# Interplay between domain walls and magnetization curling induced by chemical
modulations in cylindrical nanowires

L. Álvaro-Gómez<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS, CEA, Grenoble
INP, SPINTEC, 38000 Grenoble, France.  A. Masseboeuf  Univ. Grenoble Alpes,
CNRS, CEA, Grenoble INP, SPINTEC, 38000 Grenoble, France.  N. Mille
Synchrotron SOLEIL, l’Orme des Merisiers, Saint-Aubin, FR-91192 Gif-sur-Yvette
Cedex, France.  C. Fernández-González  Alba Synchrotron Light Facility, CELLS,
08290 Cerdanyola del Vallès, Barcelona, Spain.  S. Ruiz-Gómez  Max Planck
Institute for Chemical Physics of Solids, 01187 Dresden, Germany.  J.C
Toussaint  Univ. Grenoble Alpes, CNRS, Institut Néel, 38000 Grenoble, France.
R. Belkhou  Synchrotron SOLEIL, l’Orme des Merisiers, Saint-Aubin, FR-91192
Gif-sur-Yvette Cedex, France.  M. Foerster  Alba Synchrotron Light Facility,
CELLS, 08290 Cerdanyola del Vallès, Barcelona, Spain.  E. Pereiro  Alba
Synchrotron Light Facility, CELLS, 08290 Cerdanyola del Vallès, Barcelona,
Spain.  L. Aballe  Alba Synchrotron Light Facility, CELLS, 08290 Cerdanyola
del Vallès, Barcelona, Spain.  C. Thirion  Univ. Grenoble Alpes, CNRS,
Institut Néel, 38000 Grenoble, France.  D. Gusakova  Univ. Grenoble Alpes,
CNRS, CEA, Grenoble INP, SPINTEC, 38000 Grenoble, France.  O. Fruchart  Univ.
Grenoble Alpes, CNRS, CEA, Grenoble INP, SPINTEC, 38000 Grenoble, France.  L.
Pérez  IMDEA Nanociencia, Campus de Cantoblanco, 28049 Madrid, Spain.

###### Abstract

Cylindrical magnetic nanowires have been proposed as a means of storing and
processing information in a 3D medium, based on the motion of domain walls
(DWs). Introducing short chemical modulations in such wires would allow for
reliable digital control of DWs. Here, we outline the intricate physics of the
interaction of domain walls with modulations to control their motion,
combining micromagnetic simulations and experimental evidence. This
interaction combines a long-range moderate magnetostatic repulsion with a
local energy well. The latter depends on the respective circulation sense of
magnetization in the domain wall and modulation. We also show that a
modulation has the ability to switch the internal circulation of a DW upon its
propagation, thereby acting as a polarizing component and opening the
possibility to exploit not only the position of walls, but also their internal
structure.

IMDEA Nanociencia, Campus de Cantoblanco, 28049 Madrid, Spain. Dpto. de Física
de Materiales, Universidad Complutense de Madrid, 28040 Madrid, Spain. Dpto.
de Física de Materiales, Universidad Complutense de Madrid, 28040 Madrid,
Spain.

Cylindrical nanowires made of a soft-magnetic material are a textbook case for
investigating magnetization dynamics in a three-dimensional geometry. Their
magnetic textures are determined by the nanowire’s geometrical and
compositional features, as well as by their magnetic history. While the
uniaxial shape anisotropy favors an axial magnetization state, magnetic
charges occurring at the wire terminations can induce end domains. Theory
predicts that when the diameter is larger than circa seven times the dipolar
exchange length,
$\Delta_{\mathrm{d}}=\sqrt{\frac{2A}{\mu_{\mathrm{0}}M_{\mathrm{s}}^{2}}}$,
the magnetization rotates around the wire axis in the end domains, which is
commonly referred to as curling or vortex state1, 2, 3, 4. This configuration
decreases magnetostatic energy at the expense of exchange energy, by
converting part of the surface charges into volume charges. It is worth noting
that extended curling domains may also arise in nanowires with
magnetocrystalline anisotropy promoting azimuthal magnetization, as observed
in CoNi alloys5, single-crystalline hcp Co6, 7, 8 or single-crystal CVD Ni
wires9. However, we do not cover this specific case in the present manuscript.

There have been several experimental proofs of end curling: a decrease in
remanence with increasing diameter10, more rounded hysteresis loops measured
with the Kerr effect than with global magnetometry11, quantitative analysis
using magnetic force microscopy12, or direct imaging with photo-emission
electron microscopy13. Curling also occurs in other situations where axial
magnetization gives rise to magnetic charges. One such scenario involves
diameter modulation, as observed for nanowires made of FeCoCu14, single-
crystal Ni15, NiFe alloys 16, 17, or Ni18, 19. Another case is the axial
modulation of material, resulting in a mismatch of magnetization, such as in
Fe20Ni80/Fe80Ni2020, 21, Ni/Co22, 23, 24, or Co/Cu25. The existence of curling
textures is particularly relevant with the emergence of spintronic
investigations of nanowires26, 27, 28, 29, 21, 30, 31, 32. Indeed, the axial
charge current gives rise to an Œrsted field, which directly couples to
curling textures and may assist nucleation33 or switch the sign of curling21.

Magnetic domain walls in cylindrical nanowires can manifest as one of two
types: a Transverse-Vortex Wall (TVW) or a Bloch-Point Wall (BPW)4, 34. The
characteristic feature of the TVW is a transversal magnetization component
with respect to the wire axis, which implies the existence of two areas on the
outer surface of the wire where $\mathbf{m}\cdot\mathbf{n}=\pm 1$, where
$\mathbf{n}$ is the unit vector normal to the surface. At these locations, the
tangential magnetization takes the form of a vortex-antivortex pair35. The BPW
is characterized by a curling magnetization around the wire axis and a
magnetic singularity at its core, known as the Bloch point36, 37, 38. A
feature that clearly distinguishes its topology from that of the TVW is the
absence of any point on the wire surface with $\mathbf{m}\cdot\mathbf{n}=\pm
1$, although a radial magnetization component may arise to reduce
magnetostatic effects related to the head-to-head or tail-to-tail domains.

The first experimental evidence of a BPW and a TVW in cylindrical nanowires
was provided by shadow XMCD-PEEM microscopy in 201439. Subsequent studies
include investigations of the possible transformation of the wall type under
axial magnetic fields40 or under current pulses, where it was demonstrated BPW
velocities in excess of
$600\text{\,}\mathrm{m}\text{${}\cdot{}$}{\mathrm{s}}^{-1}$ and the switching
of BPW curling with the Œrsted field27, 29.

Therefore, both chemical modulations and BPWs involve curling of magnetization
around the wire axis, and in both cases the sign (circulation sense) can be
switched with an anti-parallel Œrsted field 21, 27, 29. The similarity of
their features naturally raises the question of how these two objects interact
with each other. This question is particularly relevant since chemical
modulations have been proposed as engineered sites to control domain walls in
nanowires. In this manuscript, we examine this interaction, both
experimentally and with micromagnetic simulations at rest and under the
stimulus of an applied axial magnetic field. We shed light on the roles of
exchange and dipolar energy, in order to understand the different mechanisms
involved in domain wall pinning and propagation through a chemical modulation.
A special focus is given to the significance of the respective circulations of
the wall and modulation, which can either be parallel ($C+C+$) or antiparallel
($C+C-$).

##  0.1 Energy landscape of a DW at a modulation

###  0.1.1 Micromagnetic simulations

In the following, we describe the energetics of a domain wall in a nanowire.
Due to the mostly one-dimensional nature of nanowires, their energy landscape
may be reduced to one dimension as in the Becker-Kondorski description41. For
nanowires with homogeneous composition, ideally, the domain wall energy does
not depend on its position, as long as it is far enough from the wire ends.
However, in practice, domain walls may be pinned at specific locations, such
as material defects or grain boundaries42, 43, 44. The introduction of
chemical modulations aims to modify the domain wall energy landscape, allowing
controlled pinning sites to be established if the artificial pinning strength
exceeds that related to material defects. In micromagnetic simulations, one
method to extract the energy landscape is to monitor the domain wall energy
during its slow drift free of external forces under large damping so that it
remains under quasistatic conditions at all times. This approach was
successfully used for a diameter-modulated nanowire 45. However, in the
present case, such a method cannot be easily applied, as the profile of the
energy landscape may be non-monotonous so that the drift does not cover all
positions. Instead, we applied an axial external magnetic field to move the
domain wall towards the chemical modulation. The damping coefficient was set
to $\alpha=1$ to be as close to equilibrium as possible and in a quasi-static
situation. Both methods are expected to coincide as long as the applied field
does not significantly affect the internal structure of the domain wall, such
as its width.

