# Configuration-Aware Safe Control for Mobile Robotic Arm
with Control Barrier Functions

Fan Ding1, Han Wang2, Jianping He1, Yi Ren3, and Yu Zheng3 1The Department of
Automation, Shanghai Jiao Tong University, Shanghai, China. E-mails:
<EMAIL_ADDRESS>2The Department of Engineering Science, University
of Oxford, Oxford, United Kingdom. E-mail<EMAIL_ADDRESS>3The
Tencent Robotics X Lab, Shenzhen, China. E-mails: {evanyren,
<EMAIL_ADDRESS>
###### Abstract

Collision avoidance is a widely investigated topic in robotic applications.
When applying collision avoidance techniques to a mobile robot, how to deal
with the spatial structure of the robot still remains a challenge. In this
paper, we design a configuration-aware safe control law by solving a Quadratic
Programming (QP) with designed Control Barrier Functions (CBFs) constraints,
which can safely navigate a mobile robotic arm to a desired region while
avoiding collision with environmental obstacles. The advantage of our approach
is that it correctly and in an elegant way incorporates the spatial structure
of the mobile robotic arm. This is achieved by merging geometric restrictions
among mobile robotic arm links into CBFs constraints. Simulations on a rigid
rod and the modeled mobile robotic arm are performed to verify the feasibility
and time-efficiency of proposed method. Numerical results about the time
consuming for different degrees of freedom illustrate that our method scales
well with dimension.

##  I Introduction

Mobile robotic arm, consisting of robotic arms and mobile or drone bases have
gained popularity in robotics community for their wide applications in
navigating and grasping tasks with obstacle avoidance requirements. Mobile
robotic arms allow more flexible operations to complete assigned tasks and, on
the other hand, introduce higher computational complexity because of their
spatial structures. Various techniques have been proposed in the past decades
for mobile robots collision avoidance problems, including Cell Decomposition
[1], Potential Field Methods [2]) and heuristic methods (Fuzzy Logic
Controller Technique [3], Neural Network Technique [4]). The readers are
referred to [5] for an overview in this area. When applying these collision
avoidance techniques to a mobile robotic arm, how to deal with its spatial
structure still remains a challenge.

Existing collision avoidance algorithms mainly use geometric envelopes to
simplify the spatial structure of the robot and reduce the complexity of the
optimization. [6] enveloped the mobile robot and moving obstacles by spheres
with different radii in and a humanoid torso robot which is abstracted as a
cylinder to navigate among modeled pedestrians was proposed in [7]. Ignoring
the spacial structured mobile robots with geometric envelopes restricts the
volume of permitted workspace and limits the close interaction between mobile
robots and obstacles in a clustered environment.

When considering the spatial structure of the mobile robotic arm, the rigid
body links are of great interest in literature, where [8] approximated the
links of the robotic arm by a set of convex polyhedra while [9] treated them
as convex ellipsoids. Caused by the high degrees of freedom, calculating the
motion control law for the entire mobile robotic arm can be computationally
expensive. In [10], a hybrid teleoperation system was proposed, where the
operator carries out general control for the mobile base while the collision
avoidance task is shifted to sensor-based robotic arm subsystem. This
decouples the motion constraints between the mobile base and the robotic arm.
Therefore, we propose to find a way of incorporating the spatial structure of
the mobile robotic arm while conducting collision avoidance task, treating the
mobile robotic arm as a whole.

Figure 1: Scenario of task scene and expected realization. The maze
environment in the right figure serves as the configuration space, in which a
series of convex ellipsoids are generated as a connected corridor from the
start place to the goal region. In the corner, for instance, where the black
arrow starts, we expect the rigid rod and the double link bypass the obstacle
without collision. Note that both the rigid rod and the double link possess
rigid body restrictions, which are considered and solved in an elegant way in
this paper.

Recently, the Control Barrier Functions (CBFs) approach is proposed as a
promising and efficient tool for safe control [11]. By using a quadratic
programming based framework the safety requirements can be encoded as affine
constraints. The CBFs method is flexible for an arbitrary nominal control law,
which determines the desired behavior. Significant success for this approach
has been shown in robotics applications, especially in collision avoidance
[12]. Extensions such as considering actuation capacity [13], safety-critical
Lagrangian systems [14], MPC-CBF [15] for collision avoidance have been
proposed. Although these methods are shown to be mathematically rigorous,
there is a theory-reality gap dual to the spatial structure of both robots and
obstacles.

In this paper, we aim at synthesizing a configuration-aware safe control law
for a mobile robotic arm with considered complex spatial structure. The
proposed safe control law is able to drive the mobile robotic arm to the goal
region while avoiding collision in a obstacle-clustered environment. The
challenges brought by the spatial structure are solved by ensuring limited
edge points of the robot to be within a series of connected convex obstacle-
free sets. By solving an online CBFs-based QP with limited number of
constraints, the synthesis approach is quite efficient and promising. The main
contributions are stated as follows:

  * •

We synthesize a configuration-aware control law that can successfully drives
our designed mobile robotic arm to the desired region while avoiding collision
with environmental obstacles.

  * •

The proposed approach incorporates the spatial structure of the mobile robotic
arm by merging geometric restrictions into CBFs constraints, thus allowing to
solve the collision avoidance problem by convex program.

  * •

Our approach is proved to scale well with dimension as the degrees of freedom
of the mobile robotic arm increases.

The remainder of this paper is organized as follows. Section II introduces the
physical structure of the mobile robotic arm. Preliminaries and reach-avoid-
grasp task are specified in Section III. Main results of this paper are shown
in Section IV. Extensive numerical simulations are presented in Section V. We
conclude the paper in Section VI.

##  II Structure of the Mobile Robotic Arm

Our designed mobile robotic arm is shown in Fig. 2. It consists of a holonomic
four-Mecanum-wheel mobile base and an on-board robotic arm, also equipped with
a stereo camera on the top of it. NVIDIA TX2 is embedded as the central
computing unit. We tailor a 3D-printed shell as the robot body, on which the
robotic arm is set to execute the grasping configuration. The equipped robotic
arm is Wlkata Robotic Arm, which possesses the standard structure of
industrial manipulators with six (plus one) revolute joints. The stereo camera
(D435) is mounted on the link between joint 3 and joint 4 of the robotic arm,
thus constituting the eye-in-hand configuration which provides higher
flexibility for robot operations and precise sights of the target [16].

Figure 2: The systematic structure of the mobile robotic arm.

This stereo camera is an RGB-D camera, from which the RGB information is
updated at 30 Hz and the depth information updated at 90 Hz. With the Intel
RealSense SDK 2.0 (a provided Intel software), the stereo camera can get on-
chip self-calibration. After that, an HSV (Hue, Saturation and Value)-based
color segmentation algorithm [17] and morphology open operation are utilized
to detect the colored environmental obstacles. Then we manage to get reliable
depth information $\mathrm{Z_{c}}$ of the object in the camera coordinate
system with a depth information filter [18]. Let pixel coordinate of point m
on the pixel plane be $(\mathrm{X_{pixel}},\mathrm{Y_{pixel}})$, its real-
world coordinate in camera coordinate system
$({\mathrm{X_{c}}},{\mathrm{Y_{c}}})$ can be calculated according to the
following equations

$\displaystyle\frac{x_{\mathrm{pixel}}-\mathrm{ppx}}{\mathrm{f_{x}}}=\frac{\mathrm{X_{c}}}{\mathrm{Z_{c}}},\quad\frac{\mathrm{Y_{pixel}}-\mathrm{ppy}}{\mathrm{f_{y}}}=\frac{\mathrm{Y_{c}}}{\mathrm{Z_{c}}},$

where $\mathrm{ppx},\mathrm{ppy},\mathrm{f_{x}},\mathrm{f_{y}}$ are the camera
intrinsic parameters. Local 3D world reconstruction [19] frame is performed in
Fig. 3.

Figure 3:  Real world reconstruction frame. The pink and purple bounding boxes
in the pixel plane are the detected results achieved by the HSV-based color
segmentation algorithm [17] and minimum volume bounding box method [20]. The
corresponding cubes are the reconstructed real-world obstacles

Thanks to the high-frequency detection rate of the camera, our mobile robotic
arm is capable of detecting the surroundings and re-localizing itself in real
time. The control flow is illustrated in Fig. 4.

Figure 4: The control framework of our designed mobile robotic arm. The
obstacle-free space $\mathcal{X}_{\mathrm{free}}$ and the obstacle space
$\mathcal{X}_{\mathrm{obs}}$ are approximated with local 3D real world
reconstructed by the stereo camera. Then the computing unit utilizes the
obstacle-free space $\mathcal{X}_{\mathrm{free}}$ and obstacle space
$\mathcal{X}_{\mathrm{obs}}$ information to solve the control law $u$ from a
QP with designed CBF constraints. The control law $u$ comprises of angle
velocities $\dot{\theta_{1}}\ldots\dot{\theta_{6}}$ of the robotic arm joints
and the velocities $v_{x},v_{y}$ of the mobile base. The mobile robotic arm
state $x_{i}$ is updated by movement of the mobile base and rotation of the
robotic arm and fed back to the stereo camera.

##  III Preliminaries and Problem Formulation

###  III-A Control Barrier Function

The set of real numbers is denoted by $\mathbb{R}$, while $\mathbb{R}^{n}$
lifts the dimension to $n$. Let $x\in\mathbb{R}^{n}$ and $u\in\mathbb{R}^{m}$
be the state vector and control input, respectively. Consider a nonlinear
control affine system as

$\dot{x}=f(x)+g(x)u,$ (1)

where we suppose that functions $f(\cdot)$ and $g(\cdot)$ are both locally
Lipschitz continuous. Consider a set $\mathcal{C}$ defined as the zero-super
level set of a continuously differentiable function $H:\mathbb{R}^{n}$
$\rightarrow$ $\mathbb{R}$ as

$\displaystyle\mathcal{C}=\\{x\in\mathbb{R}^{n}:H(x)\geq 0\\},$ (2)

and $\frac{\partial H(x)}{\partial{x}}\neq 0$ for all $x\in\\{x|H(x)=0\\}$.

######  Definition 1.

A set $\mathcal{C}$ is forward invariant with respect to system (1) if for
every initial state $x(t_{0})\in\mathcal{C}$, there exists $u$, such that the
solutions remain within $\mathcal{C}$, i.e., $x(t)\in\mathcal{C}$ for $\forall
t\geq t_{0}$.

The system (1) is safe with respect to a set $\mathcal{C}$ if $\mathcal{C}$ is
forward invariant and $\mathcal{C}\subseteq\mathcal{S}$ where $\mathcal{S}$ is
a compact set defined the safe region. Assuming that $H(x)=0$ at $t=t_{0}$, a
naturally associated condition to keep the set $\mathcal{C}$ forward invariant
is $\dot{H}(x)\geq 0$ for any $t\geq t_{0}$. By applying the chain rule,
$\dot{H}(x)$ is specified as:

$\displaystyle\dot{H}(x)=\frac{dH(x)}{dt}$ $\displaystyle=\frac{\partial
H(x)}{\partial{x}}\frac{\partial{x}}{\partial t}$
$\displaystyle=\frac{\partial H(x)}{\partial{x}}\dot{x}.$ (3)

Incorporating (1) into (3), we have

$\frac{\partial H(x)}{\partial{x}}(f(x)+g(x)u)\geq 0,$ (4)

which is regarded as the safe constraint. In the real-world applications, this
constraint is rather strict that feasible solutions can rarely be obtained and
an alternative relaxed formulation was proposed in [11].

######  Definition 2.

A continuous function $\alpha(\cdot):(-b,a)\rightarrow(-\infty,\infty)$ is
said to belong to extended class-$\mathcal{K}$ for some $a,b>0$ if it is
monotonicly increasing and $\alpha(0)=0$.

