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# Time-resolved imaging of Œrsted field induced magnetization dynamics in
cylindrical magnetic nanowires

M. Schöbitz<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS, CEA, Spintec,
38054 Grenoble, France  Friedrich-Alexander Univ. Erlangen-Nürnberg, Inorganic
Chemistry, 91058 Erlangen, Germany  Univ. Grenoble Alpes, CNRS, Institut Néel,
38042 Grenoble, France  S. Finizio  Swiss Light Source, Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  A. De Riz  Univ. Grenoble Alpes,
CNRS, CEA, Spintec, 38054 Grenoble, France  J. Hurst  Univ. Grenoble Alpes,
CNRS, CEA, Spintec, 38054 Grenoble, France  C. Thirion  Univ. Grenoble Alpes,
CNRS, Institut Néel, 38042 Grenoble, France  D. Gusakova  Univ. Grenoble
Alpes, CNRS, CEA, Spintec, 38054 Grenoble, France  J.-C. Toussaint  Univ.
Grenoble Alpes, CNRS, Institut Néel, 38042 Grenoble, France  J. Bachmann
Friedrich-Alexander Univ. Erlangen-Nürnberg, Inorganic Chemistry, 91058
Erlangen, Germany  Institute of Chemistry, Saint-Petersburg State Univ.,
198504 St. Petersburg, Russia  J. Raabe  Swiss Light Source, Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  O. Fruchart<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS, CEA, Spintec, 38054 Grenoble, France

###### Abstract

Recent studies in three dimensional spintronics propose that the Œrsted field
plays a significant role in cylindrical nanowires. However, there is no direct
report of its impact on magnetic textures. Here, we use time-resolved scanning
transmission X-ray microscopy to image the dynamic response of magnetization
in cylindrical Co$30$Ni$70$ nanowires subjected to nanosecond Œrsted field
pulses. We observe the tilting of longitudinally magnetized domains towards
the azimuthal Œrsted field direction and create a robust model to reproduce
the differential magnetic contrasts and extract the angle of tilt. Further, we
report the compression and expansion, or breathing, of a Bloch-point domain
wall that occurs when weak pulses with opposite sign are applied. We expect
that this work lays the foundation for and provides an incentive to further
studying complex and fascinating magnetization dynamics in nanowires,
especially the predicted ultra-fast domain wall motion and associated spin
wave emissions.

Continuous developments in time-resolved magnetic imaging techniques have
allowed for a shift of interest from systems that lend themselves more readily
to imaging, such as flat nanostrips1, 2, 3, to more intricate systems such as
three dimensional (3D) nanostructures, with added complexity from the volume4,
5. Such 3D nanostructures can now be fabricated with increasing ease6, 7, 8,
making the exploration of intriguing predicted magnetic configurations9, such
as domain walls (DWs) in Möbius strips or hopfions10, feasible. A textbook
case for such an investigation is provided by cylindrical magnetic nanowires
(NWs), featuring a special type of DW, the Bloch-point wall (BPW)11, which
exhibits an azimuthal curling of magnetic moments around a Bloch-point on the
NW axis12, 13. The dynamics of these walls are not yet well understood, but
stand out compared to DWs in flat nanostrips due to fascinating theoretical
predictions of fast, stable speeds14, accompanied by the controlled emission
of spin waves15. Recent experiments in NWs have shown that the Œrsted field
induced by nanosecond current pulses plays a key role in stabilising walls
exclusively of the BPW type, and further, imposes an azimuthal circulation
parallel to the field16. This allows controlling the wall structure and
enables fast DW motion with speeds
$>{\text{\unboldmath$\mathrm{600}$}}\,\text{\unboldmath$\mathrm{m/s}$}$ with
an absence of Walker breakdown. The Œrsted field induced BPW circulation
switching was further studied in a simulation and theory work, revealing a
complex mechanism of the switching process, involving nucleation and
annihilation of pairs of vortex and anti-vortex17. Further, magnetic moments
in longitudinally magnetized domains are predicted to align azimuthally with
the Œrsted field, with the degree of tilt related to a competition between
magnetic exchange and Zeeman energy. Although some works have investigated the
Œrsted field in nanowires18, 19, 20, its major influence on magnetization
dynamics in these systems has become clear rather recently and is especially
pertinent in a low current density regime where the effect of spin-transfer
torque is negligible21. However, few of the predictions have been confirmed
experimentally.

Here, we make use of time-resolved scanning transmission X-ray microscopy
(STXM) to image magnetization dynamics in NWs subjected to nanosecond pulses
of Œrsted field. Magnetically-soft Co$30$Ni$70$ NWs with diameters of $93$,
$97$ and $\mathrm{101}$ $\mathrm{nm}$ were electrodeposited in anodized
alumina templates and freed by dissolution of the template6. Wires were
dispersed onto $\mathrm{200}$ $\mathrm{nm}$ thick,
$100{\text{\unboldmath$\mathrm{\times
100}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolMicro}m^{2}}$}$ wide
X-ray transparent Si3N4 windows, suspended in a $\mathrm{5\times 5}$
$\mathrm{mm^{2}}$ intrinsic Si frame. Individual wires were lithographically
contacted with Au pads to allow for the injection of nanosecond pulses of
electric current, in turn creating an Œrsted field around the NW, as in Fig.
1a. Magnetic images were acquired using STXM at the PolLux bending magnet
beamline at the Swiss Light Source22. The sample was tilted by $\mathrm{30}$
$\mathrm{\mathrm{\SIUnitSymbolDegree}}$ with respect to the X-ray beam
direction and aligned so that the NW was oriented to be as parallel as
possible to the beam direction. Optical sensitivity to magnetization was
achieved due to the X-ray magnetic circular dichroism (XMCD) effect, whereby
circularly-polarized X-ray light is absorbed differently depending on whether
the magnetization is (anti-)parallel to the photon propagation direction.
Magnetization dynamics were observed with time-resolved STXM as shown in Fig.
1a, making use of the intrinsically pulsed nature of synchrotron radiation
(purple) and phase locking their frequency with the excitation signal (i.e.
current pulses in green)23. Time-resolved image series comprised of 1021
frames, each spaced by $\mathrm{200}$ $\mathrm{ps}$, were acquired
stroboscopically with a temporal resolution of $\mathrm{70}$ $\mathrm{ps}$ and
spatial resolution
$\approx\\!{\text{\unboldmath$\mathrm{40}$}}\,\text{\unboldmath$\mathrm{nm}$}$.
Each frame is a X-ray absorption spectroscopy (XAS) image acquired with
circularly-polarized light, with the magnetic contribution to intensity
superimposed on the spectroscopy image. Differential magnetic contrast was
revealed by a division of each XAS frame by the average of all XAS frames, the
latter essentially being a XAS image of the static magnetic state (full videos
in supplementary material).

To set ideas, consider the example of a time-resolved series with two frames
where dynamics with opposite magnetic changes cause a variation in intensity
of $\pm I_{\mathrm{D}}$. The intensity in each frame is given as
$I_{\mathrm{S}}\pm I_{\mathrm{D}}$, with $I_{\mathrm{S}}$ the static
contribution to the transmitted intensity. The average of the two frames is
simply the static part, $I_{\mathrm{S}}$, and thus the differential magnetic
contrast in each frame is given by

$(I_{\mathrm{S}}\pm I_{\mathrm{D}})/(I_{\mathrm{S}})=1\pm
I_{\mathrm{D}}/I_{\mathrm{S}}$ (1)

While static XMCD contrast (usually of the order of a few percent) arises from
the difference between XAS images taken with opposite polarizations of light,
differential magnetic contrast is much weaker and of the order of $0.1\%$. It
should be noted that if the dynamics do not lead to $\pm I_{\mathrm{D}}$ that
is symmetric about zero, great care must be taken in the analysis, as
discussed later on.