We consider a $d=$90\text{\,}\mathrm{n}\mathrm{m}$$ diameter Permalloy
(Fe20Ni80) nanowire with a $\ell=$20\text{\,}\mathrm{n}\mathrm{m}$$-long
Fe80Ni20 chemical modulation at $z_{\mathrm{DW}}=0$, in two situations:
parallel and anti-parallel curling circulations of BPW and modulation, $C+C+$
and $C+C-$ respectively. The domain wall is pushed towards the modulation
under the application of an external magnetic field oriented along
$\hat{\bm{z}}$ or $-\hat{\bm{z}}$ for z${}_{\mathrm{DW}}<0$ or
z${}_{\mathrm{DW}}>0$, respectively, with magnitude increasing in small steps.
The initial circulation of the BPW is chosen with the same sign as the
direction of motion, so that it is the stable one during dynamics and thus
does not switch spontaneously during motion46. The internal energy, i.e., the
sum of the dipolar and exchange energy:
$E_{0}=E_{\mathrm{d}}+E_{\mathrm{ex}}$, is displayed versus BPW position (zDW)
on Figure 1a. $z_{\mathrm{DW}}$ is computed as the intercept of the three iso-
surfaces of the magnetization field: $m_{\mathrm{x}}=0$, $m_{\mathrm{y}}=0$
and $m_{\mathrm{z}}=0$47. Figure 1b and Figure 1c show outer views of the
micromagnetic simulation at selected time steps. At a long distance from the
modulation, typically $250\text{\,}\mathrm{n}\mathrm{m}$ or more, both curves
overlap and reveal a repulsive interaction. To the contrary, at shorter
distances, the respective circulations matter: the repulsion keeps increasing
for opposite circulations, giving rise to an energy barrier, while the
modulation acts as a local potential well for identical circulations. This can
be understood as follows. (i) At long distances between the BPW and the
modulation the interaction is independent of their relative circulations. This
is consistent with the known fact that circulations of curling structures do
not interact with one another via dipolar interactions48, as the magnetic
potential does not depend on the sign of circulation. The repulsion results
from the interaction of the positively-charged head-to-head BPW with the
dipolar field of the modulation, whose positively-charged interface is the one
closest to the wall21. The same repulsion would apply for a negatively charged
tail-to-tail BPW, giving rise to a neighboring negatively-charged modulation
interface. (ii) At short distances, the magnetization textures of the BPW and
the modulation overlap and interact directly via exchange: magnetization
gradients decrease for parallel circulation, while they increase for
antiparallel circulations (see Figure 1b), which translates into a decrease
(resp. increase) of exchange energy. This, again, is consistent with previous
reports on the interaction of curling end structures in short nanotubes48.

Figure 1: a) Internal energy $E_{0}$ versus head-to-head BPW position (zDW) in
a $90\text{\,}\mathrm{n}\mathrm{m}$ diameter Permalloy nanowire with a
$20\text{\,}\mathrm{n}\mathrm{m}$-long chemical modulation centered at $z=0$.
Orange dashed lines indicate the two interfaces of the modulation. The wall is
driven towards the modulation by an axial quasi-static magnetic field of
rising magnitude oriented along $\hat{\bm{z}}$ or $-\hat{\bm{z}}$ for
z${}_{\mathrm{DW}}<0$ or z${}_{\mathrm{DW}}>0$, respectively. Blue stands for
parallel circulations $C+$$C+$ of the BPW and modulation, while black stands
for antiparallel $C+$$C-$ circulations. b) and c) show outer view
configurations obtained with micromagnetic simulations for selected wall
positions (zDW) for parallel and antiparallel circulations, respectively. The
color code represents $m_{\mathrm{z}}$.

###  0.1.2 Experimental evidence of DW pinning and repulsion

We have experimentally probed the effectiveness of chemical modulations as
pinning sites for domain walls, previously nucleated by applying a magnetic
field of around $1\text{\,}\mathrm{T}$ perpendicular to the wire axis or by
injecting a $10\text{\,}\mathrm{n}\mathrm{s}$-long pulse of current with
amplitude
$1\text{\times}{10}^{12}\text{\,}\mathrm{A}\text{${}\cdot{}$}{\mathrm{m}}^{-2}$.
We image the magnetization states in the wires with either full field or
scanning Transmission X-ray microscopy, as well as with X-ray ptychography,
all combined with XMCD. The combination of high spatial resolution and direct
probing of magnetization enables the extraction of unambiguous information
about the DW type and inner structure.

Figure 2: X-ray magnetic imaging of domain walls in different Permalloy
nanowires with periodic Fe80Ni20 chemical modulations. a), b), e), f), g) Top:
transmission amplitude ptychographic reconstruction image at the Fe L3 edge.
Orange dashed lines indicate the location of the chemical modulations. Blue
dotted lines indicate the location of Fe-rich minor modulations. Bottom:
corresponding reconstructed amplitude ptychographic XMCD image. c), d) Top:
TXM image at the Fe L3 edge and Fe L2 edge, respectively. Bottom:
corresponding XMCD image. Images show BPW pinning at (a), c), d)) or next to
(f), g)) a chemical modulation, and TVW pinning at (e)) or next to ( b)) a
chemical modulation. System geometry: a), b):
$d=$130\text{\,}\mathrm{n}\mathrm{m}$$,
$\ell=$60\text{\,}\mathrm{n}\mathrm{m}$$, c), d):
$d=$105\text{\,}\mathrm{n}\mathrm{m}$$,
$\ell=$100\text{\,}\mathrm{n}\mathrm{m}$$, and e), f), g):
$d=$120\text{\,}\mathrm{n}\mathrm{m}$$,
$\ell=$30\text{\,}\mathrm{n}\mathrm{m}$$.

Figure 2a shows composite transmission amplitude ptychographic reconstruction
and XMCD images at the Fe L3 edge of a
$130\text{\,}\mathrm{n}\mathrm{m}$-diameter Permalloy nanowire with periodic
$60\text{\,}\mathrm{n}\mathrm{m}$-long Fe80Ni20 chemical modulations. The
darker regions in the transmission image marked with orange dashed lines are
the Fe-rich chemical modulations (higher absorption). A closer look reveals a
depletion in the Fe content at the left of some of the chemical modulations,
extending over a few hundreds of nanometers and ending with a short Fe-rich
minor modulation (see blue dotted line, e.g., modulation $\\#4$). This
material inhomogeneity arises from an instability occurring during the
electroplating process. Based on this finding, the synthesis was revisited to
avoid this defect. Corresponding images will be shown later on, but magnetic
images of this first generation of wires are informative on their own.

The XMCD images in Figure 2a correspond to the wire axis perpendicular to the
X-ray beam (middle), and with a $10\text{\,}\mathrm{\SIUnitSymbolDegree}$
offset (bottom). The former is insensitive to the axial magnetization
component. Therefore, the neutral gray level in the Permalloy segments reveals
that they are uniformly magnetized along the axis. In contrast, a strong
dark/bright contrast arises on either side of the wire at every modulation,
indicative of magnetization curling around the axis21. Note that for a given
magnetization component, we expect XMCD to be stronger for Fe-rich areas due
to the stronger absorption13, so that the displayed contrast does not reflect
quantitatively the magnetization state. The image with the tilted sample
offers a light dark and bright contrast reflecting axial magnetization in the
segments, revealing the occurrence of three domains (black arrows). The
stronger magnetic contrast at modulations $\\#2$ and $\\#4$ can now be
understood as the signature of BPWs, while the lighter contrast at the other
modulations simply reflects spontaneous curling of magnetization due to the
mismatch of magnetization magnitude. The two DWs pinned at modulations $\\#2$
and $\\#4$ confirm the potential of modulations to pin DWs. A close look shows
that the DW is centered exactly on the main modulation, while the secondary
chemical modulation also induces a slight curling, which may be either
parallel (modulation $\\#2$) or antiparallel (modulation $\\#4$) to that of
the BPW. This complex modulations can give rise to other pinning situations,
such as illustrated in Figure 2b. Here, the DW is of TVW type and is pinned at
the secondary chemical modulation. This highlights the generality of the
pinning potential of modulations, regardless of their strength and the type of
DW. One may wonder whether this configuration results from the exchange-driven
repulsion between the modulation and the opposite circulation on the right
side of the TVW, outlined previously, however, the existence of the secondary
modulation biases the discussion.

