either drive $\mathrm{F}_{1}$ to
efficiently produce ATP or efficiently harness mechanical energy from ATP
hydrolysis.

The ${\rm F}_{\rm o}{\rm F}_{1}$ molecular motor transduces energy at high
efficiency 50. Our designed protocol may mimic an operational mechanism by
which ${\rm F}_{\rm o}$ rotates F1’s $\gamma$-shaft to synthesize ATP, driving
slower where friction is higher, which we predict would save energy compared
to constant-velocity driving. It would be interesting to probe the actual
stochastic dynamics by which $\mathrm{F}_{\mathrm{o}}$ mechanically drives
${\rm F}_{1}$ in vivo and understand its correspondence to an effective
nonequilibrium driving protocol. E.g., $\mathrm{F}_{\mathrm{o}}$ could
approximate such a protocol if its metastable rotational states are out of
phase with those of ${\rm F}_{1}$. More generally, our methodology may be
applied to identify design principles for nanoscale free-energy transduction
within and between other molecular machines.

We thank Alex Tong and Carlos Bustamante (Berkeley Physics) for enlightening
discussions, and Adrianne Zhong (Berkeley Physics) for feedback on the
manuscript. This work was supported by a Natural Sciences and Engineering
Research Council of Canada (NSERC) Discovery Grant (D.A.S.), a Tier-II Canada
Research Chair (D.A.S.), and the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - Projekt-nummer 163436311 - SFB 910 (D.G.), and was
enabled in part by support provided by BC DRI Group and the Digital Research
Alliance of Canada (www.alliancecan.ca).

Supporting Information contains further numerical simulation results and
details of simulation methods.

##  Appendix SI Supporting Information

##  Appendix SII Relating $\Gamma$ to [ATP]

Figure S1a shows experimental rotational rates from 37: The rotation rate
strongly increases with [ATP] and weakly increases with $\Delta\mu_{\rm ATP}$.
Figure S1b shows numerical Langevin simulations of the rotation rate as a
function of $\Gamma$: The rotation rate increases with $\Gamma$ and asymptotes
to the rotation rate for fast-switching dynamics (6) ($\Gamma\to\infty$). At
fixed $\Delta\mu_{\rm ATP}$, comparing Figs. S1a and b permits calibration of
switching rate $\Gamma$: $\Gamma$ ranging from 1 to 1,000 Hz reproduces
rotational rates for [ATP] ranging from 0.1 to 100 $\mu$M.

Figure S1: $\gamma$-shaft rotation rate (a) in experiment 37 as a function of
[ATP] and of chemical drive $\beta\Delta\mu_{\rm ATP}$ (inset), and (b) in
simulation as a function of switching rate $\Gamma$ for slow-switching
dynamics (5) (triangles) or fast-switching dynamics (6) (horizontal line).
Color intensity decreases with [ATP] for chemical drive $\Delta\mu_{\rm
ATP}\approx 17.6~{}k_{\rm B}T$.

##  Appendix SIII Torque fluctuations and designed protocol for
$\Gamma\to\infty$: varying $E^{{\ddagger}}$ and $\Delta\mu_{\rm ATP}$

Figures S2a-d respectively show the torque variance, torque relaxation time,
friction coefficient, and rate of change of protocol, each as a function of
trap minimum $\lambda$, for two trap strengths $E^{\ddagger}$ and three
chemical drives $\Delta\mu_{\rm ATP}$. Figure S2e shows the naive and designed
protocols obtained by integrating their respective protocol velocities (Fig.
S2d)

The torque variance empirically increases with trap strength $E^{\ddagger}$;
this is intuitive, as in a simple quadratic potential the equipartition
theorem 51 dictates that the torque variance is directly proportional to
$E^{\ddagger}$. The torque variance, torque relaxation time, and friction
coefficient each are globally maximized at the trap minimum $\lambda$ where
$U^{\rm tot}_{\rm PMF}(\theta|\lambda)$ has two metastable states (see the
leftmost potential in Fig. 2a), and have a local maximum for $\lambda$ at the
PMF’s shoulder (Fig. 1c). The friction coefficient’s maximum value decreases
with trap strength $E^{\ddagger}$, because the barrier height in the double-
well total potential $U^{\rm tot}_{\rm PMF}(\theta|\lambda)$ decreases with
$E^{\ddagger}$. Further, the global maximum shifts leftwards as
$\Delta\mu_{\rm ATP}$ increases since this reduces the $\lambda$ at which the
total potential $U^{\rm tot}_{\rm PMF}$ has two metastable states, and the
insensitivity of $U_{\rm PMF}$’s shoulder to variation of $\Delta\mu_{\rm
ATP}$ reflects that the local maximum is similarly insensitive.

Figure S2: Equilibrium torque fluctuations and designed protocol. (a) Torque
variance, (b) torque relaxation time, (c) friction coefficient, and (d)
protocol velocity, each as a function of trap minimum, $\lambda$. (e)
Protocols as a function of time. (d,e) Red lines indicate naive protocols and
curves/points indicate designed protocols. Trap strengths $E^{\ddagger}=30$
(left) and $60~{}k_{\rm B}T/{\rm rad^{2}}$ (right). Color intensity increases
with chemical drive $\Delta\mu_{\rm ATP}$.

##  Appendix SIV Excess power for $\Gamma\to\infty$: varying direction,
$E^{{\ddagger}}$, and $\Delta\mu_{\rm ATP}$

Figure S3 compares the steady-state excess power $P_{\rm ex}$ as a function of
trap minimum $\lambda$, during driven hydrolysis (increasing $\lambda$) and
synthesis (decreasing $\lambda$), for three protocol durations $t_{\rm prot}$,
two trap strengths $E^{{\ddagger}}$, and three chemical drives $\Delta\mu_{\rm
ATP}$. The excess power is qualitatively similar for the two trap strengths,
and lower for higher trap strength. Since during hydrolysis the $\gamma$-shaft
rotates in the opposite manner than during synthesis, the excess power for the
full model flips. In contrast, the linear-response approximation is the same
for driven synthesis and hydrolysis. For longer durations, the linear-response
approximation (8) closely matches the full model, as expected. For
intermediate duration, agreement is poor near the peak of the friction
coefficient (where linear response predicts that excess power peaks under
naive protocols). Agreement is better for a stronger trap since the system
mostly follows the trap for the entire duration.

Figure S3: Steady-state excess power as a function of $\lambda$, during driven
(a) hydrolysis (increasing $\lambda$) and (b) synthesis (decreasing
$\lambda$), naive (red) and designed (blue) protocols, for the full model
(points) and the linear-response approximation (8). Blue dashed box indicates
data shown in Fig. 3. Trap strengths $E^{\ddagger}=30$ and $60~{}k_{\rm
B}T/{\rm rad}^{2}$, chemical drives $\Delta\mu_{\rm ATP}=0,9,$ and
$18~{}k_{\rm B}T$ (top to bottom), and protocol durations $t_{\rm
prot}=0.125,0.5$, and $2$ s.

