and $(\mathcal{O}_{G+},\,\mathcal{O}_{G-})$ (dashed curves).

We then consider the nTGC formulation using the dimension-8 SMEFT operators as
given in Section 2 and study correlations of the sensitivity reaches between
each pair of the nTGC operators. We first study the correlations between the
pairs of nTGC operators $(\mathcal{O}_{G+},\hskip
0.85358pt\mathcal{O}_{\widetilde{B}W})$ and $(\mathcal{O}_{G+},\hskip
0.85358pt\mathcal{O}_{G-})$. We perform the $\chi^{2}$ analysis using
Eq.(4.21) and present the findings in Fig. 6 for the LHC (13 TeV) [panel (a)]
and the 100 TeV $pp$ collider [panel (b)] for a set of sample integrated
luminosities, respectively. In each panel, the $(\mathcal{O}_{G+},\hskip
0.85358pt\mathcal{O}_{\widetilde{B}W})$ correlations are shown by the contours
in solid curves, whereas the $(\mathcal{O}_{G+},\hskip
0.85358pt\mathcal{O}_{G-})$ correlations are depicted by the contours in
dashed curves. We see that the correlations of the operators
$(\mathcal{O}_{G+},\hskip 0.85358pt\mathcal{O}_{\widetilde{B}W})$ and
$(\mathcal{O}_{G+},\hskip 0.85358pt\mathcal{O}_{G-})$ are rather weak, similar
to the case of the $(h_{4},\hskip 0.85358pth_{3}^{V})$ contours in Figs. 5(a)
and (b).

The correlation features of the contours in Fig. 6 can be understood in the
following way. Using the relations in Eq.(3.6), we here denote $\hskip
0.85358pt(x,\hskip 0.85358pty)\\!=\\!(h_{4},\hskip
0.85358pth_{3}^{V})\\!=\\!(r_{4}\bar{x},\hskip
0.85358ptr_{3}^{V}\bar{y})\hskip 0.85358pt$ and $\hskip
0.85358pt(\bar{x},\hskip 0.85358pt\bar{y})=([\Lambda_{V}^{-4}],\hskip
0.85358pt[\Lambda_{G+}^{-4}])$, where $V\\!=\\!Z,A$ and
$(\Lambda_{Z}^{-4},\hskip
0.85358pt\Lambda_{A}^{-4})\\!\equiv\\!(\Lambda_{\widetilde{B}W}^{-4},\hskip
0.85358pt\Lambda_{G-}^{-4})\hskip 0.85358pt$. With these, we express the
leading cross section $\hskip 0.85358pt\sigma_{2}\hskip 0.85358pt$ in
Eqs.(4.22) and (4.23a) as follows:

$\displaystyle\sigma_{2}(\bar{x},\hskip 0.85358pt\bar{y})$
$\displaystyle=A\hskip 0.85358pt\bar{x}^{2}+B\hskip
0.85358pt\bar{y}^{2}+2\hskip 0.85358ptC\hskip 0.85358pt\bar{x}\hskip
0.85358pt\bar{y}\,=\hskip 0.85358pt\overline{\mathbb{X}}\hskip
0.85358ptV^{-1}\hskip 0.85358pt\overline{\mathbb{X}}^{T}\hskip 0.85358pt,$
(4.36a) $\displaystyle(A,\hskip 0.85358ptB,\hskip 0.85358ptC)$
$\displaystyle\equiv\left(\bar{s}^{2}\hskip
0.85358pt(r_{3}^{V})^{2}\bar{\sigma}_{2V}^{33},\hskip
0.85358pt\bar{s}^{3}\hskip 0.85358ptr_{4}^{2}\bar{\sigma}_{2}^{44},\hskip
0.85358pt\mbox{$\frac{\,{1}\,}{2}$}\bar{s}^{2}\hskip
0.85358ptr_{3}^{V}r_{4}\bar{\sigma}_{2V}^{43}\right)\hskip 0.85358pt,$ (4.36b)

where $\hskip 0.85358pt\overline{\mathbb{X}}\equiv(\bar{x},\hskip
0.85358pt\bar{y})\hskip 0.85358pt$ and the matrix $V^{-1}$ takes the form in
Eq.(4.23d). Thus, using Eq.(4.36), we compute the correlation parameter (4.32)
for $(\bar{x},\hskip 0.85358pt\bar{y})=\left([\Lambda_{V}^{-4}],\hskip
0.85358pt[\Lambda_{G+}^{-4}]\right)$ as follows:

$\displaystyle\rho\\!\left([\Lambda_{V}^{-4}],[\Lambda_{G+}^{-4}]\right)\,=\,-\text{sign}(r_{3}^{V}r_{4})\frac{\bar{\sigma}_{2V}^{43}\,\bar{s}^{-\frac{1}{2}}}{\leavevmode\nobreak\
2\sqrt{\bar{\sigma}_{2V}^{33}\hskip
0.85358pt\bar{\sigma}_{2}^{44}\,}\leavevmode\nobreak\
}\,=\,\text{sign}(r_{3}^{V}r_{4})\hskip 0.85358pt\rho(h_{3}^{V},h_{4})\hskip
0.85358pt,$ (4.37)

where the correlation parameter $\hskip
0.85358pt\rho(h_{3}^{V},h_{4})\\!<\\!0\,$ is derived in Eq.(4.33). According
to Eq.(3.7b), we have $\hskip
0.85358pt\text{sign}(r_{3}^{Z}r_{4})\\!<\\!0\hskip 0.85358pt$ and $\hskip
0.85358pt\text{sign}(r_{3}^{\gamma}r_{4})\\!>\\!0\hskip 0.85358pt$. Thus, we
can infer the signs of the corresponding correlation parameters:

$\rho\big{(}[\Lambda_{\widetilde{B}W}^{-4}],[\Lambda_{G+}^{-4}]\big{)}>0\,,\hskip
14.22636pt\rho\\!\left([\Lambda_{G-}^{-4}],[\Lambda_{G+}^{-4}]\right)<0\,.$
(4.38)

These nicely explain why in Fig. 6 the orientations of the correlation
contours (solid curves) of the operators
$(\mathcal{O}_{G+},\,\mathcal{O}_{\widetilde{B}W})$ are slightly aligned
towards to the right-hand-side of the vertical axis, whereas the orientations
of the correlation contours (dashed curves) of the operators
$(\mathcal{O}_{G+},\,\mathcal{O}_{G-})$ are slightly aligned towards to the
left-hand-side of the vertical axis. Their deviations from the vertical axis
are rather small because of the energy suppression factor $\hskip
0.85358pt1/\\!\sqrt{\bar{s}\,}\,$ in Eq.(4.37), and they become even smaller
for the contours of Fig. 6(b) at a 100 TeV $pp$ collider, as expected.

Figure 7:  Correlation contours of sensitivity reaches (95% C.L.) for the
indicated pairs of nTGC operators at the LHC (13 TeV) [ panels $(a)$ and
$(c)$] and the 100 TeV $pp$ collider [ panels $(b)$ and $(d)$]. Panels (a) and
(b) show the correlation contours of
$(\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{G-})$, whereas panels (c) and
(d) depict the correlation contours of
$(\mathcal{O}_{C+},\,\mathcal{O}_{\widetilde{B}W})$ (solid curves) and
$(\mathcal{O}_{C+},\,\mathcal{O}_{G-})$ (dashed curves).

Next, we study the correlations between the nTGC operators
$(\mathcal{O}_{\widetilde{B}W},\hskip 0.85358pt\mathcal{O}_{G-})$ and
$(\mathcal{O}_{C+},\,\mathcal{O}_{G-})$. We perform a $\chi^{2}$ analysis
using Eq.(4.21) and present the findings in Fig. 7. Using the relations
(3.6b)-(3.6c) we find $\hskip
0.85358pt[\Lambda^{-4}_{\widetilde{B}W}]\\!\propto\\!h_{3}^{Z}\hskip
0.85358pt$ and $\hskip
0.85358pt[\Lambda^{-4}_{G-}]\\!\propto\\!h_{3}^{\gamma}\hskip 0.85358pt$. So
we expect that the $(\mathcal{O}_{\widetilde{B}W},\hskip
0.85358pt\mathcal{O}_{G-})$ contour should be related to the
$(h_{3}^{Z},\hskip 0.85358pth_{3}^{\gamma})$ contour. Inspecting the contours
in Figs. 5(c)-(d) and Figs. 7(a)-(b), we see that they all exhibit significant
correlations, but in Figs. 7(a)-(b) the contours are aligned along different
directions from those of Figs. 5(c)-(d). We can understand this difference in
the following way. For convenience, we define $\hskip 0.85358pt(x,\hskip
0.85358pty)\\!=\\!(r_{3}^{Z}\tilde{x},\hskip 0.85358ptr_{3}^{\gamma}\hskip
0.85358pt\tilde{y})\hskip 0.85358pt$ with
$\widetilde{X}\equiv(\tilde{x},\hskip
0.85358pt\tilde{y})=([\Lambda^{-4}_{\widetilde{B}W}],\hskip
0.85358pt[\Lambda^{-4}_{G-}])$. With these and using Eq.(4.34), we express the
leading terms of the cross section $\hskip 0.85358pt\sigma_{2}\hskip
0.85358pt$ as follows:

$\displaystyle\sigma_{2}(\tilde{x},\hskip 0.85358pt\tilde{y})$
$\displaystyle\hskip 0.85358pt=\hskip 0.85358ptA\hskip
0.85358pt\tilde{x}^{2}\\!+B\hskip 0.85358pt\tilde{y}^{2}\\!+2\hskip
0.85358ptC\hskip 0.85358pt\tilde{x}\hskip 0.85358pt\tilde{y}\hskip
0.85358pt=\hskip 0.85358pt\widetilde{X}\hskip
0.85358ptV^{-1}\widetilde{X}^{T},$ (4.39a) $\displaystyle(A,\hskip
0.85358ptB,\hskip 0.85358ptC)$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358pt\left(\bar{s}^{2}(r_{3}^{Z})^{2}\hskip
0.85358pt\bar{\sigma}_{2Z}^{33},\,\bar{s}^{2}(r_{3}^{\gamma})^{2}\hskip
0.85358pt\bar{\sigma}_{2A}^{33},\,\mbox{$\frac{\,{1}\,}{2}$}\bar{s}^{2}r_{3}^{Z}r_{3}^{\gamma}\hskip
0.85358pt\bar{\sigma}_{2ZA}^{33}\right)\\!,$ (4.39b)

where the matrix $V^{-1}$ takes the form of Eq.(4.23d). From the above, we
compute the correlation parameter (4.32) for the operators
$(\mathcal{O}_{\widetilde{B}W},\hskip 0.85358pt\mathcal{O}_{G-})$ as follows:

$\displaystyle\rho\big{(}[\Lambda^{-4}_{\widetilde{B}W}],[\Lambda^{-4}_{G-}]\big{)}\hskip
0.85358pt=\hskip 0.85358pt-\text{sign}(r_{3}^{Z}r_{3}^{\gamma})\hskip
0.85358pt\frac{\bar{\sigma}_{2ZA}^{33}\hskip
0.85358pt\bar{s}^{0}}{\leavevmode\nobreak\ 2\sqrt{\bar{\sigma}_{2Z}^{33}\hskip
0.85358pt\bar{\sigma}_{2A}^{33}\,}\leavevmode\nobreak\
}=\,\text{sign}(r_{3}^{Z}r_{3}^{\gamma})\hskip
0.85358pt\rho(h_{3}^{Z},h_{3}^{\gamma})\hskip 0.85358pt.$ (4.40)

Because Eq.(3.7b) gives $\hskip
0.85358pt\text{sign}(r_{3}^{Z}r_{3}^{\gamma})\\!<\\!0\hskip 0.85358pt$, we
deduce $\hskip
0.85358pt\rho\big{(}[\Lambda^{-4}_{\widetilde{B}W}],[\Lambda^{-4}_{G-}]\big{)}\\!=\\!-\rho(h_{3}^{Z},h_{3}^{\gamma})\\!=\\!O(\bar{s}^{0})\\!\\!>\\!0\hskip
1.42262pt$. This explains why the contours of $\hskip
0.85358pt(\mathcal{O}_{\widetilde{B}W},\hskip 0.85358pt\mathcal{O}_{G-})\hskip
0.85358pt$ in Figs. 7(a) and (b) exhibit strong correlations [similar to those
in Figs. 5(c) and (d)], but have their orientations aligned towards the upper-
right quadrant [unlike Figs. 5(c) and (d), in which all the contours are
oriented towards the upper-left quadrant].