To get a conclusive view, we now examine the situation of symmetric chemical
modulations, i.e., after refining the synthesis process, and with a variety of
wire diameters and lengths of chemical modulation (Figure 2c-g). Domain walls
of either BPW (Figure 2c-d) or TVW type (Figure 2e) can be pinned exactly at
the modulation. Note that Figure 2c and Figure 2d are of the same modulation
imaged at the Fe L3 and Fe L2, respectively, in a case where a too high
absorption at the L3 edge resulted in saturation and thus absence of magnetic
contrast13. BPWs can also be found very close to a modulation, in situations
where the sign of circulation of the DW and modulation are opposite (Figure
2f-g). This is consistent with the micromagnetic simulations which showed that
modulations can give rise to a potential well to pin DWs, but in case of
opposite circulation situation there exists a strong repulsive potential, so
that the DW stops just before the modulation.

##  0.2 Propagation of DWs through modulations

We now consider the magnetization process of domain-wall motion across a
modulation under the stimulus of an external axial magnetic field, for both
parallel and antiparallel magnetization circulation between BPW and
modulation.

###  0.2.1 The case of identical circulations – Smooth propagation

In this section, we consider the situation where the BPW and the chemical
modulation exhibit the same sign of circulation. We start with micromagnetic
simulations of a $90\text{\,}\mathrm{n}\mathrm{m}$ diameter Permalloy
nanowire, with either a $20\text{\,}\mathrm{n}\mathrm{m}$ or a
$100\text{\,}\mathrm{n}\mathrm{m}$-long chemical modulation. The initial state
is a head-to-head BPW located at
$z_{\mathrm{DW}}=$-500\text{\,}\mathrm{n}\mathrm{m}$$, which we push towards
the modulation by applying a positive magnetic field. The strength of the
magnetic field is increased in the form of plateaus of length
$2\text{\,}\mathrm{n}\mathrm{s}$ and rising magnitude. The damping coefficient
is $\alpha=1$, to mimic a quasistatic situation. We chose a positive
circulation in both modulation and BPW, which is the stable one during motion
along $+\hat{\bm{z}}$ in a homogeneous wire 46.

Figure 3: BPW propagation through a chemical modulation in a
$90\text{\,}\mathrm{n}\mathrm{m}$-diameter Permalloy nanowire for parallel and
positive circulations $C+C+$ with respect to $\hat{\bm{z}}$. a) Internal
energy $E_{0}$ versus BPW position ($z_{\mathrm{DW}}$). Open and full blue
symbols correspond to a chemical modulation length of
$20\text{\,}\mathrm{n}\mathrm{m}$ and $100\text{\,}\mathrm{n}\mathrm{m}$,
respectively. Red symbols show the magnitude of the axial magnetic field,
applied in plateaus of duration $2\text{\,}\mathrm{n}\mathrm{s}$ and rising
magnitude. b) Outer (left) and inner (right) micromagnetic views at selected
$z_{\mathrm{DW}}$, for a modulation length of
$20\text{\,}\mathrm{n}\mathrm{m}$. Each configuration is identified with a
letter in a). The black square corresponds to the Bloch point position and the
elongated light blue rectangle to the position of the chemical modulation,
whose interfaces are indicated by the orange dashed lines. The color code
represents $m_{\mathrm{\rho}}$.

Figure 3a shows the internal energy of the system as a function of
$z_{\mathrm{DW}}$. The open blue symbols depict the scenario of a chemical
modulation with a length of $20\text{\,}\mathrm{n}\mathrm{m}$, while the
filled blue symbols depict the scenario with a chemical modulation with a
length of $100\text{\,}\mathrm{n}\mathrm{m}$. The red symbols indicate the
linearly-increasing strength of the magnetic field, with plateaus of duration
$2\text{\,}\mathrm{n}\mathrm{s}$. The two energy profiles are qualitatively
similar, with a well at the modulation and long-distance repulsion, consistent
with Figure 1a. Initially, the DW is located on the left-hand side of the
modulation in the repulsive area, and the field needs to be increased to about
$3\text{\,}\mathrm{m}\mathrm{T}$ to overcome the axial gradient of internal
energy (Note that the DW is already continuously drifting during each of the
$2\text{\,}\mathrm{n}\mathrm{s}$-long plateaus at the top of the energy
barrier, around $z_{\mathrm{DW}}=$-200\text{\,}\mathrm{n}\mathrm{m}$$, so that
the value of the magnetic field when at this position is an upper value of the
propagation field). Then, the DW remains pinned close to the bottom of the
energy well over a wide range of values of applied field, which is illustrated
by the sharp slope of the red curves. Finally, the DW depins from the
modulation and starts drifting again to the right. The depinning field is
$\approx$10\text{\,}\mathrm{m}\mathrm{T}$$ and
$\approx$14\text{\,}\mathrm{m}\mathrm{T}$$ for the short and long modulations,
respectively. The propagation and depinning fields more or less reflect the
value of steepest slope of internal energy, consistent with the Becker-
Kondorsky model. In the case of the longer modulation the internal energy is
shifted upwards even when the DW is away from the modulation, reflecting the
higher cost associated with curling in the longer segment21. The upward shift
is reduced at rest likely because most of the curling of the BPW fits within
the modulation, so that the cost of exchange required anyway by the
modulation, benefits the BPW. This results in a deeper energy well for the
longer modulation, and thus a larger depinning field. Finally, note that the
energy peak for $z_{\mathrm{DW}}>0$ is higher than for $z_{\mathrm{DW}}<0$,
contrary to the energy landscape computed in Figure 1a. This is probably
related to the slight change of the inner structure of the BPW due to the
applied field, which is different when entering versus leaving the modulation.

As the peripheral curling and the Bloch point may follow different dynamics
during the motion of a BPW49, it is interesting to track the Bloch point
during the propagation/pinning/depinning process, which is illustrated in
Figure 3b. On the left, an external view of the magnetization field at the
wire surface is shown with gray arrows. The background color codes the radial
component of magnetization $m_{\mathrm{\rho}}$, and thus the location of the
wall, as it results from the associated head-to-head charges46. Below each
view, the black square indicates the axial position of the Bloch point. On the
right side, an internal surface view through the nanowire illustrates the
radial position of the Bloch point. The Bloch point deviates only slightly in
a minor fashion from the center of mass of the wall. It is ahead of the wall
in the repulsive region, and tends to be pinned at interfaces of the
modulation. Hence, we conclude that a BPW propagates smoothly through a
chemical modulation, when their circulation senses are parallel.

###  0.2.2 Experimental evidence of BPW depinning, motion and repinning

In the following, we provide experiments supporting the micromagnetic
predictions. We consider a $130\text{\,}\mathrm{n}\mathrm{m}$-diameter
Permalloy nanowire with two $60\text{\,}\mathrm{n}\mathrm{m}$-long chemical
modulations separated by a distance of $2\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, shown in Figure 4. Note that this is a situation of asymmetric
modulations, with a Fe-depleted segment on the left-hand side of the
modulation, ending with a narrow Fe-rich secondary modulation. The initial
state is a head-to-head DW located at the right modulation, labeled $\\#2$.
There is no DW at the left modulation, labeled $\\#1$, but its circulation is
of the same sign. We apply a dc axial magnetic field, that tends to move the
DW to that second modulation. Initially, the domain wall center seems to lie
between the main and the secondary modulations of $\\#2$. At a critical field
of $9\text{\,}\mathrm{mT}$, the BPW is depinned and moves left, still being
connected to the secondary modulation. At $11\text{\,}\mathrm{mT}$, the BPW
propagates along the entire Permalloy segment until modulation $\\#1$. Upon
removal of the magnetic field, the BPW relaxes, shifting slightly to the right
side. This slight shift may be attributed to the opposing curling circulation
in the secondary modulation of $\\#1$, inducing a repulsion of the BPW towards
the opposite direction. Therefore, we have confirmed experimentally the order
of magnitude of the pinning and depinning strength at the modulations for
parallel circulations. The data also illustrates controlled pinning, stronger
than the intrinsic pinning on material defects along the Permalloy segments.

Figure 4: X-ray imaging of BPW depinning, propagation and repinning on
chemical modulations under an axial dc magnetic field. The wire of
$130\text{\,}\mathrm{n}\mathrm{m}$ diameter is made of Permalloy with periodic
$60\text{\,}\mathrm{n}\mathrm{m}$-long $\mathrm{Fe}_{65}\mathrm{Ni}_{35}$
chemical modulations, indicated with orange vertical dashed lines. Blue dotted
lines indicate the location of Fe-rich minor modulations. Imaging is performed
at the Fe L3 edge. a) Ptychographic XAS image a BPW pinned at modulation
$\\#2$. b) Corresponding XMCD images taken under a constant magnetic field,
indicated in red.