######  Definition 3.

[21] Let $\mathcal{C}\subseteq D\subset\mathbb{R}^{n}$ be the set defined by
$H(\cdot)$. Then $H$ is a control barrier function, if there exists an
extended class-$\mathcal{K}$ function $\alpha$, such that $\forall x\in D$,
$\exists u\in\mathbb{R}^{m}$, the following inequality holds:

$\sup\limits_{u\in\mathbb{R}^{m}}\Big{[}\frac{\partial
H(x)}{\partial{x}}(f(x)+g(x)u)\Big{]}\geq-\alpha(H(x)).$

Definition 3 renders the set $K_{CBF}$ of control laws $u$ that guarantees the
safe set $\mathcal{C}$ forward invariant [22]. The set of safe control law is
defined by:

$K_{\mathrm{CBF}}=\Big{\\{}u\in\mathbb{R}^{m}:\frac{\partial
H(x)}{\partial{x}}(f(x)+g(x)u)\geq-\alpha(H(x))\Big{\\}}.$ (5)

###  III-B Homogeneous Transformation Matrix

Let $P_{M}^{C}\in\mathbb{R}^{3}$ be the position vector of point $M$ in
coordinate $C$ and $P_{M}^{B}\in\mathbb{R}^{3}$ be the position vector of
point $M$ in coordinate $B$. If the origins of coordinate $B$ and $C$
coincide, we can find $\textbf{\emph{R}}_{B}^{C}\in
SO(3)\subset\mathbb{R}^{3\times 3}$, such that

$P_{M}^{C}=\textbf{\emph{R}}_{B}^{C}\times P_{M}^{B},$

where each column of $R_{B}^{C}$ represents the projections of the three axes
of coordinate $B$ on each axis of the coordinate $C$. If the origins of
coordinate $B$ and $C$ do not coincide, then we can derive a homogeneous
transformation matrix $\textbf{\emph{T}}_{B}^{C}\in
SE(3)\subset\mathbb{R}^{4\times 4}$:

$\textbf{\emph{T}}_{B}^{C}=\begin{bmatrix}\textbf{\emph{R}}_{B}^{C}&p_{B}^{C}\\\
0&1\end{bmatrix},$ (6)

where $p_{B}^{C}\in\mathbb{R}^{3}$ denotes the translation offset from the
origin of coordinate $B$ to the origin of coordinate $C$, such that

$\begin{bmatrix}P_{M}^{C}\\\
1\end{bmatrix}=\textbf{\emph{T}}_{B}^{C}\times\begin{bmatrix}P_{M}^{B}\\\
1\end{bmatrix}.$

###  III-C Reach-Avoid-Grasp Problem Formulation

The goal of this paper is to find a feasible control law that drives our
designed mobile robotic arm to the goal region while avoiding collision with
obstacles. A region is regarded as the goal region if, in which the mobile
robotic arm can execute the grasp configuration, i.e. the mounted robotic arm
could grasp the target requiring no more movement of the mobile base. More
specifically, the reach-avoid-grasp problem is posed as follows:

######  Problem 1 (Specify Goal Region).

Given a mobile robot workspace $\mathcal{X}\in\mathbb{R}^{n}$ and a goal
position $x_{\mathrm{goal}}\in\mathcal{X}$ as a priori, find the goal region
$\mathcal{X}_{\mathrm{goal}}\subset\mathcal{X}$ in which our designed mobile
robotic arm can execute the grasp configuration.

######  Problem 2 (Geometrically Describe the Mobile Robotic Arm).

Find $\mathcal{M}$ proper edge points $a_{i}$, $i=1\ldots\mathcal{M}$, and
assign $\mathcal{M}-1$ line segments $A_{i},i=1\ldots\mathcal{M}-1$, where
$A_{i}$ connects points $a_{i}$ and $a_{i+1}$ to represent the mobile robotic
arm $\mathcal{A}$:

$\mathcal{A}=\bigcup_{i=1}^{\mathcal{M}-1}A_{i}.$

######  Problem 3 (Find the Safe Control Law).

Given a mobile robot workspace $\mathcal{X}\in\mathbb{R}^{n}$, and divide it
into obstacle-occupied space $\mathcal{X}_{\mathrm{obs}}\in\mathcal{X}$ and
obstacle-free space
$\mathcal{X}_{\mathrm{free}}=\mathcal{X}\setminus\mathcal{X}_{\mathrm{obs}}$,
find a feasible control law that drives the mobile robotic arm $\mathcal{A}$
from initial state $x_{0}\in\mathcal{X}_{\mathrm{free}}$ to a state in the
goal region $\mathcal{X}_{\mathrm{goal}}\subset\mathcal{X}_{\mathrm{free}}$
while avoiding collision with the obstacles, i.e.

$\mathcal{A}(t)\subset\mathcal{X}_{\mathrm{free}}.$

##  IV Main Results

###  IV-A Goal Region and Grasping

The key to find the goal region $\mathcal{X}_{goal}$ (i.e. solve Problem 1)
for the mobile robotic arm is to specify the workspace $W$ of the robotic arm.
The workspace of a robotic arm represents the region where its end-effector
can reach without violating the joint limits. In this paper, we utilize
kinematic modeling to numerically describe the workspace of the robotic arm.
Physical model with axis assignment is shown in Fig. 5(a) and the DH (Denavit-
Hartenberg) parameters [23] are obtained according to the axis assignment,
shown in Table I.

TABLE I: Modified DH model parameters coordinate $i$ | $\theta_{i}$ | $b_{i}$ | $a_{i-1}$ | $\alpha_{i-1}$ | Rotation Range
---|---|---|---|---|---
axis1 | 0 | 80 | 0 | 0 | -100° to +100°
axis2 |  $-\pi$/2 | 0 | 32 |  $-\pi$/2 | -60° to +90°
axis3 | 0 | 0 | 108 | 0 | -180° to +50°
axis4 | 0 | 176 | 20 |  $-\pi$/2 | -180° to +180°
axis5 |  $\pi$/2 | 0 | 0 |  $\pi$/2 | -180° to +40°
axis6 | 0 | -20 | 0 |  $\pi$/2 | -180° to +180°

The workspace of the robotic arm, which is visualized by yellow point cloud
and shown in Fig. 5(b), is obtained by applying the forward kinematics
provided by Robotics Toolbox in Matlab. Since the workspace is relevant to the
state $x$ of the mobile robotic arm, we rewrite the workspace as $W(x)$. Given
the goal position $x_{goal}$, the goal region is defined as

$\mathcal{X}_{goal}=\Big{\\{}\boldsymbol{x}\in\mathcal{X}_{free}:\boldsymbol{x}_{goal}\in
W(\boldsymbol{x})\Big{\\}}.$ (7)

(a) The axis assignment

(b) The workspace

Figure 5:  The axis assignment and workspace of the physical model of our
robotic arm. In Fig. 5(a), six revolute joints are denoted where blue and red
cylinders meet. The blue and red cylinders represent the rigid link between
the joints. DH parameters $a_{1},a_{2},a_{3}$ and $b_{1},b_{4}$ are labeled
with double arrows. The yellow point cloud in Fig. 5(b) numerically describes
the workspace of the robotic arm. Note that the workspace maintains its
spatial configuration while translating and rotating with the movement of the
robotic arm base.

As for grasping configuration, we consider the homogeneous transformation
matrix [24], $\textbf{\emph{T}}_{c}^{a}$ from the camera coordinate
$(X_{c}Y_{c}Z_{c})$ to the robotic arm coordinate $(X_{a}Y_{a}Z_{a})$.
According to (6), we have

$\begin{bmatrix}X_{a}&Y_{a}&Z_{a}&1\end{bmatrix}^{T}=\textbf{\emph{T}}_{c}^{a}\begin{bmatrix}X_{c}&Y_{c}&Z_{c}&1\end{bmatrix}^{T},$
(8)

where $\textbf{\emph{T}}_{c}^{a}$ is specified as (9) calculated with the
geometric relationship in Fig. 6.

$\textbf{\emph{T}}_{c}^{a}=\begin{bmatrix}1&0&0&0\\\
0&\cos\alpha&\sin\alpha&-\sqrt{l_{1}^{2}+l_{2}^{2}}\cos(\alpha+\theta)\\\
0&-\sin\alpha&\cos\alpha&\sqrt{l_{1}^{2}+l_{2}^{2}}\sin(\alpha+\theta)\\\
0&0&0&1\end{bmatrix}.$ (9)

(a) Relative spatial pose

(b) Plane geometric relationship

Figure 6:  Geometric relationship between camera coordinate and arm
coordinate. 6(a) shows the relative spatial pose between camera coordinate and
arm coordinate on the robotic arm. Joint 1, joint 2, joint 3, joint 4 , joint
5 and joint 6 are abbreviated as J1, J2, J3, J4, J5 and J6. Geometric
relationship between these two coordinates is extracted as 6(b) where axis
$\mathrm{Z_{c}}$ and $\mathrm{Y_{c}}$ rotate to $\mathrm{Z_{c}^{\prime}}$ and
$\mathrm{Y_{c}^{\prime}}$ by rotation (defined as angle $\alpha$) of J3.

Therefore, when the object is detected by our camera, its coordinate in
robotic arm coordinate can be derived from (8) and (9). Then with inverse
kinematics tool of Robotics Toolbox in matlab, we can specify the grasp
configuration, i.e. angle assignment of each joint.

###  IV-B Solution of Reach-Avoid Problem

The reach-avoid task comprises desired position reaching and obstacle avoiding
steps. By deriving the workspace of the robotic arm, the desired position
reaching step is relaxed to desired region reaching step, which is less
computationally expensive.

We now take advantage of the control framework (shown in Fig. 4) to carry out
the reach-avoid task. When the mobile robotic arm arrives at a position
$x_{i}$, the stereo camera (D435) captures present perspective of
surroundings. By applying local 3D world reconstruction method [19], obstacle-
free space $\mathcal{X}_{\mathrm{free}}$ and obstacle-occupied space
$\mathcal{X}_{\mathrm{obs}}$ can be properly approximated. Generally speaking,
the obstacle-free space is a non-convex region [25], [26], our approach
considers a series of convex sets
$\mathcal{C}_{0},\mathcal{C}_{1},\ldots\mathcal{C}_{N}$ (i.e. ellipsoids,
convex polygons) extracted from the obstacle-free space. Based on
$\mathcal{X}_{\mathrm{free}}$ and $\mathcal{X}_{\mathrm{obs}}$, proper convex
set $\mathcal{C}_{i}$ enclosing current state $x_{i}$ can be found by solving
convex optimization problems [27]. We assign succeeded state $x_{i+1}$ of
$x_{i}$ such that the distance between $x_{i}$ and $x_{i+1}$ is maximized,
i.e. solving

$\displaystyle\max\quad$
$\displaystyle\|\boldsymbol{x}_{i}-\boldsymbol{x}_{i+1}\|_{2}$ (QP1) s.t.
$\displaystyle\boldsymbol{x}_{i+1}\in\mathcal{C}_{i}\subset\mathcal{X}_{\mathrm{free}},$

When considering the spatial structure of our designed mobile robotic arm, we
mainly focus on the rigid body links (link 1 between J1 and J2, link 2 between
J2 and J3, link 3 between J3 and J5, link 4 between J5 and J6 shown in Fig.
6(a)) of the robotic arm. To solve Problem 2, we choose the central points of
J1, J2, J3, J5, J6 as the edge points $a_{i}$, $i=1\ldots 5$. And abstract the
rigid body link of the robotic arm as line segments $A_{i},i=1,\ldots,4$,
connecting the edge points $a_{i}$ and $a_{i+1}$, to represent the robotic arm
$\mathcal{A}$, where

$\mathcal{A}=\bigcup_{i=1}^{4}A_{i}.$

In a more general sense, we denote $N_{l}$ as the number of the abstracted
line segments and $N_{ep}$ as the number of the edge points. The mobile
robotic arm is therefore generalized as:

$\mathcal{A}=\bigcup_{i=1}^{N_{l}}A_{i}.$

######  Theorem 1.

To keep the mobile robotic arm safe in the course of reach-avoid task, we only
need to keep the chosen corner points safe, i.e., the safety control is
guaranteed by the following control law

$\displaystyle
u^{*}(x)=\arg\min_{u\in\cap_{k=1,\cdot,N_{ep}}K^{(k)}_{CBF}}\quad$
$\displaystyle\|u-u^{p}(x)\|_{2},$ (QP2)

where $u^{p}(x)=\mathrm{k_{p}}(x_{i+1}-x_{i})$ and $\mathrm{k_{p}}$ is the
tuned proportional gain.

