# Azimuthal asymmetries from $\theta$ Vacuum

Weihua Yang  Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China

###### Abstract

In this note, we calculate azimuthal asymmetries in terms of fragmentation
functions in semi-inclusive electron positron annihilation process. These
fragmentation functions are divided into two parts according to the parity
even or parity odd properties. We present eight kinds of azimuthal asymmetries
and we categorize them for four categories. First, asymmetries comes from both
$\mathcal{P}$ odd FFs and $\mathcal{P}$ even FFs which have same signs (e.g.,
$H^{\perp}_{1}\bar{H}^{\perp}_{1}+\mathcal{H}^{\perp}_{1}\bar{\mathcal{H}}^{\perp}_{1}$).
Second, asymmetries comes from two interference terms between the
$\mathcal{P}$ odd FF and $\mathcal{P}$ even FF which have opposite signs
(e.g.,
$H^{\perp}_{1}\mathcal{H}^{\perp}_{1}-\mathcal{H}^{\perp}_{1}\bar{H}^{\perp}_{1}$).
Third, asymmetries comes from both $\mathcal{P}$ odd FFs and $\mathcal{P}$
even FFs which have opposite signs (e.g.,
$\mathcal{G}^{\perp}_{1T}\bar{\mathcal{D}}_{1}-D^{\perp}_{1T}\bar{D}_{1}$).
Fourth, asymmetries comes from two interference terms between the
$\mathcal{P}$ odd FF and $\mathcal{P}$ even FF which have same signs
($H_{1T}\bar{\mathcal{H}}^{\perp}_{1}+\mathcal{H}_{1T}\bar{H}_{1}^{\perp}$ ).
The $\mathcal{P}$-odd fragmentation functions, e.g.,
$\mathcal{H}_{1}^{\perp}$, are induced by $local$ $\mathcal{P}$\- and/or
$\mathcal{CP}$-odd effects which are connected to the tunneling ($\theta$
vacuum) events. Due to the $\mathcal{P}$-odd effect, azimuthal asymmetries
induced by the effect vanish if all events are summed over but survive on the
event-by-event for the interference terms with opposite signs. However,
interference terms with same signs survive and can be used to measure/study
the $\mathcal{P}$-odd fragmentation functions. Finally, we also calculate
hadron polarizations in terms of both the $\mathcal{P}$-even and
$\mathcal{P}$-odd fragmentation functions.

##  I Introduction

Parton distribution functions (PDFs) and fragmentation functions (FFs) are two
important quantities in describing the high energy reactions. Both of them are
long distance non-perturbative quantities which cannot be calculated with
perturbative theory, they are mainly calculated by phenomenological models and
parametrized by experiment data Jakob:1997wg ; Avakian:2010br ;
Bacchetta:2001di ; Amrath:2005gv ; Metz:2002iz ; Bacchetta:2007wc ;
Davidson:2001cc ; Ito:2009zc ; Matevosyan:2010hh ; Matevosyan:2011vj ;
Matevosyan:2012ga ; Dulat:2015mca ; Albino:2008fy ; canshuhua ;
deFlorian:1997zj . When three-dimensional, i.e., the transverse momentum
dependent (TMD) PDFs and FFs are considered, the sensitive quantities studied
in experiments are often different azimuthal asymmetries. In the quantum field
theoretical formulation, PDFs and/or FFs are defined by the quark-quark
correlators (correlation functions) which are defined as $4\times 4$ matrices
in Dirac space depending on the hadron state. As a result, the correlators can
be decomposed by the Dirac matrices, i.e.,
$\Gamma=\\{I,i\gamma^{5},\gamma^{\mu},\gamma^{\mu}\gamma^{5},i\sigma^{\mu\nu}\gamma^{5}\\}$.
The corresponding coefficients can be further decomposed by the Lorentz
covariants and scalar functions and these scalar functions are called PDFs and
FFs.

Generally, the quark-quark correlator satisfies two constraints, Hermiticity
and parity conservation. Hermiticity ensures that all the scalar functions are
real while parity conservation strictly limits the numbers of scalar
functions. Even though parity violating effects are not expected in
perturbative quantum chromodynamics (QCD), the non-trivial $\theta$ vacuum may
lead to the constraint breakdown tHooft:1976rip ; tHooft:1976snw ;
Jackiw:1976pf ; Callan:1976je . The $\theta$ term enters in the lagrangian as

$\displaystyle\mathcal{L}_{\theta}=\frac{\theta}{16\pi^{2}}\mathrm{Tr}\big{[}F_{\mu\nu}\tilde{F}^{\mu\nu}\big{]}$
(1)

where $F_{\mu\nu}$ is the gluon field strength tensor and
$\tilde{F}^{\mu\nu}=\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}/2$. We see
that the $\theta$ term in lagrangian violates the parity ($\mathcal{P}$)
and/or charge-parity ($\mathcal{CP}$) symmetries. However the measurements of
electric dipole moment of neutron indicate that the parity ($\mathcal{P}$)
violation is $local$ Baker:2006ts . Even though QCD is $local$ $\mathcal{P}$
violated, Kharzeev, Pisarski and Tytgat have shown it can be directly observed
Kharzeev:1998kz . In heavy ion collisions, it also leads to the famous chiral
magnetic effect Kharzeev:2004ey ; Kharzeev:2007jp ; Fukushima:2008xe .
Furthermore, Efremov, Kharzeev and Kang also discussed the $local$
$\mathcal{P}$ odd effect in fragmentation process. In this case, the
$\mathcal{P}$ odd FFs emerge and they argued that the effect can be detected
in experiments via physical observables, e.g., handedness correlation and
azimuthal asymmetries Efremov:1995ff ; Kang:2010qx .

Electron positron annihilation process is regard as the most cleanest place to
study the FFs, because in this process there are no hadronic effects in
initial states. As we mentioned before the sensitive quantities studied in
experiments are often different azimuthal asymmetries when TMD FFs are
considered. In ref. Kang:2010qx , Kang and Kharzeev derived the most general
form of the FF for quarks fragmenting into spin-0 hadrons (pseudoscalars) at
leading twist. Without the parity constraint they found two more $\mathcal{P}$
odd FFs which are related to two azimuthal asymmetries, i.e., $\cos 2\phi$ and
$\sin 2\phi$. They presented a estimation of the magnitude of the $\sin 2\phi$
asymmetry, they found $I(\bar{\theta},z_{1},z_{2})\sim 1.5\%$. This is a
relative large asymmetry and can be detected. In this note we extend the
calculation to spin-1/2 dihadron production. We calculate the azimuthal
asymmetries for unpolarized, single-spin asymmetries and double spin
asymmetries, respectively. We also calculate the hadron polarizations.

This note is organized as follows. In Sect. II we present the kinematics of
the semi-inclusive electron positron annihilation (SIA) precess and the
conventions and notations used in this note. In Sect. III we present the
decompositions of the correlation functions as well as the complete hadronic
tensor. Our calculation results are shown in Sect. IV. And a brief summary
will be given in Sect. V.

##  II Kinematics

In this section we present the kinematics of the SIA process, see Fig. 1. Here
SIA process denotes that two hadron are produced in two back-to back-jets in
electron positron annihilation preocess, respectively. In this note we only
consider the electromagnetic interaction for distinguishing the
$\mathcal{P}$-odd effect in QCD from weak interaction theory. The differential
cross section of SIA process can be written as contraction of the leptonic
tensor and hadronic tensor, i.e.,

$\displaystyle\frac{2E_{1}E_{2}d\sigma}{d^{3}\vec{p}_{1}d^{3}\vec{p}_{2}}=\frac{N_{c}\alpha^{2}_{em}e_{q}^{2}}{sQ^{4}}L_{\mu\nu}(l_{1},l_{2})W^{\mu\nu}(p_{1},p_{2}),$
(2)

where $N_{c}$ denotes the color factor, $\alpha_{em}$ is the fine structure
constant and $e_{q}$ is the electric charge of quark $q$.
$s=Q^{2}=q^{2}=(l_{1}+l_{2})^{2}$. The leptonic tensor is given by

$\displaystyle L_{\mu\nu}(l_{1},l_{2})$
$\displaystyle=\frac{1}{4}\mathrm{Tr}\left[\not{l}_{1}\gamma_{\mu}\not{l}_{2}\gamma_{\nu}\right]$
$\displaystyle=l_{1\mu}l_{2\nu}+l_{1\nu}l_{2\mu}-g_{\mu\nu}\left(l_{1}\cdot
l_{2}\right).$ (3)

The operator definition of hadronic tensor is given by

$\displaystyle W^{\mu\nu}(p_{1},p_{2})=\Delta\sum_{X}\langle
0|J^{\mu}(0)|p_{1},p_{2};X\rangle\langle p_{1},p_{2};X|J^{\nu}(0)|0\rangle$
$\displaystyle=\int d^{2}ye^{-iqy}\sum_{X}\langle
0|J^{\mu}(0)|p_{1},p_{2};X\rangle\langle p_{1},p_{2};X|J^{\nu}(y)|0\rangle,$
(4)

where $\Delta=\delta^{4}(q-p_{1}-p_{2}-X)$. Besides the Lorentz covariance,
the hadronic tensor $W_{\mu\nu}$ satisfies the general constraints imposed by
hermiticity, current conservation, and parity conservation in the
electromagnetic process, i.e.,

$\displaystyle W^{*\mu\nu}(p_{1},S_{1};p_{2},S_{2})$
$\displaystyle=W^{\nu\mu}(p_{1},S_{1};p_{2},S_{2}),$ (5) $\displaystyle
q_{\mu}W^{\mu\nu}(p_{1},S_{1};p_{2},S_{2})$
$\displaystyle=q_{\nu}W^{\mu\nu}(p_{1},S_{1};p_{2},S_{2})=0,$ (6)
$\displaystyle W^{\mu\nu}(p_{1},S_{1};p_{2},S_{2})$
$\displaystyle=W_{\mu\nu}(p^{\mathcal{P}}_{1},S^{\mathcal{P}}_{1};p^{\mathcal{P}}_{2},S^{\mathcal{P}}_{2}),$
(7)

where a vector with the superscript $\mathcal{P}$ denotes the result after
space reflection such as $p_{\mu}^{\mathcal{P}}=p^{\mu}$. From Eq. (4) we know
that the hadronic tensor can not be calculated perturbatively because it
contains the nonperturbative hadronization process. However, to obtain the
cross section, we can decompose the hadronic tensor with the basic Lorentz
tensors as shown in ref. Pitonyak:2013dsu ; Chen:2016moq . Here we will not
repeat the decompositions.

Figure 1: Semi-inclusive electron positron annihilation process. The momenta
of particles are shown in the parenthesises in the figure.
Figure 2: Semi-inclusive electron positron annihilation process in the lepton
center of mass frame.

To calculate the SIA process, we choose the lepton center-of-mass frame where
$p_{2}$ lies in the positive $z$ direction, see Fig. 2. In this frame we use
the following conventions and notation given in ref. Boer:1997mf . We define

$\displaystyle\hat{t}^{\mu}$ $\displaystyle\equiv\frac{q^{\mu}}{Q},$ (8)
$\displaystyle\hat{z}^{\mu}$ $\displaystyle\equiv\frac{Q}{p_{2}\cdot
q}p^{\mu}_{2q}=2\frac{p_{2}^{\mu}}{z_{2}Q}-\frac{q^{\mu}}{Q},$ (9)

where we have defined $p_{q}^{\mu}=p^{\mu}-\frac{p\cdot q}{q^{2}}q^{\mu}$ and
the higher twist terms ($1/Q^{2}$) is omitted. There are two projectors which
can be defined in terms of $\hat{t}$ and $\hat{z}$, they are

$\displaystyle g_{\perp}^{\mu\nu}\equiv
g^{\mu\nu}-\hat{t}^{\mu}\hat{t}^{\nu}+\hat{z}^{\mu}\hat{z}^{\nu},$ (10)
$\displaystyle\varepsilon_{\perp}^{\mu\nu}\equiv-\epsilon^{\mu\nu\rho\sigma}\hat{t}_{\rho}\hat{z}_{\sigma}=\frac{1}{p_{2}\cdot
q}\epsilon^{\mu\nu\rho\sigma}p_{2\,\rho}q_{\sigma}.$ (11)

Then we have $p_{1\perp}^{\mu}=g_{\perp}^{\mu\nu}p_{1\nu}$. For convenience,
we also define $\hat{h}^{\mu}=p_{1\perp}^{\mu}/|p_{1\perp}|$. FFs are often
define with respect to the transverse momentum of the quark, so it would be
convenient to define lightlike directions in terms of the momenta of produced
hadrons, $p_{1},p_{2}$. Then we have

$\displaystyle
p_{1}^{\mu}\equiv\frac{z_{1}Q}{\sqrt{2}}n^{\mu}+\frac{M_{1}^{2}}{z_{1}Q\sqrt{2}}\bar{n}^{\mu},$
(12) $\displaystyle
p_{2}^{\mu}\equiv\frac{M_{2}^{2}}{z_{2}Q\sqrt{2}}n^{\mu}+\frac{z_{2}Q}{\sqrt{2}}\bar{n}^{\mu},$
(13) $\displaystyle
q^{\mu}\equiv\frac{Q}{\sqrt{2}}n^{\mu}+\frac{Q}{\sqrt{2}}\bar{n}^{\mu}+q_{T}^{\mu},$
(14)

where $q_{T}^{2}\equiv-Q_{T}^{2}$, $M_{1},M_{2}$ are masses of hadron 1 and
hadron 2, respectively. Here we have assumed that $Q_{T}^{2}\ll Q^{2}$.
$\bar{n}$ and $n$ are lightlike unit vectors which satisfy $\bar{n}\cdot n=1$,
$\bar{n}^{2}=n^{2}=0$. In Eq. (9), $p_{2}$ is used to define $\hat{z}$, then
we have $p_{1\perp}^{\mu}=-z_{1}q_{T}^{\mu}$. We may also define the
projectors in terms of $\bar{n},n$,

$\displaystyle g^{\mu\nu}_{T}\equiv
g^{\mu\nu}-\bar{n}^{\mu}n^{\nu}-\bar{n}^{\nu}n^{\mu},$ (15)
$\displaystyle\varepsilon^{\mu\nu}_{T}\equiv\varepsilon^{\mu\nu\rho\sigma}\bar{n}_{\rho}n_{\sigma}.$
(16)

We see that the $perpendicular$ vectors and tensors (labeled by $\perp$) are
different from the $transverse$ vectors and tensors (labeled by $T$). They can
be connected by the following equations, e.g.,

$\displaystyle
g_{T}^{\mu\nu}=g_{\perp}^{\mu\rho}g^{\nu}_{T\rho}-\frac{Q_{T}}{Q}(\hat{z}+\hat{t})\hat{h}^{\mu},$
(17) $\displaystyle k_{\perp}^{\mu}=k_{T}^{\mu}-\frac{q_{T}\cdot
k_{T}}{Q}(\hat{z}+\hat{t}),$ (18)

where $q_{T}\cdot k_{T}=-\mathbf{q}_{T}\cdot\mathbf{k}_{T}$. Using Eq. (18),
we can transform the vectors in transverse basis into perpendicular basis to
calculate the hadronic tensor and/or cross section. Calculations will be given
in the followings.