Fig. 1: Time-resolved STXM with electrically contacted magnetic NWs. a)
Schematic of the STXM set-up, with incoming circularly-polarized X-ray photon
bunches (purple) incident at $\mathrm{30}$
$\mathrm{\mathrm{\SIUnitSymbolDegree}}$ to the normal of the SiN window. In
time-resolved mode, the frequency of the current pulses (green) inducing the
Œrsted field (blue) is phase-locked with the photon bunch frequency. b)
Voltage pulse signal measured after the NW across a $\mathrm{50}$
$\mathrm{\Omega}$ load, with $(+)$ and $(-)$ pulses of $\mathrm{3}$
$\mathrm{ns}$ applied to induce a current density of
$j={\text{\unboldmath$\mathrm{1{.}1\times{10^{12}}}$}}\,\text{\unboldmath$\mathrm{A/m^{2}}$}$
in a $\mathrm{93}$ $\mathrm{nm}$ diameter NW. c) static XMCD image of a
longitudinal domain in a $\mathrm{93}$ $\mathrm{nm}$ diameter NW. d) Average
of all XAS frames in one time-resolved image series equivalent to
$I_{\mathrm{S}}$ in eq. (1). e,f) Frames from a time-resolved image series
showing the differential magnetic contrast observed in a $\mathrm{93}$
$\mathrm{nm}$ diameter Co$30$Ni$70$ NW with the wire edges indicated by guides
to the eye. No current is flowing at time $t$ (e), but at time
$t+{\text{\unboldmath$\mathrm{0{.}8}$}}\,\text{\unboldmath$\mathrm{ns}$}$ (f)
the $\mathrm{3}$ $\mathrm{ns}$ current pulse with amplitude
$\mathrm{1{.}1\times{10^{12}}}$ $\mathrm{A/m^{2}}$ is being applied. g,h)
Illustrations of the magnetization in the NW at the time corresponding to
frames (e,f).

We first investigated the effect of the Œrsted field on uniformly-magnetized
domains in Co$30$Ni$70$ NWs. Suitable regions with
$>\\!\\!{\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolMicro}m}$}$-long
domains were detected using static XMCD STXM (Fig. 1c), after which a
$<\\!\\!{\text{\unboldmath$\mathrm{1}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolMicro}m}$}$
section was chosen within this. A repeating signal of $\mathrm{3}$
$\mathrm{ns}$ alternating positive $(+)$ and negative $(-)$ voltage pulses
with amplitude $\mathrm{1{.}1\times{10^{12}}}$ $\mathrm{A/m^{2}}$ was applied
to a $\mathrm{93}$ $\mathrm{nm}$ diameter NW (Fig. 1b). Pulses were spaced by
$\mathrm{100}$ $\mathrm{ns}$ to allow for sufficient heat dissipation. The
frames displayed in Fig. 1e,f show snapshots of the differential magnetic
contrast observed before and during the application of the $(+)$ current
pulse. Before the application of current in (e), no contrast is observed since
magnetization is at rest along the NW long axis (see the corresponding
illustration in Fig. 1g) as it is for the majority of frames in the time-
resolved series. During the application of the $\mathrm{3}$ $\mathrm{ns}$
$(+)$ current pulse (f), a bipolar contrast is observed across the NW,
indicating the tilting of magnetization to become more parallel (black) or
antiparallel (white) to the X-ray beam direction. The wire magnetization is
thus tilting towards the wire azimuthal direction (see illustration Fig. 1h),
consistent with the direction of the Œrsted field. Once the pulse has ended,
the magnetization returns to its relaxed state and the differential magnetic
contrast is no longer observed. The second pulse, with opposite polarity,
gives rise to an inverted contrast.

Fig. 2: Tilting of magnetization in NW domains as a function of current
density. a) Schematic of a NW with magnetic moment, $\mathbf{m}$, described
with spherical polar angles $\theta$ and $\phi$ in the cylindrical coordinate
system ($r$,$\varphi$,$z$). An applied Œrsted field tilts $\mathbf{m}$ from
$\hat{\mathbf{z}}$ towards $\hat{\boldsymbol{\varphi}}$ by an angle, $\theta$.
b) Average differential magnetic contrast of all 15 frames acquired during the
application of a $\mathrm{3}$ $\mathrm{ns}$ current pulse with amplitude
$\mathrm{1{.}1\times{10^{12}}}$ $\mathrm{A/m^{2}}$ in a $\mathrm{93}$
$\mathrm{nm}$ diameter NW. The width-averaged line scan is indicated by the
blue arrow. The corresponding intensity profile is shown in blue in (c). The
black curve is the fit from our model, with
$\theta_{0}={\text{\unboldmath$\mathrm{18{.}9}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolDegree}}$}$.
d) $\theta_{0}$ as a function of applied current density, $j$, in three
different NW samples with diameters $93$, $97$ and $\mathrm{101}$
$\mathrm{nm}$, and with vertical error bars. Trendlines through each data-set
and and analytical solutions (see eq. (4)) for each wire diameter are shown as
dashed and solid lines, respectively.

The signal-to-noise ratio is increased by taking the average of the 15 frames
acquired during the $\mathrm{3}$ $\mathrm{ns}$ current pulse (Fig. 2b). A
width-averaged line scan across the wire (blue area in Fig. 2b) reveals the
profile of this bipolar contrast (blue in Fig. 2c), with asymmetric peak
amplitudes $0.2$ and $\mathrm{-0{.}3}$ $\mathrm{\%}$. This is expected to be a
signature of the tilting of magnetization due to the Œrsted field, where
appropriate fitting may extract quantitative information. In the following, we
propose a model to reproduce this relatively simple physical situation and
hence fit the recorded contrast profile to estimate the tilt angle. Our model
considers the radius-dependent degree of tilt within a NW cross section, the
absorptivity of X-rays in the material and the X-ray beam spot size, which
make up the key components of magnetic transmission X-ray imaging4.

We use an Ansatz to describe the tilt by an angle $\theta$ of magnetic moment,
$\mathbf{m}$, towards the azimuthal direction when subjected to the Œrsted
field (Fig. 2a):

$\theta\left(r\right)=\theta_{0}\sin{\left(\frac{\pi}{2}\frac{r}{R}\right)}$
(2)

$\mathbf{r}$ is any point within the circular wire cross section,
$r=\absolutevalue{\mathbf{r}}$, $R$ is the wire radius and
$\theta_{0}=\theta\left(r=R\right)$. Comparisons with micromagnetic
simulations showed that this Ansatz accurately describes the tilt within the
wire17.

In magnetic materials, the absorptivity, $\mu$, or linear rate of absorption
of X-rays, depends on the chemical composition, the X-ray energy and the
magnetization direction versus the polarity of circularly-polarized X-ray
light. For a $\mathrm{100}$ $\mathrm{\%}$ circularly $(+)$ or $(-)$ polarized
X-ray beam parallel to the magnetization, the absorptivity is given as
$\mu_{\pm}$ and we additionally define the average absorptivity,
$\mu_{\mathrm{av}}=\left(\mu_{-}+\mu_{+}\right)/2$ and the difference in
absorptivity, $\Delta\mu=\mu_{-}-\mu_{+}$. The amplitude of the latter is of
particular importance, as it directly influences the strength of the XMCD
effect and thus relates to the strength of any differential magnetic contrast.
Values for $\mu_{\mathrm{av}}$ and $\Delta\mu$ of the studied Co$30$Ni$70$ NWs
at the Co L3 absorption edge can be extracted from XAS images acquired with
circularly-polarized light in static STXM. The reduced X-ray intensity behind
a NW is described by the Beer-Lambert law, linking the exponential decay of
light intensity through matter with $\mu$. Intensity profiles taken across the
NW can thus be fitted with the law, however, a non-zero X-ray spot size must
be accounted for by a convolution with a Gaussian with width $\sigma$. A
detailed analysis is shown in the supplementary material. The fitting
procedure then provides the only free parameters, absorptivity, $\mu$, and
spot size, $\sigma$. The analysis relies on the $\mathrm{30}$
$\mathrm{\mathrm{\SIUnitSymbolDegree}}$ angle between the NW magnetization and
the X-ray beam, induced by the sample holder orientation. It is also due to
this angle that a geometrical adjustment must be made to calculate $\Delta\mu$
from the extracted $\mu$ (see supplementary material).

