text:
Air India is set to finalize an order for 150 Boeing 737 Max jets
Air India was close to signing an agreement with Boeing Co to purchase 150 of the brand-new 737 Max jets. According to a media report, the Tata Group-owned Indian carrier will likely place a firm order for 50 737 Max planes with an option to purchase up to 150 aircraft. Sources in the industry reported in July that Air India was moving closer to a decision on a mega order worth $50 billion at list prices. This mega order would be split between Airbus and Boeing. This order would include up to 300 narrowbody aircraft and 70 widebody aircraft.
This would be a win for Boeing, which is planning to expand its capacity in India to meet the world's fastest-growing major aviation market's demand. Akasa, a low-cost airline in India, purchased 72 737 MAX jets for nearly $9 billion in 2021, Boeing's last mega order in India. Indian skies are dominated by low-cost carriers such as IndiGo, SpiceJet and AirAsia India, most of which operate Airbus narrowbody planes. 155 MAX planes have been ordered by SpiceJet, Boeing's biggest customer in India.
COVID infections increase in China after controls are loosened
France beat England to book a semi-final date with Morocco
NASA and Boeing will develop a more fuel-efficient aircraft design
Indian economy to grow by 6.1 to 6.3 percent in Q2 –…
World Economic Forum session emphasizes financial literacy
Economists predict more pain for the UK as the economy…

redpajama_set_name: RedPajamaCommonCrawl


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It's that time of year again, where Melbourne hosts a nine-day celebration in honour of the city's world-renowned music scene.
Melbourne Music Week 2018 is held from the 16th to the 24th of November and features intimate, interactive experiences in non-traditional venues to headline performances at the State Library of Victoria, Melbourne Music Week has performances for all! We've compiled a list of the must-see performances of the week – keep reading to find exclusive details on each event below.
Officially opening Melbourne Music Week is an all ages, free night party, featuring an all Victorian line-up led by Melbourne party starters Ara Koufax. The long-standing electronic duo are set to deliver some infectious, body-moving sound. See Kira Puru, Various Asses, Maribelle perform mind-altering performances amongst a completely transformed ACMI space. Come shake off your week, dance and celebrate the commencement of this awesome week at Federation Square, free of charge.
The second day provides a showcase of fresh music that pays respect to African sounds within the Melbourne independent music scene. Afro Hub, in collaboration with the Immigration Museum, presents a journey set to take your ears to places they've never been before. Blue Black Beats will feature Remi as the main act and an abundance of other talented artists that'll play tunes guaranteed to make your hips shake and your heart vibrate.
MSO Ensemble at the National Gallery of Victoria brings together the talents of three of Melbourne's most vibrant and versatile musicians, Monica Curro, Philip Arkinstall, and Stefan Cassomenos and since its launch in 2014, has commissioned over 100 composers, and given over 80 world premieres. Best of all, it's completely free!
Civic Sounds bring together a range of eclectic artist and sounds into the heart of Melbourne's CBD. This all-day laneway party sprawls across multiple venues with the main stage in the iconic AC/DC Lane that will play host to bands and DJs playing from midday until late.
All three of these outstanding independent artists will be performing their new releases for this exclusive MMW event. Expect music that is distinctly delivered from Ainslie Wills, bold vocals and beautiful guitar tones from Angie McMahon and strong lyrics paired with spectacular vocal ability from Gretta Ray.
Presented at the iconic Rooftop Cinema, 'NOW SOUND' is a feature length documentary exploring Melbourne's love for music. The documentary tells stories of the trials and tribulations that constantly threaten to demolish Melbourne's iconic music culture and the unending passion that sustains it. If you're a die hard music fan, this is one you won't want to miss.
Soul in the Basement is a showcase of the best soul acts and DJs from around Melbourne. Local group The Cactus Channel, who have worked with Chet Faker and Ball Park Music, will be bringing their classic soul and funk influences to the Cherry stage. Soul music enthusiast, record collector, radio presenter, Vince Peach will be spinning the best of soul and funk music from all decades. To top it off, Logan from Gold Class will be mixing all-time-favourite soul anthems. This is a groovy night you most definitely won't want to miss!
In this exclusive performance, Australian post-punk, electronica legend Ollie Olsen meets guitarist Penny Ikinger in the State Library's stunning Redmond Barry Reading Room. Jngbert Christian "Ian 'Ollie'" Olsen is a Norwegian-born Australian multi-instrumentalist, composer and sound designer who has performed, recorded and produced rock, electronic and experimental music since the mid-1970s and rock master Penny Ikinger has wowed audiences at home and abroad. Don't miss this rare opportunity to witness the pair sharing the stage in a truly spectacular setting!
Make sure to hit up one, two or all of these concerts and plan the best week imagined!


Sometimes, says Scientific American's Stuart Firestein, the more facts one gathers as a scientist, the fewer questions one is able to answer. In fact, he says, "science to many people seems to be an impenetrable mountain of facts," and as research fields become narrower and more specialized, scientists tend to cope with this information overload by ignoring most of it. "You have to know a lot to be a scientist, but knowing a lot is not what makes a scientist," Firestein says.


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I love Tina fey on screen and I had heard through social media, radio how it was a great book but it just didn't interest me. I don't know if Tina narrating it in a script like voice was off putting but it just didn't feel like they were her experiences - I get they were but her voice didn't convey them with feeling and emotion.
Gave up third of the way and couldn't bring myself to finish it.
I read the reviews, I read the summary and I still chose to give it a go, wrong choice. Soooooooooo slow and boring and I want to slap the main character . Don't know how it has high ratings, just don't know.
Another brilliant book by Nora Roberts , narrated perfectly , January lavoy deserves a special mention as she bought the characters to life in my imagination . Definitely worth reading or listening to.


text:
Tottenham and Arsenal may miss out on Vydra
Simon Bunn
Udinese coach Francesco Guidolin is convinced on-loan striker Matej Vydra will have a bright future with the club when he returns from Watford.
The Premier League has sent many admiring glances towards the Championship this season. Crystal Palace's Wilfred Zaha has already agreed to join Manchester United in the summer, and it is likely that many teams will be sniffing around Blackpool's Tom Ince in the summer.
Outside of the Championships 'big two' however much interest has also been directed at Watford's on loan forward Matej Vydra. Watford have benefited hugely from having the Pozzo family involved with the club. The wealthy Italians are patrons of Udinese and Granada and have been using Zola's team as a guinea pig factory for his large pool of young footballing talent.
Out of the mildly exaggerated 87 players that Watford have on loan this season it has been Matej Vydra that has impressed the most. The 20 year old Czech's impact should not have come as a surprise. After confirming the signing of the promising youngster in August Watford boss Gianfranco Zola went on record as saying, "He gave me good sensations - he can become a very important player for us."
"He hasn't trained very much with us but he's catching up very quickly."
The Chelsea legend's premonition was not inaccurate. Vydra has been in sensational form this season, scoring 20 goals in 29 appearances for Watford. He has also contributed 8 assists, causing the exciting youngster to be linked with clubs such as Arsenal and Tottenham. Watford were also supposedly keen to turn the loan move into a permanent one but, recent comments from Udinese coach Francesco Guidolin suggest that Vydra's future may lay back in Italy. "I expected him to do so well in England. I didn't want him to leave, as I like him and wanted him to come to Udine (following previous loan spell with Club Brugge)."
"The club decided to send him out on loan and that's fair enough, considering he is only 20 years old," Guidolin added.
With Vydra's stock having risen so rapidly during his stint at Watford it is unlikely that Udinese will want to give up such a promising talent easily.
With the exception of a ridiculously large bid, from one of the Premiership's big boys, perhaps the best chance of keeping Vydra in English football would be Watford gaining promotion this season.
Watford currently sit second in the Championship, with Vydra still only 20-years-old the Pozzo family may be willing to allow their prodigy one more season on loan, to see if he has the ability to cut it in England's top tier.
image: © Thomas-Sommernegger
More articles from Simon Bunn
I possess a swashbuckling attitude towards football writing, aiming to pillage the most exciting and thought provoking topics in the business. Blessed with savvy up to date football knowledge, a dangerous obsession for Aston Villa FC, and a mind of alternative thinking, I welcome all readers aboard.
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text:
To put it simply - we're a reliable, well-oiled and super slick business that sells a lot of furniture to a lot of people. We aim to always have an extensive range of furniture, strive to be reasonably priced in an ever-inflating market and ensure we are constantly looking for innovative ways to improve delivery.
TC Group was born from humble beginnings in 1990. Way back then we were working from an Office in Chew Magna, operating from a very small warehouse.
Well, when we say 'office', we mean 'bungalow' and when we say 'warehouse' we mean 'pig shed.' Since then, we've been pretty busy.
By 2016 both Titan and Woodstock Leabank joined the family bringing with them an additional 20 years of specialist experience, a whole load of colourful chairs and potential for so much more.
Now in 2020, we're a reliable, well oiled and super slick business, that sells a lot of furniture to a lot of people. We employ many happy, intelligent and hardworking staff, 121 to be precise, and each person lies in the heart of our business.
Jobs At TC Group
We're always on the lookout for new people to join our ever-expanding family
1300 Park Avenue Aztec West Bristol BS32 4RX United Kingdom
sales@tcgroupuk.com
Copyright © TC Group


Have you heard? Global Pet Foods in North Vancouver, BC is NOW OPEN! They are located in the Capilano Village at 801 Marine Drive, Unit 200 North Vancouver, British Columbia.
Owner, Anthony, is very excited to be in North Vancouver and has already had a very warm welcome from the community. He is eager to meet you (and your pets)! Anthony himself has three beautiful Golden Retrievers that are a part of his family.
Specializing in natural, holistic, and wholesome pet foods & supplements for every type of pet from dogs and cats, to birds, and small animals. They have the best selection of healthy pet food, toys, and accessories. Anthony, Noorhan, and the rest of his friendly staff will ensure that you get the right pet product for your specific needs!
To find a location near you, use our Location Finder.


text:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace OfficeDevPnP.PartnerPack.Infrastructure
{
    /// <summary>
    /// Defines the available scopes for a Provisioning Template definition
    /// </summary>
    public enum TemplateScope
    {
        /// <summary>
        /// Defines a Provisioning Template for a Site Collection
        /// </summary>
        Site,
        /// <summary>
        /// Defines a Provisioning Template for a Sub Site
        /// </summary>
        Web,
        /// <summary>
        /// Defines a Provisioning Template that can be applied as a partial add-on to a site
        /// </summary>
        Partial,
    }
}

redpajama_set_name: RedPajamaGithub


text:
\section{Introduction}\label{s1}
One of the greatest puzzles in the Standard Model (SM) is the discrepancy between the SM theoretical prediction and experimental data on the muon magnetic dipole moment $(g-2)_\mu$~\cite{pdg}, which is regarded as a hint towards  new physics beyond the SM (BSM). Currently, by combining the data from Brookhaven~\cite{Muong-2:2006rrc} and Fermilab~\cite{Muong-2:2021ojo}, the muon $g-2$ anomaly is given by
\begin{eqnarray}
	\Delta a_\mu = a_\mu^{\rm exp} -  a_\mu^{\rm SM} = (25.1 \pm 5.9) \times 10^{-10}\,,
\end{eqnarray}
where the latest SM calculation leads to $a_\mu^{\rm SM} = (116591810\pm 43)\times 10^{-11}$~\cite{Aoyama:2020ynm,Keshavarzi:2018mgv,Chao:2021tvp}. Already, many BSM models have been proposed in order to resolve this muon $g-2$ discrepancy (for a recent review see {\it e.g.}~\cite{Athron:2021iuf} and references therein).