On the perpendicular basis the lepton tensor can be parameterized as

$\displaystyle L_{\mu\nu}=\frac{Q^{2}}{2}$
$\displaystyle\Big{[}-g_{\perp}^{\mu\nu}+4y(1-y)\hat{z}^{\mu}\hat{z}^{\nu}-4y(1-y)\hat{x}^{\mu}\hat{x}^{\nu}$
$\displaystyle-2(1-2y)\sqrt{y(1-y)}\hat{z}^{\\{\mu}\hat{x}^{\nu\\}}\Big{]}$
$\displaystyle=Q^{2}$
$\displaystyle\Big{[}-A(y)g_{\perp}^{\mu\nu}+2B(y)\hat{z}^{\mu}\hat{z}^{\nu}-2B(y)\big{(}\hat{x}^{\mu}\hat{x}^{\nu}+\frac{1}{2}g_{\perp}^{\mu\nu}\big{)}$
$\displaystyle-C(y)D(y)\hat{z}^{\\{\mu}\hat{x}^{\nu\\}}\Big{]},$ (19)

where $\hat{x}^{\mu}=l_{\perp}^{\mu}/D(y)Q$, and

$\displaystyle A(y)$ $\displaystyle=\frac{1}{2}-y+y^{2},$ (20) $\displaystyle
B(y)$ $\displaystyle=y\,(1-y),$ (21) $\displaystyle C(y)$
$\displaystyle=1-2y,$ (22) $\displaystyle D(y)$
$\displaystyle=\sqrt{y\,(1-y)}.$ (23)

Eq. (19) can be used to calculated the cross section by contracting with the
hadronic tensor obtained in the following.

##  III Correlation functions and hadronic tensor

Figure 3: The hadronic tensor in parton model.

Because the hadronic tensor contains nonperturbative process, it can not be
calculated by perturbative theory. In this case we should resort to the parton
model Feynman:1969ej ; Feynman:1973xc ; Bjorken:1969ja . In QCD parton model,
see Fig. 3, the hadronic tensor can be rewritten as

$\displaystyle W^{\mu\nu}(p_{1},p_{2})=$
$\displaystyle\int\frac{d^{2}k_{1T}}{(2\pi)^{2}}\frac{d^{2}k_{2T}}{(2\pi)^{2}}\delta^{2}(\vec{q}_{T}-\vec{k}_{1T}-\vec{k}_{2T})$
$\displaystyle\mathrm{Tr}\left[\bar{\Delta}(p_{2},k_{2T})\gamma^{\mu}\Delta(p_{1},k_{1T})\gamma^{\mu}\right],$
(24)

where $\Delta(z_{1},k_{1T})$ and $\bar{\Delta}(z_{2},k_{2T})$ are correlation
functions and the operator definitions are given by

$\displaystyle\Delta(p_{1},k_{1T})$ $\displaystyle=\frac{1}{2\pi}\int
d\xi_{1}^{-}d^{2}\xi_{1T}e^{-ik_{1}\xi_{1}}$ $\displaystyle\sum_{X}\langle
0|\psi(0)|p_{1},X\rangle\langle
p_{1},X|\bar{\psi}(\xi_{1})|0\rangle|_{\xi_{1}^{+}=0},$ (25)
$\displaystyle\bar{\Delta}(p_{2},k_{2T})$ $\displaystyle=\frac{1}{2\pi}\int
d\xi_{2}^{-}d^{2}\xi_{2T}e^{-ik_{2}\xi_{2}}$ $\displaystyle\sum_{X}\langle
0|\bar{\psi}(0)|p_{2},X\rangle\langle
p_{2},X|\psi(\xi_{2})|0\rangle|_{\xi_{2}^{-}=0}.$ (26)

Equations (25)-(26) are not physical correlation functions, because they are
not gauge invariant since the spinor field are not located at the same point.
To obtain the gauge invariant correlation functions, gauge links should be
taken into consideration. Gauge link (also called Wilson line) is defined as

$\displaystyle\mathcal{L}(0,\infty)=\mathcal{P}exp\big{[}ig\int^{\infty}_{0}dznA(y+zn)\big{]}.$
(27)

Armed with the gauge links, the gauge invariant correlation functions are
given by,

$\displaystyle\hat{\Xi}(p_{1},k_{1T})$ $\displaystyle=\frac{1}{2\pi}\int
d\xi_{1}^{-}d^{2}\xi_{1T}e^{-ik_{1}\xi_{1}}\sum_{X}\langle
0|\psi(0)\mathcal{L}(0,\infty)$ $\displaystyle\times|p_{1},X\rangle\langle
p_{1},X|\bar{\psi}(\xi_{1})\mathcal{L}^{\dagger}(\xi,\infty)|0\rangle|_{\xi_{1}^{+}=0},$
(28) $\displaystyle\bar{\hat{\Xi}}(p_{2},k_{2T})$
$\displaystyle=\frac{1}{2\pi}\int
d\xi_{2}^{-}d^{2}\xi_{2T}e^{-ik_{2}\xi_{2}}\sum_{X}\langle
0|\bar{\psi}(0)\mathcal{L}^{\dagger}(0,\infty)$
$\displaystyle\times|p_{2},X\rangle\langle
p_{2},X|\psi(\xi_{2})\mathcal{L}(\xi,\infty)|0\rangle|_{\xi_{2}^{-}=0}.$ (29)

As we have mentioned in Sect. I, these correlation functions given by Eqs.
(28)-(29) satisfy the following constraints imposed by Hermiticity and parity
conservation, i.e.,

$\displaystyle\hat{\Xi}^{\dagger}(p,k_{T})$
$\displaystyle=\gamma^{0}\hat{\Xi}^{\dagger}(p,k_{T})\gamma^{0},$ (30)
$\displaystyle\hat{\Xi}(p,k_{T})$
$\displaystyle=\gamma^{0}\hat{\Xi}^{\dagger}(p^{\mathcal{P}},k^{\mathcal{P}}_{T})\gamma^{0}.$
(31)

Correlation functions can not be calculated with perturbative theory because
they contain the hadronization information. However, they are $4\times 4$
matrices in Dirac space and can be decomposed in terms of the $\Gamma$
matrices, i.e.,
$\Gamma=\\{I,i\gamma^{5},\gamma^{\rho},\gamma^{\rho}\gamma^{5},i\sigma^{\rho\sigma}\gamma^{5}$}.
The decomposition can be written explicitly as

$\displaystyle\hat{\Xi}=I\Xi+i\gamma^{5}\tilde{\Xi}+\gamma^{\alpha}\Xi_{\alpha}+\gamma^{\alpha}\gamma^{5}\tilde{\Xi}_{\alpha}+i\sigma^{\alpha\beta}\gamma^{5}\Xi_{\alpha\beta}.$
(32)

Furthermore, with the parity constraint the coefficients in Eq. (32) can be
rewritten as

$\displaystyle
z\Xi_{\alpha}=\bar{n}_{\alpha}\bigg{[}D_{1}+\frac{\varepsilon_{TkS}}{M}D^{\perp}_{1T}\bigg{]},$
(33) $\displaystyle z\tilde{\Xi}_{\alpha}=\bar{n}_{\alpha}\bigg{[}\lambda
G_{1L}+\frac{k_{T}\cdot S_{T}}{M}G^{\perp}_{1T}\bigg{]},$ (34) $\displaystyle
z\Xi_{\rho\alpha}=\bar{n}_{\alpha}\bigg{[}\frac{\varepsilon_{Tk\alpha}}{M}H^{\perp}_{1}+S_{T\alpha}H_{1T}+\frac{k_{T\alpha}}{M}H^{\perp}_{1S}\bigg{]},$
(35)

at leading twist. We see that $\Xi$ and $\tilde{\Xi}$ do not have leading
twist contributions. Here we have defined the shorthanded notation
$H^{\perp}_{1S}=\lambda H^{\perp}_{1L}+\frac{k_{T}\cdot
S_{T}}{M}H^{\perp}_{1T}$. If the parity constraint is released, in other words
we consider the $\mathcal{P}$ odd effects caused by the non-trivial structure
QCD $\theta$ vacuum, counterparts of the coefficients in the decompositions
can be obtained, they are

$\displaystyle
z\Xi^{P}_{\alpha}=\bar{n}_{\alpha}\bigg{[}i\gamma^{5}\mathcal{D}_{1}+\frac{k_{T}\cdot
S_{T}}{M}\mathcal{D}^{\perp}_{1T}\bigg{]},$ (36) $\displaystyle
z\tilde{\Xi}^{P}_{\alpha}=\bar{n}_{\alpha}\bigg{[}-i\gamma^{5}\lambda\mathcal{G}_{1L}+\frac{\varepsilon_{TkS}}{M}\mathcal{G}^{\perp}_{1T}\bigg{]},$
(37) $\displaystyle
z\Xi^{P}_{\rho\alpha}=\bar{n}_{\alpha}\bigg{[}\frac{k_{T\alpha}}{M}\mathcal{H}^{\perp}_{1}+\varepsilon_{TS\alpha}\mathcal{H}_{1T}+\frac{\varepsilon_{Tk\alpha}}{M}\mathcal{H}^{\perp}_{1S}\bigg{]},$
(38)

where the superscript $P$ denote the parity violating and
$\mathcal{H}^{\perp}_{1S}=\lambda\mathcal{H}^{\perp}_{1L}+\frac{k_{T}\cdot
S_{T}}{M}\mathcal{H}^{\perp}_{1T}$. Here we call
$D,G,H,\mathcal{D},\mathcal{G}$ and $\mathcal{H}$ fragmentation functions in
Eqs. (33)-(38). We also use $D,G,H$ to denote the parity conserved FFs and use
$\mathcal{D},\mathcal{G},\mathcal{H}$ to denote the parity violated FFs. The
$\mathcal{P}$-odd and even FFs can be written in a unified form, in this case
the correlation function can be written as

$\displaystyle z\Xi$
$\displaystyle=\not{\bar{n}}\bigg{[}D_{1}+\lambda\mathcal{G}_{1L}+\frac{\varepsilon_{TkS}}{M}D^{\perp}_{1T}+\frac{k_{T}\cdot
S_{T}}{M}\mathcal{D}^{\perp}_{1T}\bigg{]}$
$\displaystyle+\not{\bar{n}}\gamma^{5}\bigg{[}\mathcal{D}_{1}+\lambda
G_{1L}+\frac{\varepsilon_{TkS}}{M}\mathcal{G}^{\perp}_{1T}+\frac{k_{T}\cdot
S_{T}}{M}G^{\perp}_{1T}\bigg{]}$
$\displaystyle+i\sigma^{\rho\alpha}\gamma^{5}\bar{n}_{\rho}\bigg{[}\frac{\varepsilon_{Tk\alpha}}{M}\big{(}H^{\perp}_{1}+\mathcal{H}^{\perp}_{1S}\big{)}+S_{T\alpha}H_{1T}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{k_{T\alpha}}{M}\big{(}\mathcal{H}^{\perp}_{1}+H^{\perp}_{1S}\big{)}+\varepsilon_{TS\alpha}\mathcal{H}_{1T}\bigg{]}.$
(39)

We have 16 fragmentation functions, 8 parity conserved and 8 parity violated.
For every parity conserved one there is a parity violated one corresponding to
it, see Table. 1. Here we should know that $D_{1T}^{\perp}$ and
$\mathcal{G}_{1T}^{\perp}$ have the same forms while $G_{1T}^{\perp}$ and
$\mathcal{D}_{1T}^{\perp}$ have the same forms.

$\mathcal{P}$-even | $D_{1}$ | $D_{1T}^{\perp}$ | $G_{1L}$ | $G_{1T}^{\perp}$ | $H_{1}^{\perp}$ | $H_{1T}$ | $H_{1L}^{\perp}$ | $H_{1T}^{\perp}$
---|---|---|---|---|---|---|---|---
$\mathcal{P}$-odd | $\mathcal{D}_{1}$ | $\mathcal{D}_{1T}^{\perp}$ | $\mathcal{G}_{1L}$ |  $\mathcal{G}_{1T}^{\perp}$ |  $\mathcal{H}_{1}^{\perp}$ |  $\mathcal{H}_{1T}$ |  $\mathcal{H}_{1L}^{\perp}$ |  $\mathcal{H}_{1T}^{\perp}$
Table 1: $\mathcal{P}$-even and $\mathcal{P}$-odd FFs.

