# A versatile rotary-stage high frequency probe station for studying magnetic
films and devices

Shikun He  Data Storage Institute, A*STAR (Agency for Science, Technology and
Research), 2 Fusionopolis Way, #08-01, Innovis 138634, Singapore  Division of
Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371  Zhaoliang Meng  Data
Storage Institute, A*STAR (Agency for Science, Technology and Research), 2
Fusionopolis Way, #08-01, Innovis 138634, Singapore  Lisen Huang  Data Storage
Institute, A*STAR (Agency for Science, Technology and Research), 2
Fusionopolis Way, #08-01, Innovis 138634, Singapore  Lee Koon Yap  Data
Storage Institute, A*STAR (Agency for Science, Technology and Research), 2
Fusionopolis Way, #08-01, Innovis 138634, Singapore  Tiejun Zhou  Data Storage
Institute, A*STAR (Agency for Science, Technology and Research), 2
Fusionopolis Way, #08-01, Innovis 138634, Singapore  Christos Panagopoulos
Division of Physics and Applied Physics, School of Physical and Mathematical
Sciences, Nanyang Technological University, Singapore 637371

###### Abstract

We present a rotary-stage microwave probe station suitable for magnetic films
and spintronic devices. Two stages, one for field rotation from parallel to
perpendicular to the sample plane (Out-of-Plane) and the other intended for
field rotation within the sample plane (In-Plane) have been designed. The
sample, probes and micro positioners are rotated simultaneously with the
stages, which allows the field orientation to cover $\theta$ from 0 to
$90^{\circ}$ and $\varphi$ from 0 to $360^{\circ}$. $\theta$ and $\varphi$
being the angle between the direction of current flow and field in a Out-of-
Plane and an In-Plane rotation, respectively. The operation frequency is up to
40 GHz and the magnetic field up to 1 T. The sample holder, vision system and
probe assembly are compactly designed for the probes to land on a wafer with
diameter up to 3 cm. Using homemade multi-pin probes and commercially
available high frequency probes, several applications including 4-probe DC
measurements, the determination of domain wall velocity and spin transfer
torque ferromagnetic resonance are demonstrated.

##  I introduction

The discovery of spin transfer torque has attracted considerable attention in
the past decade.(Slonczewski, 1996; Mangin  _et al._ , 2006; Rippard  _et al._
, 2004; Houssameddine  _et al._ , 2007; Ikeda  _et al._ , 2010) Recently,
manipulation of magnetic bits through spin Hall effect(Liu  _et al._ , 2012)
and spin orbit torque(Yu  _et al._ , 2014) have also been demonstrated. The
characteristic time for current induced magnetic switching is ns and the
typical frequency of spin torque driving response is in the GHz range. These
findings have made magnetic nanodevices realistic candidates for memory and
microwave device applications. In response, there has been increasing demand
for microwave frequency probe stations capable of characterizing magnetic
films and devices.

Various probe stations with variable temperature and magnetic field are
commercially available. (Lake Shore Cryotronics, Inc. and Janis Research
Company for example.) However, for room temperature applications, it is
imperative to have the option of changing field direction with respect to the
sample. This is especially important because magnetic materials are generally
anisotropic and the effect of spin torque is angle dependent. Unfortunately,
only a few solutions for the field rotation probe station are available.
Technically, one needs to either rotate the magnet system or the sample. The
former is favorable when signal stability is a major concern. Homemade vector
magnets and projected field electromagnets (GMW associates) can provide a
rotating magnetic field, however, the field uniformity is poor and the field
strength is usually limited to a few kOe.(GMW, ) As detailed in a recent
design,(Li  _et al._ , 2012) extra care must be taken to calibrate the
magnets. Besides, close loop PID control is much more complicated for
applications require precise field values. A magnet with rotating base, on the
other hand, only enables the field rotation along a single axis. In the latter
case, the commercial products have adopted sample stage rotation design to
allow 360∘ of sample rotation within the film plane.(Lak, ; Jan, ) The
orientation of the probes and the corresponding positioners are fixed. For
devices working at low frequency ($<$1 GHz), the DC probes can always be
landed on the pads without any issue. However, in a high frequency ($\sim$10
GHz) measurement, impedance match is of primary importance. Consequently,
special probes with integrated tips such as ground-signal-ground (GSG) probes
and coplanar waveguide type device pads should be used.(Mellnik  _et al._ ,
2014) The misalignment between the probe and the pads of the device is usually
limited to 30∘ for reliable measurement.(Wang  _et al._ , 2014) Several
devices with identical pattern but different orientations must be measured to
obtain device parameters as a function of field direction in the range of 0 to
360∘. In reality, geometry as well as magnetic property variations among
sub-100 nm devices are inevitable. Furthermore, re-landing of the probes to
the device is necessary for each sample orientation, which greatly increases
the testing time, causes reliability issue and hinders fully automation.

Here, we present a compact and systematic solution for building a RF probe
station with relative field orientation capable of rotating from parallel to
perpendicular to the sample plane (out-of-plane (OP) rotation) as well as 360∘
rotation within the sample plane (in-plane (IP) rotation). The design is based
on the idea of rotating sample, probes and microwave cables simultaneously,
making the angular dependent measurement convenient and efficient. The test
platform is suitable for studying magnetic films and devices from DC to at
least 40 GHz. We also demonstrate examples of the capabilities of the
apparatus through magnetoresistance (MR) and angular magnetoresistance (AMR )
measurements on a film and two devices, the determination of domain wall
velocity and a test of spin-transfer-torque ferromagnetic resonance (STT-FMR).

##  II apparatus

###  II.1 Magnet system

Figure 1: (a) Illustration of the RF probe station with OP rotary stage
mounted. (b) Zoom in view of the OP rotary stage.

The electromagnet used for this probe station was purchased from Lake Shore
Cryotronics, Inc.. A pair of poles with 15.2 cm face diameter were selected.
The position of the poles is adjustable with a maximum gap of 17.8 cm. For a
pole gap of 10 cm, the field uniformity at the center is better than $\pm$0.5%
over a sphere of 2.5 cm diameter. The samples are placed approximately at the
center of the sample holder to further minimize the field discrepancy during
sample rotation. The magnet power supply with a current range of ±135 A is
controlled by Model 475 Gaussmeter. Magnetic field as large as 1 T can be
precisely applied. Our probe station described below is configured for this
magnet, however, the method can be applied to other commercially available
magnets as well.