In Ref.~\cite{Huang:2022}, we have demonstrated  that the muon $g-2$ anomaly can be explained by the one-loop effects induced by a new spin-2 particle $G$, which can be identified as the first Kaluza-Klein (KK) graviton in the generalized Randall-Sundrum (RS) model~\cite{Davoudiasl:1999tf,Pomarol:1999ad,Chang:1999nh,Davoudiasl:2000wi,Batell:2005wa,Batell:2006dp,Fok:2012zk,Lee:2013bua,Han:2015cty,Geng:2016xin,Falkowski:2016glr,Dillon:2016fgw,Dillon:2016tqp,Kraml:2017atm,Geng:2018hpq,Goyal:2019vsw}. By calculating all relevant one-loop Feynman diagrams, we have obtained the analytic expression of the leading-order $G$-induced contribution to the muon $g-2$, which is shown to maintain both the gauge invariance of the quantum electrodynamics and the correct divergence structure of loop integrals. Note that we have only imposed the theoretical constraint from the perturbativity on the parameter space of the spin-2 particle model in Ref.~\cite{Huang:2022}. However, another criterion to determine if the perturbative calculation remains under control is the perturbative unitarity bound, which has a long history to restrict parameters in a given model~\cite{Gell-Mann:1969cuq,Weinberg:1971fb}. Perhaps the most famous application was to limit the mass of the SM Higgs boson in Ref.~\cite{Lee:1977yc,Lee:1977eg,Durand:1989zs}. An incomplete list of the use of unitarity bounds to restrict the BSM models includes Refs.~\cite{Glashow:1976nt, Huffel:1980sk, Maalampi:1991fb, Kanemura:1993hm, Akeroyd:2000wc, Das:2015mwa, Kanemura:2015ska, Goodsell:2018tti, Appelquist:1987cf, Chaichian:1987zt, Falkowski:2016glr,Banta:2021dek}.

In the present work, we would like to derive the unitarity bounds for the spin-2 particle model in Ref.~\cite{Huang:2022}. Note that the interpretation of the muon $g-2$ anomaly involves $G$ couplings to charged leptons $\ell$ and photons $\gamma$. Thus, in order to constrain these two kinds of interactions, one needs to consider the 2-to-2 elastic scatterings of charged leptons and of photons via the mediation of $G$, with all possible initial and final helicity states. By yielding the $s$-wave projected amplitudes for all scattering processes, we can impose the unitarity bounds on the model parameters. As a result, it will be shown that the obtained unitarity bounds have significant impacts on the parameter space to explain the muon $g-2$.

The paper is organized as follows. In Sec.~\ref{model}, we briefly summarize the main results in Ref.~\cite{Huang:2022}, including the Lagrangian of the spin-2 particle model and the leading-order expression of the lepton $g-2$ contribution induced by $G$. Sec.~\ref{SecUniBound} is devoted to the calculation of amplitudes for elastic scattering processes, $\ell^- \ell^+ \to \ell^- \ell^+$ and $\gamma\gamma\to \gamma\gamma$, with all possible initial and final helicity states. From these amplitudes, we derive the unitarity bounds on the charged-lepton- and photon-$G$ couplings. We then make advantage of the obtained unitarity bounds to constrain the muon $g-2$ preferred parameter space in Sec.~\ref{SecResult}. Finally, we conclude in Sec.~\ref{Conclusion}.

\section{A Spin-2 Particle Model and its Explanation to Lepton $g-2$}\label{model}
The Lagrangian for the spin-2 particle $G$ explanation to the lepton $g-2$ is given by~\cite{Han:1998sg}
\begin{eqnarray}\label{LagG}
    {\cal L}_G = -\frac{1}{\Lambda} G_{\mu\nu} \Big[c_\gamma T_\gamma^{\mu\nu} + \sum_{\ell=e,\mu,\tau} c_\ell T_\ell^{\mu\nu}\Big]\,,
\end{eqnarray}
where the stress-energy tensors, $T_\ell^{\mu\nu}$ and $T_\gamma^{\mu\nu}$, of charged leptons  and  photons  are defined by
\begin{eqnarray}\label{DefEMT}
    T_\ell^{\mu\nu} &=& \frac{i}{4} \bar{\ell} (\gamma^\mu \partial^\nu + \gamma^\nu \partial^\mu) \ell -\frac{i}{4} (\partial^\mu \bar{\ell}\gamma^\nu + \partial^\nu \bar{\ell}\gamma^\mu) \ell \nonumber\\
    && -i\eta^{\mu\nu} [\bar{\ell} \gamma^\rho \partial_\rho \ell + im_\ell \bar{\ell}\ell -\frac{1}{2}\partial^\rho (\bar{\ell}\gamma_\rho \ell)]\,, \nonumber\\
    T_\gamma^{\mu\nu} &=& \frac{1}{4} \eta^{\mu\nu} F^{\rho\sigma}F_{\rho\sigma} - F^{\mu\rho}F^{\nu}_\rho \nonumber\\
    && -\frac{1}{\xi} \left[\eta^{\mu\nu}\left(\partial^\rho \partial^\sigma A_\sigma A_\rho + \frac{1}{2}(\partial^\rho A_\rho)^2\right)-\left(\partial^\mu \partial^\rho A_\rho A^\nu + \partial^\nu \partial^\rho A_\rho A^\mu\right)\right]\,,
\end{eqnarray}
respectively, with $\xi$ the gauge parameter for the photon field. This Lagrangian can be viewed as a part of the low-energy effective action in the generalized Randall-Sundrum model where the massive spin-2 particle is the first KK excitation of the conventional graviton~\cite{Randall:1999ee,Davoudiasl:1999tf,Pomarol:1999ad,Chang:1999nh,Davoudiasl:2000wi,Falkowski:2016glr}.

In the light of the effective interactions between $G$ and charged leptons/photons in Eq.~(\ref{LagG}), we have drawn and calculated in Ref.~\cite{Huang:2022} the one-loop Feynman diagrams contributing to the charged lepton $g-2$, with the dominant contribution given by
\begin{eqnarray}\label{G2total}
	\Delta a_{\ell}^G = \left(\frac{m_\ell^2}{\Lambda}\right)^2 \left(\frac{\Lambda}{m_G}\right)^4 \left(\frac{c_\ell^2}{48\pi^2} - \frac{c_\ell c_\gamma}{24\pi^2}\right)\,.
\end{eqnarray}
When deriving this leading-order result, we have applied the loop regularization~\cite{Wu:2002xa,Wu:2003dd} method to regularize the quartic divergences of loop integrals, which has been shown to maintain the gauge invariance of quantum electrodynamics and the correct divergence structure simultaneously.

Moreover, by requiring the validity of the perturbative expansion, {\it i.e.}, the one-loop contributions to the lepton $g-2$ should dominate over the the two-loop ones, we obtain novel constraints from the perturbativity on our non-renormalizable spin-2 particle interactions as follows
\begin{eqnarray}
	|c_\ell| < 4\pi \left(\frac{m_G}{\Lambda}\right)^2 \,,\quad \quad	|c_\gamma| < 4\pi \left(\frac{m_G}{\Lambda}\right)^2 \,,
\end{eqnarray}
which are obviously natural but non-trivial generalizations of perturbativity constraints on renormalizable operators~\cite{Nebot:2007bc}.

\section{Unitarity Bounds}\label{SecUniBound}
In this section, we
apply the tree-level unitarity bounds as our criterion to determine whether our perturbative calculations  remain under control, which can give extra constraints to our spin-2 particle model. Concretely, the unitarity of the S-matrix imposes the following bound to the $s$-wave projected amplitude $a_0(\sqrt{s})$~\cite{Lee:1977eg,Goodsell:2018tti,Banta:2021dek}
\begin{eqnarray}\label{UniBound}
	{\rm Re}(a_0)(\sqrt{s}) \leq \frac{1}{2}\,,
\end{eqnarray}
where $a_0(\sqrt{s})$ is defined as
\begin{eqnarray}\label{DefA0}
	a_0(\sqrt{s}) = \sqrt{\frac{4|{\bf p}_i| |{\bf p}_f|}{2^{\delta_i+\delta_f}s}} \frac{1}{32\pi} \int^1_{-1} d (\cos\theta) {\cal M}(i\to f) = \sqrt{\frac{4|{\bf p}_i| |{\bf p}_f|}{2^{\delta_i+\delta_f}s}} \frac{1}{16\pi} \int^0_{-s} \frac{d t}{s} {\cal M}(i\to f) \,,
\end{eqnarray}
in which the indices $\delta_{i,f}=1$ if the two particles in the initial or final states are identical to each other, otherwise $\delta_{i,f}=0$.
In our model, we have two kinds of effective vertices: the photon and lepton couplings to the spin-2 particle $G$, so we need to compute the amplitudes for $\ell^- \ell^+ \to \ell^- \ell^+$ and $\gamma\gamma \to \gamma\gamma$ of various helicity assignments to determine their respective bounds.

\subsection{$\ell^- \ell^+ \to \ell^- \ell^+$}
We begin by considering the perturbative unitarity bounds for the lepton-$G$ couplings $c_{\ell}$ with $\ell = e$, $\mu$ and $\tau$, which can be derived from the amplitudes of $\ell^- \ell^+ \to \ell^- \ell^+ $ of different $\ell^\pm$ helicity configurations. Given the Lagrangian in Eq.~(\ref{LagG}), there are two Feynman diagrams contributing to this process, which are shown in Fig.~\ref{Figll} with left and right panels corresponding to the $s$- and $t$-channels, respectively.
\begin{figure}[!htb]
	\centering
	\hspace{-5mm}
	\includegraphics[width=0.49\linewidth]{diagTEs.pdf}
	\includegraphics[width=0.49\linewidth]{diagTEt.pdf}
	\caption{Feynman diagrams which give rise to the scattering process $\ell^- \ell^+ \to \ell^- \ell^+$.}\label{Figll}
\end{figure}
Note that the lepton-$G$ interaction in Eq.~(\ref{LagG}) is $P$-invariant, {\it i.e.}, if the two scatterings of different helicity configurations are related by the parity ${P}$ transformation, then they would be equal in amplitude. For example, we have the following relations
\begin{eqnarray}
	{\cal M} (\ell_R^- \ell^+_L \to \ell^-_R \ell^+_L) = {\cal M} (\ell^-_L \ell^+_R \to \ell^-_L \ell^+_R) \,, \quad
	{\cal M} (\ell_R^- \ell^+_R \to \ell^-_R \ell^+_R) = {\cal M} (\ell^-_L \ell^+_L \to \ell^-_L \ell^+_L) \,.
\end{eqnarray}
Also, we will work in the high-energy limit in which the lepton masses can be ignored compared with the external particle momenta.
In what follows of this subsection, we shall present the detailed calculations of independent non-zero $\ell^- \ell^+ \to \ell^-\ell^+$ helicity amplitudes. By applying the unitarity bound on these amplitudes, we can obtain constraints on $c_\ell$.