##  Appendix SV Excess work driving ATP hydrolysis

Figure S4 shows the excess works by rotating the $\gamma$-shaft employing the
naive and designed protocols (see Fig. 2g) to drive ATP hydrolysis, for
different values of switching rate $\Gamma$. The linear-response approximation
is quite accurate in the long-protocol duration limit. Qualitatively, the
excess-work features during hydrolysis are similar to those during synthesis,
and the ratio of naive to designed excess work for our slowest switching rate
($\Gamma=10~{}{\rm s}^{-1}$) is almost double that in the limit
$\Gamma\to\infty$.

Figure S4: Excess work when driving hydrolysis. (a) Excess work for naive
(red) and designed (blue) protocols, and (b) ratio of naive and designed
excess works, each as a function of protocol duration, for the full model
(points) and the linear-response approximation (curves). Trap strength
$E^{\ddagger}=30~{}k_{\rm B}T/{\rm rad}^{2}$ and chemical drive
$\Delta\mu_{\rm ATP}=18~{}k_{\rm B}T$. Color intensity decreases with
switching rate $\Gamma$. In all figures, error bars show one standard error of
the mean.

##  Appendix SVI Excess work for $\Gamma\to\infty$: varying direction,
$E^{{\ddagger}}$, and $\Delta\mu_{\rm ATP}$

Figure S5 shows naive and designed excess works and their ratio, $W_{\rm
ex}^{\rm Naive}/W_{\rm ex}^{\rm Des}$ 25 App. B, during driven hydrolysis and
synthesis, for two trap strengths and three chemical drives $\Delta\mu_{\rm
ATP}$.

Longer protocol durations keep the system closer to equilibrium, increasing
the accuracy of the linear-response approximation. The designed protocol’s
excess work agrees with its linear-response approximation starting at shorter
durations than the naive protocol’s does (Fig. S5a).

For virtually all durations explored, designed protocols save energy compared
to naive protocols (Fig. S5b). The performance of designed protocols (relative
to naive), as quantified by the ratio of designed to naive excess works,
improves with protocol duration until it saturates at the linear-response
approximation. This excess-work ratio decreases with trap strength, and is
qualitatively similar in both hydrolysis and synthesis directions.

Figure S5: Excess work. (a) Excess work for naive (red) and designed (blue)
protocols, and (b) ratio of naive and designed excess works, all as a function
of protocol duration $t_{\rm prot}$, for driven hydrolysis (left) and
synthesis (right), for the full model (points) and the linear-response
approximation (curves), for trap strengths $E^{\ddagger}=30~{}k_{\rm B}T/{\rm
rad^{2}}$ (circles, solid curves) and $60~{}k_{\rm B}T/{\rm rad^{2}}$
(squares, dashed curves). Color intensity increases with the chemical drive
$\Delta\mu_{\rm ATP}$.

##  Appendix SVII Simulation methods

###  SVII.1 Torque autocovariance

For $\Gamma\to\infty$, we discretize the Langevin equation (6) to first order
in $\Delta t$ and compute $\theta_{\ell}$ ($1\leq\ell\leq t_{\rm prot}/\Delta
t$) using

$\displaystyle\theta_{\ell}=\theta_{\ell-1}+\beta D\tau^{\rm tot}_{\rm
PMF}(\theta_{\ell-1}|\lambda)~{}\Delta t+\sqrt{2D\,\Delta
t}~{}\tilde{\eta}_{\ell-1}\ ,$ (S1)

where $\tau^{\rm tot}_{\rm PMF}(\theta|\lambda)\equiv-\partial_{\theta}U^{\rm
tot}_{\rm PMF}(\theta|\lambda)$ is the sum of torques derived from the
underlying PMF and trap potential. $\tilde{\eta}_{\ell}$ is a normal random
variable with zero mean and unit variance, and $\theta_{0}$ is drawn from the
canonical distribution, i.e., proportional to $e^{-\beta U^{\rm tot}_{\rm
PMF}(\theta|\lambda)}$ for $\theta\in[-\pi/2+2\pi\lambda,\pi/2+2\pi\lambda]$.
(Notice that the trap potential $U_{\rm trap}(\theta|\lambda)$ has two minima
separated by $\pi$ rad, and we initialize the $\gamma$-shaft’s location in the
well whose minimum is at $2\pi\lambda$ rad, where $\lambda\in[0,1]$.) For each
fixed trap minimum $\lambda$, we generate a time-series of length $500$ s of
the external torque $\tau_{\ell}\equiv-\partial_{\lambda}U_{\rm
trap}(\theta_{\ell}|\lambda)=2\pi
E^{{\ddagger}}\sin~{}2(\theta_{\ell}-2\pi\lambda)$ (see main text). We discard
the initial 0.25 s (roughly 50$\times$ the torque relaxation time) of each
trajectory, and use the remaining part to compute the torque autocovariance
function $\langle\delta\tau_{0}~{}\delta\tau_{\ell}\rangle_{\lambda}$. The
torque autocovariance in Fig. 2b is an average over three such trajectories
initiated from three different initial conditions $\theta_{0}$.

For finite $\Gamma$, we again confine the bead attached to the $\gamma$-shaft
in the sinusoidal potential $U_{\rm trap}(\theta|\lambda)$ [Eq. (4)], so that
the system evolves according to (5) and switches its effective potential
$U_{n}$ following the forward (3a) and backward rates (3b). We follow the same
numerical simulation procedure as discussed above for $\Gamma\to~{}\infty$ to
compute the friction coefficient and related quantities displayed in Fig. S2.

###  SVII.2 Time-integrated average flux

During a protocol, the system in general lags behind the trap’s minimum, with
magnitude depending on the protocol duration. To quantify rotation of the
$\gamma$-shaft during a protocol completing a full rotation in a given
duration $t_{\rm prot}$, we numerically compute the time-integrated average
flux

$\displaystyle\bar{J}\equiv\dfrac{1}{2\pi}\int_{0}^{t_{\rm
prot}}~{}{\mathrm{d}}t~{}\int_{0}^{2\pi}~{}{\mathrm{d}}\theta~{}J(\theta,t)\
,$ (S2)

for probability flux $J(\theta,t)$. $\bar{J}=1$ corresponds to a complete
$\gamma$-shaft rotation over the course of the respective protocol.

For all observed protocol durations and both protocols, $\bar{J}$ eventually
approaches unity (see Sec. “Simulation parameters”), indicating that the
system reaches a nonequilibrium steady state.

###  SVII.3 Escape fraction

The external trap has two identical energy wells separated by 180∘. When the
protocol duration is sufficiently short (so driving is sufficiently rapid),
some trajectories cross the intervening barrier to the adjacent well. Our
theoretical treatment assumes that the trap is sufficiently strong to prevent
any such fluctuation, so we discard these trajectories.

Numerical simulations of the full model show that for $\Gamma\to\infty$ this
escape fraction decreases (increases) with the chemical drive $\Delta\mu_{\rm
ATP}$ when dynamically driving hydrolysis (synthesis), since the system moves
with (against) the potential gradient due to $U_{\rm PMF}(\theta)$. Similarly,
for finite $\Gamma$, larger $\Delta\mu_{\rm ATP}$ favors the $\gamma$-shaft’s
rotation in the hydrolysis direction (3) reducing the proportion of
trajectories that escape from the local well.