Finally, we examine the correlations of the fermionic contact operator
$\mathcal{O}_{C+}$ with the nTGC operators $\mathcal{O}_{\widetilde{B}W}$ and
$\mathcal{O}_{G-}$. Since $\mathcal{O}_{C+}$ is a combination of two other
operators $\hskip
0.85358pt\mathcal{O}_{C+}\\!\\!=\mathcal{O}_{G-}\\!-\mathcal{O}_{\widetilde{B}W}\hskip
0.85358pt$ viathe equation of motions (2.3a), it is connected to both of the
form factors $(h_{3}^{Z},\hskip 0.85358pth_{3}^{\gamma})$, which would
complicate the correlation analysis in the form factor formulation (4.16).
Instead, we analyze directly the contributions of the operators
$(\mathcal{O}_{C+},\hskip 0.85358pt\mathcal{O}_{G-},\hskip
0.85358pt\mathcal{O}_{\widetilde{B}W})$ to the helicity amplitudes
(2.8)-(2.9). As shown by Eq.(2.9), the operator $\mathcal{O}_{C+}$ has a
nonzero left-handed coupling $\hskip
0.85358ptc_{L(C+)}^{\prime}\\!\\!=\\!-T_{3}\hskip 1.42262pt$ only. So for
examining its correlations with $\mathcal{O}_{G-}$ and
$\mathcal{O}_{\widetilde{B}W}$, the contributions of $\mathcal{O}_{G-}$ and
$\mathcal{O}_{\widetilde{B}W}$ from the left-handed-quark couplings $\hskip
0.85358ptc_{L(G-)}^{\prime}\hskip 0.85358pt$ and $\hskip
0.85358ptc_{L(\widetilde{B}W)}^{\prime}\hskip 0.85358pt$ play key roles. Thus,
we can express as follows the relevant helicity amplitudes (2.8)-(2.9)
containing left-handed (right-handed) initial-state quarks:

$\displaystyle\mathcal{T}_{8L}$
$\displaystyle\,=\,\overline{\mathcal{T}}_{\\!8L}\\!\times\\!\big{\\{}c_{L(C+)}^{\prime}[\Lambda_{C+}^{-4}]+c_{L(\\!\widetilde{B}W)}^{\prime}[\Lambda_{\widetilde{B}W}^{-4}]+c_{L(G-)}^{\prime}[\Lambda_{G-}^{-4}]\big{\\}}$
$\displaystyle\,=\,\overline{\mathcal{T}}_{\\!8L}\hskip 0.85358pt(f_{L0}\hskip
0.85358ptx+f_{L1}\hskip 0.85358pty_{1}+f_{L2}\hskip 0.85358pty_{2})\,,$
(4.41a) $\displaystyle\mathcal{T}_{8R}$
$\displaystyle\,=\,\overline{\mathcal{T}}_{\\!8R}\\!\times\\!\big{\\{}c_{R(\\!\widetilde{B}W)}^{\prime}[\Lambda_{\widetilde{B}W}^{-4}]+c_{R(G-)}^{\prime}[\Lambda_{G-}^{-4}]\big{\\}}\hskip
0.85358pt\equiv\hskip 0.85358pt\overline{\mathcal{T}}_{\\!8R}\,(f_{R1}\hskip
0.85358pty_{1}\\!+\\!f_{R2}\hskip 0.85358pty_{2})\,,$ (4.41b)

where $\overline{\mathcal{T}}_{8L}$ (or $\overline{\mathcal{T}}_{8R}$) is the
remaining common part of the helicity amplitudes (2.8)-(2.9) after separating
out the coupling $\hskip 0.85358ptc_{Lj}^{\prime}$ (or $\hskip
0.85358ptc_{Rj}^{\prime}$) and the cutoff factor $[\Lambda_{j}^{-4}]$. In the
above, we have defined $\hskip 0.85358pt(x,\hskip 0.85358pty_{1},\hskip
0.85358pty_{2})\\!\equiv\\!([\Lambda_{C+}^{-4}],\hskip
0.85358pt[\Lambda_{\widetilde{B}W}^{-4}],\hskip
0.85358pt[\Lambda_{G-}^{-4}])\hskip 0.85358pt$ and

$\displaystyle f_{L0}$ $\displaystyle=c_{L(C+)}^{\prime}\\!=-T_{3}\hskip
0.85358pt,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
f_{L1}\\!=c_{\\!L(\widetilde{B}W)}^{\prime}\\!=T_{3}\\!-Qs_{W}^{2}\hskip
0.85358pt,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
f_{L2}\\!=c_{L(G-)}^{\prime}\\!=-Qs_{W}^{2}\hskip 0.85358pt,$ (4.42a)
$\displaystyle f_{R1}$
$\displaystyle=c_{\\!R(\widetilde{B}W)}^{\prime}\\!=-Qs_{W}^{2}\hskip
0.85358pt,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
f_{R2}=c_{R(G-)}^{\prime}\\!=c_{L(G-)}^{\prime}\\!=f_{L2}\equiv f_{2}\,.$
(4.42b)

With the above, we perform a $\chi^{2}$ analysis based upon Eq.(4.21). We
present the correlation contours of $(\mathcal{O}_{C+},\hskip
0.85358pt\mathcal{O}_{\widetilde{B}W})$ and $(\mathcal{O}_{C+},\hskip
0.85358pt\mathcal{O}_{G-})$ in Figs. 7(c) and (d) for the LHC and the 100 TeV
$pp$ collider, respectively. We find that all these contours exhibit strong
correlations. In particular, the $(\mathcal{O}_{C+},\hskip
0.85358pt\mathcal{O}_{\widetilde{B}W})$ contours (solid curves) are oriented
towards the upper-right quadrant, whereas the $(\mathcal{O}_{C+},\hskip
0.85358pt\mathcal{O}_{G-})$ contours (dashed curves) are oriented towards the
upper-left quadrant.

To understand the qualitative features of the correlation contours in Figs.
7(c) and (d), we examine the cross section $\sigma_{2}$, which contains the
squared part of the dimension-8 contributions and dominates the $\chi^{2}$
function. From Eq.(4.41), we derive the cross section $\sigma_{2}$ as follows:

$\displaystyle\sigma_{2}(x,y_{1},y_{2})$ $\displaystyle\,=\,(f_{L0}\hskip
0.85358ptx+\\!f_{L1}\hskip 0.85358pty_{1}\\!+\\!f_{2}\hskip
0.85358pty_{2})^{2}\\!\left<|\overline{\mathcal{T}}_{\\!8L}|^{\,2}\right>+(f_{R1}\hskip
0.85358pty_{1}\\!+\\!f_{2}\hskip
0.85358pty_{2})^{2}\\!\left<|\overline{\mathcal{T}}_{\\!8R}|^{\,2}\right>$
$\displaystyle\,=\,\left[(f_{L0}\hskip 0.85358ptx+\\!f_{L1}\hskip
0.85358pty_{1}\\!+\\!f_{2}\hskip 0.85358pty_{2})^{2}\\!+\\!(f_{R1}\hskip
0.85358pty_{1}\\!+\\!f_{2}\hskip
0.85358pty_{2})^{2}\right]\\!\left<|\overline{\mathcal{T}}_{\\!8}|^{\,2}\right>,$
(4.43)

where we have defined the notations
$\left<|\overline{\mathcal{T}}_{\\!8L}|^{\,2}\right>\\!=\\!\int_{\mathrm{PS}}|\overline{\mathcal{T}}_{\\!8L}|^{\,2}$
and
$\left<|\overline{\mathcal{T}}_{\\!8R}|^{\,2}\right>\\!=\\!\int_{\mathrm{PS}}|\overline{\mathcal{T}}_{\\!8R}|^{\,2}$
with $\int_{\mathrm{PS}}$ denoting the phase space integration for the final
state. From the squared term of the cross section (2.2), we can further deduce
the equality
$\left<|\overline{\mathcal{T}}_{\\!8L}|^{\,2}\right>\\!=\\!\left<|\overline{\mathcal{T}}_{\\!8R}|^{\,2}\right>\\!\equiv\\!\left<|\overline{\mathcal{T}}_{\\!8}|^{\,2}\right>$,
which is used in the last step of Eq.(4.43).

For analyzing the correlations, the overall factor
$\left<|\overline{\mathcal{T}}_{\\!8}|^{\,2}\right>$ is irrelevant. So we
define the following rescaled cross sections for the convenience of analyzing
the two-parameter correlations:

$\bar{\sigma}_{2}(x,y_{1})\equiv\sigma_{2}(x,y_{1},0)/\\!\left<|\overline{\mathcal{T}}_{\\!8}|^{\,2}\right>,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bar{\sigma}_{2}(x,y_{2})\equiv\sigma_{2}(x,0,y_{2})/\\!\left<|\overline{\mathcal{T}}_{\\!8}|^{\,2}\right>.$
(4.44)

Thus, $\hskip 0.85358pt\bar{\sigma}_{2}(x,y_{1})\hskip 0.85358pt$ and $\hskip
0.85358pt\bar{\sigma}_{2}(x,y_{2})\hskip 0.85358pt$ are expressed in the
following quadratic form:

$\displaystyle\bar{\sigma}_{2}(x,y_{1})$ $\displaystyle=\,A\hskip
0.85358ptx^{2}+B_{1}y_{1}^{2}+2\hskip 0.85358ptC_{1}\hskip 0.85358ptx\hskip
0.85358pty_{1}\equiv\hskip 0.85358pt\mathbb{X}_{1}V_{1}^{-1}\hskip
0.85358pt\mathbb{X}_{1}^{T}\,,$ (4.45a)
$\displaystyle\bar{\sigma}_{2}(x,y_{2})$ $\displaystyle=\,A\hskip
0.85358ptx^{2}+B_{2}y_{2}^{2}+2\hskip 0.85358ptC_{2}\hskip 0.85358ptx\hskip
0.85358pty_{2}\equiv\hskip 0.85358pt\mathbb{X}_{2}V_{2}^{-1}\hskip
0.85358pt\mathbb{X}_{2}^{T}\,,$ (4.45b)

where we have defined $\hskip 0.85358pt\mathbb{X}_{1}\\!\equiv\\!(x,\hskip
0.85358pty_{1})\hskip 0.85358pt$ and $\hskip
0.85358pt\mathbb{X}_{2}\\!\equiv\\!(x,\hskip 0.85358pty_{2})\hskip 0.85358pt$
as well as the following notations,

$\displaystyle(A,\hskip 0.85358ptB_{1},\hskip 0.85358ptC_{1})$
$\displaystyle\equiv(f_{L0}^{2},\,f_{L1}^{2}\\!+\\!f_{R1}^{2},\,f_{L0}f_{L1})\hskip
0.85358pt,$ (4.46a) $\displaystyle(A,\hskip 0.85358ptB_{2},\hskip
0.85358ptC_{2})$ $\displaystyle\equiv(f_{L0}^{2},\,2\hskip
0.85358ptf_{2}^{2},\,f_{L0}f_{2})\hskip 0.85358pt,$ (4.46b) $\displaystyle
V_{1}^{-1}$ $\displaystyle=\\!\left(\\!\\!\\!\begin{array}[]{cc}A&C_{1}\\\
\hskip 0.85358ptC_{1}&B_{1}\end{array}\\!\\!\right)\\!,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
V_{2}^{-1}=\\!\left(\\!\\!\\!\begin{array}[]{cc}A&C_{2}\\\ \hskip
0.85358ptC_{2}&B_{2}\end{array}\\!\\!\right)\\!.$ (4.46g)

Thus, we can deduce the following correlation parameter for the two cases:

$\displaystyle\rho_{1}(x,y_{1})$ $\displaystyle\,=\hskip
0.85358pt\frac{-\hskip
0.85358ptC_{1}}{\,\sqrt{AB_{1}}\,}=\frac{\leavevmode\nobreak\ -\hskip
0.85358pt\text{sign}(f_{L0}f_{L1})\leavevmode\nobreak\
}{\,\sqrt{1\\!+\\!f_{R1}^{2}/\\!f_{L1}^{2}\,}\,}>0\,,$ (4.47a)
$\displaystyle\rho_{2}(x,y_{2})$ $\displaystyle\,=\hskip
0.85358pt\frac{-\hskip
0.85358ptC_{2}}{\,\sqrt{AB_{2}}\,}=-\frac{1}{\sqrt{2\,}\,}\hskip
0.85358pt\text{sign}(f_{L0}f_{2})<0\,,$ (4.47b)

where $\hskip 0.85358pt(x,\hskip
0.85358pty_{1})\equiv([\Lambda_{C+}^{-4}],\hskip
0.85358pt[\Lambda_{\widetilde{B}W}^{-4}])\hskip 0.85358pt$ and $(x,\hskip
0.85358pty_{2})\equiv([\Lambda_{C+}^{-4}],\hskip
0.85358pt[\Lambda_{G-}^{-4}])$. Using the coupling formula (4.42), we derive
$\hskip 0.85358ptf_{L0}\hskip 0.85358ptf_{L1}\\!=\\!-T_{3}\hskip
0.85358pt(T_{3}-Qs_{W}^{2})\\!<0\,$ and $\hskip
0.85358ptf_{L0}f_{2}\\!=\\!T_{3}Q\hskip 0.85358pts_{W}^{2}\\!>0\,$, where each
inequality holds for both up-type and down-type quarks. From these, we deduce
that the operators $(\mathcal{O}_{C+},\hskip
0.85358pt\mathcal{O}_{\widetilde{B}W})$ are correlated positively, whereas the
operators $(\mathcal{O}_{C+},\hskip 0.85358pt\mathcal{O}_{G-})$ are correlated
negatively. Moreover, Eq.(4.47) shows that both correlation parameters are of
$\hskip 0.85358ptO(\bar{s}^{0})\hskip 0.85358pt$ and not suppressed by any
energy factor. This predicts strong correlations for the operators
$(\mathcal{O}_{C+},\hskip 0.85358pt\mathcal{O}_{\widetilde{B}W})$ and
$(\mathcal{O}_{C+},\hskip 0.85358pt\mathcal{O}_{G-})$, respectively. These
features are indeed reflected in Figs. 7(c) and (d). We see that the
correlation contours of $\,(\mathcal{O}_{C+},\,\mathcal{O}_{\widetilde{B}W})$
(solid curves) are oriented towards the upper-right quadrant due to the
positive correlation parameter $\hskip 0.85358pt\rho_{1}(x,y_{1})\\!>\\!0\,$
given by Eq.(4.47a), whereas the correlation contours of
$(\mathcal{O}_{C+},\,\mathcal{O}_{G-})$ (dashed curves) are aligned towards
the upper-left quadrant due to the negative correlation parameter $\hskip
0.85358pt\rho_{2}(x,y_{2})\\!<\\!0\,$ given by Eq.(4.47b).