Since we have all the decompositions of the correlation functions, we can
calculate the hadronic tensor. To obtain the complete hadronic tensor, we
substitute FFs decomposed in Eq. (39) into Eq. (24) (where correlation
functions should be changed to the gauge invariant ones) and carry out the
calculations. Thus the complete hadronic tensor in SIA process at leading
twist is given by

$\displaystyle
W^{\mu\nu}=\frac{4}{z_{1}z_{2}}\int\frac{d^{2}k_{1T}}{(2\pi)^{2}}\frac{d^{2}k_{2T}}{(2\pi)^{2}}\delta^{2}(\vec{q}_{T}-\vec{k}_{1T}-\vec{k}_{2T})$
$\displaystyle\times\Bigg{\\{}d_{T}^{\mu\nu}\bigg{\\{}\Big{[}D_{1}+\lambda_{1}\mathcal{G}_{1L}+\frac{\varepsilon_{Tk_{1}S_{1}}}{M}D^{\perp}_{1T}+\frac{k_{1T}\cdot
S_{1T}}{M}\mathcal{D}^{\perp}_{1T}\Big{]}\Big{[}\bar{D}_{1}+\lambda_{2}\bar{\mathcal{G}}_{1L}+\frac{\varepsilon_{Tk_{2}S_{2}}}{M}\bar{D}^{\perp}_{1T}+\frac{k_{2T}\cdot
S_{2T}}{M}\bar{\mathcal{D}}^{\perp}_{1T}\Big{]}$ $\displaystyle\hskip
22.76219pt+\Big{[}\mathcal{D}_{1}+\lambda_{1}G_{1L}+\frac{\varepsilon_{Tk_{1}S_{1}}}{M}\mathcal{G}^{\perp}_{1T}+\frac{k_{1T}\cdot
S_{1T}}{M}G^{\perp}_{1T}\Big{]}\Big{[}\bar{\mathcal{D}}_{1}+\lambda_{2}\bar{G}_{1L}+\frac{\varepsilon_{Tk_{2}S_{2}}}{M}\bar{\mathcal{G}}^{\perp}_{1T}+\frac{k_{2T}\cdot
S_{2T}}{M}\bar{G}^{\perp}_{1T}\Big{]}\bigg{\\}}$
$\displaystyle-\frac{d_{T}^{\mu\nu}k_{1T}\cdot
k_{2T}+k_{1T}^{\\{\mu}k_{2T}^{\nu\\}}}{M_{1}M_{2}}\big{(}H^{\perp}_{1}+\mathcal{H}^{\perp}_{1S}\big{)}\big{(}\bar{H}^{\perp}_{1}+\bar{\mathcal{H}}^{\perp}_{1S}\big{)}-\frac{d_{T}^{\mu\nu}\varepsilon_{Tk_{1}k_{2}}+\varepsilon_{Tk_{1}}^{~{}~{}~{}\\{\mu}k_{2T}^{\nu\\}}}{M_{1}M_{2}}\big{(}H^{\perp}_{1}+\mathcal{H}^{\perp}_{1S}\big{)}\big{(}\bar{\mathcal{H}}^{\perp}_{1}+\bar{H}^{\perp}_{1S}\big{)}$
$\displaystyle-\frac{d_{T}^{\mu\nu}k_{1T}\cdot
k_{2T}+k_{1T}^{\\{\mu}k_{2T}^{\nu\\}}}{M_{1}M_{2}}\big{(}\mathcal{H}^{\perp}_{1}+H^{\perp}_{1S}\big{)}\big{(}\bar{\mathcal{H}}^{\perp}_{1}+\bar{H}^{\perp}_{1S}\big{)}-\frac{d_{T}^{\mu\nu}\varepsilon_{Tk_{2}k_{1}}+\varepsilon_{Tk_{2}}^{~{}~{}~{}\\{\mu}k_{1T}^{\nu\\}}}{M_{1}M_{2}}\big{(}\mathcal{H}^{\perp}_{1}+H^{\perp}_{1S}\big{)}\big{(}\bar{H}^{\perp}_{1}+\bar{\mathcal{H}}^{\perp}_{1S}\big{)}$
$\displaystyle-\frac{d_{T}^{\mu\nu}\varepsilon_{Tk_{1}S_{2}}+\varepsilon_{Tk_{1}}^{~{}~{}~{}\\{\mu}S_{2T}^{\nu\\}}}{M_{1}}\big{(}H^{\perp}_{1}+\mathcal{H}^{\perp}_{1S}\big{)}\bar{H}_{1T}-\frac{d_{T}^{\mu\nu}k_{1T}\cdot
S_{2T}+k_{1T}^{\\{\mu}S_{2T}^{\nu\\}}}{M_{1}}\big{(}H^{\perp}_{1}+\mathcal{H}^{\perp}_{1S}\big{)}\bar{\mathcal{H}}_{1T}$
$\displaystyle-\frac{d_{T}^{\mu\nu}\varepsilon_{TS_{2}k_{1}}+\varepsilon_{TS_{2}}^{~{}~{}~{}\\{\mu}k_{1T}^{\nu\\}}}{M_{1}}\big{(}\mathcal{H}^{\perp}_{1}+H^{\perp}_{1S}\big{)}\bar{\mathcal{H}}_{1T}-\frac{d_{T}^{\mu\nu}k_{1T}\cdot
S_{2T}+k_{1T}^{\\{\mu}S_{2T}^{\nu\\}}}{M_{1}}\big{(}\mathcal{H}^{\perp}_{1}+{H}^{\perp}_{1S}\big{)}\bar{H}_{1T}$
$\displaystyle-\frac{d_{T}^{\mu\nu}\varepsilon_{Tk_{2}S_{1}}+\varepsilon_{Tk_{2}}^{~{}~{}~{}\\{\mu}S_{1T}^{\nu\\}}}{M_{2}}H_{1T}\big{(}\bar{H}^{\perp}_{1}+\bar{\mathcal{H}}^{\perp}_{1S}\big{)}-\frac{d_{T}^{\mu\nu}k_{2T}\cdot
S_{1T}+k_{2T}^{\\{\mu}S_{1T}^{\nu\\}}}{M_{2}}H_{1T}\big{(}\bar{\mathcal{H}}^{\perp}_{1}+\bar{H}^{\perp}_{1S}\big{)}$
$\displaystyle-\frac{d_{T}^{\mu\nu}\varepsilon_{TS_{1}k_{2}}+\varepsilon_{TS_{1}}^{~{}~{}~{}\\{\mu}k_{2T}^{\nu\\}}}{M_{2}}\mathcal{H}_{1T}\big{(}\bar{\mathcal{H}}^{\perp}_{1}+\bar{H}^{\perp}_{1S}\big{)}-\frac{d_{T}^{\mu\nu}k_{2T}\cdot
S_{1T}+k_{2T}^{\\{\mu}S_{1T}^{\nu\\}}}{M_{2}}\mathcal{H}_{1T}\big{(}\bar{H}^{\perp}_{1}+\bar{\mathcal{H}}^{\perp}_{1S}\big{)}$
$\displaystyle-\big{(}d_{T}^{\mu\nu}S_{1T}\cdot
S_{2T}+S_{1T}^{\\{\mu}S_{2T}^{\nu\\}}\big{)}H_{1T}\bar{H}_{1T}-\big{(}d_{T}^{\mu\nu}\varepsilon_{TS_{2}S_{1}}-\varepsilon_{TS_{2}}^{~{}~{}~{}\\{\mu}S_{1T}^{\nu\\}}\big{)}H_{1T}\bar{\mathcal{H}}_{1T}$
$\displaystyle-\big{(}d_{T}^{\mu\nu}S_{1T}\cdot
S_{2T}+S_{1T}^{\\{\mu}S_{2T}^{\nu\\}}\big{)}\mathcal{H}_{1T}\bar{\mathcal{H}}_{1T}-\big{(}d_{T}^{\mu\nu}\varepsilon_{TS_{2}S_{1}}-\varepsilon_{TS_{1}}^{~{}~{}~{}\\{\mu}S_{2T}^{\nu\\}}\big{)}\mathcal{H}_{1T}\bar{H}_{1T}\bigg{\\}}\Bigg{\\}},$
(40)

where $d_{T}^{\mu\nu}=-g_{T}^{\mu\nu}$. Equation (40) is the complete hadronic
tensor which is given in terms of both the $\mathcal{P}$ even and
$\mathcal{P}$ odd fragmentation functions in SIA process. It can be easily
checked that the hadronic tensor given in Eq. (40) satisfies the current
coservation constraint, i.e., $q_{\mu}W^{\mu\nu}=q_{\nu}W^{\mu\nu}=0$.
Substituting Eq. (40) and Eq. (19) into Eq. (2), we can obtain the complete
cross section in SIA process at leading twist.

##  IV Cross sections and azimuthal asymmetries

In the previous section, we show the complete hadronic tensor at leading twist
in SIA process with both parity conserved FFs and parity violated FFs’s
contributions. In this section, we present the results by dividing the cross
section into three parts according to polarizations of the produced dihadron,
i.e., unpolarized part, single hadron polarized part and double hadron
polarized part, respectively. The hadronic tensor shown in Eq. (40) is given
in the transverse basis, to make Lorentz contraction with the leptonic tensor
shown in Eq. (19), we should first transform it into the perpendicular bases.
Here we will not present the calculation procedure and only show the final
results.

###  IV.1 Unpolarized cross section and azimuthal asymmetries

First of all we consider the unpolarized case where the hadron produced in
final state are pseudoscalar particles or spin-0 particles. The differential
cross section of it is given by

$\displaystyle\frac{2E_{1}E_{2}d\sigma}{d^{3}p_{1}d^{3}p_{2}}$
$\displaystyle=\frac{8N_{c}\alpha^{2}_{em}e_{q}^{2}}{Q^{4}z_{1}z_{2}}\bigg{[}A(y)\Big{(}\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]\Big{)}$
$\displaystyle-B(y)\cos
2\phi~{}\mathcal{C}\Big{[}w_{12}\frac{H^{\perp}_{1}\bar{H}^{\perp}_{1}+\mathcal{H}^{\perp}_{1}\bar{\mathcal{H}}^{\perp}_{1}}{M_{1}M_{2}}\Big{]}$
$\displaystyle-B(y)\sin
2\phi~{}\mathcal{C}\Big{[}w_{12}\frac{H^{\perp}_{1}\bar{\mathcal{H}}^{\perp}_{1}-\mathcal{H}^{\perp}_{1}\bar{H}^{\perp}_{1}}{M_{1}M_{2}}\Big{]}\bigg{]},$
(41)

where the abbreviation $\mathcal{C}[D_{1}\bar{D}_{1}]$ denotes a convolution,

$\displaystyle\mathcal{C}[D_{1}\bar{D}_{1}]=\int\frac{d^{2}k_{1T}}{(2\pi)^{2}}\frac{d^{2}k_{2T}}{(2\pi)^{2}}\delta^{2}(\vec{q}_{T}-\vec{k}_{1T}-\vec{k}_{2T})D_{1}\bar{D}_{1}.$
(42)

The wight function
$w_{12}=2\hat{\mathbf{h}}\cdot\mathbf{k}_{1T}\hat{\mathbf{h}}\cdot\mathbf{k}_{2T}-\mathbf{k}_{1T}\cdot\mathbf{k}_{2T}/M_{1}M_{2}$.
From Eq. (41) we can see that the $\cos 2\phi$ terms corresponds to the
Collins effect which is the reflection of the Collins function $H^{\perp}_{1}$
($\bar{H}^{\perp}_{1}$). Apart from the contribution of the Collins function,
there is also contribution of $\mathcal{P}$ odd Collins-type function
$\mathcal{H}_{1}^{\perp}$ ($\bar{\mathcal{H}}_{1}^{\perp}$). This azimuthal
asymmetry can be written as

$\displaystyle A_{UU}^{\cos
2\phi}=-\frac{B(y)}{A(y)}\frac{\mathcal{C}\big{[}w_{12}\big{(}H^{\perp}_{1}\bar{H}^{\perp}_{1}+\mathcal{H}^{\perp}_{1}\bar{\mathcal{H}}^{\perp}_{1}\big{)}\big{]}}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]},$
(43)

where the subscript $UU$ denotes unpolarized cases. The first $U$ denotes
hadron 1 is unpolarized, the same to the second hadron. Often we define the
weighted functions, i.e.,

$\displaystyle D_{1}(z)=\int\frac{d^{2}k_{T}}{(2\pi)^{2}}D_{1}(z,k_{T}),$ (44)
$\displaystyle
H_{1}^{\perp}(z)=\int\frac{d^{2}k_{T}}{(2\pi)^{2}}\frac{|k_{T}|}{M}H_{1}^{\perp}(z,k_{T}).$
(45)

Then we have

$\displaystyle\tilde{A}_{UU}^{\cos
2\phi}=-\frac{B(y)}{A(y)}\frac{H^{\perp}_{1}\bar{H}^{\perp}_{1}+\mathcal{H}^{\perp}_{1}\bar{\mathcal{H}}^{\perp}_{1}}{D_{1}\bar{D}_{1}-\mathcal{D}_{1}\bar{\mathcal{D}}_{1}}.$
(46)

We notice that the sign before two term in the numerator are the same, which
means both the Collins function and Collins-type function have positive
contributions to the $\cos 2\phi$ azimuthal asymmetry. In other words Collins
effect ($\cos 2\phi$) may have alternative origins except for the Collins
function as indicated in ref. Kang:2010qx . The $\mathcal{P}$ odd FFs will
complete the extraction of Collins function, in other words the Collins
function extracted from experiments data contains $\mathcal{H}^{\perp}_{1}$
which is not easily to excluded.