For $\Gamma\to\infty$, in the fastest simulated protocols dynamically driving
hydrolysis (synthesis) and for the lowest (highest) $\Delta\mu_{\rm ATP}$, we
find that $0.005\%$ ($0.5\%$) of trajectories escape the local trap. The
fraction decreases with protocol duration, as expected. For the stronger trap
($E^{\ddagger}=60~{}k_{\rm B}T/{\rm rad^{2}}$), no trajectories escape across
all observed durations.

###  SVII.4 Excess work

To numerically compute the excess work, we first discretize each protocol for
a given protocol duration $t_{\rm prot}$. We compute the external work during
a single trajectory by evaluating the net change in internal energy due to the
control-parameter change from an initial value $\lambda_{\rm i}=0$ to
$\lambda_{\rm f}=1$ in $t_{\rm prot}$:

$\displaystyle w=\sum_{\ell=0}^{t_{\rm prot}/\Delta t-1}~{}[U_{\rm
trap}(\theta_{\ell},\lambda_{\ell+1})-U_{\rm
trap}(\theta_{\ell},\lambda_{\ell})]\ ,$ (S3)

where $\theta_{\ell}$ is obtained from the discretized version of the Langevin
equation at fixed $\lambda_{\ell}$, (S1) for $\Gamma\to\infty$ and analogously
for finite $\Gamma$. When driving hydrolysis, we advance $\lambda$ from
$\lambda_{\rm i}$ to $\lambda_{\rm f}$, whereas when driving synthesis, we
reduce it from $\lambda_{\rm f}$ to $\lambda_{\rm i}$. The excess work is the
work minus the equilibrium free-energy difference: $w_{\rm ex}\equiv w-\Delta
F$. Here $\Delta F=\mp 3\Delta\mu_{\rm ATP}$, where the ‘-’ (‘+’) sign
corresponds to hydrolysis (synthesis), and the factor of 3 represents the 3
ATP hydrolyzed or synthesized per complete rotation. Finally, we average over
the $\mathcal{N}_{\rm R}$ realizations that stay in the trap’s local well (see
‘Escape fraction’ section) to get the excess work $W_{\rm ex}\equiv\langle
w_{\rm ex}\rangle$. To exclude transient relaxation to the nonequilibrium
steady state, we compute the excess work starting after two full rotations
(Figs. 4 and S4) or four full rotations (Fig. S5). Similar to the procedure in
this section for excess work, we compute time-integrated average flux, escape
fraction, and excess power in the numerical simulations.

###  SVII.5 Simulation parameters

In our model, the system relaxation time in the trap is (to a harmonic
approximation) $t_{\rm trap}\equiv\dfrac{k_{\rm
B}T}{DE^{\ddagger}}=1.2-2.4\times 10^{-3}$ s for trap strengths
$E^{\ddagger}=30~{}k_{\rm B}T/{\rm rad}^{2}$ and $60~{}k_{\rm B}T/{\rm
rad}^{2}$, and diffusion constant $D=13.7$ rad2/s. Therefore, for each
numerical simulation we generally choose discretization timestep $\Delta
t=10^{-5}$ s, at least two orders of magnitude shorter than the relaxation
time. We average over 3 independent trajectories to obtain the friction
coefficient and related observables in Figs. 2(b-g) and S2.

For computing excess work, time-integrated average flux, and escape fraction
when $\Gamma\to\infty$, we rotate the $\gamma$-shaft with protocol durations
as fast as $t_{\rm prot}=2^{-4}$ s (i.e., 62.5 ms), $\approx$12.5$\times$
slower than the maximum torque relaxation time ($\sim 5$ ms). The data shown
in Figs. 3, S3, and S5 are obtained by averaging over $10^{5}$ trajectories
during a complete rotation of the $\gamma$-shaft following four complete
rotations to allow the system to reach steady state.

For the computation of excess work for finite $\Gamma$ (Figs. 4 and S4), we
rotate the $\gamma$-shaft in the range of protocol durations $2^{-1}-2^{4}$ s,
where the fastest protocol has duration $\approx$5$\times$ the maximum of the
torque relaxation time for the slowest potential-switching rate
($\Gamma=10~{}{\rm s}^{-1}$). We average over $10^{3}$ trajectories during a
complete rotation of the $\gamma$-shaft following two complete rotations to
allow the system to reach steady state. In order to increase the accuracy of
the Langevin simulations, we used the discretization time $\Delta t=10^{-6}$
s.

For the largest finite $\Gamma$ and longest protocol durations, this timestep
still produces discretization artifacts (seen in the modest disagreement
between the full model and the linear-response approximation in Figs. 4 and
S4), but an even shorter timestep is computationally prohibitive.

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††thanks: Present Address: Department of Physics, IIT Delhi, Hauz Khas New
Delhi-110016, India

# Exploring the sensitivity of hadron colliders to non-universality in heavy
neutral currents

F. A. Conventi1,2<EMAIL_ADDRESS>G. D’Ambrosio1
<EMAIL_ADDRESS>A.M. Iyer3 a.iyer<EMAIL_ADDRESS>E.
Rossi1,4<EMAIL_ADDRESS>1INFN-Sezione di Napoli, Via Cintia, 80126
Napoli, Italy;
2Università degli Studi di Napoli Parthenope, Napoli, Italy;
3Univ. Lyon, Universite Claude Bernard Lyon 1, CNRS/IN2P3, UMR5822 IP2I,
F-69622, Villeurbanne, France;
4 Università degli Studi di Napoli ’Federico II’, Dipartimento di Fisica
“Ettore Pancini”, Via Cintia, 80126 Napoli, Italy.

###### Abstract

We present sensitivity projections for discovering a heavy resonance decaying
to electron and muon pairs and for probing the charged lepton non-universality
in such decays at the HL-LHC and FCC-hh. The analysis takes into account the
expected differences in the reconstruction efficiencies and the dilepton mass
resolutions for dielectron and dimuon final states. We demonstrate how the
analyses at HL-LHC naturally paves the way for a FCC-hh machine thereby
underlining its importance.

The Standard Model (SM) of particle physics has withstood the test of
experimental validation to a significant extent. In particular, the
electroweak sector is characterized by a well defined pattern of couplings
which manifests in terms of accurate predictions for several processes. Any
departure from this paradigm implies the presence of New Physics (NP) effects.
One of the most interesting observations in this direction corresponds to the
observation of flavour non-universality in terms of the theoretically clean
ratios: $R_{K}$ and $R_{K^{*}}$. Recent results obtained by the LHCb
Collaboration are compatible with the standard model at the level of 2.5
standard deviations Aaij  _et al._ (2019, 2017), still leaving room for
studies on flavour non-universality. Anyway, independently of these anomalies,
it is instructive to investigate the potential of the direct searches in
measuring deviations from universality.

In a direct search, non-universality between a set of final states would
manifest in the form of correspondingly different yields in the detector. In
this paper we present sensitivity projections for testing charged lepton
flavor non-universality in dilepton decays of a new heavy boson that should be
discovered at current and future $pp$ colliders. The analysis uses a simple
test statistic to estimate the significance of these departure from the
universality case.