###  4.7 Comparison with the Existing LHC Bounds on nTGCs

In this subsection, we make direct comparison with the published LHC
measurements of nTGCs through the reaction $\hskip 0.85358ptp{\hskip
0.85358pt}p{\hskip 0.85358pt}(q{\hskip
0.85358pt}\bar{q})\\!\rightarrow\\!Z\gamma\hskip 0.85358pt$ with
$Z\\!\\!\rightarrow\\!\nu{\hskip 0.85358pt}\bar{\nu}{\hskip 0.85358pt}\hskip
0.85358pt$ by the ATLAS [19] and CMS [18] collaborations using the
conventional nTGC form factor formula (3.1). The CMS collaboration analyzed
19.6 fb-1 of Run-1 data at $\sqrt{s\,}\\!=\\!8$ TeV [18], whereas the ATLAS
collaboration analyzed 36.1 fb-1 of Run-2 data at $\sqrt{s\,}=13$ TeV [19].
They obtained the following sensitivity bounds (95% C.L.) on the form factors:

CMS: $\displaystyle h_{3}^{Z}\in(-1.5,\,1.6)\\!\times\\!10^{-3},$
$\displaystyle h_{3}^{\gamma}\in(-1.1,\,0.9)\\!\times\\!10^{-3},$
$\displaystyle h_{4}^{Z}\in(-3.9,\,4.5)\\!\times\\!10^{-6},$ $\displaystyle
h_{4}^{\gamma}\in(-3.8,\,4.3)\\!\times\\!10^{-6};$ (4.48a) ATLAS:
$\displaystyle h_{3}^{Z}\in(-3.2,\,3.3)\\!\times\\!10^{-4},$ $\displaystyle
h_{3}^{\gamma}\in(-3.7,\,3.7)\\!\times\\!10^{-4},$ $\displaystyle
h_{4}^{Z}\in(-4.5,\,4.4)\\!\times\\!10^{-7},$ $\displaystyle
h_{4}^{\gamma}\in(-4.4,\,4.3)\\!\times\\!10^{-7}.$ (4.48b)

We see that the CMS and ATLAS analyses both obtained much stronger bounds on
$(h_{4}^{Z},\,h_{4}^{\gamma})$ than on $(h_{3}^{Z},\,h_{3}^{\gamma})$, i.e.,
by factors $\sim\\!(210-380)$ at CMS (Run-1) and $\sim\\!(710-860)$ at ATLAS
(Run-2). In comparison, we see in Table 7 using our SMEFT form factor
formulation (3.5) that the LHC sensitivity bounds on $h_{4}^{V}$ are stronger
than those on $\hskip 0.85358pth_{3}^{V}$ only by factors of about $20\hskip
0.85358pt$. Our Table 8 further demonstrates that using the conventional form
factor formulation (3.1) would generate spuriously stronger $h_{4}^{V}$ bounds
(marked in blue) at the LHC (13 TeV) than the SMEFT bounds (marked in red) by
a factor of about $20$, and thus much stronger than the $h_{3}^{V}$ bounds by
a large factor of $\sim\\!20\\!\times\\!20=400\hskip 0.85358pt$, which agrees
with the ATLAS results in Eq.(4.48) within a factor of $2\hskip
1.42262pt$.555Since our analyses in Tables 7-8 have used as input the full
Run-2 integrated luminosity of $140\,$fb-1 as well as different kinematic cuts
for each bin, unlike the experimental analyses of ATLAS [19] and CMS [18],
such a minor difference in the bounds could be expected. Unfortunately, this
means that the strong experimental bounds (4.48) on
$(h_{4}^{Z},\,h_{4}^{\gamma})$ are unreliable because they were obtained by
using the conventional form factor formulation (3.1), which does not respect
the SM electroweak gauge symmetry of SU(2)${}_{\mathrm{L}}\otimes\hskip
0.85358pt$U(1)Y as incorporated in the SMEFT.

To study quantitatively the conventional parametrization (3.1) including the
nTGC form factors $(h_{3}^{V},\hskip 0.85358pth_{4}^{V})$ only, we denote
their contributions to the total cross section by
$\,\widetilde{\sigma}(Z\gamma)\\!=\sigma_{0}+\widetilde{\sigma}_{1}+\widetilde{\sigma}_{2}\,$,
where $\widetilde{\sigma}_{1}$ is the interference term and
$\widetilde{\sigma}_{2}$ is the squared contribution. This is similar to what
we did around Eq.(4.18). We find that $\widetilde{\sigma}_{2}$ always
dominates over $\widetilde{\sigma}_{1}$ for both the LHC and the 100 TeV $pp$
collider. Using the conventional form factor formula (3.1), we derive the
squared contribution $\widetilde{\sigma}_{2}$ as follows:

$\widetilde{\sigma}_{2}=\frac{\leavevmode\nobreak\
e^{4}(X_{LR}^{V})^{2}(\hat{s}-M_{Z}^{2})^{3}\\!\left[4(h_{3}^{V})^{2}\\!\left(\\!\frac{M_{Z}^{2}}{\hat{s}}+1\\!\right)\\!+4\hskip
0.85358pth_{3}^{V}h_{4}^{V}\\!\left(\\!1\\!-\\!\frac{\hat{s}}{M_{Z}^{2}}\\!\right)\\!+\\!(h_{4}^{V})^{2}\\!\left(\\!1\\!-\\!\frac{\hat{s}}{M_{Z}^{2}}\\!\right)^{\\!\\!2}\right]\leavevmode\nobreak\
}{768\hskip 0.85358pt\pi\hskip
0.85358pts_{W}^{2}c_{W}^{2}M_{Z}^{6}\,\hat{s}},$ (4.49)

where the coupling factor $(X_{LR}^{V})^{2}$ is defined as

$\displaystyle(X_{LR}^{V})^{2}\equiv(X_{L}^{V})^{2}\\!+\\!(X_{R}^{V})^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
X_{L,R}^{Z}\equiv x_{L,R}^{Z}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
X_{L,R}^{\gamma}\equiv-\frac{c_{W}}{s_{W}}x_{L,R}^{A}\,,\hskip 28.45274pt$
(4.50a) $\displaystyle(x_{L}^{Z},\hskip
0.85358ptx_{R}^{Z})=(T_{3}\\!-\\!Qs_{W}^{2},\hskip 0.85358pt-Qs_{W}^{2})\hskip
0.85358pt,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (x_{L}^{A},\hskip
0.85358ptx_{R}^{A})=-Qs_{W}^{2}(1,\hskip 0.85358pt1)\hskip 0.85358pt.$ (4.50b)

Defining a scaled dimensionless energy parameter
$\,\bar{s}=\hat{s}/M_{Z}^{2}\,$ and making the high-energy expansion for
$\,\bar{s}\\!\gg\\!1\,$, we can compare the leading energy-dependence of each
term of $\,\widetilde{\sigma}_{2}$ with that of $\,\sigma_{2}\hskip
0.85358pt$, as follows:

$\displaystyle\widetilde{\sigma}_{2}$
$\displaystyle\,\approx\,\frac{\leavevmode\nobreak\
e^{4}(X_{LR}^{V})^{2}\\!\left[(h_{4}^{V})^{2}\hskip
0.85358pt\bar{s}^{4}\\!-4\hskip
0.85358pth_{4}^{V}h_{3}^{V}\bar{s}^{3}\\!+4(h_{3}^{V})^{2}\hskip
0.85358pt\bar{s}^{2}\hskip 0.85358pt\right]\leavevmode\nobreak\ }{768\hskip
0.85358pt\pi\hskip 0.85358pts_{W}^{2}\hskip 0.85358ptc_{W}^{2}M_{Z}^{2}}\hskip
0.85358pt,$ (4.51a) $\displaystyle\sigma_{2}$
$\displaystyle\,\approx\,\frac{\leavevmode\nobreak\
e^{4}\\!\left[T_{3}^{2}(h_{4}^{Z})^{2}\hskip 0.85358pt\bar{s}^{3}\\!+8\hskip
0.85358ptT_{3}X_{L}^{V}h_{4}^{Z}h_{3}^{V}\hskip
0.85358pt\bar{s}^{2}\\!+\\!4\hskip
0.85358pt(X_{LR}^{V})^{2}(h_{3}^{V})^{2}\hskip 0.85358pt\bar{s}^{2}\hskip
0.85358pt\right]\leavevmode\nobreak\ }{\leavevmode\nobreak\ 768\hskip
0.85358pt\pi s_{W}^{2}\hskip 0.85358ptc_{W}^{2}M_{Z}^{2}\,}\hskip 0.85358pt,$
(4.51b)

where the cross section $\,\sigma_{2}\hskip 0.85358pt$ is given by our SMEFT
form factor formula (3.5). We note that the form factors
$(h_{4}^{Z},\,h_{4}^{\gamma})$ in the above cross section $\hskip
0.85358pt\sigma_{2}\hskip 0.85358pt$ should obey the condition (3.4b) due to
the underlying electroweak gauge symmetry of the SM that is respected by the
corresponding dimension-8 nTGC operators. We have used the relation (3.4b) to
combine the $\hskip 0.85358pth_{4}^{\gamma}\hskip 0.85358pt$ contribution with
that of $\hskip 0.85358pth_{4}^{Z}\hskip 0.85358pt$. To examine the
correlation of $(h_{3}^{\gamma},\,h_{4}^{\gamma})$ from Eq.(4.51b), we can use
Eq.(3.4b) to replace $\hskip 0.85358pth_{4}^{Z}\hskip 0.85358pt$ by $\hskip
0.85358pth_{4}^{\gamma}\hskip 0.85358pt$. Inspecting Eq.(4.51), we see that
both the $(h_{4}^{V})^{2}$ and $(h_{4}^{V}h_{3}^{V})$ terms in $\hskip
0.85358pt\widetilde{\sigma}_{2}$ have higher energy dependences than those of
$\sigma_{2}$ by an extra factor $\hskip 0.85358pt\bar{s}^{1}$, which leads
erroneously to much stronger bounds on $h_{4}^{V}\hskip 0.85358pt$.

We first make a one-parameter analysis and derive the bound on each form
factor coefficient $\,h_{j}^{V}$ individually (where $j\\!=\\!3,4\hskip
0.85358pt$ and $\hskip 0.85358ptV\\!=\\!Z,\gamma\,$) using the conventional
form factor parametrization (3.1). To make a more precise comparison with the
ATLAS bounds (4.48), we adopt the same kinematic cut on the transverse
momentum of the final-state photon, $P_{T}^{\gamma}$ > 600 GeV, and the same
integrated luminosity $\hskip 0.85358pt\mathcal{L}=36.1$ fb-1 as in the ATLAS
analysis [19]. For illustration, we ignore the other detector-level cuts and
the systematic errors, and choose a typical detection efficiency
$\,\epsilon\\!=\\!75\%\hskip 1.42262pt$.666We thank our ATLAS colleague Shu Li
for discussing the typical detection efficiency of the ATLAS detector [19].
With these, we derive the following bounds on the nTGCs (95% C.L.) when using
the conventional form factor parametrization (3.1):

$|h_{3}^{Z}|<3.0\\!\times\\!10^{-4},\hskip
14.22636pt|h_{3}^{\gamma}|<3.4\\!\times\\!10^{-4},\hskip
14.22636pt|h_{4}^{Z}|<4.4\\!\times\\!10^{-7},\hskip
14.22636pt|h_{4}^{\gamma}|<4.9\\!\times\\!10^{-7},$ (4.52)

and note that the squared nTGC contributions dominate the sensitivity.
Comparing the above estimated bounds (4.52) with the ATLAS experimental bounds
(4.48), we see that they agree well with each other: the agreements for
$h_{4}^{Z}$ are within about $2\%\hskip 1.42262pt$ and the agreements for
$(h_{4}^{\gamma},h_{3}^{Z},\hskip 0.85358pth_{3}^{\gamma})$ are within about
$(8\\!-\\!13)\%\hskip 1.42262pt$. This means that by making plausible
simplifications we can reproduce quite accurately the experimental bounds
(4.48) established by the ATLAS collaboration [19] using the conventional form
factor formulation in Eq.(3.1).

Next, we analyze the correlation contours for $(h_{3}^{\gamma},\hskip
0.85358pth_{4}^{\gamma})$ and $(h_{3}^{Z},\hskip 0.85358pth_{4}^{Z})$,
respectively, using the conventional form factor parametrization (3.1), which
can be compared to the correlation contours obtained by using our SMEFT form
factor formulation (3.5). Fig. 8 displays the correlation contours at 95% C.L.
for LHC Run-2. Panels (a) and (b) show the correlation contours based on the
SMEFT form factor formula (3.5), where the blue (red) contours correspond to
inputting integrated LHC luminosities of $36.1\hskip 0.85358pt$fb-1
($140\hskip 0.85358pt$fb-1). Panels (c) and (d) present the correlation
contours based on the conventional form factor parametrization (3.1), where
the red and blue contours are given by our theoretical analysis with the
assumed detection efficiencies $\hskip 0.85358pt\epsilon\\!=\\!100\%\hskip
0.85358pt$ and $\hskip 0.85358pt\epsilon\\!=\\!75\%\hskip 0.85358pt$
respectively. For comparison, we show in panels (c) and (d) the experimental
contours as extracted from the ATLAS results [19] based on the conventional
form factor formula (3.1), where the black solid curves depict the observed
bounds and the black dashed curves show the expected limits. It is impressive
to see in panels (c) and (d) that our theoretical contours agree well with the
experimental contours obtained by using the conventional form factor
parametrization (3.1).

Figure 8:  Correlation contours of the sensitivity reaches (95% C.L.) for the
indicated pairs of nTGC form factors at the LHC (13 TeV). Panels (a) and (b)
present the correlation contours for $(h_{3}^{\gamma},\,h_{4}^{\gamma})$ and
$(h_{3}^{Z},\,h_{4}^{Z})$ respectively, by using our SMEFT form factor formula
(3.5), where in each panel the red contour inputs the full integrated
luminosity 140 fb${}^{-1}\\!\\!$ of Run-2 and the blue contour inputs a
partial integrated luminosity 36.1 fb${}^{-1}\\!$ as in the ATLAS analysis
[19]. Panels (c) and (d) compare the theoretical correlation contours (red and
blue colors) with the experimental contours (black color) from the ATLAS
analysis [19], where we derived the red and blue contours by using the
conventional form factor formula (3.1) and by assuming an ideal detection
efficiency $\epsilon\\!=\\!100\%$ (for red contours) or a reduced detection
efficiency $\hskip 0.85358pt\epsilon\\!=\\!75\%$ (for blue contours). The
ATLAS contours are shown by the black solid curves (observed) and the black
dashed curves (expected).