\subsubsection{$\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L$}
Let us begin by computing the amplitude of $\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L$. In the massless limit, the momenta of external particles in the center-of-mass (com) frame are given by:
\begin{eqnarray}
	&{\rm Incoming}:&\quad	\ell^-_R: \quad p_1 = (E,0,0, E)\,, \quad \quad \ell^+_L: \quad p_2 =(E,0,0,-E)\, \\
	& {\rm Outgoing}:&\quad	\ell^-_R: \quad k_1 = (E,E\sin\theta, 0, E\cos\theta)\,, \quad \quad \ell^+_L:\quad k_2 = (E, -E\sin\theta,0,-E\cos\theta)\,, \nonumber
\end{eqnarray}
while the polarization vectors are denoted as
\begin{eqnarray}
	&{\rm Incoming}&:	u_R(p_1) = \sqrt{2E} \left(\begin{array}{c}
		0\\
		0\\
		1\\
		0
	\end{array}\right)\,, \quad
	v_L(p_2) = \sqrt{2E} \left(\begin{array}{c}
		0\\
		0\\
		0\\
		-1\\
	\end{array}\right)\,,\nonumber\\
	&{\rm Outgoing}&: u_R(k_1) = \sqrt{2E} \left(\begin{array}{c}
		0\\
		0\\
		\cos\frac{\theta}{2}\\
		\sin\frac{\theta}{2}
	\end{array}\right)\,, \quad
	v_L(k_2) = \sqrt{2E} \left(\begin{array}{c}
		0\\
		0\\
		\sin\frac{\theta}{2}\\
		-\cos\frac{\theta}{2}\\
	\end{array}\right)\,.
\end{eqnarray}

The total amplitude is composed of the $s$- and $t$-channel ones. The $s$-channel amplitude is given by
\begin{eqnarray}
	i{\cal M}_s(\ell_R^- \ell^+_L \to \ell^-_R \ell^+_L) &&= -\frac{ic_\ell^2}{32\Lambda^2} \bar{v}_L (p_2) [\gamma^\mu (p_{1}^\nu - p_2^\nu) + \gamma^\nu (p_1^\mu - p_2^\mu) - 2\eta^{\mu\nu} (\slashed{p}_1 - \slashed{p}_2 - 2m_\ell) ] u_R(p_1)	\nonumber\\
	&& \bar{u}_R(k_1)  [\gamma^\rho (k_{1}^\sigma - k_2^\sigma) + \gamma^\sigma (k_1^\rho - k_2^\rho) - 2\eta^{\rho\sigma} (\slashed{k}_1 - \slashed{k}_2 - 2m_\ell) ] v_L(k_2) \frac{B_{\mu\nu, \rho\sigma}}{Q^2-m_G^2}\,.\nonumber\\
\end{eqnarray}
By using the external particles' equations of motion $\slashed{p}_1 u_R(p_1) = m_\ell u_R(p_1) \simeq 0$ and $\slashed{p}_2  v_L(p_2) = - m_\ell v_L(p_2) \simeq 0$ where the symbol $\simeq$ represents the massless limit, the above $s$-channel amplitude can be simplified to
\begin{eqnarray}\label{sllR}
	i{\cal M}_s(\ell_R^- \ell^+_L \to \ell^-_R \ell^+_L) &=& -\frac{ic_\ell^2}{8\Lambda^2} \big\{ (p_1-p_2)(k_1-k_2) [\bar{u}_R(k_1) \gamma^\mu v_L(k_2)] [\bar{v}_L(p_2) \gamma_\mu u_R(p_1)] \nonumber\\
	&& +  [\bar{u}_R(k_1) (\slashed{p}_1 - \slashed{p}_2) v_L(k_2)][\bar{v}_L(p_2)(\slashed{k}_1 -\slashed{k}_2) u_R(p_1)] \big\} \frac{1}{Q^2-m_G^2} \,.
\end{eqnarray}
Now we turn to compute the fermion current in this $s$-channel $\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L$ process. Note that
\begin{eqnarray}
	\gamma^0 \gamma^\mu = \left(\begin{array}{cc}
		0 & 1 \\
		1 & 0
	\end{array}\right) \left(\begin{array}{cc}
		0 & {\sigma}^\mu \\
		\bar{\sigma}^\mu & 0
	\end{array}\right) = \left(\begin{array}{cc}
		\bar{\sigma}^\mu & 0 \\
		0 & \sigma^\mu
	\end{array}\right)\,.
\end{eqnarray}
Thus, for a vector-like current there are only two kinds of nonzero chirality configurations for  $(\ell^-, \ell^+)$ pairs: $(L,R)$ and $(R,L)$, while other two configurations like $(L,L)$ and $(R,R)$ are zero. Hence, for a $\ell^-_R \ell^+_L$ pair, the corresponding current given by
\begin{eqnarray}
	\bar{v}_L(p_2) \gamma^\mu u_R(p_1)  &=& (2E) (0,-1) \sigma^\mu \left(\begin{array}{c}
		1 \\
		0
	\end{array}\right) = (2E)(0,-1,-i,0)\,, \nonumber\\
	\bar{v}_L(k_2) \gamma^\mu u_R(k_1) &=& (2E) (\sin \frac{\theta}{2}, -\cos\frac{\theta}{2}) \sigma^\mu \left(\begin{array}{c}
		\cos\frac{\theta}{2} \\
		\sin\frac{\theta}{2}
	\end{array}\right) = (2E) (0, -\cos\theta, -i, \sin\theta)\,,
\end{eqnarray}
so that
\begin{eqnarray}
	[\bar{u}_R(k_1) \gamma^\mu v_L(k_2)] [\bar{v}_L(p_2) \gamma_\mu u_R(p_1)] =-(2E)^2(1+\cos\theta) = 2u\,,
\end{eqnarray}
\begin{eqnarray}
	\bar{v}_L(p_2)  (\slashed{k}_1 - \slashed{k}_2) u_R(p_1) = (2E)^2 \sin\theta = s \sin\theta \,, \nonumber\\
	\bar{u}_R(k_1) (\slashed{p}_1 - \slashed{p}_2) v_L(k_2) = -(2E)^2 \sin\theta = - s \sin\theta\,.
\end{eqnarray}
Consequently, the s-channel amplitude can be given by
\begin{eqnarray}
	i{\cal M}_s = -\frac{4i c_\ell^2}{32 \Lambda^2}\frac{\left(2u(u-t)-s^2\sin^2\theta\right) }{s-m_G^2}= -\frac{ic_\ell^2}{4 \Lambda^2}\frac{u(u-3t)}{s-m_G^2} = -\frac{ic_\ell^2}{4 \Lambda^2}\frac{(s+t)(s+4t)}{s-m_G^2}\,,
\end{eqnarray}
where in the last equality we have used the following relations
\begin{eqnarray}
	s(1+\cos\theta) = - 2u\,,\quad s(1-\cos\theta) = -2t, \quad 4ut = s^2 \sin^2\theta\,.
\end{eqnarray}

Now we come to compute $t$-channel contribution, with the amplitude given by
\begin{eqnarray}
	i{\cal M}_t (\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L) &&= -\frac{i c_\ell^2}{32\Lambda^2} \bar{v}_L(p_2) [\gamma^\mu (-p_{2}^\nu -k_2^\nu)+\gamma^\nu (-p_2^\mu -k_2^\mu)-2\eta^{\mu\nu}(-\slashed{p}_2 - \slashed{k}_2-2m_\ell)]v_L(k_2) \nonumber\\
	&& \bar{u}_R(k_1)[\gamma^\rho (p_1^\sigma+k_1^\sigma)+\gamma^\sigma(p_1^\rho+k_1^\rho)-2\eta^{\rho\sigma}(\slashed{p}_1+\slashed{k}_1-2m_\ell)]u_R(p_1) \frac{B_{\mu\nu, \rho\sigma}}{q^2-m_G^2}\,.
\end{eqnarray}
By using the lepton on-shell conditions, the $t$-channel amplitude can be reduced into
\begin{eqnarray}
	i{\cal M}_t (\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L) &=& \frac{4ic_\ell^2}{32\Lambda^2}\frac{1}{q^2-m_G^2} \big\{ (k_1+p_1)(k_2+p_2) [\bar{u}_R(k_1) \gamma^\mu u_R(p_1)] [\bar{v}_L(p_2) \gamma_\mu v_L(k_2)] \nonumber\\
	&& + [\bar{u}_R(k_1) (\slashed{k}_2+\slashed{p}_2)u_R(p_1) ][\bar{v}_L(p_2)(\slashed{k}_1 + \slashed{p}_1)v_L(k_2)]
	\big\}  \,.
\end{eqnarray}
In the center of mass frame, the t-channel amplitude can be further simplified to
\begin{eqnarray}
	i{\cal M}_t (\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L)  =\frac{ic_\ell^2}{4\Lambda^2} \frac{u(u-3s)}{t-m_G^2} = \frac{ic_\ell^2}{4\Lambda^2} \frac{(t+s)(t+4s)}{t-m_G^2}\,.
\end{eqnarray}
Therefore, the total amplitude is given by summing over the $s$- and $t$-channel amplitudes:
\begin{eqnarray}
	i{\cal M}  (\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L) &=& -\frac{ic_\ell^2}{4\Lambda^2} \left[\frac{u(u-3t)}{s-m_G^2} - \frac{u(u-3s)}{t-m_G^2}\right] \nonumber\\
	&=& -\frac{ic_\ell^2}{4\Lambda^2} \left[\frac{(s+t)(s+4t)}{s-m_G^2}-\frac{(t+s)(t+4s)}{t-m_G^2}\right] \,.
\end{eqnarray}
Here the minus sign between the two terms in the bracket can be understood to come from the interchange the two external fermion particles of momenta $k_1$ and $p_2$.