Since the leptons are very clean objects in a detector, the developed strategy
will be used to study flavour non-universality using a simplified model with
an additional heavy state. Without loss of generality, we consider a heavy
vector boson ($Z^{\prime}$) decaying into a pair of leptons. These states are
a characteristic of several models beyond the SM: for instance scenarios with
additional $U(1)$ Donini  _et al._ (1997), extra-dimensional frameworks with
bulk gauge fields Gherghetta and Pomarol (2000) constitute some of the most
obvious extensions. The scenarios with an additional heavy vector were also
found to be useful in the context of flavour physics Gauld  _et al._ (2014);
Glashow  _et al._ (2015); Bhattacharya  _et al._ (2015); Crivellin  _et al._
(2015a, b); Aristizabal Sierra  _et al._ (2015); Crivellin  _et al._ (2015c);
Celis  _et al._ (2015); Bélanger  _et al._ (2015); Gripaios  _et al._ (2016);
Allanach  _et al._ (2016); Fuyuto  _et al._ (2016); Chiang  _et al._ (2016);
Boucenna  _et al._ (2016a, b); Celis  _et al._ (2017); Altmannshofer  _et al._
(2016); Crivellin  _et al._ (2017); Garcia Garcia (2017); Bečirević  _et al._
(2016); Bhattacharya  _et al._ (2017); Bhatia  _et al._ (2017); Cline  _et
al._ (2017). The analysis in this paper, however, can be trivially extended to
the $s$-channel decay of any heavy resonance like gravitons, heavy scalars
etc. The production mechanism is irrelevant for our analysis, therefore,
without loss of generality, we assume the $Z^{\prime}$ being predominantly
produced by light quarks. From a model point of view, denoting the coupling of
the leptons to the vector boson as $g_{l}$, the goal of this paper can be
restated in terms of extracting the sensitivity of the direct searches to
explore the difference $g_{e}-g_{\mu}$ and its deviations from 0. Similar
analyses exist for the Z-boson couplings from LEP Schael  _et al._ (2006).
111The analysis presented in this paper can be easily applied to test
universality of couplings for the SM Z boson as well.

The paper is organized as follows: we begin with the traditional bump hunt
searches of heavy neutral resonances decaying into a di-lepton (di-muon and
di-electron) final state. In this section we point out the role of the
different reconstruction resolutions between different flavour leptons in the
eventual computation of the discovery significance. This is then followed by
the description of the analysis and the estimation of the sensitivity to non-
universality of the HL-LHC collider. We note that the study at HL-LHC
naturally paves the way for an FCC-hh machine which is characterized by
significantly enhanced sensitivities to even smaller deviations from
universality. We conclude the paper with the prospects of including tau as a
part of future analysis to complete the picture.

##  I Bump Hunt Searches

The search for a heavy neutral resonance decaying into a di-lepton final state
is one of the most prominent channels being probed at LHC and there exist
relatively strong bounds on $\sigma\times\mathcal{B}_{ll}$ Sirunyan  _et al._
(2018); Aad  _et al._ (2019). A standard search strategy focuses on the
possibility for observing an excess of events over the Standard Model (SM)
prediction, where the SM background is mainly due to the universal coupling of
the $\gamma^{*}/Z$ to leptons.

In this analysis, we consider the production of a heavy $Z^{\prime}$ decaying
into muons and electrons. For the purpose of simulation, we use the Lagrangian
of the Sequential Standard Model (SSM). Using the model file from FEYNRULES
Alloul  _et al._ (2014), the matrix element for the process is produced using
MADGRAPH Alwall  _et al._ (2014) at a centre of mass energy of 14 TeV.
Showering and hadronization are described using PYTHIA 8 Sjostrand  _et al._
(2008). CMS cards of DELPHES 3.4 de Favereau  _et al._ (2014) is used for
detector simulation at the LHC. The efficiencies estimated from the simulation
are then used for different values of $\sigma\mathcal{B}$.

Event selection: In order to identify the leptons from the $Z^{\prime}$, the
following selection criteria have been applied:

  * •

two isolated leptons (electrons or muons) with a $p_{T}\geq 50~{}GeV$ and
$|\eta|<2.5$;

  * •

$\not{E}_{T}<10GeV$.

The main source of background is represented by the $pp\rightarrow
Z/\gamma^{*}\rightarrow ll$ where $l=e,\mu$.

Independently of the relative sizes of the coupling with the vector boson (SM
or beyond), the leptons are characterized by different detector acceptances
and mass reconstruction resolution. The acceptance efficiency ($\epsilon$) is
mass dependent: For instance for $m_{Z^{\prime}}=3$ TeV, we estimate
$\epsilon_{e}=0.46$ and $\epsilon_{\mu}=0.61$. While for 5 TeV the
corresponding values for electrons and muons are 0.48 and 0.35 respectively.
The mass reconstruction resolution for $m_{ll}>1$ TeV is much better for the
di-electron final state. The mass reconstruction resolution for di-leptons is
shown in Fig. 1 for a 5 TeV narrow resonance with a generated mass width
$\Gamma$ of $50$ GeV. The different mass reconstruction resolution can be
attributed to fact that the momentum of the electrons and muons are measured
differently: the former due to deposition in the E-cal and the latter due to
the bending in the tracker.222Under the assumption of enough statistics (not
necessarily equal) for either lepton, the asymmetry in the reconstruction
between the electrons and muons progressively increases with the resonance
mass. The smearing increases with the $p_{T}$ of the di-muons.

Figure 1: Mass reconstruction resolution of the di-electron (in pink-dashed)
and the di-muon (in red-solid) pairs for $M_{Z^{\prime}}=5$ TeV.

To calculate the expected significance for $Z^{\prime}\rightarrow ee$ and
$Z^{\prime}\rightarrow\mu\mu$ at LHC, we use a binned likelihood fit
$L(\mu_{e},\mu_{\mu})$.

In the case where background is well known we can evaluate the expected
significance as the probability of background only hypothesis
($\mu_{e}=\mu_{\mu}=0$) using the two dimensional profiled likelihood ratio
test Cowan  _et al._ (2011):

$q_{0}=-2\log\left[\frac{L(0,0)}{L(\hat{\mu}_{e},\hat{\mu}_{\mu})}\right]$ (1)

where $\hat{\mu}$ is the best value of $\mu$ estimated by fitting to the data
for both the electron and the muon. The signal discovery significance Z can be
evaluated as:

$Z_{tot}=\sqrt{q_{0}}.$ (2)

and for sufficiently large background we can use the asymptotic formula:

$Z_{tot}=\sqrt{q_{0}}=\sqrt{\sum\limits_{i=1;j=e,\mu}^{N_{e,\mu}}\left(2(s^{j}_{i}+b^{j}_{i})\log\left[1+\frac{s^{j}_{i}}{b^{j}_{i}}\right]-2s^{j}_{i}\right)}$
(3)

where the sum runs over the bins, $s^{j}_{i}$ and $b^{j}_{i}$ are the expected
numbers for signal and background events in the $i^{th}$ bin for $j=e,\mu$.
Note that the total number of bins $N_{e,\mu}$ are in general different for
the electron and the muon. It is important to stress that Eq. 3 just gives the
local significance. We account for the look elsewhere effect which leads to
the modification of local p-value corresponding to a given $Z_{tot}$ E.Gross
and Vitells (2010).