We note that the correlation contours of panels (a) and (b) in Fig. 8 have
very different features from those of panels (c) and (d), which can be
understood as follows. For convenience, we denote $\hskip
0.85358pt\mathbb{X}\\!=\\!(x,\,y)\\!\equiv\\!(h_{3}^{V},\hskip
0.85358pth_{4}^{Z})$. Thus, we can express the cross sections of Eqs.(4.51a)
and (4.51b) as follows:

$\displaystyle\widetilde{\sigma}_{2}$
$\displaystyle\,\propto\,\widetilde{A}{x}^{2}\\!+\widetilde{B}{y}^{2}\\!+2\hskip
0.85358pt\widetilde{C}{x}\hskip 0.85358pt{y}\hskip 0.85358pt\equiv\hskip
0.85358pt\mathbb{X}\hskip 0.85358pt\widetilde{V}^{-1}\hskip
0.85358pt\mathbb{X}^{T},$ (4.53a) $\displaystyle\sigma_{2}$
$\displaystyle\,\propto\,Ax^{2}\\!+By^{2}\\!+2\hskip 0.85358ptCx\hskip
0.85358pty\hskip 0.85358pt\equiv\hskip 0.85358pt\mathbb{X}\hskip
0.85358ptV^{-1}\hskip 0.85358pt\mathbb{X}^{T},$ (4.53b)

where we have defined the following notations,

$\displaystyle(\widetilde{A},\hskip 0.85358pt\widetilde{B},\hskip
0.85358pt\widetilde{C})$ $\displaystyle\,=\hskip 0.85358pt(4\hskip
0.85358pt\bar{s}^{2},\hskip 0.85358pt\bar{s}^{4},\hskip
0.85358pt-2\bar{s}^{3})\hskip 0.85358pt,$ (4.54a) $\displaystyle(A,\hskip
0.85358ptB,\hskip 0.85358ptC)$ $\displaystyle\,=\hskip 0.85358pt\left(4\hskip
0.85358pt\bar{s}^{2}(X_{LR}^{V})^{2},\hskip
0.85358pt\bar{s}^{3}T_{3}^{2},\hskip 0.85358pt4\hskip
0.85358pt\bar{s}^{2}X_{L}^{V}T_{3}\right),$ (4.54b)
$\displaystyle\widetilde{V}^{-1}$
$\displaystyle=\\!\left(\\!\\!\\!\begin{array}[]{cc}\widetilde{A}&\widetilde{C}\\\\[1.13809pt]
\hskip
0.85358pt\widetilde{C}&\widetilde{B}\end{array}\\!\right)\\!,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
V^{-1}=\\!\left(\\!\\!\\!\begin{array}[]{cc}A&C\\\\[0.85358pt] \hskip
0.85358ptC&B\end{array}\\!\right)\\!.$ (4.54g)

With these we can compute the correlation parameter of the form factors in
each case:

$\displaystyle\widetilde{\rho}\hskip 0.85358pt=\frac{-\hskip
0.85358pt\widetilde{C}}{\,\sqrt{\widetilde{A}\hskip
0.85358pt\widetilde{B}\,}\,}=1\\!>0\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \rho\hskip 0.85358pt=\frac{-\hskip
0.85358ptC}{\,\sqrt{AB\,}\,}=-\frac{\,2\hskip
1.42262pt\text{sign}(T_{3})X_{L}^{V}\,}{|X_{LR}^{V}|\,}\hskip
0.85358pt\bar{s}^{-\frac{1}{2}}\\!<0\,.$ (4.55)

The fact of $\hskip
0.85358pt\widetilde{\rho}\\!=\\!O(\bar{s}^{0})\\!\\!>\\!0\hskip 0.85358pt$
explains why the $\hskip 0.85358pt(h_{3}^{V},\hskip 0.85358pth_{4}^{V})\,$
contours in Figs. 8(c) and (d) exhibit strong correlations and have their
orientations aligned towards the upper-right quadrant. On the other hand, from
Eq.(4.50) we find that $\hskip 0.85358pt\text{sign}(T_{3})X_{L}^{V}\\!>\\!0\,$
holds for the initial-state quarks being either up-type or down-type, and thus
Eq.(4.55) gives $\hskip 0.85358pt\rho\\!<\\!0\hskip 1.42262pt$. This means
that the $\hskip 0.85358pt(h_{3}^{V},\hskip 0.85358pth_{4}^{V})\,$ contours in
Figs. 8(a) and (b) should have their orientations towards the upper-left
quadrant, but this correlation is almost invisible because $\hskip
0.85358pt\rho\\!=\\!O(\bar{s}^{-\frac{1}{2}})\hskip 0.85358pt$ receives a
large energy-suppression factor at the LHC. Thus, the correlation features of
the $\hskip 0.85358pt(h_{3}^{V},\hskip 0.85358pth_{4}^{V})\,$ contours are
well understood both for Figs. 8(a)-(b) [based on the SMEFT form factor
formula (3.5)] and for Figs. 8(c)-(d) [based on the conventional form factor
formula (3.1)].

Our quantitative comparisons in Figs. 8 are instructive and encouraging. We
suggest that the ATLAS and CMS colleagues perform a systematic nTGC analysis
based on the new SMEFT form factor formula (3.5), using the full Run-2 data
set. Moreover, we note that in Refs. [18]-[19] the CMS and ATLAS
collaborations analyzed the correlations between the form factors
$(h_{3}^{V},\,h_{4}^{V})$ and found strong correlations. We have reproduced
this feature in Figs. 8(c)-(d), but we note that those correlation contours
differ substantially from our new correlation contours in Figs. 8(a)-(b).
Based upon the above analysis, we suggest that the CMS and ATLAS
collaborations should make updated analyses on the $(h_{3}^{V},\,h_{4}^{V})$
correlations using our new SMEFT form factor formulation with their full Run-2
data sets. We anticipate that such new analyses should yield results similar
to the theoretical predictions for LHC Run-2 given in Table 7 and Figs.
8(a)-(b).

##  5 Comparison with Probes of nTGCs at Lepton Colliders

In this Section we first summarize the sensitivity reaches of nTGC new physics
scales at high-energy $e^{+}e^{-}$ colliders found in our previous work [5].
Then we analyze the sensitivity reaches of the nTGC form factors at these
$e^{+}e^{-}$ colliders. Finally, we compare these sensitivity limits with
those obtained for the hadron colliders as given in Section 4 of the present
study.

At high-energy $e^{+}e^{-}$ colliders, we found in Ref. [5] that the reaction
$\,e^{+}e^{-}\\!\\!\rightarrow\\!Z\gamma\,$ with hadronic decays $\hskip
0.85358ptZ\\!\\!\rightarrow\\!q\hskip 0.85358pt\bar{q}\,$ gives greater
sensitivity reach than the leptonic and invisible decays $\hskip
0.85358ptZ\\!\rightarrow\\!\ell\hskip 0.85358pt\bar{\ell},\nu\hskip
0.85358pt\bar{\nu}\hskip 0.85358pt$. Therefore we choose for comparison the
sensitivity reaches obtained using hadronic $Z$ decays, and consider the
$e^{+}e^{-}$ collision energies $\sqrt{s}=(0.25,\hskip 0.85358pt0.5,\hskip
0.85358pt1,\hskip 0.85358pt3,\hskip 0.85358pt5)$ TeV with a benchmark
integrated luminosities $\mathcal{L}=5\hskip 0.85358pt$ab-1. These results are
summarized in the upper half of Table 9 for the new physics scale
$\,\Lambda\,$ of each dimension-8 nTGC operator or related contact operator
$(\mathcal{O}_{G+},\,\mathcal{O}_{G-},\,\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{C+})$
at the $2{\hskip 0.85358pt}\sigma$ level, where each entry has two limits
which correspond to the (unpolarized, polarized) $e^{\mp}$ beams. For the
polarized $e^{\mp}$ beams, we choose the benchmark polarizations
$(P_{L}^{e},\,P_{R}^{\bar{e}})=(0.9,\,0.65)$. For comparison, we summarize in
the lower half of Table 9 the sensitivity reaches of $\,\Lambda\,$ via the
reaction $\,pp(q{\hskip 0.85358pt}\bar{q})\\!\rightarrow\\!Z\gamma\,$ with
$\hskip 0.85358ptZ\\!\rightarrow\\!\ell\hskip 0.85358pt\bar{\ell},\nu\hskip
0.85358pt\bar{\nu}\hskip 0.85358pt$ at the LHC (13 TeV) and the 100 TeV
$\\!pp$ collider, based on Tables 4 and 5 of Section 3.

From the comparison in Table 9, we see that the the sensitivity reaches for
the nTGC operator $\mathcal{O}_{G+}$ (and also the contact operator
$\mathcal{O}_{C-}$) at the LHC (13 TeV) with integrated luminosities
$\mathcal{L}\\!=\\!(0.14,\,0.3,\,3)$ ab-1 are higher than those of
$e^{+}e^{-}$ colliders with collision energies $\sqrt{s}=(250,\hskip
0.85358pt500)$ GeV, and are comparable to those of an $e^{+}e^{-}$ collider of
energy $\sqrt{s}=1$ TeV, but much lower than that of the CLIC with
$\sqrt{s}=(3-5)$ TeV. On the other hand, the sensitivity reaches of the 100
TeV $pp$ collider with an integrated luminosity $\mathcal{L}=3\hskip
1.42262pt$ab-1 can surpass those of all the $e^{+}e^{-}$ colliders with
collision energies up to $(3-5)$ TeV.

$\sqrt{s\,}$ (TeV) |  $\mathcal{L}$ (ab-1) | $\Lambda_{G+}$ | $\Lambda_{G-}$ | $\Lambda_{\\!\widetilde{B}W}$ | $\Lambda_{C+}$
---|---|---|---|---|---
$e^{+}e^{-}$ (0.25) | 5 | (1.3, 1.6) | (0.90, 1.2) | (1.2, 1.3) | (1.2, 1.6)
$e^{+}e^{-}$ (0.5) | 5 | (2.3, 2.7) | (1.4, 1.7) | (1.8, 1.9) | (1.8, 2.2)
$e^{+}e^{-}$ (1) | 5 | (3.9, 4.7) | (1.9, 2.5) | (2.5, 2.6) | (2.6, 2.9)
$e^{+}e^{-}$ (3) | 5 | (9.2, 11.0) | (3.4, 4.3) | (4.3, 4.5) | (4.4, 5.2)
$e^{+}e^{-}$ (5) | 5 |  (13.4, 15.9) |  (4.4, 5.6) |  (5.7, 5.9) |  (5.7, 6.8)
| 0.14 |  3.3 |  1.1 |  1.3 |  1.4
LHC (13) | 0.3 |  3.6 |  1.2 |  1.4 |  1.5
| 3 |  4.2 |  1.4 |  1.7 |  1.7
| 3 |  23 |  4.6 |  5.6 |  5.9
$pp$ (100) | 10 |  26 |  5.1 |  6.1 |  6.5
| 30 |  28 |  5.5 |  6.7 |  7.1
Table 9: Comparisons of $\,2{\hskip 0.85358pt}\sigma$ sensitivities to the new
physics scale $\,\Lambda\,$ (in TeV) for each dimension-8 nTGC operator or
related contact operator
$(\mathcal{O}_{G+},\,\mathcal{O}_{G-},\,\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{C+})$,
at $e^{+}e^{-}\\!$ colliders of different collision energies, and at the LHC
and the $pp$ (100 TeV) collider. The reactions
$\,e^{-}e^{+}\\!\\!\rightarrow\\!Z\gamma\\!\\!\rightarrow\\!\\!q\bar{q}\gamma$
and $\,pp\hskip 0.85358pt(q{\hskip
0.85358pt}\bar{q})\\!\\!\rightarrow\\!\\!Z\gamma\\!\\!\rightarrow\\!\\!\ell\bar{\ell}\gamma,\hskip
0.85358pt\nu\bar{\nu}\gamma$ are analyzed for the lepton and hadron colliders
respectively. For the $\hskip 0.85358pte^{+}e^{-}\\!$ colliders, each entry
corresponds to (unpolarized, polarized) $e^{\mp}$ beams, where we choose the
benchmark $e^{\mp}$ beam polarizations as $(P_{L}^{e},\hskip
0.85358ptP_{R}^{\bar{e}})=(0.9,\,0.65)$.

We consider next the other three dimension-8 operators
$(\mathcal{O}_{G-},\,\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{C+})$. Table
9 shows that the LHC has sensitivities to $\Lambda$ that are comparable to
those of $e^{+}e^{-}$ colliders with $\sqrt{s\,}=(250,\hskip 0.85358pt500)$
GeV, but are clearly lower than those of $e^{+}e^{-}$ colliders with collision
energies $\sqrt{s\,}\\!\geqslant\\!1$ TeV. On the other hand, we find that the
sensitivities of the 100 TeV $\\!pp$ collider with an integrated luminosity
$\mathcal{L}\\!=\\!3\hskip 0.85358pt$ab-1 are significantly greater than those
of the $e^{+}e^{-}$ colliders with energy $\sqrt{s\,}\\!\leqslant\\!3$ TeV.
Moreover, a 100 TeV $\\!pp$ collider with an integrated luminosity $\hskip
0.85358pt\mathcal{L}=(10-30)\hskip 0.85358pt$ab-1 has sensitivities comparable
to those of an $e^{+}e^{-}$ collider with $\sqrt{s\,}=5\hskip 0.85358pt$ TeV,
while a 100 TeV $\\!pp$ collider with an integrated luminosity of $30\hskip
0.85358pt$ab-1 would have higher sensitivities than an $e^{+}e^{-}$ collider
with $\sqrt{s\,}=5$ TeV. In passing, we find that our collider limits given in
Table 9 are much stronger than the unitarity limits of Table 1 and Fig. 1.
This shows that the perturbation expansion in the SMEFT formulation is well
justified for the present collider analyses of probing the nTGCs.