\subsubsection{$\ell_R^-  \ell_L^+ \to \ell^-_L \ell_R^+$}
Now we consider another kinematic configuration $\ell_R^- \ell_L^+ \to \ell_L^- \ell_R^+$, in which the four momenta are denoted by
\begin{eqnarray}
	p_1^\mu = (E,0,0,E)\,, &\quad& p_2^\nu = (E,0,0,-E)\,, \nonumber\\
	k_1^\rho = (E, E\sin\theta, 0, E\cos\theta)\,,  &\quad& k_2^\sigma = (E, -E\sin\theta, 0, -E\cos\theta)\,,
\end{eqnarray}
while their polarization spinors are given by
\begin{eqnarray}
	u_L(k_1) = \sqrt{2E} \left(\begin{array}{c}
		\xi_L \\
		0
	\end{array}\right)
	=\sqrt{2E} \left(\begin{array}{c}
		-\sin\frac{\theta}{2}\\
		\cos\frac{\theta}{2}\\
		0\\
		0
	\end{array} \right)\,, \,
	v_R(k_2) =\sqrt{2E} \left(\begin{array}{c}
		\eta_L \\
		0
	\end{array}\right) = \sqrt{2E} \left(\begin{array}{c}
		\cos\frac{\theta}{2} \\
		\sin\frac{\theta}{2}\\
		0\\
		0
	\end{array}
	\right)\,.
\end{eqnarray}
For the $s$-channel, the amplitude after taking into account the on-shell condition can also be reduced to Eq.~(\ref{sllR}), so that by using the explicit expressions for momenta and polarization spinors of external particles, we have
\begin{eqnarray}
	\bar{v}_L(p_2) \gamma^\mu u_R(p_1) &=& (2E) (0,-1) \sigma^\mu \left(\begin{array}{c}
		1\\
		0
	\end{array}\right) = (2E) (0,-1,-i,0)^\mu\,,\nonumber\\
	\bar{u}_L(k_1)\gamma^\mu v_R(k_2) &=& (2E) (-\sin\frac{\theta}{2}, \cos\frac{\theta}{2}) \bar{\sigma}^\mu \left(\begin{array}{c}
		\cos\frac{\theta}{2} \\
		\sin\frac{\theta}{2}
	\end{array}\right) = (0,-\cos\theta, -i, \sin\theta)^\mu\,,
\end{eqnarray}
and
\begin{eqnarray}
	\left[ \bar{u}_L(k_1) \gamma^\mu v_R(k_2) \right] \left[\bar{v}_L(p_2) \gamma_\mu u_R(p_1)\right] &=& (2E)^2 (1-\cos\theta) = -2t\,,\nonumber\\
	\bar{u}_L(k_1) (\slashed{p}_1-\slashed{p}_2) v_R(k_2) &=& -(2E)^2 \sin\theta\,,\nonumber\\
	\bar{v}_L(p_2) (\slashed{k}_1-\slashed{k}_2) u_R(p_1) &=& (2E)^2 \sin\theta\,.
\end{eqnarray}
Therefore, the $s$-channel amplitude is given by
\begin{eqnarray}
	i{\cal M}_s(\ell_R^- \ell_L^+ \to \ell_L^- \ell_R^+) = -\frac{ic_\ell^2}{4\Lambda^2} \frac{t(t-3u)}{s-m_G^2}\,.
\end{eqnarray}
Note that the bi-spinor forms like $\bar{u}_L(k_1)\gamma^\mu u_R(p_1)=0$ or $\bar{v}_L(p_2) \gamma^\mu v_R(k_2) = 0$ vanish, so that we cannot write down the $t$-channel amplitude. Thus, the total amplitude of $\ell_R^- \ell_L^+ \to \ell_L^- \ell_R^+$ is given by its $s$-channel one, with the following final result
\begin{eqnarray}
	i{\cal M}(\ell_R^- \ell_L^+ \to \ell_L^- \ell_R^+) = 	i{\cal M}_s(\ell_R^- \ell_L^+ \to \ell_L^- \ell_R^+)  = -\frac{ic_\ell^2}{4\Lambda^2} \frac{t(t-3u)}{s-m_G^2}\,.
\end{eqnarray}

\subsubsection{$\ell_R^- \ell_R^+ \to \ell_R^- \ell_R^+$}
Since we cannot write the bi-spinors, such as $\bar{v}_R(p_2) \gamma^\mu u_R(p_1)$ or $\bar{u}_R(k_1) \gamma^\mu v_R(k_2)$, the $s$-channel diagram cannot provide any contribution to this kinematic configuration. However, the $t$-channel Feynman diagram does contribute, and in the following we would like to compute it. By using the on-shell conditions, we can simplify the $t$-channel amplitude into the following form
\begin{eqnarray}
	i{\cal M}_t(\ell_R^- \ell_R^+ \to \ell_R^- \ell_R^+) =&& \frac{i c_\ell^2}{8\Lambda^2} \frac{1}{q^2 - m_G^2} \bigg\{ (k_1+p_1) (k_2+p_2) [\bar{u}_R(k_1) \gamma^\mu u_R(k_2)][\bar{v}_R(p_2) \gamma_\mu v_R(k_2)] \nonumber\\
	&+& [\bar{u}_R(k_1)(\slashed{p}_2 + \slashed{k}_2) u_R(p_1)][\bar{v}_R(p_2) (\slashed{p}_1+\slashed{k}_1)v_R(k_2)]	\bigg\}\,.
\end{eqnarray}
Here by taking the explicit form of momenta and spinors in the com frame, we can obtain
\begin{eqnarray}
	\bar{u}_R(k_1) \gamma^\mu u_R(p_1) &=& (2E) (\cos\frac{\theta}{2}, \sin\frac{\theta}{2}) \sigma^\mu \left(\begin{array}{c}
		1 \\
		0
	\end{array}\right) = (2E) \left(\cos\frac{\theta}{2}, \sin\frac{\theta}{2}, i\sin\frac{\theta}{2}, \cos\frac{\theta}{2} \right)^\mu\,, \nonumber\\
	\bar{v}_R(p_2) \gamma^\mu v_R(k_2) &=&  (2E) (1,0) \bar{\sigma}^\mu \left(\begin{array}{c}
		\cos\frac{\theta}{2} \\
		\sin\frac{\theta}{2}
	\end{array}\right) = (2E) \left(\cos\frac{\theta}{2}, -\sin\frac{\theta}{2}, i\sin\frac{\theta}{2}, -\cos\frac{\theta}{2}\right)^\mu\,.
\end{eqnarray}
so that
\begin{eqnarray}
	[\bar{u}_R(k_1) \gamma^\mu u_R(p_1)][\bar{v}_R(p_2) \gamma_\mu v_R(k_2)] &=& 2(2E)^2 = 2s\,, \nonumber\\
	\bar{u}_R(k_1) (\slashed{p}_2 + \slashed{k}_2) u_R(p_1) &=& 2 (2E)^2 \cos\frac{\theta}{2}\,, \nonumber\\
	\bar{v}_R(p_2) (\slashed{p}_1 + \slashed{k}_1) v_R(k_2) &=&  2 (2E)^2 \cos\frac{\theta}{2}\,.
\end{eqnarray}
It turns out that the final expression for the amplitude of $\ell_R^- \ell_R^+ \to \ell_R^- \ell_R^+$ is given by
\begin{eqnarray}
	i{\cal M}(\ell_R^- \ell_R^+ \to \ell_R^- \ell_R^+) = i{\cal M}_t(\ell_R^- \ell_R^+ \to \ell_R^- \ell_R^+) = \frac{ic_\ell^2}{4\Lambda^2} \frac{s(s-3u)}{t-m_G^2}\,.
\end{eqnarray}

\subsubsection{Unitarity Bounds for $\ell^- \ell^+ \to \ell^- \ell^+$}
In the above discussion, we obtains all of the nonzero $\ell^-\ell^+ \to \ell^- \ell^+$ amplitudes of different helicity configurations, which are summarized again as follows
\begin{eqnarray}
	{\cal M} (\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L) &=& {\cal M} (\ell^-_L \ell^+_R \to \ell^-_L \ell^+_R) = -\frac{c_\ell^2}{4\Lambda^2}\left[ \frac{u(u-3t)}{s-m_G^2} - \frac{u(u-3s)}{t-m_G^2} \right]\,,\nonumber\\
	{\cal M}	 (\ell^-_R \ell^+_R \to \ell^-_R \ell^+_R) &=& {\cal M} (\ell^-_L \ell^+_L \to \ell^-_L \ell^+_L) = \frac{c_\ell^2}{4\Lambda^2} \frac{s(s-3u)}{t-m_G^2}\,, \nonumber\\
	{\cal M} (\ell^-_R \ell^+_L \to \ell^-_L \ell^+_R) &=& {\cal M} (\ell^-_L \ell^+_R \to \ell^-_R \ell^+_L) = -\frac{c_\ell^2}{4\Lambda^2} \frac{t(t-3u)}{s-m_G^2}\,,
\end{eqnarray}
where the two amplitudes in a single line are $P$-symmetric to one another, while other helicity configurations would lead to vanishing amplitudes. By using the definition of $a_0(\sqrt{s})$ in Eq.~(\ref{DefA0}), we can obtain the $s$-wave projected amplitudes, given by
\begin{eqnarray}
	&& a_0(\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L) = a_0(\ell^-_L \ell^+_R \to \ell^-_L \ell^+_R) \nonumber\\
	&=& -\frac{1}{16\pi s} \frac{c_\ell^2}{4\Lambda^2} \left[-\frac{s(28s^2 - 21 m_G^2 s -6 m_G^4)}{6(s-m_G^2)} + (4s+m_G^2) (s+m_G^2) \ln\left(\frac{s+m_G^2}{m_G^2}\right)  \right] \nonumber\\
	&\approx & \frac{c_\ell^2}{16\pi} \frac{s}{4\Lambda^2} \left( \frac{14}{3} - 4\ln\frac{s}{m_G^2} \right) \sim \frac{c_\ell^2}{16\pi} \left(\frac{14}{3} - 4\ln \frac{4\Lambda^2}{m_G^2}\right)\,,\nonumber\\
	&& a_0 (\ell^-_R \ell^+_R \to \ell^-_R \ell^+_R) = a_0 (\ell^-_L \ell^+_L \to \ell^-_L \ell^+_L) \nonumber\\
	&=& \frac{1}{16\pi} \frac{c_\ell^2}{4\Lambda^2} \left[ 3s -(4s +3m_G^2) \ln \left( \frac{s+m_G^2}{m_G^2} \right) \right] \approx \frac{c_\ell^2}{16\pi} \frac{s}{4\Lambda^2} \left( 3 - 4\ln\frac{s}{m_G^2} \right) \nonumber\\
	&\sim& \frac{c_\ell^2}{16\pi} \left( 3-4\ln\frac{4\Lambda^2}{m_G^2} \right)\,, \nonumber\\
	&& a_0 (\ell^-_R \ell^+_L \to \ell^-_L \ell^+_R) = a_0 (\ell^-_L \ell^+_R\to \ell^-_R \ell^+_L) \nonumber\\
	&=& \frac{1}{16\pi}\frac{c_\ell^2}{4\Lambda^2}\frac{s^2}{6(s-m_G^2)} \approx  \frac{1}{16\pi}\frac{c_\ell^2}{6} \frac{s}{4\Lambda^2} \sim \frac{c_\ell^2}{96\pi}\,.
\end{eqnarray}
where the symbol $\sim$ denotes the high energy limit with $s \to 4\Lambda^2$. By requiring the $s$-wave projected amplitudes to satisfy the unitarity bound ${\rm Re} [a_0(\sqrt{s})] \leq 1/2 $, we can obtain the following constraints on the Wilsonian coefficient $c_\ell$
\begin{eqnarray}
	\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L : \quad	c_\ell &\leq& \sqrt{\frac{8\pi}{\left| \frac{14}{3} - 4\ln\frac{4\Lambda^2}{m_G^2} \right|}}\,,\\
	\ell^-_R \ell^+_L \to \ell^-_L \ell^+_R : \quad		c_\ell &\leq& \sqrt{48\pi}\,,\\
	\ell^-_R \ell^+_R \to \ell^-_R \ell^+_R : \quad
	c_\ell &\leq & \sqrt{\frac{8\pi}{\left| 3 - 4\ln\frac{4\Lambda^2}{m_G^2} \right|}}\,. \label{UBLepton}
\end{eqnarray}
In Fig.~\ref{UniBoundLep}, we plot the upper limits on $c_\ell$ derived from different helicity states, which are functions of the spin-2 particle mass $m_G$ when $\Lambda = 1$~TeV. It is seen that the most stringent constraint on $c_\ell$ is given by the channel $\ell^-_R \ell^+_R \to \ell^-_R \ell^+_R$ ($\ell^-_L \ell^+_L \to \ell^-_L \ell^+_L$) in the spin-2 particle mass range from $100$~GeV to 500~GeV, which is of great interest to in the interpretation of the muon $g-2$ anomaly.