Using this fitting procedure on multiple XAS images, we determined
$\mu_{\mathrm{av}}={\text{\unboldmath$\mathrm{0{.}006{\,}\pm
0.002}$}}\,\text{\unboldmath$\mathrm{nm^{-1}}$}$ and also found an average
$\sigma={\text{\unboldmath$\mathrm{50\pm{4}}$}}\,\text{\unboldmath$\mathrm{nm}$}$,
where the uncertainty is the standard deviation of the data sets. For
comparison, the theoretical absorptivity for Co$30$Ni$70$ at the Co L3 edge is
$\mu_{\mathrm{av,th}}={\text{\unboldmath$\mathrm{0{.}019}$}}\,\text{\unboldmath$\mathrm{nm^{-1}}$}$,
which is a factor 3 larger than our extracted values24. This is a common
feature in STXM imaging, partly related to background intensity incident on
the X-ray detector. We expect this amounts to
$\approx\\!{\text{\unboldmath$\mathrm{33}$}}\,\text{\unboldmath$\mathrm{\%}$}$
at the PolLux beamline STXM and arises from higher-order light from the
monochromating mirror
($\approx\\!{\text{\unboldmath$\mathrm{15}$}}\,\text{\unboldmath$\mathrm{\%}$}$)25
and leakage of the zone plate center stop
($\lesssim{\text{\unboldmath$\mathrm{18}$}}\,\text{\unboldmath$\mathrm{\%}$}$).
Accounting for this via a subtraction from the static XAS images, we correct
$\mu_{\mathrm{av}}={\text{\unboldmath$\mathrm{0{.}011{\,}\pm
0.005}$}}\,\text{\unboldmath$\mathrm{nm^{-1}}$}$ which is closer to the
theoretical value. The reason for the remaining difference is unclear. We
similarly correct $\Delta\mu$ from $\mathrm{0{.}002{\,}\pm 0.001}$
$\mathrm{nm^{-1}}$ to $\mathrm{0{.}003{\,}\pm 0.002}$ $\mathrm{nm^{-1}}$ (the
uncertainty also being the standard deviation). Accounting for the
$\approx\\!{\text{\unboldmath$\mathrm{50}$}}\,\text{\unboldmath$\mathrm{\%}$}$
degree of circular polarization of X-ray light in this experiment, we find a
$\Delta\mu=0.006\pm{\text{\unboldmath$\mathrm{0{.}004}$}}\,\text{\unboldmath$\mathrm{nm^{-1}}$}$.
This can be compared to the theoretical value, which again for the Co L3 edge
is
$\Delta\mu_{\mathrm{th}}={\text{\unboldmath$\mathrm{0{.}01}$}}\,\text{\unboldmath$\mathrm{nm^{-1}}$}$24.
Our calculated value is again lower than theory, but still in reasonable
agreement considering the uncertainty. The background subtraction was
therefore applied to all acquired STXM images. It should be noted that
considering the derived values of $\mu_{\mathrm{av}}$ and $\Delta_{\mu}$ as
effective, allows extracting the exact magnetization direction if the values
were calibrated on uniform magnetization at $\mathrm{30}$
$\mathrm{\mathrm{\SIUnitSymbolDegree}}$.

To now fit the bipolar contrast profile in Fig. 2c, we re-use the Beer-Lambert
law and include non-uniform magnetization, such as described by our Ansatz in
(2):

$I\left(x\right)=I_{0}\exp{-\int{\left(\mu_{\mathrm{av}}+\frac{1}{2}\Delta\mu\;\hat{\mathbf{k}}\cdot\mathbf{m}\right)\mathrm{d}\ell}}$
(3)

This describes the progressive absorption of X-rays through each elementary
segment with length, $\mathrm{d}\ell$, and with
$\hat{\mathbf{k}}\cdot\mathbf{m}$ the component of magnetization along the
X-ray beam direction, $\hat{\mathbf{k}}$. $I$ depends on
$\mathbf{m}\left(r\right)$ which itself depends on $\theta_{0}$.

The intensity profile is then convoluted with the Gaussian to account for the
finite spot size, already determined from the static XAS image analysis. By
performing the same image calculation as in (1) with the dynamic
($\theta_{0}\neq 0$) and the static ($\theta_{0}=0$) intensity profile, the
bipolar differential magnetic contrast profiles can be reproduced. The only
free parameter for the fit is $\theta_{0}$, while all other variables are
fixed as previously determined from the XAS image analysis. We revisit now the
magnetic image in Fig. 2b and the corresponding contrast profile in blue in
Fig. 2c, which is fit very well ($r^{2}=0.94$) by the black curve. The model
also reproduces the asymmetric signal, which originates in part from the non-
linear change in $\hat{\mathbf{k}}\cdot\mathbf{m}$ due to the geometry of the
$\mathrm{30}$ $\mathrm{\mathrm{\SIUnitSymbolDegree}}$ sample holder and in
part from the non-linear X-ray absorption due to the exponential nature of the
Beer-Lambert law. From the fit we determine the tilt of magnetic moments on
the surface as
$\theta_{0}={\text{\unboldmath$\mathrm{18{.}9}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolDegree}}$}$,
caused by the current-pulse-induced Œrsted field. This value comes with
multiple sources of uncertainty for the magnitude of the extracted
$\theta_{0}$. First, we determined that the uncertainty in the NW diameter
measured from scanning electron microscopy images and the uncertainty in the
spot size extracted from the XAS analysis translate to an uncertainty of
$\lesssim\\!{\text{\unboldmath$\mathrm{10}$}}\,\text{\unboldmath$\mathrm{\%}$}$
in $\theta_{0}$. This remains moderate compared with the last source, the
uncertainty in $\mu_{\mathrm{av}}$ and $\Delta\mu$, for which the impact is
critical due to the aforementioned exponential in (3). We repeat the fitting
of the data in Fig. 2c using values for $\mu_{\mathrm{av}}$ of $0.006$ and
$\mathrm{0{.}016}$ $\mathrm{nm^{-1}}$ and $\Delta\mu$ of $0.001$ and
$\mathrm{0{.}004}$ $\mathrm{nm^{-1}}$, which correspond to an uncertainty of
one standard deviation (see supplementary material Fig. S5). This gives
$\theta_{0,\mathrm{max}}={\text{\unboldmath$\mathrm{47{.}8}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolDegree}}$}$
and
$\theta_{0,\mathrm{min}}={\text{\unboldmath$\mathrm{13{.}2}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolDegree}}$}$,
which is strongly asymmetric about
$\theta_{0}={\text{\unboldmath$\mathrm{18{.}9}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolDegree}}$}$
due to the exponential in (3). These values of $\theta_{0,\mathrm{max}}$ and
$\theta_{0,\mathrm{min}}$ hence provide the range of uncertainty that we
expect for our extracted value of $\theta_{0}$. However, for the case of the
intensity profile produced with $\theta_{0,\mathrm{max}}$, the fit to the data
is quite poor ($r^{2}=0.83$) indicating that the original fit with
$\theta_{0}={\text{\unboldmath$\mathrm{18{.}9}$}}\,\text{\unboldmath$\mathrm{\mathrm{\SIUnitSymbolDegree}}$}$
is more appropriate.

This fitting analysis to determine $\theta_{0}$ was applied to multiple time-
resolved image series from three wire diameters and several applied current
densities, $j$, with the results plotted in Fig. 2d. The vertical error bars
for each data point represent the range of uncertainty calculated as described
above. The most significant result is that $\theta_{0}$ increases linearly
with $j$, as also shown by the dashed trendline (constrained to pass through
the origin) for each data set. While the linear dependence of the $93$ and
$\mathrm{97}$ $\mathrm{nm}$ diameter wires is clear, the $\mathrm{101}$
$\mathrm{nm}$ diameter wire exhibits a larger spread. The reason most likely
being a variation in the imaging conditions from one time-resolved series to
another (when settings were changed), resulting in a change in the incident
X-ray intensity. This directly impacts the signal-to-noise ratio in the
differential images and possibly the applied background subtraction, as the
latter may not scale linearly with the incident light intensity. In both
instances the result would be a vertical offset (of unknown direction) of the
data. Imaging conditions (settings) were kept constant during the acquisition
of each of the other two sample data sets, however, were adjusted between the
samples. For any $j$, $\theta_{0}$ increases from the $97$ to $93$ to
$\mathrm{101}$ $\mathrm{nm}$ diameter wire and using the trendline, we find a
tilt rate in $\theta_{0}$ equivalent to $10.7$, $16.0$ and $\mathrm{27{.}4}$
$\mathrm{\mathrm{\SIUnitSymbolDegree}}$ per $\mathrm{10^{12}}$
$\mathrm{A/m^{2}}$, respectively.