###  II.2 Probe station with OP rotary stage

A schematic of the setup with the OP rotary stage installed is shown in Fig. 1
(a). The probe station consists of three main parts: the above mentioned
magnet, a vision system and a stage assembly driven by a motorized rotary
positioner. Auxiliary parts including supporting structures for the stage and
Hall sensor mounting kit are also necessary. The vision system rests on a XY
positioner in order to zoom in a device anywhere in a circle of 5 cm diameter.
The camera and the lens provide a maximum total magnification of 70. The heavy
duty rotary positioner (RM-5 Newmark systems, INC.) has a maximum output
torque of 10.2 N$\cdotp$m. One end of the stage was mounted on the positioner
and the other end with a guiding tube ran through a bearing. With this
configuration, the stage can be oriented with an accuracy of $0.02{}^{\circ}$.

Fig. 1 (b) is a closer view of the OP stage. The sample holder with a diameter
of 4 cm is equipped with a spring-loaded clip for easy and reliable sample
mounting as well as sample position adjustment. The springs are made of BeCu
to avoid magnetic impurities. Due to limited space, we used low profile non-
magnetic XYZ micro positioners (Quater Research and Development) which have a
travel distance of 0.75 cm along each axis. The positioners are partially
exposed to the magnetic field. Instead of using vacuum and magnetic mounting
base, our positioners were mounted on the stage through rails and locked by
set screws. This method improves the stability and allows an additional
position adjustment of 2 cm. Up to 4 positioners can be mounted to the stage,
which enables a standard four probe low frequency measurement. As can be seen
from Fig. 1 (b), probe adapters were used to secure the preferred RF probes
and the homemade multi-pin probe. Nonmagnetic flexible RF cables with
appropriate length were used. To minimize the vibration of probes due to cable
bending, flange connector assemblies and cable clamp at the edge of the stage
is crucial. Note that we have minimized the usage of magnetic components for
the stage.

Figure 2: Schematic of the probe station with IP rotary stage mounted.

The home position of the stage was set for the sample surface parallel to the
applied magnetic field. ($\theta_{H}$=0 as illustrated in Fig. 3 (a).) It is
necessary to land the probes on the pads (with the guidance of the vision
system) at the home position before performing an actual experiment.

###  II.3 IP rotary stage

The probe station with the IP rotary stage mounted is shown in Fig. 2. The
sample holder placed at the field center, the XYZ micro positioners mounted on
the stage and the probes are identical to the ones used in the previous OP
stage. The rotary positioner was mounted on a heavy base and the axis of
rotation (Z) was aligned to the field center. The circular shape rotary stage,
at a height convenient for operating the micro positioners, was mounted
through three stainless steel 316 tubes. The probe arm assemblies made of
Polyether ether ketone (PEEK) are of essential to reduce the pole gap. Here,
with two positioners mounted, the minimum pole gap is 7.5 cm in order to
perform an IP rotation. Whenever a RF probe is used, the other end of the RF
cable is connected to a RF flange connector fixed at the top of an Al tube.
Note that a right angle RF probe (with connector along Z axis which can
further reduce the pole gap) is more favorable than the 45 degree probe
illustrated in Fig. 2.

##  III Experimental test and applications

In this section, we present several sets of experimental data to demonstrate
the capability of the apparatus. The coordinate system of the probe station is
schematically shown in Fig. 3 (a) and (b). $\theta_{H}$ is the angle between
magnetic field and the sample film plane in a OP rotation whereas
$\varphi_{H}$ corresponds to the angle between the magnetic field and the
direction of current flow in an IP rotation.

Figure 3: Illustrations of the measurement configurations using the RF probe
station. The stage rotation is equivalent to a field rotation in the sample
coordinate system for OP stage (a) and IP stage (b). $\theta_{H}$ and
$\varphi_{H}$ are defined as the angle between field and the direction of
current flow. (c) The film level measurement using homemade multi-pin probe.
(d) GSG probe landed on a magnetic tunnel junction device. (e) GSG probe
landed on a coplanar waveguide which is connected to a stripe device.

To minimize cable bending in an IP rotation, the rotary stage was configured
to rotate between 0 and 180∘. The field polarity is reversed to cover
$\varphi_{H}$ in the range between 180 and 360∘. We assume that the
magnetization always follows the orientation of field
($\varphi_{H}=\varphi_{M}=\varphi$) in an IP rotation due to the small in
plane anisotropy of our samples. Fig. 3 (c) illustrates the film level
measurement using a homemade multi-pin probe. Five 45∘ Tungsten needles with
12 $\mu$m tip diameter were separated by 2 mm. For device level measurement,
various probe types such as GSG and ground-signal (GS) RF probes with proper
pitch size can be used. Fig. 3 (d) and (e) are two examples using a GSG probe
with 250 $\mu$m pitch size.

###  III.1 Microwave responses

Figure 4: Two ports scattering parameters with 0 dBm input for the OP rotary
stage in an external magnetic field of 2 kOe. (a) Photograph of the
measurement configuration with two GSG probes mounted. The inset is a
schematic of the probes and a calibration device. The two GSG probes were
separated by 500$\,\mu$m in a coplanar waveguide with nominal impedance of 50
ohm. (b) Scattering parameters as a function of frequency measured with
$\theta=6\mbox{0}^{\text{$\circ$}}$. (c) Normalized scattering parameters at
10 GHz as a function of $\theta$.

The high frequency responses of the stages including all flexible cables were
characterized by a vector network analyzer. The setup configuration of the
calibration is shown in Fig. 4 (a). Two-port scattering parameters (S11 and
S21 being the reflection coefficient and transmission coefficient,
respectively) of the OP rotary stage were measured with a 500 $\mu$m long
coplanar waveguide. The microwave power is 0 dBm. The magnetic field was kept
at 2 kOe. Panel (b) is a plot of S11 and S21 as a function of frequency. S21
at 40 GHz is about -12.6 dB which is suitable for techniques such as
ferromagnetic resonance(He and Panagopoulos, 2016) and pulse measurements for
most magnetic materials. The stage rotation changes the status of cable
bending, resulting in a variation in the S parameters. However, as can be seen
in Fig. 4 (c), the variation in S parameters between different orientations is
normally less than 1$\%$. The frequency was fixed at 10 GHz during angle
sweep. Similarly, for the IP rotary stage, S11 in ST-FMR configuration only
varies about 1$\%$ for $\varphi$ in the range between 0 and
180${}^{\text{$\circ$}}$ (not shown). Therefore, the rotating stage method
causes negligible influence on the data analysis relevant to measurements of
the angular dependence.