###  0.2.3 The case of opposite circulations – Modulations as a polarizing
component

In addition to the above, we now consider the propagation of a BPW through a
chemical modulation, with initially anti-parallel circulation. We expect a
behavior different from the case of identical circulation, as the modulation
should act as an energy barrier to the domain wall (Figure 1). The initial
configuration is a head-to-head BPW at
$z_{\mathrm{DW}}=$-500\text{\,}\mathrm{n}\mathrm{m}$$ and a chemical
modulation at z${}_{\mathrm{DW}}=0$. The magnetic field strength is increased
linearly, and a damping coefficient $\alpha=1$ is used, in order to mimic a
quasistatic situation. The circulation is initially positive in the BPW, which
is the stable one during motion along $+\hat{\bm{z}}$ in a homogeneous wire.

Figure 5a shows the DW internal energy during its propagation through the
modulation, along with selected micromagnetic views and tracking of the Bloch
point. The left and right panels correspond to a chemical modulation length of
$20\text{\,}\mathrm{n}\mathrm{m}$ and $100\text{\,}\mathrm{n}\mathrm{m}$,
respectively. The energy profiles are similar to those in Figure 1 for
$z_{\mathrm{DW}}\lesssim$200\text{\,}\mathrm{n}\mathrm{m}$$, i.e., an energy
barrier due to the magnetostatic-driven repulsion. However, the situation
changes drastically at later stages, although reminiscent of a potential well
for $z_{\mathrm{DW}}\gtrsim$200\text{\,}\mathrm{n}\mathrm{m}$$. First, the
profiles are largely asymmetric when comparing the left and right sides of the
modulation. Second, discontinuities of the internal energy are observed.
Third, the propagation field is considerably larger than for identical
circulations $C+C+$, e.g., $18\text{\,}\mathrm{mT}$ for a
$20\text{\,}\mathrm{n}\mathrm{m}$-long modulation. For the case of a
$100\text{\,}\mathrm{n}\mathrm{m}$-long modulation, the depinning field
reaches $32\text{\,}\mathrm{mT}$. All this hints at a drastically-different
propagation mechanism than in the case of $C+C+$, which we examine now in more
detail.

Figure 5: BPW $C+$ propagation through a chemical modulation of opposite
circulation $C-$ for a $90\text{\,}\mathrm{n}\mathrm{m}$-diameter Permalloy
nanowire. a) Internal energy $E_{0}$ versus BPW position $z_{\mathrm{DW}}$.
Left and right panels illustrate the case of chemical modulation lengths of
$20\text{\,}\mathrm{n}\mathrm{m}$(open symbols) and
$100\text{\,}\mathrm{n}\mathrm{m}$(filled symbols), respectively. Red symbols
denote the amplitude of the axial magnetic field, increased every 2 ns. Blue,
black and green symbols represent the internal energy for DW configurations of
BPW $C+$, TVW and BPW $C-$, respectively. b) Simulated micromagnetic states
shown from outer and inner perspectives at selected $z_{\mathrm{DW}}$
indicated by black dashed lines in a). The color code denotes
$m_{\mathrm{\varphi}}$ and $m_{\mathrm{\rho}}$, respectively. The black square
corresponds to the Bloch point position, while the elongated blue rectangle
indicates the chemical modulation.

Figure 5b displays the micromagnetic simulations outer views and surface inner
views. The color code denotes the azimuthal $m_{\mathrm{\varphi}}$ and radial
magnetization $m_{\mathrm{\rho}}$, respectively. The black square indicates
the Bloch point position, while the elongated blue rectangle denotes the
chemical modulation. Labels on the left correspond to configurations at
various zDW values indicated by the black dashed lines in Figure 5a. Upon
analyzing the temporal evolution of these configurations, several key
observations emerge: initially, the BPW and modulation display opposite
$m_{\mathrm{\varphi}}$ contrast due to their antiparallel circulations
(labeled as $a$). At this point the inner view presents $m_{\mathrm{\rho}}=\pm
1$ isolines in the bulk connecting the surface to the Bloch point. As the BPW
approaches the modulation it shrinks while increasing in energy (blue symbols
in Figure 5a). At the point labeled as $b$, a new isoline
$m_{\mathrm{\rho}}=1$ crosses the bulk, ending by the disappearance of the
blue isoline $m_{\mathrm{\rho}}=-1$ with the Bloch point, resulting in a TVW-
type DW (labeled as $c$). Note that between $b$ and $c$ the topology of the
wall remains that of a BPW 29, 50. The TVW-type configuration is represented
by the black symbols in Figure 5a. Following this, the evolution diverges for
short and large modulation lengths. For short modulations of
$20\text{\,}\mathrm{n}\mathrm{m}$ length (left panel), the energy increases
reaching a maximum of $2.11\text{\times}{10}^{-16}\text{\,}\mathrm{J}$, at
which point the Bloch point reappears in the volume and pins onto the chemical
modulation (labeled as $d$). At this stage, the Bloch point lags behind the
BPW (see state $d$ in the outer-micromagnetic view), indicating a state far
from equilibrium. The equal $m_{\mathrm{\varphi}}$ contrast in wall and
modulation illustrates a reversal of BPW circulation. The BPW configuration
with negative circulation is represented by the green symbols in Figure 5a,
which indicate the subsequent decrease in energy associated with depinning
(labeled as $e$), followed by a potential well for parallel circulations, as
illustrated in Figure 1. In contrast, for a $100\text{\,}\mathrm{n}\mathrm{m}$
modulation, the energy increases further, up to
$2.79\text{\times}{10}^{-16}\text{\,}\mathrm{J}$, until the TVW-type
configuration propagates through the modulation (labeled as $d$). This is
followed by a decrease in energy until the TVW depins from the modulation and
propagates freely along $\hat{\bm{z}}$ (labeled as $e$). Subsequently, the
Bloch point re-enters the volume, and a BPW $C-$ of reversed circulation keeps
propagating (not shown). The larger deppining field induces a high excitation
to the system causing the re-injection of the Bloch point in the volume (or
transformation to BPW $C-$) at
$z_{\mathrm{DW}}>$600\text{\,}\mathrm{n}\mathrm{m}$$.

To better understand the microscopic mechanism of the domain wall
transformation, and link it to the non-trivial energy evolution in Figure 5,
it is helpful to perform a topological analysis50. To mimic a more realistic
situation, the simulations were conducted with an $\alpha=0.02$ and an axial
magnetic field of $50\text{\,}\mathrm{mT}$. The tracking of the volume and
surface singularities was performed by using a post-processing tool. Figure 6a
displays the outer micromagnetic view of the initial state. The color code
corresponds to the azimuthal magnetization, being red for the BPW curling
along $+m_{\mathrm{\upvarphi}}$ and blue for the chemical modulation
magnetized along $-m_{\mathrm{\upvarphi}}$. The three-dimensional view in
Figure 6b displays the trace of the Bloch point (in black) and surface vortex-
antivortex pairs (in colors). The chemical modulation is represented by the
grey disk. The displacement of the Bloch point occurs along the wire axis as
is well known for domain wall motion in an homogeneous nanowire51. Upon
approaching the chemical modulation, the Bloch point travels towards the
surface until it annihilates with a surface vortex or antivortex (labeled
$out$). Subsequently, an azimuthal displacement of the latter occurs until a
Bloch point is injected into the volume (labeled $in$) and propagates trough
the chemical modulation.

The evolution of the two vortex-antivortex pairs is displayed in Figure 6d
along the axial $\hat{\bm{z}}$ and azimuthal $\hat{\bm{\upvarphi}}$
directions. It is important to note that here we depict a zoom of the critical
events, while the Bloch point (represented by the black symbols) has existed
in the system since $t=0$. The first pair nucleated V${}_{1}^{+}$ \-
AV${}_{1}^{+}$ shows positive polarity (red and orange lines), whereas the
second pair V${}_{2}^{-}$ \- AV${}_{2}^{-}$ shows negative polarity (blue and
green lines). Within each V-AV pair, the vortex and antivortex move away from
each other along the azimuthal direction. We believe that the polarity of the
first pair appearing can be understood as it matches the polarity of the head-
to-head BPW, and also the one favored by the chirality of the
Landau–Lifshitz–Gilbert (LLG) equation via $\mathbf{H}\times\mathbf{M}$. The
reason for the opposite polarity of the second pair is probably to reduce the
global magnetostatic energy. Overall, the detailed process involves the Bloch
point reaching the surface at the moment when the vortex with negative
polarity V${}_{2}^{-}$ (blue) and the antivortex with positive polarity
AV${}_{1}^{+}$ (orange) merge. At this precise moment, only V${}_{1}^{+}$ and
AV${}_{2}^{-}$ remain in the system far from each other. Later, a Bloch Point
with negative polarity re-enters in the volume as V${}_{1}^{+}$ and
AV${}_{2}^{-}$ annihilate.