Figure 2:  Contours in the total di-lepton significance ($Z_{tot}$) for
$m_{Z^{\prime}}=5$ TeV decaying into electrons $(\sigma\mathcal{B})_{e}$ and
muons $(\sigma\mathcal{B})_{\mu}$. The diagonal dotted line corresponds to the
lepton flavour universality case
$\left((\sigma\mathcal{B})_{e}=(\sigma\mathcal{B})_{\mu}\right)$. The
asymmetry in the expected significance is due to different mass reconstruction
resolution (Fig.1).

Fig. 2 gives contours in the total di-lepton significance as a function of
cross-section times the branching fractions of the $Z^{\prime}$ decaying into
electrons $(\sigma\mathcal{B})_{e}$ and muons $(\sigma\mathcal{B})_{\mu}$. The
diagonal dotted line corresponds to the lepton flavour universality case
$\left((\sigma\mathcal{B})_{e}=(\sigma\mathcal{B})_{\mu}\right)$. The points
are scanned such that
$(\sigma\mathcal{B})_{e}+(\sigma\mathcal{B})_{\mu}\leq(\sigma\mathcal{B})_{max}$,
with the outer edge corresponding to $(\sigma\mathcal{B})_{max}$. For the LHC,
we choose $(\sigma\mathcal{B})_{max}$ as the upper bound obtained on
$(\sigma\mathcal{B})_{tot}$ from direct searches in di-lepton final state
Sirunyan  _et al._ (2018); Aad  _et al._ (2019). Lines parallel to the outer
edge represent contours of some constant
$(\sigma\mathcal{B})_{tot}<(\sigma\mathcal{B})_{max}$, decreasing
progressively as one moves inwards. For any given $(\sigma\mathcal{B})_{tot}$,
the scan over $(\sigma\mathcal{B})_{e}$ and $(\sigma\mathcal{B})_{\mu}$ is
done such that
$(\sigma\mathcal{B})_{e}+(\sigma\mathcal{B})_{\mu}=(\sigma\mathcal{B})_{tot}$
The asymmetric behaviour of the contour plot is due to the different mass
resolutions shown in Fig. 1. Thus, a larger coupling to the electrons leads to
a larger evaluated value for the total signal sensitivity. These
considerations lead to the following questions:

  1. 1.

does the absence of a signal imply no NP or a larger coupling to the muons?

  2. 2.

what are the prospects for unearthing non-universality at the HL-LHC and
future colliders?

##  II Non-universality test

In real life experiments, the statistic $q_{0}$ in Eq. 1 is minimized at the
best fit value of $\sigma\mathcal{B}$ for the leptons. Fig. 3 shows the
distributions of the test statistic $q_{0}$ under two different assumptions:
the left plot corresponds to the universal coupling case where
$(\sigma\mathcal{B})_{e}=(\sigma\mathcal{B})_{\mu}$ while the right plot
illustrates the $(\sigma\mathcal{B})_{e}<(\sigma\mathcal{B})_{\mu}$ case and
hence non-universality. The different widths of the parabola reflect the
differences in the mass reconstruction resolutions between the leptons. The
black line represents the 1 $\sigma$ measurement uncertainty.

|
---|---
Figure 3: Distribution of test statistic under the assumption of universal
($(\sigma\mathcal{B})_{e}=(\sigma\mathcal{B})_{\mu}$) (left plot) and non-
universal ($(\sigma\mathcal{B})_{e}<(\sigma\mathcal{B})_{\mu}$) (right plot)
couplings. The different widths are a consequence of different mass
reconstruction resolution for the electrons (blue-thick) and muons (orange-
dashed).

The departure from the universality hypothesis can be quantified by the
following asymmetry variable:

$\hat{A}=\frac{(\sigma\mathcal{B})_{\mu}-(\sigma\mathcal{B})_{e}}{(\sigma\mathcal{B})_{\mu}+(\sigma\mathcal{B})_{e}}\in[-1,1].$
(4)

The two extremities $\hat{A}=-1$ and $\hat{A}=1$ correspond to a very large
signal in the electron channel
($\sigma\mathcal{B})_{e}\gg(\sigma\mathcal{B})_{\mu}$ and muon channel
($\sigma\mathcal{B})_{\mu}\gg(\sigma\mathcal{B})_{e}$ respectively. In
general, $\hat{A}$ divides the phase space into two specific regions:
$\hat{A}>0$ corresponds to the case where couplings to muons is larger and is
called the Pro-muon region; $\hat{A}<0$ corresponds to the case where
couplings to electrons is larger and is called the Pro-electron region. Thus,
a measurement corresponding to $\hat{A}\neq 0$ could be a hint of non-
universality. An estimate for the significance in the measurement of $\hat{A}$
must also account for the individual uncertainties in the extraction of
$\sigma\mathcal{B}_{e,\mu}$ which correspond to the widths in Fig. 3.

The significance in the measurement of $\hat{A}$ can be quantified by using a
two dimensional profiled likelihood ratio test similar to Eq. 1 and defined as

$q=-2\log\left[\frac{L(\hat{A}=0)}{L(\hat{A})}\right]$ (5)

treating $(\sigma\mathcal{B})$ as a nuisance parameter. The measured values of
$(\sigma\mathcal{B})$ for the electrons and muons are related to
$(\sigma\mathcal{B})_{tot}$ as:
$(\sigma\mathcal{B})_{e,\mu}=\hat{\mu}^{m}_{e,\mu}(\sigma\mathcal{B})_{tot}$.

Fig. 4 shows the typical behaviour of $q$ for two different values of non-
universality: $\mu^{m}_{e}=0.7;\mu^{m}_{\mu}=0.3$ (orange) and
$\mu^{m}_{e}=0.3;\mu^{m}_{\mu}=0.7$ (blue). We use two different benchmark
values of the total cross-section: $(\sigma\mathcal{B})_{Tot}=0.01$ fb (top
row) and $(\sigma\mathcal{B})_{Tot}=0.025$ fb (bottom row) for
$M_{Z^{\prime}}=3,5$ TeV. The plots quantify the departure of the universality
hypothesis ($\hat{A}=0$ or $\hat{\mu}_{e}=\hat{\mu}_{\mu}=0.5$). The solid
black line corresponds to the $2\sigma$ intercept. Scanning for different
values of $(\sigma\mathcal{B})_{Tot}$, we obtain the asymmetry-sensitivity
plots shown in Fig. 5. Moving along either curve, from the bottom to the top,
corresponds to increasing values of $(\sigma\mathcal{B})_{Tot}$ and, hence,
$Z_{tot}$. With respect to bounds from direct searches Aad  _et al._ (2019),
we must note that they are obtained under a lepton flavour universality
assumption. Estimation of a bound under the non-universality hypothesis will
require a recast of the entire analysis and is out of scope of the paper.
However, we naively calculate the $Z_{tot}$ from the upper bound on
$(\sigma\mathcal{B})_{tot}$ represented by the lower boundary of the orange-
shaded region in Fig. 5.