###  III.2 DC measurement on films and devices

For DC and low frequency measurements, a Keithley model 6221 current source is
used for current injection whereas either a Keithley model 2182A nanovoltmeter
or a Stanford Research model SR830 lock in amplifier is used to record the
voltage drop across the sample. Fig. 5 (b) is the magnetoresistance (MR) loop
taken from a film with stack structure NiTa(150)/Pt(10)/IrMn(8)/Co
(3)/Cu(3)/Co(0.5)/CoIr(3)/NiTa(5)/Pt(10) (numbers are nominal thicknesses in
nm). The setup configuration using the OP rotary stage is illustrated in Fig.
5 (a). The measurement also requires four adjacent tips of the multi-pin probe
as shown in Fig. 3 (c). The data was taken at $\theta$=0 with a DC current of
1 mA. We observe a sudden resistance jump close to zero field which
corresponds to the free layer switch and a broader jump around 500 Oe related
to the reference layer switch.(Baibich  _et al._ , 1988) A similar MR minor
loop on a CoFeB/MgO/CoFeB(Meng  _et al._ , 2012) based magnetic tunnel
junction (MTJ) device is shown in the inset of Fig. 5 (c). The loop was taken
at $\theta$=90∘ since the device has perpendicular magnetic anisotropy.(Ikeda
_et al._ , 2010; Sato  _et al._ , 2012) Resistance as a function of field
orientation $\theta$ of the MTJ is shown in Fig. 5 (c). The amplitude of the
field was 1.5 kOe. Note that the device was originally in the antiparallel
high resistance state at $\theta$=90∘. Both free layer and reference layer
were gradually tilted towards the film plane upon changing field orientation,
hence the observed decrease in the device resistance. Tilting a free layer is
necessary to optimize the performance of a device such as a spin torque
diode.(Miwa  _et al._ , 2014)

Anisotropy magnetoresistance (AMR) measurement with the IP rotary stage is
also easy to perform. Shown in Fig. 5 (e) is an example of a 60
$\mu$m$\times$30 $\mu$m wide stripe patterned on Py(6 nm)/Pt(10 nm) film. The
corresponding measurement configuration is schematically shown in Fig. 5 (d).
The field was kept at 1 kOe. Clear oscillation of the resistance as a function
of $\varphi$ was observed. The solid line is the calculated curve using
$R=43.41+0.195\cos^{2}\varphi$,(McGuire and Potter, 1975) where the numbers
are the fitting parameters. The fit gives a AMR ratio of
$[R(0^{\circ})-R(90^{\circ})]/R(90^{\circ})$=0.45%, smaller than in Py
films(Lee  _et al._ , 2000) due to the shunting effect of Pt. MR loops of the
same device at various field orientations are shown in Fig. 5 (f). Sweeping
the magnetic field leads to rotation of the magnetization, which gives rise to
the low-field MR and hysteresis. The features, however, are heavily dependent
on the field direction which is indicative of magnetic anisotropy,(Fan  _et
al._ , 2015) including the shape anisotropy of the rectangular stripe.

Figure 5: (a) Side view schematic of the sample orientation using OP rotary
stage. $\theta_{\textrm{H}}$ is defined as the relative angle between the
magnetic field and the sample film plane. (b) MR loop of a Co/Cu/CoIr based
film with spin valve structure. (c) Resistance as a function of $\theta$ for a
CoFeB/MgO/CoFeB based magnetic tunneling junction device. The field is 1.5
kOe. The device is initially at high resistance states. Inset shows the minor
loop of the device. Big and small arrows represent the magnetization
orientation of the reference and free layer, respectively. (d) Schematic view
of the sample orientation using IP rotary stage. $\varphi_{\textrm{H}}$ is
defined as the relative angle between the magnetic field and the direction of
current flow. (e) AMR of a Py/Pt stripe device measured with IP rotary stage.
The field is fixed at 1 kOe. Square and solid line are the data and the fit to
$R=R_{0}+C\cos^{2}\varphi$, respectively. (f) MR loops of a Py/Pt device
measured at selected field orientations. Curves are shifted up by 0.2 for
clarity.

###  III.3 Pulse measurement

Figure 6: Domain wall motion detection of a 300 nm$\times 60$$\mu$m nanowire
device patterned from a CoFe/Pd multilayer. The measurement was on the OP
rotary stage with $\theta=90^{\circ}$. (a) Hall voltage as a function of the
integrated pulse duration for selected driving current values. The inset is a
schematic of the device and multi-pin probes. (b) The dependence of the
averaged DW velocity on driving current. Solid line is a linear fit to the
data.

The bandwidth of the setup depends on the cables and probes being used. The RF
cables and probes are suitable for studying the time dependent response of a
magnetic device to pulses as short as 50 ps. On the other hand, using the
homebuilt multi-pin probes, measurements using pulses in the ~ns range are
also possible. A benefit of our design is that up to 10 signal connections are
available.

The inset in Fig. 6 (a) shows a schematic of a domain wall motion measurement
using pulse current.(Parkin  _et al._ , 2008) The stack structure of the 300
nm wide nanowire device is Si/ SiO2/Ta(3)/Pd(3)/[CoFe(0.16)/Pd(0.22)]8/Ta(3),
where thehe number in brackets is the thickness of the layers in nm. The
nanowire was modified with a triangular contact pad and one Hall cross for
magnetotransport measurements. In order to create a single DW in the nanowire,
the magnetization of the device was first saturated in one direction using a
large positive magnetic field (1 kOe, $\theta=90^{\circ}$). Next, a large
pulse current (50 mA, 50 ns) was applied through the Au electrode ( number 5
and 6) to generate a local Oersted field. The DW was then created and pinned
at the joint between the triangular contact pad and nanowire.

The DW motion in the nanowire was detected by means of the anomalous Hall
effect.(Koyama  _et al._ , 2011) CoFe/Pd multilayers exhibit strong
perpendicular anisotropy.(Meng  _et al._ , 2016) For relatively small magnetic
field values in this study, the Hall voltage is nearly proportional to the
magnetization at the Hall cross. Therefore, a sudden change in Hall voltage is
an indication of magnetization reversal due to domain wall motion. A
continuous sine-wave current was injected into the nanowire (electrodes 1 and
2) to measure the Hall resistance through pins 3 and 4 (inset of Fig. 6 (a)).
Note that the amplitude ( i${}_{\textrm{ac}}$=10 $\mu$A) is too small to drive
the DW motion. Two steps are repeated in the measurement. First, a 1 $\mu$s
wide pulse current with desired amplitude from the pulse generator was
injected into the nanowire to drive the DW. In a second step, the lock-in
amplifier, which was locked to the frequency of i${}_{\textrm{ac}}$ and with a
time constant of 0.1 s, was used to measure the Hall resistance
(R${}_{\textrm{Hall}}$). Fig. 6 (a) shows R${}_{\textrm{Hall}}$ as a function
of $t{}_{\textrm{int}}$ for three different driving currents. The sudden jump
in R${}_{\textrm{Hall}}$ is clearly visible , indicating that the DW have
passed through the Hall cross due to the spin polarized current. The DW
velocity is therefore defined as the distance from the electrode to the Hall
cross (30$\mu$m) divided by the integrated time ($t{}_{\textrm{int}}$) of
current pulses at which the jump was observed. The extracted averaged DW
velocity ($v$) as a function of driving current (I) is plotted in Fig. 6 (b).
Notably the DW velocity increases linearly with pulse current.