Figure 6: Topology of field-driven BPW circulation switching prior to
propagate through a chemical modulation of antiparallel circulation. a) Outer
micromagnetic view of the initial state, with a BPW of positive circulation
and a modulation of negative circulation with respect to $\hat{\bm{z}}$. The
color code represents $m_{\mathrm{\varphi}}$. (b) Black: Bloch point
trajectory while crossing the modulation under an axial field of
$50\text{\,}\mathrm{mT}$. Color: generated pairs of vortex and antivortex on
the surface. (c) Unrolled surface of the wire showing the vortex/antivortex
pair formation to the left of the modulation. The color code represents
$m_{\mathrm{\rho}}$. (d) Temporal evolution along $\hat{\bm{z}}$ and
$\hat{\bm{\upvarphi}}$ of Bloch point (black) and vortex-antivortex pairs
(colored curves).

The phenomenon of BPW circulation switching upon crossing a chemical
modulation under a magnetic field represents a novel discovery. Nevertheless,
the underlying mechanism mirrors the one reported in ref Riz et al. 29 for BPW
switching in an homogeneous nanowire driven by an anti-parallel Oersted field,
sharing the same topological features. We can draw an analogy between this
phenomenon and the mechanisms of coherent reversal versus nucleation-
propagation for the magnetization reversal of magnetic systems: the switching
of BPW circulation via the homogeneous rise of $m_{\mathrm{\rho}}$ to $\pm 1$
on the wire periphery would imply a substantial dipolar energy cost. Instead,
here we observe the local nucleation of an area with $m_{\mathrm{\rho}}=\pm
1$, which then propagates around the perimeter through the motion of surface
vortex-antivortex pairs along opposite azimuthal directions.

Regarding the switching of BPW in a homogeneous or in chemically-modulated
system, it is important to note the distinctions between the two scenarios.
The unrolled map of Figure 6c illustrates that the vortex-antivortex pairs
nucleate always on the same side of the BPW, between the BPW interface and the
chemical modulation. This contrasts with the scenario of the Œrsted field
induced BPW switching on a homogeneous nanowire (without chemical modulation),
where one vortex or antivortex of same polarity appears at each side of the
wall29. Here, the strong exchange interaction with the opposite circulation of
the modulation acts in a very similar fashion to an Œrsted field, but on one
side only of the DW.

##  0.3 Conclusion

In summary, we have experimentally outlined and analyzed, through
micromagnetic simulations, the interaction between higher-magnetization
chemical modulations and Bloch Point walls (BPWs) in cylindrical nanowires.
These modulations act as local potential wells, that effectively pin domain
walls due to their shared azimuthal curling, thereby lowering the total energy
of the system. However, the mechanism of propagation through a modulation is
complex. At distances larger than the BPW width, the interaction is primarily
repulsive, driven by magnetostatics. As the distance decreases, exchange
interactions become significant, breaking the degeneracy between cases of
identical or opposite circulation senses in the wall and the modulation. In
the former case, the BPW can propagate through the modulation relatively
unchanged, although the depinning field is higher than the field required to
push the wall into the energy well of the modulation. For opposite
circulations, the modulation forces the BPW to change its circulation before
propagation can occur, effectively acting as a polarizing device. This process
involves the annihilation/re-nucleation of a Bloch point and a transient
topology state of a transverse-vortex wall, a phenomenon reported here for the
first time. This process is similar, and thus consistent, with the change of
domain wall type in homogeneous nanowires driven by field or Œrsted-field,
shedding light on the generality of the phenomenon. These findings open the
possibility of utilizing chemical modulations as a polarizing component,
thereby allowing the control of the internal structure of domain walls.

##  0.4 Methods

Permalloy (Fe20Ni80) cylindrical nanowires with Fe-rich chemical modulations
were synthesized by single-bath template-assisted electrochemical deposition.
Nanoporous anodic aluminum oxide (AAO) templates were prepared by hard
anodization of Al disks (Goodfellow, $99.999\text{\,}\%$ in purity) in a
water-based solution of oxalic acid ($0.3\text{\,}\mathrm{M}$) and ethanol
($0.9\text{\,}\mathrm{M}$), applying $140\text{\,}\mathrm{V}$ of anodization
voltage at
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}1\text{\,}\mathrm{\SIUnitSymbolCelsius}$
for $2.5\text{\,}\mathrm{h}$. The remaining Al was etched with an aqueous
solution of CuCl2 ($0.74\text{\,}\mathrm{M}$) and HCl
($3.25\text{\,}\mathrm{M}$), the oxide barrier was removed and the pores
opened to a final diameter of $\approx$$120\text{\,}\mathrm{n}\mathrm{m}$ with
H3PO4 (5 % vol.).

The growth conditions were similar to those described in ref Ruiz-Gómez et al.
52. For chemical modulations with a length below
$60\text{\,}\mathrm{n}\mathrm{m}$, the composition was Fe65Ni35, while for
lengths above $60\text{\,}\mathrm{n}\mathrm{m}$, the composition was Fe80Ni20.
After the electrochemical growth, the AAO templates were dissolved in
H3PO4($0.4\text{\,}\mathrm{M}$) and H2CrO4($0.2\text{\,}\mathrm{M}$). Then,
the nanowires were dispersed on $20\text{\,}\mathrm{n}\mathrm{m}$ thick Si3N4
windows to allow for transmission microscopy. To nucleate domain walls,
nanowires were contacted electrically using laser lithography in order to
allow the injection of $\text{\,}\mathrm{GHz}$ bandwidth electric pulses.
Resistivity values obtained are around
$20\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{${}\cdot{}$}\mathrm{cm}$.
To prevent sample oxidation and to increase thermal transfer to the Si3N4, a
$20\text{\,}\mathrm{n}\mathrm{m}$-thick layer of Al2O3 was deposited on the
device by Atomic Layer Deposition (ALD). Additionally,
$100\text{\,}\mathrm{n}\mathrm{m}$ of Al was deposited on the backside of the
substrate.

The wires were imaged with X-ray Magnetic Circular Dichroism (XMCD) coupled to
Transmission X-ray Microscopy (TXM) 53 at the MISTRAL beamline of ALBA
Synchrotron and by X-ray ptychography54, 55, 56 at HERMES STXM beamline of
SOLEIL synchrotron. The photon energy was set to the Fe L3 absorption edge,
and the sample holder could be rotated from the plane normal to the X-ray
incidence in order to be sensitive to both axial and azimuthal magnetization
components. During TXM imaging, series of about 64 images per polarity were
were acquired, registered and averaged in order to increase the signal-to-
noise ratio, while keeping under control the loss of spatial resolution due to
time drift. Ptychographic reconstruction was carried out using the open-source
PyNX software developed at the European Synchrotron Radiation Facility 57. If
not initially present, DWs were nucleated in situ by applying a
$10\text{\,}\mathrm{n}\mathrm{s}$-long pulse of current with amplitude
$1\text{\times}{10}^{12}\text{\,}\mathrm{A}\text{${}\cdot{}$}{\mathrm{m}}^{-2}$.
We performed micromagnetic simulations with the mumax3 code58. The system
considered is a cylindrical Permalloy nanowire of
$90\text{\,}\mathrm{n}\mathrm{m}$ diameter and
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ length, with a
$20\text{\,}\mathrm{n}\mathrm{m}$-long or
$100\text{\,}\mathrm{n}\mathrm{m}$-long Fe80Ni20 chemical modulation at its
center. In order to mimic an infinite wire, the magnetic charges at the wire
ends were removed numerically. The mesh size used was
$2\text{\,}\mathrm{n}\mathrm{m}$. The following material parameters were
used59: spontaneous magnetization
$M_{1}=$8\text{\times}{10}^{5}\text{\,}\mathrm{A}\mathrm{/}\mathrm{m}$$ for
the Permalloy segments,
$M_{2}=$14\text{\times}{10}^{5}\text{\,}{}\mathrm{A}\mathrm{/}\mathrm{m}$$ for
the chemical modulation, single exchange stiffness for all materials
$A=$1.3\text{\times}{10}^{-11}\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}$$ and
zero magnetocrystalline anisotropy. The choice of an identical value for
exchange stiffness stems from the similarity of ordering temperature for the
two compositions. The resulting dipolar exchange lengths are
$5.6\text{\,}\mathrm{n}\mathrm{m}$ and $3.3\text{\,}\mathrm{n}\mathrm{m}$,
respectively. To study static or quasi-static dynamics a damping term of
$\alpha=1$ was used. The initial state considered was a BPW at z $=$\pm
500\text{\,}\mathrm{n}\mathrm{m}$$ and a chemical modulation centered at z
$=0$. Circulations of both the BPW and the modulation are defined arbitrarily
with respect to $+\hat{\bm{z}}$, and labeled $C+$ for positive and $C-$ for
negative.