To compare the experimental $\theta_{0}$ we use an analytical model developed
by A. De Riz et al17, balancing the competition between exchange and Zeeman
Œrsted energies to describe $\theta_{0}\left(j\right)$ in longitudinal domains
in NWs. To first order, this reads:

$\theta_{0}\approx\frac{1}{3.16}\frac{\mu_{0}jM_{\mathrm{s}}R^{3}}{\pi A}$ (4)

with $\mu_{0}=4\pi\times 10^{-7}$ the vacuum magnetic permeability,
$M_{\mathrm{s}}$ the spontaneous magnetization of the material and $A$ the
exchange stiffness. De Riz et al. found that for
$j<{\text{\unboldmath$\mathrm{3\times{10^{12}}}$}}\,\text{\unboldmath$\mathrm{A/m^{2}}$}$,
the model is an accurate description and matches well with simulations,
meaning that it is appropriate to compare to the results presented here. Using
magnetic parameters for Co$30$Ni$70$ NWs
($M_{\mathrm{s}}={\text{\unboldmath$\mathrm{0{.}77}$}}\,\text{\unboldmath$\mathrm{MA/m}$}$26
and
$A={\text{\unboldmath$\mathrm{1{.}5\times{10^{-11}}}$}}\,\text{\unboldmath$\mathrm{J/m}$}$27),
(4) is plotted as solid line for each NW diameter in Fig. 2d. The theory
confirms the experimentally observed linear dependence of $\theta_{0}$ on $j$,
however, predicts larger tilt rates. These are $\theta_{0}=23.4$, $26.5$ and
$\mathrm{29{.}9}$ $\mathrm{\mathrm{\SIUnitSymbolDegree}}$ per
$\mathrm{10^{12}}$ $\mathrm{A/m^{2}}$ for $93$, $97$ and $\mathrm{101}$
$\mathrm{nm}$ diameter wires, respectively. Even though there is
$\lesssim\\!{\text{\unboldmath$\mathrm{55}$}}\,\text{\unboldmath$\mathrm{\%}$}$
discrepancy between the theory and experiment, the results are promising as
they indicate that the analysis is appropriate within its range of
uncertainty. The experiment fails to reproduce the theoretically predicted
$1/R^{3}$ dependence, however, we expect this to be linked to systematic
errors such as the changes in imaging conditions discussed previously.
Further, local inhomogeneities in the nanowire (e.g. small changes in diameter
or crystal grains) can also lead to a systematic offset of the data both along
$\theta_{0}$ or $j$ which cannot be accounted for.

Fig. 3: BPW in a $\mathrm{101}$ $\mathrm{nm}$ diameter NW subjected to an
Œrsted field. Average of all wavelet filtered time-resolved differential
magnetic contrast frames acquired during (a) the application of a $\mathrm{3}$
$\mathrm{ns}$ $(+)$ current pulse with amplitude
$\mathrm{7{.}5\times{10^{11}}}$ $\mathrm{A/m^{2}}$, (b) the $\mathrm{100}$
$\mathrm{ns}$ rest period, (c) the $\mathrm{3}$ $\mathrm{ns}$ $(-)$ current
pulse and (d) the $\mathrm{100}$ $\mathrm{ns}$ rest period. The BPW position
is indicated by orange circles. The micromagnetic simulations in (e-h)
correspond to the images in (a-d) and show a tail-to-tail BPW in its
compressed (a), expanded (g) and relaxed (f,h) state. The direction of the
applied Œrsted field is indicated where applicable. Simulated time-resolved
differential magnetic contrast of a compressed (i) and expanded (j) BPW in a
$\mathrm{100}$ $\mathrm{nm}$ diameter NW. Each frame was blurred with a
Gaussian. Contrast profiles for the case of a compressed (k) and expanded (l)
BPW, corresponding to line scans left (solid lines) and right (dashed lines)
of the wall center. The true line scans from the image in (a) and (c) are
shown in blue, while the fit from our model is shown in black.

We now turn to the imaging of BPWs under the influence of the Œrsted field in
the same NWs. DW positions were determined using static XMCD, after which a
current pulse of sufficient amplitude
($j\approx{\text{\unboldmath$\mathrm{1\times{10^{12}}}$}}\,\text{\unboldmath$\mathrm{A/m^{2}}$}$)
was sent through the NW to ensure a) the BPW DW type by transforming walls of
the transverse-vortex kind to BPWs17 and b) that the BPW is sufficiently
pinned on an extrinsic pinning site to allow for a reproducible magnetization
process over millions of pulses. Multiple time-resolved series were acquired
while applying a similar voltage pulse signal as in Fig. 1b, with different
$j$ around $j_{c}$, the critical current density expected for BPW circulation
switching17, 16.

We imaged a tail-to-tail BPW in a $\mathrm{101}$ $\mathrm{nm}$ diameter NW,
first in the regime when $j<j_{c}$ by applying a current density of
$\pm{\text{\unboldmath$\mathrm{7{.}5\times{10^{11}}}$}}\,\text{\unboldmath$\mathrm{A/m^{2}}$}$.
Fig. 3a-d show the frame average of the differential magnetic contrast frames
acquired during (a) the application of the $\mathrm{3}$ $\mathrm{ns}$ $(+)$
current pulse, (b) the first $\mathrm{100}$ $\mathrm{ns}$ rest period, (c) the
$\mathrm{3}$ $\mathrm{ns}$ $(-)$ current pulse and (d) the second
$\mathrm{100}$ $\mathrm{ns}$ rest period. All frames were filtered with a hat
wavelet filter to improve the signal-to-noise ratio. In addition to the Œrsted
field tilting observed within the domains during the pulse application, all
four images contain a strong contrast at the BPW location (orange circle).

As expected for this regime, in Fig. 3a,c no specific differential magnetic
contrast is visible at the wall center, indicating that the sign of azimuthal
curling directly around the Bloch-point remains unchanged for the duration of
the image series. The applied Œrsted field pulses (see black arrows in Fig. 3e
and f) are thus once parallel and once antiparallel to the wall circulation,
so that different dynamics are expected during each of the two pulses. Indeed,
in (a) and (c) we observe two different stronger contrasts around the wall
center. In (a) there are four small symmetric lobes of differential magnetic
contrast with a stronger intensity than seen in the longitudinal domains far
from the wall. The bipolar contrast to either side of the wall center is
indicative of a change of circulation that is now opposed to the static
circulation (i.e. state $I_{\mathrm{S}}$), which can be understood as follows:
the bipolar contrast, albeit stronger, matches that within the domains,
suggesting a tilt towards the Œrsted field direction. This tilt direction
opposes the intrinsic BPW circulation and as the BPW does not reverse its sign
of circulation, the observed change therefore reflects rather a compression of
the wall. Such a compression was predicted by micromagnetic simulations (Fig.
3e) of a tail-to-tail BPW in a $\mathrm{100}$ $\mathrm{nm}$ diameter
Co$30$Ni$70$ NW, subjected to an antiparallel Œrsted field induced by a
current with amplitude $\mathrm{8\times{10^{11}}}$ $\mathrm{A/m^{2}}$. These
simulations were obtained with our home-made finite element freeware
FeeLLGood28, based on the Landau-Lifshitz-Gilbert equation. The simulated
distribution of magnetization was further used to simulate XAS images29
expected for an absorption according to eq. (3) and governed by our
experimentally-determined values of $\mu_{\mathrm{av}}$ and $\Delta\mu$.
Importantly, the $\mathrm{30}$ $\mathrm{\mathrm{\SIUnitSymbolDegree}}$ sample
holder alignment was accounted for in the simulated imaging. Differential
magnetic contrast images were calculated by applying eq. (1) to simulated XAS
images of a dynamic and static BPW. A Gaussian filter was applied to reproduce
the effect of a finite width spot size as in the experiment (unfiltered images
are shown in the supplementary material). The differential magnetic contrast
simulated for a compressed tail-to-tail BPW is shown in Fig. 3i. The
similarity with the image in (a) is striking, thus confirming the qualitative
explanation of a BPW compression. Minor differences, e.g. the size of certain
features compared to in (a), are likely related to the Gaussian filter applied
to the simulation.

Conversely, in (c) there are only two large lobes of opposite differential
magnetic contrast, suggesting that the circulation of the static state is now
being enhanced. This should be the result of an expansion of the BPW (see
simulation in Fig. 3g) as the applied Œrsted field is parallel to the wall
circulation. The differential magnetic contrast from the simulated expanded
wall (g) is shown in Fig. 3j, with key contrast features again matching with
those observed in (c). This combination between time-resolved imaging and
simulated imaging provides a powerful tool to explain observed contrasts.