$\sqrt{s\,}$ (TeV) |  $\mathcal{L}$ (ab-1) | $|h_{4}|$ | $|h_{3}^{Z}|$ | $|h_{3}^{\gamma}|$
---|---|---|---|---
$e^{+}e^{-}$ (0.25) | 5 |  (3.9, 2.0)$\times 10^{-4}$ |  (2.7, 2.3)$\times 10^{-4}$ |  (4.9, 1.6)$\times 10^{-4}$
$e^{+}e^{-}$ (0.5) | 5 |  (3.8, 1.9)$\times 10^{-5}$ |  (6.2, 5.2)$\times 10^{-5}$ |  (10, 3.7)$\times 10^{-5}$
$e^{+}e^{-}$ (1) | 5 |  (4.5, 2.3)$\times 10^{-6}$ |  (1.5, 1.2)$\times 10^{-5}$ |  (2.3, 1.0)$\times 10^{-5}$
$e^{+}e^{-}$ (3) | 5 |  (1.6, 0.84)$\times 10^{-7}$ |  (1.7, 1.4)$\times 10^{-6}$ |  (2.5, 1.0)$\times 10^{-6}$
$e^{+}e^{-}$ (5) | 5 |  (3.6, 1.8)$\times 10^{-8}$ |  (5.8, 4.9)$\times 10^{-7}$ |  (8.9, 3.4)$\times 10^{-7}$
| 0.14 |  9.6$\times 10^{-6}$ |  1.9$\times 10^{-4}$ |  2.2$\times 10^{-4}$
LHC (13) | 0.3 |  7.5$\times 10^{-6}$ |  1.5$\times 10^{-4}$ |  1.8$\times 10^{-4}$
| 3 |  3.8$\times 10^{-6}$ |  0.80$\times 10^{-4}$ |  0.97$\times 10^{-4}$
| 3 |  4.0$\times 10^{-9}$ |  6.1$\times 10^{-7}$ |  7.2$\times 10^{-7}$
$pp$ (100) | 10 |  2.6$\times 10^{-9}$ |  4.2$\times 10^{-7}$ |  4.9$\times 10^{-7}$
| 30 |  1.9$\times 10^{-9}$ |  3.0$\times 10^{-7}$ |  3.5$\times 10^{-7}$
Table 10:  Sensitivity reaches on the nTGC form factors at the $2\hskip
0.85358pt\sigma$ level of $e^{+}e^{-}\\!\\!$ colliders with different
collision energies, compared with those of the LHC and the $pp$ (100 TeV)
collider. The reactions
$\,e^{-}e^{+}\\!\\!\rightarrow\\!Z\gamma\\!\rightarrow\\!q\bar{q}\gamma$ and
$\,pp(q{\hskip
0.85358pt}\bar{q})\\!\rightarrow\\!Z\gamma\\!\rightarrow\\!\ell\bar{\ell}\gamma,\hskip
0.85358pt\nu\bar{\nu}\gamma\hskip 0.85358pt$ are considered for the lepton and
hadron colliders respectively. For the $e^{+}e^{-}$ colliders, each entry
corresponds to (unpolarized, polarized) $e^{\mp}$ beams. As benchmarks for the
$e^{\mp}$ beam polarizations we choose
$(P_{L}^{e},\,P_{R}^{\bar{e}})=(0.9,\,0.65)$.

Next, we analyze the probes of nTGCs at $e^{+}e^{-}$ colliders using the form
factor formulation we described in Section 3. According to the relations we
derived in Eq.(3.6), can translate our sensitivity reaches on the new physics
scale $\Lambda_{j}$ of each dimension-8 operator $\mathcal{O}_{j}$ to that of
the related form factor $h_{j}^{V}$. The corresponding sensitivities on the
form factors $(h_{4},\hskip 0.85358pth_{3}^{Z},\hskip
0.85358pth_{3}^{\gamma})$ are presented in the upper half of Table 10. For
comparison, we also show the sensitivities of the LHC (13 TeV) and a 100 TeV
$\\!pp$ collider in the lower half of Table 10.

We see from Table 10 that the LHC has sensitivities for the form factor
$|h_{4}|$ that are higher than those of the $e^{+}e^{-}$ colliders with
$\sqrt{s\,}=(250,\hskip 0.85358pt500)$ GeV by a factor of
${O}(10\\!-\\!10^{2})$, but has comparable sensitivities to that of an
$e^{+}e^{-}$ collider with $\sqrt{s\,}=1$ TeV, whereas the LHC sensitivities
are lower than those of the $e^{+}e^{-}$ colliders with $\sqrt{s\,}=(3-5)$ TeV
by a factor of $O(10-10^{2})$. On the other hand, a 100 TeV $pp$ collider
would have much higher sensitivities than all the $e^{+}e^{-}$ colliders with
$\sqrt{s}\leqslant\\!5$ TeV, by factors ranging from $O(10-\\!10^{5})$. We
also see that a 100 TeV $\\!pp$ collider has a sensitivity for probing the
form factor $h_{4}$ that is better than that of the LHC by a factor
${O}(10^{3})\hskip 0.85358pt$.

Similar features hold for the form factors $(h_{3}^{Z},\hskip
0.85358pth_{3}^{\gamma})$, as can be seen by inspecting Table 10. We find that
an $e^{+}e^{-}$ collider of any given collision energy $\sqrt{s\,}\,$ has
comparable sensitivities for probes of $(h_{3}^{Z},\hskip
0.85358pth_{3}^{\gamma})$, with the differences being less than a factor of 2
. We see also that the sensitivities improve from $O(10^{-4})$ to $O(10^{-7})$
when the collider energy increases from $\sqrt{s\,}=0.25$ TeV to 5 TeV. We
further note that the LHC and 100 TeV $\\!pp$ colliders have comparable
sensitivities to $(h_{3}^{Z},\hskip 0.85358pth_{3}^{\gamma})$ for any given
integrated luminosity. When the integrated luminosity of the LHC (or the 100
TeV $pp$ collider) increases over the range from
$\,\mathcal{L}\\!=\\!(0.14-3)\hskip 0.85358pt$ab-1 [or $\hskip
0.85358pt\mathcal{L}\\!=\\!(3-30)\hskip 0.85358pt$ab-1], we see that the
sensitivities to the form factors $(h_{3}^{Z},\hskip 0.85358pth_{3}^{\gamma})$
increase by about a factor of $2\hskip 0.85358pt$. Comparing the sensitivity
reaches of the $e^{+}e^{-}$ and hadron colliders in Table 10, we find that the
sensitivities of the LHC are comparable to those of a 0.25 TeV $e^{+}e^{-}$
collider, but lower than those of $e^{+}e^{-}$ colliders with
$\sqrt{s\,}\\!=\\!(0.5-1)$ TeV by a factor of $O(10)$, and lower than those of
$e^{+}e^{-}$ colliders with $\sqrt{s}\\!=\\!(3-5)\hskip 0.85358pt$TeV by
factors of ${O}(10^{2}-10^{3})$. On the other hand, the sensitivities of the
$pp\hskip 0.85358pt$(100 TeV) collider for probing $(h_{3}^{Z},\hskip
0.85358pth_{3}^{\gamma})$ are generally higher than those of the 250 GeV
$e^{+}e^{-}$ collider by a factor of ${O}(10^{3})$, higher than those of the
0.5 TeV to 1 TeV $e^{+}e^{-}$ colliders by a factor of ${O}(10^{2})$, and
higher than those of the 3 TeV $e^{+}e^{-}$ collider by a factor of ${O}(10)$,
while they are comparable to those of a 5 TeV $e^{+}e^{-}$ collider.

Figure 9: Normalized angular distributions in $\phi_{*}$ for $\hskip
0.85358pte^{+}\hskip 0.85358pte^{-}\\!\\!\rightarrow\\!Z\gamma\hskip
0.85358pt$ with $Z\\!\\!\rightarrow\\!d\hskip 0.85358pt\bar{d}\hskip
0.85358pt$, as generated by $h_{4}$ in our form factor formulation (3.5) in
panels $(a)$ and $(b)$, and as generated by $(h_{4}^{Z},\hskip
0.85358pth_{4}^{\gamma})$ in the conventional form factor formulation (3.1)
with $h_{5}^{V}\\!=0\hskip 0.85358pt$ in panels $(c)$ and $(d)$. The panels
$(a)$ and $(c)$ correspond to the $e^{+}e^{-}$ colliders with
$\sqrt{s}\\!=\\!250\,$GeV and the panels $(b)$ and $(d)$ correspond to
$\sqrt{s}=3\,$TeV.

Finally, it is instructive to present the $\phi_{*}$ angular distributions for
the form factor $h_{4}^{V}$. In the gauge-invariant form factor formulation
given in Eq.(3.5), we have imposed the constraints (3.4a)-(3.4b). Hence, the
form factor $\hskip 0.85358pth_{5}^{V}\hskip 0.85358pt$ is not independent,
and should be replaced by $\hskip 0.85358pth_{5}^{V}\\!=-h_{4}^{V}/2\hskip
0.85358pt$, according to Eq.(3.4a). Moreover, Eq.(3.4b) shows that $\hskip
0.85358pth_{4}^{\gamma}\hskip 0.85358pt$ is not independent, so the form
factors $(h_{4}^{V},\,h_{5}^{V})$ reduce to a single parameter $\hskip
0.85358pth_{4}\hskip 0.85358pt(\equiv h_{4}^{Z})$ as shown below Eq.(3.4). We
can then derive the interference cross section $\sigma_{1}$ contributed by
$\hskip 0.85358pth_{4}\hskip 0.85358pt$ and the normalized angular
distribution $\hskip 0.85358ptf^{1}_{\phi_{*}}\hskip 0.85358pt$ as follows:

$\displaystyle\sigma_{1}$ $\displaystyle\,=\,\frac{\leavevmode\nobreak\
e^{2}\\!\left(-\mbox{$\frac{\,{1}\,}{2}$}+s_{W}^{2}\right)\\!(s\\!-\\!M_{Z}^{2})\leavevmode\nobreak\
}{8\pi s_{W}c_{W}v^{2}\,s}h_{4}\,,$ (5.1a) $\displaystyle f_{\phi_{*}}^{1}$
$\displaystyle\,=\,\frac{1}{\,2\pi\,}-\frac{\,3\pi(f_{L}^{2}\\!-\\!f_{R}^{2})(M_{Z}^{2}+5\hskip
0.85358pts)\cos\phi_{*}\,}{256(f_{L}^{2}\\!+\\!f_{R}^{2})M_{Z}\sqrt{s}}+\frac{\,s\cos
2\phi_{*}\,}{8\pi M_{Z}^{2}}\,.$ (5.1b)

We see that the interference cross section scales as
$\,\sigma_{1}\\!\propto\\!E^{0}\,$, while the angular distribution $\hskip
0.85358ptf^{1}_{\phi_{*}}\hskip 0.85358pt$ has the leading term $\cos
2\phi_{*}$ enhanced by $E^{2}$ and the subleading term $\cos\phi_{*}$ enhanced
by $E^{1}$ for large energy $\sqrt{s\,}\\!=\\!E\hskip 0.85358pt$. We plot the
angular distribution $f_{\phi_{*}}^{1}$ in Fig. 9(a) and 9(b) for the
$e^{+}e^{-}$ collider energies $\sqrt{s\,}=250\,$GeV and $3\,$TeV,
respectively. In each panel, the $\hskip 0.85358pth_{4}\hskip 0.85358pt$
contribution is depicted by the red solid curve, and the SM contribution is
shown as the black dashed curve which is almost flat. We also observe that
$\cos\phi_{*}$ and $\cos 2\phi_{*}$ terms in the function $f_{\phi_{*}}^{1}$
in Eq.(5.1b) have opposite signs. They are comparable for lower collision
energy $\sqrt{s\,}=250\,$GeV, but $\cos(2\phi_{*})$ becomes dominant for a
large collision energy $\sqrt{s\,}=3\,$TeV. We can evaluate the numerical
coefficients of $f_{\phi_{*}}^{1}$, as follows:

$\displaystyle f_{\phi_{*}}^{1}$
$\displaystyle=\,\frac{1}{\,2\pi\,}-0.485\cos\phi_{*}+0.299\cos
2\phi_{*}\hskip 0.85358pt,$ $\displaystyle\text{for}\leavevmode\nobreak\
\sqrt{s\,}=250\hskip 0.85358pt\text{GeV},$ (5.2a) $\displaystyle
f_{\phi_{*}}^{1}$
$\displaystyle=\,\frac{1}{\,2\pi\,}-5.67\cos\phi_{*}+43.1\cos 2\phi_{*}\hskip
0.85358pt,$ $\displaystyle\text{for}\leavevmode\nobreak\ \sqrt{s\,}=3\hskip
0.85358pt\text{TeV}.$ (5.2b)

This explains why panel (a) of Fig. 9 exhibits a significant cancellation
between the $\cos\phi_{*}$ and $\cos(2\phi_{*})$ terms, whereas in panel (b)
the $\cos(2\phi_{*})$ term dominates and thus the red curve exhibits
interesting $\cos(2\phi_{*})$ behavior.

$\sqrt{s\,}$ (TeV) | 0.25 | 0.5 | 1 | 3 | 5
---|---|---|---|---|---
$|h_{4}|$ | $3.9\\!\times\\!10^{-4}$ | $3.8\\!\times\\!10^{-5}$ | $4.5\\!\times\\!10^{-6}$ | $1.6\\!\times\\!10^{-7}$ | $3.6\\!\times\\!10^{-8}$
$|h_{4}^{Z}|$ | $8.9\\!\times\\!10^{-5}$ | $4.2\\!\times\\!10^{-6}$ | $2.5\\!\times\\!10^{-7}$ | $3.0\\!\times\\!10^{-9}$ | $3.9\\!\times\\!10^{-10}$
$|h_{4}^{\gamma}|$ | $6.7\\!\times\\!10^{-4}$ | $3.2\\!\times\\!10^{-5}$ | $1.9\\!\times\\!10^{-6}$ | $2.3\\!\times\\!10^{-8}$ | $2.9\\!\times\\!10^{-9}$
Table 11: Comparisons of the $2\hskip 0.85358pt\sigma$ sensitivities to
probing the form factor $\hskip 0.85358pth_{4}\hskip 0.85358pt$ of our SMEFT
formulation (3.5) (marked in red color) and the conventional form factors
$(h_{4}^{Z},\,h_{4}^{\gamma})$ that take into account only U(1)em gauge
invariance (marked in blue color), as derived by analyzing the reaction
$\,e^{+}\hskip
0.85358pte^{-}\\!\\!\rightarrow\\!Z\gamma\\!\rightarrow\\!q\hskip
0.85358pt\bar{q}\gamma$ at various $e^{+}e^{-}\\!$ colliders with
$\mathcal{L}\\!=5\,$ab${}^{-1}\\!$ and unpolarized $e^{\mp}$ beams. As
discussed in the text, the conventional form-factor limits (blue color) are
included for illustration only, as they do not respect the full SM gauge
symmetry, and hence are invalid.

Figure 10: Sensitivity reaches for the new physics scale $\Lambda$ of the nTGC
operators at the hadron colliders LHC (13 TeV) and $pp$ (100 TeV) in plot (a)
and $e^{+}e^{-}$ colliders with collision energies
$\sqrt{s\,}\\!=\\!(0.25,\,0.5,\,1,\,3,\,5)$ TeV in plot (b). In each plot, the
(${\hskip 1.42262pt}2{\hskip 0.85358pt}\sigma,\,5{\hskip 0.85358pt}\sigma$)
sensitivities are shown in (light, heavy) colors, respectively.