When the tracking of topological elements characterizing the type of wall is
required (Bloch point for BPW, surface vortex/antivortex for the TVW), we use
an in-house post-processing tool, developed specifically for tetrahedron-based
finite-elements 50. Therefore, in this case the micromagnetic simulations were
conducted with feeLLGood code 60, 61, 62, 63, a home-made micromagnetic code
based on finite-element methods.

This work received financial support from the French RENATECH network,
implemented at the Upstream Technological Platform in Grenoble PTA
(ANR-22-PEEL-0015), and from the French National Research Agency (Grant No.
ANR-17-CE24-0017 MATEMAC-3D; Grant ANR-22-CE24-0023 DIWINA), from the Spanish
(MCIN/AEI/10.13039/501100011033 through Projects PID2020-117024GB-C43,
TED2021-130957B-C52 and CEX2020- 001039-S). S. R-G acknowledges support from
the Humboldt foundation grant 1223621 and Marie Curie fellowship grant
GAP-101061612. M. F. acknowledges the funding from MICIN through grant number
PID2021-122980OB-C54. We acknowledge support from the team of the Nanofab
platform (CNRS Néel institut), from the ALBA in-house research program and
MISTRAL and CIRCE beamlines and from the SOLEIL HERMES beamline.

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# Survival of New Physics: An Anomaly-free Neutral Gauge Boson at the LHC

Ying Zhang1111Email address<EMAIL_ADDRESS>Qing Wang2,3222Email
address<EMAIL_ADDRESS>1School of Science, Xi’an Jiaotong
University, Xi’an, 710049, P.R.China
2Department of Physics, Tsinghua University, Beijing 100084, P.R.China
3Center for High Energy Physics, Tsinghua University, Beijing 100084,
P.R.China

###### Abstract

An anomaly-free $U(1)^{\prime}$ effective Lagrangian as the most common new
physics beyond the standard model is proposed to survey the maximal parameter
space constrained by electroweak precise measurements at the LEP and direct
detection in dilepton decay channel at the LHC at $\sqrt{s}=7$ TeV. By the
global fit of effective couplings of $Z$ boson to the SM fermions,
$\Delta_{11},\Delta_{21},g_{2}\Delta_{31}$ related to mixings and $r$ related
to $U(1)^{\prime}$ charge assignment are bounded. The allowed areas are
plotted in the not only $r$-$g_{2}$ but also $m_{Z^{\prime}}$-$g_{2}$ planes,
which show that a sub-TeV $Z^{\prime}$ is still permissible as long as the
coupling $g_{2}\sim 0.01$. The results provides a prime requirement to an
extra $U(1)^{\prime}$ gauge boson and hinds the direction of possible new
physics beyond the standard model. The possible signal in dilepton decay
channel at LHC at $\sqrt{s}=14$ TeV is also provided.

##  I motivation

When people were exciting for the found of Higgs-like particle with about
125GeV mass at the LHC, we have to worry about the survival of new physics
(NP) right away. Almost all experiments are proving the Standard Model (SM),
the space left to NP is less and less. There is an impending question we are
dying to know that how much space can NP survive. It is a good checking point
to choose a neutral gauge boson, which often appears in GUT and superstring
model associated with $U(1)^{\prime}$ group, as a popular candidate of NP
beyond the SM.

There are many relative issues summarized by A. Leike and P. Langacker
LeikeReview ; LangackerReview . However, duo to different motivations,
$Z^{\prime}$ interactions to the SM fermions is set by models, which makes the
results are highly model-dependent. The minimal mass of the vector boson is
limited at about $1.8$ TeV. In the paper, we relax any possible motivations
and roles coming from underlying theory or phenomenology, and construct a
model-independent effective Lagrangian to describe $U(1)^{\prime}$ gauge boson
$Z^{\prime}$ only following the requirement of gauge symmetry. In bosonic
sector, all possible mass and kinetic mixings meeting gauge symmetry are
investigated. And in fermionic sector, anomaly-free charge assignments is
required to satisfy gauge symmetry. The interactions to fermions are dominated
by a global coupling $g_{2}$ and a charge assignment parameter $r$. They are
keys to realize model-independent description. We consider constrains from not
only electroweak precise observables but also direct detection at LHC, and
then decide the possible parameter space left to the simplest NP particle.

The article is organized as following: firstly, anomaly-free $Z^{\prime}$
effective Lagrangian is constructed. And then, we diagonalize gauge bosons
mixing matrixes to obtain mass eigenvalues of neutral bosons. The limit to
parameters is studied based on electroweak precise measurements in LEP and
direct search at the LHC at $\sqrt{s}=7$ TeV. The allowed area is shown in
$m_{Z^{\prime}}$-$g_{2}$ plane. Finally, the possible signal in dilepton final
states at $\sqrt{s}=14$ TeV at the LHC is predicted.

##  II effective Lagrangian

To construct-independent $Z^{\prime}$ effective theory, we will focus on
parameterizations in $Z^{\prime}$ mixings and interactions. Denote
$U(1)^{\prime}$ gauge eigenstate as $X_{\mu}$. On gauge eigenstates base
$(W_{\mu}^{3},B_{\mu},X_{\mu})^{T}$, the covariant derivative has the form

$D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+igW_{\mu}\hat{U}-ig^{\prime}\hat{U}\frac{\tau_{3}}{2}B_{\mu}-i{g^{\prime\prime}}\hat{U}X_{\mu}.$

Here, $\hat{U}$ is non-linear realization of Goldstone bosons. $W_{\mu}$ and
$B_{\mu}$ are respectively gauge field of $SU(2)_{L}$ and $U(1)_{Y}$ with
gauge coupling $g$ and $g^{\prime}$. Using $SU(2)_{L}$ covariant building
blocks $T\equiv\hat{U}\tau_{3}\hat{U}^{\dagger}$ and
$V_{\mu}\equiv(D_{\mu}\hat{U})\hat{U}^{\dagger}$, mass terms arise from 4
operators in $p^{2}$ order:
$tr[V_{\mu}V^{\mu}],tr[TV_{\mu}]^{2},tr[V_{\mu}]^{2}$ and
$tr[TV_{\mu}]tr[V^{\mu}]$. The first operator corresponds to the electroweak
standard model. The second one provides an extra mass correction to isospin
third-component $W^{3}_{\mu}$. The third one generates non-standard mixing
between $B_{\mu}$ and $W^{3}_{\mu}$. And the last one parameterizes
$Z$-$Z^{\prime}$ mixing. However, the second and third one can be absorbed in
the re-definition of gauge couplings $g$ and $g^{\prime}$, and are not
independent OurCPC2012 . Similarly, kinetic mixing terms are also controlled
by 4 operators: $tr[TW_{\mu\nu}]^{2}$, $tr[TW_{\mu\nu}]B^{\mu\nu}$,
$tr[TW_{\mu\nu}]X^{\mu\nu}$ and $B_{\mu\nu}X^{\mu\nu}$ OurJHEP2008 ;
OurJHEP2009 . The first operator corresponds a correction to $W_{\mu}^{3}$
kinetic term. The second one yields kinetic mixing between $W^{3}_{\mu}$ and
$B_{\mu}$. The third and forth ones cause $U(1)^{\prime}$ boson $X_{\mu}$
kinetic mixings with $W^{3}_{\mu}$ and $B_{\mu}$, respectively. The first two
operators are non-standard term beyond the SM and there is no any reason to
neglect these invariant ones EWCL .

Expressing these operators in obvious gauge fields, Lagrangian related to
mixings is written as

$\displaystyle\mathcal{L}_{mix}$ $\displaystyle=$
$\displaystyle\frac{m_{0}^{2}}{2}(c_{W}W_{\mu}^{3}-s_{W}B_{\mu})^{2}+\frac{m_{1}^{2}}{2}X_{\mu}X^{\mu}+2\beta
m_{0}m_{1}X_{\mu}(c_{W}W_{\mu}^{3}-s_{W}B_{\mu})$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{1}{4}(1-\alpha_{b})(W_{\mu\nu}^{3})^{2}$
$\displaystyle+\frac{1}{2}\alpha_{a}B_{\mu\nu}W_{\mu\nu}^{3}+\alpha_{c}X^{\mu\nu}W_{\mu\nu}^{3}+\alpha_{d}X^{\mu\nu}B_{\mu\nu}$

with mass mixing $\beta$ OurCPC2012 . Here, $m_{0}$ and $m_{1}$ are $Z$ and
$Z^{\prime}$ mass in gauge eigenstates and $s_{W},c_{W}$ are sine and cosine
of Weinberg angle.