Apart from the $\cos 2\phi$ azimuthal asymmetry, we also obtain one special
azimuthal asymmetry at leading twist, i.e., $\sin 2\phi$ asymmetry. $\sin
2\phi$ azimuthal asymmetry arise from the interference of $H^{\perp}_{1}$
($\bar{H}^{\perp}_{1}$) and $\bar{\mathcal{H}}^{\perp}_{1}$
($\mathcal{H}^{\perp}_{1}$), i.e.,

$\displaystyle\tilde{A}_{UU}^{\sin
2\phi}=-\frac{B(y)}{A(y)}\frac{H^{\perp}_{1}\bar{\mathcal{H}}^{\perp}_{1}-\mathcal{H}^{\perp}_{1}\bar{H}^{\perp}_{1}}{D_{1}\bar{D}_{1}-\mathcal{D}_{1}\bar{\mathcal{D}}_{1}}.$
(47)

$\tilde{A}_{UU}^{\sin 2\phi}$ vanishes if only $\mathcal{P}$-even FFs are
taken into consideration as discussed in ref. Chen:2016moq . The first term in
the numerator corresponds to the interference between Collins function
$H_{1}^{\perp}$ and $\mathcal{P}$-odd (anti) Collins-type FF
$\bar{\mathcal{H}}^{\perp}_{1}$ fragmented by antiquark while the second terms
corresponds to the interference between (anti) Collins function
$\bar{H}_{1}^{\perp}$ and $\mathcal{P}$ odd Collins-type FF
$\mathcal{H}^{\perp}_{1}$ fragmented by quark. We also notice that the signs
before these two terms in the numerator are opposite, which means that the
effect will vanish when sum over many events. In other words we conclude that
the $\mathcal{P}$ odd effect only survive on the event-by-event basis.

###  IV.2 Single hadron polarized cross section and asymmetries

For the single hadron polarized case, we assume that hadron 1 is polarized
while hadron 2 is unpolarized, or hadron 1 is spin-1/2 particle while hadron 2
is spin-0 particle. For simplicity, we omit those term which have been shown
in Eq. (41). Thus the cross section is given by

$\displaystyle\frac{2E_{1}E_{2}d\sigma}{d^{3}p_{1}d^{3}p_{2}}$
$\displaystyle=\frac{8N_{c}\alpha^{2}_{em}e_{q}^{2}}{Q^{4}z_{1}z_{2}}\Bigg{\\{}A(y)\Big{(}\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]\Big{)}$
$\displaystyle+A(y)$
$\displaystyle\bigg{\\{}\lambda_{1}\Big{[}\mathcal{C}[\mathcal{G}_{1L}\bar{D}_{1}]-\mathcal{C}[{G}_{1L}\bar{\mathcal{D}}_{1}]\Big{]}$
$\displaystyle+$
$\displaystyle|S_{1T}|\Big{\\{}\cos(\phi-\phi_{S_{1}})\mathcal{C}\big{[}w_{1}\big{(}\mathcal{D}^{\perp}_{1T}\bar{D}_{1}-G^{\perp}_{1T}\bar{\mathcal{D}}_{1}\big{)}\big{]}$
$\displaystyle\hskip
14.22636pt+\sin(\phi-\phi_{S_{1}})\mathcal{C}\big{[}w_{1}\big{(}\mathcal{G}^{\perp}_{1T}\bar{\mathcal{D}}_{1}-D^{\perp}_{1T}\bar{D}_{1}\big{)}\big{]}\Big{\\}}\bigg{\\}}$
$\displaystyle-B(y)$ $\displaystyle\bigg{\\{}\lambda_{1}\sin
2\phi\big{[}w_{12}\big{(}H^{\perp}_{1L}\bar{H}_{1}^{\perp}-\mathcal{H}^{\perp}_{1L}\mathcal{H}^{\perp}_{1}\big{)}\big{]}$
$\displaystyle+\lambda_{1}\cos
2\phi\big{[}w_{12}\big{(}\mathcal{H}^{\perp}_{1L}\bar{H}_{1}^{\perp}+H^{\perp}_{1L}\mathcal{H}^{\perp}_{1}\big{)}\big{]}$
$\displaystyle+$
$\displaystyle|S_{1T}|\Big{\\{}\cos(\phi+\phi_{S_{1}})\mathcal{C}\big{[}w_{2}\big{(}H_{1T}\bar{\mathcal{H}}^{\perp}_{1}+\mathcal{H}_{1T}\bar{H}_{1}^{\perp}\big{)}\big{]}$
$\displaystyle\hskip
14.22636pt+\sin(\phi+\phi_{S_{1}})\mathcal{C}\big{[}w_{2}\big{(}H_{1T}\bar{H}_{1}^{\perp}-\mathcal{H}_{1T}\bar{\mathcal{H}}^{\perp}_{1}\big{)}\big{]}$
$\displaystyle+\cos 2\phi$
$\displaystyle\cos(\phi-\phi_{S_{1}})\mathcal{C}\big{[}w_{12}w_{1}\big{(}\mathcal{H}^{\perp}_{1T}\bar{H}_{1}^{\perp}+H^{\perp}_{1T}\bar{\mathcal{H}}^{\perp}_{1}\big{)}\big{]}$
$\displaystyle+\sin 2\phi$
$\displaystyle\cos(\phi-\phi_{S_{1}})\mathcal{C}\big{[}w_{12}w_{1}\big{(}\mathcal{H}^{\perp}_{1T}\bar{\mathcal{H}}^{\perp}_{1}-H^{\perp}_{1T}\bar{H}_{1}^{\perp}\big{)}\big{]}\Big{\\}}\bigg{\\}}\Bigg{\\}},$
(48)

where the weighted functions
$w_{1}=\hat{\mathbf{h}}\cdot\mathbf{k}_{1T}/M_{1}$,
$w_{2}=\hat{\mathbf{h}}\cdot\mathbf{k}_{2T}/M_{2}$. From Eq. (48) we can see
that the transverse polarization part vanish when integrate over the
transverse momenta. However, we here need to the consider azimuthal
asymmetries and the transverse momenta should not be integrated over except
for weighted integrals. Here we have,

$\displaystyle
A_{TU}^{\sin(\phi-\phi_{S_{1}})}=\frac{\mathcal{C}\big{[}w_{1}\big{(}\mathcal{G}^{\perp}_{1T}\bar{\mathcal{D}}_{1}-D^{\perp}_{1T}\bar{D}_{1}\big{)}\big{]}}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]},$
(49) $\displaystyle
A_{TU}^{\cos(\phi-\phi_{S_{1}})}=\frac{\mathcal{C}\big{[}w_{1}\big{(}\mathcal{D}^{\perp}_{1T}\bar{D}_{1}-G^{\perp}_{1T}\bar{\mathcal{D}}_{1}\big{)}\big{]}}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]}.$
(50)

From Eq. (49) we can see that $A_{UL}^{\sin(\phi-\phi_{S_{1}})}$ corresponds
to the Sivers effect in fragmentation process since this azimuthal asymmetry
comes from the FFs $D^{\perp}_{1T}\bar{D}_{1}$. But we see the two terms in
the numerator have opposite signs which means that the $\mathcal{P}$ odd FFs
$\mathcal{G}^{\perp}_{1T}\bar{\mathcal{D}}_{1}$ have negative contribution to
this asymmetry relative to $D^{\perp}_{1T}\bar{D}_{1}$. Thus the $\mathcal{P}$
odd FFs can reduce the magnitude of the asymmetry. However, azimuthal
asymmetry $A_{UL}^{\cos(\phi-\phi_{S_{1}})}$ comes from the interference of
the $\mathcal{P}$-odd FF(s) $\mathcal{D}^{\perp}_{1T}(\bar{\mathcal{D}}_{1})$
and $\mathcal{P}$-even FF(s) $\bar{D}_{1}(G^{\perp}_{1T})$ and it only survive
event-by-event basis as we have discussed for Eq. (46).

Besides, we can also calculate the other two azimuthal asymmetries which are
generated from the chiral-odd FFs. (The following two azimuthal asymmetries
are not the right simplified ones, we show them here for direct impression.)
They are given by,

$\displaystyle
A_{TU}^{\sin(\phi+\phi_{S_{1}})}=-\frac{\mathcal{C}\big{[}w_{2}\big{(}H_{1T}\bar{H}_{1}^{\perp}-\mathcal{H}_{1T}\bar{\mathcal{H}}^{\perp}_{1}\big{)}\big{]}}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]},$
(51) $\displaystyle
A_{TU}^{\cos(\phi+\phi_{S_{1}})}=-\frac{\mathcal{C}\big{[}w_{2}\big{(}H_{1T}\bar{\mathcal{H}}^{\perp}_{1}+\mathcal{H}_{1T}\bar{H}_{1}^{\perp}\big{)}\big{]}}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]}.$
(52)

We can see these two asymmetries are related to the transverse polarization
transformation function $H_{1T}$ and Collins function $H_{1}^{\perp}$. Similar
to $A_{LU}^{\sin(\phi-\phi_{S_{1}})}$, the contribution of $\mathcal{P}$ odd
FFs $\mathcal{H}_{1T}\bar{\mathcal{H}}^{\perp}_{1}$ can reduce the
contribution from $\mathcal{P}$ even FFs $H_{1T}\bar{H}_{1}^{\perp}$, since
they have different signs. It is interesting to see that the interference two
terms contributing to azimuthal asymmetry $A_{LU}^{\cos(\phi+\phi_{S_{1}})}$
have both positive contributions and they can survive even summing over many
events. At leading twist, $A_{TU}^{\cos(\phi+\phi_{S_{1}})}$ vanishes for
parity violated reason, see ref Chen:2016moq . Thus
$A_{TU}^{\cos(\phi+\phi_{S_{1}})}$ is important for determining the
$\mathcal{P}$-odd FFs by measuring the azimuthal asymmetry at leading twist.

###  IV.3 Double hadron polarized cross section and asymmetries

For the double hadron polarized case, we consider two spin-1/2 hadrons in the
final states, there are 54 terms. For simplicity, we only consider the
integrated cross section and only a few of them are left which are given by

$\displaystyle\frac{d\sigma}{dz_{1}dz_{2}dy}=\frac{4\pi
N_{c}e_{q}^{2}\alpha_{em}^{2}}{Q^{2}}\Big{[}A(y)\big{(}D_{1}\bar{D}_{1}-\mathcal{D}_{1}\bar{\mathcal{D}}_{1}\big{)}$
$\displaystyle\hskip
71.13188pt-A(y)\lambda_{1}\lambda_{2}\big{(}G_{1L}\bar{G}_{1L}-\mathcal{G}_{1L}\bar{\mathcal{G}}_{1L}\big{)}$
$\displaystyle-B(y)|S_{1T}||S_{2T}|\cos(\phi_{S_{1}}+\phi_{S_{2}})\big{(}H_{1T}\bar{H}_{1T}+\mathcal{H}_{1T}\bar{\mathcal{H}}_{1T}\big{)}$
$\displaystyle-B(y)|S_{1T}||S_{2T}|\sin(\phi_{S_{1}}+\phi_{S_{2}})\big{(}H_{1T}\bar{\mathcal{H}}_{1T}-\mathcal{H}_{1T}\bar{H}_{1T}\big{)}\Big{]}.$
(53)

Here we have used

$\displaystyle d^{3}p_{1}d^{3}p_{2}/E_{1}E_{2}$
$\displaystyle=(dz_{1}/z_{1})(z_{2}Q^{2}dz_{2}/4)d\mathbf{p}_{1\perp}d\Omega_{2}$
$\displaystyle=\pi Q^{2}z_{1}z_{2}dz_{1}dz_{2}dyd^{2}\mathbf{q}_{T}.$ (54)

In Eq. (53), there are two azimuthal asymmetries which correspond to the
double spin asymmetries.

$\displaystyle\tilde{A}_{TT}^{\cos(\phi_{S_{1}}+\phi_{S_{2}})}=-\frac{B(y)}{A(y)}\frac{H_{1T}\bar{H}_{1T}+\mathcal{H}_{1T}\bar{\mathcal{H}}_{1T}}{D_{1}\bar{D}_{1}-\mathcal{D}_{1}\bar{\mathcal{D}}_{1}},$
(55)
$\displaystyle\tilde{A}_{TT}^{\sin(\phi_{S_{1}}+\phi_{S_{2}})}=-\frac{B(y)}{A(y)}\frac{H_{1T}\bar{\mathcal{H}}_{1T}-\mathcal{H}_{1T}\bar{H}_{1T}}{D_{1}\bar{D}_{1}-\mathcal{D}_{1}\bar{\mathcal{D}}_{1}}.$
(56)

First of all we consider the asymmetry
$\tilde{A}_{TT}^{\cos(\phi_{S_{1}}+\phi_{S_{2}})}$. We find that both
$H_{1T}\bar{H}_{1T}$ and $\mathcal{H}_{1T}\bar{\mathcal{H}}_{1T}$ have
contributions to this asymmetry which means
$\tilde{A}_{TT}^{\cos(\phi_{S_{1}}+\phi_{S_{2}})}$ has one more origin and it
will complicate the extraction of the transverse polarization transformation
function $H_{1T}$. This is the same to Collins asymmetry,
$\tilde{A}_{UU}^{\cos 2\phi}$. The asymmetry
$\tilde{A}_{TT}^{\sin(\phi_{S_{1}}+\phi_{S_{2}})}$, similar to
$\tilde{A}_{UU}^{\sin 2\phi}$, arise from the two interference terms with
opposite signs. So it can only survive on event-by-event basis.

So far we present eight kinds of azimuthal asymmetries. We can divide them
into four parts according to the FFs’ contributions.

  * •

$\tilde{A}_{UU}^{\cos 2\phi}$ and
$\tilde{A}_{TT}^{\cos(\phi_{S_{1}}+\phi_{S_{2}})}$. In this part, both the
$\mathcal{P}$ odd FFs and $\mathcal{P}$ even FFs have positive contributions
which means these kinds of azimuthal asymmetries have two origins.

  * •

$\tilde{A}_{UU}^{\sin 2\phi}$, $\tilde{A}_{TU}^{\cos(\phi-\phi_{S_{1}})}$ and
$\tilde{A}_{TT}^{\sin(\phi_{S_{1}}+\phi_{S_{2}})}$. In this part, asymmetries
comes from the interference between the $\mathcal{P}$ odd FF and $\mathcal{P}$
even FF and terms in the numerator have opposite signs which means these
azimuthal asymmetries only survive event-by-event.