###### Proof.

First we prove that the edge points are safe by applying the safe control law.
Let $x^{(k)}$ be the state of the edge point k, extended from (5), we obtain
the $K_{CBF}^{(k)}$ with respect to $x^{(k)}$ as:

$\displaystyle K^{(k)}_{CBF}=$
$\displaystyle\Big{\\{}u\in\mathbb{R}^{m}:\Big{[}\frac{\partial H(x)}{\partial
x}(f(x)+g(x)u)$ (10) $\displaystyle+\alpha(H(x))\Big{]}\Big{|}_{x=x^{(k)}}\geq
0\Big{\\}}.$

Given the states $x^{(k)}$, $x^{(k+1)}$ of edge point $k$, edge point $k+1$, a
set $\mathcal{C}$ by (2) and $u\in K_{CBF}^{(k)}\bigcap K_{CBF}^{(k+1)}$, then
$\mathcal{C}$ is ensured forward invariant with (5) [22]. According to
Definition 1, we can derive that $x^{(k)}$ and $x^{(k+1)}$ are ensured within
set $\mathcal{C}$ (i.e. edge point $k$ and edge point $k+1$ are safe). Then we
prove that the mobile robotic arm $\mathcal{A}$ is safe given all the edge
points safe. For a convex set $\mathcal{C}_{i}\subset\mathbb{R}^{n}$,
$\forall$ $y^{(1)},y^{(2)}\in\mathcal{C}_{i}$, equation

$\lambda\boldsymbol{y}^{(1)}+(1-\lambda)\boldsymbol{y}^{(2)}\in\mathcal{C}_{i},\forall\lambda\in[0,1]$

holds according to convex set definition. This indicates that if points
$y^{(1)}$ and $y^{(2)}$ are within the convex set, then the the generalized
line segment connecting them is ensured in this convex set. Earlier in this
subsection, we get a series of convex sets
$\mathcal{C}_{0},\ldots\mathcal{C}_{N}$ from obstacle free space
$\mathcal{X}_{\mathrm{free}}\subset\mathbb{R}^{n}$ and now we substitute
$\mathcal{C}$ with one of the convex set. Therefore, given
$a_{i}\in\mathcal{C},i=1,\ldots,N_{ep}$ (edge points are safe), line segments
$A_{i}\subset\mathcal{C}$(safe), $i=1,\ldots\,N_{l}$ (line segments are safe),
hence $\mathcal{A}=\bigcup_{i=1}^{N_{l}}A_{i}\subset\mathcal{C}$ (the mobile
robotic arm is safe).

∎

Based on the extracted convex set $\mathcal{C}_{0},\ldots\mathcal{C}_{N}$, the
corresponding control barrier function $H_{0}(x),\ldots,H_{N}(x)$ can be
specified by

$\mathcal{C}_{i}=\\{\boldsymbol{x}\in\mathbb{R}^{n}:H_{i}(\boldsymbol{x})\geq
0\\},i=0,\ldots,N.$

The corresponding safe control law $u^{*}(x)$ is computed by solving (QP2)
with substituting $H(x)$ by the specific $H_{i}(x)$, where $i$ depends on the
current located safe set. For now we have solved Problem 3.

##  V Numerical Simulation

In this section, simulation of a rigid rod and a mobile robotic arm in a
complex maze is performed to verify the proposed algorithm. The QP solver is
CVXGEN [28].

Firstly we apply our method on a rigid rod, aiming at driving the rigid rod
from the initial configuration to the goal region without collision with the
rectangle obstacles in a maze environment. The maze environment introduces a
start position on the bottom left and a goal position on the top right, shown
in Fig. 7.

(a) $l=1$

(b) $l=1.4$

(c) $l=1.8$

Figure 7:  Simulation results of a rigid rod in a maze environment with length
$l=1$ in Fig. 7(a), $l=1.4$ in Fig. 7(b) and $l=1.8$ in Fig. 7(c). The rigid
rod (the thick pink segment with a black vertex and a red vertex) aims at
navigating from start position to the goal position without collision with
obstacles in the maze environment. Slim red curve and black curve are the
trajectories of the red vertex and the black vertex. In both cases with
different initial pose, the rigid rod successfully and safely reaches the goal
region.

Then we conduct simulations on our designed mobile robotic arm. While
describing the workspace of the robotic arm, we notice that J4 and J6 play
little role in forming the workspace, both of which are for adjusting the pose
of the end-effector to better execute the grasp configuration. Therefore, J4
and J6 make little difference in the collision avoidance problem and we focus
on J1, J2, J3 and J5 while modeling the mobile robotic arm as shown in Fig. 8.
When four joints are considered, the modeled mobile robotic arm is
theoretically close to our designed mobile robotic arm in real world.

(a) One joint

(b) Two joints

(c) Three joints

(d) Four joints

Figure 8:  Models of mobile robotic arm considering different amount of
rotating joints. Mobile base is modeled as a grey cube and rigid body links
are modeled as blue line segments in Fig. 8(a)-8(d). Red points represent the
edge points on the mobile robotic arm, and one edge point is sheltered in the
mobile base. One joint (J1), two joints (J1, J2), three joints (J1, J2, J3)
and four joints (J1, J2, J3, J5) are respectively considered in Fig. 8(a),
8(b), 8(c) and 8(d).

According to Theorem 1, each joint corresponds to _one_ CBF constraint in a
convex quadratic programming framework, the algorithm therefore scales well
with dimension. The time consuming of our approach is enumerated as
0.0220$(s)$, 0.0245$(s)$, 0.0279$(s)$ and 0.0292$(s)$ when one, two, three and
four joint(s) are respectively considered in our simulation.

Now we present the simulation results of considering three-joint and four-
joint mobile robotic arm model. We refine the state vector $x$ and input
vector $u$ in the state space model of the mobile robotic arm as:

$x=\begin{bmatrix}X&Y&Z\end{bmatrix}^{T},u=\begin{bmatrix}\dot{\theta}_{1}&\dot{\theta}_{2}&\dot{\theta}_{3}&\dot{\theta}_{5}&v_{x}&v_{y}\end{bmatrix},^{T}$

where $X,Y,Z$ denote the coordinate of the end-effector ,
$\dot{\theta}_{1},\dot{\theta}_{2},\dot{\theta}_{3},\dot{\theta}_{5}$ are
angular velocities of J1, J2, J3, J5 and $v_{x},v_{y}$ are velocities of the
mobile base. When specifying the state space model of our mobile robotic arm,
we only take the most complex situation (four joints) for example, shown as
follow

$\begin{split}\begin{bmatrix}\dot{X}\\\ \dot{Y}\\\
\dot{Z}\end{bmatrix}=&\begin{bmatrix}f_{11}&f_{12}&f_{13}&f_{14}&1&0\\\
f_{21}&f_{22}&f_{23}&f_{24}&0&1\\\
0&f_{32}&f_{33}&f_{34}&0&0\end{bmatrix}\begin{bmatrix}\dot{\theta}_{1}\\\
\dot{\theta}_{2}\\\ \dot{\theta}_{3}\\\ \dot{\theta}_{5}\\\ v_{x}\\\ x_{y}\\\
\end{bmatrix}\\\ \end{split},$ (11)

where,

$\displaystyle\begin{aligned}
f_{11}=&-a_{2}s_{1}s_{2}-d_{4}s_{1}c_{23}+d_{6}s_{1}c_{235},\end{aligned}$
$\displaystyle\begin{aligned}
f_{12}=&-a_{2}c_{1}c_{2}-d_{4}c_{1}s_{23}+d_{6}c_{1}s_{235},\end{aligned}$
$\displaystyle\begin{aligned}
f_{13}=&-d_{4}c_{1}s_{23}+d_{6}c_{1}s_{235},\end{aligned}$
$\displaystyle\begin{aligned} f_{14}=d_{6}c_{1}s_{235},\end{aligned}$
$\displaystyle\begin{aligned}
f_{21}=&-a_{2}c_{1}s_{2}+d_{4}c_{1}s_{23}+d_{6}s_{1}s_{235},\end{aligned}$
$\displaystyle\begin{aligned}
f_{22}=a_{2}s_{1}c_{2}-d_{4}s_{1}s_{23}+d_{6}s_{1}s_{235},\end{aligned}$
$\displaystyle\begin{aligned}
f_{23}=&-d_{4}s_{1}s_{23}+d_{6}s_{1}s_{235},\end{aligned}$
$\displaystyle\begin{aligned} f_{24}=d_{6}s_{1}s_{235},\end{aligned}$
$\displaystyle\begin{aligned}
f_{32}=&-a_{2}s_{2}-d_{4}c_{23}+d_{6}c_{235},\end{aligned}$
$\displaystyle\begin{aligned} f_{33}=&-d_{4}c_{23}+d_{6}c_{235},\end{aligned}$
$\displaystyle\begin{aligned} f_{34}=d_{6}c_{235}.\end{aligned}$

($s_{1}$ denotes $\sin\theta_{1}$, $c_{1}$ denotes $\cos\theta_{1}$ and
$s_{23}$ denotes $\sin(\theta_{2}+\theta_{3}$). $a_{2},d_{1},d_{4},d_{6}$ are
DH parameters in Table I.)

The coordinates of the rest edge points can be calculated according to the
rigid body restrictions of the mobile robotic arm. Driven by the safe control
law solved by (QP2), our mobile robotic arm can successfully avoid collision
with obstacles and navigate to the desired region in the environment where
several obstacles are randomly set, shown in Fig. 9. Fig. 10 introduces the
minimum distance to the safe set boundary as the indicator of safety and the
edge points are ensured in the safe set, hence ensuring the whole mobile
robotic arm in the safe set as depicted in Theorem 1.

(a) Simulating process of three-joint mobile robotic arm

(b) Simulating process of four-joint mobile robotic arm

Figure 9: Simulating process overview of the modeled mobile robotic arm. By
applying the safe control law solved by (QP2), the mobile robotic arm manages
to reach the goal region while avoiding collision with environmental obstacles
both in Fig. 9(a) (three-joint mobile robotic arm) and Fig. 9(b) (four-joint
mobile robotic arm).

(a) three-joint mobile robotic arm simulation result

(b) four-joint mobile robotic arm simulation result

Figure 10: The minimum distance to the safe set boundary of 4 edge points (see
Fig. 10(a)) and 5 edge points (see Fig. 10(b)) with respect to time are
depicted as curves of different colors. Even though the edge point of the
mobile robotic arm may get close to the safe set boundary, the edge point is
still ensured safe without collision with the obstacles, also shown in the top
view figure in Fig. 9.