For comparison, we consider the conventional form factor formulation (3.1)
with $h_{5}^{V}\\!=0\hskip 0.85358pt$, where $(h_{4}^{Z},\hskip
0.85358pth_{4}^{\gamma})$ are treated as two independent parameters. In this
case we derive the following interference cross sections
$(\tilde{\sigma}_{1}^{Z},\hskip 0.85358pt\tilde{\sigma}_{1}^{A})$ contributed
by $(h_{4}^{Z},\hskip 0.85358pth_{4}^{\gamma})$ and their normalized angular
distributions $\hskip 0.85358pt(\tilde{f}^{1Z}_{\phi_{*}},\hskip
0.85358pt\tilde{f}^{1A}_{\phi_{*}})\hskip 0.85358pt$:

$\displaystyle(\tilde{\sigma}_{1}^{Z},\,\tilde{\sigma}_{1}^{A})$
$\displaystyle\,=\,\frac{\leavevmode\nobreak\
e^{4}(s\\!-\\!M_{Z}^{2})^{2}\leavevmode\nobreak\ }{128\pi
M_{Z}^{4}\,s}\left(\\!\frac{\,1\\!-\\!4s_{W}^{2}\,}{s_{W}^{2}c_{W}^{2}}h_{4}^{Z},\leavevmode\nobreak\
\frac{2}{\,s_{W}c_{W}\,}h_{4}^{\gamma}\\!\right)\\!,$ (5.3a)
$\displaystyle\tilde{f}_{\phi_{*}}^{1Z}$
$\displaystyle\,=\,\frac{1}{\,2\pi\,}-\frac{\leavevmode\nobreak\
3\pi(f_{L}^{2}\\!-\\!f_{R}^{2})(3s\\!+\\!M_{Z}^{2})\leavevmode\nobreak\
}{\leavevmode\nobreak\
128M_{Z}(f_{L}^{2}\\!+\\!f_{R}^{2})\sqrt{s}\leavevmode\nobreak\
}\frac{\leavevmode\nobreak\
1\\!-\\!4s_{W}^{2}\\!+\\!8s_{W}^{4}\leavevmode\nobreak\
}{1\\!-\\!4s_{W}^{2}}\cos\phi_{*}\hskip 0.85358pt,$ (5.3b)
$\displaystyle\tilde{f}_{\phi_{*}}^{1A}$
$\displaystyle\,=\,\frac{1}{2\pi}-\frac{\leavevmode\nobreak\
3\pi(f_{L}^{2}\\!-\\!f_{R}^{2})(3s\\!+\\!M_{Z}^{2})\leavevmode\nobreak\
}{\leavevmode\nobreak\
128M_{Z}(f_{L}^{2}+f_{R}^{2})\sqrt{s}\leavevmode\nobreak\
}(1\\!-\\!4s_{W}^{2})\cos\phi_{*}\hskip 0.85358pt.$ (5.3c)

We see that the interference cross sections in (5.3) scale as
$\,(\tilde{\sigma}_{1}^{Z},\hskip
0.85358pt\tilde{\sigma}_{1}^{A})\\!\propto\\!E^{2}\,$, while the angular
distributions $\hskip 0.85358pt(f^{1Z}_{\phi_{*}},\hskip
0.85358ptf^{1A}_{\phi_{*}})\hskip 0.85358pt$ have leading terms
$\propto\cos\phi_{*}$ enhanced by $E^{1}$ for large energy
$\sqrt{s\,}\\!=\\!E^{1}\hskip 0.85358pt$. We also note that the distribution
$f^{1A}_{\phi_{*}}$ is much suppressed relative to $f^{1Z}_{\phi_{*}}$ due to
the small factor $\,(1-4s_{W}^{2})\\!\ll\\!1\hskip 0.85358pt$. We plot the
angular distributions $\hskip 0.85358pt(f^{1Z}_{\phi_{*}},\hskip
1.42262ptf^{1A}_{\phi_{*}})\hskip 0.85358pt$ of Eq.(5.3c) as the blue solid
curves in Fig. 9(c)-(d), while the squared distributions $\hskip
0.85358pt(f^{2Z}_{\phi_{*}},\hskip 0.85358ptf^{2A}_{\phi_{*}})\hskip
0.85358pt$ and the SM distribution $\hskip 0.85358ptf^{0}_{\phi_{*}}\hskip
0.85358pt$ are plotted as the green solid curves and black dashed curves,
respectively. As expected, the distributions $\hskip
0.85358pt(f^{2Z}_{\phi_{*}},\hskip 0.85358ptf^{2A}_{\phi_{*}})\hskip
0.85358pt$ and $\hskip 0.85358ptf^{0}_{\phi_{*}}\hskip 0.85358pt$ are
dominated by the constant term and thus nearly flat.

Figure 11:  Sensitivity reaches for the nTGC form factors
$(h_{4},\,h_{3}^{Z},\,h_{3}^{\gamma})$ at the hadron colliders LHC (13 TeV)
and $pp$ (100 TeV) in plot (a) and at $e^{+}e^{-}$ colliders with collision
energies $\sqrt{s\,}\\!=\\!(0.25,\,0.5,$ $\,1,\,3,\,5)$ TeV in plot (b). In
each plot, the ($\,2{\hskip 0.85358pt}\sigma,\,5{\hskip 0.85358pt}\sigma$)
sensitivities are shown in (heavy, light) colors, respectively.

We stress that Eq.(5.3) and the corresponding Figs. 9(c)-(d) are incorrect
because the conventional form factor formulation (3.1) with
$\,h_{5}^{V}\\!=0\,$ does not obey the consistency conditions (3.4) imposed by
the spontaneous breaking of the electroweak gauge symmetry
SU(2)${}_{L}\otimes$U(1)Y of the SMEFT. In the following we further show that
the conventional form factor formulation also leads to erroneously strong
sensitivity limits on the form factors $(h_{4}^{Z},\,h_{4}^{\gamma})$.

Following the steps leading to Eq.(4.11), we construct the following
observables $(\mathbb{O}_{1},\,\mathbb{\widetilde{O}}_{1})$ for probing the
form factors:

$\displaystyle\mathbb{O}_{1}$ $\displaystyle\,=\,\left|\hskip
0.85358pt\sigma_{1}\\!\int\\!\\!\mathrm{d}\phi_{*}\,f_{\phi_{*}}^{1}\\!\times\text{sign}(\cos
2\phi_{*})\right|,$ (5.4a) $\displaystyle\mathbb{\widetilde{O}}_{1}$
$\displaystyle\,=\,\left|\hskip
0.85358pt\tilde{\sigma}_{1}\\!\int\\!\\!\mathrm{d}\phi_{*}\,\tilde{f}_{\phi_{*}}^{1}\\!\times\text{sign}(\cos\phi_{*})\right|.$
(5.4b)

From the above formulae (5.1a) and (5.3), we deduce that energy-dependences of
the observables are $\hskip
0.85358pt\mathbb{O}_{1}(h_{4})\\!\propto\\!E^{2}\hskip 0.85358pt$ and $\hskip
0.85358pt\mathbb{\widetilde{O}}_{1}(h_{4}^{V})\\!\propto\\!E^{3}\hskip
0.85358pt$. This shows that in the conventional form factor formulation
$\mathbb{\widetilde{O}}_{1}^{c}$ has an erroneously large energy-dependence
($E^{3}$ instead of $E^{2}$), leading to incorrectly strong sensitivities to
the form factors $(h_{4}^{Z},\,h_{4}^{\gamma})$. We present these incorrect
sensitivities in blue color in Table 11. For comparison, we also show in this
Table the correct sensitivities (red color) to the form factors $h_{4}$, as
derived within our consistent form factor formulation (3.5) with the
constraints (3.4a) and (3.4b). From this comparison, we see that for the form
factor $\hskip 0.85358pth_{4}^{Z}\hskip 0.85358pt$ the conventional
sensitivities (blue color) are erroneously stronger than our new sensitivities
(red color) by a factor of $5$ for the collider energy $\sqrt{s\,}=250$ GeV,
by a factor of $\,{O}(10-20)$ for the collider energies $\sqrt{s\,}=(0.5-1)$
TeV, and by a factor of ${O}(10^{2})$ for the collider energies
$\sqrt{s\,}=(3-5)$ TeV.

Finally, for an intuitive comparison and overview, we summarize in Fig. 10 the
sensitivity reaches for the new physics scales of the dimension-8 nTGC
operators at the hadron colliders [panel (a)] and the $e^{+}e^{-}$ colliders
[panel (b)] from Tables 4-5 and Table 9. We present these limits at both the
$(2{\hskip 0.85358pt}\sigma,\hskip 0.85358pt5{\hskip 0.85358pt}\sigma)$
levels, which are indicated by the (light, heavy) colors respectively. In Fig.
10(b) we only plot the sensitivity reaches for $e^{-}e^{+}$ collisions with
unpolarized electron/positron beams. We note that according to Table 9, adding
the $e^{-}/e^{+}$ beam polarizations can increase the sensitivity reaches on
the new physics scale by about $20\%$ for $\mathcal{O}_{G+}$ and
$\mathcal{O}_{C+}$, and by about $5\%$ for $\mathcal{O}_{\widetilde{B}W}$.
Also, we summarize in Fig. 11 the sensitivity reaches for probing the nTGC
form factors $(h_{4},\hskip 0.85358pth_{3}^{Z},\hskip
0.85358pth_{3}^{\gamma})$ at the hadron colliders [panel (a)] and the
$e^{+}e^{-}$ colliders [panel (b)] from Table 7 and Table 10, where the
$(2{\hskip 0.85358pt}\sigma,\hskip 0.85358pt5{\hskip 0.85358pt}\sigma)$ limits
are marked by the (light, heavy) colors, respectively. In Fig. 10(b) we
present only the sensitivity reaches for $e^{-}e^{+}$ collisions with
unpolarized electron/positron beams. We note that according to Table 10,
adding the $e^{-}/e^{+}$ beam polarizations can increase the sensitivity
reaches for the nTGC form factors by about $100\%$ for $h_{4}$, by about
$20\%$ for $h_{3}^{Z}$, and by about $160\%$ for $h_{3}^{\gamma}$.

The reason that the effects of beam polarization for probing the nTGC form
factors in $e^{-}e^{+}$ collisions appear much stronger than those for probing
the new physics cutoff scales of the dimension-8 nTGC operators can be
understood as follows. We note that the relation between the polarized and
unpolarized cross sections of the SM backgrounds is given by [5]:

$\displaystyle\sigma_{0}(P_{L}^{e},P_{R}^{\bar{e}})$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle
4\frac{\,P_{L}^{e}P_{R}^{\bar{e}}c_{L}^{2}\\!+\\!(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})c_{R}^{2}\,}{c_{L}^{2}\\!+c_{R}^{2}}\,\sigma_{0}(0.5,0.5)\,,$
(5.5)

where $P_{L}^{e}$ $(P_{R}^{\bar{e}})$ denotes the fraction of left-handed
(right-handed) electrons (positrons) in the $e^{-}$ ($e^{+}$) beam and $\hskip
0.85358ptP_{L}+\\!P_{R}=\\!1\hskip 0.85358pt$ holds for both $e^{-}$ and
$e^{+}$ beams.777Note that the degree of longitudinal beam polarization for
$e^{-}$ or $e^{+}$ is defined as $\,\widehat{P}\\!=\\!P_{R}\\!-\\!P_{L}$ [5].
Thus, the left-handed and right-handed fractions of $e^{-}$ and $e^{+}$ in the
beam can be expressed as
$\,P_{L,R}^{e}\\!=\\!\mbox{$\frac{\,{1}\,}{2}$}(1\mp\widehat{P}^{e})$ and
$P_{L,R}^{\bar{e}}\\!=\\!\mbox{$\frac{\,{1}\,}{2}$}(1\mp\widehat{P}^{\bar{e}})$,
respectively. For instance, unpolarized $e^{-}$ and $e^{+}$ beams have
vanishing degrees of polarization $\,(\widehat{P}^{e}\\!,\hskip
0.85358pt\widehat{P}^{\bar{e}})=0$ , whereas a polarized $e^{-}$ beam with
fraction $P_{L}^{e}=90\%$ has $\,\widehat{P}^{e}\\!=\\!-0.8$ and a polarized
$e^{+}$ beam with fraction $P_{R}^{\bar{e}}=65\%$ has
$\,\widehat{P}^{\bar{e}}\\!=\\!0.3$ .