\begin{figure}[!htb]
	\centering
	\hspace{-5mm}
	\includegraphics[width=0.7\linewidth]{UBl.pdf}
	\caption{The unitarity bounds on the lepton-$G$ Wilsonian coefficients $c_\ell$  for the three independent helicity configurations: $\ell^-_R \ell^+_L \to \ell^-_R \ell^+_L$, $\ell^-_R \ell^+_L \to \ell^-_L \ell^+_R$ and $\ell^-_R \ell^+_R \to \ell^-_R \ell^+_R$, which are labeled as RL, LR, and RR in the legend. } \label{UniBoundLep}
\end{figure}

\subsection{$\gamma\gamma\to \gamma\gamma$}
In order to obtain the unitarity bounds on the photon-$G$ coupling $c_\gamma$, we have to calculate the $\gamma\gamma \to \gamma\gamma$ amplitudes of various polarization configurations. The relevant Feynman diagrams are shown in Fig.~\ref{Figgg}, which correspond to the $s$-, $t$-, and $u$-channels, respectively. As the photon-$G$ vertex in Eq.~(\ref{LagG}) preserves the spatial parity $P$ symmetry, the amplitudes of different photon polarizations are the same to each other when they are related by parity transformation, which greatly reduces the number of independent photon 2-to-2 scattering amplitudes.
In this subsection, we compute these independent nonzero photon scattering amplitudes, based on which we can yield the associated unitarity bounds on $c_\gamma$.
\begin{figure}[!htb]
	\centering
	\hspace{-5mm}
	\includegraphics[width=0.32\linewidth]{diagTGs.pdf}
	\includegraphics[width=0.32\linewidth]{diagTGt.pdf}
	\includegraphics[width=0.32\linewidth]{diagTGu.pdf}
	\caption{Feynman diagrams for the photon-photon scattering process of $\gamma\gamma\to \gamma\gamma$.}\label{Figgg}
\end{figure}

\subsubsection{$\gamma_R\gamma_R \to \gamma_R \gamma_R$}
Firstly, we consider the amplitude of $\gamma_R \gamma_R \to \gamma_R \gamma_R$, where the four external photons are all right-handed.
In the com frame, the two incoming photons are specified by
\begin{eqnarray}\label{gIn}
	p_1^\mu = (E, 0, 0, E)\,, &\quad& \epsilon^\mu_R (p_1) = \frac{1}{\sqrt{2}} (0,-1,-i, 0)\,, \nonumber\\
	p_2^\nu = (E,0,0,-E)\,, &\quad& \epsilon^\nu_R(p_2) = \frac{1}{\sqrt{2}} (0,1, -i, 0)\,,
\end{eqnarray}
while the two outgoing photons are represented by
\begin{eqnarray}\label{gOut}
	k_1^\rho = (E, E\sin\theta, 0 , E\cos\theta)\,, &\quad& \epsilon^*_R(k_1) = \frac{1}{\sqrt{2}} (0, -\cos\theta, i, \sin\theta)\,, \nonumber\\
	k_2^\sigma = (E, -E\sin\theta, 0, -E\cos\theta)\,, &\quad& \epsilon^*_R(k_2) = \frac{1}{\sqrt{2}} (0,\cos\theta, i, -\sin\theta)\,.
\end{eqnarray}
The amplitude constitutes three Feynman diagrams shown in Fig.~\ref{Figgg}, corresponding to $s$-, $t$- and $u$-channels, respectively. In the following, we shall calculate them one by one. According to the Feynman rules, the $s$-channel amplitude is given by
\begin{eqnarray}
	i{\cal M}_s (\gamma_R \gamma_R \to \gamma_R \gamma_R) &=& -\frac{ic_\gamma^2}{2\Lambda^2} \epsilon_R^\mu (p_1) \epsilon_R^\nu(p_2) \epsilon_R^*(k_1) \epsilon^*_R (k_2) [(p_1\cdot p_2) C_{\alpha\beta, \mu\nu} + D_{\alpha\beta, \mu\nu}(p_1, p_2)] \nonumber\\
	&& \frac{B^{\alpha \beta, \lambda\delta}(Q)}{Q^2-m_G^2}  [(k_1\cdot k_2) C_{\lambda\delta, \rho\sigma} + D_{\lambda\delta, \rho\sigma}(k_1, k_2)]\,.
\end{eqnarray}
Note that, by using the relation $Q= p_1 +p_2 = k_1 + k_2$, we find that terms proportional to four- and two-powers of $1/m_G$ are canceled. By further making use of the specific momenta and polarization vectors of the four external particles in Eqs.~(\ref{gIn}) and (\ref{gOut}), the $s$-channel amplitude vanishes, while the $t$- and $u$-channel amplitudes are given by
\begin{eqnarray}
	i{\cal M}_t (\gamma_R \gamma_R \to \gamma_R \gamma_R) &=& -\frac{ic_\gamma^2}{2\Lambda^2} \epsilon_R^\mu (p_1) \epsilon_R^\nu(p_2) \epsilon_R^*(k_1) \epsilon^*_R (k_2) [(-p_1\cdot k_1) C_{\alpha\beta, \mu\rho} + D_{\alpha\beta, \mu\rho}(p_1, -k_1)] \nonumber\\
	&& \frac{B^{\alpha \beta, \lambda\delta}(q)}{q^2-m_G^2}  [(-p_2\cdot k_2) C_{\lambda\delta, \nu\sigma} + D_{\lambda\delta, \nu\sigma}(p_2, -k_2)] \nonumber\\
	&=& -\frac{ic_\gamma^2}{2\Lambda^2} \frac{2s^2}{t-m_G^2}\,, \\
	i{\cal M}_u (\gamma_R \gamma_R \to \gamma_R \gamma_R) &=& -\frac{ic_\gamma^2}{2\Lambda^2} \epsilon_R^\mu (p_1) \epsilon_R^\nu(p_2) \epsilon_R^*(k_1) \epsilon^*_R (k_2) [(-p_1\cdot k_2) c_{\alpha\beta, \mu\rho} + D_{\alpha\beta, \mu\rho}(p_1, -k_2)] \nonumber\\
	&& \frac{B^{\alpha \beta, \lambda\delta}(q^\prime)}{q^{\prime \, 2}-m_G^2}  [(-p_2\cdot k_1) C_{\lambda\delta, \nu\sigma} + D_{\lambda\delta, \nu\sigma}(p_2, -k_1)] \nonumber\\
	&=& -\frac{ic_\gamma^2}{2\Lambda^2} \frac{2s^2}{u-m_G^2}\,,
\end{eqnarray}
where $q\equiv p_1 - k_1 = k_2 -p_2$ and $q^\prime = p_1 - k_2 = k_1 -p_2$, so that $q^2 = t$ and $q^{\prime \, 2} = u$. Therefore, the total amplitude of $\gamma_R \gamma_R \to \gamma_R \gamma_R$ is written as
\begin{eqnarray}\label{gg0}
	i{\cal M}(\gamma_R \gamma_R \to \gamma_R \gamma_R) = -\frac{iC_\gamma^2}{2\Lambda^2} \left(\frac{2s^2}{t-m_G^2}+ \frac{2s^2}{u-m_G^2}\right)\,.
\end{eqnarray}

\subsubsection{$\gamma_R\gamma_L \to \gamma_R \gamma_L$}
We now turn to the process of $\gamma_R \gamma_L \to \gamma_R \gamma_L$. In the com frame, we can define the polarization vectors of the incoming particles as follows
\begin{eqnarray}
	\epsilon_R^\mu(p_1) = \frac{1}{\sqrt{2}} (0,-1,-i,0)^\mu\,, \quad \epsilon_L^\nu = \frac{1}{\sqrt{2}} (0,-1,-i,0)^\nu\,,
\end{eqnarray}
while those of the outgoing particles are given by
\begin{eqnarray}
	\epsilon_R^* (k_1) = \frac{1}{\sqrt{2}} (0,-\cos\theta, i, \sin\theta)\,, \quad \epsilon^*_L (k_2) = \frac{1}{\sqrt{2}} (0, -\cos\theta, i, \sin\theta)\,.
\end{eqnarray}
By taking the above expressions of the polarization vectors and momenta of the external particles into formulas for the $s$-, $t$- and $u$-channels, we can easily obtain the following partial amplitudes
\begin{eqnarray}
	i{\cal M}_s (\gamma_R \gamma_L \to \gamma_R \gamma_L) &=& -\frac{ic_\gamma^2}{2\Lambda^2} \frac{2u^2}{s-m_G^2}\,, \nonumber\\
	i{\cal M}_t (\gamma_R \gamma_L \to \gamma_R \gamma_L) &=& -\frac{ic_\gamma^2}{2\Lambda^2} \frac{2u^2}{t-m_G^2}\,, \nonumber\\
	i{\cal M}_u (\gamma_R \gamma_L \to \gamma_R \gamma_L) &=& 0\,.
\end{eqnarray}
Thus, the total amplitude is the summation over the $s$-, $t$- and $u$-channels
\begin{eqnarray}
	i{\cal M} (\gamma_R \gamma_L \to \gamma_R \gamma_L) = -\frac{ic_\gamma^2}{2\Lambda^2} \left(\frac{2u^2}{s-m_G^2} + \frac{2u^2}{t-m_G^2}\right)\,.
\end{eqnarray}