##  VI Conclusion and Future Direction

In this paper, we solved the problem of configuration-aware safe control for a
designed mobile robotic arm with Control Barrier Functions. We firstly propose
to geometrically describe our designed mobile robotic arm and mean to fully
cooperate the spatial structure of the mobile robotic arm. Then we merge these
geometric restrictions into CBFs constraints and solve the safe control law
from a QP with designed CBFs constraints. This safe control law can
successfully navigate the modeled mobile robotic arm to a desired region while
avoiding collision with environmental obstacles. We show that our method is
promising and computationally efficient by conducting numerical simulations.
In the future we will apply our approach to physical test-bed and explore
higher dimensional arms.

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correspondence between occurrence nets and prime event structures is due to
Winskel’s seminal work (Win:ES, ). We have introduced a new class of Petri
nets, called causal nets, and showed their correspondence with prime event
structures (Theorem 6.9). We have then presented the reversible variant of
causal nets (rcns), and showed that they are an operational counterpart of the
reversible prime event structures of (PU:jlamp15, ) (Theorem 7.19). Finally,
we showed that causal nets are tightly connected to occurrence nets (Theorem
8.13), and give a revisitation of Winskel’s correspondence via Theorem 8.14.
To give a complete picture of known results, we depict the extension of
occurrence nets into reversible occurrence nets proposed in (MMPPU:RC20, );
where the latter ones correspond to a proper subclass of $r\textsc{pes}$ that
just accounts for causally-consistent reversibility (crpes).

Besides contributing to the well-established line of work initiated in
(NPW:PNES, ; Win:ES, ) that links classes of nets with classes of event
structures, our work explains the reversing mechanisms of rpeses by relying on
a well-studied concurrent model. In this respect, our contribution constitutes
a step forward with respect to (MMPPU:RC20, ), which just deals with the
subclass of causal-reversible rpeses which, e.g., do not provide out-of-
causal-order reversibility. Remarkably, our reversible causal nets do not
appeal to original elements to achieve reversibility as, e.g., the bonds of
(PhilippouP18, ). As a matter of fact, causal nets do not rely either on the
information (i.e., colours) carried on by tokens (as done in (BarylskaGMPPP18,
; revPT, )).

The characterisation of rpeses as restrictions over the structure of the nets
enables the study of less constrained models. For instance, an undoable event
in an rpes can be undone just by performing one reversing event. This excludes
the possibility of defining alternative conditions for reversing an event. We
can overcome this limitation in the operational model just by dropping the
requirement of non-overlapping backward transitions (i.e.,
$[\\![{\\!~{}^{\bullet}{{\underline{T}}}}]\\!]={\\!~{}^{\bullet}{{\underline{T}}}}$).
This kind of analysis seems an interesting line to explore in future work.

Causal nets and their reversible versions play a role analogous to occurrence
nets (in their connection with peses). Consequently, we may ask ourselves
about the class of (reversible) nets whose unfoldings are (reversible) causal
nets, or, alternatively, which are the suitable categories that would allow
us, if possible, to extend Winskel’s chain of coreflection to reversible event
structures.

## Acknowledgment

This work has been partially supported by the BehAPI project funded by the EU
H2020 RISE under the Marie Sklodowska-Curie action (No: 778233), by the
Italian PRIN 2020 project NiRvAna – Noninterference and Reversibility Analysis
in Private Blockchains, the Italian PRIN 2022 project DeKLA – Developing
Kleene Logics and their Applications, the INdAM-GNCS E53C22001930001 project
RISICO – Reversibilità in Sistemi Concorrenti: Analisi Quantitative e
Funzionali, and the European Union - NextGenerationEU SEcurity and RIghts in
the CyberSpace (SERICS) Research and Innovation Program PE00000014, projects
STRIDE and SWOP.

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# Effective electromagnetic actions for Lorentz violating theories exhibiting
the axial anomaly

Andrés Gómez<EMAIL_ADDRESS>Facultad de Ciencias, Universidad
Nacional Autónoma de México,
04510 México, Distrito Federal, México  A. Martín-Ruiz
<EMAIL_ADDRESS>Luis F. Urrutia<EMAIL_ADDRESS>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
04510 México, Distrito Federal, México

###### Abstract

The CPT odd contribution to the effective electromagnetic action deriving from
the vacuum polarization tensor in a large class of fermionic systems
exhibiting Lorentz invariance violation (LIV) is calculated using thermal
field theory methods, focusing upon corrections depending on the chemical
potential. The systems considered exhibit the axial anomaly and their
effective actions are described by axion electrodynamics whereby all the LIV
parameters enter in the coupling $\Theta(x)$ to the unmodified Pontryagin
density. A preliminary application to type-I tilted Weyl semimetals is briefly
presented.

Lorentz violation, Vacuum polarization, Thermal field theory, Axial anomaly

##  I Introduction

The study of quantum corrections in the QED fermionic sector of the Standard
Model Extension (SME) Kostelecky0 ; Kostelecky1  due to the electromagnetic
interaction was sparkled in Ref. Kostelecky1  which posed the problem of
obtaining corrections to the Chern-Simons interaction
$\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}{\Delta
k}_{\mu}A_{\nu}F_{\alpha\beta}$ as arising from the one loop vacuum
polarization tensor of fermions with the additional coupling
${\bar{\Psi}}\gamma^{5}\gamma^{\mu}b_{\mu}\Psi$. Since then, a large number of
authors have carried the calculation of the CPT odd contribution to the vacuum
polarization obtaining a result that can be presented as $\Delta
k^{\mu}=\zeta\,(e^{2}/\pi^{2})\,b^{\mu}$, where $\zeta$ is always a finite
coefficient adopting different values, including $\zeta=0$, according to the
regularization prescription chosen to deal with the superficially linear
divergence of the vacuum polarization tensor $\Pi^{\mu\nu}$ CG ; JK ; PV ;
CHUNG1 ; CHUNG2 ; CHEN ; BONNEAU ; CHAICHIAN ; PV1 ; BATTISTEL ; ANDRIANOV ;
BALT1 ; BALT2 ; ALFARO . A survey of the principal approaches regarding this
issue can be found in Refs. PV1 ; ALFARO ; JACKIW2 ; BALT3 . The search of a
physical criterion to select one specific value of $\zeta$ has been
unsuccessful and the current understanding is that $\zeta$ would be fixed
either by an experimental condition or by a more fundamental theory. In order
to briefly summarize some of the previous results we assume that $b_{\mu}$ is
timelike. The case $\zeta=0$ is obtained by demanding gauge invariance of the
effective Lagrangian density CG , and has the drawback of eliminating the
Chern-Simons contribution from the outset since this term changes by a total
derivative under gauge transformations. Using the full non-perturbative
expression in $b_{\mu}$ for the fermion propagator and different regulators
consistent with symmetric integration in four dimensions, the authors of Refs.
JK ; PV ; CHUNG1 ; CHUNG2  find $|\zeta|={3}/{16}$. In an attempt to elucidate
a further significance of the non-perturbative approach, which demands the use
of a unique regulator for all the contributions of $\Pi^{\mu\nu}$, the CPT
even contribution to $\Pi^{\mu\nu}$ (second order in $b_{\mu}$) was calculated
in the $m=0$ case, finding that gauge invariance is violated BALT1 . The use
of Pauli-Villars regularization was subsequently introduced in the non-
perturbative calculation of the massive case, yielding gauge invariance to
second order but two possibilities for the CPT odd contribution: either the
expected value $\zeta=0$ or $|\zeta|=3/8$ BALT2 . Again, there is no unique
way of fixing the $\zeta$ contribution by demanding gauge invariance of
$\Pi^{\mu\nu}$. An alternative regularization prescription based upon the
maximal residual symmetry group of the vector $b_{\mu}$, proposed in Ref.
ANDRIANOV , provides a definition on how to perform the loop integration
$d^{4}k$ in $\Pi^{\mu\nu}$. For example, in the timelike case, the maximal
residual symmetry group is $SO(3)$ which calls for a spherically symmetric
integration in the tri-momentum ${\mathbf{k}}$ instead of on the four-momentum
$k^{\mu}$. This prescription yields $|\zeta|=1/4$.

In this work we extend previous calculations of effective electromagnetic
action induced by radiative correction in the SME by including additional
terms of the fermionic sector in the minimal QED extension of the SME TABLE .
Our starting point is the action

$\displaystyle S=\int
d^{4}x\,\bar{\Psi}\left(\Gamma^{\mu}i\partial_{\mu}-M-e\Gamma^{\mu}A_{\mu}\right)\Psi,$
(1)

which is coupled to the electromagnetic field
$A^{\mu}=(A^{0},A^{i})=(A^{0},{\mbox{\boldmath$A$}})$, with the notation
${\mbox{\boldmath$A$}}=(A_{x},\,A_{y},\,A_{z})$ in terms of the Cartesian
components. The metric is $\eta_{\mu\nu}=\mbox{diag}(+,-,-,-)$ and we use
$\hbar=c=1$ units henceforth. We set
$m=m_{5}=H_{\mu\nu}=e_{\lambda}=f_{\lambda}=g_{\lambda\kappa\nu}=0$, in the
notation of Table XVI of Ref. TABLE . Thus we restrict ourselves to

$\Gamma^{\mu}=c^{\mu}{}_{\nu}\gamma^{\nu}+d^{\mu}{}_{\nu}\gamma^{5}\gamma^{\nu},\qquad
M=a_{\mu}\gamma^{\mu}+b_{\mu}\gamma^{5}\gamma^{\mu},$ (2)

with $\gamma^{\mu}$ being the standard gamma matrices. The term
$c^{\mu}{}_{\nu}$ already includes the $\delta^{\mu}_{\nu}$ contribution
corresponding to the free Dirac action. The terms
${\bar{\Psi}}\,\Gamma^{\mu}i\partial_{\mu}\Psi$ are CPT even, while those in
${\bar{\Psi}}M\Psi$ are CPT odd. Still, since $\gamma^{5}$ is PT odd, each of
them separately contains a mixture of PT even and odd terms.

Let us emphasize that the methods and techniques employed in the calculation
of these extended effective actions in high energy physics, besides being
valuable in their own, can be of relevance in some areas of condensed matter
physics. Indeed, the identification of fermionic quasiparticles of Dirac
and/or Weyl type in the linearized approximation of Hamiltonians in
topological phases of matter provides the opportunity of studying them under
the perspective of the SME. Such approach has been particularly fruitful in
the case of Weyl semimetals (WSMs) whose electronic Hamiltonians naturally
include some of the LIV terms considered in the fermion sector of the SME.
Nevertheless, in this case, the LIV parameters need not be highly suppressed,
since they are determined by the electronic structure of the material and are
subjected to experimental determination GRUSHIN ; Burkov ; Goswami ; TI ;
REVLANDS ; Kostelecky3 . Our choice of the LIV parameters in Eq. (2) are those
relevant for the description of a general WSM. The simplest example arises in
the Hamiltonian of a Weyl semimetal with no tilting and with an isotropic
Fermi velocity, which can be embedded in the fermionic action

$\displaystyle S=\int
d^{4}x\,{\bar{\Psi}}(x)\left(i\gamma^{\mu}\partial_{\mu}-{{\tilde{b}}_{\mu}\gamma^{5}\gamma^{\mu}}\right)\Psi(x),$
(3)

where ${\tilde{b}}^{\mu}=({\tilde{b}}^{0},{\mbox{\boldmath${\tilde{b}}$}})$.
Starting from the action (3) coupled to an electromagnetic field, the chiral
rotation method of Ref. Burkov , which provides an alternative quite different
from the standard vacuum polarization calculation, yields

$\displaystyle S_{{\rm{eff}}}=\frac{e^{2}}{32\pi^{2}}\int
d^{4}x\,\Theta(x)\,\epsilon^{\alpha\beta\mu\nu}F_{\alpha\beta}F_{\mu\nu},\quad\Theta(x)=2{\tilde{b}}_{\mu}x^{\mu},$
(4)

which determines $|\zeta|=1/4$. Another outstanding example relevant in
condensed matter is the case of the superfluid 3He-A, where the value
$\zeta=1/2$ is reported VOLOVIK .