According to Ref. [5], we construct the following three kinds of
$\mathbb{O}_{1}$ observables and extract the different signal terms of
$O(\Lambda^{-4})$ in the differential cross section:

$\displaystyle\mathbb{O}_{A}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\int\\!\\!\mathrm{d}\theta\mathrm{d}\theta_{*}\mathrm{d}\phi_{*}\,\frac{d^{3}\sigma_{1}}{\mathrm{d}\theta\mathrm{d}\theta_{*}\mathrm{d}\phi_{*}}\text{sign}(\cos\\!\phi_{*})\,,$
(5.6a) $\displaystyle\mathbb{O}_{B}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\int\\!\\!\mathrm{d}\theta\mathrm{d}\theta_{*}\mathrm{d}\phi_{*}\,\frac{d^{3}\sigma_{1}}{\mathrm{d}\theta\mathrm{d}\theta_{*}\mathrm{d}\phi_{*}}\text{sign}(\cos\\!\phi_{*})\text{sign}(\cos\\!\theta)\text{sign}(\cos\\!\theta_{*})\,,$
(5.6b) $\displaystyle\mathbb{O}_{C}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\int\\!\\!\mathrm{d}\theta\mathrm{d}\theta_{*}\mathrm{d}\phi_{*}\,\frac{d^{3}\sigma_{1}}{\mathrm{d}\theta\mathrm{d}\theta_{*}\mathrm{d}\phi_{*}}\text{sign}(\cos\\!2\phi_{*})\,.$
(5.6c)

For these observables, we can deduce the following:

$\displaystyle\mathbb{O}_{A}(G+)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathbb{A}\times\mbox{$\frac{\,{1}\,}{2}$}c_{L}P_{L}^{e}P_{R}^{\bar{e}}(5s\\!+\\!M_{Z}^{2})\Lambda^{-4}_{G+}\,,$
(5.7a) $\displaystyle\mathbb{O}_{A}(j)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathbb{A}\times
3\\!\left[c_{L}x_{L}P_{L}^{e}P_{R}^{\bar{e}}\\!+\\!c_{R}x_{R}(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})\right]\\!(s\\!+\\!M_{Z}^{2})\Lambda^{-4}_{j}\,,$
(5.7b) $\displaystyle\mathbb{O}_{B}(G+)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathbb{B}\times\mbox{$\frac{\,{1}\,}{2}$}c_{L}P_{L}^{e}P_{R}^{\bar{e}}(5s\\!+\\!M_{Z}^{2})\Lambda^{-4}_{G+}\,,$
(5.7c) $\displaystyle\mathbb{O}_{B}(j)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathbb{B}\times
3\\!\left[c_{L}x_{L}P_{L}^{e}P_{R}^{\bar{e}}\\!-\\!c_{R}x_{R}(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})\right]\\!(s\\!+\\!M_{Z}^{2})\Lambda^{-4}_{j}\,,$
(5.7d) $\displaystyle\mathbb{O}_{C}(G+)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathbb{C}\times\mbox{$\frac{\,{1}\,}{2}$}c_{L}P_{L}^{e}P_{R}^{\bar{e}}\,s\,\Lambda^{-4}_{G+}\,,$
(5.7e) $\displaystyle\mathbb{O}_{C}(j)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathbb{C}\times\\!\left[c_{L}x_{L}P_{L}^{e}P_{R}^{\bar{e}}\\!-\\!c_{R}x_{R}(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})\right]\\!M_{Z}^{2}\Lambda^{-4}_{j}\,,$
(5.7f)

where the index $j$ denotes the operators $(\mathcal{O}_{G-},\hskip
0.85358pt\mathcal{O}_{\widetilde{B}W},\hskip 0.85358pt\mathcal{O}_{C+})$,
respectively. The values of the coefficients $(\mathbb{A},\hskip
0.85358pt\mathbb{B},\hskip 0.85358pt\mathbb{C})$ in these formulae are given
by the numerical results for the observables in Eq.(5.6). The dependence of
each sensitivity limit on the polarization choice is determined by the
relation between the left- and right-handed couplings. The most sensitive
observable for probing $\mathcal{O}_{G+}\,(h_{4})$ is $\mathbb{O}_{C}$, while
the most sensitive observable for probing
$\mathcal{O}_{\widetilde{B}W}\,(h_{3}^{Z})$ and $\mathcal{O}_{C+}$ is
$\mathbb{O}_{A}$. For probing $\mathcal{O}_{G-}\,(h_{3}^{\gamma})$, the most
sensitive observable is $\mathbb{O}_{B}$ in the case of unpolarized beams, and
is $\mathbb{O}_{A}$ in the case of polarized beams [for the choice
$(P_{L}^{e},\hskip 0.85358ptP_{R}^{\bar{e}})\\!=\\!(0.9,\hskip
0.85358pt0.65)$]. We note that the sensitivity limits for the nTGC form
factors scale as $\,(h_{3}^{V},\hskip
0.85358pth_{4}^{V})\\!\propto\\!\mathbb{O}_{X}$ (where $X\\!=\\!A,B,C$), and
that the new physics cutoff reaches of the dimension-8 nTGC operators behave
like $\Lambda\\!\propto\\!\mathbb{O}_{X}^{1/4}$. Hence, the improvements from
the beam polarizations can be significant for the form factors, but become
rather mild for the cutoff scales of the dimension-8 operators.

For convenience, we express a given observable
$\,\mathbb{O}_{X}\\!\equiv\overline{\mathbb{O}}_{X}/\Lambda_{j}^{4}\,$ for the
dimension-8 operator formulation and
$\,\mathbb{O}_{X}\\!\equiv\overline{\mathbb{O}}_{X}h_{i}^{V}\,$ for the form
factor formulation. We may estimate the significance by
$\,\mathcal{Z}\\!\simeq\\!S/\\!\sqrt{B\,}$. If we require the significances of
the polarized and unpolarized cases to be equal,
$\mathcal{Z}_{\mathrm{pol}}\\!=\\!\mathcal{Z}_{\mathrm{unpol}}\hskip
0.85358pt$, we can derive the following ratio of the polarized/unpolarized
limits on the dimension-8 cutoff scales and on the form factors, respectively:

$\displaystyle\mathcal{R}_{\Lambda_{j}}=\frac{\Lambda_{j}(\mathrm{pol})}{\,\Lambda_{j}(\mathrm{unpol})\,}=\big{[}\overline{\mathcal{R}}_{X}(P_{L}^{e},P_{R}^{\bar{e}})\big{]}^{\\!1/4},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{R}_{h_{i}^{V}}=\frac{\,h_{i}^{V}\\!(\mathrm{unpol})\,}{h_{i}^{V}\\!(\mathrm{pol})}=\overline{\mathcal{R}}_{X}(P_{L}^{e},P_{R}^{\bar{e}})\hskip
0.85358pt,$ (5.8)

where the ratio $\overline{\mathcal{R}}_{X}(P_{L}^{e},P_{R}^{\bar{e}})$ is
defined as

$\displaystyle\overline{\mathcal{R}}_{X}(P_{L}^{e},P_{R}^{\bar{e}})\hskip
0.85358pt\equiv\hskip
0.85358pt\frac{\,\overline{\mathbb{O}}_{X}(P_{L}^{e},P_{R}^{\bar{e}})\,}{\leavevmode\nobreak\
\overline{\mathbb{O}}_{X}(0.5,0.5)\,}\sqrt{\frac{\,\sigma_{0}(0.5,0.5)\,}{\sigma_{0}(P_{L}^{e},P_{R}^{\bar{e}})}\,}\hskip
0.85358pt.$ (5.9)

From the above, we derive the following estimate of the ratio
$\overline{\mathcal{R}}_{X}$ for each observable $\mathbb{O}_{X}$:

$\displaystyle\overline{\mathcal{R}}_{A}\\!$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\,2\hskip
0.85358pt\big{|}c_{L}x_{L}P_{L}^{e}P_{R}^{\bar{e}}+c_{R}x_{R}(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})\big{|}\,}{|c_{L}x_{L}+c_{R}x_{R}|}\sqrt{\frac{c_{L}^{2}\\!+c_{R}^{2}}{\,P_{L}^{e}P_{R}^{\bar{e}}c_{L}^{2}\\!+\\!(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})c_{R}^{2}\,}}\hskip
0.85358pt,\hskip 34.1433pt$ (5.10a)
$\displaystyle\overline{\mathcal{R}}_{B,C}\\!$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\,2\hskip
0.85358pt\big{|}c_{L}x_{L}P_{L}^{e}P_{R}^{\bar{e}}-c_{R}x_{R}(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})\big{|}\,}{|c_{L}x_{L}-c_{R}x_{R}|}\sqrt{\frac{c_{L}^{2}\\!+c_{R}^{2}}{\,P_{L}^{e}P_{R}^{\bar{e}}c_{L}^{2}+(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})c_{R}^{2}\,}},$
(5.10b)

where $(c_{L},\hskip 0.85358ptc_{R})\\!=\\!(T_{3}-Qs_{W}^{2},-Qs_{W}^{2})$
denote the $Z$ coupling factors with the (left, right)-handed electrons, and
the coupling coefficients $(x_{L},\hskip 0.85358ptx_{R})$ are given by

$\displaystyle(x_{L},\,x_{R})$ $\displaystyle\,=\,-Qs_{W}^{2}(1,\,1)\hskip
0.85358pt,$ $\displaystyle(\hskip 0.85358pt\text{for}\leavevmode\nobreak\ $
$\displaystyle\mathcal{O}_{G-}\hskip 0.85358pt)\hskip 0.85358pt,$ (5.11a)
$\displaystyle(x_{L},\,x_{R})$
$\displaystyle\,=\,(T_{3}\\!-\\!Qs_{W}^{2},\,-Qs_{W}^{2})\hskip 0.85358pt,$
$\displaystyle(\hskip 0.85358pt\text{for}\leavevmode\nobreak\ $
$\displaystyle\mathcal{O}_{\\!\widetilde{B}W}\hskip 0.85358pt)\hskip
0.85358pt,$ (5.11b) $\displaystyle(x_{L},\,x_{R})$
$\displaystyle\,=\,-T_{3}(1,0),$ $\displaystyle(\hskip
0.85358pt\text{for}\leavevmode\nobreak\ $
$\displaystyle\mathcal{O}_{G+},\mathcal{O}_{C+}\hskip 0.85358pt)\hskip
0.85358pt.$ (5.11c)

Because the coupling coefficient $x_{R}\\!=0\hskip 0.85358pt$ for
$\mathcal{O}_{G+}(h_{4})$ and $\mathcal{O}_{C+}$, we can reduce the
significance ratio (5.10) to the following form and compute its value for
$(P_{L}^{e},P_{R}^{\bar{e}})\\!=\\!(0.9,0.65)$:

$\displaystyle\overline{\mathcal{R}}_{X}(P_{L}^{e},P_{R}^{\bar{e}})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!2P_{L}^{e}P_{R}^{\bar{e}}\sqrt{\frac{c_{L}^{2}\\!+c_{R}^{2}}{\,P_{L}^{e}P_{R}^{\bar{e}}c_{L}^{2}\\!+\\!(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})c_{R}^{2}\,}},$
(5.12a) $\displaystyle\overline{\mathcal{R}}_{X}(0.9,0.65)\\!\\!\\!$
$\displaystyle\simeq$ $\displaystyle\\!\\!2.0\hskip 0.85358pt,$ (5.12b)

where $X\\!=\\!A,B,C$. Thus, we deduce the following ratios for the operators
$(\mathcal{O}_{G+},\mathcal{O}_{C+})$ and the form factor $h_{4}$:

$\displaystyle\overline{\mathcal{R}}_{\Lambda_{G+}}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\overline{\mathcal{R}}_{\Lambda_{C+}}\\!=\,\big{[}\overline{\mathcal{R}}_{X}(0.9,0.65)\big{]}^{1/4}\simeq
1.2\hskip 0.85358pt,$ (5.13a)
$\displaystyle\overline{\mathcal{R}}_{h_{4}}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\overline{\mathcal{R}}_{X}(0.9,0.65)\simeq 2.0\hskip
0.85358pt.$ (5.13b)

This means that the $e^{-}/e^{+}$ beam polarizations can enhance the
sensitivity reach for the cutoff scale $\Lambda_{G+}$ by about 20%, and
enhance the sensitivity reach for the form factor $h_{4}$ much more
significantly, namely by about 100%, which explains the features shown in
Tables 9-10 and Figs. 10-11.

We further note that the coupling coefficients $x_{L,R}\\!=\\!c_{L,R}$ for the
nTGC operator $\mathcal{O}_{\widetilde{B}W}$ and form factor $h_{3}^{Z}$. We
find that to enhance the polarization effects for probing
$\mathcal{O}_{\widetilde{B}W}$ and $h_{3}^{Z}$, the most sensitive observable
is $\mathbb{O}_{A}$. Thus, we simplify the significance ratio (5.10) to the
following form and compute its value for
$(P_{L}^{e},P_{R}^{\bar{e}})\\!=\\!(0.9,0.65)$:

$\displaystyle\overline{\mathcal{R}}_{A}(P_{L}^{e},P_{R}^{\bar{e}})$
$\displaystyle=\,2\hskip
0.85358pt\sqrt{\frac{\,P_{L}^{e}P_{R}^{\bar{e}}c_{L}^{2}\\!+\\!(1\\!-\\!P_{L}^{e})(1\\!-\\!P_{R}^{\bar{e}})c_{R}^{2}\leavevmode\nobreak\
}{c_{L}^{2}\\!+c_{R}^{2}}},$ (5.14)
$\displaystyle\overline{\mathcal{R}}_{A}(0.9,0.65)$ $\displaystyle\simeq
1.2\,.$ (5.15)

With these, we deduce the following ratios for the operator
$\mathcal{O}_{\widetilde{B}W}$ and the form factor $h_{3}^{Z}$:

$\displaystyle\overline{\mathcal{R}}_{\Lambda_{\widetilde{B}W}}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\big{[}\overline{\mathcal{R}}_{A}(0.9,0.65)\big{]}^{1/4}\simeq
1.05\hskip 0.85358pt,$ (5.16a)
$\displaystyle\overline{\mathcal{R}}_{h_{3}^{Z}}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\overline{\mathcal{R}}_{X}(0.9,0.65)\simeq 1.2\,.$
(5.16b)

This shows that the beam polarizations can increase mildly the sensitivity
reach for the cutoff scale $\Lambda_{\widetilde{B}W}$ by about 5%, and
increase the sensitivity reach for the form factor $h_{3}^{Z}$ by a larger
amount of 20%, which agree with the features shown in Tables 9-10 and Figs.
10-11. Finally, we note that the enhancement ratio (5.9) does not apply to the
cases of $\mathcal{O}_{G-}$ and $h_{3}^{\gamma}$ because there
$\mathbb{O}_{B}$ is the most sensitive observable for the unpolarized case and
$\mathbb{O}_{A}$ is the most sensitive observable for the polarized case.
Thus, we define the corresponding ratio
$\mathcal{R}_{AB}(P_{L}^{e},P_{R}^{\bar{e}})$ of significances between the
polarized and unpolarized cases and compute its value
$\mathcal{R}_{AB}(0.9,0.65)$:

$\displaystyle\overline{\mathcal{R}}_{AB}(P_{L}^{e},P_{R}^{\bar{e}})$
$\displaystyle\simeq\frac{\,\overline{\mathbb{O}}_{A}(P_{L}^{e},P_{R}^{\bar{e}})\,}{\leavevmode\nobreak\
\overline{\mathbb{O}}_{B}(0.5,0.5)\,}\sqrt{\frac{\,\sigma_{0}(0.5,0.5)\,}{\sigma_{0}(P_{L}^{e},P_{R}^{\bar{e}})}\,}\hskip
0.85358pt,$ (5.17a) $\displaystyle\overline{\mathcal{R}}_{AB}(0.9,0.65)$
$\displaystyle\simeq 2.6\,.$ (5.17b)

From these we derive the significance ratios for the operator
$\mathcal{O}_{G-}$ and the form factor $h_{3}^{\gamma}$:

$\displaystyle\mathcal{R}_{\Lambda_{G-}}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\big{[}\overline{\mathcal{R}}_{AB}(0.9,0.65)\big{]}^{1/4}\simeq
1.27\hskip 0.85358pt,$ (5.18a) $\displaystyle R_{h_{3}^{\gamma}}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\overline{\mathcal{R}}_{X}(0.9,0.65)\simeq 2.6\,.$
(5.18b)

We see that the beam polarization effects can raise the sensitivity reach for
the cutoff scale $\Lambda_{G-}$ by about 27%, and raise the sensitivity reach
for the form factor $h_{3}^{\gamma}$ by about 160%, which agree with the
results presented in Tables 9-10 and Figs. 10-11.