###  III.4 Spin transfer torque ferromagnetic resonance

Figure 7: ST-FMR results for a 30$\mu$m$\times$80$\mu$m Py (6 nm)/Pt(10 nm)
stripe taken at $\varphi=120^{\circ}$. (a) Mix voltage spectra for selected
frequencies. (b) Resonance frequency as a function of resonance field. (c) FMR
linewidth as a function of frequency. In all cases, the circles and the solid
lines correspond to the raw data and the fit, respectively.

We have also tested our setup with spin transfer torque ferromagnetic
resonance (ST-FMR) measurement.(Liu  _et al._ , 2011; Wang  _et al._ , 2014)
The device is the same Py(6 nm)/Pt(10 nm) stripe as for Fig. 5 (c) and (d).
The device, as a load, was integrated to a coplanar waveguide of nominal
impedance 50 Ohm as shown in Fig. 3 (e). A bias Tee was used to inject the RF
current to the device and measure the DC voltage generated in the device
simultaneously.(Liu  _et al._ , 2011) The microwave output power was kept at
10 dBm while $\varphi$=$120{}^{\circ}$. Shown in Fig. 7 (a) is the mixing
voltage(Liu  _et al._ , 2011) as a function of magnetic field for selected
microwave frequencies. Resonance peaks are clearly identified up to 30 GHz.
All curves are well fitted by the equation:(Liu  _et al._ , 2011)

$V_{\textrm{mix}}=V_{S}\cdot F_{{\rm sym}}+V_{A}\cdot F_{{\rm asym}}$ (1)

, where $F_{{\rm sym}}=(\frac{\Delta H}{2})^{2}/[(H-H_{{\rm
res}})^{2}+(\frac{\Delta H}{2})^{2}]$ and $F_{{\rm asym}}=\frac{\Delta
H}{2}(H-H_{{\rm res}})/[(H-H_{\textrm{res}})^{2}+(\frac{\Delta H}{2})^{2}]$
represent the symmetric and antisymmetric Lorentz component of the resonance,
respectively. $V_{S}$ and $V_{A}$ are the amplitudes of each component.
$H_{{\rm res}}$ is the resonance field and $\Delta H$ the full width at half
maximum (FWHM) of a resonance spectrum. The fitted parameters are useful for
determining important material parameters such as anisotropy and spin Hall
angle.(Mellnik  _et al._ , 2014; Wang  _et al._ , 2014) Shown in Fig. 7 (b) is
the Kittel fit to the FMR frequency as a function of resonance magnetic field.
For IP magnetic field,
$f=\frac{\gamma}{2\pi}\sqrt{H_{\textrm{res}}(H_{\textrm{res}}+4\pi\textrm{$M_{eff}$})}$,
where $\gamma$ is the gyromagnetic ratio. The effective magnetization
4$\pi\textrm{$M_{eff}$}$=0.82$\pm 0.01$T agrees well with the expected value
for Py.(Wang  _et al._ , 2014) Furthermore, the linewidth increases linearly
with applied microwave frequency as shown in Fig. 7 (c). The damping
$\alpha=0.018$ is obtained by fitting the data to $\Delta
H=\frac{4\pi}{\gamma}\alpha f+\Delta H_{0}$. The relatively large damping
coefficient compared to that measured in Permalloy film(Fuchs  _et al._ ,
2007; He and Panagopoulos, 2016) is attributed to the spin pumping effect due
to the thick Pt layer.(Tserkovnyak  _et al._ , 2002)

Figure 8: Angular dependence of the ST-FMR for a Py (6 nm)/Pt(10 nm) stripe
measured in the IP rotary stage. (a) The resonance field as a function of
$\varphi$ at $f=8\mbox{\,}$GHZ. The solid line is the calculated curve using
Eqn. 2 with $4\pi M_{{\rm eff}}$=0.84$\pm 0.01$ T and $H_{2\parallel}=79\pm 2$
Oe. (b) Symmetric and antisymmetric component of ST-FMR spectra as a function
of $\varphi$. Solid lines are the fit to a
$\textrm{cos}{}^{2}\varphi\textrm{sin}\varphi$ function.

Compared to the measurement at a fixed angle, we are more interested in the
angular dependence of the ST-FMR spectrum. Fig. 8 shows the IP angular
dependence of the resonance field and resonance amplitudes measured at 8 GHz.
The data points are extracted by fitting 72 spectra collected at a list of
field orientations. For this particular sample, besides the perpendicular
uniaxial anisotropy which is included in $4\pi M_{{\rm eff}}$, an additional
in-plane uniaxial anisotropy (effective field $H_{2\parallel}$) with easy axis
along the long-side is assumed. Using the coordinates defined in Fig. 3, the
resonance condition for any given IP field orientation can be
derived:(Maksymowicz and Leaver, 1973; Liu  _et al._ , 2003)

$2\pi f=\gamma\sqrt{(H_{{\rm res}}+4\pi M_{{\rm
eff}}+H_{2\parallel}\cos^{2}\varphi_{M})(H_{{\rm res}}+H_{2\parallel}\cos
2\varphi_{M})}$ (2)

Thus, $4\pi M_{{\rm eff}}$=0.84$\pm 0.01$T and $H_{2\parallel}$=62$\pm$2 Oe .
Py/Pt is a well investigated bilayer for the spin Hall effect (SHE).(Sinova
_et al._ , 2015) The antisymmetric and symmetric voltage components are
attributed to field-like torque and spin torque generated by the SHE,
respectively. The output DC voltages arise from the combined effect of AMR and
magnetization precession driven by these torques.(Azevedo  _et al._ , 2011) A
$\textrm{cos}{}^{2}\varphi\textrm{sin}\varphi$ relation is expected if no
other effect is involved.(Harder  _et al._ , 2011; Skinner  _et al._ , 2014;
Mecking  _et al._ , 2007) As can be seen in Fig. 8 (b), both components can be
fitted well by this relation, adding credence to the suitability of the IP
rotary stage for angular dependent studies. Note that the measurement only
takes 4 h with a single program sequence. In contrast, using probe stations
equipped with rotary sample stages, each data point shown in Fig. 8 requires a
separate probe landing and stage rotating procedures. It can take several days
to obtain the same curve. Therefore, the method of rotating sample and probes
simultaneously is much more efficient.