The action (4) captures the electromagnetic response of WSMs. While the axion
coupling $\Theta(x)$ arises from the nontrivial topology of the band structure
of the material, the appearance of the abelian Pontryagin density (APD)
$\epsilon^{\alpha\beta\mu\nu}F_{\alpha\beta}F_{\mu\nu}$ is related to the
axial anomaly

$\displaystyle\partial^{\mu}J_{\mu}^{5}=-\frac{e^{2}}{16\pi^{2}}\,\epsilon^{\alpha\beta\mu\nu}F_{\alpha\beta}F_{\mu\nu}.$
(5)

From Eq. (4) we obtain the effective sources FRANZ

$\rho=\frac{\delta S_{{\rm eff}}}{\delta
A_{0}}=\frac{e^{2}}{2\pi^{2}}{\mbox{\boldmath$\tilde{b}$}}\cdot{\mbox{\boldmath$B$}},\qquad{\mbox{\boldmath$J$}}=\frac{\delta
S_{{\rm
eff}}}{\delta{\mbox{\boldmath$A$}}}=\frac{e^{2}}{2\pi^{2}}({\mbox{\boldmath$\tilde{b}$}}\times{\mbox{\boldmath$E$}}-{\tilde{b}}_{0}{\mbox{\boldmath$B$}}),$
(6)

where $E$ and $B$ are the electromagnetic fields. In our index notation the
above current is $J_{i}=\delta S_{\rm eff}/\delta A^{i}$. In particular, for
${\tilde{b}}_{0}=0$, this yields $J_{x}=\sigma_{xy}E_{y}$ with the transverse
anomalous Hall conductivity
$\sigma_{xy}=-\frac{e^{2}}{2\pi^{2}}{\tilde{b}}_{z}$, a distinguishing feature
of Weyl semimetals BURKOVBALENTS . As it will become clear in Section III.1,
the knowledge of this quantity allows to unambiguously fix the otherwise
arbitrary coefficient $\zeta$ in this case. This additional information is a
general consequence of the underlying microscopic theory describing the
material, as opposed to the lack of a more fundamental theory containing the
SME.

Recently, the appearance of the APD in effective actions arising from (1) has
promoted the use of anomaly calculations to obtain them. For example, in the
path integral approach, $S_{\rm eff}$ was obtained by introducing the
electromagnetic coupling in Eq. (3) and subsequently eliminating the fermionic
term proportional to ${\tilde{b}}_{\mu}$ through a chiral rotation.
Nevertheless, this produces an electromagnetic contribution to the action
arising from the nonzero Jacobian of the chiral rotation which is proportional
to the Pontryagin density Burkov . Following this idea, the Fujikawa
prescription to obtain the chiral anomalies Fujikawa ; Bertlement  has also
been used to calculate the effective electromagnetic action of different
materials in Refs. Goswami ; TI ; REVLANDS  as well as to give an alternative
explanation of the indeterminacy of the coefficient $\zeta$ related to some
freedom in the definition of the fermionic axial vector current CHUNG3 .

Nevertheless, as pointed out in Ref. LUAGA , the anomaly does not directly
yield the effective action. Also, the method of eliminating the modifying
fermionic contributions via a chiral rotation cannot be easily extended to
deal with the more complicated configurations envisaged in the action (1).

The reasons indicated above suggest the convenience of applying and extending
the quantum field theory methods developed in high energy physics to obtain
the required effective electromagnetic actions corresponding to fermionic
systems described by the action (1) which are of interest in condensed matter
physics. In particular this procedure should clarify how the LIV corrections
enter in the effective action, while the chiral anomaly remains insensitive to
them LUAGA ; FIDEL ; SALVIO ; Scarpelli . Motivated by the inclusion of
temperature effects in the odd contribution of the vacuum polarization tensor
arising from the $b_{\mu}$ coupling in the action (3), reported in Refs. EBERT
; TEMP1 ; TEMP2 ; TEMP3 ; TEMP4 ; TEMP5 , we provide the first steps to
incorporate thermal field theory in the case of the more general LIV couplings
defined in Eqs. (2). In this way, we incorporate non-zero chemical potential
effects, but still remain in the zero temperature limit. We follow the
conventions of Ref. PESKIN .

##  II  The effective action

Integrating the fermions in Eq. (1) defines the effective action

$\displaystyle\exp(iS_{\rm
eff})=\det\Big{[}{\bar{\Psi}}(x)\,\Big{(}\Gamma^{\mu}i\partial_{\mu}-M-e\Gamma^{\mu}A_{\mu}\Big{)}\,\Psi(x)\Big{]},$
(7)

which we write as

$\displaystyle S_{\rm
eff}^{(2)}(A)=\frac{1}{2}\int\frac{d^{4}p}{\left(2\pi\right)^{4}}\,A_{\mu}(-p)\,\Pi^{\mu\nu}(p)\,A_{\nu}(p),$
(8)

to second order in the electromagnetic potential. This introduces the vacuum
polarization tensor $\Pi^{\mu\nu}(p)$

$\displaystyle
i\Pi^{\mu\nu}(p)=e^{2}\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\,\mbox{tr}\left[S(k-p)\Gamma^{\mu}S(k)\Gamma^{\nu}\right],$
(9)

where $S(k)=i/(\Gamma^{\mu}k_{\mu}-M)$ is the exact fermion propagator in
momentum space including LIV modifications. Since $S_{\rm eff}^{(2)}$ is real
we must have $\Pi^{*}_{\mu\nu}(p)=\Pi_{\nu\mu}(p)$. As we will show, the new
effective action (8) will keep the form of Eq. (4) with all the modifications
entering through a new vector ${\cal B}_{\lambda}$ to be determined. The link
between both expressions is accomplished with the identifications

$\displaystyle\Pi^{\mu\nu}(p)=-i\frac{e^{2}}{2\pi^{2}}\,{\cal
B}_{\lambda}p_{\kappa}\epsilon^{\mu\nu\lambda\kappa},\quad\Theta(x)=2{\cal
B}_{\lambda}x^{\lambda}.$ (10)

In the following we calculate the CPT odd contribution of the effective action
(8). In the massless case ($m=0$), $\Gamma^{\mu}k_{\mu}-M$ is linear in
$\gamma^{\mu}$ and $\gamma^{5}\gamma^{\mu}$. The appearance of the matrix
$\gamma^{5}$ suggests the convenience of using left and right chiral
projectors in order to evaluate $\gamma^{5}=\pm 1$ BALT1 ; SALVIO ; TEMP1 .
Therefore, we can work in the chiral basis, where the decomposition of the
Dirac spinor into the right and left Weyl spinors is manifest. These latter
are eigenspinors of $\gamma^{5}$ with the eigenvalues $\pm 1$. We now define
the projection operators

$\displaystyle P_{R}=\frac{1+\gamma^{5}}{2},\qquad
P_{L}=\frac{1-\gamma^{5}}{2},\qquad\gamma_{5}^{2}=1,$ (11)

which project onto right- and left-handed spinors, respectively. Note that
$\gamma^{\mu}P_{L}=P_{R}\gamma^{\mu}$. The projectors (11) allow us to define
the matrices $\Gamma_{R}^{\mu}$ such that

$\displaystyle\Gamma^{\mu}P_{R}=\left(c^{\mu}{}_{\nu}-d^{\mu}{}_{\nu}\right)\gamma^{\nu}P_{R}\equiv\Gamma_{R}^{\mu}P_{R},$
(12)

which identifies $\Gamma^{\mu}_{R}=r^{\mu}{}_{\nu}\gamma^{\nu}$ with
$r^{\mu}{}_{\nu}=c^{\mu}{}_{\nu}-d^{\mu}{}_{\nu}$. In an analogous way we
define the left-handed part of the matrices $\Gamma^{\mu}$ as

$\displaystyle\Gamma^{\mu}P_{L}=\Gamma_{L}^{\mu}P_{L},\qquad\Gamma_{L}^{\mu}=l^{\mu}{}_{\nu}\gamma^{\nu},\qquad
l^{\mu}{}_{\nu}=c^{\mu}{}_{\nu}+d^{\mu}{}_{\nu}\,.$ (13)

Following the same idea, we can also split the fermion propagator $S(k)$ into
its right- and left-handed parts, i.e.

$\displaystyle\frac{i}{\Gamma^{\mu}k_{\mu}-M}\gamma^{\alpha}P_{R}=P_{R}S_{R}(k)\gamma^{\alpha},\quad
S_{R}(k)=\frac{i}{\big{(}k_{\mu}r^{\mu}{}_{\nu}-a_{\nu}+b_{\nu}\big{)}\gamma^{\nu}},$
$\displaystyle\frac{i}{\Gamma^{\mu}k_{\mu}-M}\gamma^{\alpha}P_{L}=P_{L}S_{L}(k)\gamma^{\alpha},\quad
S_{L}(k)=\frac{i}{\big{(}k_{\mu}l^{\mu}{}_{\nu}-a_{\nu}-b_{\nu}\big{)}\gamma^{\nu}}.$
(14)

Note that the propagators $S_{L}$ and $S_{R}$, having the generic form
$i/(Z_{\nu}\gamma^{\nu})$, can be readily rationalized as
$i(Z_{\nu}\gamma^{\nu})/Z^{2}$.

To proceed forward with the calculation we now split the combination
$T^{\mu\nu}(k,p)=S(k-p)\Gamma^{\mu}S(k)\Gamma^{\nu}$ under the trace in
$\Pi^{\mu\nu}(p)$ into its left- and right-handed parts, i.e.

$\displaystyle T_{L(R)}^{\mu\nu}(k,p)$
$\displaystyle=S(k-p)\Gamma^{\mu}S(k)\Gamma^{\nu}P_{L(R)},$ (15)

which implies that the vacuum polarization can be written as the sum
$\Pi^{\mu\nu}(p)=\Pi_{L}^{\mu\nu}(p)+\Pi_{R}^{\mu\nu}(p)$, where

$\displaystyle i\Pi_{L(R)}^{\mu\nu}(p)$
$\displaystyle=e^{2}\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\,\mbox{tr}\left[T_{L(R)}^{\mu\nu}(k,p)\right]$
(16)

is the vacuum polarization for a left-(right-)handed massless fermion. We now
concentrate in the calculation of $\Pi_{L(R)}^{\mu\nu}(p)$. Using Eqs.
(12)-(14) and the cyclic property of the trace, the left-handed part
$\Pi_{L}^{\mu\nu}$ can be written as

$\displaystyle i\Pi_{L}^{\mu\nu}(p)$
$\displaystyle=e^{2}l^{\mu}{}_{\beta}l^{\nu}{}_{\alpha}\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\,\mbox{tr}\left[S_{L}(k-p)\gamma^{\beta}S_{L}(k)\gamma^{\alpha}P_{L}\right].$
(17)

A similar procedure yields the right-handed part $\Pi^{\mu\nu}_{R}(p)$ by
means of the replacements $L\to R$, $l^{\mu}{}_{\nu}\to r^{\mu}{}_{\nu}$,
$b_{\mu}\to-b_{\mu}$, $P_{L}\to P_{R}$ in Eq. (17). In the following we
restrict ourselves to the axial contributions $\Pi^{\mu\nu}_{A,L}$ and
$\Pi^{\mu\nu}_{A,R}$ of the left- and right-handed terms, respectively, which
are obtained by isolating the terms $P_{R}\to+\gamma^{5}/2$ and
$P_{L}\to-\gamma^{5}/2$ in the corresponding expressions for the vacuum
polarization. Both expressions can be summarized in the general form

