aainstitutetext: Centre for Cosmology, Particle Physics and Phenomenology
(CP3), Universite Catholique de Louvain

# Impact of Dimension-8 SMEFT operators on Diboson Productions

Céline Degrande a Hao-Lin Li<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We for the first time identify all the dimension-8 (dim-8) SMEFT operators
that can have an interference with the SM with $E^{4}/\Lambda^{4}$ enhancement
in the high energy limit for the processes $q\bar{q}\to WW/WZ$. Our results
therefore explicitly show that the non-interference observed for the
dimension-six does not extend to dimension-eight. We compute the contributions
of those dimension-8 operators to the cross-section at the 14 TeV Large Hadron
Collider and compare the results with dimension-6 (dim-6) originated
corrections at the order of dim-6-SM interference and dim-6 squared. We find
one (two) dim-8 operator(s) can generate amplitudes of a similar order of
magnitude compared with their dim-6 squared counterparts assuming unity
dimensionless Wilson coefficients. During the study, new non-interference
scenarios are found due to the selection rule of angular momentum as well as
strong suppression due to the symmetric initial state for the proton-proton
collider.

††preprint: IRMP-CP3-23-15

##  1 Introduction

Since no undisputed evidence of new physics (NP) has been found at the Large
Hadron Collider (LHC) since the discovery of the Higgs, precision measurements
have become a major goal for the High-Luminosity LHC. In the hypothesis where
the non-observation of NP signal is due to the relatively high NP scale, the
Standard Model effective field theory (SMEFT) provides a suitable framework to
parameterize deviations from the SM predictions according to the decoupling
theorem Appelquist:1974tg . In the SMEFT framework, the NP effects are encoded
in the Wilson coefficients of an infinite series of higher dimensional
operators that are constructed with the SM field contents and obey the SM
gauge symmetries:

$\displaystyle{\cal L}_{\rm SMEFT}={\cal L}_{\rm
SM}+\sum_{d,i}\frac{C^{d}_{i}}{\Lambda^{d-4}}{\cal O}^{(d)}_{i},$ (1)

where $\Lambda$ is the characteristic NP scale, $d$ corresponds to the mass
dimension of the operator starting from 5. By simple dimensional analysis, one
insertion of dimension-$d$ SMEFT operators in the Feynman diagram will
suppress the amplitudes by $(E/\Lambda)^{(d-4)}$, with $E$ an energy scale of
the considered process. Therefore, below $\Lambda$, the leading NP effects are
expected to start from the lowest dimensional operators. Complete and
independent sets of dimension-5 (dim-5) and dimension-6 (dim-6) operators are
found in the Ref. Weinberg:1979sa  and Grzadkowski:2010es . Recently, the
complete and independent set of dimension-8 (dim-8) operators has been found
by the traditional method Murphy:2020rsh  and by the systematic Young tableau
method Li:2020gnx . Therefore in the SMEFT framework, the amplitude of any
process can be expanded in the following form:

$\displaystyle{\cal A}={\cal A}_{\rm
SM}+\sum_{i}\frac{C^{(6)}_{i}}{\Lambda^{2}}{\cal
A}_{i}^{(6)}+\sum_{j}\frac{C^{(8)}_{j}}{\Lambda^{4}}{\cal
A}_{j}^{(8)}+\sum_{i,j}\frac{C^{(6)}_{i}C^{(6)}_{j}}{\Lambda^{4}}{\cal
A}_{ij}^{(8)^{\prime}}+\dots,$ (2)

where we have neglected contributions from dim-5 and dim-7 operators which
violate lepton or baryon numbers and are not our focus in this work. We also
explicitly factor out the dependence on the Wilson coefficients and the NP
scale $\Lambda$ in front of ${\cal A}^{(6)}$, ${\cal A}^{(8)}$ and ${\cal
A}^{(8)^{\prime}}$, where ${\cal A}^{(8)^{\prime}}$ represents the amplitude
that contains two insertions of dim-6 operators and behaves like a dim-8
amplitude.

However, physical observables are obtained by amplitudes squared rather than
amplitudes, in this sense, we can expand observables such as scattering cross-
section with the inverse power of $\Lambda$ as well:

$\displaystyle d\sigma$ $\displaystyle\sim|{\cal A}_{\rm
SM}|^{2}+\sum_{i}\frac{C^{(6)}_{i}}{\Lambda^{2}}2\operatorname{Re}\left[{\cal
A}_{i}^{(6)}{\cal A}^{*}_{\rm
SM}\right]+\sum_{i,j}\frac{C^{(6)}_{i}C^{(6)*}_{j}}{\Lambda^{4}}{\cal
A}^{(6)}_{i}{\cal A}^{(6)^{*}}_{j}$ (3)
$\displaystyle+\sum_{i}\frac{C^{(8)}_{i}}{\Lambda^{4}}2\operatorname{Re}\left[{\cal
A}_{i}^{(8)}{\cal A}^{*}_{\rm
SM}\right]+\sum_{i,j}\frac{C^{(6)*}_{i}C^{(6)*}_{j}}{\Lambda^{4}}2\operatorname{Re}\left[{\cal
A}^{(8)^{\prime}}{\cal A}^{*}_{\rm SM}\right]+\dots.$

From the above formula one can find that the leading contribution to the
observable is coming from the interference between the dim-6 and the SM
amplitudes provided that ${\cal A}_{\rm SM}$ is non-zero, then the dim-6
squared amplitude and the interference between dim-8 or double insertion of
dim-6 interactions and SM amplitudes are at the same order in terms of
expansion of inverse power of $\Lambda$. It is pointed out in Ref.
Azatov:2016sqh  that for a large class of 4-point amplitudes in the high
energy limit, the interference between the dim-6 and SM is suppressed by the
helicity selection rule originated from the supersymmetric property of the SM
amplitude. Consequently, the next-to-leading order corrections including the
interference between dim-8 and SM amplitudes are expected to have a
significant impact on theoretical predictions of observables in the high
energy limit. The reader may be confused by the simultaneous use of the high
energy limit and the SMEFT framework, we clarify here that, the high energy
limit should be understood as the energy regime that is large compared with
the SM electroweak scale but still relatively small compared with the NP
scale. This is closely related to the validity problem of using the EFT
framework in collider physics, where the center of mass energy of the
scattering process in the experiments should not be higher than the heavy NP
scale $\Lambda$. At hadron colliders, this hypothesis is hard to check
especially in the high energy tails of distributions. However, the fact that
such processes/tails are strongly suppressed by the parton luminosity
functions helps. In fact, in the SMEFT framework, the parton level NP effects
are more prominent at higher energy, while in the meantime, theoretical
predictions in such a regime suffer from a relatively large uncertainty due to
the EFT expansion, so it is an art to balance these two competing effects in
order to quantify the sensitivity of the LHC Farina:2016rws . The community
came up with several proposals to evaluate the EFT validity problems, but no
common agreement has been reached Brivio:2022pyi . In Ref. Cohen:2021gdw , a
simplified model has been studied to demonstrate the EFT validity and the
corresponding unitarity bound when taking into account the parton distribution
function. The validity and consistency issues and the related dimension-8
effect of concrete NP models such as two Higgs doublet model and vector-like
quark model have also been studied in Ref. Dawson:2021xei ; Dawson:2022cmu .
In this paper, we study the validity issues within the SMEFT framework and
quantitatively investigate the dim-8 effects on the diboson production at the
LHC.

The dim-6 correction to the diboson production at the LHC has been extensively
studied in the literature Butter:2016cvz ; Green:2016trm ; Zhang:2016zsp ;
Baglio:2017bfe ; Ellis:2018gqa ; Baglio:2018bkm ; Baglio:2019uty ;
Baglio:2020oqu ; Bellan:2021dcy ; Almeida:2021asy , while the study of the
effect of dim-8 operators in the SMEFT framework is still limited. For
example, Refs. Degrande:2013kka ; Bellazzini:2018paj  studied the dim-8 SMEFT
contribution to the process $q\bar{q}\to ZZ/Z\gamma$, and Ref.
Bellazzini:2018paj  also discussed the UV origins of the relevant dim-8
operators. Ref. Azatov:2017kzw  estimates a part of the dim-8 contribution to
$q\bar{q}\to WW/WZ$ and compares it with the dim-6 originated contribution,
but no complete operator basis is mentioned. After the discovery of the
complete and independent dim-8 operator basis, no comprehensive study of the
dim-8 correction to the diboson production of $WW/WZ$ channel exists,
therefore in this work we find out all dim-8 operators that generate contact
interactions for the production channels $q\bar{q}\to WW/WZ$ and are able to
induce an enhancement of $E^{4}/\Lambda^{4}$ in the amplitude in the high
energy regime. We find that the operator ${\cal O}_{10}$ in eq. (13) has a
significant impact on the cross-section of $WW$ production compared with the
dim-6 contribution from triple gauge boson operator ${\cal O}_{W}$, while
${\cal O}_{12}$ and ${\cal O}_{13}$ in general have non-negligible
contributions to the cross-section of $WZ$ final state. Through the analysis
of interference amplitudes, we also find new non-interference scenarios that
are caused by the angular momentum selection rule and the scenario where the
angular differential cross-section gets suppressed because the $q\bar{q}$ and
$\bar{q}q$ initiated channels strongly cancel each other in the proton-proton
(pp) collider.

Our paper is organized as follows. In section 2, we identify all the dim-8
operators contributing contact diagrams in the quark-initiated diboson
production. In section 3, we describe three different classes of non-
interference that appear in the study of SMEFT contribution to diboson
production. In section 4, we present the analytical and numerical results for
the dim-8 contribution to the diboson production at LHC and compare their
effects to the dim-6 originated corrections. Finally, in section 5, we make a
conclusion and point out several possible future directions following this
work.

##  2 The operators

For dim-6 SMEFT operators, corrections to the diboson production occur through
their modification to triple gauge couplings (TGC) and to quark gauge
interactions as shown in figure 1(a). However, starting from dim-8, the
contact interaction for $q\bar{q}\to VV$ appears as in figure 1(b). Moreover,
among the complete and independent dim-8 operators given in Ref. Li:2020gnx ,
we find that only operators that can generate contact interactions for diboson
processes can maximize the energy dependence of amplitudes to $E^{4}$ in the
high energy regime, and corrections to TGC and quark gauge couplings from
dim-8 operators all need insertions of the Higgs vacuum expectation value
(vev), which lowers the power of energy present in the amplitude by
dimensional analysis. For this reason, we focus on the dim-8 operators that
can generate contact interaction and study their interference effect with the
SM amplitude in diboson production.

(a) dim-6 vertex corrections

(b) dim-8 contact corrections

Figure 1: Corrections that contribute to diboson processes from SMEFT
operators.

Following the notation in the package ABC4EFT Li:2022tec , we first identify
three dim-8 Lorentz classes that are capable of generating $q\bar{q}\to VV$
contact interactions — $F_{L}F_{R}\psi\psi^{\dagger}D$,
$F_{L}F_{L}\psi\psi^{\dagger}D$111Here we only list independent complex
Lorentz classes, the Lorentz class $F_{R}F_{R}\psi\psi^{\dagger}D$ should be
thought of as the conjugate Lorentz class of $F_{L}F_{L}\psi\psi^{\dagger}D$.
The same concept applies to the operator level in eq. (5). and
$\psi\psi^{\dagger}\phi^{2}D^{3}$, where $\psi$ and $\psi^{\dagger}$ represent
left-handed and right-handed fermion fields, $\phi$ is a scalar corresponding
to the Higgs doublet or its conjugate in the SMEFT. $F_{L}$ and $F_{R}$ are
gauge boson field strength tensors that transform as $(2,0)$ and $(0,2)$
representations of the Lorentz group, and they are related to the ordinary
field strength tensor $F^{\mu\nu}$ in following formulae:

$\displaystyle F_{L/R}^{\mu\nu}=\frac{1}{2}\left(F^{\mu\nu}\mp
i\tilde{F}^{\mu\nu}\right),\quad\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}.$
(4)

The reason that $\psi\psi^{\dagger}\phi^{2}D^{3}$ can generate an enhancement
of $E^{4}/\Lambda^{4}$ in the high energy regime can be more easily understood
in the Feynman gauge, where additional derivatives acting on the Goldstones in
the Higgs doublet increases the power of energy dependence in the amplitude.
This observation also tells us that the enhancement must occur in the
longitudinal-longitudinal component of the helicity amplitude as we will
confirm in section 4.

After filling the $F$’s and $\psi^{(\dagger)}$’s with the concrete fields in
the SM, we obtain the following independent operators for Lorentz classes
$F_{L}F_{R}\psi\psi^{\dagger}D$, and $F_{L}F_{L}\psi\psi^{\dagger}D$:

$\displaystyle\mathcal{Q}_{1}=iB_{R}^{\lambda\mu}B_{L}{}_{\lambda\nu}(d_{{}_{\mathbb{C}}}{}_{p}\sigma_{\mu}\overleftrightarrow{D}^{\nu}d_{{}_{\mathbb{C}}}^{\dagger}{}_{r})$
$\displaystyle\mathcal{Q}_{2}=iB_{R}^{\lambda\mu}B_{L}{}_{\lambda\nu}(u_{{}_{\mathbb{C}}}{}_{p}\sigma_{\mu}\overleftrightarrow{D}^{\nu}u_{{}_{\mathbb{C}}}^{\dagger}{}_{r})$
$\displaystyle\mathcal{Q}_{3}=iB_{R}^{\lambda\mu}B_{L}{}_{\lambda\nu}(Q_{p}\sigma_{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}_{r})$
$\displaystyle\mathcal{Q}_{4}=i\tau^{Ii}_{j}W_{R}^{I\lambda\mu}B_{L}{}_{\lambda\nu}(Q_{pi}\sigma_{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{rj})$
$\displaystyle\mathcal{Q}_{5}=i\tau^{Ii}_{j}W^{I\lambda\mu}_{L}B_{L\lambda\nu}(Q_{pi}\sigma_{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{rj})$
$\displaystyle\mathcal{Q}_{6}=iW_{R}^{I\lambda\mu}W^{I}_{L\lambda\nu}(d_{{}_{\mathbb{C}}}{}_{p}\sigma_{\mu}\overleftrightarrow{D}^{\nu}d_{{}_{\mathbb{C}}}^{\dagger}{}_{r})$
$\displaystyle\mathcal{Q}_{7}=iW_{R}^{I\lambda\mu}W^{I}_{L\lambda\nu}(u_{{}_{\mathbb{C}}}{}_{p}\sigma_{\mu}\overleftrightarrow{D}^{\nu}u_{{}_{\mathbb{C}}}^{\dagger}{}_{r})$
$\displaystyle\mathcal{Q}_{8}=iW_{R}^{I\lambda\mu}W^{I}_{L\lambda\nu}(Q_{p}\sigma_{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{r})$
$\displaystyle\mathcal{Q}_{9}=i\epsilon^{IJK}\tau^{Ki}_{j}W_{R}^{I\lambda\mu}W^{J}_{L\lambda\nu}(Q_{pi}\sigma_{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{rj})$
$\displaystyle\mathcal{Q}_{10}=i\epsilon^{IJK}\tau^{Ki}_{j}W_{L}^{I\lambda\mu}W^{J}_{L\lambda\nu}(Q_{pi}\sigma_{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{rj}),$
(5)

where $W$ and $B$ are the gauge field strength tensors of $SU(2)_{L}$ and
$U(1)_{Y}$ gauge group, respectively, the double arrow on the covariant
derivative is defined as $X\overleftrightarrow{D}Y=X(DY)-(DX)Y$, $Q$ is the
left-handed quark doublets, $d_{{}_{\mathbb{C}}}$ and $u_{{}_{\mathbb{C}}}$
are charge conjugate of the right-handed down-type and up-type quarks, which
also transform as left-handed fermion fields. All these fermion fields are in
the forms of 2-component Weyl spinors, and their relations to the
corresponding 4-component Dirac fields are manifested in appendix A. In this
way, one can find the transformations between the 2-component and 4-component
fermion tensors in eq. (5) in the following formulae:

$\displaystyle J^{\mu\nu}_{d,pr}\equiv
i(d_{{}_{\mathbb{C}}}{}_{p}\sigma^{\mu}\overleftrightarrow{D}^{\nu}d_{{}_{\mathbb{C}}}^{\dagger}{}_{r})=i\bar{d}_{{\rm
R}p}\gamma^{\mu}\overleftrightarrow{D}^{\nu}d_{{\rm R}r},$ $\displaystyle
J^{\mu\nu}_{u,pr}\equiv
i(u_{{}_{\mathbb{C}}}{}_{p}\sigma^{\mu}\overleftrightarrow{D}^{\nu}u_{{}_{\mathbb{C}}}^{\dagger}{}_{r})=i\bar{u}_{{\rm
R}p}\gamma^{\mu}\overleftrightarrow{D}^{\nu}u_{{\rm R}r},$ $\displaystyle
J^{\mu\nu,I}_{(Q,3),rp}\equiv
i\tau^{Ii}_{j}(Q_{pi}\sigma^{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{rj})=i\tau^{Ii}_{j}\bar{q}_{{\rm
L}r}^{j}\gamma^{\mu}\overleftrightarrow{D}^{\nu}q_{{\rm L}pi},$ $\displaystyle
J^{\mu\nu}_{(Q,1),rp}\equiv
i(Q_{pi}\sigma^{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{ri})=i\bar{q}_{{\rm
L}r}^{i}\gamma^{\mu}\overleftrightarrow{D}^{\nu}q_{{\rm L}pi}.$ (6)

Pay attention that for the last equation, the two fermion fields exchange
their positions on the right-hand side. One can also verify that the
conjugates of these tensor currents are equivalent to swapping the flavor
indices of the fermions, i.e. $J^{\mu\nu\dagger}_{f,pr}=J^{\mu\nu}_{f,rp}$,
and for the special case $p=r$ the fermion tensor is hermitian. In this work,
for simplicity, we only focus on the diagonal flavor components which have the
largest interference with the SM as it is not suppressed by CKM.

The bosonic parts of the operators in eq. (5) are expressed in terms of the
chiral eigenstates $F_{L/R}$ for the sake of the advantage in enumerating the
independent Lorentz structures using the Young tableau method. However, it is
better for the phenomenological purpose to convert them into the two real
degrees of freedom $F$ and $\tilde{F}$, such that the construction of a
Hermitian Lagrangian is more convenient. To achieve this goal, the following
identities involving two field strength tensors are useful:

$\displaystyle F_{\rm{L}\mu\rho}F_{\rm{R}}{}^{\rho\nu}$
$\displaystyle=\frac{1}{2}F_{\mu\rho}F^{\rho\nu}+\frac{1}{8}F^{2}\delta^{\nu}_{\mu}$
(7)
$\displaystyle\epsilon^{IJK}F^{I}_{\rm{L}\mu\rho}F^{J}_{\rm{R}}{}^{\rho\nu}$
$\displaystyle=\epsilon^{IJK}\frac{1}{4}\left(iF^{I}{}^{\nu\rho}\tilde{F}^{J}_{\rho\mu}+iF^{I}_{\mu\rho}\tilde{F}^{J\rho\nu}\right),$
(8) $\displaystyle F_{1\rm{L}\mu\rho}F_{2\rm{R}}{}^{\rho\nu}$
$\displaystyle=\frac{1}{4}\left\\{F_{1\mu\rho}F_{2}{}^{\rho\nu}+F_{2\mu\rho}F_{1}{}^{\rho\nu}+\frac{1}{2}[(F_{1}\cdot
F_{2})-i(F_{1}\cdot\tilde{F}_{2})]\delta^{\nu}_{\mu}\right.$ (9)
$\displaystyle\left.-i(\tilde{F}_{1}^{\nu\rho}{F}_{2\rho\mu}+\tilde{F}_{1\mu\rho}F_{2}^{\rho\nu})\right\\},$
$\displaystyle F_{1\rm{L}\mu\rho}F_{2\rm{L}}{}^{\rho\nu}$
$\displaystyle=\frac{1}{4}\left(F_{1\mu\rho}F_{2}{}^{\rho\nu}-F^{\nu\rho}_{1}F_{2\rho\mu}+iF_{1}{}^{\nu\rho}\tilde{F}_{2\rho\mu}-iF_{1\mu\rho}\tilde{F}_{2}{}^{\rho\nu}\right.$
(10) $\displaystyle\left.-\frac{1}{2}[(F_{1}\cdot
F_{2})-i(F_{1}\cdot\tilde{F}_{2})]\delta^{\nu}_{\mu}\right)\;,$ $\displaystyle
F_{1\rm{R}\mu\rho}F_{2\rm{R}}{}^{\rho\nu}$
$\displaystyle=\frac{1}{4}\left(2F_{1\mu\rho}F_{2}{}^{\rho\nu}-F^{\nu\rho}_{1}F_{2\rho\mu}-iF_{1}{}^{\nu\rho}\tilde{F}_{2\rho\mu}+iF_{1\mu\rho}\tilde{F}_{2}{}^{\rho\nu}\right.$
(11) $\displaystyle\left.-\frac{1}{2}[(F_{1}\cdot
F_{2})+i(F_{1}\cdot\tilde{F}_{2})]\delta^{\nu}_{\mu}\right)\;.$

For example, using eq. (7), the $B^{\lambda\mu}_{R}B_{L\lambda\nu}$ in ${\cal
Q}_{1-3}$ and $W^{I\lambda\mu}_{R}W^{I}_{L\lambda\nu}$ in ${\cal Q}_{6-8}$
yield two different substructures for two identical gauge bosons $B$ and
$W^{I}$ respectively, while the second term on the right-hand side of eq. (7)
proportional to $\delta_{\nu}^{\mu}$ contracts the $\sigma_{\mu}D^{\nu}$
forming an EOM of the corresponding fermions, which can be converted to the
operator of other types.

For the Lorentz class $\psi\psi^{\dagger}\phi^{2}D^{3}$, we have following
operators in 2-component Weyl fermion notation:

$\displaystyle{\cal
Q}_{11}=i(u_{{}_{\mathbb{C}}p}\sigma^{\mu}\overleftrightarrow{D}^{\nu}u_{{}_{\mathbb{C}}}^{\dagger}{}^{r})(D^{\mu}H^{\dagger}D^{\nu}H),$
$\displaystyle{\cal
Q}_{12}=i(d_{{}_{\mathbb{C}}p}\sigma^{\mu}\overleftrightarrow{D}^{\nu}d_{{}_{\mathbb{C}}}^{\dagger}{}^{r})(D^{\mu}H^{\dagger}D^{\nu}H),$
$\displaystyle{\cal
Q}_{13}=i(Q_{p}\sigma^{\mu}\overleftrightarrow{D}^{\nu}Q^{\dagger}{}^{r})(D^{\mu}H^{\dagger}D^{\nu}H),$
$\displaystyle{\cal
Q}_{14}=i(Q_{pi}\sigma^{\mu}\overleftrightarrow{D}^{\nu}d_{{}_{\mathbb{C}}}^{\dagger}{}^{rj})(D^{\mu}H^{\dagger
i}D^{\nu}H_{j}),$ $\displaystyle{\cal
Q}_{15}=i(u_{{}_{\mathbb{C}}p}\sigma^{\mu}\overleftrightarrow{D}^{\nu}d_{{}_{\mathbb{C}}}^{\dagger}{}^{r})\epsilon^{ij}(D^{\mu}H_{i}D^{\nu}H_{j}),$
(12)

where the last one, ${\cal Q}_{15}$, is an operator of the complex type for
which the conjugated operator counts as an independent operator.

Using identities in eq. (6)-(11) and the Fierz identity of the $SU(N)$ gauge
group, the operators in eq. (5) and (12), and their complex conjugate can be
converted to the following independent operators:

$\displaystyle\mathcal{O}_{1}=iB^{\mu}{}{}_{\nu}B^{\nu}{}{}_{\lambda}(\bar{d}_{{\rm
R}p}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}d_{{\rm R}r}),$
$\displaystyle\mathcal{O}_{2}=iB^{\mu}{}{}_{\nu}B^{\nu}{}{}_{\lambda}(\bar{u}_{{\rm
R}p}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}u_{{\rm R}r}),$
$\displaystyle\mathcal{O}_{3}=iB^{\mu}{}{}_{\nu}B^{\nu}{}{}_{\lambda}\left(\bar{q}_{{\rm
L}p}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}q_{{\rm L}r}\right),$
$\displaystyle\mathcal{O}_{4}=iW^{I\mu}{}_{\lambda}B^{\nu\lambda}\left(\bar{q}^{i}_{{\rm
L}p}\gamma_{\nu}\left(\tau^{I}\right)_{i}{}^{j}\overleftrightarrow{D}_{\mu}q_{{\rm
L}rj}\right),$
$\displaystyle\mathcal{O}_{5}=iW^{I\mu}{}_{\lambda}\tilde{B}^{\nu\lambda}\left(\bar{q}^{i}_{{\rm
L}p}\gamma_{\nu}\left(\tau^{I}\right)_{i}{}^{j}\overleftrightarrow{D}_{\mu}q_{{\rm
L}rj}\right),$
$\displaystyle\mathcal{O}_{6}=iW^{I\nu}{}_{\lambda}B^{\mu\lambda}\left(\bar{q}^{i}_{{\rm
L}p}\gamma_{\nu}\left(\tau^{I}\right)_{i}{}^{j}\overleftrightarrow{D}_{\mu}q_{{\rm
L}rj}\right),$
$\displaystyle\mathcal{O}_{7}=iW^{I\nu}{}_{\lambda}\tilde{B}^{\mu\lambda}\left(\bar{q}^{i}_{{\rm
L}p}\gamma_{\nu}\left(\tau^{I}\right)_{i}{}^{j}\overleftrightarrow{D}_{\mu}q_{{\rm
L}rj}\right),$
$\displaystyle\mathcal{O}_{8}=iW^{I}{}^{\mu}{}{}_{\nu}W^{I}{}^{\nu}{}{}_{\lambda}(\bar{d}_{{\rm
R}p}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}d_{{\rm R}r}),$
$\displaystyle\mathcal{O}_{9}=iW^{I}{}^{\mu}{}{}_{\nu}W^{I}{}^{\nu}{}{}_{\lambda}(\bar{u}_{{\rm
R}p}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}u_{{\rm R}r}),$
$\displaystyle\mathcal{O}_{10}=iW^{I}{}^{\mu}{}{}_{\nu}W^{I}{}^{\nu}{}{}_{\lambda}\left(\bar{q}_{{\rm
L}r}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}q_{{\rm L}p}\right),$
$\displaystyle\mathcal{O}_{11}=i\epsilon^{IJK}W^{I}{}^{\mu}{}{}_{\nu}W^{J}{}^{\nu}{}{}_{\lambda}\left(\bar{q}^{i}_{{\rm
L}p}\gamma^{\lambda}\left(\tau^{K}\right)_{i}{}^{j}\overleftrightarrow{D}_{\mu}q_{{\rm
L}rj}\right),$
$\displaystyle\mathcal{O}_{12}=i\epsilon^{IJK}\tilde{W}^{I}{}^{\mu}{}{}_{\nu}W^{J}{}^{\nu}{}{}_{\lambda}\left(\bar{q}^{i}_{{\rm
L}p}\gamma^{\lambda}\left(\tau^{K}\right)_{i}{}^{j}\overleftrightarrow{D}_{\mu}q_{{\rm
L}rj}\right),$
$\displaystyle\mathcal{O}_{13}=i\epsilon^{IJK}W^{I}{}^{\mu}{}{}_{\nu}\tilde{W}^{J}{}^{\nu}{}{}_{\lambda}\left(\bar{q}^{i}_{{\rm
L}p}\gamma^{\lambda}\left(\tau^{K}\right)_{i}{}^{j}\overleftrightarrow{D}_{\mu}q_{{\rm
L}rj}\right),$ $\displaystyle\mathcal{O}_{14}=i\left(\bar{u}_{{\rm
R}r}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}u_{{\rm
R}p}\right)\left(D_{\lambda}H^{\dagger}D^{\mu}H\right),$
$\displaystyle\mathcal{O}_{15}=i\left(\bar{d}_{{\rm
R}r}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}d_{{\rm
R}p}\right)\left(D_{\lambda}H^{\dagger}D^{\mu}H\right),$
$\displaystyle\mathcal{O}_{16}=i\left(\bar{q}_{{\rm
L}r}\gamma^{\lambda}\overleftrightarrow{D}_{\mu}q_{{\rm
L}p}\right)\left(D_{\lambda}H^{\dagger}D^{\mu}H\right),$
$\displaystyle\mathcal{O}_{17}=i\left(\bar{q}_{{\rm
L}p}\gamma^{\lambda}\tau^{K}\overleftrightarrow{D}_{\mu}q_{{\rm
L}r}\right)\left(D_{\lambda}H^{\dagger}\tau^{K}D^{\mu}H\right),$
$\displaystyle\mathcal{O}_{18}=i(\bar{u}_{{\rm
R}p}\gamma^{\mu}\overleftrightarrow{D}^{\nu}{d}_{{\rm
R}r})\epsilon^{ij}(D^{\mu}H_{i}D^{\nu}H_{j}),$ (13)

which revive the results in Ref. Li:2020gnx . Restricting to the flavor
diagonal components we have the following relations between ${\cal Q}$ and
${\cal O}$ operators:

$\displaystyle\begin{pmatrix}{\cal O}_{1},{\cal O}_{2},{\cal O}_{3},{\cal
O}_{8},{\cal O}_{9},{\cal O}_{10},{\cal O}_{14},{\cal
O}_{15}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}{\cal Q}_{1},{\cal Q}_{2},{\cal
Q}_{3},{\cal Q}_{6},{\cal Q}_{7},{\cal Q}_{8},2{\cal Q}_{11},2{\cal
Q}_{12}\end{pmatrix},$ (14) $\displaystyle\begin{pmatrix}{\cal O}_{4}\\\ {\cal
O}_{5}\\\ {\cal O}_{6}\\\ {\cal
O}_{7}\end{pmatrix}=\begin{pmatrix}1&1&-1&-1\\\ i&-i&i&-i\\\ 1&1&1&1\\\
i&-i&-i&i\end{pmatrix}\begin{pmatrix}{\cal Q}_{4}\\\ {\cal Q}^{*}_{4}\\\ {\cal
Q}_{5}\\\ {\cal Q}^{*}_{5}\end{pmatrix},\quad\begin{pmatrix}{\cal O}_{11}\\\
{\cal O}_{12}\\\ {\cal O}_{13}\end{pmatrix}=\begin{pmatrix}0&1&1\\\ 2i&i&-i\\\
-2i&i&-i\end{pmatrix}\begin{pmatrix}{\cal Q}_{9}\\\ {\cal Q}_{10}\\\ {\cal
Q}^{*}_{10}\end{pmatrix},$ (15) $\displaystyle\begin{pmatrix}{\cal O}_{16}\\\
{\cal O}_{17}\end{pmatrix}=\begin{pmatrix}1&0\\\
-1/2&1\end{pmatrix}\begin{pmatrix}{\cal Q}_{13}\\\ {\cal
Q}_{14}\end{pmatrix}.$ (16)

##  3 Different classes of non-interfering effect

The non-interference between the SM amplitude and the amplitude with one
insertion of dim-6 SMEFT operator for various of 2-to-2 processes is one of
our motivations for the study of the dim-8 operators. The original observation
in Ref. Azatov:2016sqh  is mainly based on the helicity selection rule in the
massless SM, which states that the interference between the helicity
amplitudes — ${\cal A}^{\rm SM}_{\\{h_{i}\\}}{\cal A}^{\rm
EFT*}_{\\{h_{i}\\}}$ for the 2-to-2 scattering processes involving $VVVV$,
$VV\psi\psi$, $V\psi\psi\phi$ and $VV\phi\phi$ vanishes for each set of
helicity configuration $\\{h_{i}\\}$ in the massless limit, where
$V,\psi,\phi$ represents spin-1,1/2,0 particle respectively. The reference
also mentioned the possible selection rule due to the conservation of the
$SU(2)$ isospin, which can be in principle extended to the $SU(3)$ color gauge
group. Such a selection rule also exists in the relevant loop calculation,
which is used to account for some vanishing entries in the anomalous dimension
matrix of the Renormalization group equation of the dim-6 SMEFT operators
Craig:2019wmo . Furthermore, in Ref. Jiang:2020rwz ; Shu:2021qlr  more
sophisticated selection rules based on the argument of the partial wave
expansion and the principle of the conservation of angular momentum help to
explain more vanishing entries in the anomalous dimension matrix that Ref.
Craig:2019wmo  cannot explain. As we will show later, such a selection rule
used for loop calculation can also benefit the analysis of the non-
interference in the scattering process, as the tree-level interference cross-
section for certain scattering processes can be obtained by cutting the
internal propagators in the relevant loop amplitudes.

On the other hand, researchers in the community are also interested in
reviving the non-interference effect. For example, by measuring the azimuthal
angle distribution of the decay product of the intermediate massive particles,
one can restore the interference effects from the triple gauge dim-6 operators
Panico:2017frx ; Azatov:2017kzw . In the meantime, a method of quantitative
evaluation of the upper bound of the interfering effect that can be revived
has been proposed in Ref. Degrande:2020tno . Recently, Ref. Hwang:2023wad
proposed a new kinematic variable — VBFhardness to help to resurrect the non-
interference in dilepton final states in probing anomalous triple gauge
couplings. In this sense, it is useful to classify different types of non-
interfering effects between the SM and the SMEFT amplitudes in advance, which
helps people working on resurrection programmes to identify their target
operators and processes. Here we classify the non-interfering effects for the
2-to-2 scattering in the following types:

  * •

Non-interference due to the selection rule of helicities.

As mentioned above, this results from the fact that certain types of 4-point
amplitudes generated by the massless SM have definite helicity combinations,
which mismatch those generated by one insertion of dim-6 operators. Such a
selection rule is only valid in the massless limit, and mostly for 4-point
amplitudes, therefore the interfering total cross-section $\sigma^{\rm int}$
can in principle be revived by the mass effect of vector bosons and quarks or
higher order corrections. On the other hand, even in the massless limit, the
interference effect can be resurrected by analyzing the differential cross-
section $d\sigma^{\rm int}$ for the 2-to-4 process with successive decay of
the unstable particle produced in the 2-to-2 processes Panico:2017frx ;
Azatov:2017kzw .

  * •

Non-interference due to CP symmetries.

Since some of our operators are odd under CP transformation, an amplitude with
a single insertion of a vertex from those operators, $\mathcal{A}^{CP}_{d8}$,
has a weak CP phase of $\pi/2$, i.e. the amplitude for the CP conjugate
process differs by a sign,

$\mathcal{A}^{CP}_{d8}\left(a\to
b\right)=-\mathcal{A}^{CP}_{d8}\left(\bar{a}\to\bar{b}\right).$ (17)

On the contrary, the SM model amplitude is CP-even if we neglect its strongly
suppressed Jarlskog invariant. As a result, the interference between the SM
and the dim-8 amplitudes will change sign under CP transformation, and it will
not contribute to CP even observables such as the $W^{+}W^{-}$ total cross-
section, invariant mass distribution, etc. Since the process is C-invariant,
the interference can only contribute to observables that violate parity. In
$W^{+}W^{-}$, this requires involving the helicities of the gauge bosons. The
situation is more complicated when the process is not C-invariant such as
$W^{+}Z$ as a C transformation turns it into $W^{-}Z$. This issue could be
overcome by summing up the two processes. However, at the LHC, the two
processes have different cross-sections due to the C-breaking initial state
(pp) which manifests itself through different luminosities of the initial
partons. Nevertheless, a similar phase difference appears in the first
approximation between $\mathcal{A}^{CP}_{d8}$ and the SM amplitude and can be
understood in the following way. At the tree level, and far away from any
resonance, one can relate the complex conjugate of an amplitude ${\cal A}(i\to
f)$ — ${\cal A}^{*}(i\to f)$ with the amplitude of the reversed process ${\cal
A}(f\to i)$ due to the optical theorem at leading order. On the other hand,
the reversed amplitude can be related to the original one with the time
reversal transformation due to its antiunitary property. According to the CPT
theorem, the T transformation is equivalent to the CP transformation, so one
can derive the following relation Jacob:1959at ; Tung:1985na ;
Degrande:2021zpv ; Azatov:2019xxn :

$\displaystyle{\cal A}(a\to b)=\lambda_{CP}{\cal A}^{*}(a\to b),$ (18)

with $\lambda_{CP}=\pm 1$ for CP-even and CP-odd amplitudes
respectively222More precisely, the CP-violating amplitude can be thought of as
the tree-level amplitude obtained by any odd number of insertion of CP-
violating interactions in the perturbation theory.. Therefore, if the SM
amplitude is CP-preserving and the NP amplitude is CP-violating, then their
interference ${\cal A}_{\rm SM}{\cal A}^{*}_{\rm NP}+c.c.$ is zero for any
helicity combinations. Such an argument is very strong as one does not require
initial and final states to be CP eigenstates. This non-interference effect
cannot be revived at the total cross-section level but is possible to revive
at the differential cross-section level by again studying the azimuthal angle
distribution of decay products. In addition, the inclusion of the width effect
of the massive gauge bosons or higher order corrections in the ${\cal A}_{\rm
SM}$ which generates the imaginary part of the amplitude can also resurrect
this non-interference effect provided that $a$ and $b$ are not CP eigenstates.

  * •

Non-interference due to the selection rule of Lorentz and gauge quantum
numbers.

As pointed out in Ref. Li:2022abx , the amplitude generated by the J-basis
operator in the SMEFT can be factorized into the following form333This can
also be thought of as the definition of the J-basis operators — they generate
local on-shell amplitudes for the specific scattering channel of quantum
number J.:

$\displaystyle{\cal A}_{\rm EFT}(i\to f)\sim\langle
f|O^{J,\mathbf{R}}|i\rangle\sim{\cal T}_{\rm EFT}^{J,\mathbf{R}}\langle
f|J,\mathbf{R}\rangle\langle J,\mathbf{R}|i\rangle,$ (19)

which only contains the generalized partial wave amplitude of the angular
momentum $J$ and the gauge quantum number $\mathbf{R}$. In the above formula,
we have suppressed any additional labels for the spin and gauge indices
associated with the representation $J$ and $\mathbf{R}$, ${\cal
T}^{J,\mathbf{R}}$ encodes the dynamic information from the operator
$O^{J,\mathbf{R}}$, and $\langle f|J,\mathbf{R}\rangle$ and $\langle
J,\mathbf{R}|i\rangle$ are Poincaré CG coefficients. Similarly, one can do the
partial wave expansion for the SM amplitude:

$\displaystyle{\cal A}_{\rm SM}(f\to i)\sim\sum_{J,\mathbf{R}}{\cal T}_{\rm
SM}^{J,\mathbf{R}}\langle f|J,\mathbf{R}\rangle\langle J,\mathbf{R}|i\rangle,$
(20)

and therefore, the interference term can be written as:

$\displaystyle{\cal A}_{\rm SM}{\cal A}^{*}_{\rm
EFT}\sim\sum_{J^{\prime},\mathbf{R}^{\prime}}{\cal T}_{\rm
SM}^{J^{\prime},\mathbf{R}^{\prime}}{\cal T}_{\rm EFT}^{*J,\mathbf{R}}\langle
J,\mathbf{R}|f\rangle\langle f|J^{\prime},\mathbf{R}^{\prime}\rangle\langle
J^{\prime},\mathbf{R}^{\prime}|i\rangle\langle i|J,\mathbf{R}\rangle,$ (21)

where we deliberately separate the two CG coefficients in ${\cal A}^{*}_{\rm
EFT}$ such that the bras and kets of initial and final states are next to each
other. From eq. (21), we can see that if we sum over the gauge configurations
of the initial or final states, then the inner product of state vectors for
the gauge space will collapse to form
$\langle\mathbf{R}|\mathbf{R}^{\prime}\rangle\sim\delta_{\mathbf{R}\mathbf{R}^{\prime}}$,
as a consequence, if SM cannot generate the generalized partial wave amplitude
of gauge quantum number $\mathbf{R}$ in ${\cal A}_{\rm SM}$, then it does not
interfere with the EFT amplitude. For the same reason, integrating over the
kinematic configuration of the final state (corresponds to integrating over
the solid angle in the 2-to-2 scattering process) will generate the inner
product between eigenstates of angular momentum $\langle
J|J^{\prime}\rangle\sim\delta_{JJ^{\prime}}$, which selects out the specific
partial wave of $J^{\prime}=J$ in the SM scattering amplitude. If one cannot
differentiate the components of a gauge multiplet in the experiment, then such
a non-interference effect due to the selection rule for the gauge quantum
number is not revivable, while for the case of angular momentum, the non-
interference is in principle possible to be revived by analyzing the angular
distribution for the 2-to-2 process unless there are strong cancellation
between different processes as we will discuss in the next section.

##  4 Result

In this section, we present the analytical and numerical results for the dim-8
operators discussed in section 2 for different diboson final states. For each
final state, we will compare the hadronic cross-section of the dim-8 and SM
interference to those of the dim-6 square and dim-6-SM interference. We
restrict our study to a single dim-6 operator ${\cal O}_{W}$:

$\displaystyle{\cal
O}_{W}=\epsilon^{IJK}W^{I\nu}_{\mu}W^{J\rho}_{\nu}W^{K\mu}_{\rho},$ (22)

which is known to be able to generate diboson amplitudes with
$E^{2}/\Lambda^{2}$ enhancement in the high energy limit but with vanishing
interference amplitude with the SM counterparts. For dim-8 operators, we
summarize their possible contribution to different final states in table 1.

operator | channels
---|---
${\cal O}_{1}$ | $\bar{d}d\to\gamma\gamma/\gamma Z/ZZ$
${\cal O}_{2}$ | $\bar{u}u\to\gamma\gamma/\gamma Z/ZZ$
${\cal O}_{3}$ |  $\bar{u}u\to\gamma\gamma/\gamma Z/ZZ$ $\bar{d}d\to\gamma\gamma/\gamma Z/ZZ$
${\cal O}_{4}$, ${\cal O}_{5}$, ${\cal O}_{6}$, ${\cal O}_{7}$ |  $\bar{u}u\to\gamma\gamma/\gamma Z/ZZ$ $\bar{d}d\to\gamma\gamma/\gamma Z/ZZ$ $\bar{u}d\to\gamma W/WZ$
${\cal O}_{8}$ |  $\bar{d}d\to\gamma\gamma/\gamma Z/ZZ/WW$
${\cal O}_{9}$ |  $\bar{u}u\to\gamma\gamma/\gamma Z/ZZ/WW$
${\cal O}_{10}$ |  $\bar{u}u\to\gamma\gamma/\gamma Z/ZZ/WW$ $\bar{d}d\to\gamma\gamma/\gamma Z/ZZ/WW$
${\cal O}_{11}$, ${\cal O}_{12}$, ${\cal O}_{13}$ |  $\bar{u}u\to WW$ $\bar{d}d\to WW$ $\bar{u}d\to\gamma W/WZ$
${\cal O}_{14}$ |  $\bar{u}u\to WW/ZZ$
${\cal O}_{15}$ |  $\bar{d}d\to WW/ZZ$
${\cal O}_{16}$ |  $\bar{u}u\to WW/ZZ$ $\bar{d}d\to WW/ZZ$
${\cal O}_{17}$ |  $\bar{u}u\to WW/ZZ$ $\bar{d}d\to WW/ZZ$ $\bar{u}d\to WZ$
${\cal O}_{18}$ |  $\bar{u}d\to WZ$
Table 1: Diboson modes that dim-8 operators can generates.