##  IV conclusion

We have developed a probe station to study magnetic films and devices with
controllable field orientation. The sample holder, probe assembly and
microwave cables are rotated simultaneously with two separate motorized
stages. The sample rotation is equivalent to a field rotation which covers
$\theta$ from 0 to $90^{\circ}$ and $\varphi$ from 0 to $360^{\circ}$. We have
also shown novel designs of a vision system assembly, sample holder with
spring-loaded clip, micro positioner mounting and multi-pin probe.
Furthermore, a range of DC and RF measurement techniques incorporated to this
probe station were employed to demonstrate the versatility of this probe
station. The angular-resolved RF probe station is useful for determining
magnetic anisotropy, evaluating device performance and addressing the physics
of spin torque.

###### Acknowledgements.

The authors are grateful to Yi Wang, Sze Ter Lim, Cheow Hin Sim and Ruisheng
Liu for fruitful discussions. We thank HongJing Chung for preparing films. We
acknowledge Singapore Ministry of Education (MOE), Academic Research Fund Tier
2 (Reference No: MOE2014-T2-1-050) and National Research Foundation (NRF) of
Singapore, NRF-Investigatorship (Reference No: NRF-NRFI2015-04) for the
funding of this research.

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# Kervaire conjecture on weight of group via fundamental group of ribbon
sphere-link

Akio KAWAUCHI
Osaka central Advanced Mathematical Institute, Osaka Municipal University
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
<EMAIL_ADDRESS>
( )

###### ABSTRACT

Kervaire conjecture that the weight of the free product of every non-trivial
group and the infinite cyclic group is not one is confirmed by confirming
Conjecture Z of a knot exterior.

x

Keywords: Weight , Kervaire conjecture , Conjecture Z, Whitehead aspherical
conjecture, Ribbon sphere-link,

Mathematics Subject classification 2020: 20F06, 57M05

## 1\. Introduction

A weight system of a group $G$ is a system of elements
$w_{i},\,(i=1,2,\dots,n)$ of $G$ such that the normal closure of the elements
$w_{i},\,(i=1,2,\dots,n)$ (=: the smallest normal subgroup generated by
$w_{i},\,(i=1,2,\dots,n)$) is equal to $G$. The weight of a group $G$ is the
least cardinal number of a weight system of $G$. Kervaire’s conjecture on the
weight of a group is the following conjecture (see Kervaire [6], Magnus-
Karrass-Solitar [8, p. 403]):

x

Kervaire Conjecture. The free product $G*{\mathbf{Z}}$ of every non-trivial
group $G$ and the infinite cyclic group ${\mathbf{Z}}$ is not of weight one.

x

Some partial confirmations of this conjecture are known. In this paper, the
result of Klyachko [7] that $G*{\mathbf{Z}}$ is not of weight one for every
non-trivial torsion-free group $G$ is used. González-Acuña and Ramírez in [2]
showed that it is sufficient to confirm this conjecture for every non-trivial
finitely presented group, and further showed that this conjecture is
equivalent to it Conjecture Z of a knot exterior. To explain this conjecture,
some terminologies are fixed. A knot exterior is a compact 3-manifold
$E=\mbox{cl}(S^{3}\setminus N(k))$ for a tubular neighborhood $N(k)$ of a
polygonal knot $k$ in the 3-sphere $S^{3}$. Let $F$be a compact connected
orientable non-separating proper surface of $E$. Let
$E(F)=\mbox{cl}(E\setminus F\times I)$ be the compact piecewise-linear
3-manifold for a normal line bundle $F\times I$ of $F$ in $E(F)$ where
$I=[-1,1]$. Let $\hat{E}(F)$ be the 3-complex obtained from $E(F)$ by adding
the cone $\mbox{Cone}(v,F\times\partial I)$ over the base $F\times\partial I$
with vertex $v$. The 3-complex $\hat{E}(F)$ is also obtained from $E$ by
shrinking the normal line bundle $F\times I$ into the vertex $v$.

x

Conjecture Z. The fundamental group $\pi_{1}(\hat{E}(F),v)$ is always
isomorphic to ${\mathbf{Z}}$.

x

See [1, 2, 9] for some investigations of this surface $F$ and some partial
confirmations. In this paper, Kervaire conjecture is confirmed by confirming
this Conjecture Z.

x

Theorem. Conjecture Z is true. .

x

The proof of Theorem is done by showing that the fundamental group
$\pi_{1}(\hat{E}(F),v)$ is always torsion-free, which comes from torsion-
freeness of the fundamental group of a ribbon $S^{2}$-link claimed in [4].
Conceptially, it can be understood that this result comes from Whitehead
aspherical conjecture saying that every connected subcomplex of every
aspherical 2-complex is aspherical (see [4, 5]). Then
$\pi_{1}(\hat{E}(F),v)\cong{\mathbf{Z}}$ is concluded by the result of [7].

In the first draft of this research, the author tried to show that every
finitely presented group $G$ with free product group $G*{\mathbf{Z}}$ of
weight one is torsion-free. This trial succeeds for a group $G$ of deficiency
$0$, but failed for a group $G$ of negative deficiency. The main point of this
failure is the attempt to construct a finitely presented group of deficiency
$0$ from the group of negative deficiency, which forced the author to show
that $G$ is torsion-free while the deficiency remains negative. Fortunately,
in Conjecture Z, the 3-complex $\hat{E}(F)$ which is the object to be
considered is simple, so it could be done.

The proof of Theorem is done in Section 2.

x

## 2\. Proof of Theorem

The proof of Theorem is done as follows, but the main point of the proof is
how a finite contractible 2-complex is constructed to contain, as a
subcomplex, a 2-complex which is homotopy equivalent to the 3-complex
$\hat{E}^{\prime}$.

x

Proof of Theorem. Let $\hat{E}(F)^{\prime}$ be the 3-complex obtained from
$E(F)$ by shrinking $F\times 1$ and $F\times(-1)$ into distinct vertexes
$v_{+}$ and $v_{-}$, respectively. The 3-complex $\hat{E}(F)$ is homotopy
equivalent to a bouquet $X=\hat{E}(F)^{\prime}\vee S^{1}$ of
$\hat{E}(F)^{\prime}$ and the circle $S^{1}$. Let $Y$ be the 3-complex
obtained from $X$ by attaching of a 2-cell arising from attaching a meridian
disk to $E$. Then $\pi_{1}(Y,v)$ is obtained from
$\pi_{1}(X,v)=\pi_{1}(\hat{E},v)*{\mathbf{Z}}$ by adding a relation and
$\pi_{1}(Y,v)=\\{1\\}$. If $\pi_{1}(\hat{E}(F),v)$ is torsion-free, then
$\pi_{1}(\hat{E}(F)^{\prime},v)$ is torsion-free. By the result of [7],
$\pi_{1}(\hat{E}(F)^{\prime},v)=\\{1\\}$ and
$\pi_{1}(\hat{E}(F),v)\cong{\mathbf{Z}}$. To complete the proof, it sufficies
to show that $\pi_{1}(\hat{E}(F),v)$ is torsion-free, Collapse $F$ into a
triangulated graph $\gamma$ by using that $F$ is a bounded surface. Enlarge
the fiber $I$ of a normal line bundle $F\times I$ of $F$ in $E$ into a fiber
$\bar{I}$ of a normal line bundle $F\times\bar{I}$ of $F$ in $E$ so that
$I\subset\bar{I}\setminus\partial\bar{I}$. Let
$\bar{I}^{c}=\mbox{cl}(\bar{I}\setminus I)$. Let
$E(F)^{-}=\mbox{cl}(E\setminus F\times\bar{I})$. Collapse $F\times\bar{I}^{c}$
into $\gamma\times\bar{I}^{c}$. Triangulate $\gamma\times\bar{I}^{c}$ without
introducing new vertexes. The 3-complex $\hat{E}(F)$ is collapsed into a
finite 3-complex