$\displaystyle i\Pi_{A,\,\chi}^{\mu\nu}(p)$
$\displaystyle=\frac{{\chi}}{2}e^{2}m^{\mu}{}_{\beta}m^{\nu}{}_{\alpha}\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\mbox{tr}\left[S_{{\chi}}(k-p)\gamma^{\beta}S_{{\chi}}(k)\gamma^{\alpha}\gamma^{5}\right],$
(18)

with the following assignments

$\displaystyle R$ $\displaystyle:$ $\displaystyle{\chi=+1}\qquad
m^{\mu}{}_{\nu}=r^{\mu}{}_{\nu},\qquad C_{\rho}=a_{\rho}-b_{\rho},$
$\displaystyle L$ $\displaystyle:$ $\displaystyle{\chi=-1}\qquad
m^{\mu}{}_{\nu}=l^{\mu}{}_{\nu},\qquad C_{\rho}=a_{\rho}+b_{\rho}.$ (19)

Clearly, the full axial contribution to the vacuum polarization is the sum of
the $L$ and $R$ parts, i.e.
$\Pi_{A}^{\mu\nu}=\Pi_{A,L}^{\mu\nu}+\Pi_{A,R}^{\mu\nu}$. For the sake of
simplicity, we do not introduce an additional subindex $\chi$ either in the
matrix $m^{\alpha}{}_{\beta}$ or in the vector $C_{\rho}$, which are to be
restored at the end of the calculations according to Eq. (19). The calculation
in Eq. (18) proceeds as follows: to simplify the notation we introduce the
primed vectors $q_{\nu}^{\prime}=q_{\mu}m^{\mu}{}_{\nu}$ maintaining the
original measure $d^{4}k$. Also we rationalize the denominators in the
propagators and take the trace
$\mbox{tr}(\gamma^{\rho}\gamma^{\beta}\gamma^{\sigma}\gamma^{\alpha}\gamma^{5})=-4i\epsilon^{\rho\beta\sigma\alpha}$,
with $\epsilon^{0123}=+1$. We use the antisymmetry of the Levi-Civita symbol
to eliminate the term
$\epsilon^{\alpha\beta\rho\sigma}(k^{\prime}-C)_{\rho}(k^{\prime}-C)_{\sigma}$
in the numerator. Since we are interested in the contribution to the effective
action including the product of two electromagnetic tensors without additional
derivatives, we calculate the integral Eq. (18) only to first order in
$p_{\alpha}$. Finally, and assuming that $m^{\mu}{}_{\nu}$ is invertible, the
identity
$m^{\mu}{}_{\beta}\,m^{\nu}{}_{\alpha}\,m^{\kappa}{}_{\sigma}\epsilon^{\beta\alpha\rho\sigma}=(\det
m)\,(m^{-1})^{\rho}{}_{\lambda}\epsilon^{\mu\nu\lambda\kappa}$ yields the
further simplification

$\displaystyle\Pi_{A,\,\chi}^{\mu\nu}(p)=-2{\chi}e^{2}(\det
m)(m^{-1})^{\rho}{}_{\lambda}\epsilon^{\mu\nu\lambda\kappa}p_{\kappa}\,I_{\rho}(C),$
(20)

where we have defined the integral

$\displaystyle
I_{\rho}(C)=\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\,g_{\rho}(k_{0},{\mbox{\boldmath$k$}})\,,\quad
g_{\rho}(k_{0},{\mbox{\boldmath$k$}})=\frac{\left(k^{\prime}-C\right)_{\rho}}{\left[\left(k^{\prime}-C\right)^{2}\right]^{2}},$
(21)

and we recall that ${k^{\prime}}_{\mu}={k}_{\alpha}m^{\alpha}{}_{\mu}$.
Previous to regularization, the above expression is our final result for the
vacuum polarization tensor in Minkowski spacetime, which can be evaluated for
the left- and right-handed fermions according to Eq. (20) with the assignments
in Eq. (19). The result (20) holds for arbitrary LIV terms
$c^{\mu}{}_{\nu},\,d^{\mu}{}_{\nu},\,a_{\mu}$ and $b_{\mu}$ as long as these
produce invertible matrices $l^{\mu}{}_{\nu}$ and $r^{\mu}{}_{\nu}$.

##  III The regularization

As a first step in the inclusion of thermal field theory methods in the
description of the radiative corrections to the fermionic action (1) under the
new conditions (3), here we shall consider the case of a non-zero chemical
potential $\mu$ but still remain in the zero-temperature limit. The inclusion
of both will be published elsewhere. To this end we adopt the finite
temperature approach in the imaginary time formulation KAPUSTA  , which we
recover after the substitution BERNARD ; DITTRICH ; DAS

$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\ \ \rightarrow\ \
+iT\sum_{n=-\infty}^{\infty}\int\frac{d^{3}{\mbox{\boldmath$k$}}}{(2\pi)^{3}},$
(22)

with $k_{0}\to k_{0}=i\omega_{n}+\Lambda$.

In order to have a specific system which determines the choice of the finite
but undetermined parameters, together with a better physical understanding of
the subsequent steps, we introduce the Hamiltonian

$\displaystyle
H_{\chi}(\mbox{\boldmath$p$})={\mbox{\boldmath$v$}}_{\chi}\cdot({\mbox{\boldmath$p$}}+\chi{\mbox{\boldmath$\tilde{b}$}})-\chi{\tilde{b}}_{0}+\chi
v_{F}{\mbox{\boldmath$\sigma$}}\cdot({\mbox{\boldmath$p$}}+\chi{\mbox{\boldmath$\tilde{b}$}}),$
(23)

borrowed from condensed matter, which describes a Weyl semimetal with two
linear 3D band crossings of chirality $\chi=\pm 1$ close to
$\pm{\mbox{\boldmath$\tilde{b}$}}$ in momentum space and $\pm{\tilde{b}}_{0}$
in energy. Here $v_{F}$ is the isotropic Fermi velocity at each band crossing,
${\mbox{\boldmath$\sigma$}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the triplet
of spin-$1/2$ Pauli matrices, ${\mbox{\boldmath$v$}}_{\chi}$ is the tilting
parameter and $p$ is the momentum. In the Weyl basis for the gamma matrices

$\gamma^{0}=\begin{pmatrix}0&\sigma_{0}\\\ \sigma_{0}&0\end{pmatrix},\hskip
14.22636pt\gamma^{i}=\begin{pmatrix}0&\sigma^{i}\\\
-\sigma^{i}&0\end{pmatrix},\hskip
14.22636pt\gamma^{5}=\begin{pmatrix}-\sigma_{0}&0\\\
0&\sigma_{0}\end{pmatrix},$ (24)

both $2\times 2$ Hamiltonians in Eq. (23) can be embedded in the action (1)
with the choices $c^{0}{}_{0}=1,c^{0}{}_{i}=d^{0}{}_{i}=d^{0}{}_{0}=0$, i.e.
$\Gamma^{0}=\gamma^{0}$ together with $\Gamma^{i}$ and $M$ being determined by
the parameters in (23). For simplicity, the anisotropy in the Fermi velocities
is not included in (23) but enters naturally in our final result through the
coefficients $c^{i}{}_{j}$ and $d^{i}{}_{j}$. The Hamiltonian (23) is realized
in terms of chiral fermions whose dispersion relation behaves linearly, with
band crossing points localized both in momentum and energy, as schematically
indicated in the Fig. 1(a). As usual, the position of the chemical potential
determines most of the transport properties of a given material. In a metal
for instance, it has to be measured from the gap closing, since it represents
the filling of either the conduction or the valence bands. This suggests that
in WSMs, the transport properties will depend on the band filling, i.e. on the
chemical potential as measured from the node of the cones. Therefore, in order
to apply the finite temperature approach (22) to WSMs, the parameter $\Lambda$
in Eq. (22) has to be understood as the chemical potential measured from the
band-crossing points, as shown in Fig. 1(a). To be precise, $\Lambda=\mu-
E_{\chi}({\mbox{\boldmath$p$}}_{\chi})$, where ${\mbox{\boldmath$p$}}_{\chi}$
is the location in momentum of the node with chirality $\chi$, and
$E_{\chi}({\mbox{\boldmath$p$}}_{\chi})$ the corresponding energy. Both
${\mbox{\boldmath$p$}}_{\chi}$ and $E_{\chi}({\mbox{\boldmath$p$}}_{\chi})$
will be determined later for the problem at hand. In the simplified model (3),
it is clear that
${\mbox{\boldmath$p$}}_{\chi}=\chi{\mbox{\boldmath$\tilde{b}$}}$ and
$E_{\chi}({\mbox{\boldmath$p$}}_{\chi})=-\chi{\tilde{b}}_{0}$.

The sum in Eq. (22) is over the Matsubara frequencies $\omega_{n}=(2n+1)\pi T$
required to produce anti-periodic boundary conditions for the fermions KAPUSTA
. Next we focus in Eq. (21) and we make use of the additional relation KAPUSTA

$\displaystyle\lim_{T\to
0}T\sum_{n=-\infty}^{\infty}g_{\rho}(k_{0}=i\omega_{n}+{\Lambda},{\mbox{\boldmath$k$}})$
$\displaystyle=\frac{1}{2\pi
i}\int_{-i\infty}^{i\infty}dk_{0}\,g_{\rho}(k_{0},{\mbox{\boldmath$k$}})\,+$
$\displaystyle\phantom{=}+\frac{1}{2\pi
i}\oint_{\Omega}dk_{0}\,g_{\rho}(k_{0},{\mbox{\boldmath$k$}}),$ (25)

where the contour $\Omega$ is shown in the Fig. 1(b).

Figure 1: (a) The effective chemical potential. (b) The contour of integration
in the $k_{0}$-plane.

###  III.1 The $\mu$-independent contribution

The first term in the right hand side of Eq. (25) reduces to the standard
zero-temperature, zero-chemical potential contribution, which has been
previously discussed in the literature as extensively reported in section I.
Going back to Eq. (20), this corresponds to the direct evaluation of the
integral $I_{\rho}(C)$ after the change of integration variables
$k^{\prime}_{\mu}=k_{\nu}m^{\nu}{}_{\mu}$. Since the only vector at our
disposal is $C_{\mu}$ we have

$I^{(1)}_{\rho}(C)=\frac{i}{(\det
m)}\,{\bar{N}}C_{\rho},\quad{\bar{N}}=\frac{1}{C^{2}}\left[\int\frac{d^{4}k^{\prime}}{\left(2\pi\right)^{4}}\;\frac{\left(k^{\prime}-C\right)\cdot
C}{\left[\left(k^{\prime}-C\right)^{2}\right]^{2}}\right]_{E},$ (26)

where the integral inside the square brackets is in Euclidean space and the
factor $+i$ comes from the Wick rotation. The factor ${\bar{N}}$ is
regularization dependent and could only be a function of the magnitude of the
four-vector $C_{\mu}$. However, a change of scale $C_{\sigma}\to\lambda
C_{\sigma}$ followed by an additional change of variables
$k^{\prime\prime}_{\mu}=\lambda k^{\prime}_{\mu}$ shows that ${\bar{N}}$ is
just a numerical factor, independent of $C_{\mu}$. Therefore, ${\bar{N}}$ is
the same for both left- and right-handed fermions, i.e.
${\bar{N}}_{L}={\bar{N}}_{R}={\bar{N}}$. In this way, the total contribution
to the vacuum polarization in this case is summarized in the four-vector

$\displaystyle{\cal
B}^{(1)}_{\lambda}=-4\pi^{2}{\bar{N}}\,\Big{[}C_{L\rho}\,(l^{-1})^{\rho}{}_{\lambda}-C_{R\rho}\,(r^{-1})^{\rho}{}_{\lambda}\Big{]},$
(27)

according to Eq. (10). As shown previously in the literature the factor
${\bar{N}}$ is finite but undetermined. In the case of WSMs, its dependence
upon the regularization procedure has been studied in Ref. Goswami  and the
final choice is made by selecting the anomalous Hall conductivity
$\sigma_{xy}=-e^{2}{\tilde{b}}_{z}/(2\pi^{2}$) as the physical quantity to be
reproduced in the zero-tilting limit of the Hamiltonian (23) Burkov ; Goswami
; BURKOVBALENTS . To this end it is necessary to take a cut-off in the
direction of the spatial component of $C$, with the result
${\bar{N}}=-1/(8\pi^{2})$. The suitability of this choice will be demonstrated
later by comparing our field-theoretic results with those obtained from a
semiclassical Boltzmann approach, which has proven to be successful in the
study the transport properties of WSMs.