\subsubsection{$\gamma_R \gamma_L \to \gamma_L \gamma_R$}
In the case of $\gamma_R \gamma_L \to \gamma_L \gamma_R$, the polarization vectors for the external photons in the com frame are given by
\begin{eqnarray}
	&& \epsilon_R (p_1) = \frac{1}{\sqrt{2}} (0, -1, -i, 0)\,,\quad \epsilon_L (p_2) = \frac{1}{\sqrt{2}} (0, -1, -i, 0) \,, \nonumber\\
	&& \epsilon^*_L(k_1) = \frac{1}{\sqrt{2}} (0, \cos\theta, i -\sin\theta)\,, \quad \epsilon_R^* (k_2) = \frac{1}{\sqrt{2}} (0, \cos\theta, i, -\sin\theta)\,.
\end{eqnarray}
Following exactly the same procedure in the previous subsections, we can derive the following amplitudes for the $s$-, $t$- and $u$-channels of $\gamma_R \gamma_L \to \gamma_L \gamma_R$:
\begin{eqnarray}
	i{\cal M}_s (\gamma_R \gamma_L \to \gamma_L \gamma_R) &=& -\frac{ic_\gamma^2}{2\Lambda^2} \frac{2t^2}{s-m_G^2}\,, \nonumber\\
	i{\cal M}_t (\gamma_R \gamma_L \to \gamma_L \gamma_R) &=& 0 \,, \nonumber\\
	i{\cal M}_u (\gamma_R \gamma_L \to \gamma_L \gamma_R) &=& -\frac{ic_\gamma^2}{2\Lambda^2} \frac{2t^2}{u-m_G^2}\,.
\end{eqnarray}
It turns out that the total amplitude of $\gamma_R \gamma_L \to \gamma_L \gamma_R$ is given by
\begin{eqnarray}
	i{\cal M} (\gamma_R \gamma_L \to \gamma_L \gamma_R) = -\frac{ic_\gamma^2}{2\Lambda^2} \left( \frac{2t^2}{s-m_G^2} + \frac{2t^2}{u-m_G^2} \right)\,.
\end{eqnarray}

\subsubsection{Unitarity Bounds for $\gamma\gamma \to \gamma\gamma$}
Here we summarize the nonzero independent $\gamma\gamma\to \gamma\gamma$ amplitudes of different polarizations as follows
\begin{eqnarray}
	i{\cal M} (\gamma_R \gamma_L \to \gamma_R \gamma_L) &=& i{\cal M} (\gamma_L\gamma_R \to \gamma_L \gamma_R) = -\frac{ic_\gamma^2}{2\Lambda^2} \left( \frac{2u^2}{s-m_G^2} + \frac{2u^2}{t-m_G^2} \right)\,,\nonumber\\
	i{\cal M}  (\gamma_R \gamma_L \to \gamma_L \gamma_R) &=& i{\cal M} (\gamma_L \gamma_R \to \gamma_R \gamma_L)  =  -\frac{ic_\gamma^2}{2\Lambda^2} \left( \frac{2t^2}{s-m_G^2} + \frac{2t^2}{u-m_G^2} \right)\,,\nonumber\\
	i{\cal M}  (\gamma_R \gamma_R \to \gamma_R \gamma_R) &=& i{\cal M} (\gamma_L \gamma_L \to \gamma_L \gamma_L)  =  -\frac{ic_\gamma^2}{2\Lambda^2} \left( \frac{2s^2}{t-m_G^2} + \frac{2s^2}{u-m_G^2} \right)\,,
\end{eqnarray}
while all other polarization amplitudes vanish identically. According to the definition of $a_0(\sqrt{s})$ in Eq.~(\ref{DefA0}), the $s$-wave projected amplitudes for the nontrivial polarization assignments are given by
\begin{eqnarray}
	&& a_0(\gamma_R \gamma_R \to \gamma_R \gamma_R) = a_0 (\gamma_L\gamma_L \to \gamma_L\gamma_L) \nonumber\\
	&=& \frac{c_\gamma^2 }{4\pi} \frac{s}{4\Lambda^2} \ln \frac{s+m_G^2}{m_G^2} \approx \frac{c_\gamma^2 }{4\pi} \frac{s}{4\Lambda^2} \ln \frac{s}{m_G^2}
	\sim  \frac{c_\gamma^2}{4\pi} \ln \frac{4\Lambda^2}{m_G^2}\,, \nonumber\\
	&& a_0(\gamma_R \gamma_L \to \gamma_R \gamma_L) = a_0(\gamma_R \gamma_L \to \gamma_L \gamma_R) = a_0(\gamma_L \gamma_R \to \gamma_L \gamma_R) = a_0(\gamma_L \gamma_R \to \gamma_R \gamma_L) \nonumber\\
	&=& -\frac{1}{32\pi s}\frac{c_\gamma^2}{\Lambda^2} \left[ \frac{s(11s^2 - 3 m_G^2 s-6m_G^4)}{6(s-m_G^2)} -(s+m_G^2) \ln\frac{s+m_G^2}{m_G^2} \right] \nonumber\\
	&\approx & -\frac{c_\gamma^2}{8\pi}\frac{s}{4\Lambda^2} \left(\frac{11}{6} - \ln\frac{s}{m_G^2}\right) \sim -\frac{c_\gamma^2}{8\pi} \left( \frac{11}{6} - \ln\frac{4\Lambda^2}{m_G^2} \right)\,,
\end{eqnarray}
where we have taken the high-energy limit with $s\sim 4\Lambda^2 \gg m_G^2$.
Therefore, the unitarity bounds for these channels are given by
\begin{eqnarray}
	&& \gamma_R \gamma_R \to \gamma_R \gamma_R: \quad c_\gamma \lesssim \sqrt{\frac{2\pi}{\left|\ln\frac{4\Lambda^2}{m_G^2}\right|}}\,, \label{UBGamma} \\
	&& \gamma_R \gamma_L \to \gamma_R \gamma_L: \quad  c_\gamma \lesssim \sqrt{\frac{4\pi}{\left| \frac{11}{6} - \ln\frac{4\Lambda^2}{m_G^2} \right|}}\,.
\end{eqnarray}

\begin{figure}[!htb]
	\centering
	\hspace{-5mm}
	\includegraphics[width=0.7\linewidth]{UBg.pdf}
	\caption{The unitarity bounds on the photon-$G$ Wilsonian coefficient $c_\gamma$ for two independent helicity configurations: $\gamma_R \gamma_R \to \gamma_R \gamma_R$ and $\gamma_R \gamma_L \to \gamma_R \gamma_L$, which are labeled as RR and RL in the legend.  } \label{UniBoundGam}
\end{figure}
In Fig.~\ref{UniBoundGam}, we make plots for these unitarity bounds, which are set on the coupling $c_\gamma$ as a function of the spin-2 particle mass $m_G$ with a fixed cutoff scale $\Lambda = 1$~TeV. It is clear that the constraint from the channel $\gamma_R \gamma_R \to \gamma_R \gamma_R$ is much stronger than that from $\gamma_R\gamma_L\to \gamma_R \gamma_L$ for $m_G>100$~GeV. Hence, we will apply the upper limit from $\gamma_R \gamma_R \to \gamma_R \gamma_R$ as our unitarity bound in the following numerical analysis.

\section{Numerical Studies}\label{SecResult}
In this section, we study the constraints on the model parameter space by exploiting the unitarity bounds in Eqs.~(\ref{UBLepton}) and (\ref{UBGamma}), which sets upper limits on the massive spin-2 particle couplings $c_\ell$ and $c_\gamma$. Note that the spin-2 particle model in Ref.~\cite{Huang:2022} has been proposed to explain the long-standing lepton $g-2$ anomaly. Currently, the most precise value of the muon $g-2$ is provided by the combined data from Brookhaven and Fermilab, which deviates the SM predictions by 4.25 $\sigma$~CL. On the other hand, the measurements of the fine structure constant $\alpha$ at Laboratoire Kastler Brossel (LKB)~\cite{Morel:2020dww} and  Berkeley~\cite{Parker:2018vye} have given two latest SM predictions~\cite{Aoyama:2012wj,Aoyama:2019ryr} on the electron anomalous magnetic dipole moment, $a_e^{\rm LKB}$ and $a_e^{\rm B}$, with their differences from the experimental result $a^{\rm exp}_e$~\cite{Hanneke:2008tm} given by
\begin{eqnarray}\label{g2Electron}
	\Delta a_e^{\rm LKB} &=& a_e^{\rm exp} - a_e^{\rm LKB} = (4.8\pm 3.0) \times 10^{-13}\,, \nonumber\\
	\Delta a_e^{\rm B} &=& a_e^{\rm exp} - a_e^{\rm B} = (-8.8 \pm 3.6) \times 10^{-13}\,.
\end{eqnarray}
Based on the general lepton $g-2$ formula in Eq.~(\ref{G2total}), we can also discuss the implications of our spin-2 particle model on the electron $g-2$. Since the data in Eq.~(\ref{g2Electron}) given at LKB and  Berkeley are incompatible with each other, we would like to discuss them separately.

\begin{figure}[!htb]
	\centering
	\hspace{-5mm}
	\includegraphics[width=0.5\linewidth]{M200L1000.pdf}
	\includegraphics[width=0.5\linewidth]{M500L1000.pdf}
	\caption{The parameter space in the $c_\ell$-$c_\gamma$ plane for the cutoff scale fixed at $\Lambda=1$~TeV and the spin-2 particle mass at $m_G = 200$~GeV (left panel) and $500$~GeV (right panel). The blue and yellow shaded regions show the parameter space that can explain the $\Delta a_\mu$ and $\Delta a_e^{\rm LKB}$ anomalies in $2\sigma$ range, while the areas colored in red and gray are excluded by the theoretical constraints from perturbativity and unitarity, respectively.} \label{FigL1000}
\end{figure}

In Fig.~\ref{FigL1000},  the parameter spaces explaining the Muon $(g-2)_\mu$ and LKB $(g-2)_e$ data at $2\sigma$~CL are plotted as the blue and yellow shaded regions in the $c_\ell$-$c_\gamma$ plane, where the cutoff scale is fixed to be $\Lambda =1$~TeV and the massive graviton mass to be $m_G = 200$~GeV (left panel) and 500~GeV (right panel). In the same plots, we also lay out the constraints of the perturbativity and unitarity, and the excluded regions are colored in red and gray, respectively. As a result, it is seen from Fig.~\ref{FigL1000} that, in spite of the strong theoretical perturbativity and unitarity bounds, there is still a substantial portion of parameter space in both plots to explain the $\Delta a_\mu$ and $\Delta a_e^{\rm LKB}$ discrepancies. In particular, it is interesting to note that the muon $g-2$ signal regions are all located in the LBK allowed parameter space, which implies that the current data supports the scenario in which the spin-2 field $G$ couples to all SM charged leptons via a universal coupling, {\it i.e.}, $c_\ell= c_e = c_\mu = c_\tau$. This lepton universality is shown in Ref.~\cite{Huang:2022} to help avoid the strong constraints from the charged-lepton-flavor-violation and $CP$-violation experiments. Furthermore, the comparison of the two plots in Fig.~\ref{FigL1000} indicates that, as the mass of $G$ increases, the unitarity bounds becomes more and more important than the perturbativity ones in limiting the parameter space. Especially, when $m_G = 500$~GeV, the unitarity bounds dominate the theoretical constraint, and shrink the muon $(g-2)$ preferred region to be in a small corner with $1<c_\ell <2$ and $-1.5<c_\gamma<0$.