$E(F)^{-}\cup\gamma\times\bar{I}^{c}\cup\mbox{Cone}(v,\gamma\times\partial I)$

and thus collapsed into a finite 2-complex

$P=P^{-}\cup\gamma\times\bar{I}^{c}\cup\mbox{Cone}(v,\gamma\times\partial I)$

for any 2-complex $P^{-}$ collapsed from $E(F)^{-}$. This 2-complex $P$ is a
subcomplex of a 3-complex

$Q=\mbox{Cone}(v,P^{-}\cup\gamma\times\bar{I}^{c}).$

Since every 2-complex of $\gamma\times\bar{I}^{c}$ contains at most one
1-simplex of $\gamma\times\partial I$, every 3-simplex of
$\mbox{Cone}(v,\gamma\times\bar{I}^{c})$ contains at most one 2-simplex of
$\mbox{Cone}(v,\gamma\times\partial I)$. Collapse every 3-simplex of
$\mbox{Cone}(v,\gamma\times\bar{I}^{c})$ from a 2-face containing $v$ and not
belonging to $\mbox{Cone}(v,\gamma\times\partial I)$. Then collapse every
3-simplex of $\mbox{Cone}(v,P^{-})$ from any 2-face containing $v$. Thus, the
3-complex $Q$ is collapsed into a finite 2-complex $C$ containing the
2-complex $P$ as a subcomplex. Since $Q$ is contractible, $C$ is a finite
contractible 2-complex. By the argument of [4], the group $\pi_{1}(P,v)$ is a
torsion-free group, actually a ribbon $S^{2}$-knot group for
$H_{1}(P;{\mathbf{Z}})\cong{\mathbf{Z}}$. Thus, the group
$\pi_{1}(\hat{E}(F),v)$ is torsion-free since $\pi_{1}(\hat{E}(F),v)$ is
isomorphic to $\pi_{1}(P,v)$. This completes the proof of Theorem. $\square$

x

The result that the fundamental group of a ribbon $S^{2}$-link is torsion-free
can be shown without use of the smooth 4D Poincaré conjecture and the smooth
unknotting conjecture for an $S^{2}$-link as it is observed in [5], although
the proof in [4] uses them. On the other hand, the result of [3, Lemma 3.4] is
still basic to this proof.

x

## Acknowledgements

This work was partly supported by JSPS KAKENHI Grant Numbers JP19H01788,
JP21H00978 and Osaka central Advanced Mathematical Institute, Osaka
Metropolitan University (MEXT Joint Usage/ Research Center on Mathematics and
Theoretical Physics JPMXP 0619217849).

## References

  * [1] M. Eudave-Muñoz, On knots with icon surfaces, Osaka Journal of Mathematics, 50 (2013), 271–285.
  * [2] F. González-Acuña and A. Ramírez, A knot-theoretic equivalent of the Kervaire conjecture, J. Knot Theory Ramifications, 15 (2006), 471–478.
  * [3] A. Kawauchi, Classical Poincaré Conjecture via 4D topology, Available from: https://sites.google.com/view/ kawauchiwriting
  * [4] A. Kawauchi, Ribbonness of Kervaire’s sphere-link in homotopy 4-sphere and its consequences to 2-complexes. Available from: https://sites.google.com/view/ kawauchiwriting
  * [5] A. Kawauchi, Whitehead aspherical conjecture via ribbon sphere-link. Available from: https://sites.google.com/view/ kawauchiwriting
  * [6] M. A. Kervaire, On higher dimensional knots, in: Differential and combinatorial topology, Princeton Math. Ser. 27 (1965), 105-119, Princeton Univ. Press.
  * [7] Ant. A. Klyachko, A funny property of a sphere and equations over groups, Comm. Algebra 21 (1993), 2555–2575.
  * [8] W. Magunus, A. Karrass and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publisheres (1966).
  * [9] J. Rodríguez-Viorato and F. González-Acuña, On pretzel knots and Conjecture Z, Journal of Knot Theory and Its Ramifications, 25 (2016) 1650012.


# The $D_{s}$, $D^{+}$, $B_{s}$ and $B$ decay constants from $2+1$ flavor
lattice QCD

A. Bazavova, C. Bernardb, C. Bouchardc, C. DeTard, D. Due, A.X. El-Khadrae, J.
Foleyd, E.D. Freelandf, E. Gámizg, Steven Gottliebh, U.M. Helleri, J. Kimj, J.
Komijanib, A.S. Kronfeldk, J. Laihol,m, L. Levkovad, P.B. Mackenziek, D.
Mohlerk, E.T. Neilk,n,o,α, M.B. Oktayd, S. Qiud, ,k,β, R.L. Sugarp, D.
Toussaintj, R.S. Van de Waterk, and R. Zhouh,k
aPhysics Department, Brookhaven National Laboratory, Upton NY 11973, USA
bDepartment of Physics, Washington University, St. Louis, MO 63130, USA
cDepartment of Physics, The Ohio State University, Columbus, OH 43210, USA
dPhysics Department, University of Utah, Salt Lake City, UT 84112, USA
ePhysics Department, University of Illinois, Urbana, IL 61801, USA
fLiberal Arts Department, School of the Art Institute of Chicago, Chicago, IL
60603, USA
gCAFPE and Depto. de Física Teórica y del Cosmos, Universidad de Granada,
Granada, España
hDepartment of Physics, Indiana University, Bloomington, IN 47405, USA
iAmerican Physical Society, One Research Road, Ridge, NY 11961, USA
jDepartment of Physics, University of Arizona, Tucson, AZ 85721, USA
kFermi National Accelerator Laboratory, Batavia, IL 60510, USA
lSUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12
8QQ, UK
mDepartment of Physics, Syracuse University, Syracuse, NY 13244, USA
nDepartment of Physics, University of Colorado, Boulder, CO 80309, USA
oRIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973,
USA
pDepartment of Physics, University of California, Santa Barbara, CA 93106, USA
Email: α β Fermilab Lattice and MILC Collaborations