  * •

$\tilde{A}_{TU}^{\sin(\phi-\phi_{S_{1}})}$ and
$A_{TU}^{\sin(\phi+\phi_{S_{1}})}$. These asymmetries are from both the
$\mathcal{P}$ odd FFs and $\mathcal{P}$ even FFs but these two terms have
opposite sign. In other words, the $\mathcal{P}$ odd FFs can reduce the effect
generated by $\mathcal{P}$ even FFs.

  * •

$A_{TU}^{\cos(\phi+\phi_{S_{1}})}$. This kind of azimuthal asymmetry is very
special. The interference two terms contributing to azimuthal asymmetry
$A_{LU}^{\cos(\phi+\phi_{S_{1}})}$ have both positive contributions and they
can survive even summing over many events. This is important for determining
the $\mathcal{P}$-odd FFs by measuring the azimuthal asymmetry at leading
twist.

###  IV.4 Hadron polarizations

In the previous sections we considered azimuthal asymmetries which can be
detected directly in SIA experiments. If the spin of hadrons measured in the
final state is not zero, the hadron polarization can be measured. For
simplicity, in this part we only consider the one hadron polarized case. The
hadron polarizations can be calculated in three steps. The first one is to
illustrate the probability interpretation, the second is to calculate
eigenvalues and the third is the to calculate hadron polarizations. In the
following we will take $\langle S_{T}^{x}\rangle$ for example to illustrate
the calculations.

Hadron polarization $\langle S_{T}^{x}\rangle$, in general, denotes the
probability of the hadrons in the spin $S_{T}^{x}$ state. For a system of
hadrons, we use spin density matrix to describe the polarization. The spin
density matrix can be written as

$\displaystyle\rho=\sum_{m,n}|n\rangle\langle n|~{}\rho~{}|m\rangle\langle
m|=\sum_{m,n}\mathrm{Tr}\big{[}\rho|m\rangle\langle n|\big{]}|n\rangle\langle
m|,$ (57)

where $|m\rangle,|n\rangle$ are eigenstates of corresponding spin operators
and $P\equiv\mathrm{Tr}\big{[}\rho|m\rangle\langle n|\big{]}$ is the
probability of finding one of these states. The spin operator can be defined
in arbitrary direction, i.e.,

$\displaystyle\Sigma^{i}\hat{n}^{i}=\Sigma^{x}\sin\theta\cos\phi+\Sigma^{y}\sin\theta\sin\phi+\Sigma^{z}\cos\theta,$
(58)

where $\theta$ and $\phi$ denote the polar and azimuthal angle. Thus the
polarization of a system can be calculated as

$\displaystyle O=\langle\mathcal{O}\rangle=\mathrm{Tr}[\mathcal{O}\rho],$ (59)

where $\mathcal{O}$ denotes any spin operator. Using Eq. (57)-(59), we obtain

$\displaystyle\langle
S_{T}^{x}\rangle=P(1,\frac{\pi}{2},0)-P(-1,\frac{\pi}{2},0),$ (60)

where $P=P(m,\theta,\phi)$ where $m$ denotes the eigenvalue of the
corresponding state. Do not be confused with the eigenstate $|m\rangle$. So
$\langle S_{T}^{x}\rangle$ denote the difference of the two probabilities. We
can also calculate

$\displaystyle\langle\lambda\rangle=P(1,0,0)-P(-1,0,0),$ (61)
$\displaystyle\langle
S_{T}^{y}\rangle=P(1,\frac{\pi}{2},\frac{\pi}{2})-P(-1,\frac{\pi}{2},\frac{\pi}{2}).$
(62)

Similarly, all the eigenvalues of the system on eigenstates of
$\lambda,S_{T}^{x},S_{T}^{y}$ can be calculated. They are given in Table. 2.

|  $|\psi_{z+}\rangle$ |  $|\psi_{z-}\rangle$ |  $|\psi_{y+}\rangle$ |  $|\psi_{y-}\rangle$ |  $|\psi_{x+}\rangle$ |  $|\psi_{x-}\rangle$
---|---|---|---|---|---|---
$\lambda$ | $1$ | $-1$ | $0$ | $0$ | $0$ | $0$
$~{}~{}S_{T}^{x}~{}~{}$ | $0$ | $0$ | $1$ | $-1$ | $0$ | $0$
$S_{T}^{y}$ | $0$ | $0$ | $0$ | $0$ | $1$ | $-1$
Table 2: Eigenvalues of the system on eigenstates of
$\lambda,S_{T}^{x},S_{T}^{y}$.

To calculate the hadron polarization, we consider the single hadron polarized
case. First of all, we consider the longitudinal polarization. Using Eq. (61)
and substituting the corresponding eigenvalues in Table. 2 into Eq. (48), we
can obtain the longitudinal polarization of hadron in SIA process,

$\displaystyle\langle\lambda_{1}\rangle=\frac{\mathcal{C}[\mathcal{G}_{1L}\bar{D}_{1}]-\mathcal{C}[{G}_{1L}\bar{\mathcal{D}}_{1}]}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]}.$
(63)

From Eq. (63)we see the longitudinal polarization in SIA process arising form
the interference of the $\mathcal{P}$ odd FF(s) $\mathcal{G}_{1L}$
($\bar{\mathcal{D}}_{1}$) and the $\mathcal{P}$ even FF(s) $\bar{D}_{1}$
(${G}_{1L}$). The signs of the terms in the numerator are opposite which means
that the $\langle\lambda_{1}\rangle$ only survives on event-bey-event basis
because the averaged over cross section of many event vanishes.

According to the cross section shown in Eq. (48) we know the transverse
polarization cannot be calculate directly. In Fig. 2, we define the plane
determined by momenta $l_{1},l_{2}$ and $p_{2}$ as $x-O-z$ plane or lepton-
hadron plane. $p_{2}$ lies in z-direction while the x-direction is determined
by the transverse momenta of leptons as shown. Here we say transverse
polarization is the polarization defined with respect to the lepton hadron
plane. Chen $et~{}al.$ argued in ref. Chen:2016moq ; Yang:2017sxz , the
transverse hadron polarization $\langle S_{T}^{x}\rangle$ corresponds to
$|S_{T}|\cos\phi_{S}$ while $\langle S_{T}^{y}\rangle$ corresponds to
$|S_{T}|\sin\phi_{S}$ and these polarizations vanish in Eq. (48). In fact, the
transverse polarization vanish at leading twist but they survive at twist-3
level. However, in ref. Chen:2016moq ; Yang:2017sxz , Chen $et~{}al.$ also
discussed the transverse polarizations with respect to hadron-hadron place
which is defined by the momenta of the two hadrons as shown in Fig. 2. In the
case we can define unit vector $\vec{e}_{n}$ and $\vec{e}_{t}$ to label the
transverse directions. Here
$\vec{e}_{n}=\vec{p}_{1}\times\vec{p}_{2}/|\vec{p}_{1}\times\vec{p}_{2}|=(-\sin\phi,\cos\phi)$
and $\vec{e}_{t}=\vec{p}_{1T}/|\vec{p}_{1T}|=(\cos\phi,\sin\phi)$, i.e., the
normal and tangent of the hadron-hadron plane, respectively. Thus the
transverse hadron polarization $S_{T}^{n}$ corresponds to
$|S_{T}|\sin(\phi_{S}-\phi)$ while $S_{T}^{t}$ corresponds to
$|S_{T}|\cos(\phi_{S}-\phi)$. So we can calculate the transverse polarizations
with respect to hadron hadron plane.

$\displaystyle\langle
S_{T}^{n}\rangle=-\frac{\mathcal{C}\Big{[}w_{1}\big{(}\mathcal{G}^{\perp}_{1T}\bar{\mathcal{D}}_{1}-D^{\perp}_{1T}\bar{D}_{1}\big{)}\Big{]}}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]},$
(64) $\displaystyle\langle
S_{T}^{t}\rangle=\frac{\mathcal{C}\Big{[}w_{1}\big{(}\mathcal{D}^{\perp}_{1T}\bar{D}_{1}-G^{\perp}_{1T}\bar{\mathcal{D}}_{1}\big{)}\Big{]}}{\mathcal{C}[D_{1}\bar{D}_{1}]-\mathcal{C}[\mathcal{D}_{1}\bar{\mathcal{D}}_{1}]}.$
(65)

For $\langle S_{T}^{n}\rangle$, we see it reflects the effect of Sivers-type
FF $D_{1T}^{\perp}$. But the two terms in the numerator have opposite signs,
which means that the contribution from $\mathcal{P}$-odd FFs
$\mathcal{G}^{\perp}_{1T}\bar{\mathcal{D}}_{1}$ will dilute the contribution
from $\mathcal{P}$-even FFs $D^{\perp}_{1T}\bar{D}_{1}$. However the
transverse polarization $\langle S_{T}^{t}\rangle$ comes from the interference
of the $\mathcal{P}$-odd FF(s)
$\mathcal{D}^{\perp}_{1T}(\bar{\mathcal{D}}_{1})$ and $\mathcal{P}$-even FF(s)
$\bar{D}_{1}(G^{\perp}_{1T})$ and it only survive event-by-event basis, just
like $\langle\lambda_{1}\rangle$, see ref Chen:2016moq .

##  V Summary

Fragmentation functions are important quantities in high energy reactions and
they describe the fragmentation and/or hadronization processes. FFs can be
defined by the quark-quark correlation functions which satisfy the hermiticity
and parity conservation constraints, in general. When the non-trivial QCD
topological structure is taken into consideration, QCD theory can be local
parity violated. In this case, the parity conservation constraint will break
down. As a result, FFs can be $\mathcal{P}$-odd in addition to the
$\mathcal{P}$-even properties. We apply this argument into the SIA process in
order to calculate the azimuthal asymmetries and hadron polarizations which
are expressed in terms of FFs including both the $\mathcal{P}$-odd ones and
the $\mathcal{P}$-even ones.

We calculate eight kinds of azimuthal asymmetries and category them into four
parts. The first part is $\tilde{A}_{UU}^{\cos 2\phi}$ and
$\tilde{A}_{TT}^{\cos(\phi_{S_{1}}+\phi_{S_{2}})}$. In this part, both the
$\mathcal{P}$-odd FFs and $\mathcal{P}$-even FFs have positive contributions
which means these kinds of azimuthal asymmetries have two origins. The second
part is $\tilde{A}_{UU}^{\sin 2\phi}$,
$\tilde{A}_{TU}^{\cos(\phi-\phi_{S_{1}})}$ and
$\tilde{A}_{TT}^{\sin(\phi_{S_{1}}+\phi_{S_{2}})}$. In this part, asymmetries
comes from the interference between the $\mathcal{P}$-odd FF and
$\mathcal{P}$-even FF and terms in the numerator have opposite signs which
means these azimuthal asymmetries only survive event-by-event. The third part
is $\tilde{A}_{TU}^{\sin(\phi-\phi_{S_{1}})}$ and
$A_{TU}^{\sin(\phi+\phi_{S_{1}})}$. These asymmetries are from both the
$\mathcal{P}$-odd FFs and $\mathcal{P}$-even FFs but these two terms have
opposite sign. In other words, the $\mathcal{P}$-odd FFs can reduce the effect
generated by $\mathcal{P}$-even FFs. The fourth part is
$A_{TU}^{\cos(\phi+\phi_{S_{1}})}$. This kind of azimuthal asymmetry is very
special. The interference two terms contributing to azimuthal asymmetry
$A_{LU}^{\cos(\phi+\phi_{S_{1}})}$ have both positive contributions and they
can survive even summing over many events. This is important for determining
the $\mathcal{P}$-odd FFs by measuring the azimuthal asymmetry at leading
twist.

We also calculate the hadron polarizations in which we assume that hadron 1 is
polarized and hadron 2 is unpolarized. We find that the longitudinal
polarization in SIA process arising form the interference of the
$\mathcal{P}$-odd FF(s) $\mathcal{G}_{1L}$ ($\bar{\mathcal{D}}_{1}$) and the
$\mathcal{P}$-even FF(s) $\bar{D}_{1}$ (${G}_{1L}$). The signs of the terms in
the numerator are opposite which means that the $\langle\lambda_{1}\rangle$
only survives on event-bey-event basis because the averaged over cross section
of many event vanishes. For $\langle S_{T}^{n}\rangle$, we see it reflects the
effect of Sivers-type FF $D_{1T}^{\perp}$. But the two terms in the numerator
have opposite signs, which means that the contribution from $\mathcal{P}$-odd
FFs $\mathcal{G}^{\perp}_{1T}\bar{\mathcal{D}}_{1}$ will dilute the
contribution from $\mathcal{P}$-even FFs $D^{\perp}_{1T}\bar{D}_{1}$. However
the transverse polarization $\langle S_{T}^{t}\rangle$ comes from the
interference of the $\mathcal{P}$-odd FF(s)
$\mathcal{D}^{\perp}_{1T}(\bar{\mathcal{D}}_{1})$ and $\mathcal{P}$-even FF(s)
$\bar{D}_{1}(G^{\perp}_{1T})$ and it only survive event-by-event basis.