We now return to our model for a quantitative verification of the qualitative
explanation. We plot in blue in Fig. 3k,l contrast profiles from line scans
taken across the wire, through the regions of contrast left (solid curve) and
right (dashed curve) of the BPW center. Fig. 3k and (l) correspond to the
images in (a) and (c), respectively. In this case, fitting is achieved because
the static ($I_{\mathrm{S}}$) and the dynamic, either compressed or expanded,
state intensity profiles are defined by separate $\theta_{0}$ and are non-
zero. It must be noted that the convolution with the Gaussian spot size along
$x$ only is now slightly less valid because the magnetization is no longer
homogeneous along $z$, and a finer analysis should convolute along both the
$x$ and $z$ direction. Still, the black curves in Fig. 3k,l are fits to the
contrast profiles and show an excellent agreement on either side of the wall
center. The BPW compression proposed to explain (a) is confirmed numerically
with the fits in (k): left and right of the wall center, $\theta_{0}$ tilts
from $16$ to $\mathrm{-17}$ $\mathrm{\mathrm{\SIUnitSymbolDegree}}$ and from
$28$ to $\mathrm{-8}$ $\mathrm{\mathrm{\SIUnitSymbolDegree}}$, respectively.
The Œrsted field reverses the sign of circulation close to the wall center,
compressing the wall and giving rise to the bipolar contrast and hence four
contrast lobes around the wall center. Similarly, the fits for (l) reveal
$\theta_{0}$ tilts from $17$ to $\mathrm{45}$
$\mathrm{\mathrm{\SIUnitSymbolDegree}}$ and from $20$ to $\mathrm{41}$
$\mathrm{\mathrm{\SIUnitSymbolDegree}}$, left and right of the wall center,
respectively, confirming the enhancement of the static circulation, or an
expansion of the BPW. (a) and (c) together show the breathing of the BPW,
predicted only by simulations until now17. The differential magnetic contrast
patterns were observed in multiple image series and are inverted in the case
of a BPW with opposite static circulation (see supplementary material).

For the interpulse periods displayed in Fig. 3b,d a strong white or black
contrast, respectively, is observed at the DW location. This should be a
result of small scale BPW motion of the order of
$270\pm{\text{\unboldmath$\mathrm{30}$}}\,\text{\unboldmath$\mathrm{nm}$}$,
however, intricacies of this contrast, such as the direction of motion and a
time evolution of the contrast are not yet understood (see supplementary
material for further discussion).

We finally mention the case when $j>j_{c}$, for which the BPW switches its
sense of circulation16, 17. This could not be observed experimentally with
time-resolved STXM because DWs disappeared for $j\geq j_{c}$, which we
attribute to heat assisted DW depinning. Future measurements, possibly with
engineered DW pinning sites are required to observe this effect with temporal
resolution.

In conclusion, we have used time-resolved STXM to image dynamic changes of
magnetic textures in cylindrical NWs. We observe the effect of the Œrsted
field on longitudinally magnetized domains and evidence the breathing of a BPW
when subjected to pulses with opposite sign. A quantitative analysis of the
differential magnetic contrast is provided by a robust model based on the
absorptivity of X-rays and a description of the magnetization in a NW cross
section. This highlights the depth of information obtainable with time-
resolved magnetic imaging and that a direct comparison of the observed
dynamics with simulations and theory is possible. Further work with this
technique will significantly improve our understanding of magnetic 3D
nanosized systems and enable better control over them.

See supplemental material for the following: full time-resolved image series
from which still frames are shown in this text are shown in Fig. S1 and Fig.
S2; calculation of the projection of magnetization in a tilted NW; detailed
explanation of the XAS analysis, impact of the uncertainty in absorptivity on
the asymmetric error bars on $\theta_{0}$; unfiltered simulated images of a
breathing BPW; images of breathing of a BPW with switched circulation; and an
explanation of differential magnetic contrast of BPW motion.

M. S. acknowledges a grant from the Laboratoire d’excellence LANEF in Grenoble
(ANR-10-LABX-51-01). The project received financial support from the French
National Research Agency (Grant No. JCJC MATEMAC-3D). This work was partly
supported by the French RENATECH network, and by the Nanofab platform
(Institut Néel), whose team is greatly acknowledged for technical support.
Part of the work was performed at the PolLux STXM endstation of the Swiss
Light Source, Paul Scherrer Institut, Villigen PSI, Switzerland, financed by
the German Bundesministerium für Bildung und Forschung (BMBF) through
contracts 05K16WED and 05K19WE2.

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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#  A bound on Universal Extra Dimension Models
from up to $2\,\text{fb}^{-1}$ of LHC Data at 7 TeV

Kenji Nishiwaki, Kin-ya Oda,
Naoya Okuda, and Ryoutaro Watanabe
∗Department of Physics, Kobe University, Kobe 657-8501, Japan
${}^{\dagger}{}^{\ddagger}{}^{\S}$Department of Physics, Osaka University,
Osaka 560-0043, Japan
E-mail<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS>E-mail:
<EMAIL_ADDRESS>
###### Abstract

The recent up to $\sim 2\,\text{fb}^{-1}$ of data from the ATLAS and CMS
experiments at the CERN Large Hadron Collider at 7 TeV put an upper bound on
the production cross section of a Higgs-like particle. We translate the
results of the $H\to WW\to l\nu l\nu$ and $H\to\gamma\gamma$ as well as the
combined analysis by the ATLAS and CMS into an allowed region for the Kaluza-
Klein (KK) mass $M_{\text{KK}}$ and the Higgs mass $M_{H}$ for all the known
Universal Extra Dimension (UED) models in five and six dimensions. Our bound
is insensitive to the detailed KK mass splitting and mixing and hence
complementary to all other known signatures.

KOBE-TH-11-07
OU-HET-722/2011

##  1 Introduction

The ATLAS and CMS experiments at the CERN Large Hadron Collider (LHC) have
presented their latest results for the ${\color[rgb]{0,0,0}\lesssim
2}\,\text{fb}^{-1}$ of data at the center of mass energy 7 TeV at the XXV
International Symposium on Lepton Photon Interactions at High Energies (Lepton
Photon 11), Mumbai, India, 22–27 August 2011. One of the most remarkable among
them is the bound on the Higgs mass in the Standard Model (SM). A combined
analysis of the ATLAS experiment excludes the existence of the SM Higgs in
mass ranges $146\,\text{GeV}<M_{H}<232\,\text{GeV}$,
$256\,\text{GeV}<M_{H}<282\,\text{GeV}$, and
$296\,\text{GeV}<M_{H}<466\,\text{GeV}$ within the 95% Confidence Level (CL)
based on 1.0–$2.3\,\text{fb}^{-1}$ data [1] and that of the CMS experiment
excludes $145\,\text{GeV}<M_{H}<216\,\text{GeV}$,
$226\,\text{GeV}<M_{H}<288\,\text{GeV}$, and
$310\,\text{GeV}<M_{H}<400\,\text{GeV}$ within the 95% CL based on
1.1–$1.7\,\text{fb}^{-1}$ data [2]. Further the production cross section of a
Higgs-like particle, a particle that decays the same way as the SM Higgs, is
severely constrained by these data in the still-allowed regions, namely light
$115\,\text{GeV}<M_{H}<145\,\text{GeV}$, middle
$288\,\text{GeV}<M_{H}<296\,\text{GeV}$, and heavy $M_{H}>466\,\text{GeV}$
windows.

In this Letter, we translate the above constraint on the production cross
section into that on the Kaluza-Klein (KK) scale of various 5-Dimensional (5D)
and 6D Universal Extra Dimension (UED) models, namely the minimal UED (mUED)
[3] and the Dirichlet Higgs (DH) [4, 5] models in 5D and the 6D UED models on
$T^{2}/Z_{2}$ [3], $T^{2}/Z_{4}$ [6, 7], $T^{2}/(Z_{2}\times Z^{\prime}_{2})$
[8], ${RP}^{2}$ [9], $S^{2}/Z_{2}$ [10], Projective Sphere (PS) [11], and
$S^{2}$ [12].111 In [11] the terminology “real projective plane” is employed
for a sphere with its antipodal points being identified. In order to
distinguish [11] from [9], we call the former the Projective Sphere (PS). We
note that the PS and $S^{2}$ UED models have no orbifold fixed point and hence
no localized interaction on it. Concretely, we bound the UED parameter space
of $M_{\text{KK}}$ (first-level KK mass) and $M_{H}$ (zero mode Higgs mass)
based on the leading ATLAS and CMS constraints on the total cross section and
on that of each channel [1, 2].

One of the biggest advantages of the UED models is the existence of a natural
Dark Matter (DM) candidate, the Lightest KK Particle (LKP) [13]. The 6D UED
models have further advantages of the requirement of the number of generations
to be (zero modulo) three [14] and the assurance of the proton stability [15].