###### Abstract:

We present a study of the $D$ and $B$ leptonic decay constants on the MILC
$N_{f}=2+1$ asqtad gauge ensembles using asqtad-improved staggered light
quarks and clover heavy quarks in the Fermilab interpretation. Our previous
analysis [1] computed the decay constants at lattice spacings $a\approx
0.14,0.11$ and $0.083\,\textrm{fm}$. We have extended the simulations to finer
$a\approx 0.058$ and $0.043\,\textrm{fm}$ lattice spacings, and have also
increased statistics; this allows us to address many important sources of
uncertainty. Technical advances include a two-step two-point fit procedure,
better tuning of the heavy quark masses and a better determination of the
axial-vector current matching. The present analysis remains blinded, so here
we focus on the improvements and their predicted impact on the error budget
compared to the prior analysis.

##  1 Introduction

Decays of $B$ and $D$ mesons provide an important testing ground for the CKM
paradigm for flavor-changing weak interactions in the Standard Model. The $D$
and $B$ decay constants, which encapsulate the role of QCD interactions in
these decay processes, are a crucial theoretical input. Precise calculation of
these strong-coupling quantities, enabled by lattice simulations, is very
important in predicting experimental rates for rare decays, such as the
process $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ observed recently by LHCb and CMS [2],
where the $B$-meson decay constants enter into the Standard Model rates. In
addition, precise knowledge of the $B^{+}$-decay constant in combination with
the observed $B^{+}\to\tau^{+}\nu$ decay rate probes the $V$-$A$ structure of
the $Wub$ vertex and helps in understanding the tension between inclusive and
exclusive determinations of $|V_{ub}|$.

id |  $\mathbf{a}$ [fm] | beta | $\mathbf{m_{l}/m_{h}}$ | $\mathbf{am_{h}}$ | $\mathbf{m_{h}/m_{s}}$ | $\mathbf{r_{1}/a}$ | $\mathbf{N_{\mathit{config}}}$ | $\mathbf{N_{\mathit{tsrc}}}$
---|---|---|---|---|---|---|---|---
A | 0.043 | 7.81 | 0.2 | 0.014 | 1.079 | 7.208 | 801 | 4
B | 0.059 | 7.46 | 0.1 | 0.018 | 1.019 | 5.307 | 827 | 4
C | 0.058 | 7.465 | 0.139 | 0.018 | 1.024 | 5.330 | 801 | 4
D | 0.058 | 7.47 | 0.2 | 0.018 | 1.028 | 5.353 | 673 | 8
E | 0.058 | 7.48 | 0.4 | 0.018 | 1.037 | 5.399 | 593 | 4
F | 0.083 | 7.075 | 0.05 | 0.031 | 1.255 | 3.738 | 791 | 4
G | 0.083 | 7.08 | 0.1 | 0.031 | 1.256 | 3.755 | 1015 | 4
H | 0.083 | 7.085 | 0.15 | 0.031 | 1.262 | 3.772 | 984 | 4
I | 0.082 | 7.09 | 0.2 | 0.031 | 1.267 | 3.789 | 1931 | 4
J | 0.081 | 7.11 | 0.4 | 0.031 | 1.290 | 3.858 | 1996 | 4
K | 0.11 | 6.76 | 0.1 | 0.05 | 1.489 | 2.739 | 2099 | 4
L | 0.11 | 6.76 | 0.14 | 0.05 | 1.489 | 2.739 | 2110 | 4
M | 0.11 | 6.76 | 0.2 | 0.05 | 1.489 | 2.739 | 2259 | 4
N | 0.11 | 6.79 | 0.4 | 0.05 | 1.534 | 2.821 | 2052 | 4
O | 0.14 | 6.572 | 0.2 | 0.0484 | 1.156 | 2.222 | 631 | 24
Table 1: MILC asqtad ensembles and parameters.

This study uses simulations on the ensembles listed in Table 1. Since Ref. [1]
we have added ensembles labeled A through E, at two finer lattice spacings,
and ensemble F at a sea-quark mass $m_{l}=0.05m_{h}$, nearer the physical
mass. Statistical accuracy is also better, with about $3.6\times$ more
$N_{\mathit{config}}\cdot N_{\mathit{tsrc}}$ combinations than in Ref. [1].
Better statistics, finer lattice spacings and a nearly physical sea quark mass
all help to control the leading systematic uncertainties observed in Ref. [1]
that arise from heavy- and light-quark discretization effects and the (chiral)
extrapolation. This analysis also benefits from reduced uncertainty from the
input charm and bottom quark masses, due to a retuning of the masses with
improved techniques and higher statistics, and a reanalysis of the
nonperturbative matching of the flavor conserving heavy and light vector
currents, again with higher statistics.

##  2 Two-point fits

We use a two-stage procedure for performing the two-point fits. In the first
stage, plots of the effective mass are inspected for a stable “plateau” at
large values of the source-sink separation $t$. A range
$[t_{p,{min}},t_{p,{max}}]$ is then chosen based on the correlator signal-to-
noise ratio (SNR), maintaining SNR $\geq 10$ for all correlators and holding
the SNR range approximately fixed over different ensembles; this translates to
fits at approximately equal physical distances on the different lattice
spacings. The first-stage fit is carried out to a standard two-state
functional form (one oscillating), with no excited states included.

For the second stage, the results of the first-stage fit are used to set
empirical Bayesian priors; best-fit values give the prior means, and the width
is set equal to the best-fit one-sigma error estimate times an inflation
factor of $3$, to ensure that the second-stage fit parameters are not
overconstrained. We have tested that increasing the inflation factor beyond
this point has negligible effect on the resulting final classical error
estimates; however, the use of this two-stage procedure serves to stabilize
the fits performed over bootstrap resampled data, by reducing the occurrence
of outliers. We fit a basis of four (or five) smeared and local source two-
point correlators including the two correlators having an
$\mathcal{O}(a)$-improved axial-current at the sink in stage two. We find good
isolation of the ground state when including four to five states (plus an
equal number of oscillating states) and fitting down to $t_{{min}}=2$.