## References

  * (1) R. Jakob, P. J. Mulders and J. Rodrigues, Nucl. Phys. A 626, 937 (1997) doi:10.1016/S0375-9474(97)00588-5 [hep-ph/9704335].
  * (2) H. Avakian, A. V. Efremov, P. Schweitzer and F. Yuan, Phys. Rev. D 81, 074035 (2010) doi:10.1103/PhysRevD.81.074035 [arXiv:1001.5467 [hep-ph]].
  * (3) A. Bacchetta, R. Kundu, A. Metz and P. J. Mulders, Phys. Lett. B 506, 155 (2001)
  * (4) D. Amrath, A. Bacchetta and A. Metz, Phys. Rev. D 71, 114018 (2005)
  * (5) A. Metz, Phys. Lett. B 549, 139 (2002)
  * (6) A. Bacchetta, L. P. Gamberg, G. R. Goldstein and A. Mukherjee, Phys. Lett. B 659, 234 (2008)
  * (7) R. M. Davidson and E. Ruiz Arriola, Acta Phys. Polon. B 33, 1791 (2002) [hep-ph/0110291].
  * (8) T. Ito, W. Bentz, I. C. Cloet, A. W. Thomas and K. Yazaki, Phys. Rev. D 80, 074008 (2009)
  * (9) H. H. Matevosyan, A. W. Thomas and W. Bentz, Phys. Rev. D 83, 074003 (2011)
  * (10) H. H. Matevosyan, W. Bentz, I. C. Cloet and A. W. Thomas, Phys. Rev. D 85, 014021 (2012) doi:10.1103/PhysRevD.85.014021 [arXiv:1111.1740 [hep-ph]].
  * (11) H. H. Matevosyan, A. W. Thomas and W. Bentz, Phys. Rev. D 86, 034025 (2012)
  * (12) S. Dulat et al., Phys. Rev. D 93, no. 3, 033006 (2016) doi:10.1103/PhysRevD.93.033006 [arXiv:1506.07443 [hep-ph]].
  * (13) S. Albino, B. A. Kniehl and G. Kramer, Nucl. Phys. B 803, 42 (2008) doi:10.1016/j.nuclphysb.2008.05.017 [arXiv:0803.2768 [hep-ph]].
  * (14) S. Albino, B. A. Kniehl and G. Kramer, http://lapth.cnrs.fr/ffgenerator/
  * (15) D. de Florian, M. Stratmann and W. Vogelsang, Phys. Rev. D 57, 5811 (1998) doi:10.1103/PhysRevD.57.5811 [hep-ph/9711387].
  * (16) G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976). doi:10.1103/PhysRevLett.37.8
  * (17) G. ’t Hooft, Phys. Rev. D 14, 3432 (1976) Erratum: [Phys. Rev. D 18, 2199 (1978)]. doi:10.1103/PhysRevD.18.2199.3, 10.1103/PhysRevD.14.3432
  * (18) R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37, 172 (1976). doi:10.1103/PhysRevLett.37.172
  * (19) C. G. Callan, Jr., R. F. Dashen and D. J. Gross, Phys. Lett. B 63, 334 (1976) [Phys. Lett. 63B, 334 (1976)]. doi:10.1016/0370-2693(76)90277-X
  * (20) C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006) doi:10.1103/PhysRevLett.97.131801 [hep-ex/0602020].
  * (21) D. Kharzeev, R. D. Pisarski and M. H. G. Tytgat, Phys. Rev. Lett. 81, 512 (1998) doi:10.1103/PhysRevLett.81.512 [hep-ph/9804221].
  * (22) D. Kharzeev, Phys. Lett. B 633, 260 (2006) doi:10.1016/j.physletb.2005.11.075 [hep-ph/0406125].
  * (23) D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A 803, 227 (2008) doi:10.1016/j.nuclphysa.2008.02.298 [arXiv:0711.0950 [hep-ph]].
  * (24) K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 78, 074033 (2008) doi:10.1103/PhysRevD.78.074033 [arXiv:0808.3382 [hep-ph]].
  * (25) A. Efremov and D. Kharzeev, Phys. Lett. B 366, 311 (1996) doi:10.1016/0370-2693(95)01329-6 [hep-ph/9506412].
  * (26) Z. B. Kang and D. E. Kharzeev, Phys. Rev. Lett. 106, 042001 (2011) doi:10.1103/PhysRevLett.106.042001 [arXiv:1006.2132 [hep-ph]].
  * (27) D. Pitonyak, M. Schlegel and A. Metz, Phys. Rev. D 89, no. 5, 054032 (2014) doi:10.1103/PhysRevD.89.054032 [arXiv:1310.6240 [hep-ph]].
  * (28) K. b. Chen, W. h. Yang, S. y. Wei and Z. t. Liang, Phys. Rev. D 94, no. 3, 034003 (2016) doi:10.1103/PhysRevD.94.034003 [arXiv:1605.07790 [hep-ph]].
  * (29) D. Boer, R. Jakob and P. J. Mulders, Nucl. Phys. B 504, 345 (1997) doi:10.1016/S0550-3213(97)00456-2 [hep-ph/9702281].
  * (30) R. P. Feynman, Phys. Rev. Lett. 23, 1415 (1969). doi:10.1103/PhysRevLett.23.1415
  * (31) R. P. Feynman, “Photon-hadron interactions,” Reading 1972, 282p
  * (32) J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969). doi:10.1103/PhysRev.185.1975
  * (33) W. h. Yang, K. b. Chen and Z. t. Liang, Phys. Rev. D 96, no. 5, 054016 (2017) doi:10.1103/PhysRevD.96.054016 [arXiv:1707.00402 [hep-ph]].


# Towards Integrated Traffic Control
with Operating Decentralized Autonomous Organization

Shengyue Yao1, Jingru Yu1, Yi Yu1, Jia Xu1, Xingyuan Dai2,
Honghai Li3, Fei-Yue Wang4, Fellow, IEEE, Yilun Lin1,∗, Member, IEEE *This
work was supported by the Shanghai Artificial Intelligence
Laboratory.*Corresponding author: Yilun Lin (linyilun@pjlab.org.cn)1Shengyue
Yao (yaoshengyue@pjlab.org.cn), Jingru Yu (yujingru@pjlab.org.cn), Yi Yu
(yuyi@pjlab.org.cn), Jia Xu<EMAIL_ADDRESS>and Yilun Lin
<EMAIL_ADDRESS>are with Urban Computing, Shanghai AI Lab, Shanghai,
China.2Xingyuan Dai<EMAIL_ADDRESS>is with the The State Key
Laboratory for Management and Control of Complex Systems, Institute of
Automation, Chinese Academy of Sciences, Beijing, 100190, China.3Honghai Li
<EMAIL_ADDRESS>is with the Research and Development Center of Transport
Industry of Autonomous Driving Technology, RIOH High and Technology
Group.3Fei-Yue Wang<EMAIL_ADDRESS>is with the Institute of
Automation, Chinese Academy of Sciences, Beijing, China, and the Macau
Institute of Systems Engineering, Macau University of Science and Technology,
Macau, China.

###### Abstract

With a growing complexity of the intelligent traffic system (ITS), an
integrated control of ITS that is capable of considering plentiful
heterogeneous intelligent agents is desired. However, existing control methods
based on the centralized or the decentralized scheme have not presented their
competencies in considering the optimality and the scalability simultaneously.
To address this issue, we propose an integrated control method based on the
framework of Decentralized Autonomous Organization (DAO). The proposed method
achieves a global consensus on energy consumption efficiency (ECE), meanwhile
to optimize the local objectives of all involved intelligent agents, through a
consensus and incentive mechanism. Furthermore, an operation algorithm is
proposed regarding the issue of structural rigidity in DAO. Specifically, the
proposed operation approach identifies critical agents to execute the smart
contract in DAO, which ultimately extends the capability of DAO-based control.
In addition, a numerical experiment is designed to examine the performance of
the proposed method. The experiment results indicate that the controlled
agents can achieve a consensus faster on the global objective with improved
local objectives by the proposed method, compare to existing decentralized
control methods. In general, the proposed method shows a great potential in
developing an integrated control system in the ITS.

##  I INTRODUCTION

The development of the intelligent traffic system (ITS) has long been focused
in academia and industry over the past decades, which has evolved into a
complex system with plentiful intelligent agents, including connected and
automated vehicles (CAV) and multiple intelligent traffic control measures
(e.g., traffic signal, variable speed limits, perimeter gating, etc.). It is
foreseeable that the ITS will become increasingly complex with the surge of
intelligent agents, thus the control of ITS becomes increasingly essential.
Specifically, the control of ITS should consider all involved intelligent
agents which are heterogeneous in different scales, such as in different
responding frequency; as well as with diverse objectives, such as mitigating
congestion, guaranteeing safety, reducing emission and energy consumption.

Existing optimal control methods of individual agents in ITS are theoretically
mature and have evolved towards more expeditious, agile, and adaptive with the
aid of AI-based approaches [1, 2, 3]. However, given the complex nature of
ITS, these methods are incapable of performing as expected in practice, while
a wide deployment of intelligent agents has frequently trapped the entire ITS
into the Braess paradox [4, 5]. Therefore, investigating an integrated control
method, which is capable of coordinating the behaviours of heterogeneous
agents in ITS renders a non-trivial task to be tackled.

Recent research focuses on enhancing scalability and optimality in integrated
traffic control, for which a centralized or decentralized control scheme is
adopted, respectively. The centralized scheme assumes perfect knowledge of all
agents’ states and optimizes global objectives, while the decentralized scheme
optimizes local objectives and achieves Pareto optimality[6]. However, the
centralized control scheme has its limit in coordinating heterogeneous agents,
whereas the decentralized scheme is inefficient in synchronizing agents’
behaviours. Therefore, neither the centralized nor the decentralized scheme
presents its competency in the integrated control of plentiful heterogeneous
intelligent agents.

Fortunately, the emergence of Blockchain technology and Decentralized
Autonomous Organization (DAO) provide opportunities for developing integrated
traffic control with scalability and optimality considerations[7, 8]. DAO
controls a complex system of heterogeneous agents using smart contracts,
ensuring control optimality through a proposal-voting-action-incentive
mechanism. In addition, the autonomous and independent execution of smart
contracts ensures scalability and secures data transmission. While the
potential of DAO-based control in ITS has been discussed, few studies have
evaluated its performance in practice. Specifically, the feasible design of
smart contracts and endogenous governing defects of DAO, especially the
structural rigidity[9], pose challenges in its application in practice,
particularly in complex systems with numerous intelligent agents.

With the consideration of the issues above, a consensus and incentive
mechanism for DAO-based ITS control and an operation on DAO are proposed in
this research. Specifically, the proposed mechanism focuses on optimizing the
energy consumption efficiency (ECE) in ITS as mentioned by Wang [10]. The ECE
represents the rate of improved local objectives and changed control effort
magnitude, which should be maximized and balanced through all involved agents.
Meanwhile, the operation on DAO unveils critical agents in ITS to execute
smart contracts in DAO, in order to avoid frequent code altering in deployed
smart contracts or deploying new smart contracts. In addition, the operation
on DAO is inspired by recent studies in the dense reinforcement learning and
the message transaction optimization for Blockchain-enabled Internet-of-
Vehicles [11, 12].

The rest of this paper is organized as follows. Section II investigates the
related work regarding existing integrated control schemes and DAO-based
control mechanism. Section III elaborates the methodology including the
problem formulation, the solving framework, the detailed algorithms of the
operation on DAO and the consensus and incentive mechanism. Section IV
discusses the application of the proposed integrated control method in ITS, as
well as presents examination results of the proposed method in a numerical
experiment. Finally, this research is concluded in section V

##  II RELATED WORK

###  II-A Integrated traffic control in ITS

In this section, existing integrated control schemes in ITS (centralized and
decentralized schemes) are reviewed. Specifically, the application scenario
and the defects of centralized and decentralized control scheme are
elaborated, respectively.

####  II-A1 Centralized control method

The centralized scheme operates under the assumption of a perfect knowledge of
all controlled agents’ states. This knowledge is then leveraged to regulate
the agents’ behaviors, optimizing a global objective that encapsulates the
objectives of all agents.

In ITS, the centralized control scheme is commonly employed in managing
multiple homogeneous agents, such as CAV platoon control [13, 14, 15], traffic
signal coordination [16], and highway control measures coordination [17, 18].
However, the centralized scheme exhibits limitations in scalability and is
incompetent in coordinating heterogeneous intelligent agents. Additionally,
the assumption of perfect knowledge raises various concerns about
communication reliability and security [19, 20].

####  II-A2 Decentralized control method

The decentralized scheme operates by regulating intelligent agents based on
local information as well as information received from their connected agents.
This approach optimizes their behaviors towards their local objectives while
achieving a consensus to achieve a Pareto optimality.

In ITS, the decentralized scheme is frequently employed in coordinating
behaviors of heterogeneous agents, such as coordinating multiple urban traffic
control measures with diverse objectives (e.g., mitigating traffic congestion
and reducing evacuation time after an event [21]) or in different scales [22,
23]. However, the defects in achieving optimization and synchronization
impedes the decentralized scheme from being adopted to regulate a large number
of intelligent agents [6, 18].

In summary, neither the centralized nor the decentralized scheme has
demonstrated its competence in the integrated control of numerous
heterogeneous intelligent agents. An integrated ITS control which can achieve
both the optimality and the scalability is desired, as it is illustrated in
Fig. 1.

Figure 1: Existing integrated control scheme capability and unsolved issue
Figure 2: DAO-based control mechanism

###  II-B DAO and related control methods

The emergence of Blockchain technology and decentralized autonomous
organization (DAO) provides a promising avenue for developing an integrated
traffic control system that considers both the scalability and the optimality.
It is worth noting that the deployment of smart contracts on the Blockchain
can further guarantee secure and private-preserving data transmission,
although this aspect is not discussed in this study [24, 25].

The decentralized and autonomous nature of the DAO has spurred interests in
applying it to the integrated control of ITS. [7, 8]. In addition, the
framework of Dao-based control in ITS has been extensively discussed, which
can be used in integrated control of urban traffic system [25], crowd-sensing
systems [26], vehicle parking recommendation systems[27], etc. In general,
agents in physical world are enecrypted into Blocks in Blockchians, which can
automatically and independently execute smart contracts that are encoded and
deployed in advance. Moreover, the smart contracts stimulates agents to
achieve a consensus through a proposal-voting-action-incentive mechanism [28].
The DAO-based control framework can be summarized in Fig. 2.