##  6 Conclusions

Neutral triple-gauge couplings (nTGCs) provide an important window for probing
new physics beyond the SM. In this work, we have studied systematically the
prospective experimental sensitivities to nTGCs at the 13 TeV LHC and a future
100 TeV $pp$ collider, using the SMEFT approach to classify and characterize
the nTGCs that can arise from gauge-invariant dimension-8 operators.

In Section 2.1 we first considered a set of CP-conserving dimension-8 nTGC
operators and the related contact operators in Eq.(2.2). Then, in Section 2.2
we derived their contributions to the scattering amplitudes of the partonic
process $\bar{q}{\hskip 0.85358pt}q\\!\rightarrow\\!Z\gamma\hskip 0.85358pt$
in Eqs.(2.7) and (2.8). With these, we computed the corresponding total cross
sections including the SM contribution, the interference term of
$O(1/\Lambda^{4})$, and the squared term of $O(1/\Lambda^{8})$, as in
Eqs.(2.2) and (2.2), where $\Lambda$ is the new physics cutoff scale defined
in Eq.(2.1). We further presented in Eq.(2.14), Eq.(2.17), and Fig. 3 their
contributions to the differential angular distributions, in comparison with
that of the SM. In Section 2.3 we analyzed the perturbative unitarity bounds
on the nTGCs, as shown in Table 1 and Fig. 1, which are much weaker than the
collider limits presented in Sections 4-5. Hence, the perturbation expanison
is well justified for the current collider analyses.

In Section 3 we presented a new form factor formulation of the neutral triple
gauge vertices (nTGVs) $Z\gamma V^{*}$ (with $V\\!=\\!Z,\gamma$), by mapping
them to the dimension-8 nTGC operators of the SMEFT that incorporate the
spontaneously-broken electroweak gauge symmetry
SU(2)${}_{\mathrm{L}}\otimes$U(1)Y of the SM. This differs from the
conventional form factor parametrization of nTGCs that takes into account only
the unbroken U(1)em gauge symmetry [3][4]. Using the SMEFT approach, we have
found that a new momentum-dependent nTGC term with form factor $h_{5}^{V}$ has
to be added and the mapping with the dimension-8 SMEFT interactions enforces
new nontrivial relations (3.4a)-(3.4b) between the form factors
$(h_{4}^{V},\hskip 0.85358pth_{5}^{V})$ and between the form factors
$(h_{4}^{Z},\,h_{4}^{\gamma})$. The new form factor $h_{5}^{V}$ was not
included in all the previous form factor analyses of nTGVs. We have
demonstrated that including the new form factor $h_{5}^{V}$ is crucial for a
fully consistent form factor formulation of nTGVs and ensures the exact
cancellation of the spuriously large unphysical terms of $O(E^{5})$ in the
scattering amplitudes of $\hskip 0.85358ptq{\hskip
0.85358pt}\bar{q}\\!\rightarrow\\!Z\gamma\hskip 0.85358pt$, as shown in
Eqs.(3.8b) and (3.10). In consequence, among the six general nTGC form factors
$(h_{3}^{V},\,h_{4}^{V},\,h_{5}^{V})$ in Eq.(3.2), we have proven that only
three of them, $(h_{3}^{Z},\,h_{3}^{\gamma},\,h_{4})$ with $\hskip
0.85358pth_{4}\\!\equiv\\!h_{4}^{Z}\hskip 0.85358pt$, are independent, and the
correct nTGC form factor formula is given by Eq.(3.5). We have further
presented the explicit correspondence between the nTGC form factors and the
cutoff scales of the dimension-8 nTGC operators in Eqs.(3.6)-(3.7).

In Section 4, we have systematically studied the sensitivity reaches for
probing the new physics scales of the nTGC operators and for probing the nTGC
form factors in the reactions $\,pp\hskip 0.85358pt(q{\hskip
0.85358pt}\bar{q})\\!\rightarrow\\!Z\gamma\\!\rightarrow\\!\ell\bar{\ell}\gamma,\hskip
0.85358pt\nu\bar{\nu}\gamma$ at the LHC and the future $pp$ (100 TeV)
collider. We have presented analyses of sensitivity reaches using the
interference contributions of $O(\Lambda^{-4})$ in Section 4.2 and including
the squared contributions up to $O(\Lambda^{-8})$ in Section 4.3. We have
evaluated the prospective $2{\hskip 0.85358pt}\sigma$ and $5{\hskip
0.85358pt}\sigma$ sensitivities of the LHC and the future 100 TeV $pp$
collider to the different nTGCs, and have combined the sensitivity reaches of
the leptonic decay channel $Z\\!\rightarrow\\!\ell^{+}\ell^{-}$ (Secs.
4.2-4.3) and the invisible decay channel $Z\\!\rightarrow\\!\nu{\hskip
0.85358pt}\bar{\nu}\hskip 0.85358pt$ (Sec. 4.4). We have presented our
findings in Tables 2 to 4 for the dimension-8 operator $\hskip
0.85358pt\mathcal{O}_{G+}$ and the equivalent operator $\hskip
0.85358pt\mathcal{O}_{C-}$, and in Table 5 for the other dimension-8 operators
$(\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{G-},\,\mathcal{O}_{C+})$. These
sensitivity reaches are further summarized in our Fig. 10(a). From Table 4, we
see that the $2{\hskip 0.85358pt}\sigma$ ($5{\hskip 0.85358pt}\sigma$)
sensitivity to the scale of the operator $\mathcal{O}_{G+}$ could reach 4.4
TeV (3.9 TeV) at the 13 TeV LHC with 3 ab-1 integrated luminosity, and reach
30 TeV (26 TeV) at the 100 TeV $pp$ collider with 30 ab-1, whereas the
estimated sensitivity reaches on the scales of the dimension-8 operators
$(\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{G-},\,\mathcal{O}_{C+})$ shown
in Table 5 are somewhat smaller. Then, in Section 4.5 we have presented the
LHC sensitivity reaches on the three independent form factors
$(h_{4},\,h_{3}^{Z},\,h_{3}^{\gamma})\hskip 0.85358pt$ in Table 7, with a
summary of these sensitivities given in Fig. 11(a). We see that the
sensitivities for probing the form factor $h_{4}$ are generally higher than
those of the other two form factors $(h_{3}^{Z},\,h_{3}^{\gamma})$ by about a
factor of $5\\!\times\\!10^{-2}$ at the LHC and by about a factor of $10^{-2}$
at the 100 TeV $pp$ collider. We emphasise that if the dimension-8 SMEFT
relations between the different form factors are not taken into account, one
would find unrealistically strong sensitivities due to the uncancelled large
unphysical energy-dependent terms associated with the form factor $h_{4}\hskip
0.85358pt$, as seen by comparing Eq.(4.20) with Eq.(4.17). Then, we explicitly
demonstrated in Table 8 that the sensitivities to $h_{4}^{Z}$ and
$h_{4}^{\gamma}$ in the conventional form factor approach (marked in blue
color) are (erroneously) higher than the correct sensitivities (marked in red
color and extracted from Table 7) by about a factor of $5\\!\times\\!10^{-2}$
at the LHC and by about a factor of $10^{-2}$ at the $pp$ (100 TeV) collider.
Hence, it is important to use the consistent form factor approach for the nTGC
analysis as we advocated in Section 3. After these comparisons, in Section 4.6
we analyzed the 2-parameter correlations for both the nTGC form factors and
for the nTGC dimension-8 operators. We presented in Fig.5 the correlations of
each pair of the form factors $(h_{4},\hskip 0.85358pth_{3}^{V})$ and
$(h_{3}^{Z},\hskip 0.85358pth_{3}^{\gamma})$ at hadron colliders, where the
$(h_{4},\hskip 0.85358pth_{3}^{V})$ contours in the plots (a) and (b) have
rather weak correlations due to the extra energy-suppression factor of
Eq.(4.33), and the plots (c) and (d) demonstrate large correlations between
the form factors $(h_{3}^{Z},\hskip 0.85358pth_{3}^{\gamma})$. Then, we
presented the correlations of each pair of the nTGC operators
$(\mathcal{O}_{G+},\,\mathcal{O}_{\widetilde{B}W})$ and
$(\mathcal{O}_{G+},\,\mathcal{O}_{G-})$ in Figs. 6(a)-(b) which are suppressed
by large energy factor $\hskip 0.85358pt1/\\!\sqrt{\bar{s}\,}\,$ as shown in
Eq.(4.37). The correlations of each pair of the nTGC operators
$(\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{G-})$,
$(\mathcal{O}_{C+},\,\mathcal{O}_{\widetilde{B}W})$, and
$(\mathcal{O}_{C+},\,\mathcal{O}_{G-})$ are presented in Figs. 7(a) to (d).
These correlations are not suppressed by any energy factor and are thus
significant at both the LHC and the 100 TeV $pp$ collider. We demonstrated in
Figs. 7(c)-(d) that the correlations of the operators
$(\mathcal{O}_{C+},\,\mathcal{O}_{\widetilde{B}W})$ and
$(\mathcal{O}_{C+},\,\mathcal{O}_{G-})$ are particularly strong. Finally, in
Section 4.7 we have made direct comparison with the published LHC measurements
on nTGCs in the reaction $\hskip 0.85358ptp{\hskip 0.85358pt}p{\hskip
0.85358pt}(q{\hskip 0.85358pt}\bar{q})\\!\rightarrow\\!Z\gamma\hskip
0.85358pt$ (with $Z\\!\\!\rightarrow\\!\nu{\hskip 0.85358pt}\bar{\nu}{\hskip
0.85358pt})$ by the CMS [18] and ATLAS [19] collaborations. Using the same
kinematic cuts and integrated luminosity together with an estimated detection
efficiency as in the ATLAS analysis [19], we have applied the conventional
form factor formula (3.1) to reproduce the nTGC bounds in Eq.(4.52) and the
strong correlations of $(h_{3}^{V},\hskip 0.85358pth_{4}^{V})$ in Figs.
8(c)-(d), which agree well with the ATLAS results [19]. However, the
$(h_{3}^{V},\hskip 0.85358pth_{4}^{V})$ contours of Figs. 8(c)-(d) differ
substantially from those contours of Figs. 8(a)-(b), which exhibit rather weak
correlations as predicted using our new SMEFT form factor formula (3.5).
Hence, it is important to use the SMEFT form factor formulation described in
Section 3 to analyze the LHC bounds on nTGCs.

We presented in Section 5 systematic comparisons of the sensitivity reaches
for the nTGCs between the hadron colliders (the LHC and the 100 TeV $\\!pp$
collider) and $e^{+}e^{-}$ colliders with different energies. Table 9
summarizes the comparisons for probing the nTGCs of dimension-8 operators
$(\mathcal{O}_{G+},\,\mathcal{O}_{\widetilde{B}W},\,\mathcal{O}_{G-},\,\mathcal{O}_{C+})$,
whereas Table 10 summarizes the comparisons for probing the nTGC form factors
$(h_{4},\,h_{3}^{Z},\,h_{3}^{\gamma})$. We have summarized the above
comparisons of sensitivity reaches between the hadron colliders and lepton
colliders in Figs. 6 and 10. Then, in Table 11, we have further demonstrated
that using naively the conventional form factor formula without including the
nontrivial constraints of the dimension-8 SMEFT approach would cause erroneous
sensitivities to $(h_{4}^{Z},\,h_{4}^{\gamma})$ (marked in blue color) that
are stronger than the correct sensitivities (marked in red color and extracted
from Table 10) at the $e^{+}e^{-}$ colliders by a factor of $\hskip
0.85358ptO(10)\hskip 0.85358pt$ for the collision energy
$\sqrt{s\,}\\!\leqslant\\!1$ TeV and by a factor of $\hskip
0.85358ptO(10^{2})\hskip 0.85358pt$ for $\sqrt{s\,}=(3-5)\hskip 0.85358pt$TeV.
Hence, it is important to use the consistent form factor approach of Section 3
for nTGC analyses at $e^{+}e^{-}$ colliders. In general, from the comparisons
of Tables 9-10 and Figs. 10-11, we find that the LHC sensitivity reaches on
the nTGCs are similar to those at the $e^{+}e^{-}$ colliders with collision
energy $\sqrt{s\,}\\!\leqslant\\!1$ TeV [5]. On the other hand, a higher-
energy $e^{+}e^{-}$ collider with $\sqrt{s\,}=\\!(3\\!-\\!5)\hskip
0.85358pt$TeV would have greater sensitivities than the LHC to probing the new
physics scales of the nTGC operators and the corresponding nTGC form factors.
However, we have shown that the sensitivity reaches of the 100 TeV $pp\hskip
0.85358pt$ collider would be even higher.

Overall, we have found that nTGCs provide a powerful means for probing any
possible new physics beyond the SM that could generate the dimension-8 nTGC
operators in the SMEFT. We have found that both $\hskip 0.85358ptpp\hskip
0.85358pt$ and $e^{+}e^{-}$ colliders have significant roles to play. We
advocate as a first step that the ATLAS and CMS experiments at the LHC apply
the dimension-8 SMEFT approach proposed here to analyze the nTGCs, in
preference to the conventional form factor approach that does not take into
account the full electroweak gauge symmetry SU(2)${}_{\mathrm{L}}\otimes\hskip
0.85358pt$U(1)Y of the SM.

Acknowledgements
The work of JE was supported in part by United Kingdom STFC Grants
ST/P000258/1 and ST/T000759/1, in part by the Estonian Research Council via a
Mobilitas Pluss grant, and in part by the TDLI distinguished visiting fellow
programme. The work of HJH and RQX was supported in part by the National NSF
of China (under grants Nos. 11835005 and 12175136) and by the National Key R &
D Program of China (No. 2017YFA0402204). RQX has been supported by an
International Postdoctoral Exchange Fellowship.

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