There exist several bounds on the 5D mUED model, within which the brane-
localized interactions are assumed to be vanishing at the 5D Ultra-Violet (UV)
cutoff scale ${\color[rgb]{0,0,0}\Lambda_{5D}}$. The latest analysis on DM
relic abundance including the effects from second KK resonances gives the
preferred KK scale at around $M_{\text{KK}}\sim 1.3\,\text{TeV}$ [16].  It is
noted that the first KK charged Higgs becomes the LKP when $M_{H}\gtrsim
240$–300 GeV, depending on the KK scale [17].  The electroweak precision data
suggests that the KK scale should be $M_{\text{KK}}\gtrsim 800\,\text{GeV}$
($300\,\text{GeV}\lesssim M_{\text{KK}}\lesssim 400\,\text{GeV}$) at the 95%
CL for $M_{H}=115$ (700) GeV [18, 19, 20]. The observed branching ratio of
$B_{d}\rightarrow X_{s}\gamma$ confines the KK scale as
$M_{\text{KK}}>600\,\text{GeV}$ [21] at the $95\%\,\text{CL}$. Recent study
puts a constraint ${M_{\text{KK}}>{\color[rgb]{0,0,0}600}\,\text{GeV}}$ for
$10<\Lambda_{{\color[rgb]{0,0,0}5D}}/M_{\text{KK}}<40$ at the
$95\%\,\text{CL}$ {} [22], from the ATLAS SUSY search result in
multijet$+E_{\text{T}}^{\text{miss}}$ with $1\,\text{fb}^{-1}$ data [23].222
Inclusion of the decay channel into KK Higgs, if allowed, might significantly
affect the result. We thank K. Tobioka on this point.  We see that current LHC
bound from jets plus missing $E_{T}$ is not severe even for the most
constraint mUED. This is because we have typically smaller mass splitting
between the LKP and other new particles than the one between the lightest
supersymmetric particle and other sparticle in the minimal supersymmetric
standard model.

We note that all of these bounds are strongly dependent on the mass splitting
and mixing within the first KK level and therefore on the boundary mass
structure which is derived from the above-mentioned assumption that all of
them are zero at the 5D UV cutoff scale. The bound on the KK scale put in this
Letter is complementary to them in the sense that this is depending only on
the Higgs mass. That is, our bound is insensitive to the boundary masses if
they are smaller than the KK scale, as is necessary to have a higher
dimensional picture at all.

##  2  Procedure to obtain the bound

The ATLAS and CMS groups have shown the results for the combined analyses for
the ratio $\sigma^{95\%}_{pp\to H}/\sigma^{\text{SM}}_{pp\to H}$ as a function
of the Higgs mass $M_{H}$, where $\sigma^{95\%}_{pp\to H}$ is an upper bound
on the production cross section of a particle that decays the same as the SM
Higgs, at the 95% CL [1, 2]. In our case the constrained production cross
section is that of the UED Higgs. In UED models, a process can be affected by
KK-loops when it is loop-induced in the SM. In particular, the dominant Higgs
production channel via the gluon fusion process can be greatly enhanced, see
[12] and references therein. For middle and heavy Higgs mass regions, the
constraint is mainly from the $H\to WW$ and $ZZ$ channels, which are dominated
by the tree-level SM processes and therefore the result of the combined
analysis can be applied directly.

For the light Higgs mass region, the severest bound on $\sigma_{pp\to
H}^{\text{95\%}}/\sigma_{pp\to H}^{\text{SM}}$ comes from $H\to WW\to l\nu
l\nu$ or $H\to\gamma\gamma$. The latter is loop-induced in the SM and can be
affected by the KK-loops. Further, the loop-induced decay into gluons is not
negligible in this region in computation of the total decay width. Therefore,
we cannot trust the combined analysis which assumes that the branching ratios
are not changed from the SM. In the light mass range, we apply the CMS bounds
on $\sigma_{pp\to H\to\gamma\gamma}^{\text{95\%}}/\sigma_{pp\to
H\to\gamma\gamma}^{\text{SM}}$ and $\sigma_{pp\to H\to
WW}^{\text{95\%}}/\sigma_{pp\to H\to WW}^{\text{SM}}$ [2].

For the Higgs production, we focus on the the gluon fusion process via the
(KK) top quark loops, which is the dominant Higgs production channel in the SM
and the UED models [12, 24, 25, 26]. The parton level cross section of each
model $\hat{\sigma}^{\text{model}}_{gg\to H}$ is

$\displaystyle\hat{\sigma}^{\text{model}}_{gg\to H}(\hat{s})$
$\displaystyle={\pi^{2}\over 8M_{H}}\,\Gamma_{H\to
gg}^{{\text{model}}}(M_{H})\,{\delta(\hat{s}-M_{H}^{2})},$ (1)

where $\hat{s}$ is the parton level center-of-mass-energy-squared and
$\Gamma_{H\to gg}^{{\text{model}}}$ is the partial decay width into a pair of
gluons in each model:

$\displaystyle\Gamma^{\text{model}}_{H\to gg}(M_{H})$
$\displaystyle={\color[rgb]{0,0,0}K}{\alpha_{s}^{2}\over
8\pi^{3}}{M_{H}^{3}\over
v_{\text{EW}}^{2}}\,\left|J_{t}^{\text{model}}\\!\left({\color[rgb]{0,0,0}M_{H}^{2}}\right)\right|^{2}{\color[rgb]{0,0,0},}$
(2)

where $\alpha_{s}$ is the QCD coupling strength and $v_{\text{EW}}$ is the
Higgs vacuum expectation value $\simeq 246\,\text{GeV}$ and $K$ is the
$K$-factor to take into account the higher order QCD corrections, whose NNLO
value is $\simeq 1.9$ at the relevant energies, see e.g. Ref. [27]. When we
consider a ratio such as $\sigma_{pp\to H}^{\text{95\%}}/\sigma_{pp\to
H}^{\text{SM}}$ from the gluon fusion process, the overall K-factor does not
influence the result. However it contributes to the decay branching ratios of
the light Higgs boson non-negligibly.

For each model, the loop function $J_{t}^{\text{model}}$ describes the
contributions of all the zero and KK modes for the top quark in the triangle
loops:

$\displaystyle J_{t}^{\text{SM}}(\hat{s})$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right),$ (3) $\displaystyle
J_{t}^{\text{mUED}}(\hat{s})$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}I\\!\left({m_{t(n)}^{2}\over\hat{s}}\right)\right\\},$ (4)
$\displaystyle J_{t}^{\text{DH}}(\hat{s})$
$\displaystyle=\sqrt{2}\varepsilon_{1}\left[\left|I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}I\\!\left(m_{t(n)}^{2}\over\hat{s}\right)\right|^{2}+\left|2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}\tilde{I}\\!\left(m_{t(n)}^{2}\over\hat{s}\right)\right|^{2}\right]^{1/2},$
(5) $\displaystyle{J_{t}^{T^{2}/Z_{2}}(\hat{s})=J_{t}^{{RP}^{2}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{{m+n\geq
1\atop\text{or\ }m=-n\geq 1}}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t(m,n)}^{2}\over\hat{s}\right)\right\\},$
(6) $\displaystyle{J_{t}^{T^{2}/Z_{4}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq
1,n\geq 0}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t({m,n})}^{2}\over\hat{s}\right)\right\\},$
(7) $\displaystyle{J_{t}^{T^{2}/(Z_{2}\times Z^{\prime}_{2})}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq
0,n\geq 0,\atop(m,n)\not=(0,0)}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\left(m_{t(m,n)}^{2}\over\hat{s}\right)\right\\},$ (8)
$\displaystyle J_{t}^{S^{2}/Z_{2}}(\hat{s})$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j\geq
1}\left({m_{t}\over
m_{t(j)}}\right)^{2}n^{\color[rgb]{0,0,0}S^{2}/Z_{2}}(j)\,I\left(m_{t(j)}^{2}\over\hat{s}\right)\right\\},$
(9) $\displaystyle J_{t}^{\text{PS}}(\hat{s})={J_{t}^{S^{2}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j\geq
1}\left({m_{t}\over
m_{t(j)}}\right)^{2}(2j+1)\,I\\!\left(m_{t(j)}^{2}\over\hat{s}\right)\right\\},$
(10)

where $I$ and $\tilde{I}$ are given by

$\displaystyle I(\lambda)$
$\displaystyle={\color[rgb]{0,0,0}-2\lambda+\lambda(1-4\lambda)}\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right],$ (11)
$\displaystyle\tilde{I}(\lambda)$
$\displaystyle={\color[rgb]{0,0,0}(+\lambda)}\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right],$ (12)

the $n^{\text{model}}(j)$ counts the number of degeneracy:

$\displaystyle n^{S^{2}/Z_{2}}(j)$ $\displaystyle=\begin{cases}j+1,\\\
j,\end{cases}$ $\displaystyle n_{\text{even}}^{\text{PS}}(j)$
$\displaystyle=\begin{cases}2j+1,\\\ 0,\end{cases}$ $\displaystyle
n_{\text{odd}}^{\text{PS}}(j)$ $\displaystyle=\begin{cases}0,&\quad\text{for
}j=\text{{\color[rgb]{0,0,0}even}},\\\ 2j+1,&\quad\text{for
}j=\text{{\color[rgb]{0,0,0}odd}},\end{cases}$ (13)

and we write the KK top and $W$ masses ($X=t,W$)

$\displaystyle m_{{\color[rgb]{0,0,0}X}(n)}$
$\displaystyle\equiv\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{n^{2}}{R^{2}}}}=\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+n^{2}{M_{\text{KK}}^{2}}},$
(14) $\displaystyle m_{{\color[rgb]{0,0,0}X}(m,n)}$
$\displaystyle\equiv\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{m^{2}+n^{2}}{R^{2}}}}=\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\left(m^{2}+n^{2}\right){M_{\text{KK}}^{2}}}},$
(15) $\displaystyle m_{{\color[rgb]{0,0,0}X}(j)}$
$\displaystyle\equiv\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{j(j+1)}{R^{2}}}}=\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{j(j+1){M_{\text{KK}}^{2}}}{2}}},$
(16)

with ${M_{\text{KK}}}$ being the first KK mass: $M_{\text{KK}}=1/R$ for the
$S^{1}/Z_{2}$ (mUED), an interval (DH), and $T^{2}$-based compactifications
(namely $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$, $T^{2}/Z_{4}$ and
$RP^{2}$) and being ${M_{\text{KK}}}=\sqrt{2}/R$ for the $S^{2}$-based ones
(namely $S^{2}/Z_{2}$, PS and $S^{2}$). The range of the KK summation reflects
the structure of each extra dimensional background.333 The origin of the
factor 2 in front of each KK summation is the fact that there are both left
and right handed (namely, vector-like) KK modes for each chiral quark zero
mode corresponding to a SM quark.

The factor $\sqrt{2}\varepsilon_{1}$ in Eq. (5) is equal to $2\sqrt{2}/\pi\sim
0.90$. Readers who want more explanations on the above expressions should
consult Ref. [12].

As is mentioned above, we compute the decay rate into a photon pair, following
[26]. The result is

$\begin{split}&\Gamma^{\text{model}}_{H\to\gamma\gamma}(M_{H})={G_{F}\over
8\sqrt{2}\pi}M_{H}^{3}\cdot{\alpha^{2}\over\pi^{2}}\left|J_{W}^{\text{model}}\\!\left(M_{H}^{2}\right)+{4\over
3}J_{t}^{\text{model}}\\!\left(M_{H}^{2}\right)\right|^{2},\end{split}$ (17)

where

$\displaystyle J_{W}^{\text{SM}}(M_{H}^{2})$ $\displaystyle=L\\!\left({1\over
2},3,3,6,0;{M_{W}^{2}\over M_{H}^{2}},{M_{W}^{2}\over M_{H}^{2}}\right),$ (18)
$\displaystyle J_{W}^{\text{mUED}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{n=1}^{\infty}L\\!\left({1\over
2},4,4,8,1;{M_{W}^{2}\over M_{H}^{2}},{M_{W(n)}^{2}\over M_{H}^{2}}\right),$
(19) $\displaystyle J_{W}^{T^{2}/Z_{4}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{m\geq 1,n\geq
0}L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (20) $\displaystyle
J_{W}^{T^{2}/(Z_{2}\times Z^{\prime}_{2})}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{m\geq 0,n\geq
0\atop(m,n)\neq(0,0)}L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (21) $\displaystyle
J_{W}^{T^{2}/Z_{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{{m+n\geq 1\atop\text{or\
}m=-n\geq 1}}L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (22) $\displaystyle
J_{W}^{{RP}^{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{(m,n)}^{A}L\\!\left({1\over
2},4,4,8,1;{M_{W}^{2}\over M_{H}^{2}},{M_{W(m,n)}^{2}\over
M_{H}^{2}}\right)+\sum_{(m,n)}^{B}L\\!\left(0,1,0,2,0;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (23) $\displaystyle
J_{W}^{S^{2}/Z_{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}n^{S^{2}/Z_{2}}(j)\,L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right),$ (24) $\displaystyle
J_{W}^{S^{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}(2j+1)\,L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right),$ (25) $\displaystyle
J_{W}^{\text{PS}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}\Bigg{[}n_{\text{even}}^{\text{PS}}(j)\,L\\!\left({1\over
2},4,4,8,1;{M_{W}^{2}\over M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right)$
$\displaystyle\phantom{=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}\Bigg{[}}+n_{\text{odd}}^{\text{PS}}(j)\,L\\!\left(0,1,0,2,0;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right)\Bigg{]},$ (26)

with

$\displaystyle L(a,b,c,d,e;\lambda_{1},\lambda_{2})$
$\displaystyle=a+b\lambda_{1}-\left[\lambda_{1}\\!\left(c-d\lambda_{2}\right)-e\lambda_{2}\right]\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda_{2}}+1-i\epsilon\right].$ (27)

The $A$-summation for $RP^{2}$ are over the region that satisfies both $m\geq
1$ and $n\geq 1$ as well as over the ranges $(m,n)=(0,2),(0,4),(0,6),\dots$
and $(m,n)=(2,0),(4,0),(6,0),\dots$. Similarly, the $B$-summation are over
$m\geq 1$ and $n\geq 1$ as well as over $(m,n)=(0,1),(0,3),(0,5),\dots$ and
$(m,n)=(1,0),(3,0),(5,0),\dots$. The Dirichlet Higgs model only allows the
heavy mass region in which $H\to\gamma\gamma$ is irrelevant and hence we do
not compute the process for it.

In six dimensional UED models, KK summation in Eqs. (6)–(10) and (20)–(26)
must be terminated by a UV cutoff, for which we take the maximum and minimum
possible values consistent with the Naive Dimensional Analysis (NDA), shown in
Table 1. Let us briefly explain this treatment hereafter. For more details,
see Ref. [12]. Since the electroweak symmetry is broken by the Higgs mechanism
in the SM and UED models (except for the DH model), the gluon fusion process
is described by a dimension-six operator at lowest in 4D point of view after
KK expansion. This means that the calculation is UV logarithmic-divergent
(convergent) in six (five) dimensions.444 Of course 5D UED is non-
renormalizable and we have to introduce a cutoff scale in theory. However,
this does not appear in our analysis because we can calculate the gluon fusion
process with no UV divergence in 5D UEDs.  Therefore we need to put an upper
limit of the summation over KK indices in 6D. We do it by adopting the NDA. In
both the $T^{2}$ and $S^{2}$-based geometries, the most stringent bound turns
out to be the one from the perturbativity of the $U(1)_{Y}$ gauge interaction,
which results in the following allowed regions of KK indices [12]:

$\displaystyle m^{2}+n^{2}$ $\displaystyle\lesssim{\color[rgb]{0,0,0}30},$
$\displaystyle\text{for $T^{2}$-case}\
\left(m_{(m,n)}^{2}=\frac{m^{2}+n^{2}}{R^{2}}\right),$ (28) $\displaystyle j$
$\displaystyle\lesssim{\color[rgb]{0,0,0}9},$ $\displaystyle\text{for
$S^{2}$-case}\ \left(m_{(j,m)}^{2}=\frac{j(j+1)}{R^{2}}\right),$ (29)

where the index $m$ for the $S^{2}$-case discriminates the degenerate states
of each $j$-th level. The cutoff scale of 6D UED theory $\Lambda_{6D}$ must be
lower than that in Eq. (28) or (29). In Table 1, we list the values that we
take.

Based on the knowledge sketched above, we can evaluate the total cross section
of the Higgs production of the UED models $\sigma_{pp\to H}^{\text{model}}$
and the ratio to that of the SM $\sigma_{pp\to
H}^{\text{model}}/\sigma^{\text{SM}}_{pp\to H}$ to be compared to the
experimental result.