In an ideal scenario, for any given Z’ mass we can expect to be sensitive to
tiny deviations from universality i.e. $\hat{A}\rightarrow 0$ as
$(\sigma\mathcal{B})$ becomes very large or $Z_{tot}$ increase. However HL-LHC
sensitivity is limited by existing bounds on $(\sigma\mathcal{B})_{tot}$ as
well as a finite total integrated luminosity. The ruled out region for 3 and 5
TeV masses are illustrated by the pink band in Fig.5. Taking this into
account, Fig.6 illustrates the expected limits on $|\hat{A}|$ as a function of
Z’ mass for the full LHC dataset at $\mathcal{L}=3$ab-1. The flat behaviour is
due to the fact that the bounds on direct searches becomes progressively
stronger in going from 1 to 5 TeV.

|
---|---
|
Figure 4: Test statistic $q$ in Eq. 5 with a benchmark of
$\sigma\mathcal{B}_{Tot}=0.01$ fb (top row) and $0.025$ fb ( bottom row). Left
(right) column corresponds to $M_{Z^{\prime}}=3(5)$ TeV. The orange and blue
curves correspond to two different hypothesis for $\mu^{m}_{e}$ and
$\mu^{m}_{\mu}$ (See text for details).

Having laid out our strategy for extracting non-universality, we find it
relevant to draw the attention to Fig. 1 of Greljo and Marzocca (2017) which
uses differential LFU (Lepton flavour universality) ratios to extract non-
universality. Similar ratios are also employed by experiments Aad  _et al._
(2020) and a combination with the proposed analysis in this paper could
possibly reveal more information.

---
Figure 5: 2$\sigma$ (Green lines) and 3$\sigma$ (Red solid lines) asymmetry
sensitivity plot for the full LHC dataset at $\mathcal{L}=3$ab-1. The pink-
shaded region is the upper bound on $(\sigma\mathcal{B})_{tot}$ from direct
searches at LHC.
---
Figure 6: Summary of exclusion in $\hat{A}$ for different masses corresponding
to the rescaled LHC bounds for the electron channel. The red(green-dashed)
lines represent 3(2)$\sigma$ exclusion for the full LHC dataset at
$\mathcal{L}=3$ab-1.

The current non universality tests at LHC, while being powerful are limited on
the following accounts: 1) reduced sensitivity to heavier masses 2) Reduced
sensitivity to minor deviations from universality. These considerations
naturally lead to evaluate and study the possible improvements with future
colliders as it will be discussed below.

##  III Future Colliders

The advent of the FCChh is expected to provide continuity from the tail end of
the sensitivity of the HL-LHC. The higher energy and integrated luminosity of
FCChh will allow one to extend the discovery reach toward both higher masses
and smaller couplings, and to increase the sensitivity for charged lepton
flavour non-universality. This makes a future collider all the more relevant
not only for explorations deeper in the UV regions of phase space but also
provides enhanced sensitivity to minor deviations from non-universality.

We first begin with the discovery prospect of such states at the FCC. For the
purpose of FCC studies, we use the FCC-hh card reported in this reference de
Favereau  _et al._ (2014). One notable difference between the HL-LHC detectors
and those contemplated for FCC-hh is that the latter are expected to have
relatively similar reconstruction resolutions for electron and muon momenta.
This is particularly true for lower masses as compared to the heavier masses
as shown in Fig. 7 where the effect of the Z’ mass resolution on the expected
signal sensitivity for the FCC have been reported.

|
---|---
Figure 7: Contours of total signal significance as a function of branching
fraction into the leptons for 5 TeV (left) and 15 TeV (right) computed at
$\mathcal{L}=10$ ab-1 for the FCC.

The philosophy is exactly similar to the corresponding plot for the LHC in
Fig. 2. All the point on the edge of the plot satisfy
$(\sigma\mathcal{B})_{e}+(\sigma\mathcal{B})_{\mu}=0.1$ fb. The behavior of
increasing asymmetry between the leptons is similar to that of the LHC albeit
at much higher masses: at the LHC one expects a symmetric reconstruction at
around the scale of the $Z$ boson mass with the asymmetry increasing
progressively.

Our procedure to evaluate the signal discovery significance for FCC will be
exactly similar to the one shown for HL-LHC. For the purpose of continuity we
begin with $M_{Z^{\prime}}=5$ TeV and compare the FCC results with that at HL-
LHC as shown in Fig.8.

The results are compared at the end of the expected run of the corresponding
machines: 3 ab-1 for HL-LHC and 30 ab-1 for FCC. All the lines represent 3
$\sigma$ sensitivity to non-universality. The FCC curve is characterized by
two distinct features: A) more symmetric sensitivity on either side of
$\hat{A}$ owing to symmetric reconstruction for 5 TeV $Z^{\prime}$ mass and B)
higher sensitivity to minor deviations from non-universality corresponding to
the regions around $\hat{A}=0$.

---
Figure 8: 3$\sigma$ asymmetry sensitivity plot with the respect to expected
discovery significance in the electron channel for FCC (blue-solid) at a
luminosity of 30 ab-1 and for LHC (orange-dashed)) at a luminosity of 3 ab-1
for a $Z^{\prime}$ mass of 5 TeV.

FCC results for a $Z^{\prime}$ mass of 5 and 10 TeV are shown in Fig. 9.

As noted before, the strength of the FCC at 30 ab-1 is demonstrated by its
ability to probe regions very close to $\hat{A}=0$.

---
Figure 9: 3 $\sigma$ (red shadowed region) and 2 $\sigma$ (green lines)
asymmetry sensitivity plot for the FCC with the respect to expected discovery
significance in the electron channel for a $Z^{\prime}$ mass of 5 and 10 TeV
with a luminosity of 30 ab-1. The red shadowed region illustrates regions of
non-universality that can be excluded at $3\sigma$ at this luminosity.

Table 1 summarize the A bounds for a Z’ of 5 TeV and 10 TeV for Zee equal to
10 and 15. for HL-LHC and FCC.

$m_{Z^{\prime}}$ | $Z_{tot}$ | HL-LHC | FCC
---|---|---|---
$5$ TeV | 10 | $(-0.95,0.76)$ | $(-0.53,0.52)$
| 15 | $--$ | $(-0.36,0.35)$
$10$ TeV | 10 | – | $(-0.63,0.55)$
| 15 | – | $(-0.42,0.38)$

Table 1: 3$\sigma$ $\hat{A}$ bounds for a $5$ and $10$ TeV $Z^{\prime}$ at
$Z_{ee}=10,15$ level for HL-LHC and FCC. We do not quote the sensitivity for
10 TeV at HL-LHC as it is out of reach for practical values of
$\sigma\mathcal{B}$

.

##  IV Conclusions

In this work, using a simple test statistic, we present sensitivity
projections for testing charged lepton flavour universality in dilepton decays
of a neutral heavy boson, should it be discovered at HL-LHC or FCC-hh. While
being powerful HL-LHC limits show reduced sensitivity to heavier Z’ masses and
minor deviations from leptons universality. This motivates a detailed analysis
in the future FCC-hh machines where the differences between the electrons and
the muons are ironed out for relatively lighter masses. Furthermore, this
machine is sensitive to minor deviations from non-universality. This work also
offers a nice complementarity between the observations in flavour factories
and direct searches. This strategy can also be extended to tau final states
which are mainly identified by their hadronic decays. Using the techniques
introduced in this paper and adapting improved identification criteria for the
tau, will enable us to get a complete picture of (non-)universality in the
neutral current sector.