###  III.2 The $\mu$-dependent contribution

Next we consider the second term in the right-hand side of Eq. (25) and
calculate

$\displaystyle
I^{(2)}_{\rho}(C)=\int\frac{d^{3}{\mbox{\boldmath$k$}}}{(2\pi)^{3}}\,\frac{1}{2\pi
i}\,\oint_{\Omega}dk_{0}\,g_{\rho}(k_{0},{\mbox{\boldmath$k$}}).$ (28)

Let us start by finding the double poles of the integrand in the
$k_{0}$-plane. They are located at the two points
$k^{\prime}_{0\pm}=C_{0}\pm|{\mbox{\boldmath$k$}}^{\prime}-{\mbox{\boldmath$C$}}|$,
where we recall the relations $k^{\prime}_{0}=k_{\mu}m^{\mu}{}_{0}$ and
$k^{\prime}_{i}=k_{0}m^{0}{}_{i}+k_{j}m^{j}{}_{i}$. In the general case the
solution for $k_{0}$ involves a rather cumbersome second order equation which
will not be very illuminating for our purposes. In order to illustrate the
full procedure we restrict ourselves to the simpler situation where
$m^{0}{}_{\nu}=\delta^{0}_{\nu}$. Under this assumption the poles are located
at

$\displaystyle
k_{0\xi}=-k_{j}m^{j}{}_{0}+C_{0}+\xi\,|{\mbox{\boldmath$k$}}^{\prime}-{\mbox{\boldmath$C$}}|,\quad\xi=\pm
1,\quad k^{\prime}_{j}=k_{i}\,m^{i}{}_{j},$ (29)

which corresponds to the dispersion relation of the model, wherefrom we can
determine the position in momentum and energy of the Dirac/Weyl node. Clearly,
the band touching point (i.e. the node) occurs at
${\mbox{\boldmath$k$}}^{\prime}={\mbox{\boldmath$C$}}$, and the corresponding
energy is $E_{0}\equiv
k_{0\xi}({\mbox{\boldmath$k$}}^{\prime}={\mbox{\boldmath$C$}})$, which can be
explicitly expressed as

$E_{0}=-\mathcal{V}^{j}C_{j}+C_{0}=\mbox{\boldmath$\mathcal{V}$}\cdot{\mbox{\boldmath$C$}}+C_{0},\quad\mathcal{V}^{j}=(m^{-1})^{j}{}_{i}\,m^{i}{}_{0}.$
(30)

Applying the residue theorem we obtain

$\displaystyle\frac{1}{2\pi i}\oint_{\Omega}dk_{0}\,g_{\rho}=\sum_{\xi=\pm
1}\mathrm{Res}(g_{\rho},k_{0}=k_{0\xi})H(k_{0\xi})H(\Lambda-k_{0\xi}),$ (31)

where the Heaviside functions $H$ guarantee that the poles $k_{0\xi}$ fall
inside the contour $\Omega$ of Fig. 1, i.e. $0<k_{0\xi}<\Lambda$, where
$\Lambda=\mu-E_{0}$, with $E_{0}$ given by Eq. (30). The residues are

$\mathrm{Res}(g_{\rho},k_{0}=k_{0\xi})=-\xi\frac{1}{4}\delta^{i}_{\rho}\frac{k^{\prime}_{i}-C_{i}}{|{\mbox{\boldmath$k$}}^{\prime}-{\mbox{\boldmath$C$}}|^{3}}.$
(32)

Substitution of the residues (32) in Eq. (28) determines $I^{(2)}_{\rho}(C)$,
and combining the result with Eq. (20) it follows that the vacuum polarization
becomes

$\displaystyle{\Pi}_{A,\,\chi}^{\mu\nu}(p)$
$\displaystyle=i\chi\,\frac{e^{2}}{2}(\det
m)(m^{-1})^{i}{}_{\lambda}p_{\kappa}\epsilon^{\mu\nu\lambda\kappa}$
$\displaystyle\phantom{=}\times\sum_{\xi=\pm
1}\xi\int\frac{d^{3}{\mbox{\boldmath$k$}}}{(2\pi)^{3}}\frac{k^{\prime}_{i}-C_{i}}{|{\mbox{\boldmath$k$}}^{\prime}-{\mbox{\boldmath$C$}}|^{3}}H(k_{0\xi})H(\Lambda-
k_{0\xi}),$ (33)

which includes the factor $+i$ arising from the Eq. (22). It is convenient to
make the change of variables

$\displaystyle k^{\prime\prime}_{i}=k^{\prime}_{i}-C_{i},\quad
d^{3}{\mbox{\boldmath$k$}}^{\prime\prime}=d^{3}{\mbox{\boldmath$k$}}^{\prime}=\det(m)\,d^{3}{\mbox{\boldmath$k$}}$
(34)

where we used that $\det(m^{i}{}_{j})=\det(m^{\mu}{}_{\nu})=\det(m)$ since
$m^{0}{}_{\nu}=\delta^{0}_{\nu}$. This yields

$\displaystyle{\Pi}^{\mu\nu}_{A,\,\chi}(p)=i\chi\frac{e^{2}}{2}(m^{-1})^{i}{}_{\lambda}\,p_{\kappa}\,\epsilon^{\mu\nu\lambda\kappa}\,\big{(}I_{i}^{+}-I_{i}^{-}\big{)}$
(35)

with

$I^{\xi}_{i}=\int\frac{d^{3}{\mbox{\boldmath$k$}}^{\prime\prime}}{(2\pi)^{3}}\frac{k^{\prime\prime}_{i}}{|{\mbox{\boldmath$k$}}^{\prime\prime}|^{3}}\,H(k_{0\xi})\,H(\Lambda-
k_{0\xi}).$ (36)

In the new double-primed variables $k^{\prime\prime}$ the constraints
$0<k_{0\xi}<\Lambda$ read

$\displaystyle
0<-(k^{\prime\prime}_{j}+C_{j})(m^{-1})^{j}{}_{i}\,m^{i}{}_{0}+C_{0}+\xi\,|{\mbox{\boldmath$k$}}^{\prime\prime}|<\Lambda.$
(37)

From the identifications in Eq. (30) these constrains become

$\displaystyle
0<\mbox{\boldmath$\mathcal{V}$}\cdot{\mbox{\boldmath$k$}}^{\prime\prime}+\xi\,|{\mbox{\boldmath$k$}}^{\prime\prime}|+E_{0}<\Lambda.$
(38)

It is convenient to perform the 3-momentum integration in spherical
coordinates. Hence we present the condition (38) as

$\displaystyle
0<|{\mbox{\boldmath$k$}}^{\prime\prime}|\,\big{(}|\mbox{\boldmath$\mathcal{V}$}|\cos\theta+\xi\big{)}+E_{0}<\Lambda,$
(39)

Writing $I_{i}^{\xi}$ as a linear combination of the vectors ${\cal V}_{i}$
and $C_{i}$, one can show that the projection upon $C_{i}$ is zero, so that

$\displaystyle I^{\xi}_{i}=N^{\xi}\,\mathcal{V}_{i},$ (40)

where $N^{\xi}$ depends only on
$(|\mbox{\boldmath$\mathcal{V}$}|,E_{0},\Lambda)$ as

$\displaystyle
N^{\xi}=\frac{1}{(2\pi)^{2}|\mbox{\boldmath$\mathcal{V}$}|}\int_{0}^{\pi}\\!\\!d\theta\sin\theta\cos\theta\int_{0}^{\infty}\\!\\!d|{\mbox{\boldmath$k$}}^{\prime\prime}|\,H(k_{0\xi})H(\Lambda-
k_{0\xi}).$ (41)

Note that for $|\mbox{\boldmath$\mathcal{V}$}|\geq 1$ there exists an angular
direction for which $|\mbox{\boldmath$\mathcal{V}$}|\cos\theta+\xi$ is zero
whereby the integral over $|{\mbox{\boldmath$k$}}^{\prime\prime}|$ diverges
and which would imply the need of a nontrivial renormalization procedure.
Thus, from now on we take the simpler convergent case
$|\mbox{\boldmath$\mathcal{V}$}|<1$ (relevant for type-I WSMs) which yields
$|\mbox{\boldmath$\mathcal{V}$}|\cos\theta+\xi$ with definite sign: positive
(negative) for $\xi=+1(\xi=-1)$. This in turn fixes the inequalities indicated
in Eq. (39) as

$\displaystyle\xi=+1$
$\displaystyle:\quad\frac{-E_{0}}{|\mbox{\boldmath$\mathcal{V}$}|\cos\theta+1}<|{\mbox{\boldmath$k$}}^{\prime\prime}|<\frac{\Lambda-
E_{0}}{|\mbox{\boldmath$\mathcal{V}$}|\cos\theta+1}$ (42)
$\displaystyle\xi=-1$ $\displaystyle:\quad\frac{\Lambda-
E_{0}}{|\mbox{\boldmath$\mathcal{V}$}|\cos\theta-1}<|{\mbox{\boldmath$k$}}^{\prime\prime}|<\frac{-E_{0}}{|\mbox{\boldmath$\mathcal{V}$}|\cos\theta-1}$
(43)

The above equations demand us to distinguish the signs of $E_{0}$ and of
$\Lambda-E_{0}$, since we must ensure that
$|{\mbox{\boldmath$k$}}^{\prime\prime}|\geq 0$. This leaves us with four cases
according to the combinations ${\Lambda-E_{0}}\gtrless 0$ and $E_{0}\gtrless
0$. Here we make use of the assumption that $\Lambda>0$, as it constrains the
possibilities of these four cases. Under this condition a detailed analysis
yields the global result

$\displaystyle
I^{+}_{i}-I^{-}_{i}=\frac{\Lambda}{\pi^{2}}N_{\chi}{\mathcal{V}}_{i},\quad
N_{\chi}=\frac{1}{2|\mbox{\boldmath$\mathcal{V}$}|^{3}}\Big{(}|\mbox{\boldmath$\mathcal{V}$}|-\mathrm{arctanh}(|\mbox{\boldmath$\mathcal{V}$}|)\Big{)}.$
(44)

The resulting contribution to the vacuum polarization is

$\Pi^{\mu\nu}_{A}=i\frac{e^{2}}{2\pi^{2}}p_{\kappa}\,\epsilon^{\mu\nu\lambda\kappa}\,\Big{[}\Lambda_{R}\,N_{R}\,{\mathcal{V}}_{Ri}\,(r^{-1})^{i}{}_{\lambda}-\Lambda_{L}\,N_{L}\,{\mathcal{V}}_{Li}\,(l^{-1})^{i}{}_{\lambda}\Big{]},$
(45)

where $\Lambda$ has acquired a chirality dependence through the splitting of
the zero excitation energy modes given by

$E_{0L(R)}=\mbox{\boldmath$\mathcal{V}$}_{L(R)}\cdot{\mbox{\boldmath$C$}_{L(R)}}+C_{0L(R)},\quad\Lambda_{L(R)}=\mu-
E_{0L(R)}.$ (46)

We thus obtain the additional contribution

$\displaystyle{\cal
B}^{(2)}_{\lambda}=-\Big{[}\Lambda_{R}\,N_{R}\,{\mathcal{V}}_{Ri}\,(r^{-1})^{i}{}_{\lambda}-\Lambda_{L}\,N_{L}\,{\mathcal{V}}_{Li}\,(l^{-1})^{i}{}_{\lambda}\Big{]}$
(47)

to the coupling $\Theta(x)$ in Eq. (10).