\begin{figure}[!htb]
	\centering
	\hspace{-5mm}
	\includegraphics[width=0.48\linewidth]{cG05L1000.pdf}
	\includegraphics[width=0.48\linewidth]{cG03L1000.pdf}
	\includegraphics[width=0.48\linewidth]{cG0L1000.pdf}
	\includegraphics[width=0.48\linewidth]{cGn1L1000.pdf}
	\caption{The parameter space in the $m_G$-$c_l$ plane with $\Lambda=1$~TeV and $c_\gamma=0.5$ (top-left panel), 0.3 (top-right panel), 0 (bottom-left panel), and $-1$ (bottom-right panel). The color coding is the same as that in Fig.~\ref{FigL1000} } \label{cGplot}
\end{figure}

We can also show the relevant parameter space in the $m_G$-$c_\ell$ plane as in Fig.~\ref{cGplot}, when the cutoff scale is still fixed to be $\Lambda=1$~TeV and the photon-$G$ coupling is chosen as $c_\gamma = 0.5$, 0.3, 0, and $-1$, respectively. From these plots, it is clear that the unitarity bounds give the stronger constraint than the perturbativity ones in the large spin-2 mass regions. Remarkably, when $c_\gamma\gtrsim 0.5$, the muon $g-2$ signal regions with large $m_G$ values which were open under the perturbativity constraints are now ruled out by the unitarity bounds. However, as the photon-$G$ coupling $c_\gamma$ decreases, more parameter spaces are now allowed by the unitarity. In one special case with $c_\gamma = 0$ where the Barr-Zee-type diagrams~\cite{Bjorken:1977vt,Barr:1990vd} give a vanishing contribution to the lepton $g-2$, the spin-2 particle mass is limited to be below 500~GeV as shown in the lower-left panel in Fig.~\ref{cGplot}. When $c_\gamma$ further drops to take negative values, more allowed parameter regions  now open with $m_G$ extending to even larger values beyond 500~GeV, as shown  in the lower-right panel of Fig.~\ref{cGplot}.

\begin{figure}[!htb]
	\centering
	\hspace{-5mm}
	\includegraphics[width=0.5\linewidth]{M200L1000B.pdf}
	\includegraphics[width=0.5\linewidth]{M500L1000B.pdf}
	\caption{The parameter space in the $c_\ell$-$c_\gamma$ plane for the cutoff scale fixed at $\Lambda = 1$~TeV and the spin-2 particle mass at $m_G=200$~GeV (left panel) and $500$~GeV (right panel). The color coding is the same as that in Fig.~\ref{FigL1000}, except that the yellow region now represents the parameter space predicted by the Berkeley data of $\Delta a_e^{\rm B}$.} \label{FigBerkeley}
\end{figure}
Finally, we consider the simultaneous interpretation of the $\Delta a_\mu$ anomaly and the Berkeley $\Delta a_e^{\rm B}$ data,
with the numerical studies in Fig.~\ref{FigBerkeley}. It is remarkable to see that the parameter spaces allowed by the Berkeley $(g-2)_e$ data are completely disfavored by the perturbativity and unitarity constraints. This result can be understood as follows. First of all, it is worth mentioning that the Berkeley data
 prefers a negative $\Delta a_e^{\rm B}$, which is compared with the positive value  $\Delta a_\mu$. Hence, as argued in Ref.~\cite{Huang:2022}, the opposite sign between the muon and electron $g-2$ anomalies can only be achieved by making the Barr-Zee Feynman diagrams~\cite{Bjorken:1977vt,Barr:1990vd} dominate the total lepton $g-2$ formula in Eq.~(\ref{G2total}), since the combination $c_\gamma c_\ell$ in the Barr-Zee contribution can take any sign. On the other hand, the Berkeley result of $\Delta a_e^{\rm B}$ is so large that the coupling $|c_\gamma|$ should take a significant value, which has been strongly ruled out by the perturbativity and unitarity bounds. Especially, in the case with $m_G = 500$~GeV and $\Lambda = 1$~TeV, the unitarity bounds derived in the present paper further strengthen the limits from the perturbativity, which shows the significance of unitarity bounds in constraining our spin-2 particle model.

\section{Conclusions}\label{Conclusion}
In Ref.~\cite{Huang:2022}, we explain the long-standing discrepancy between the experimental measurements and the SM prediction of the muon $g-2$ in terms of a new massive spin-2 particle, which can be easily derived in the generalized RS models. Note that we have only considered the perturbativity constraints in Ref.~\cite{Huang:2022} on the model parameter space. In the present paper, we have further investigated unitarity bounds on this spin-2 particle model.
We have obtained the bounds by computing the $s$-wave projected amplitudes for two-body elastic scatterings of charged leptons and photons in the high-energy limit for all possible initial and final helicity states. As a result, we have found that the most stringent constraints on the lepton-$G$ coupling $c_\ell$ and on the photon-$G$ coupling $c_\gamma$ are yielded via the processes $\ell^-_R \ell^+_R \to \ell^-_R \ell^+_R $ and $\gamma_R \gamma_R \to \gamma_R \gamma_R$, respectively. We have applies the obtained unitarity bounds to numerically explore the spin-2 particle parameter space again. It turns out that the tree-level $S$-matrix unitarity gives useful constraints on the muon $g-2$ preferred parameter regions, which is complementary to the perturbativity.
In particular, it is seen that the unitarity bounds give the strongest limits on $c_\ell$ and $c_\gamma$ in the large $m_G$ region. Nevertheless, despite the stringent constraints imposed by the perturbativity and unitarity, there is still a considerable portion of parameter spaces which can explain the muon $g-2$ anomaly. By further extending this analysis to the electron $g-2$, we have shown that the LKB measurement of $\Delta a_e^{\rm LKB}$ can be compatible with the latest $\Delta a_\mu$ data, while $\Delta a_e^{\rm B}$ obtained at Berkeley cannot be explained by the present spin-2 particle model due to the strong constraints from the perturbativity and unitarity.

\section*{Acknowledgements}
This work is supported in part by the National Key Research and Development Program of China (Grant No. 2020YFC2201501 and No. 2021YFC2203003), the National Natural Science Foundation of China (NSFC)  (Grant No. 12005254 and No. 12147103), and the Key Research Program of Chinese Academy of Sciences  (No. XDPB15).

redpajama_set_name: RedPajamaArXiv


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text:
Press Releases Hartwig campaign: Sarah Godlewski lacks ethics, complaints filed related to campaign finance...
Hartwig campaign: Sarah Godlewski lacks ethics, complaints filed related to campaign finance violations and electioneering
FOR IMMEDIATE RELEASE: Tuesday, October 16, 2018, 2:01 PM
Contact: Travis Hartwig – [email protected], 262-441-9062
(MADISON, WI) – The Hartwig campaign has been notified of two separate ethics complaints filed against Sarah for Wisconsin.
The first complaint alleges Sarah Godlewski "Knowingly and willingly violated Wisconsin's prohibition on electioneering" by campaigning for additional votes outside of her polling location on August 14, 2018.
The second complaint alleges Sarah Godlewski violated campaign finance law by accepting a campaign contribution of $150,000 from an individual contributor. State of Wisconsin law mandates contribution limits of $20,000 for the State Treasurer's race. It appears this contribution has since been amended, changing the name of the contributor to the candidate. Which begs the question "Who signed the check and actually made this contribution?"
"I am not sure if Sarah is unaware of the laws here in Wisconsin after living out of state until this spring, or if she just thinks that she is above the law." said Hartwig. "But Wisconsinites deserve better than Sarah Godlewski."
So far in the race for State Treasurer, Sarah has done very little to earn the trust of voters. She has actively mislead voters about her residency, hosted lavish out-of-state fund raisers for special interests groups and campaigned on a pledge to invest our schools trust funds in riskier investments.
"Sarah's behavior reflects the Washington D.C. swamp that she is from" said Hartwig. "She has pushed the limits of our campaign finance system and has violated basic electioneering laws."
Hartwig, who grew up in East Troy, WI, holds a double major in Finance and Business Economics from Carroll University. He was a Mutual Fund Administrator at U.S. Bank Fund Services in downtown Milwaukee and owns a home in Oak Creek, WI.
Travis Hartwig is running for State Treasurer as a life-long conservative that believes Wisconsin taxpayers need a watchdog. He will use his experience to make the office more efficient and find ways to cut government spending because he believes that you do a better job of spending your hard-earned money than the government does


Cart Ruts Malta – the islands of Malta and Gozo are the most famous location of these mysterious and ancient paired grooves in the rock (also known as Cart Tracks).
The most popular area in Malta for them is Clapham Junction (Misrah Ghar il-Kbir as it is known by the Maltese) due to the high number in total, multiple pairs beside each other for some unknown reason and that they even intersect each other in crazy spaghetti junctions.
There are more Cart Ruts being discovered in other countries in Europe and America.
What are the Cart Ruts?
What are the Cart Ruts? These are deep ruts, rilles, tracks, grooves, channels left in Malta's limestone but in such numbers, variety and confusion that they leave more questions than answers.
They are located all over the islands of Malta and Gozo and were even said to have been found on the tiny island of Fifla!
If they were created and used by some form of vehicle then how did they move along some of these when the pairs of ruts are nearly a meter deep?
Why do the Cart Ruts on the islands of Malta and Gozo lead straight off high cliffs and isolated headlands?
Why are some of them found leading into the sea and on the seabed floor?
Why do the Malta Cart Tracks areas have curious geology like the Clapham Junction Triangle?
If these apparant ancient green roads and tracks were man made then for what purpose? Why were the Cart Ruts created?
Where in Malta and the world?
There are also other Cart Ruts locations around the world, with the internet helping othes be 'discovered' or brought to peoples attention. These can be similarly confusing and chaotic but some of them seem to have more substantial evidence that they were man made.
How were Malta's Cart Ruts created?
How were Cart Ruts in Malta formed? Were they made by vehicles like wooden sleds or wheeled carts? Are they natural? Were they created for or by aliens/UFO's? There are no records or mythology left by those who were around when these meandering paired tracks were created.
Who made the Cart Ruts?
Who made the Cart Ruts? Was it the Maltese Temple Builders, Romans, Phoenicians, Malta's Bronze Age people or other original Maltese civilisation and people?
How old are Malta's Cart Tracks and those in other countries?
How old are the Maltese Cart Ruts?
When were the Cart Tracks started and used? When were other countries Cart Ruts sites developed?
This site investigates the Cart Ruts mysteries and has unique photo guides to the main Maltese Cart Ruts sites, with more being added.
Visit the Cart Ruts Blog for a list of all the posts about Malta and Gozo's Cart Tracks and also those that are found around the rest of the planet.
Cart Ruts mysteries and evidence?
With Maltese Cart Ruts going off high cliff tops, Cart Ruts located on the sea floor, Cart Tracks in mass numbers intersecting each other in total chaos as seen at Clapham Junction and San Gwann Junction.
Even the Ghar Zerriegha site that appears to show defining evidence that the Malta Cart Ruts were man made is perhaps not so 100% conclusive when you consider about what is in the area and if these were formed at the same time or for the same reason.
Below are some of the unique photograph guides and investigations to the site on Malta and Gozo. These are the main sites. For a full list of our Malta photo guides click here or for those in other countries click here.
Clapham Junction Malta – most famous Cart Ruts site of the island of Malta and the largest number of Cart Tracks.
Naxxar Gap Cart Ruts Naxxar Gap Cart Ruts – Cart Tracks curving down the Great Fault escarpment in sweeping curves.
If the Cart Ruts went down, how and where did they come up the Great Fault Line?
Ghar Zerriegha Cart Tracks – Cart Ruts going off a Cliff? Where did they lead to?
Man made evidence with steps in the middle of the Cart Ruts?
What? When? Who? Why? Where on Malta and around the world?
Made by ancient man? By aliens or for UFO's? Signs of Gods?
Unique photo investigation of Malta's Cart Ruts sites. Also other countries.