##  V Acknowledgements

We are grateful to M. Mangano for his continuous suggestions throughout the
course of the project. F.C. and A.I would wish to thank Antonio Giannini for
his help with the computation of signal sensitivities in the initial stages of
the project. G.D and A.I. wish to thank useful discussions with Alberto Orso
Maria Iorio. AI wishes to thank Michael Winn for useful observations during
the GdR-InF 2019 meeting. We wish to thank Sabyasachi Chakraborty, Seema
Sharma and Tuhin Roy for a careful reading of the manuscript and several
useful comments. A.I would like to thank CEFIPRA under the project “Composite
Models at the Interface of Theory and Phenomenology” (Project No. 5904-C).
G.D. was supported in part by MIUR under Project No. 2015P5SBHT and by the
INFN research initiative ENP. G.D. thanks “Satish Dhawan Visiting Chair
Professorship” at the Indian Institute of Science.

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# Magnetic dilution by severe plastic deformation

Martin Stückler
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences
Jahnstraße 12, 8700 Leoben, Austria
<EMAIL_ADDRESS>Lukas Weissitsch
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences
Jahnstraße 12, 8700 Leoben, Austria
Stefan Wurster
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences
Jahnstraße 12, 8700 Leoben, Austria
Peter Felfer
Department of Materials Science, Friedrich-Alexander-Universität Erlangen-
Nürnberg
Martensstraße 5, 91058 Erlangen, Germany Heinz Krenn
Institute of Physics, University of Graz
Universitätsplatz 5, 8010 Graz, Austria Reinhard Pippan
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences
Jahnstraße 12, 8700 Leoben, Austria Andrea Bachmaier
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences
Jahnstraße 12, 8700 Leoben, Austria

###### Abstract

Mixtures of Fe and Cu powders are cold-compacted and subsequently deformed
with severe plastic deformation by high-pressure torsion, leading to bulk
samples. The dilution of Fe in the Cu matrix is investigated with SQUID-
magnetometry, whereas the magnetic properties change as a function of Fe-
content from a frustrated regime to a thermal activated behaviour. The
magnetic properties are correlated with the microstructure, investigated by
synchrotron X-ray diffraction and atom probe tomography. Annealing of the as-
deformed states leads to demixing and grain growth, with the coercivity as a
function of annealing temperature obeying the random anisotropy model. The
presented results show that high-pressure torsion is a technique capable to
affect the microstructure even on atomic length scales.

##  1 Introduction

Metastable phases exhibit a high potential in application, since attractive
functional and magnetic properties were observed in such non-equilibrium
states [1, 2, 3]. In particular, the immiscible Fe-Cu system is of great
interest, as Fe, mixed with a low amount of Cu, exhibits desirable
magnetostrictive properties [4]. On the other hand, Cu mixed with low amounts
of Fe shows the granular giant magnetoresistive effect [5, 6]. But all of the
mentioned studies have in common, that sample preparation routes were used,
which are hardly accessible for industrial applications in terms of bulk
products. Whereas magnetron sputtering or vapour depositions turn out to be
not feasible on large scales, upscaling is possible in the case of mechanical
alloying or rapid solidification. Anyway, one has to prepare a bulky sample,
either from powder or thin ribbons. Therefore, sample preparation directly in
bulk form is desirable. Techniques of severe plastic deformation, such as
high-pressure torsion (HPT), are capable to prepare such metastable phases
directly in bulk form and can also be used at large scales. However the
achievable degree of intermixing has to be examined. Therefore, non-
equilibrium Fe-Cu samples are processed with HPT in this study, whereas
intermixing is investigated for various Fe-contents with advanced
microstructural characterization techniques in combination with SQUID-
magnetometry.

##  2 Experimental

Powder mixtures, consisting of high purity powders (Fe: MaTeck 99.9% -100 +200
mesh, Cu: AlfaAesar 99.9% -170 +400 mesh), were prepared with three different
Fe-contents, namely Fe07wt.%-Cu, Fe14wt%-Cu and Fe25wt.%-Cu, and
hydrostatically compacted at 5 GPa. The cylindrical samples (d=8 mm,
h$\sim$0.5 mm) were subsequently deformed with HPT at 5 GPa for 100 turns at
room temperature. The applied amount of shear strain at $r$=3 mm was
$\gamma{\sim}$3000\. More details on sample preparation can be found elsewhere
[7]. All compositions were exposed to annealing treatments at 150°C, 250°C and
500°C for 1 h each, which were conducted in vacuum ($p{\leq}$10-3 mbar) to
prevent oxidation. Microstructural analysis was carried out with a scanning
electron microscope (SEM; Zeiss LEO 1525), with attached electron backscatter
diffraction (EBSD; Bruker Nano eFlashFS), in samples tangential direction.
Investigation on crystallographic compositions was carried out with
synchrotron X-ray diffraction at DESY (beam energy: 100 kV; beam size: 0.2x0.2
mm2). The beam was oriented in axial direction of the sample at $r$=3 mm.
Further microstructural analysis was performed via atom probe tomography (APT)
experiments on a LEAP4000 X at 50 K in laser mode with a pulse repetition rate
of 250 kHz. Magnetic properties were investigated using a SQUID-magnetometer
(Quantum Design MPMS-XL-7).

##  3 Results and Discussion

###  3.1 Microstructural Characterization

HPT-processing applies a shear strain onto the sample, which increases with
increasing radius[8], causing a microstructural evolution, which strongly
affects the mechanical properties. To verify that a steady state is reached,
Vickers hardness testing is carried out. The hardness shows a saturating
behaviour at $r{\geq}$1 mm for 100 turns and therefore further investigations
are carried out at large radii. Fig. 1 shows synchrotron XRD-patterns of as-
deformed samples, as well as the patterns of Fe14wt.%-Cu in annealed states as
an example. The as-deformed samples exhibit very prominent fcc-Cu peaks, but
tiny bcc-Fe peaks can also be identified. The latter are two orders in
magnitude smaller and reveal the presence of residual Fe-particles. Annealed
samples show an increasing intensity for bcc-Fe, indicating demixing. While
the peak-width of the 150°C-annealed sample does not change with respect to
the as-deformed state, the 250°C- and the 500°C-annealed sample both show a
narrowing of peak-width arising from grain growth. Annealed Fe07wt.%-Cu and
Fe25wt.%-Cu samples exhibit the same behaviour.

Figure 1: Synchrotron-XRD diffraction patterns of as-deformed samples and of
Fe14wt.%-Cu in annealed states, measured in transmission mode.

###  3.2 Magnetic Properties

Figure 2: Specific magnetization $m$ versus the applied field $H$ as a
function of Fe-concentration, measured at 300 K and 8 K. Note: the saturation
magnetization of bulk Fe is 220 emu/g. Figure 3: Coercivity $H_{C}$ as a
function of Fe-concentration and annealing temperature. The coercivity shows
the same behaviour in all investigated compositions. Filled bars represent
measurements at 300 K, whereas open bars represent measurements at 8 K. For
the 150°C annealed state of Fe25wt.%-Cu the coercivity is below the
measurements resolution.