For analytical results, we use FeynRules Alloul:2013bka  to generate the
FeynArts Hahn:2000kx  model file and then use FeynArts and FormCalc
Hahn:1998yk  to generate tree level quark initiated diboson production
amplitudes. For numerical results, we use the input scheme of
$\alpha,G_{F},M_{Z}$, with their values listed below:

$\displaystyle\alpha^{-1}=127.95,\quad G_{F}=1.16638\times 10^{-5}\ {\rm
GeV}^{-2},\quad M_{Z}=91.1876\ {\rm GeV}.$ (23)

We use the Mathematica package ManeParse Clark:2016jgm  to read the PDF data
set in the LHAPDF6 format Buckley:2014ana . The PDF set we use to calculate
the hadronic cross-section at the 14TeV LHC is nCTEQ15FullNuc_1_1. In the
following calculation, we assume that the CKM matrix is diagonal and all the
light quarks $u,d,c,s$ are massless. Yukawa couplings of light quarks are also
set to zero. For each dim-8 operator, we assume the Wilson coefficient is non-
vanishing only for the flavor diagonal component of the first generation of
quark. We compare our analytical results obtained by the FeynArts and FormCalc
at both amplitude level and cross-section level with those obtained by
MadGraph Alwall:2014hca  and find that they are in agreement within machine
precision.

###  4.1 $WW$ final state

We summarize the scaling behavior of the interference amplitudes and those
after integrating over the solid angle in the high energy limit in table 2,
and we save the results for those coefficients in the center of mass energy
expansion, e.g. $a_{i},b_{i},c_{i}$, together with the full analytical
expression for the helicity amplitudes in Mathematica files in the link
ampresult . As one can see only ${\cal O}_{8}$, ${\cal O}_{9}$ and ${\cal
O}_{11}$ cannot generate interference amplitudes that scale as $S^{2}$, with
$S$ the square of center of mass energy of the parton system. ${\cal O}_{11}$
is CP odd as illustrated in appendix B. Following the discussion in section 3,
the amplitude it generates is purely imaginary, multiplied by the purely real
SM CP conserving amplitude444neglecting CP violation by the CKM matrix, the
resulting interference vanishes for all helicity configurations. The inclusion
of the width effect in the propagator in the SM amplitudes does not help to
revive this non-interference effect when summing over the helicity
configurations of the final states, because in this case the initial and final
states are CP eigenstates, and the cancellations occur between the CP
conjugate helicity configurations. However, when taking into account the decay
of the unstable vector bosons, the interference between different
polarizations will contribute to the full amplitude, and thus is possible to
revive the non-interference effects.

The operator ${\cal O}_{8}$ and ${\cal O}_{9}$ in contrast generate
interference amplitudes scaling linearly with $S$, and the reason can be
understood with the analysis of the helicities of $W$ bosons in the final
state. As discussed in section 2, operators ${\cal O}_{8}$ and ${\cal O}_{9}$
are derived from ${\cal Q}_{6}$ and ${\cal Q}_{7}$ respectively, which only
contain the contraction of field strength tensors of opposite chirality —
$W_{L}W_{R}$, and thus the vector bosons they produce in the high energy limit
should be of opposite helicities with total angular momentum at least equal to
2. Meanwhile, the fermion part of the two operators only involves the right-
handed up-type or down-type quarks, therefore the corresponding SM helicity
amplitude only receives the contribution from the s-channel diagram with the
exchange of $Z/\gamma$ boson, which cannot generate a diboson final state with
a total angular momentum higher than $1$. In other words, the SM amplitude for
diboson of opposite helicities is zero. Consequently, the non-vanishing
interference only occurs for the diboson helicity configurations
$\\{--,-0,+0,++,0+,0-,00\\}$, which can be converted from the configurations
of opposite helicities $\\{+-,-+\\}$ by either one insertion of the gauge
boson-goldstone vertices or one insertion of gauge boson mass, and each
insertion reduces the momentum dependence by respectively one or two power of
the mass. Finally combining with the SM amplitude which at most scales as
$1/\sqrt{S}$ results in the lower power dependence on $S$. We tabulate the
helicity amplitudes for ${\cal O}_{8}$ and the SM counterparts in table 3,
where $\theta$ is the scattering angle between the quark and the $W^{+}$ in
the center of mass frame of quark system as pictorially shown in figure 2.

Similar to ${\cal O}_{8}$ and ${\cal O}_{9}$, ${\cal O}_{14}$ and ${\cal
O}_{15}$ also contain the same right-handed fermion tensor structure, on the
contrary, their interference amplitudes scale as $S^{2}$ in the high energy
limit. The reason is that they directly produce the longitudinal modes of
gauge bosons that are acted with additional derivatives — $\partial
V_{L}\partial V_{L}$, which is confirmed by the formula for the $(0,0)$
polarization in table 4. Combined with the constant scaling for the SM
counterpart, the overall scaling of the interference amplitude remains as
$S^{2}$.

Even though the interference amplitudes for ${\cal O}_{8,9}$ and ${\cal
O}_{14,15}$ do not vanish, when integrated over the solid angle they become
zero, this can be explained by the third non-interference scenario we
discussed in the previous section. Using the package ABC4EFT Li:2022tec , we
perform the J-basis analysis for the operators, which tells us that they
coincidence with the J-basis operator for the scattering channel $q\bar{q}\to
WW$ with angular momentum $J=2$. However, as pointed out in the previous
paragraph, the SM amplitude gets contribution only from the s-channel diagram
of exchanging $Z/\gamma$, which corresponds to a partial wave of $J=1$.
Therefore, the angular part of the NP amplitude and the corresponding SM
counterpart should be proportional to the Wigner $d$-matrix of the following
forms:

$\displaystyle{\cal A}^{\rm NP}\sim
d^{2}_{(h_{W^{-}}-h_{W^{+}}),(h_{q}-h_{\bar{q}})},\quad{\cal A}^{\rm SM}\sim
d^{1}_{(h_{W^{-}}-h_{W^{+}}),(h_{q}-h_{\bar{q}})}.$ (24)

By the orthogonality of the $d$-matrix, the interference amplitude vanishes
after integrating over the solid angle.

Another interesting observation is that, for ${\cal O}_{16}$ and ${\cal
O}_{17}$, the integrated interference amplitude scales as $S$, rather than
$S^{2}$ before the integration. This can also be understood with the help of
partial wave decomposition. From the J-basis analysis, both of the operators
generate scattering amplitude for $q\bar{q}\to G^{+}G^{-}$ with angular
momentum $J=2$, where $G^{\pm}$ is the Goldstone corresponds to the
longitudinal mode of $W^{\pm}$. However, if one expands the $(0,0)$
polarization component of the SM amplitude in a series of powers of $S$, one
can find that the coefficient for $S^{0}$ term is proportional to $\cos\theta$
which corresponds to a partial wave of $J=1$, the $J=2$ partial wave is
contained in the term of expansion order of $S^{-1}$. Such a property is more
transparent in the Feynman gauge, where the $(0,0)$ polarization is generated
by the s-channel diagram with the virtual gauge boson decaying to a pair of
charged Goldstone bosons, which contributes only to the $J=1$ partial wave.
The SM t-channel diagram which contains the partial wave of $J=2$ involves
only the transverse polarization final states555The fermion to Goldstone
couplings are suppressed by Yukawa couplings., and to convert it into the
$(0,0)$ polarization state two insertions of Goldstone boson-gauge boson
vertices are needed, which decrease the dependence on the scattering energy by
$1/S$.

Operator | $2\operatorname{Re}({\cal A}^{\rm SM}{\cal A}^{\rm NP*})$ | $2\int d\Omega\operatorname{Re}({\cal A}^{\rm SM}{\cal A}^{\rm NP*})$
---|---|---
${\cal O}_{8}$ | $d\bar{d}:\ b_{8}S+c_{8}$ | 0
${\cal O}_{9}$ | $\bar{u}u:\ b_{9}S+c_{9}$ | 0
${\cal O}_{10}$ |  $u\bar{u}/d\bar{d}:\ a_{10}\cdot S^{2}+b_{10}\cdot S+c_{10}$ | $u\bar{u}/d\bar{d}:\ \overline{a}_{10}\cdot S^{2}+\overline{b}_{10}\cdot S+\overline{c}_{10}$
${\cal O}_{11}$ |  0 | 0
${\cal O}_{12}$ |  $u\bar{u}:\ a^{u}_{12}S^{2}+b^{u}_{12}S+c^{u}_{12}$ |  $u\bar{u}:\ \overline{a}^{u}_{12}S^{2}+\overline{b}^{u}_{12}S+\overline{c}^{u}_{12}+\overline{D}^{u}_{12}\log{S}$
$d\bar{d}:\ a^{d}_{12}S^{2}+b^{d}_{12}S+c^{d}_{12}$ |  $d\bar{d}:\ \overline{a}^{d}_{12}S^{2}+\overline{b}^{d}_{12}S+\overline{c}^{d}_{12}+\overline{D}^{d}_{12}\log{S}$
${\cal O}_{13}$ |  $u\bar{u}:\ a^{u}_{13}S^{2}+b^{u}_{13}S+c^{u}_{13}$ |  $u\bar{u}:\ \overline{a}^{u}_{13}S^{2}+\overline{b}^{u}_{13}S+\overline{c}^{u}_{13}+\overline{D}^{u}_{13}\log{S}$
$d\bar{d}:\ a^{d}_{13}S^{2}+b^{d}_{13}S+c^{d}_{13}$ |  $d\bar{d}:\ \overline{a}^{d}_{13}S^{2}+\overline{b}^{d}_{13}S+\overline{c}^{d}_{13}+\overline{D}^{d}_{13}\log{S}$
${\cal O}_{14}$ | $u\bar{u}:\ a_{14}S^{2}+b_{14}S+c_{14}$ | 0
${\cal O}_{15}$ | $d\bar{d}:\ a_{15}S^{2}+b_{15}S+c_{15}$ | 0
${\cal O}_{16}$ |  $u\bar{u}:\ a^{u}_{16}S^{2}+b^{u}_{16}S+c^{u}_{16}$ |  $u\bar{u}:\ \overline{b}^{u}_{16}S+\overline{c}^{u}_{16}+\overline{D}^{u}_{16}\log{S}$
$d\bar{d}:\ a^{d}_{16}S^{2}+b^{d}_{16}S+c^{d}_{16}$ |  $d\bar{d}:\ \overline{b}^{d}_{16}S+\overline{c}^{d}_{16}+\overline{D}^{d}_{16}\log{S}$
${\cal O}_{17}$ |  $u\bar{u}:\ a^{u}_{17}S^{2}+b^{u}_{17}S+c^{u}_{17}$ |  $u\bar{u}:\ \overline{b}^{u}_{17}S+\overline{c}^{u}_{17}+\overline{D}^{u}_{17}\log{S}$
$d\bar{d}:\ a^{d}_{17}S^{2}+b^{d}_{17}S+c^{d}_{17}$ |  $d\bar{d}:\ \overline{b}^{d}_{17}S+\overline{c}^{d}_{17}+\overline{D}^{d}_{17}\log{S}$
Table 2: Scaling of $q\bar{q}\to WW$ interference amplitude after summing and averaging over spins and helicities. $(h_{W^{+}},h_{W^{-}})$ | ${\cal A}_{h_{i}}^{\rm 8}/\frac{C_{8}}{\Lambda^{4}}$ | ${\cal A}_{h_{i}}^{\rm SM}$
---|---|---
$-,+$ | $S\sin^{2}\left(\frac{\theta}{2}\right)\sin(\theta)\left(S-2M_{W}^{2}\right)\delta_{ab}$ | 0
$-,-$ | $S\sin(\theta)\cos(\theta)M_{W}^{2}\delta_{ab}$ | $\frac{4\pi\alpha\sin(\theta)M_{Z}^{2}\delta_{ab}\sqrt{1-\frac{4M_{W}^{2}}{S}}}{3\left(S-M_{Z}^{2}\right)}$
$-,0$ | $\frac{S^{3/2}\sin^{2}\left(\frac{\theta}{2}\right)(2\cos(\theta)+1)M_{W}^{2}\delta_{ab}}{\sqrt{2}M_{W}}$ | $\frac{4\pi\alpha\sin^{2}\left(\frac{\theta}{2}\right)M_{Z}^{2}\delta_{ab}\sqrt{2S-8M_{W}^{2}}}{3SM_{W}-3M_{W}M_{Z}^{2}}$
$+,+$ | $S\sin(\theta)\cos(\theta)M_{W}^{2}\delta_{ab}$ | $\frac{4\pi\alpha\sin(\theta)M_{Z}^{2}\delta_{ab}\sqrt{1-\frac{4M_{W}^{2}}{S}}}{3\left(S-M_{Z}^{2}\right)}$
$+,-$ | $-2S\sin\left(\frac{\theta}{2}\right)\cos^{3}\left(\frac{\theta}{2}\right)\delta_{ab}\left(S-2M_{W}^{2}\right)$ | 0
$+,0$ | $\frac{S^{3/2}\cos^{2}\left(\frac{\theta}{2}\right)(2\cos(\theta)-1)M_{W}^{2}\delta_{ab}}{\sqrt{2}M_{W}}$ | $\frac{4\pi\alpha\cos^{2}\left(\frac{\theta}{2}\right)M_{Z}^{2}\delta_{ab}\sqrt{2S-8M_{W}^{2}}}{3SM_{W}-3M_{W}M_{Z}^{2}}$
$0,+$ | $-\frac{S^{3/2}\sin^{2}\left(\frac{\theta}{2}\right)(2\cos(\theta)+1)M_{W}^{2}\delta_{ab}}{\sqrt{2}M_{W}}$ | $\frac{4\sqrt{2}\pi\alpha\sin^{2}\left(\frac{\theta}{2}\right)M_{Z}^{2}\delta_{ab}\sqrt{S-4M_{W}^{2}}}{3M_{W}M_{Z}^{2}-3SM_{W}}$
$0,-$ | $\frac{S^{3/2}\cos^{2}\left(\frac{\theta}{2}\right)(1-2\cos(\theta))M_{W}^{2}\delta_{ab}}{\sqrt{2}M_{W}}$ | $\frac{4\sqrt{2}\pi\alpha\cos^{2}\left(\frac{\theta}{2}\right)M_{Z}^{2}\delta_{ab}\sqrt{S-4M_{W}^{2}}}{3M_{W}M_{Z}^{2}-3SM_{W}}$
$0,0$ | $-S\sin 2\theta M_{W}^{2}\delta_{ab}$ | $\frac{2\pi\alpha\sin(\theta)M_{Z}^{2}\delta_{ab}\left(2M_{W}^{2}+S\right)\sqrt{1-\frac{4M_{W}^{2}}{S}}}{3M_{W}^{2}\left(M_{Z}^{2}-S\right)}$
Table 3: Helicity amplitudes for $d\bar{d}\to WW$ for $h_{d}=1$ and $h_{\bar{d}}=-1$, where ${\cal A}_{h_{i}}^{\rm 8}$ is generated by ${\cal O}_{8}$. $(h_{W^{+}},h_{W^{-}})$ | ${\cal A}_{h_{i}}^{\rm 15}/\frac{C_{15}}{\Lambda^{4}}$
---|---
$-,+$ | $S\sin^{2}\left(\frac{\theta}{2}\right)\sin(\theta)M_{W}^{2}\delta_{ab}$
$-,-$ | $-S\sin^{2}\left(\frac{\theta}{2}\right)\sin(\theta)M_{W}^{2}\delta_{ab}$
$-,0$ | $\frac{S^{3/2}\sin^{2}\left(\frac{\theta}{2}\right)\cos(\theta)M_{W}\delta_{ab}}{\sqrt{2}}$
$+,+$ | $S\sin(\theta)\cos^{2}\left(\frac{\theta}{2}\right)M_{W}^{2}\delta_{ab}$
$+,-$ | $-S\sin(\theta)\cos^{2}\left(\frac{\theta}{2}\right)M_{W}^{2}\delta_{ab}$
$+,0$ | $\frac{S^{3/2}\cos^{2}\left(\frac{\theta}{2}\right)\cos(\theta)M_{W}\delta_{ab}}{\sqrt{2}}$
$0,+$ | $-\frac{S^{3/2}\sin^{2}(\theta)M_{W}\delta_{ab}}{2\sqrt{2}}$
$0,-$ | $\frac{S^{3/2}\sin^{2}(\theta)M_{W}\delta_{ab}}{2\sqrt{2}}$
$0,0$ | $-\frac{1}{8}S^{2}\sin(2\theta)\delta_{ab}$
Table 4: Helicity amplitudes for $d\bar{d}\to WW$ for $h_{d}=1$ and
$h_{\bar{d}}=-1$, where ${\cal A}_{h_{i}}^{\rm 15}$ is generated by ${\cal
O}_{15}$. Figure 2: The coordinate system in studying the $q\bar{q}\to WW$
process, $\theta$ is defined to be and angle between by the positive
directions of momenta of $W^{+}$ and the quark $q$ in the bosons center of
mass frame.