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Home POLITICS 'You are not a pastor but a short devil' - FFK
'You are not a pastor but a short devil' - FFK
Former Aviation Minister, Femi Fani-Kayode, aka FFK, has accused a certain pastor of plotting against the National Leader of the All Progressives Congress (APC), Asiwaju Bola Tinubu as well as working against former Finance Minister, Kemi Adeosun and Deji Adeyanju.
Though he did not mention the name of the pastor he was referring to, FFK stated that the pastor's aim is to become president come 2023 and that is the reason he plotting against Tinubu.
In a series of tweets on his verified handle, FFK said:
"I disagree profoundly with the world view of men like Zangon Daura, Mamman Daura, Abba Kyari and Lawal Daura, but I would not describe them as evil. I have known them all for over 30 years and I can testify to that.
The evil one is the short Pastor who pretends to be an angel. All u do is plot how to jail ur fellow Yorubas even when they did no wrong. U locked up @adeyanjudeji; u are behind the incarceration & politically-motivated prosecutions of SW leaders and u pushed for the sacking of Kemi Adeosun and wanted her prosecuted and jailed for 2 years.
Worse of all, u are now plotting and planning against ur own leader Bola Tinubu just because you want to be President in 2023 and you want him weakened, discredited and out of ur way. Anyone that believes the only way up is to destroy others will end in destruction & shame…
You are a short, evil and wicked man and you are not a real Pastor. Your heart is as dark as hell and by the time this is all over, Nigerians will know that you are the most duplicitious creature and evil beast that they have ever had the misfortune of having to put up with!"
While many are of the belief that FFK is subtly throwing shades at Vice President Yemi Osinbajo, no one can really say who the pastor he is referring to is.
See the tweets here:
Femi Fani-Kayode on Twitter
The word "Yoruba" did not even exist until the 18th century and even then most of the tribes of the SW, including the Oyo's, rejected it due to its origin and meaning. The word "Yoruba" is alien to our culture and not known in the Anago language. Oduduwans please take note.
Any Omo Karo Jire or Oduduwan that continues to call himself a "Yoruba" is lost and does not know the implications of what he is doing to his own people. He is simply affirming and confirming an insulting label which has deep sinister, mystical and spiritual connotations…
I am not a "Yariba" or "Yoruba" but an "Omo Karo Jire" or an "Oduduwan" and my lanuage is not "Yoruba" but "Anago". We are what we call ourselves. We are not "shady and uneliable"(Yariba) and we must not accept names that are given to us by our historical adversaries…
'You are not a pastor but a short devil' - FFK Reviewed by JNJ GLOBAL GIST on October 21, 2019 Rating: 5


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26 Songs To Listen To If You're In The Middle Of A Full-Fledged Quarter-Life Crisis (Like Me)
I did this post last year with 25 songs, but I have to admit, they all definitely still apply. While...
Miranda Lambert, Charley Crockett & More To Perform At NASCAR Street Race Weekend In Chicago
What a lineup… NASCAR just announced that Miranda Lambert and The Chainsmokers will be headlining Street Race weekend, along with other...
Miranda Lambert And Jelly Roll Might Just Be The Duo We Didn't Know We Needed
The duo we didn't know we needed. Miranda Lambert met up with Nashville native and new top name in the...
Kat Hasty Says Her New Music Is Heavily Inspired By Miranda Lambert's 'Palomino'
Get ready because Kat Hasty has some new music on the way. You may know the Texas singer from her...
Miranda Lambert Says She Learned Quickly That "Hollywood Is Not Anything I Want To Be Part Of" While Living In L.A. When Blake Shelton Began 'The Voice'
Miranda Lambert's most-recent record, Palomino, was easily one of my favorites from last year. And in an interview with the LA Times...
Chris Stapleton & Miranda Lambert Wrote A Song Called "Somebody Winds Up Pregnant," And I Really Hope They Release It
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Miranda Lambert On Cutting Her Teeth At The Local Texas Honky Tonk As A Teenager: "It Was The Best Schooling I Could Have Asked For"
These days, Miranda Lambert plays to thousands of people in arenas across the country. She's actually in the middle of her...
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Miranda Lambert always brings it at her live shows… And apparently, so do her fans, but not necessarily always in a...
Miranda Lambert On Writing "Over You" With Blake Shelton About His Late Brother: "I'm Glad We Shared That Song… To Have That Together"
Miranda Lambert and ex-husband Blake Shelton have had their fair share of negative press when it comes to their relationship....
Pistol Annies' A Cappella Rendition Of "Auld Lang Syne" Is The Perfect Song To Ring In The New Year
Happy New Years Eve. As we count down the hours to 2023, we also close out 2022 and celebrate making...
It Doesn't Get Much Better Than George Strait, Chris Stapleton, Miranda Lambert, McConaughey & More On One Stage
Everything's bigger in Texas… like the star power. Back in 2017, after the Hurricanes Harvey, Irma, and eventually Maria wreaked...
Six Of My Favorite, Must-Listen Country Deep Cuts Released In 2022 From Lainey Wilson, Ben Burgess And More
Well, a great year of country music is almost officially in the books. There was so much great music released...


Today at Casa del Pox I had more than a couple of delightful things happen.
First, the squawking of the birds that have been hanging out in my eaves had a new noise — very high pitched squeeks. The eggs have hatched, and we have new little ones on the balcony. This will make the kitties forays onto the balcony more restricted – especially as the babies grow and need to learn to fly. I'm a little concerned that this batch has been hatched this late – I hope they will be strong enough to go whereever these birds go when it is time.
Second, I had a visitor who crossed the death threshold. My cousin Marie I guess is convinced that I have the chicken pox (she's had it) and not some other crazy virus and brought me some home-cooked food, kitty food and litter, and leftover blueberry pancakes from the place she and my Mom went to lunch. All good. And it was nice to actually talk to someone in person rather than on the phone. Family rocks!
Third, I'm still tired, but this afternoon, I got a bit of my knitting mojo back – I've started the Candelabra Socks (rav link) by Chrissy Gardiner. The yarn is Miss Babs Yummy in the Kiwi colorway.
I decided to follow another knitter's suggestion and use highlighters to mark the different cables and special stitches. In this case, I copied the chart on my printer/scanner/copier with the zoom up to about 115%, and then hand-colored with highlighter pens. It really helps!
Fourth, I have to admit that probably one of the events that I was looking forward to most at Sock Summit was the Flash Mob. I had learned the routine, done a practice with my friends, was planning impromptu teach ins upon request. And all the looks of it – it turned out to be a great time. Ah well, at least I got to see a YouTube video. This one is fun because you can hear hizKnits laugh at the end!
and finally, a lovely reader in Germany has offered to procur me a skein of Wollmeise at the annual sale after reading about my cancelled trip to Sock Summit. This is a lovely and amazing offer that I will definitely not turn down – and I will report when the goodie arrives to celebrate with everyone. Thanks, Liesl!


BOSTON (AP) — Political leaders seem split on how to address the internet-fueled changes roiling the Massachusetts economy.
At the Statehouse, lawmakers have largely embraced major online players. They've passed ride-booking regulations, but agreed not to require drivers for Uber and Lyft to be fingerprinted. They abandoned a proposal that would have applied the state's hotel and motel taxes to online lodging services such as Airbnb.
And new legislation guarantees the biggest of online sellers — Amazon — won't have to pay workers at a planned distribution center in Fall River overtime on Sundays.
Democratic Sen. Elizabeth Warren has warned about the downside of the so-called "gig economy" for workers denied access to traditional employment benefits like paid vacation, health coverage, workers compensation and retirement plans.


I'm soundproofing a small room in the basement of a 3 story house. we have the walls covered. My question is how best to insulate air conditioning and heating ducts that run along the sides of the room. We plan to drop the ceiling below where these ducts are now, but I also think we need to wrap them in some real good material to make sure they don't carry sound throughout the whole house. Right now, I'd say the heatings ducts are the primary reason why sound carries through the house.
Any ideas what the best materials are to insulate the heating ducts and big pipes in this room?
to clarify, we're putting up drywall w/spacing and curved insulation between the spacing of the drywall on all of the walls. I believe it's about 6" thick all around. we'll probably do the same on the ceiling, boxing in the air ducts that go into the room. However, we'd rather wrap the duct work now than have to go in after we've built out the room to retrofit it. Within reason, this is a job where doing it right is a bigger priority than doing it as cheap as possible.


Siroflex Instant Paintable Caulk is a high quality acrylic sealant specifically developed to be overpainted with water based and synthetic paints instantly after application, without cracking or discolouration. This product is for interior use and adheres perfectly without the use of a primer. This product has low odour, is easy to apply and not corrosive towards metals.
Siroflex Instant Paintable Caulk is ideal for the sealing and filling of gaps and cracks where movement can occur, such as those between plasterwork and window frames, door frames, skirting boards, windowsills and architraves. For filling cracks and screw and nail holes in walls and ceilings. Use prior to painting and wall papering to prevent unsightly cracks developing later. Has good adhesion without primer on concrete, brickwork, plaster and painted wood.
Surfaces must be clean and solid. Use diluted product as a primer for highly porous surfaces. Cut the cartridge seal and screw on the nozzle which should be cut to the required size. Using a skeleton gun, apply the sealant into the gap or joint ensuring a good solid fill. If required, place masking tape along edges removing within 5 minutes of application. Always tool using a spatula or stick wetted with water within 5 minutes of application. Uncured sealant can be removed with water. Can be over painted with water based and synthetic paints immediately providing a skin has formed.