However, the detailed design of the smart contract in ITS control has received
little attention, particularly the formulation of the consensus and incentive
mechanism. Another critical issue is the structural rigidity. Specifically,
altering the code in the smart contract once the system is operating can be
extremely difficult, while deploying new smart contracts can be expensive [9,
29]. The issue of structural rigidity poses a significant challenge when
applying the DAO in a complex system with a growing number of intelligent
agents, such as in ITS.

Despite extensive research on integrated traffic control, the research gap
still lies in the lack of a suitable control design that can achieve both
scalability and optimality while regulating numerous heterogeneous intelligent
agents. Particularly, the detailed DAO-based control design addressing the
formulation of the consensus and incentive mechanism and the structural
rigidity issue is desired.

##  III METHODOLOGY

###  III-A Problem definition

The problem of an integrated ITS control, considering plentiful heterogeneous
intelligent agents, is defined in this section.

The problem is defined under the following assumptions: (1) The intelligent
agents in ITS are connected through Peer-to-Peer (P2P) communication and the
communication delay is negligible (in milliseconds) compare to the control
response time (in seconds). (2) The connection intensities, which reflect the
exchange rates of control effort between agents, are known a-priori through
learning the history data.

Based on the assumptions above, we focus on developing a DAO-based integrated
control, which aims to increase and balance the ECE of all intelligent agents
in ITS, upon the optimization of their local objectives.

For a road network contains a set of $N$ intelligent agents
$V_{N}=\\{v_{1},...v_{n}\\}$ with different response time $\tau_{i}$, its
communication topology can be represented by a graph $G$, that
$G=\\{V_{N},E,A_{N\times N}\\}$. $E\subseteq V_{N}\times V_{N}$ denotes the
set of edges $e_{ij}$, which indicates the communication exists between agents
$i$ and $j$, $i\subseteq N$, $j\subseteq N$. $A_{N\times N}$ is the adjacent
matrix, that $a_{ij}\subseteq\mathbb{R},\forall e_{ij}\subseteq E$ and
$a_{ij}=0$ otherwise. In addition, $a_{ii}=0$. The value of $a_{ij}$ indicates
the linking intensity, that $a_{ij}=\frac{u_{j}}{u_{i}}$. The dynamic of agent
$v_{i}$ can be represented by an ODE in Eq. (1).

$\dot{x_{i}}=f(x_{i},\textbf{u}_{i})$ (1a)
$\textbf{u}_{i}=u_{i}+\sum_{j=1}^{N}a_{ji}u_{j}$ (1b)

where $x_{i}$ is the observed state of agent $v_{i}$, and the control input
$u_{i}$ is optimized towards its local objective $J_{i}=\min g(x_{i},u_{i})$.
For the integrated control of the defined road network system, all intelligent
agents in the road network are controlled towards their local objectives.
Meanwhile, a global objective of maximizing the cumulative ECE, meanwhile
achieving a consensus on the ECE, is desired, as represented by Eq. (2).

$J_{G}^{cumulative}=\max\int_{t=0}^{\infty}\sum_{i}^{N}r_{i}$ (2a)
$J_{G}^{balance}=\lim_{t\to\infty}|r_{i}-r_{j}|=0,\forall i\subseteq N,\forall
j\subseteq N$ (2b) s.t. $r_{i}=\frac{\dot{g_{i}}}{|\dot{u_{i}}|}$ (2c)

where $|\cdot|$ denotes the absolute value. Note that the ECE of an agent
$r_{i}$ represents the rate of its changed local objectives and its changed
control effort magnitude. Hence a higher value of ECE indicates that the agent
is able to achieve a better performance with less consumed energy in changing
its control effort. Therefore, the ECE should be maximized and balanced among
all agents in the controlled system with an integrated control, as indicated
by Eq. (2)

As it is mentioned in section II, solving the defined problem in a centralized
scheme is intricate due to the heterogeneity of agents, as well as the
difficulties in summarizing Eq. (2a) and Eq. (2b). In addition, the defined
problem cannot be effectively solved in a decentralized scheme due to the
synchronization issue. Even with the DAO-based control, the issue of
structural rigidity with the growing size of set $A$ and $E$ posses a critical
challenge. Therefore, instead of analytically solving the defined problems, a
solving method based on reiteratively executing a DAO-based control protocol
and operating the DAO is proposed.

###  III-B Solving framework

Addressing the issues of heterogeneity, synchronization and DAO’s structural
rigidity, the problem defined in section III-A can be solved by a framework as
in Fig. 3. In general, the operation on DAO extracts a sub-graph
$G^{s}=\\{V^{s}_{N^{s}},E^{s}_{N^{s}},A^{s}_{N^{s}\times N^{s}}\\}$ from $G$
with a frequency of $\tau_{o}$ by identifying critical agents, which will be
elaborated in section III-C. Afterwards, a DAO-based control protocol encoded
in the smart contract will be executed in $G^{s}$. Specifically, the DAO-based
control considers consensus among all involved agents (Eq. 2b) and improving
local objectives. Meanwhile, the global objective in Eq. (2a) can be improved
by distributing incentives, which will be elaborated in section III-D.

Figure 3: Problem solving framework

###  III-C Operation on DAO

According to Fig. 3, the operation on DAO aims to identify critical agents by
extracting $G^{s}$ from $G$, in order to avoid deploying new smart contracts
to accommodate the growing amount of agents in DAO. Given the deployed smart
contract capability $\Phi$ (i.e., the amount of intelligent agents in ITS that
the deployed smart contract is capable of effectively executing an integrated
control), and the tolerance level $\Psi$ (i.e. the magnitude of global
optimization objective that can be compromised during the operation), the
objective of the operation on DAO can be formulated in Eq. (3).

$J^{s}=\min(|E^{s}_{N^{s}}|)$ (3a) s.t. $W_{i}(G^{s})\leq\Psi,\forall
i\subseteq N^{s}$ (3b) $W_{i}(G^{s})=\sum_{j\subseteq
N}a_{ij}\cdot|r_{i}-r_{j}|-\sum_{j\subseteq N^{s}}a_{ij}\cdot|r_{i}-r_{j}|$
(3c) $N^{s}\leq\Phi$ (3d)

A heuristic algorithm is proposed in Algorithm 1 to find an approximate
solution for achieving Eq.(3). Note that if a feasible $G^{s}$ is not found by
Algorithm 1, the proposed operation on the existing DAO may reach its capacity
limit and new smart contracts are necessary to be deployed.

Algorithm 1 DAO operation algorithm

0: $G$, $\Phi$,$\Psi$

0: $G^{s}$

1: Initialization $G^{s}=G$, $N^{s}=N$, $\textbf{N}^{s}=\\{1,2,...N\\}$

2: while Maximum iteration is not reached do

3: while $N^{s}>\Phi$ do

4: Randomly select $i\subseteq\textbf{N}^{s}$

5: if $\forall j,e_{ij}\notin E^{s}$ then

6: $\textbf{N}^{s}=\textbf{N}^{s}/\\{i\\}$

7: $N^{s}=N^{s}-1$

8: end if

9: $w_{i}=\min(a_{ij}\cdot|r_{i}-r_{j}|)$, $\forall j$ that $e_{ij}\subseteq
E^{s}$

10: $\tilde{G^{s}}=\\{V^{s},E^{s}/\\{e_{ij}\\},\\{A^{s}:a_{ij}=0\\}\\}$

11: if $W_{i}(\tilde{G})\leq\Psi$ then

12: $G^{s}=\tilde{G^{s}}$

13: else

14: break

15: end if

16: end while

17: return $G^{s}$

18: end while

19: return did not find $G^{s}$

###  III-D Consensus and incentive mechanism

Having the critical agents identified with $G^{s}$, an integrated control of
$v_{i}\subseteq V_{i}^{s}$ is achieved through a consensus and incentive
mechanism. Initially, a control protocol is designed in Eq. (4), which is
inspired by the protocol formulated by Xie et al. [30].

$u_{i}=\sigma_{\Delta}(-d_{i}-\alpha_{i}\bigtriangledown_{u_{i}}g(x_{i},u_{i})-\beta_{i}\sum_{j=1}^{N^{s}}a_{ij}(r_{i}-r_{j}))$
(4a) s.t.
$\dot{d_{i}}=\alpha_{i}\beta_{i}\sum_{j=1}^{N^{s}}a_{ij}(r_{i}-r_{j}),d_{i}(0)=0$
(4b) $\sigma_{\Delta}(u)=\text{sgn}(u)\min\\{|u|,\Delta\\}$ (4c)

where $N^{s}$ is the amount of agents in $V^{s}$,
$\bigtriangledown_{u_{i}}g(x_{i},u_{i})$ is the local objective optimization
term,
$\sum_{j=1}^{N^{s}}a_{ij}(\frac{r_{i}}{\tau_{i}}-\frac{r_{i}}{\tau_{i}})$ is
the consensus term, and $-d_{i}$ is the stabilizing term that maintains the
consensus at the optimal point of local objectives. $\sigma_{\Delta}$ is the
saturation function bounding the value of control input with a level $\Delta$,
we set $\Delta=\min\\{|u_{i}^{min}|,|u_{i}^{max}|\\}$ for simplicity. In
addition, $\alpha_{i},\beta_{i}\subseteq\mathbb{R}^{+}$ are weighting
coefficients with respect to local and global objectives, respectively.

According to the framework of DAO-based control in Fig. 2, the value of
$\alpha_{i}$ and $\beta_{i}$ represent the voting power of agent $i$, which
can be proportional to the mining workload [31], the reputation of following
the traffic rules[32], or the discrepancy between its current state and the
consensus goal[33]. In this study, the Proof-of-Stack (PoS) scheme by Zhu et
al. [33] is adopted and the value of $\alpha_{i}$ and $\beta_{i}$ are defined
by Eq. (5).

$\alpha_{i}=\gamma_{i}\exp({k_{1}\tilde{r_{i}}})$ (5a)
$\beta_{i}=\gamma_{i}\exp{-(k_{2}\tilde{r_{i}})}$ (5b) s.t.
$\tilde{r_{i}}=\sum_{j=1}^{N^{s}}a_{ij}(r_{i}-\frac{r_{i}+r_{j}}{2})$ (5c)

$k_{1},k_{2}\subseteq\mathbb{R}^{+}$ are positive constants.
$\gamma_{i}\subseteq\mathbb{R}^{+}$ is the weighting coefficient. Following
the process of DAO-based control, $\gamma_{i}$ updates according to the
incentives received by agent $i$ after $u_{i}$ is executed, that:

$\gamma_{i}^{update}=\gamma_{i}^{old}(\frac{2*\arctan(h_{i})}{\pi}+1)$ (6a)
s.t.
$h_{i}=k_{3}\cdot(\tilde{r_{i}}-\tilde{r_{i}}^{old})+k_{4}\cdot\sum_{j=1}^{N^{s}}a_{ij}(g_{j}-g_{j}^{old})$
(6b)

$k_{3}$,$k_{4}\subseteq\mathbb{R}^{+}$ are positive constants. The term
$k_{3}\cdot(\tilde{r_{i}}-\tilde{r_{i}}^{old})$ presents the reward from
achieving a consensus, whereas the term
$k_{4}\cdot\sum_{j=1}^{N^{s}}a_{ij}(g_{j}-g_{j}^{old})$ presents the ’support’
or ’objection’ from the connected agents of $v_{i}$ in $V^{s}$.

With the proposed solving method, the development of an integrated control in
ITS can follow the process of repeatedly deploying smart contract and
operating DAO with the growth of ITS system complexity, as presented in Fig.4.
Following this process, the capability of DAO-based ITS control in achieving
optimality and scalability can be extensively enhanced.

Figure 4: Integrated control development process

##  IV CASE STUDY AND RESULTS DISCUSSION

###  IV-A Application scenario in ITS

Fig. 5(a) illustrated a typical scenario that can apply the proposed
integrated control method. Specifically, a single-lane arterial road with two
three-arm intersections is depicted. There exists multiple intelligent agents
in the depicted scenario, including CAVs, intelligent traffic lights, variable
speed limits (VSL), perimeter gating, etc, which exhibits significant
heterogeneity in terms of their scales and objectives. For instance, the
acceleration of a CAV can be decided in milliseconds, with the objectives of
reducing travel delay and acceleration oscillation; whereas the control of
gated flow by the perimeter gating control aims to decrease the total travel
delay of vehicles in the controlled network, with a response time in minutes.

With the proposed method, intelligent agents can be controlled via
reiteratively operating DAO and executing a DAO-based control through a
consensus and incentive mechanism, as presented in Fig. 5(b). It is worthwhile
to note that except for addressing the issue of heterogeneity and
synchronization, the proposed method has practical significance. For instance,
if CAV $v_{9}$ encounters an emergency, the agents $v_{1}$, $v_{3}$, $v_{5}$,
$v_{9}$ may execute excessive control to improve their own objectives, thus a
coordination among them is needed to alleviate the ECE; meanwhile other
agents’ impact on reaching the consensus is minor.

(a) A typical scenario of integrated control in ITS by the proposed method

(b) Integrated control framework in the described scenario

Figure 5: Application scenario in ITS

###  IV-B Numerical experiment

(a) Convergence time

(b) Evaluation criteria value

Figure 6: Experiment results. (a) presents the convergence curve of the
maximum discrepancy in ECE among all agents, which are calculated by
$\max|r_{i}-r_{j}|,\forall i,j\subseteq N$. (b) presents results of three
evaluation criteria: (1)The reached consensus on ECE at the end of simulation.
(2) The accumulated magnitude of improved local objectives, which are
represented by $\Delta J$, that $\Delta
J=\sum_{i}(g_{i}(t_{end})-\sum_{i}g_{i}(t_{start}))$. (3) The cumulative ECE,
which is calculated by Eq. (2a). Note that infinity values are excluded.

A numerical experiment is designed in this section to present the feasibility
and efficiency of the proposed consensus and incentive mechanism and the
operation of DAO. In this experiment, the traditional decentralized consensus
control, the PoS-based consensus control, the DAO based control without
operation on DAO, as well as the proposed method are examined.