In figure 3, we plot the hadronic differential cross-section with respect to
the scattering angle in the diboson center of mass frame, $\theta$, at the 14
TeV LHC. We set the new physics scale at $\Lambda=1$ TeV as an example and set
all the dimensionless Wilson coefficients equal to 1. We also add as a
comparison the contributions from dim-6 interference and dim-6 square
differential cross-section for the operator ${\cal O}_{W}$. In principle, the
$O_{W}$ operator can only be generated through loop-level matching. As a
result, the dimensionless Wilson coefficient tends to receive a
$1/(16\pi^{2})$ suppression. Nevertheless, we still set $c_{i}$ to unity to
compare the relative contribution from different operators, the concrete scale
of each Wilson coefficient depends on the physical assumption of the UV model,
which is beyond the scope of our study in this paper. In the plot, we
conservatively rescale the dim-6 square contribution by a factor of 0.1 to
show its shape. In fact, if the UV theory responsible for the ${\cal O}_{W}$
is weakly coupled, then the dim-6 square contribution will at least be
suppressed by a loop factor $1/(16\pi^{2})\sim 0.01$ compared to the dim-8 and
SM interference by naive dimension analysis.

Figure 3: Differential cross-sections for dimension-8 and dimension-6
operators. The solid lines are for interference cross-sections for the dim-8
operators, while dash-dotted brown and blue lines are for the ${\rm
BSM}\times{\rm SM}$ interference and the ${\rm BSM}^{2}$ generated by the
dimension-6 ${\cal O}_{W}$ respectively. The results for $O_{8,9}$ operators
are scaled by a factor of $100$. The result for ${\rm BSM}^{2}$ is rescaled by
a factor of $0.1$, and taking into account the loop-generated nature of
$O_{W}$, in reality it could be further suppressed compared to the
$\text{dim-8}\times{\rm SM}$ contributions due to an additional loop factor.

The dominance of the dim-6 square contribution is coming from the enhancement
in the amplitude scaling as $S^{2}$ in the high energy limit. In the left
panel, the three solid lines in green, red, and purple represent the
differential interference cross-sections for operators ${\cal O}_{10}$, ${\cal
O}_{12}$ and ${\cal O}_{13}$ respectively, and they all have different angular
distributions comparing with each other and with the dim-6 originated
contributions. Therefore in principle, one can measure the angular
distribution of the vector boson in the diboson production to differentiate
effects from dim-8 SMEFT operators with dim-6 ones. On the other hand, the
distributions for ${\cal O}_{12}$ and ${\cal O}_{13}$ look similar except for
a global sign, thus presenting a difficulty in discerning the two operators.

In the right panel in the figure 3, we plot the similar differential cross-
section for the operators that contain the Higgs field — ${\cal O}_{14-17}$.
One can find that the scales of the interference cross-sections are generally
smaller than those generated by operators without Higgs field by one to two
orders of magnitude for $O_{16,17}$ compared to $O_{10-13}$. The reason is
that only the $(0,0)$ polarized final state gets enhanced by $S^{2}$ in the
high energy regime for ${\cal O}_{16,17}$, while the SM amplitude of two
longitudinal polarized $W$ bosons for the left-handed initiated process is
generally smaller than those of the $(+,-)$ or $(-,+)$ polarized final states
as one can see from the left panel in figure 4, where we plot the SM parton
level cross-sections of each helicity configurations as functions of the
center of mass energy.

Figure 4: SM parton level cross-section as a function of the center of mass
frame energy for different helicity of W bosons. The upper panel is for the
$u$ quark-initiated process, and the bottom panel is for the $d$ quark-
initiated process. Different colors represent different helicities of $W^{+}$
and $W^{-}$, respectively. In the right panel, the cross-section for helicity
states $-+$ and $+-$ for right-handed quark-initiated processes are zero and
thus are not visible on the log-scale plots.

On the contrary, the corresponding differential interference cross-section for
${\cal O}_{14,15}$ and ${\cal O}_{8,9}$ are at a similar order even though for
the right-handed quark-initiated process, the amplitude of the $(0,0)$
polarized final state is generally larger than those of the final states with
transverse polarization as one can see from the right panel in figure 4. This
results from the fact that the $q\bar{q}$ and $\bar{q}q$ processes get strong
cancellations for these four operators, such that a simple order of magnitude
comparison for individual amplitude is not valid anymore. The relation between
the total pp differential cross-section and the separated partonic cross-
sections are illustrated in figure 5. The definition of the differential
cross-section with respect to the pseudo-rapidity of $W^{+}$ in the COM frame
of the pp system is detailed in appendix C. To illustrate the cancellation
between $q\bar{q}$ and $\bar{q}q$ channels, we plot in figure 6 the
interference differential cross-sections
$d\sigma_{q\bar{q}}/d\eta(x_{1},x_{2},\eta)$ and
$d\sigma_{\bar{q}q}/d\eta(x_{1},x_{2},\eta)$ for $q\bar{q}$ and $\bar{q}q$
processes respectively, where $x_{1}$ and $x_{2}$ indicate the momentum
fractions for $q$ and $\bar{q}$ for the $q\bar{q}$ process and vice verse for
the $\bar{q}q$ process, and $\eta$ is the pseudo-rapidity of the $W^{+}$ in
the lab frame. From figure 6, one can find that in the center of mass frame of
the parton system, the cross-section from $q\bar{q}$ and $\bar{q}q$ channels
cancel each other exactly, and as one boosts along the beam direction, such
cancellation is restored by the difference between the probabilities of
finding a $d$-quark and $\bar{d}$-quark in the proton as shown in the left and
the right plots in the figure.

Figure 5: The pictorial illustration for the relation between the total
differential cross-section and the partonic differential cross-sections Figure
6: Comparison of the differential cross-section for $q\bar{q}$ and $\bar{q}q$
channel for $O_{8}$ and $O_{15}$, as one can see from the middle plot, the
total differential cross-section vanished in the COM frame of partons as
$q\bar{q}$ and $\bar{q}q$ contributions cancel exactly.

In figure 7, we demonstrate the relative scale of the diboson cross-section
from the dim-8-SM interference term to those from the dim-6 square term
(dim-6-SM interference term) in the top (bottom) row. In each plot, each curve
is obtained by equating the magnitudes of the two cross-sections in the
comparison. Different colored lines represent different cuts on the diboson
invariant mass $M_{WW}$, and the dashed lines all have a fixed lower bond as
the kinematic threshold for the diboson production while varying the upper
bounds with the NP scale $\Lambda$. The solid lines, however, also vary the
lower bounds with respect to the NP scale $\Lambda$. From the plot, one can
deduce that the effect from ${\cal O}_{10}$ is larger than those from ${\cal
O}_{12}$ and ${\cal O}_{13}$ with equal dimensionless Wilson coefficients.
This can be seen from the fact that one needs a relatively smaller ratio of
dimensionless Wilson coefficients $|C_{10}/C^{2}_{W}|\sim 3$ for ${\cal
O}_{10}$ to achieve the equal magnitude of the interference cross-section
comparing to the dim-6 square counterpart, while for ${\cal O}_{12}$ or ${\cal
O}_{13}$ a ratio of order ${\cal O}(10)$ is needed. This fact indicates that
when people include the dim-6 square effects in their global analysis in the
SMEFT framework, they should also take into account at least the effect from
${\cal O}_{10}$. Furthermore, as discussed before, if one assumes the operator
${\cal O}_{W}$ is generated by a weakly coupled UV theory, then $O_{i}$ with
$|C_{i}/C^{2}_{W}|\sim 100$ should also be considered to be important.

From the plots in the first row in figure 7, one can also observe that the
dashed lines become flat at a high NP scale, which can be understood as
follows. Both $\sigma_{i}^{\rm int}$ and $\sigma_{W}^{\rm EFT2}$ can be
factorized as following forms:

$\displaystyle\sigma_{i}^{\rm int}=\frac{C_{i}}{\Lambda^{4}}f_{i}(M^{\rm
min}_{WW},M^{\rm max}_{WW}),$ (25) $\displaystyle\sigma_{W}^{\rm
EFT2}=\frac{C_{W}^{2}}{\Lambda^{4}}f_{\rm EFT2}(M^{\rm min}_{WW},M^{\rm
max}_{WW}),$ (26)

where $M^{\rm min}_{WW}$ and $M^{\rm max}_{WW}$ are cuts on the diboson
invariant mass, as a consequence, $|\sigma_{i}^{\rm int}/\sigma_{W}^{\rm
EFT2}|=1$ indicates:

$\displaystyle\left|\frac{C_{i}}{C_{W}^{2}}\right|=\left|\frac{f_{\rm
EFT2}(M^{\rm min}_{WW},M^{\rm max}_{WW})}{f_{i}(M^{\rm min}_{WW},M^{\rm
max}_{WW})}\right|.$ (27)

For the fixed $M_{WW}^{\rm min}$, increasing the $M_{WW}^{\rm max}$ saturates
the values of both $f_{i}$ and $f_{\rm EFT2}$ because the contribution from
the high center of mass energy is suppressed by the parton luminosity despite
an enhancement in the parton level cross-section due to its scaling with $S$.
Furthermore, one can also find that the solid lines that cut on a relatively
high diboson invariant mass tend to decrease as $\Lambda$ increase, this is
because the up-type and down-type quark-initiated processes have different
coefficients in the $S$ expansion and the scaling behavior for the
corresponding parton luminosity functions for $u\bar{u}$ and $d\bar{d}$ are
different. Suppose we parameterize the parton level cross-sections for the
dim-8 interference and dim-6 square in the high energy limit in the following
forms:

$\displaystyle\hat{\sigma}^{\rm int}_{u}\sim\frac{C_{i}}{\Lambda^{4}}k^{\rm
int}_{u}S,\quad\hat{\sigma}^{\rm int}_{d}\sim\frac{C_{i}}{\Lambda^{4}}k^{\rm
int}_{d}S,$ (28) $\displaystyle\hat{\sigma}^{\rm
EFT2}_{u}\sim\frac{C_{W}^{2}}{\Lambda^{4}}k^{\rm
EFT2}_{u}S,\quad\hat{\sigma}^{\rm
EFT2}_{d}\sim\frac{C_{W}^{2}}{\Lambda^{4}}k^{\rm EFT2}_{d}S,$ (29)

where we factor out the Wilson coefficient dependence, and $k$’s are some
constants. This time $|\sigma_{i}^{\rm int}/\sigma_{W}^{\rm EFT2}|=1$
indicates:

$\displaystyle\left|\frac{C_{i}}{C_{W}^{2}}\right|$
$\displaystyle=\left|\frac{\int dS\ L_{u\bar{u}}(S)k^{\rm
EFT2}_{u}S+L_{d\bar{d}}(S)k^{\rm EFT2}_{d}S}{\int dS\ L_{u\bar{u}}(S)k^{\rm
int}_{u}S+L_{d\bar{d}}(S)k^{\rm int}_{d}S}\right|,$ (31)
$\displaystyle=\left|\frac{k^{\rm EFT2}_{u}}{k^{\rm
int}_{u}}\left(1+\frac{\int dS\ L_{d\bar{d}}(S)\left(k^{\rm EFT2}_{d}/k^{\rm
EFT2}_{u}-k^{\rm int}_{d}/k^{\rm int}_{u}\right)S}{\int dS\
L_{u\bar{u}}(S)S+L_{d\bar{d}}(S)k^{\rm int}_{d}/k^{\rm
int}_{u}S}\right)\right|,$

where the $L_{u\bar{u}}$ and $L_{d\bar{d}}$ are the parton luminosity
functions for $u\bar{u}$ and $d\bar{d}$. The second term in the parenthesis in
eq. (31) accounts for the dependence of $\Lambda$ for solid lines through the
$\Lambda$ dependence in the integral upper limit.