To $Z^{\prime}$ interactions to EW gauge bosons $W^{\mu}$ and $Z$, some
independent parameters control them, which may come from underlying theory and
not vanish even though no $Z^{\prime}$ mixings. For example the decay channel
$\Gamma(Z^{\prime}\rightarrow W^{+}W^{-})$ may arise from $Z$-$Z^{\prime}$
mixing or high order operator $X_{\mu\nu}tr[T[V^{\mu},V^{\nu}]]$. The former
is suppressed by small $Z$-$Z^{\prime}$ mixing angle, and the later stands for
possible NP which has no any promption from high energy experiments. For the
similar reason, all these decays of $Z^{\prime}$ to EW bosons are expected to
be small.

The extra neutral current interaction is introduced like

$\mathcal{L}_{int}=-g_{2}\sum_{f}\bar{f}\gamma_{\mu}(y^{\prime}_{Lf}P_{L}+{y^{\prime}_{Rf}}P_{R})fX^{\mu}$

with left-handed (right-handed) fermionic $U(1)^{\prime}$ charge
$y^{\prime}_{Lf}$ ($y^{\prime}_{Rf}$). To keep gauge symmetry, $U(1)^{\prime}$
charge assignments must be anomaly-free. For the family universal case, there
are 6 independent charges: $y^{\prime}_{l}$ and $y^{\prime}_{q}$ for left-
handed leptons and quarks,
$y^{\prime}_{u},y^{\prime}_{d},y^{\prime}_{\nu},y^{\prime}_{e}$ for right-
handed up-quark, down-quark, neutrino and electron, respectively. From
$[SU(3)_{C}]^{2}U(1)^{\prime}$, $[SU(2)]_{L}^{2}U(1)^{\prime}$ and
$[U(1)_{Y}]^{2}U(1)^{\prime}$ cancellation requirements, we have

$\displaystyle
y^{\prime}_{l}=-3y^{\prime}_{q},~{}~{}~{}y^{\prime}_{d}=2y^{\prime}_{q}-y^{\prime}_{u},~{}~{}~{}y^{\prime}_{e}=-2y^{\prime}_{q}-y^{\prime}_{u}.$

$U(1)_{Y}[U(1)^{\prime}]^{2}$ anomaly is cancelled automatically. If and only
if the number of right-handed neutrinos is 3, $[U(1)^{\prime}]^{3}$ anomaly
and gravitational-gauge mixing anomaly can be satisfied simultaneously and the
charge is $y^{\prime}_{\nu}=-4y^{\prime}_{q}+y^{\prime}_{u}$. Without loss of
generality, the coupling $g_{2}$ is normalized so that $y^{\prime}_{u}=1$.
Now, these couplings are dominated by two free parameters: coupling $g_{2}$
that controls global intensity and charge ratio $r(\equiv
y^{\prime}_{q}/y^{\prime}_{u})$ that assigns flavor charges

$\displaystyle
y^{\prime}_{l}=-3r,~{},y^{\prime}_{q}=1,~{}y^{\prime}_{u}=1,~{}y^{\prime}_{d}=2r-1,~{}y^{\prime}_{e}=-2r-1,~{}y^{\prime}_{\nu}=-4r+1.$

Briefly, anomaly-free $Z^{\prime}$ effective theory, inspired by
$U(1)^{\prime}$ gauge symmetry, can be parameterized by mass mixing $\beta$,
kinetic mixing $\alpha_{c},\alpha_{d}$ in bosonic sector ($\alpha_{a}$ and
$\alpha_{b}$ respectively parameterize non-standard
$W^{3}_{\mu\nu}W^{3\mu\nu}$ and $W^{3}_{\mu\nu}B^{\mu\nu}$ electroweak bososic
kinetic mixing terms, which are not relative to $Z^{\prime}$ boson), coupling
$g_{2}$ and the charge ratio $r$ in fermionic sector. The full Lagrangian for
$U(1)^{\prime}$ boson is

$\mathcal{L}_{Z^{\prime}}=\mathcal{L}_{mix}+\mathcal{L}_{int}.$

##  III diagonalization matrix

To obtain mass eigenstates, let’s make a rotation $U$

$\displaystyle\left(\begin{array}[]{c}W^{3}_{\mu}\\\ B_{\mu}\\\
X_{\mu}\end{array}\right)=U\left(\begin{array}[]{c}Z_{\mu}\\\ A_{\mu}\\\
Z^{\prime}_{\mu}\end{array}\right)$ (7)

to diagonalize mass and kinetic mixings simultaneously. Considering rotation
$U$ reducing to Weinberg’s rotation when no $Z^{\prime}$ mixings, $U$ may be
expressed as the sum of Weinberg’s rotation and $Z^{\prime}$ mixing
corrections

$\displaystyle
U=\left(\begin{array}[]{ccc}c_{W}+\Delta_{11}&s_{W}+\Delta_{12}&\Delta_{13}\\\
-s_{W}+\Delta_{21}&c_{W}+\Delta_{12}&\Delta_{23}\\\
\Delta_{31}&\Delta_{32}&1+\Delta_{33}\end{array}\right).$

Notice that 9 $\Delta_{ij}$ ($i=1..3,j=1..3$) are not independent, they are
determined by only 7 phenomenological parameters: mass $m_{1}$ and $m_{2}$,
mass mixing $\beta$, and kinetic mixings $\alpha_{a,b,c,d}$, i.e.
$\Delta_{ij}=\Delta_{ij}(m_{1},m_{2},\beta,\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d})$.
So, we must find two constraint conditions on $\Delta$s. One is
$s_{W}\Delta_{22}=c_{W}\Delta_{12}$, which results from the requirement for a
massless photon. Another constraint is $\Delta_{32}=0$ due to the requirement
of keeping photon coupling vector-type StueckelbergNote .

After the rotation (III), the mass eigenvalues of $Z$ and $Z^{\prime}$ are

$\displaystyle m_{Z}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}(1+2c_{W}\Delta_{11}-2s_{W}\Delta_{21})+4\beta
m_{0}m_{1}\Delta_{31}+\mathcal{O}(\Delta^{2})$ $\displaystyle
m_{Z^{\prime}}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}(1+2\Delta_{33})+4\beta
m_{0}m_{1}(c_{W}\Delta_{13}-s_{W}\Delta_{23})+\mathcal{O}(\Delta^{2}).$

##  IV constraints in LEP

Due to the good fit of the SM to electroweak precise observables, $Z^{\prime}$
effective theory must be constrained by precise electroweak experiments. The
heavy neutral boson contributes to low energy observables by two fashions:
mixings and $Z^{\prime}$ exchange. They often coexist in phenomenology.
However, corrections to fermionic couplings only come from mixings. A
theoretical observables can be divided into the SM part and $Z^{\prime}$
contribution

$\mathcal{O}_{th}=\mathcal{O}_{SM}+\Delta\mathcal{O}_{Z^{\prime}}.$

Due to the triumph of the SM, $\Delta\mathcal{O}_{Z^{\prime}}$ must be very
small. Generally, it’s a safe precession to neglect high order effect of
$Z^{\prime}$ LeikeReview . Using rotation (7), vector and axial-vector
couplings in weak neutral current are given by

$\displaystyle g^{f}_{V,A}=g^{f,0}_{V,A}+\Delta g^{f}_{V,A}$

$g^{f,0}_{V,A}$ is the SM couplings in tree level,
$g^{f,0}_{V}=t_{3L}^{f}-2q^{f}s_{W}^{2}$ and $g^{f,0}_{A}=t_{3L}^{f}$. The
effective couplings with radiative corrections to propagators and vertices in
the SM can be found in LEP report LEP2006 . The $Z^{\prime}$ corrections are

$\displaystyle\Delta
g^{f}_{V,A}=c_{W}\Delta_{11}t_{3iL}+s_{W}\Delta_{21}(y_{iL}\pm
y_{iR})+\frac{s_{W}c_{W}}{e}g_{2}\Delta_{31}(y^{\prime}_{iL}\pm
y^{\prime}_{iR}).$