Summarizing, the full effective electromagnetic action of the system described
by the fermionic action (1), with the only restrictions
$c^{0}{}_{\nu}=\delta^{0}_{\nu}$, $d^{0}{}_{\nu}=0$, $l^{\mu}{}_{\nu}$ and
$r^{\mu}{}_{\nu}$ invertible and $|\mbox{\boldmath$\mathcal{V}$}|<1$, is given
by the action (4) with $\Theta(x)=x^{\lambda}\big{(}{\cal
B}^{(1)}_{\lambda}+{\cal B}^{(2)}_{\lambda}\big{)}$. Notice that the
contribution proportional to $\mu$ is not regularization dependent. The
coupling $\Theta$ is CPT and PT odd as reflected in Eqs. (6) with
${\tilde{b}}_{\mu}\rightarrow{\cal B}_{\mu}$ since ${\cal B}_{i}$ breaks T but
not P, while ${\cal B}_{0}$ does the opposite. Then, even if we start with
only the CPT even part of the Lagrangian in Eq. (1) (the $\Gamma^{\mu}$ term )
by setting $a_{\mu}=b_{\mu}=0$ it is not surprising that we obtain a non-zero
PT odd $\Theta$ because $\Gamma^{\mu}$ already includes both PT even and odd
contributions. In this particular case ${\cal
B}^{(1)}_{\lambda}=0,\,\Lambda_{\chi}=\mu$, but ${\cal B}^{(2)}_{\lambda}\neq
0$ because ${\mbox{\boldmath$\cal V$}}_{R}$ and ${\mbox{\boldmath$\cal
V$}}_{L}$ remain arbitrary. In other words, the source of the PT (CPT) odd
effective electromagnetic action here is the $\gamma^{5}$ PT odd contribution
in $\Gamma^{\mu}$.

One particularly interesting and simple system takes place when we take
$c^{j}{}_{i}=\delta^{j}{}_{i}$ and $d^{j}{}_{i}=0$, which corresponds to the
case of arbitrary tilting $\mbox{\boldmath$V$}_{R}$ and
$\mbox{\boldmath$V$}_{L}$, but with equal isotropic Fermi velocity $v_{F}=1$
at each node. That is, we take

$\displaystyle C_{R\nu}$ $\displaystyle=a_{\nu}-b_{\nu},\qquad\qquad\quad
C_{L\nu}=a_{\nu}+b_{\nu},$ (48) $\displaystyle r^{\mu}{}_{\nu}$
$\displaystyle=\delta^{\mu}{}_{\nu}+V^{i}_{R}\,\delta^{\mu}{}_{i}\,\delta^{0}{}_{\nu},\qquad
l^{\mu}{}_{\nu}=\delta^{\mu}{}_{\nu}+V^{i}_{L}\,\delta^{\mu}{}_{i}\,\delta^{0}{}_{\nu},$
(49) $\displaystyle(r^{-1})^{\mu}{}_{\nu}$
$\displaystyle=\delta^{\mu}_{\nu}-V^{i}_{R}\delta^{\mu}{}_{i}\delta^{0}{}_{\nu},\qquad(l^{-1})^{\mu}{}_{\nu}=\delta^{\mu}_{\nu}-V^{i}_{L}\delta^{\mu}{}_{i}\delta^{0}{}_{\nu}.$
(50)

Under these conditions ${\cal V}^{i}=V^{i}$. Putting together the
contributions for ${\cal B}_{\lambda}={\cal B}^{(1)}_{\lambda}+{\cal
B}^{(2)}_{\lambda}$ we obtain

$\displaystyle{\cal B}_{0}$
$\displaystyle=b_{0}+{\mbox{\boldmath$b$}}\cdot{\mbox{\boldmath$U$}}_{+}+{\mbox{\boldmath$a$}}\cdot{\mbox{\boldmath$U$}}_{-}-\sum_{\chi=\pm
1}\frac{\chi\Lambda_{\chi}}{2|{\mbox{\boldmath$V$}}_{\chi}|}\,\Big{(}|{\mbox{\boldmath$V$}}_{\chi}|-\mathrm{arctanh}(|{\mbox{\boldmath$V$}}_{\chi}|)\Big{)},$
$\displaystyle{\cal B}^{i}$ $\displaystyle=b^{i}-\sum_{\chi=\pm
1}(V_{\chi})^{i}\,\frac{\chi\Lambda_{\chi}}{2|{\mbox{\boldmath$V$}}_{\chi}|^{3}}\,\Big{(}|{\mbox{\boldmath$V$}}_{\chi}|-\mathrm{arctanh}(|{\mbox{\boldmath$V$}}_{\chi}|)\Big{)},$
(51)

with
${\mbox{\boldmath$U$}}_{\pm}=\frac{1}{2}({\mbox{\boldmath$V$}}_{-}\pm{\mbox{\boldmath$V$}}_{+})$
and $\Lambda_{\chi}=\mu-E_{0\chi}$ according to Eq. (46). We recall that
$\chi=1$($\chi=-1$) denote the $R(L)$ contributions. Consistently with the
property that the axial anomaly is insensitive to LIV modifications LUAGA ;
FIDEL ; SALVIO ; Scarpelli , our results (51) show that the Pontryagin density
remains unchanged, and that the additional LIV terms in Eq. (2) which defines
the fermionic action (1), as well as the chemical potential, modify only the
$\Theta$ coupling.

In order to check the consistency of our results with a condensed matter
approach, we first establish the conditions under which our general model
reduces to that of a WSM as described by the Hamiltonian (23). Indeed, the
equivalence is achieved by setting
${\mbox{\boldmath$V$}}_{\chi}={\mbox{\boldmath$v$}}_{\chi},\,{\mbox{\boldmath$U$}}_{\chi}={\mbox{\boldmath$u$}}_{\chi}$,
and

${\mbox{\boldmath$a$}}=0,\quad{\mbox{\boldmath$b$}}={\mbox{\boldmath$\tilde{b}$}},\quad
a_{0}=-{\mbox{\boldmath$\tilde{b}$}}\cdot{\mbox{\boldmath$u$}}_{-},\quad
b_{0}={\tilde{b}}_{0}-{\mbox{\boldmath$\tilde{b}$}}\cdot{\mbox{\boldmath$u$}}_{+}$
(52)

such that $\Lambda_{\chi}=\mu+\chi{\tilde{b}}_{0}$. In particular, for a
type-I Weyl semimetal (i.e. with $v_{\chi}<v_{F}=1$) with the tilting
${\mbox{\boldmath$v$}}_{\chi}$ parallel to $\tilde{b}$, the semiclassical
Boltzmann approach Kubo2  leads to the same effective action (4) with the Weyl
node separation shifted by ENPREP

$\displaystyle{{\tilde{b}}}\;\to\;{\tilde{b}}\,-\sum_{\chi=\pm
1}\frac{\chi\Lambda_{\chi}}{2|{v}_{\chi}|^{2}}\,\Big{[}v_{\chi}-{\rm
arctanh}(v_{\chi})\Big{]},$ (53)

where ${\tilde{b}}=|\mbox{\boldmath$\tilde{b}$}|$ and
$\Lambda_{\chi}=\mu\,+\chi{\tilde{b}}_{0}>0$ is the chemical potential
measured from the nodal point. Clearly, the spatial components in Eq. (51)
successfully simplify to the result of Eq. (53).

##  IV Summary and conclusions

We extend the vacuum polarization method of high energy physics, used in the
calculation of the electromagnetic response of fermionic systems, to a large
class of fermionic couplings included in the Standard Model Extension (SME)
describing Lorentz symmetry violations Kostelecky0 . Emphasis is made in the
CPT odd contribution to the effective action which exhibits remnants of the
abelian chiral anomaly due to the appearance of the Pontryagin density
$\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$. Our approach does not
rest in a direct manipulation of chiral transformations which induce the
chiral anomaly via the corresponding Jacobian Fujikawa . Rather, it is based
in the standard perturbative expansion in powers of the electromagnetic
potential of the determinant resulting from the integration of the fermions in
the functional integral corresponding to the action (1) plus the specific
choices (2) . This amounts to the calculation of the vacuum polarization
tensor to second order in $A_{\mu}$ starting from the exact modified fermion
propagators obtained from the SME. A distinguishing feature of the method is
the inclusion of corrections depending upon the chemical potential $\mu$,
which enlarge the range of possible phenomenological applications. This is
possible through a systematic application of thermal field theory methods
KAPUSTA , which is exemplified here in the case of the zero temperature limit.
The fermionic systems we consider exhibit the axial anomaly and the resulting
electromagnetic actions are fully described by axion electrodynamics in the
form of Eq. (4). All corrections induced by the different parameters in the
SME enter only in the coupling $\Theta(x)=2{\cal B}_{\lambda}x^{\lambda}$ to
the Pontryagin density, as shown in our general expressions (27) and (47).
This is consistent with the property that the axial anomaly is insensitive to
the Lorentz invariance violating parameters of the SME LUAGA ; FIDEL ; SALVIO
; Scarpelli . Along the text we have insisted in the advantage of extending
these high energy physics methods to condensed matter physics. In fact we
envisage interesting applications in the realm of topological quantum matter,
where a rich phenomenology in the electromagnetic response can be explored.
Our general results can be applied to type-I Weyl semimetals (WSM) with
arbitrary tilting and anisotropies. A simpler system results from the
restriction to a WSM with arbitrary tilting but equal isotropic Fermi velocity
at each node, described by the Hamiltonian (23). In this setting, our result
(51) for the vector contribution ${\cal B}^{i}$ to the $\Theta$ coupling has
been validated by a direct calculation of the conductivity using the Boltzmann
semiclassical approach, as shown in Eq. (53). The calculation and proper
regularization of the corrections for the case
$|\mbox{\boldmath$\mathcal{V}$}|\geq 1$ (relevant for type-II WSM) is left
unsettled for future research. Pending also is the introduction of a chiral
chemical potential which is allowed by the chiral structure of the Hamiltonian
describing a WSM with two band-crossings, i.e. each node can be held at
different chemical potential. We find appealing also to pursue the comparison
between the quantum field theory approach and the semiclassical Boltzmann
formalism. In particular, it would be interesting to elucidate the role of the
Berry phase in the first strategy ENPREP . Finally, the recent calculation of
the one-loop Heisenberg-Euler effective action for zero chemical potential in
two of the most studied minimal Lorentz-violating extensions of QED PETROV ,
motivates the challenge of extending the thermal field theory approach to the
calculation of the CPT even contributions to the effective electromagnetic
action arising from the fermionic sector of the SME considered in Eq. (2).

## Acknowledgements

L.F.U. and A.M.-R. acknowledge support from the project CONACYT (México) #
CF-428214. A.M.-R. has been partially supported by DGAPA-UNAM Projects #
IA101320 and # IA102722. L.F.U. and A.G.A. were supported in part by Project
DGAPA-UNAM # 103319.

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