From the plots of the bottom row in figure 7, one can find that in the high
energy regime the dim-8-SM interference cross-section is much larger than
dim-6-SM interference cross-section, as a small ratio of $|C_{i}/C_{W}|<{\cal
O}(1)$ is needed to equate the two cross-sections. This is expected, since at
the high energy, the dim-6-SM interference is suppressed by $1/S$, while
dim-8-SM interference is enhanced by $S$. In contrast, one can find that the
dashed lines increase very fast as one increases the NP scale $\Lambda$, this
is a manifest result of improving EFT validity by increasing the NP scale.
Again one can also understand this feature by factorizing the $\sigma_{W}^{\rm
int}$ into the following form:

$\displaystyle\sigma_{W}^{\rm int}=\frac{C_{W}}{\Lambda^{2}M_{W}^{2}}f_{\rm
W}(M^{\rm min}_{WW},M^{\rm max}_{WW}),$ (32)

where we extract out an additional factor of $M^{2}_{W}$ such that the
dimension of $f_{W}$ matches that of $f_{i}$ in eq. (25). This time
$|\sigma_{i}^{\rm int}/\sigma_{W}^{\rm int}|=1$ yields:

$\displaystyle\left|\frac{C_{i}}{C_{W}}\right|=\left|\frac{\Lambda^{2}f_{W}(M^{\rm
min}_{WW},M^{\rm max}_{WW})}{M_{W}^{2}f_{i}(M^{\rm min}_{WW},M^{\rm
max}_{WW})}\right|.$ (33)

Therefore the Wilson coefficients ratio increases with $\Lambda^{2}$, provided
that the ratio of $f_{W}$ and $f_{i}$ saturates for fixed $M_{WW}^{\rm
min}=2M_{W}$ and a large enough $M_{WW}^{\rm max}$.

Figure 7: Comparisons of the interference cross-sections of dim-8 operator
${\cal O}_{10}$, ${\cal O}_{12}$ and ${\cal O}_{13}$ respectively from left to
right with respect to the cross-section from the dim-6 amplitude square (top)
and dim-6 interference (bottom). Different colors represent different cuts on
the diboson invariant mass, the lower bonds for the dashed lines are fixed at
the diboson production threshold. The dim-8 interference cross-section and
dim-6 square cross-section (top) or dim-6 interference cross-section (bottom)
have equal magnitude for each point on the lines.

###  4.2 $WZ$ Final State

In this section, We also examine the operators that contribute to the $WZ$
final state, namely ${\cal O}_{4-7}$, ${\cal O}_{11-13}$, and $O_{18}$. Among
these operators, $O_{5,7,11}$ are CP-odd operators, as explained in appendix
B, and result in zero interference amplitudes at leading order due to the
second non-interference scenario discussed in Section 3. However, unlike in
the $WW$ final state, this non-interference effect can be resurrected by
including the gauge boson width in propagators, which results in the amplitude
after summing over the gauge boson helicities being proportional to
$\Gamma_{W}/M_{W}$. This is because the CP conjugate of the $W^{+}Z$ final
state is $W^{-}Z$, and the cancellation between the CP conjugate processes by
summing over the helicity configurations, as done for the $WW$ final state,
does not apply here. On the other hand, $O_{18}$ generates amplitudes only
involving the right-handed quark, contrary to the SM, where only the left-
handed quark contributes, thus resulting in vanishing interference amplitudes.
Table 5 lists the scaling behavior of the interference amplitudes for all the
dim-8 operators that contribute to $u\bar{d}\to W^{+}Z$. The relevant
analytical results can be found in ampresult .

Operator | $2\operatorname{Re}({\cal A}^{\rm SM}{\cal A}^{\rm NP*})$ | $2\int d\Omega\operatorname{Re}({\cal A}^{\rm SM}{\cal A}^{\rm NP*})$
---|---|---
${\cal O}_{4}$ | $u\bar{d}:\ e_{4}S^{2}+f_{4}S+g_{4}$ | $\overline{e}_{4}S^{2}+\overline{f}_{4}S+\overline{g}_{4}$
${\cal O}_{5}$ | $u\bar{d}:\ \frac{\Gamma_{W}}{M_{W}}(f_{5}S+g_{5})$ | $\frac{\Gamma_{W}}{M_{W}}(\overline{f}_{5}S+\overline{g}_{5})$
${\cal O}_{6}$ | $u\bar{d}:\ e_{6}S^{2}+f_{6}S+g_{6}$ | $\overline{e}_{6}S^{2}+\overline{f}_{6}S+\overline{g}_{6}$
${\cal O}_{7}$ | $u\bar{d}:\ \frac{\Gamma_{W}}{M_{W}}(f_{7}S+g_{7})$ | $\frac{\Gamma_{W}}{M_{W}}(\overline{f}_{7}S+\overline{g}_{7})$
${\cal O}_{11}$ | $u\bar{d}:\ g_{11}\frac{\Gamma_{W}}{M_{W}}$ | 0
${\cal O}_{12}$ | $u\bar{d}:\ e_{12}S^{2}+f_{12}S+g_{12}$ | $\overline{e}_{12}S^{2}+\overline{f}_{12}S+\overline{g}_{12}$
${\cal O}_{13}$ | $u\bar{d}:\ e_{13}S^{2}+f_{13}S+g_{13}$ | $\overline{e}_{13}S^{2}+\overline{f}_{13}S+\overline{g}_{13}$
${\cal O}_{18}$ | $u\bar{d}:\ 0$ | 0
Table 5: Scaling of $q\bar{q}\to WZ$ interference amplitude.

In figure 8, we present the hadronic angular distribution for the $W^{+}$
boson in the production process $pp\to W^{+}Z$, where $\theta$ is defined as
the angle between the $W^{+}$ and the beam axis in the diboson frame. It can
be observed that the differential interference cross-sections of ${\cal O}{4}$
and ${\cal O}{6}$ are smaller compared to those of ${\cal O}{12}$ and ${\cal
O}{13}$. Furthermore, the angular distributions for ${\cal O}{4}$ and ${\cal
O}{6}$ are highly similar, as can be seen from the plot where they overlap
with each other to a significant extent, making it difficult to differentiate
between the effects of these two operators in the $WZ$ production channel
based on this distribution alone.

Figure 8: Differential cross-section for dimension-8 and dimension-6 operators
for $pp\to W^{+}Z$. The dotted cyan line and dashed yellow line for ${\cal
O}_{4}$ and ${\cal O}_{6}$ respectively are almost overlapped with each other,
they are also scaled by a factor of 10. The brown and blue dash-dotted lines
are the ${\rm BSM}\times{\rm SM}$ interference and the ${\rm BSM}^{2}$
contributions for the dimension-6 ${\cal O}_{W}$ respectively.

In the first row of figure 9, we plot the ratio of the dim-8 dimensionless
Wilson coefficient $C_{i}$ to $C_{W}$ against the NP scale $\Lambda$, ensuring
that the dim-8 interference cross-sections are equal to the dim-6 square
(dim-6 interference) cross-sections. The colors in the plot represent various
cuts on the invariant mass of the diboson system. In contrast to the $WW$
final state, we observe that the solid lines, which pertain to the high
invariant mass region, approach a constant. This behavior arises because the
$WZ$ final state only involves the $u\bar{d}$ initiated process (ignoring the
$c\bar{s}$ initiated process). Therefore, only terms proportional to
$L_{u\bar{d}}\cdot S$ exist within the integral in both numerators and the
denominator of equation (31), canceling out with each other and leaving a
constant ratio.

In the second row of figure 9, we compare the dim-8 and dim-6 interference
cross-sections. Similar to the $WW$ final state, the increasing dashed line
indicates an improvement in the convergence of the EFT as the NP scale
increases. The decreasing solid lines reflect that at high energy, the
interference effect of dim-8 dominates over that of dim-6 due to the faster
growth of the dim-8 interference amplitude with increasing energy.

Figure 9: Comparisons of interference cross-sections of the dim-8 operator
${\cal O}_{4}$, ${\cal O}_{6}$, ${\cal O}_{12}$ and ${\cal O}_{13}$
respectively from left to right with respect to the cross-section from the
dim-6 amplitude square (top) and dim-6 interference (bottom) for $pp\to
W^{+}Z$. Different colors represent different cuts on the diboson invariant
mass. Each line represents the ratio of the dim-8 and dim-6 Wilson
coefficients such that the dim-8 interference cross-section and the dim-6
square cross-section (top) or dim-6 interference cross-section (bottom) have
the same magnitude.

##  5 Conclusion and Outlook

We investigate the dim-8 SMEFT contributions to diboson production with
$WW/WZ$ final states. We classify all the operators that are capable of
generating $E^{4}/\Lambda^{4}$ enhancement in the interference amplitude, and
quantitatively compare their values with the dim-6 interference and dim-6
square amplitudes. We find that the non-interference found for dimension-six
operators is not observed for dimension-eight and therefore further limit the
legitimacy of constraining operators mainly from the square of the dimension-
six amplitudes. Preliminary result seems to indicate that this is also true
for the other electroweak diboson processes. We find that the dim-8 operator
${\cal O}_{10}$ can generate the interference cross-section for the process
$pp\to W^{+}W^{-}$ that is comparable with the dim-6 interference cross-
section from the operator ${\cal O}_{W}$ assuming equal dimensionless Wilson
coefficients. For the $WZ$ final state, the interference cross-section
generated by ${\cal O}_{12}$ or ${\cal O}_{13}$ is also comparable with the
dim-6 interference cross-section from ${\cal O}_{W}$. We also point out that
at the total cross-section level, the non-interference can occur due to the
selection rule of the angular momentum of the scattering process, while
suppression of the interference cross-section can also result from the
cancellation between the $q\bar{q}$ and $\bar{q}q$ initiated channels in the
pp collider. Both of these effects can be analyzed with the J-basis operators
prior to the derivation of Feynman rules and the calculation of amplitudes.

Finally, we note that precise bounds on the Wilson coefficients of various
operators require dedicated collider simulations. We anticipate that the
semileptonic decay channel, where one of the vector bosons decays leptonicly,
will have better sensitivity in the moderate invariant mass regime. This is
due to the lepton signature which can effectively reduce the large QCD
background. For the higher invariant mass regime, we expect the sensitivity
for fully hadronic decay to be better by reconstructing two opposite fat jets
and a larger hadronic decay branching ratio. It is also important to consider
detector effects in determining the extent to which we can exploit the
information in the angular distribution shown in figure 3 and 8 to enhance the
signal sensitivity and discerning effects from different operators.
Furthermore, when considering the decay, additional dim-8 operators such as
$\bar{\psi}\psi\bar{\psi}\psi F^{\mu\nu}$ can interfere with the SM full
amplitude. We acknowledge that further studies are needed to address these
issues.

###### Acknowledgements.

HL thanks Rafael Aoude, Olivier Mattelaer, Zhe Ren, Ming-Lei Xiao and Yu-Hui
Zheng for the helpful discussion. HL is supported by the 4.4517.08
IISN-F.N.R.S convention.

##  Appendix A Spinor Conventions

4-component Dirac field $\Psi$ can be expressed in terms of two 2-component
left-handed Weyl fermions $\xi$ and $\chi$ by:

$\displaystyle\Psi=\left(\begin{array}[]{c}\xi_{\alpha}\\\
\chi^{\dagger\dot{\alpha}}\end{array}\right),\quad\bar{\Psi}=\Psi^{\dagger}\gamma^{0}=\left(\chi^{\alpha},\;\xi^{\dagger}_{\dot{\alpha}}\right)\;.$
(36)

Conventions between the 2-component fields and 4-component field in the SM are
summarized in the following formulae:

$\displaystyle q_{\rm{L}}=\begin{pmatrix}Q\\\ 0\end{pmatrix},\quad
u_{\rm{R}}=\left(\begin{array}[]{c}0\\\
u_{{}_{\mathbb{C}}}^{\dagger}\end{array}\right),\quad d_{\rm
R}=\left(\begin{array}[]{c}0\\\
d_{{}_{\mathbb{C}}}^{\dagger}\end{array}\right),\quad l_{\rm
L}=\left(\begin{array}[]{c}L\\\ 0\end{array}\right),\quad e_{\rm
R}=\left(\begin{array}[]{c}0\\\
e_{{}_{\mathbb{C}}}^{\dagger}\end{array}\right).$ (45)
$\displaystyle\bar{q}_{\rm{L}}=\left(0\,,\,Q^{\dagger}\right),\quad\bar{u}_{\rm{R}}=\left(u_{{}_{\mathbb{C}}}\,,\,0\right),\quad\bar{d}_{\rm
R}=\left(d_{{}_{\mathbb{C}}}\,,\,0\right),\quad\bar{l}_{\rm
L}=\left(0\,,\,L^{\dagger}\right),\quad\bar{e}_{\rm
R}=\left(e_{{}_{\mathbb{C}}}\,,\,0\right).$ (46)

$\sigma^{\mu}$ and $\bar{\sigma}^{\mu}$ is defined by
$\sigma^{\mu}_{\alpha\dot{\alpha}}=(\mathbbm{1}_{\alpha\dot{\alpha}},{\bm{\sigma}}^{i}_{\alpha\dot{\alpha}})^{\mu}$,
$\bar{\sigma}^{\mu\dot{\alpha}\alpha}=(\mathbbm{1}^{\dot{\alpha}\alpha},-{\bm{\sigma}}^{i\dot{\alpha}\alpha})^{\mu}$,
with $\bm{\sigma}^{i}$ ordinary Pauli matrices. We have the relation between
$\sigma^{\mu}$ and $\bar{\sigma}^{\mu}$:

$\bar{\sigma}^{\mu\dot{\alpha}\alpha}=\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}\sigma^{\mu}_{\beta\dot{\beta}}.$
(47)

where the Levi-Civita tensors have conventions
$\epsilon^{12}=\epsilon_{21}=+1$. Charge conjugate matrix
$C=i\gamma^{0}\gamma^{2}$, Dirac matrices $\gamma^{\mu}$ and the Lorentz
generator $\sigma^{\mu\nu}$ have following forms in the chiral representation:

$\displaystyle C=\begin{pmatrix}\epsilon_{\alpha\beta}&0\\\
0&\epsilon^{\dot{\alpha}\dot{\beta}}\end{pmatrix}=\begin{pmatrix}-\epsilon^{\alpha\beta}&0\\\
0&-\epsilon_{\dot{\alpha}\dot{\beta}}\end{pmatrix},\quad\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}_{\alpha\dot{\beta}}\\\
\bar{\sigma}^{\mu\dot{\alpha}\beta}&0\end{pmatrix},$
$\displaystyle\sigma^{\mu\nu}=\dfrac{i}{2}[\gamma^{\mu},\gamma^{\nu}]=\begin{pmatrix}\left(\sigma^{\mu\nu}\right)_{\alpha}{}^{\beta}&0\\\
0&\left(\bar{\sigma}^{\mu\nu}\right)^{\dot{\alpha}}{}_{\dot{\beta}}\end{pmatrix}$
(48)

We have the following useful identities changing the $\bar{\sigma}$ to
$\sigma$ in the fermion current:

$\displaystyle\xi^{\dagger}_{1\dot{\alpha}}\bar{\sigma}^{\mu\dot{\alpha}\alpha}\xi_{2\alpha}=-\xi_{2}^{\alpha}\sigma^{\mu}_{\alpha\dot{\alpha}}\xi^{\dagger\dot{\alpha}}.$
(49)

We also list the conversion between two-component and four-component spinors
present in the Feynman rule or scattering amplitude in table 6. Various
definitions related to spinors are defined as follows. We first introduce two
eigenvectors of the helicity operators
$\hat{h}=\mathbf{p}\cdot\bm{\sigma}/|\mathbf{p}|$ for momentum $\mathbf{p}$ in
the direction
$\hat{\mathbf{p}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$:

$\displaystyle\chi^{-}_{\alpha}=\begin{pmatrix}-e^{-i\phi}\sin(\theta/2)\\\
\cos(\theta/2)\end{pmatrix},\quad\chi^{+\dot{\alpha}}=\begin{pmatrix}\cos(\theta/2)\\\
e^{i\phi}\sin(\theta/2)\end{pmatrix},$ (50)

with $\pm$ indicating their eigenvalue of the helicity operator. then we fix
the momentum dependence of the massless spinor variables to the following
forms:

$\displaystyle\lambda_{\alpha}=\sqrt{2E}\chi^{-}_{\alpha},\quad\tilde{\lambda}^{\dot{\alpha}}=\sqrt{2E}\chi^{+\dot{\alpha}}.$
(51)

One can easily verify that:

$\displaystyle(\lambda^{\alpha})^{*}=\tilde{\lambda}^{\dot{\alpha}}=\epsilon^{\dot{\alpha}\dot{\beta}}\tilde{\lambda}_{\dot{\beta}},\quad(\tilde{\lambda}_{\dot{\alpha}})^{*}={\lambda}_{\alpha}=\epsilon_{\dot{\alpha}\dot{\beta}}{\lambda}^{\beta}.$
(52)

We also define the helicity eigenstates of the massive Dirac 4-component
spinors in the following forms:

$\displaystyle u^{h}(p)=\begin{pmatrix}\sqrt{E-h|\mathbf{p}|}\chi^{(h)}\\\
\sqrt{E+h|\mathbf{p}|}\chi^{(h)}\end{pmatrix},\quad
v^{h}(p)=\begin{pmatrix}-\sqrt{E+h|\mathbf{p}|}\chi^{(-h)}\\\
\sqrt{E-h|\mathbf{p}|}\chi^{(-h)}\end{pmatrix}$ (53)

Table 6: Relations between 2- and 4-component notation for massless fermion fermion type | state type | helicity | notaion
---|---|---|---
Particle | In-coming | $+$ |  ${\rm P}_{\rm R}u=u^{+}=\tilde{\lambda}^{\dot{\alpha}}$, $\overline{v}{\rm P}_{\rm R}=\overline{v^{+}}=\tilde{\lambda}_{\dot{\alpha}}$
$-$ |  ${\rm P}_{\rm L}u=u^{-}={\lambda}_{\alpha}$, $\overline{v}{\rm P}_{\rm L}=\overline{v^{-}}={\lambda}^{\alpha}$
Out-going | $+$ |  $\overline{u}{\rm P}_{\rm L}=\overline{u^{+}}={\lambda}^{\alpha}$, ${\rm P}_{\rm L}v={v^{+}}=-{\lambda}_{\alpha}$
$-$ |  $\overline{u}{\rm P}_{\rm R}=\overline{u^{-}}=\tilde{\lambda}_{\dot{\alpha}}$, ${\rm P}_{\rm R}v={v^{-}}=\tilde{\lambda}^{\dot{\alpha}}$
Anti-Particle | In-coming | $+$ |  $\overline{v}{\rm P}_{\rm R}=\overline{v^{+}}=\tilde{\lambda}_{\dot{\alpha}}$, ${\rm P}_{\rm R}u={u^{+}}=\tilde{\lambda}^{\dot{\alpha}}$
$-$ |  $\overline{v}{\rm P}_{\rm L}=\overline{v^{-}}={\lambda}^{\alpha}$, ${\rm P}_{\rm L}u={u^{-}}={\lambda}_{\alpha}$
Out-going | $+$ |  ${\rm P}_{\rm L}v={v^{+}}={-\lambda}_{\alpha}$, $\overline{u}{\rm P}_{\rm L}=\overline{u^{+}}={\lambda}^{\alpha}$
$-$ |  ${\rm P}_{\rm R}v={v^{-}}=\tilde{\lambda}^{\dot{\alpha}}$, $\overline{u}{\rm P}_{\rm R}=\overline{u^{-}}=\tilde{\lambda}_{\dot{\alpha}}$

##  Appendix B CP Properties of operators

Following the convention in eq.(13.9) and eq.(13.10) in Ref. Branco:1999fs ,
the CP transformation of the $W$ and $Z$ bosons can be expressed as:

$\displaystyle({\cal CP})W^{+\mu}(t,\mathbf{r})({\cal
CP})^{\dagger}=-e^{i\xi_{W}}W^{-}_{\mu}(t,-\mathbf{r})$ (54)
$\displaystyle({\cal CP})W^{-\mu}(t,\mathbf{r})({\cal
CP})^{\dagger}=-e^{-i\xi_{W}}W^{+}_{\mu}(t,-\mathbf{r})$ (55)
$\displaystyle({\cal CP})Z^{\mu}(t,\mathbf{r})({\cal
CP})^{\dagger}=-Z_{\mu}(t,-\mathbf{r}),$ (56)

from which by setting the artibrary phase $\xi_{W}=0$ one can deduce that:

$\displaystyle({\cal CP})W^{1\mu}(t,\mathbf{r})({\cal
CP})^{\dagger}=-W_{\mu}^{1}(t,-\mathbf{r})$ (57) $\displaystyle({\cal
CP})W^{2\mu}(t,\mathbf{r})({\cal CP})^{\dagger}=W_{\mu}^{2}(t,-\mathbf{r})$
(58) $\displaystyle({\cal CP})W^{3\mu}(t,\mathbf{r})({\cal
CP})^{\dagger}=-W_{\mu}^{3}(t,-\mathbf{r}),$ (59)

which can be collectively expressed as $({\cal
CP})W^{I\mu}(t,\mathbf{r})({\cal
CP})^{\dagger}=-s^{I}W_{\mu}^{I}(t,-\mathbf{r})$ with $s^{1,3}=1,\ s^{2}=-1$.
Under such a convention one can derive the CP transformation properties of the
field strength tensor of the $SU(2)$ gauge boson $W_{\mu\nu}^{I}$:

$\displaystyle
W_{\mu\nu}^{I}=\partial_{\mu}W_{\nu}^{I}-\partial_{\nu}W_{\mu}^{I}+g\epsilon^{IJK}W_{\mu}^{J}W_{\nu}^{K}$
$\displaystyle,\quad\tilde{W}_{\mu\nu}^{I}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}W^{I\rho\sigma}$
(60) $\displaystyle({\cal CP})\partial_{\mu}W_{\nu}^{I}(t,\mathbf{r})({\cal
CP})^{\dagger}$ $\displaystyle=-s^{I}[\partial^{\mu}W^{I\nu}](t,-\mathbf{r})$
(61) $\displaystyle({\cal
CP})\epsilon^{IJK}W_{\mu}^{J}W_{\nu}^{K}(t,\mathbf{r})({\cal CP})^{\dagger}$
$\displaystyle=\epsilon^{IJK}s^{J}s^{K}[W^{J\mu}W^{K\nu}](t,-\mathbf{r})$ (62)
$\displaystyle=-s^{I}\epsilon^{IJK}[W^{J\mu}W^{K\nu}](t,-\mathbf{r})$
$\displaystyle({\cal CP})W_{\mu\nu}^{I}(t,\mathbf{r})({\cal CP})^{\dagger}$
$\displaystyle=-s^{I}W^{I\mu\nu}(t,-\mathbf{r}).$ (63)

In the above equations, the square bracket on the right-hand side reminds that
the derivative operator with upper indices on the right should reads
$\partial^{\mu}[t,-\mathbf{r}]=(\partial^{t},\partial^{-\mathbf{r}})$ which is
equivalent to the derivative $\partial_{\mu}[t,\mathbf{r}]$ on the left with
lower indices, that is the operator $({\cal CP})$ only acts on the fields. In
eq. (62), one can easily verify that
$\epsilon^{IJK}s^{J}s^{K}=-s^{I}\epsilon^{IJK}$, without summation of the
repeated indices. Similarly, the CP transformation property of the dual field
strength tensor $\tilde{W}_{\mu\nu}^{I}$ can also be easily derived with eq.
(63), we work on the situations $(\mu=0,\nu=i)$ and $(\mu=i,\nu=j)$ with
$i,j=(1,2,3)$ separately and verify that they lead to a coherent expression:

$\displaystyle({\cal CP})\tilde{W}_{0i}^{I}(t,\mathbf{r})({\cal
CP})^{\dagger}$ $\displaystyle=\frac{1}{2}\epsilon_{0ijk}({\cal
CP})W_{ji}^{I}(t,\mathbf{r})({\cal CP})^{\dagger}$ (64)
$\displaystyle=\frac{(-s^{I})}{2}\epsilon_{0ijk}\tilde{W}^{Ijk}(t,-\mathbf{r})$
$\displaystyle=\frac{(s^{I})}{2}\epsilon^{0ijk}W^{Ijk}(t,-\mathbf{r})=s^{I}\tilde{W}^{I0i}(t,-\mathbf{r})$
$\displaystyle({\cal CP})\tilde{W}_{ij}^{I}(t,\mathbf{r})({\cal
CP})^{\dagger}$ $\displaystyle=\epsilon_{ij0k}({\cal
CP})W_{0k}^{I}(t,\mathbf{r})({\cal CP})^{\dagger}$ (65)
$\displaystyle=(-s^{I})\epsilon_{ij0k}W^{I0k}(t,-\mathbf{r})$
$\displaystyle=(-s^{I})\epsilon^{ij0k}W^{I0k}(t,-\mathbf{r})=s^{I}\tilde{W}^{Iij}(t,-\mathbf{r})$
$\displaystyle({\cal CP})\tilde{W}_{\mu\nu}^{I}(t,\mathbf{r})({\cal
CP})^{\dagger}$ $\displaystyle=s^{I}\tilde{W}_{\mu\nu}^{I}(t,-\mathbf{r}).$
(66)

We can check that the dimension 6 SMEFT operator ${\cal
O}_{\tilde{W}}=\epsilon^{IJK}W^{I}W^{J}\tilde{W}^{K}$ is indeed a CP violating
operator:

$\displaystyle({\cal CP})\epsilon^{IJK}W^{I}W^{J}\tilde{W}^{K}({\cal
CP})^{\dagger}$
$\displaystyle=\epsilon^{IJK}(-s^{I}W^{I})(-s^{J}W^{J})(s^{K}\tilde{W}^{K})$
(67) $\displaystyle=-s^{I}\epsilon^{IJK}s^{I}(W^{I})(W^{J})(\tilde{W}^{K})$
$\displaystyle=-\epsilon^{IJK}W^{I}W^{J}\tilde{W}^{K}.$

For the fermion part, we also adopt the convention in Ref. Branco:1999fs ,
where under the CP transformation, the fermion bi-linears transform as
follows:

$\displaystyle({\cal CP})\overline{\psi_{L}}\gamma^{\mu}\chi_{L}({\cal
CP})^{\dagger}=-e^{i(\xi_{\psi}-\xi_{\chi})}\overline{\psi_{L}}\gamma_{\mu}\chi_{L}$
(68) $\displaystyle({\cal CP})\overline{\psi_{R}}\gamma^{\mu}\chi_{R}({\cal
CP})^{\dagger}=-e^{i(\xi_{\psi}-\xi_{\chi})}\overline{\psi_{R}}\gamma_{\mu}\chi_{R}.$
(69)

If we consider the flavor diagonal effective operators, then $\chi=\psi$, and
the phase factor on the right-hand side vanish. With these conventions, we can
analyse the CP transformation properties of the dimesnion 8 operators ${\cal
O}_{1\sim 18}$. As we have already obtained the CP transformation properties
of the field strength tensors of the gauge fields in eq. (63) and (66), we
only left with the same analysis for the fermion tensors. To proceed, we need
to first specify the meaning of the covariant with double arrow present in the
fermion tensors as follows:

$\displaystyle\overline{\psi}_{i}\gamma^{\mu}\overleftrightarrow{D}^{\nu}\chi_{j}$
$\displaystyle\equiv\overline{\psi}_{i}\gamma^{\mu}(D^{\nu}\chi_{j})-\overline{D^{\nu}\chi}_{j}\gamma^{\mu}\psi_{i},$
(70) with
$\displaystyle\overline{D^{\nu}\chi}_{j}\equiv(D^{\nu}\chi_{j})^{\dagger}\gamma^{0}$
(71)
$\displaystyle=(\partial^{\nu}\chi_{j}+igT^{I}_{jk}A^{I\nu}\chi_{k})^{\dagger}\gamma^{0}$
$\displaystyle=(\partial^{\nu}\overline{\chi}_{j}-ig\overline{\chi}_{k}T^{I}_{kj}A^{I\nu}),$

where $i,j,k$ are the gauge indices of certain gauge group, and $T^{I}$ and
$A^{I}$ are the generators and the gauge field of the corresponding gauge
group respectively. Therefore the CP conjugate of the first term on the right-
hand side of eq. (70) gives:

$\displaystyle({\cal
CP})\overline{\psi}_{i}\gamma^{\mu}(D^{\nu}\chi_{j})({\cal CP})^{\dagger}$
$\displaystyle=({\cal
CP})\overline{\psi}_{i}\gamma^{\mu}(\partial^{\nu}\chi_{j})({\cal
CP})^{\dagger}+igT^{I}_{jk}({\cal
CP})\overline{\psi}_{i}\gamma^{\mu}A^{I\nu}\chi_{k}({\cal CP})^{\dagger}$ (72)
$\displaystyle=-\partial_{\nu}\overline{\chi}_{j}\gamma_{\mu}\psi_{i}-igT^{I}_{jk}\overline{\psi}_{j}\gamma_{\mu}(-s^{I}A^{I}_{\nu})\psi_{i}$
$\displaystyle=-\partial_{\nu}\overline{\chi}_{k}\gamma_{\mu}\psi_{i}-igs^{I}T^{I}_{kj}\overline{\chi}_{k}\gamma_{\mu}(-s^{I}A^{I}_{\nu})\psi_{i}$
$\displaystyle=-\partial_{\nu}\overline{\chi}_{k}\gamma_{\mu}\psi_{i}+igT^{I}_{kj}\overline{\chi}_{k}\gamma_{\mu}A^{I}_{\nu}\psi_{i}=-\overline{D_{\nu}\chi}_{j}\gamma_{\mu}\psi_{i},$

where in the second line we have used eq. (68) and (69) ignoring additional
phases, while for the gauge field $A^{I}$ we have $({\cal
CP})A^{I\mu}(t,\mathbf{r})({\cal
CP})^{\dagger}=-s^{I}A_{\mu}^{I}(t,-\mathbf{r})$ with $s^{I}=1$ for
$T^{I*}=(T^{I})^{\rm T}=T^{I}$, and $s^{I}=-1$ for $T^{I*}=(T^{I})^{\rm
T}=-T^{I}$, which is valid for the $SU(2)$ and $SU(3)$ gauge group we
considered in the SMEFT. From the second line to the third line, we have used
$T_{jk}^{I}=s^{I}T_{jk}^{I}$. Similar to eq. (72), one can derive the CP
conjugation of $\overline{D^{\nu}\chi}_{j}\gamma^{\mu}\psi_{i}$, and they
together gives:

$\displaystyle({\cal
CP})\overline{\psi}_{i}\gamma^{\mu}\overleftrightarrow{D}^{\nu}\chi_{j}({\cal
CP})^{\dagger}=\overline{\chi}_{j}\gamma^{\mu}\overleftrightarrow{D}^{\nu}\psi_{i}.$
(73)

Therefore, for the fermion tensors $J_{d}$, $J_{u}$, and $J_{Q,1}$, one can
easily see that:

$\displaystyle({\cal CP})J^{\mu\nu}_{d,pr}(t,\mathbf{r})({\cal CP})^{\dagger}$
$\displaystyle=(J_{d,rp})_{\mu\nu}(t,-\mathbf{r}),$ (74) $\displaystyle({\cal
CP})J^{\mu\nu}_{u,pr}(t,\mathbf{r})({\cal CP})^{\dagger}$
$\displaystyle=(J_{u,rp})_{\mu\nu}(t,-\mathbf{r}),$ (75) $\displaystyle({\cal
CP})J^{\mu\nu}_{(Q,1),pr}(t,\mathbf{r})({\cal CP})^{\dagger}$
$\displaystyle=(J_{(Q,1),rp})_{\mu\nu}(t,-\mathbf{r})$ (76)

while for the $J^{I}_{(Q,3)\mu\nu}$, we have:

$\displaystyle({\cal CP})J^{I\mu\nu}_{(Q,3),pr}({\cal CP})^{\dagger}$
$\displaystyle=({\cal CP})i\tau^{I}_{ji}\bar{q}_{{\rm
L}p}^{j}\gamma^{\mu}\overleftrightarrow{D}^{\nu}q_{{\rm L}ri}({\cal
CP})^{\dagger}$ (77) $\displaystyle=i\tau^{I}_{ji}\bar{q}_{{\rm
L}ri}\gamma^{\mu}\overleftrightarrow{D}^{\nu}q_{{\rm L}pj}$
$\displaystyle=is^{I}\tau^{I}_{ij}\bar{q}_{{\rm
L}ri}\gamma^{\mu}\overleftrightarrow{D}^{\nu}q_{{\rm
L}pj}=s^{I}(J^{I}_{(Q,3),rp})_{\mu\nu}.$

With the aforementioned transformation properties, we can find that, for the
flavor diagonal components, only three operators ${\cal O}_{5}$, ${\cal
O}_{7}$ and ${\cal O}_{11}$ are CP violating, operators ${\cal O}_{12}$ and
${\cal O}_{13}$ even though containing $\tilde{W}$ are still CP conserving,
because under the transformation the fermion tensor $J^{I\mu\nu}_{(Q,3)}$ does
not contribute additional minus sign as $W^{I}$ in does eq. (67).

##  Appendix C Differential Cross-Section in the Non-COM Frame

The differential cross-section for the 2-to-2 process can be expressed in the
following form:

$\displaystyle d\sigma_{1+2\to
3+4}=\frac{1}{4F(2\pi)^{2}}\frac{d^{3}p_{3}}{2E_{3}}\frac{d^{3}p_{4}}{2E_{4}}\delta^{(4)}(p_{3}+p_{4}-p_{1}-p_{2})\left|{\cal
M}\right|^{2}_{1+2\to 3+4},$ (78)

where $F$ is the Lorentz invariant Møller flux factor defined by:

$\displaystyle F=E_{1}E_{2}v_{12}=\sqrt{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}},$ (79)

such that the whole formula in eq (78) is Lorentz invariant. However, the
partially integrated differential cross-section, e.g. the differential cross-
section with respect to the solid angle $d\sigma/d\Omega$, may not be Lorentz
invariant as the integration can break the Lorentz invariance. For the pp
collider, the COM frame of the parton system generally is not the COM frame of
the proton system. Since the lab frame coincides with the COM of the proton
system, it is convenient to derive a formula for the differential cross-
section with respect to the solid angle in the COM frame of the proton system
– $d\sigma/d\Omega^{*}$ as follows. Since the amplitude square $\left|{\cal
M}\right|^{2}$ and the flux factor $F$ are Lorentz invariant, we focus on the
simplification of the phase space and the $\delta$-function:

$\displaystyle\frac{d^{3}p_{3}}{2E_{3}}\frac{d^{3}p_{4}}{2E_{4}}\delta^{(4)}(p_{3}+p_{4}-p_{1}-p_{2})$
$\displaystyle=$ $\displaystyle
d^{4}p_{3}\delta(p_{3}^{2}-m_{3}^{2})\Theta(E_{3})d^{4}p_{3}\delta(p_{4}^{2}-m_{4}^{2})\Theta(E_{4})\delta^{(4)}(p_{3}+p_{4}-p_{1}-p_{2})$
$\displaystyle=$ $\displaystyle
d^{4}p_{3}\delta(p_{3}^{2}-m_{3}^{2})\Theta(E_{3})d^{4}p_{3}\delta((p_{1}+p_{2}-p_{3})^{2}-m_{4}^{2})\Theta(E_{1}+E_{2}-E_{3})$
$\displaystyle=$ $\displaystyle
dE_{3}d\Omega\frac{\sqrt{E_{3}^{2}-m_{3}^{2}}}{2}\delta(S-2(p_{1}+p_{2})\cdot
p_{3}+m_{3}^{2}-m_{4}^{2})\Theta(E_{1}+E_{2}-E_{3}),$ (80)

where $S$ is the COM energy square of the parton system. Up to now we have not
specified the reference frame yet, here we assume that the two initial partons
are massless and their momenta point along with the z-axis in the COM frame of
proton, thus their 4-momentum can be expressed as:

$\displaystyle p_{1}=(x_{1}\sqrt{s}/2,0,0,x_{1}\sqrt{s}/2),\quad
p_{2}=(x_{2}\sqrt{s}/2,0,0,-x_{2}\sqrt{s}/2),$ (81)

where $\sqrt{s}$ is the center of mass energy of the proton system, and
$x_{1}$ and $x_{2}$ are the factions of the proton energy that are carried by
the corresponding partons, and indeed the COM energy of the parton system can
be expressed as $\sqrt{S}=\sqrt{x_{1}x_{2}s}$. Suppose the momentum of the
final state particle 3 pointing in the direction with polar angle $\theta$
with respect to the z-axis, then the $\delta$-function in the last line in eq.
(80) fixes the dependence of $E_{3}$ on the scattering angle $\theta$ by the
following formula:

$\displaystyle
x_{1}x_{2}s+m_{3}^{2}-m_{4}^{2}+(x_{1}-x_{2})\sqrt{s}\cos\theta\sqrt{E_{3}^{2}-m_{3}^{2}}-(x_{1}+x_{2})\sqrt{s}E_{3}=0,$
(82)

which gives the solution to $E_{3}$:

$\displaystyle E_{3}(x_{1},x_{2},\theta)=$
$\displaystyle\Bigl{[}\left(x_{1}-x_{2}\right)\cos(\theta)\sqrt{\left(m_{3}^{2}s\left(x_{1}-x_{2}\right){}^{2}\cos^{2}(\theta)-m_{3}^{2}\left(2m_{4}^{2}+s\left(x_{1}^{2}+x_{2}^{2}\right)\right)+\left(m_{4}^{2}-sx_{1}x_{2}\right){}^{2}+m_{3}^{4}\right)}$
$\displaystyle+\left(x_{1}+x_{2}\right)\left(m_{3}^{2}-m_{4}^{2}+sx_{1}x_{2}\right)\Bigr{]}/\left[\sqrt{s}\left(\left(x_{1}+x_{2}\right){}^{2}-\left(x_{1}-x_{2}\right){}^{2}\cos^{2}(\theta)\right)\right].$
(83)

One can easily verify that the above result reproduces expression in the
center of mass frame of parton by setting $x_{1}=x_{2}$:

$\displaystyle E_{3}^{\rm
COM}=\frac{m_{3}^{2}-m_{4}^{2}+Sx_{1}^{2}s}{2\sqrt{s}x_{1}}=\frac{m_{3}^{2}-m_{4}^{2}+S}{2S},$
(84)

where we have used the relation $S=x_{1}^{2}s$ when $x_{1}=x_{2}$. Therefore,
using eq. (80) and the solution (83) one can define the differential cross-
section in the pp COM reference frame as:

$\displaystyle\frac{d\sigma_{1+2\to
3+4}}{d\Omega}(x_{1},x_{2},\theta)=\frac{1}{4x_{1}x_{2}s^{3/2}(2\pi)^{2}}\frac{|\mathbf{p}_{3}|^{2}}{\left|(x_{1}-x_{2})E_{3}\cos\theta-(x_{1}+x_{2})|\mathbf{p}_{3}|\right|}\left|{\cal
M}\right|^{2}_{1+2\to 3+4},$ (85)

where $|\mathbf{p}_{3}|=\sqrt{E_{3}^{2}-m_{3}^{2}}$ with $E_{3}$ to be the
solution in eq. (83), and we also set the $F={x_{1}x_{2}s}/2$ for massless
initial partons. Furthermore, the existence of the solution of $E_{3}$ will
put a constraint on the $x_{1}$ vs $x_{2}$ plane when fixing the scattering
angle $\theta$ of the particle 3 ($W^{+}$ in our process), which is summarized
in the following inequalities for $\theta\neq\pi/2$:

$\displaystyle\left(m_{3}^{2}s\left(x_{1}-x_{2}\right){}^{2}\cos^{2}(\theta)-m_{3}^{2}\left(2m_{4}^{2}+s\left(x_{1}^{2}+x_{2}^{2}\right)\right)+\left(m_{4}^{2}-sx_{1}x_{2}\right){}^{2}+m_{3}^{4}\right)>0,$
(86) $\displaystyle
x_{1}x_{2}s+m_{3}^{2}-m_{4}^{2}+(x_{1}-x_{2})\sqrt{s}\sqrt{E_{3}^{2}-m_{3}^{2}}\cos\theta>0,$
(87)
$\displaystyle\left[(x_{1}+x_{2})\sqrt{s}E_{3}-(x_{1}x_{2}s+m_{3}^{2}-m_{4}^{2})\right]/(x_{1}-x_{2})\sqrt{s}\cos\theta>0,$
(88)

where the first inequality comes from the requirement of expression under the
square root to be larger than zero, the second and the third ones are obtained
by demanding $E_{3}$ and $\sqrt{E_{3}^{2}-m3^{2}}$ larger than zero. For
$\theta=\pi/2$, the eq. (82) becomes a linear equation, so the only
requirement is $E_{3}>m_{3}$.

We can now use eq. (85) to derive the hadronic differential cross-section with
respect to the solid angle in the pp COM frame:

$\displaystyle\frac{d\sigma_{p+p\to 3+4}}{d\Omega}=$
$\displaystyle\sum_{q_{1},q_{2}}\int dx_{1}dx_{2}\
\left[f_{q_{1}}(x_{1})f_{q_{2}}(x_{2})\frac{d\sigma_{1+2\to
3+4}}{d\Omega}(x_{1},x_{2},\theta)\right.$
$\displaystyle\quad\quad\left.+f_{q_{2}}(x_{1})f_{q_{1}}(x_{2})\frac{d\sigma_{2+1\to
3+4}}{d\Omega}(x_{1},x_{2},\theta)\right]/(1+\delta_{q_{1}q_{2}})$
$\displaystyle=$ $\displaystyle\sum_{q_{1},q_{2}}\int
dx_{1}dx_{2}\frac{1}{2x_{1}x_{2}s^{3/2}(2\pi^{2})}\frac{|\mathbf{p}_{3}|^{2}}{\left|(x_{1}-x_{2})E_{3}\cos\theta-(x_{1}+x_{2})|\mathbf{p}_{3}|\right|}$
$\displaystyle\left[f_{q_{1}}(x_{1})f_{q_{2}}(x_{2})\left|{\cal
M}\right|^{2}_{1+2\to 3+4}+f_{q_{2}}(x_{1})f_{q_{1}}(x_{2})\left|{\cal
M}\right|^{2}_{2+1\to 3+4}\right]/(1+\delta_{q_{1}q_{2}}),$

where $\delta_{q_{1}q_{2}}$ equals $1$ for $q_{1}=q_{2}$ and zero otherwise,
and the invariant scattering amplitudes $\left|{\cal M}\right|^{2}_{2+1\to
3+4}$ and $\left|{\cal M}\right|^{2}_{1+2\to 3+4}$ are related by exchange of
the momenta $p_{1}$ and $p_{2}$, which yields following equations:

$\displaystyle\left|{\cal M}\right|^{2}_{1+2\to 3+4}\equiv\left|{\cal
M}\right|^{2}_{1+2\to 3+4}(S,T),$ (90) $\displaystyle\left|{\cal
M}\right|^{2}_{2+1\to 3+4}=\left|{\cal M}\right|^{2}_{1+2\to 3+4}(S,U),$ (91)

where $S,T,U$ are Mandelstam variables for the parton system defined by:

$\displaystyle S=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2},$ (92) $\displaystyle
T=(p_{1}-p_{3})^{2}=(p_{2}-p_{4})^{2},$ (93) $\displaystyle
U=(p_{1}-p_{4})^{2}=(p_{2}-p_{3})^{2}.$ (94)

If one uses the pseudo-rapidity $\eta$ as the angular variable instead of the
polar angle, then the differential cross-section with respect to the pseudo-
rapidity $\eta$ is related to the differential cross-section of the polar
angle $\theta$ in the following formulas:

$\displaystyle\eta=-\ln(\tan\frac{\theta}{2}),$ (95)
$\displaystyle\frac{d\sigma}{d\eta}=\frac{d\sigma}{d\theta}\frac{d\theta}{d\eta}=\frac{d\sigma}{d\theta}\times\sin\theta.$
(96)

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