$Z^{\prime}$ corrections are constrained by $Z$-pole observables. The validity
of constraint depends on two factors: experiments precision and calculation
precision based on the SM. Although there are 14 observables in LEP/SLD at
$Z$-pole, the SM calculation can be parameterized into only 4 radiate
correction factors: $\Delta\rho,\Delta r_{W},\Delta\kappa$ and
$\Delta\rho_{b}$ (or express into 4 new parameters
$\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{b}$ introduced by Altarelli,
et.al. LEP2006 ; EWepsilon ). Considering the number of independent
measurements in experiment and the SM calculation, it’s a balanced treatment
to choice pseudo observables, 8 effective coupling constants $g_{V,A}^{f}$ for
$f=l,\nu,c,b$, to limit $Z^{\prime}$ parameters. We minimize

$\displaystyle\chi^{2}=\sum_{f}\frac{(g_{V,A}^{f,exp}-g_{V,A}^{f,SM}-\Delta
g_{V,A}^{f})^{2}}{(\delta g_{V,A}^{f,exp})^{2}}$

where supercript exp, SM respectively denote the corresponding experiment
values and the SM fit values, and $\delta g_{V,A}^{f,exp}$ are their
experimental errors. The four free parameters are
$\Delta_{11},\Delta_{21},g_{2}\Delta_{31}$ and $r$. As we have mentioned,
$\Delta_{ij}$ are the functions of mixing parameters. We must keep the fit
parameters independent. It can be proved by calculating rotation matrixes
invoked by mass mixing $\beta$, and kinetic mixings $\alpha_{c,d}$,
respectively, even if EW boson kinetic mixing $\alpha_{a}$ and $\alpha_{b}$
vanishing. After detailed calculations, we arrive at the global fit results in
Table. 1. $Z^{\prime}$ slight improves fit confidence level from 93% (about
$1.8\sigma$) to 96% (about $2.1\sigma$). The parameter ranges are shown in
Table. 2.

Table 1: LEP experiment results on the effective coupling constants and the SM Z-pole fit. Data come from Table 7.9 and Table G.3 in literature LEP2006 . The last row represents the corresponding C.L.. coupling | exp. | SM fit |  $Z^{\prime}$ fit
---|---|---|---
$g_{A}^{\nu}$ | $+0.50075\pm 0.00077$ | $+0.50199\pm^{0.00017}_{0.00020}$ | $+0.50063$
$g_{A}^{l}$ | $-0.50125\pm 0.00026$ | $-0.50127\pm^{0.00020}_{0.00017}$ | $-0.50116$
$g_{A}^{b}$ | $-0.5144\pm 0.0051$ | $-0.49856\pm^{0.00041}_{0.00020}$ | $-0.49845$
$g_{A}^{c}$ | $+0.5034\pm 0.0053$ | $+0.50144\pm^{0.00017}_{0.00020}$ | $+0.50013$
$g_{V}^{\nu}$ | $+0.50075\pm 0.00077$ | $+0.50199\pm^{0.00017}_{0.00020}$ | $+0.50063$
$g_{V}^{l}$ | $-0.03753\pm 0.00037$ | $-0.03712\pm 0.00032$ | $-0.03751$
$g_{V}^{b}$ | $-0.3220\pm 0.0077$ | $-0.34372\pm^{0.00049}_{0.00028}$ | $-0.34267$
$g_{V}^{c}$ | $+0.1873\pm 0.0070$ | $+0.19204\pm 0.00023$ | $+0.19185$
$\chi^{2}$/dof | - | $24.6/8$ | $20.1/8$
P value | - | $93\%$ | $96\%$
Table 2: global fit results. The corresponding errors come from diagonal elements of the inverse of Hessian matrix. The ranges in $2\sigma$ C.L. are listed in the last column. quantity | fit result | range in $2\sigma$
---|---|---
$\Delta_{11}$ | $-0.00067\pm 0.00040$ | $(-0.00147,0.00013)$
$\Delta_{21}$ | $0.0017\pm 0.0076$ | $(-0.0135,0.0169)$
$g_{2}\Delta_{31}$ | $-0.00044\pm 0.0018$ | $(-0.00404,0.00316)$
$r$ | $-0.015\pm 1.1$ | $(-2.215,2.185)$

Notice that $Z^{\prime}$ mass does not been limited by effective couplings.
Generally, $m_{Z^{\prime}}$ can be limited by $\rho$ or $Z$ mass correction by
$Z-Z^{\prime}$ mixing. There are enough more results on the issue in
literature zprimebound  which shown that a small mass mixing corresponds a
heavy $Z^{\prime}$ and vice versa. For the typical value $\beta\sim 10^{-3}$,
$m_{Z^{\prime}}$ is several TeV.

##  V search at the LHC

The LHC has searched a vector resonance decaying into dilepton final states at
$\sqrt{s}=7$ TeV ATLASZprime . No statistically significant excess above the
SM expectation is observed, which strictly limits $Z^{\prime}$ couplings to
fermions. Figure.1 shows the 95% C.L. allowed areas in $g_{2}$-$r$ plane with
$m_{Z^{\prime}}=0.8,~{}1.0,~{}1.5,~{}2.0$ TeV respectively. The theoretical
cross sections are calculated by Madgrapha5 ver1.5.12. Compared with observed
limits on $\sigma(pp\rightarrow Z^{\prime}\rightarrow ll)$ at ATLAS, the
values of $g_{2}$ and $r$ can be determined with fixed $m_{Z^{\prime}}$. It
indicate that a light $Z^{\prime}$ with enough small coupling is not
eliminated.

Figure 1: $95\%$ C.L. possible allowed area in the $g_{2}$ vs $r$ plane at 7
TeV LHC. Exclusion lines correspond to excepted $2\sigma$ signal regions at
ATLAS in dilepton decay channel in Fig.3 of ATLASZprime . Black vertical lines
denote $r$ fit ranges in $1\sigma$ and $2\sigma$ C.L., respectively.

More clearly, Fig. 2 plots $7$ TeV ATLAS allowed parameter space in the plane
of $Z^{\prime}$ mass and coupling $g_{2}$, corresponding the different $r$
C.L.. A sub-TeV $Z^{\prime}$ is allowed when the coupling $g_{2}\sim 0.01$.

Figure 2: allowed $Z^{\prime}$ mass by $7$ TeV ATLAS in
$m_{Z^{\prime}}$-$g_{2}$ plane in different charge ratio $r$ fit range
corresponding $1\sigma$, $2\sigma$ and $>2\sigma$ C.L..

A possible $\sigma(pp\rightarrow Z^{\prime}\rightarrow
e^{+}e^{-},\mu^{+}\mu^{-})$ signal in dilepton final state at $\sqrt{s}=14$
TeV LHC is also calculated by Madgraph5, which is shown in Fig.3. For a
significant coupling $g_{2}=0.05$, $Z^{\prime}$ may still exist at about $1.2$
TeV. Even for $g_{2}=0.1$, $Z^{\prime}$ with more that $1.6$ TeV mass is not
eliminated. On the other hand, a light $Z^{\prime}$ at sub-TeV would be
permissible as long as the coupling $g_{2}$ is enough small.

Figure 3: theoretical signal for $\sigma(pp\rightarrow Z^{\prime}\rightarrow
ll)$ at 14TeV LHC. The colorful massive shadow areas correspond $2\sigma$ fit
range of charge ratio $r$. The red solid area on the top is eliminated by
ATLAS direct detection in dilepton final states at $\sqrt{s}=7$ TeV.

##  VI conclusion

In conclusion, a model-independent effective theory for anomaly-free neutral
boson has been presented. Based on electroweak precise measurements in LEP,
four parameters related to $Z^{\prime}$ mixings and charge assignment have
been constrained. Especially, the charge ratio $r$ has range $(-2.2,2.2)$ at
95% C.L.. To consistent with the LHC direct detection in dilepton decay
channel at $7$ TeV, the limit areas to fixed $Z^{\prime}$ mass are shown in
$r$-$g_{2}$ plane. More clearly, we map the possible parameter space to the
plane of $m_{Z^{\prime}}$-$g_{2}$ at $2\sigma$ C.L. of $r$. in Fig. 2. The
results show that a sub-TeV $Z^{\prime}$ with coupling $g_{2}\sim 0.01$ is
still not eliminated in $2\sigma$ C.L. of $r$. It suggests a prime requirement
to extra vector boson in NP. Further, a possible theoretical
$\sigma(pp\rightarrow Z^{\prime}\rightarrow ll)$ signal at $\sqrt{s}=14$ TeV
LHC is also calculated.

## Acknowledgments

This work was supported by National Science Foundation of China (NSFC) under
Grant No.11005084 and partly by the Fundamental Research Funds for the Central
University. We thank Rong Li for useful discussions.

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