##  3 Results

|  $T^{2}$-based |  $S^{2}$-based
---|---|---
| max | min | max | min
KK index | $m^{2}+n^{2}<28$ | $m^{2}+n^{2}\leq 10$ | $j(j+1)\leq 90$ | $j(j+1)\leq 30$
UV cutoff | $\Lambda_{6D}\sim 5M_{\text{KK}}$ | $\Lambda_{6D}\sim 3M_{\text{KK}}$ | $\Lambda_{6D}\sim 7M_{\text{KK}}$ | $\Lambda_{6D}\sim 4M_{\text{KK}}$
Table 1: Our choices of maximum and minimum upper bounds for KK indices and
for the corresponding UV cutoff scale.

First, we show our results for the light region:
$115\,{\text{GeV}}<M_{H}<145\,\text{GeV}$. We apply the CMS bounds on
$H\to\gamma\gamma$ and $H\to WW$ channels that are dominant in this range. In
Fig. 1, we list the contour plots for the excluded region in the
$M_{\text{KK}}$-$M_{H}$ plane for various UED models in 5 and 6 dimensions.
Plots for the maximum and minimum choices of the UV cutoff scale are presented
for the 6D UED models. In general, UED models enhance Higgs production via
gluon fusion and reduce the Higgs decay into a pair of photons. Therefore,
$\sigma^{\text{UED}}_{pp\to H\to\gamma\gamma}$ receives nontrivial
contributions from such effects. Typically, the enhancement of Higgs
production cross section overcomes the suppression of the di-photon branching
ratio in the $H\to\gamma\gamma$ excluded range (with orange and red colors),
whereas the region for smaller $M_{\text{KK}}$ is not excluded because of the
suppression of the di-phton branching ratio. (For example, in the case of
$S^{2}$ UED, the di-photon cross section is suppressed for
$M_{\text{KK}}\lesssim 400\,\text{GeV}$.) We find that all the suppressed
region is already excluded by $WW$ channel. It is natural that the lower the
cutoff scale becomes, the more the allowed parameter region is enlarged since
smaller numbers of KK tops contribute to the process. In 6D, we have more
light KK top quarks running in the loop, and get stronger constraints than in
5D. The $\operatorname{BR}(H\to WW)$ is also affected by the enhancement of
the total Higgs decay rate due to the increase of $H\to gg$.

Second, we move on to the middle region:
$288\,\text{GeV}<M_{H}<296\,\text{GeV}$. The Standard Model is still allowed
in this range whereas we find that all the UED models below $M_{KK}=1.4$ TeV
is excluded.

Finally, let us discuss the heavy region: $M_{H}>446\,\text{GeV}$. We choose
severer bound on $\sigma^{\text{95\%}}_{pp\to H}/\sigma^{\text{SM}}_{pp\to H}$
between ATLAS and CMS data for each $M_{H}$. That is, we use ATLAS and CMS
bounds for $M_{H}<500\,\text{GeV}$ and $M_{H}\geq 500\,\text{GeV}$,
respectively. In Fig. 2 we plot our results. We note that in all the allowed
region, we get $M_{H}<2M_{\text{KK}}$ and hence the Higgs does not decay into
a pair of KK particles.

Now let us comment on the DH model. In this model, the bound is put only on
$M_{\text{KK}}$ ($=M_{H}$) and the theoretical value of the ratio
$\sigma_{pp\to H}^{\text{DH}}/\sigma^{\text{SM}}_{pp\to H}$ decreases when one
increases $M_{\text{KK}}$, while the experimental upper bound
$\sigma_{95\%}/\sigma_{\text{SM}}$ is an increasing function of $M_{H}$ in the
high-mass region. The cross-over occurs at $M_{\text{KK}}=480\,\text{GeV}$
which gives $\sigma_{pp\to H}^{\text{DH}}/\sigma^{\text{SM}}_{pp\to H}\simeq
1.2$ and we conclude that the allowed parameter region of $M_{\text{KK}}$ is:

$\displaystyle M_{\text{KK}}>480\,\text{GeV}\quad(\text{95\% CL in Dirichlet
Higgs model}).$ (30)

In the $M_{\text{KK}}$ region of
$110\,{\text{GeV}}<M_{\text{KK}}<149\,\text{GeV}$ and
$206\,{\text{GeV}}<M_{\text{KK}}<300\,\text{GeV}$, the value of $\sigma_{pp\to
H}^{\text{DH}}/\sigma^{\text{SM}}_{pp\to H}$ grows significantly and thereby
these regions are rejected by the CMS result at the 95% CL. Noting that the
indirect electroweak constraint gives
$430\,\text{GeV}<M_{\text{KK}}<500\,\text{GeV}$ at the 90% CL [4], the allowed
region of $M_{\text{KK}}$ roughly lies between $480\,\text{GeV}\lesssim
M_{\text{KK}}\lesssim 500\,\text{GeV}$.

##  4 Summary and Discussions

In this Letter we have constrained the UED models in 5D and 6D by use of the
latest ATLAS and CMS bounds on the Higgs production cross section. The bound
on 6D UED is severer than that on 5D UED because 6D KK mass spectrum is denser
than that in 5D and therefore the KK top modes contribute to the gluon fusion
process larger. The KK (Higgs) mass of the Dirichlet Higgs model is pinned
down at around 500 GeV.

In the light mass range $115\,\text{GeV}<M_{H}<145\,\text{GeV}$, one of the
dominant constrains on the Higgs production cross section is coming from the
$H\to\gamma\gamma$ decay, which is reduced when KK scale is not large, due to
the interference between the SM gauge boson and KK top loops, with some
corrections from SM top and KK gauge bosons. We have taken into account the
CMS constraints from $H\to\gamma\gamma$ and $H\to WW\to l\nu l\nu$ instead of
the combined analysis. We find in this light region that the Higgs mass above
140 GeV is already excluded in the 5D and 6D $T^{2}/Z_{4}$ UED models, while
the mass above 130 GeV is ruled out in other 6D UED models, both within a
reasonably small KK scale $<1.4\,\text{TeV}$.

Our analyses on the 6D UED models cannot evade ambiguities from the NDA, but
the plots in Figs. 1 and 2 imply that the dependence on the cutoff is rather
mild $\lesssim 10\%$. For a low cutoff scale, there can also be contributions
from higher dimensional operators that must be taken into account. We have
ignored these possible contributions in our analysis.

In both 5D and 6D cases, the bound is insensitive to the detailed boundary
mass structure. In this sense this constraint is complementary to other ones
such as the relic abundance of the LKP and the $M_{T2}$ analysis of the decay
of the colored KK into the LKP.

When the KK scale is not much heavier than the weak scale $\simeq
246\,\text{GeV}$, the UED models tend to prefer much heavier Higgs mass than
in the SM in order to cancel the KK top loops in the $T$-parameter. (This
contribution has the same origin as the gluon fusion process discussed in this
Letter.) In this regard it would be important to put an experimental bound for
the Higgs mass beyond 600 GeV.

There is the triviality bound if the Higgs is heavy and the UED scale is light
which are being studied by the authors, along with the vacuum stability bound
for the case of light Higgs, and will be presented in a separate publication.

### Acknowledgment

We are grateful to Abdelhak Djouadi, Koichi Hamaguchi and Kohsaku Tobioka for
helpful communications. We also thank the Yukawa Institute for Theoretical
Physics for hosting the workshop “Field Theory and String Theory,”
YITP-W-11-05, during which this work was initiated. K.O. is partially
supported by Scientific Grant by Ministry of Education and Science (Japan),
Nos. 23104009, 23740192 and 20244028.

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Figure 1:  Excluded regions in $M_{\text{KK}}$-$M_{H}$ plane by the CMS
constraints on $\sigma^{95\%}_{pp\to H\to WW}/\sigma^{\text{SM}}_{pp\to H\to
WW}$ and $\sigma^{95\%}_{pp\to H\to\gamma\gamma}/\sigma^{\text{SM}}_{pp\to
H\to\gamma\gamma}$ for the light Higgs.  The bound from $WW$ channel (cyan and
blue respectively for minimum and maximum UV cutoffs, former of which is
superimposed on the latter) is superimposed on that from $\gamma\gamma$
channel (orange and red, the same as above) leaving its outline. The UV cutoff
scales of our choice are summarized in Table 1.

Figure 2:  Constraints from the combined analysis from each ATLAS and CMS
experiment in the heavy mass region, drawn the same as Fig. 1 with maximum UV
cutoff (blue) being superimposed by the minimum UV cutoff (cyan).