##  3 Chiral fits

The decay constant $f_{H_{q}}$ for a meson $H_{q}$ is related to
$\phi_{H_{q}}=f_{H_{q}}\sqrt{M_{H_{q}}}$, where

$r_{1}^{3/2}\phi_{H_{q}}=\left(\frac{r_{1}}{a}\right)^{3/2}\sqrt{Z_{V^{4}_{QQ}}\,Z_{V^{4}_{qq}}}\;\rho_{A^{4}_{Qq}}\;\;a^{3/2}\phi_{H_{q}}^{\mathit{lat}}\quad.$
(1)

Values for $r_{1}/a$ are shown in Table 1. The flavor-conserving vector-
current renormalization factors $Z_{V^{4}_{QQ}}$ and $Z_{V^{4}_{qq}}$ are
found nonperturbatively, while $\rho_{A^{4}_{Qq}}=1+\mathcal{O}(\alpha_{V})$
is known to one-loop order and is near unity. The
$a^{3/2}\phi_{H_{q}}^{\mathit{lat}}$ are determined by fitting two-point
functions.

Guided by heavy meson staggered chiral perturbation theory [3], we fit to the
function

$\phi_{H_{q}}\left(m_{q},m_{h},m_{l}\right)=\Phi_{H}\left[1+\Delta
f_{H_{q}}\left(m_{q},m_{h},m_{l}\right)+P\left(m_{q},m_{h},m_{l}\right)+K\left(am_{Q}\right)+c_{a}a^{2}\right]\qquad.$
(2)

Term $\Delta f_{H_{q}}$, parameterizing the NLO chiral logarithms, includes
hyperfine splitting effects. They are corrected for staggered taste effects at
finite lattice spacing and for finite volume. The polynomial $P$ includes
analytic terms up to second order in the quark masses. The $K$ terms, which
parametrize the leading-order heavy quark discretization effects, are
constrained in fits according to power counting estimates [1]. The residual
heavy-quark discretization error for the physical $\phi_{H_{q}}$ is then
incorporated into the overall statistical error.

Figure 1: Fit of all $D$ system partially quenched data to the $S\raise
1.72218pt\hbox{$\chi$}PT$ model function. Figure 2: Fit of all $B$ system
partially quenched data to the $S\raise 1.72218pt\hbox{$\chi$}PT$ model
function.

Figure 1 shows a preliminary fit for the model in Equation (2) with all the
$D$-meson $\phi_{H_{q}}$ simulation results at the five lattice spacings
listed in Table 1. Figure 2 is the corresponding plot for the $B$-meson
system. The figures indicate that the model adequately represents the
simulation results. Points differing only by valence light-quark mass are
shown in the individual plot subpanels. A reasonable fit is obtained even when
the fit curve may visibly deviate from simulation results on a particular
ensemble since points there are strongly correlated.

---
Figure 3: Combined chiral continuum extrapolation for the $B$ (top) and $D$
(bottom) systems. The curves and statistical error bands in the $a\to 0$ limit
at physical quark masses are labeled “a0.00”. The extrapolated values (and
errors) are the points labeled by triangles at the physical pion mass. Note:
these results are blinded by normalization factors known only to a few
collaboration members, who are outside the analysis group. Note: the subset of
points shown for each of the $B_{s}$ and $D_{s}$ extrapolations serve merely
as a guide since their valence masses only approximate the physical strange
quark mass: $r_{1}|m_{q}-m_{s}|<0.07$.

Figure 3 shows the combined continuum and chiral extrapolation curve (and
error band) for the $D$\- and $B$-meson systems. Shown overlaying each of the
extrapolated $B$ and $D$ fit curves are the (finite $a$) “full QCD” points
where $m_{q}=m_{l}$. No points correspond exactly to the tuned strange quark
mass, consequently, we show a subset of points with roughly $m_{q}\approx
m_{s}$ overlaying each of the $B_{s}$ and $D_{s}$ curves.

##  4 Predicted errors and outlook

Our results remain blinded, hence, we do not quote values for the decay
constants here. We will continue to use the blinded analysis to understand
systematic effects. In Figure 4 we compare predicted errors in this study to
Ref. [1]. This analysis has higher statistics and includes results at finer
$a\approx 0.058$ and $0.043\,\textrm{fm}$ lattice spacings leading to improved
estimates for discretization effects modeled in the chiral fit function. We
anticipate this will lead to a reduction in the residual discretization errors
for the physical decay constants. The addition of results with
$m_{l}=0.05m_{h}$, nearer to the physical quark (ensemble F in Table 1),
narrows the extent of the chiral extrapolation, which is expected to reduce
the residual chiral extrapolation uncertainty.

Figure 4: The relative contributions to the error in quadrature from sources
of uncertainty. On the left a summary of our previous analysis [1]. On the
right are our predictions for this analysis.

The current study includes several technical improvements compared to Ref.
[1]. We have introduced a two-step procedure for two-point fitting which
allows us to stably model more excited states and, hence, better utilize two-
point data at small times where the signal-to-noise ratio is larger. This fit
procedure also better preserves expected (significant) correlations among
points by reducing the likelihood of finding outliers in output bootstrap
distributions that tend to wash out correlations. The new procedure and better
statistics help to directly reduce statistical errors as well as the “2-pt
fit” uncertainty previously estimated from plausible variations in fitting
procedures. We have reduced the error in the decay constants due to the input
heavy quark mass through improvements to the tuning process used to determine
the charm and bottom quark masses: a) We have four times the statistics than
in prior tuning runs. b) We employ priors in energy-momentum dispersion
relation fits which help stabilize the kinetic masses that are matched to the
physical $D_{s}$ and $B_{s}$ masses. c) We compensate for mistuning of the
strange sea-quark masses in simulations. d) We smoothly extend the charm and
bottom tunings from the subset of ensembles used for tuning to all other
ensembles. Uncertainties in the decay constants due to the flavor-conserving
matching factors $Z_{V^{4}}$ for both clover and staggered currents have been
reduced by new determinations using better stochastic color wall meson sources
and higher statistics.

From Figure 4, we anticipate that the total error in quadrature for the decay
constants in this study will be around half of the error found in Ref. [1]. We
are continuing to refine this analysis before unblinding the values for the
decay constants.

Acknowledgments
This work was supported by the U.S. Department of Energy, the National Science
Foundation, and the URA Visiting Scholars’ Program. Fermilab and BNL are
operated under contracts De-AC02-07CH11359 and DE-AC02-98CH10886 respectively,
with the DOE. Computations were carried out at the Argonne Leadership
Computing Facility, the National Center for Atmospheric Research, the National
Center for Supercomputing Resources, the National Energy Resources
Supercomputing Center, the National Institute for Computational Sciences, the
Texas Advanced Computing Center, and the USQCD facilities at Fermilab, under
grants from the NSF and DOE.

## References

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  * [2] CMS and LHCb Collaborations, Combination of results on the rare decays $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ from the CMS and LHCb experiments, Tech. Rep. CMS-PAS-BPH-13-007. CERN-LHCb-CONF-2013-012.
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