Similar to Fig. 5(a), a total of 10 intelligent agents in four different
scales are simulated, that $V=\\{v_{1},...v_{10}\\}$, the adjacent matrix of
the simulated system is:

$\small{A=\begin{bmatrix}0&0&0.1&0&0&0.2&0&0&0&0\\\
0&0&0.3&0.1&0.1&0&0&0&0&0\\\ -0.2&0.2&0&0&0.1&0&0&0&0.03&0.1\\\
0&-0.1&0&0&0&0.5&0.2&0&0&0\\\ 0&0&-0.1&-0.03&0&0&0&0&0.4&0\\\
-0.02&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0.2&0&0&0&0.4&0&0\\\
0&0&0&0.1&0.2&0&-0.1&0&0.3&0\\\ 0&0&-0.1&0&0&0&0&0.2&0&0\\\
0&0&0&0.05&0&0&0&0&0&0\\\ \end{bmatrix}}$ (7)

For simplicity, the detailed system dynamic of each agent is not modeled,
while hypothetical functions are adopted to represent their system dynamics,
as in Eq. 8.

$f(x_{i},\textbf{u}_{i})=x_{i}\cdot\sin{i}+\textbf{u}_{i}\cdot\cos{i}$ (8)

with $u_{i}\subseteq\\{-3,3\\}$. The local objectives $g_{i}\\{x_{i},u_{i}\\}$
are designed as:

$g_{i}\\{x_{i},u_{i}\\}=i\cdot\sin{(x_{i})}+u_{i}^{2}\cdot\cos{i}$ (9)

For the response time of each agents and the operation on DAO, $\tau_{1}$ is
10 seconds, $\tau_{2}$, $\tau_{3}$ are 2 seconds, $\tau_{4}$ to $\tau_{6}$ are
1 seconds, $\tau_{7}$ to $\tau_{10}$ are 0.5 seconds, and $\tau_{o}$ is 5
seconds. In addition, for the decentralized consensus control, $\alpha_{i}$
and $\beta_{i}$ are time-invariant coefficients that $\alpha=2$ and $\beta=1$.
For the PoS-based control, $\gamma_{i}$ is time-invariant that $\gamma_{i}=1$.
In addition, $k_{1}=2$, $k_{2}=5$, $k_{3}=0.3$, $k_{4}=0.1$, and $\Psi=2$,
$\Phi=4$. A total time of 500 seconds are simulated. The initial state $x_{i}$
are randomly selected in $[-20,20]$ and the initial control effort $u_{i}$ are
randomly selected in $[-3,3]$.

The performances of four considered methods are presented in Fig. 6. It can be
observed from Fig.6(a) that the proposed consensus and incentive mechanism,
together with the operation on DAO boosts the speed of achieving consensus
significantly. In addition, by operating the DAO reiteratively, the
convergence speed is able to improve around 20% comparing to the DAO-based
control without such an operation.

It is observed from Fig. 6(b) that the reached consensus on the ECE is
significantly improved by applying the proposed integrated control method.
Additionally, the proposed method achieved a significant improvement in the
accumulated value of local objectives by around 50% comparing to the PoS-based
control. Although a compromise of the overall ECE can be observed when
applying the proposed method, the compromise is negligible.

The evaluation results suggest that the cooperation and synchronization of
heterogeneous agents in ITS can be extensively achieved by the proposed
consensus and incentive mechanism. Specifically, the local objectives can be
significantly improved without compromising global objectives, which is
realized through balancing the ECE. The performance can be further improved by
operating the DAO, which suggests that the capability of DAO-based control in
ITS can be enhanced via the proposed operation method. Consequently, the
frequency of altering code in the deployed smart contract or deploying new
smart contracts can be ultimately reduced with a growing complexity of ITS.

##  V CONCLUSION

This paper presents a DAO-based control method that can realize an integrated
control of plentiful heterogeneous agents in a complex system, which is
applicable in ITS. Specifically, a consensus and incentive control protocol
and an operation algorithm on DAO are proposed. By extending the PoS based
consensus control protocol, the proposed control protocol aims to increase the
cumulative energy consumption efficiency of all intelligent agents within the
system. With the consideration of incentives from the rewards of approaching a
consensus and the ’support/objection’ from connected agents, the system is
able to achieve a consensus with improved local objectives. In addition, a
heuristic algorithm is proposed to identify critical agents within DAO, in
order to extend the capability of the proposed control protocol.

The experiment results show that the controlled system can achieve a faster
consensus, resulting in a higher improved value of local objectives with minor
compromise in the global objective, compared to existing decentralized control
methods. These results indicate that the proposed method can be applied in
practice, with an extensive significance in improving the control performance
and extending the control capability of DAO. Moreover, the proposed DAO
operation algorithm can be adopted as a quantitative measurement of the DAO-
based control’s capability, which shows its potential as a guidance in
developing the DAO-based integrated control system with ITS becoming growing
complex.

However, the proposed method is examined by a numerical experiment, whereas
its performance in practice needs to be evaluated in a traffic simulation
platform, with the consideration of several practical issues, such as
communication failures. Further, the proposed method is based on the
assumption of static connection intensities known a-priori, which can be time-
variant in practice. These issues can be further investigated by future
research includes applying the proposed method in a traffic simulation
platform, as well as developing a distributed learning algorithm to obtain the
time-variant adjacent matrix in ITS.

## References

  * [1] Y. Han, A. Hegyi, L. Zhang, Z. He, E. Chung, and P. Liu, “A new reinforcement learning-based variable speed limit control approach to improve traffic efficiency against freeway jam waves,” _Transportation research part C: emerging technologies_ , vol. 144, p. 103900, 2022.
  * [2] L. Li, Y. Lv, and F.-Y. Wang, “Traffic signal timing via deep reinforcement learning,” _IEEE/CAA Journal of Automatica Sinica_ , vol. 3, no. 3, pp. 247–254, 2016.
  * [3] F. Belletti, D. Haziza, G. Gomes, and A. M. Bayen, “Expert level control of ramp metering based on multi-task deep reinforcement learning,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 19, no. 4, pp. 1198–1207, 2017.
  * [4] C. Diakaki, M. Papageorgiou, I. Papamichail, and I. Nikolos, “Overview and analysis of vehicle automation and communication systems from a motorway traffic management perspective,” _Transportation Research Part A: Policy and Practice_ , vol. 75, pp. 147–165, 2015.
  * [5] D. H. Wolpert and K. Tumer, “Collective intelligence, data routing and braess’ paradox,” _Journal of Artificial Intelligence Research_ , vol. 16, pp. 359–387, 2002.
  * [6] L. Baskar, B. De Schutter, and H. Hellendoorn, “Decentralized traffic control and management with intelligent vehicles,” in _Proceedings of the 9th TRAIL Congress, Delft, The Netherlands_ , vol. 1, 2006, p. 3.
  * [7] Q. Miao, W. Zheng, Y. Lv, M. Huang, W. Ding, and F.-Y. Wang, “DAO to HANOI via DeSci: AI Paradigm Shifts from AlphaGo to ChatGPT,” _IEEE/CAA Journal of Automatica Sinica_ , vol. 10, no. 4, pp. 877–897, Apr. 2023.
  * [8] M. B. Mollah, J. Zhao, D. Niyato, Y. L. Guan, C. Yuen, S. Sun, K.-Y. Lam, and L. H. Koh, “Blockchain for the Internet of Vehicles Towards Intelligent Transportation Systems: A Survey,” _IEEE Internet of Things Journal_ , vol. 8, no. 6, pp. 4157–4185, Mar. 2021.
  * [9] U. W. Chohan, “The decentralized autonomous organization and governance issues,” _Available at SSRN 3082055_ , 2017.
  * [10] F.-Y. Wang, “The DAO to MetaControl for MetaSystems in Metaverses: The System of Parallel Control Systems for Knowledge Automation and Control Intelligence in CPSS,” _IEEE/CAA Journal of Automatica Sinica_ , vol. 9, no. 11, pp. 1899–1908, Nov. 2022.
  * [11] S. Feng, H. Sun, X. Yan, H. Zhu, Z. Zou, S. Shen, and H. X. Liu, “Dense reinforcement learning for safety validation of autonomous vehicles,” _Nature_ , vol. 615, no. 7953, pp. 620–627, Mar. 2023.
  * [12] V. Sharma, “An energy-efficient transaction model for the blockchain-enabled internet of vehicles (iov),” _IEEE Communications Letters_ , vol. 23, no. 2, pp. 246–249, 2018.
  * [13] S. Yao and B. Friedrich, “Managing connected and automated vehicles in mixed traffic by human-leading platooning strategy: a simulation study,” in _2019 IEEE Intelligent Transportation Systems Conference (ITSC)_. IEEE, 2019, pp. 3224–3229.
  * [14] V. Milanés, S. E. Shladover, J. Spring, C. Nowakowski, H. Kawazoe, and M. Nakamura, “Cooperative adaptive cruise control in real traffic situations,” _IEEE Transactions on intelligent transportation systems_ , vol. 15, no. 1, pp. 296–305, 2013.
  * [15] H. Shi, Y. Zhou, K. Wu, X. Wang, Y. Lin, and B. Ran, “Connected automated vehicle cooperative control with a deep reinforcement learning approach in a mixed traffic environment,” _Transportation Research Part C: Emerging Technologies_ , vol. 133, p. 103421, 2021.
  * [16] M. Liu, S. Hoogendoorn, and M. Wang, “Receding horizon cooperative platoon trajectory planning on corridors with dynamic traffic signal,” _Transportation Research Record_ , vol. 2674, no. 12, pp. 324–338, 2020.
  * [17] C. Roncoli, M. Papageorgiou, and I. Papamichail, “Traffic flow optimisation in presence of vehicle automation and communication systems–part ii: Optimal control for multi-lane motorways,” _Transportation Research Part C: Emerging Technologies_ , vol. 57, pp. 260–275, 2015.
  * [18] A. Fares, M. A. Khamis, and W. Gomaa, “Marl-fwc: Optimal coordination of freeway traffic control measures,” in _The 8th International Conference on Advanced Machine Learning and Technologies and Applications (AMLTA2022)_. Springer, 2022, pp. 643–656.
  * [19] R. A. Shet and S. Yao, “Cooperative driving in mixed traffic: An infrastructure-assisted approach,” _IEEE Open Journal of Intelligent Transportation Systems_ , vol. 2, pp. 429–447, 2021.
  * [20] G. Raja, K. Kottursamy, K. Dev, R. Narayanan, A. Raja, and K. B. V. Karthik, “Blockchain-integrated multiagent deep reinforcement learning for securing cooperative adaptive cruise control,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 23, no. 7, pp. 9630–9639, 2022.
  * [21] F.-Y. Wang, “Parallel control and management for intelligent transportation systems: Concepts, architectures, and applications,” _IEEE transactions on intelligent transportation systems_ , vol. 11, no. 3, pp. 630–638, 2010.
  * [22] I. Gerostathopoulos and E. Pournaras, “Trapped in traffic? a self-adaptive framework for decentralized traffic optimization,” in _2019 IEEE/ACM 14th International Symposium on Software Engineering for Adaptive and Self-Managing Systems (SEAMS)_. IEEE, 2019, pp. 32–38.
  * [23] D. Li and Z. Hou, “Perimeter control of urban traffic networks based on model-free adaptive control,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 22, no. 10, pp. 6460–6472, 2020.
  * [24] S. Hassan and P. De Filippi, “Decentralized Autonomous Organization,” _Internet Policy Review_ , vol. 10, no. 2, Apr. 2021.
  * [25] S. Wang, W. Ding, J. Li, Y. Yuan, L. Ouyang, and F.-Y. Wang, “Decentralized autonomous organizations: Concept, model, and applications,” _IEEE Transactions on Computational Social Systems_ , vol. 6, no. 5, pp. 870–878, 2019\.
  * [26] Z. Zhu, X. Wang, Y. Zhao, S. Qiu, Z. Liu, B. Chen, and F.-Y. Wang, “Crowdsensing Intelligence by Decentralized Autonomous Vehicles Organizations and Operations,” _IEEE Transactions on Intelligent Vehicles_ , vol. 7, no. 4, pp. 804–808, Dec. 2022.
  * [27] Q. Ren, K. L. Man, M. Li, B. Gao, and J. Ma, “Intelligent design and implementation of blockchain and Internet of things–based traffic system,” _International Journal of Distributed Sensor Networks_ , vol. 15, no. 8, p. 155014771987065, Aug. 2019.
  * [28] C. Bellavitis, C. Fisch, and P. P. Momtaz, “The rise of decentralized autonomous organizations (DAOs): A first empirical glimpse,” _Venture Capital_ , vol. 25, no. 2, pp. 187–203, Apr. 2023.
  * [29] Q. DuPont, “Experiments in Algorithmic Governance: A History and Ethnography of ’The DAO,’ a failed Decentralized Autonomous Organization,” _Bitcoin and beyond_ , 2017.
  * [30] Y. Xie and Z. Lin, “Global optimal consensus for multi-agent systems with bounded controls,” _Systems & Control Letters_, vol. 102, pp. 104–111, 2017.
  * [31] W. Al Amiri, M. Baza, K. Banawan, M. Mahmoud, W. Alasmary, and K. Akkaya, “Privacy-preserving smart parking system using blockchain and private information retrieval,” in _2019 international conference on smart applications, communications and networking (SmartNets)_. IEEE, 2019, pp. 1–6.
  * [32] Y. Wang, Z. Su, and N. Zhang, “Bsis: Blockchain-based secure incentive scheme for energy delivery in vehicular energy network,” _IEEE Transactions on Industrial Informatics_ , vol. 15, no. 6, pp. 3620–3631, 2019.
  * [33] J. Zhu, H. Deng, X. Li, Y. Yuan, and F. Wang, “Blockchain-based Consensus Study on Distributed Control Systems,” in _2021 IEEE 1st International Conference on Digital Twins and Parallel Intelligence (DTPI)_. Beijing, China: IEEE, July 2021, pp. 164–167.