The magnetization versus the applied magnetic field is measured in the as-
deformed state for all compositions at 300 K and at 8 K (Fig. 2). The
hysteresis, measured at 300 K, do not saturate, even at the highest applied
field of 70 kOe. The observed high-field susceptibility is expected to arise
from diluted Fe with suppressed long-range interaction. A non-zero high-field
susceptibility was also observed in measurements at 8 K. Although the
hysteresis show paramagnetic features, non-zero coercivities are observed.
Fig. 3 shows the coercivites deduced from hysteresis measurements for all
investigated configurations. The coercivity as a function of annealing
temperature shows the same development in all investigated compositions. This
behaviour can be correlated with the microstructural states in this system. It
is known, that HPT-processed samples exhibit a high amount of residual
stresses $\sigma$ in the as-deformed state [9]. The high magnetostriction
$\lambda_{S}$ of Fe gives then rise to an enhancement of anisotropy by
magnetoelastic anisotropy, which reads
$K_{\sigma}{=}3/2\cdot\sigma\lambda_{S}$. Approximating the coercivity with
the anisotropy field, as proposed in the Stoner-Wohlfarth model,
$H_{C}{\propto}K_{eff}/M_{S}$, this further leads to an increase in coercivity
[10]. Upon annealing at 150°C neither grain growth nor demixing was observed,
but recovery mechanisms lead to a reduction in residual stresses [11], which
therefore lowers the coercivity. The coercivity of the as-deformed, as well as
of the 150°C-annealed sample arises therefore from random anisotropy [12].
Annealing at higher temperatures leads to demixing and grain growth. The
coercivity as a function of annealing temperature increases for the
250°C-annealed sample to around 400 Oe. The random anisotropy model peaks at
$d{=}\sqrt{A/K}$ and breaks down for larger grain sizes $d$. Using typical
values for bcc-Fe [13], the peak coercivity can be calculated to
$H_{C}{\approx}$600 Oe, in the order of the measured value. Annealing at 500°C
leads to a drop in coercivity, indicating a crossover from single-domain to
multi-domain behaviour with a classical $1/d$-scaling law. Fig.4 shows the
phase map derived from an EBSD scan of the 500°C-annealed sample (Fe14wt%-Cu).
The Fe grain size was analyzed to $d$=(210$\pm$80) nm. The corresponding
coercivity can be calculated, leading to $H_{C}^{th}$=(100$\pm$50) Oe, higher
than the coercivity determined from hysteresis measurements ($H_{C}$=60 Oe).
For HPT-processed samples, grains elongate in radial direction, therefore the
grain size is underestimated in the direction observed in the EBSD scan, which
might explain the deviation.

Figure 4: Post-processed EBSD scan of Fe14wt.%-Cu, annealed at 500∘C. Red
regions represent Cu grains, while green regions represent Fe grains. Lines
mark grain boundaries. The mean grain size of Fe is determined to (210$\pm$80)
nm. The shearing direction is parallel to the horizontal axis.

Additionally, low-field magnetic measurements are carried out. In Zero-Field
Cooling (ZFC) measurements, the demagnetized sample is cooled in the absence
of an external field. Starting from the lowest observed temperature, the
magnetic moment is recorded during heating. In Field-Cooling (FC)
measurements, the magnetic moment is recorded during cooling in an external
field. Fig. 5 shows ZFC/FC-measurements for the as-deformed states. For
reasons of comparability, the data are normalized, i.e. the susceptibility at
300 K is subtracted from all values. Herein, Fe25wt.%-Cu shows a large
splitting and a broad peak in the ZFC-curve, which is a typical behaviour for
thermal activation[14]. The ZFC/FC-curves of Fe14wt.%-Cu also shows splitting
of both curves, but in the FC-curve also a maximum at around 50 K can be
identified. A local maximum in the FC-curve is also observed in Fe07wt.%-Cu.
In the literature this feature is often attributed to a spin-glass behaviour
of diluted Fe [15, 16, 17].

Figure 5: Normalized zero-field cooling and field-cooling curves of as-
deformed samples as a function of Fe-concentration. In all measurements a
field of $H$=50 Oe was applied. Open symbols represent FC-curves, whereas
filled symbols represent ZFC-curves.

###  3.3 Atom probe tomography

To investigate the distribution of Fe on the atomic scale, APT experiments are
carried out. As mentioned above, the splitting in the ZFC/FC-curve is
attributed to thermal activation, and therefore to the presence of Fe
particles, small enough for thermal relaxation. Thus, the measured APT data is
analyzed with respect to Fe-clusters, using Delaunay triangulation. Details on
the used clustering algorithm can be found elsewhere [18]. In Fig. 6(a)-(c)
the atom probe reconstruction of Fe07wt.%-Cu, Fe14wt.%-Cu and Fe25wt.%-Cu
respectively is plotted, showing the distribution of Fe. Therein, Fe25wt.%-Cu
shows large volumes of clustered Fe-atoms, as expected from the observed
ZFC/FC-curve. In Fig. 6(d)-(f) the results from the clustering algorithm are
shown. Qualitative differences arise, as the formation of Fe-clusters depends
on the present local crystallographic defects. The cluster sizes span over a
wide range, whereas the cluster size for Fe25wt.%-Cu shows the broadest
distribution. This can also be attributed to the observed maxima in the
ZFC/FC-measurements (Fig. 5), which are expected to arise from magnetic
blocking. Anyway, in every sample about 95% of all Fe-atoms are rejected by
the clustering algorithm, meaning the majority of Fe-atoms is diluted in the
Cu-matrix, which explains the large high-field susceptibility in Fig. 2. One
important parameter to achieve accurate spatial reconstruction is the
detection efficiency of the used APT system. In [19] it has been shown, that
the cluster size distribution is systematically underestimated by the
detection efficiency.

Figure 6: APT reconstruction data for Fe07wt.%-Cu (a), Fe14wt.%-Cu (b) and
Fe25wt.%-Cu (c). 105 Fe-Atoms are plotted (blue dots). Red dots represent
clustered Fe-Atoms. (d)-(f) show the Fe-cluster size distributions of
Fe07wt.%-Cu, Fe14wt.%-Cu and Fe25wt.%-Cu, respectively.

##  4 Conclusion

In this study, the magnetic and the microstructural properties of HPT-
processed binary Fe-Cu samples, in the as-deformed as well as in annealed
states, are correlated, whereas a specific focus is on the investigation of
the amount of intermixing. In the as-deformed states, Fe is present in two
different configurations, namely diluted in the fcc-Cu matrix, as well as in
form of clusters. The ratio of clustered and diluted Fe varies as a function
of composition and gives rise to either thermal relaxation (in case of
Fe25wt.%-Cu), magnetic frustration (in case of Fe07wt.%-Cu) or a superposition
of both, which is true for Fe14wt.%-Cu. The results further prove that
intermixing of Fe and Cu can be obtained with HPT on an atomic scale. The
coercivity as a function of the annealing temperature complies the random
anisotropy model, showing Fe-particle sizes below the exchange length persist
in annealed states, which further gives rise to magnetic tunability.

##  5 acknowledgments

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant No. 757333). Synchrotron measurements have been performed at PETRA III:
P07 at DESY Hamburg (Germany), a member of the Helmholtz Association (HGF). We
gratefully acknowledge the assistance by N. Schell. The authors thank F.
Spieckermann and C. Gammer for their help with data processing.

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