Like a novel-length episode of Black Mirror, Dave Eggers' The Circle paints a twisted and frighteningly relatable picture of technology and human nature gone wrong. The story beings with Mae Holland, a recent college graduate, bored, broke and overqualified at her nightmare of a job in public utility. She's eventually recruited into the Circle, an exciting, all-encompassing tech company like Google, Facebook and Apple cranked to 11.
It looks like a dream job, at first. But what seems like an exceptionally accommodating work environment eventually becomes overbearing and unsettling. Are the constant parties and the gamification of workplace participation a team building strategy, or is it an attention draining trap? Are the campus dorms, stores, and health coverage simply convenient, or are employees being made dependent and cut off from the outside world? The longer she stays, the more the Circle and its utopian vision blurs the line where it starts to resemble a cult.
While the Circle's technological advances have expressed positive impacts, they each make the Circle more powerful, and in a better position to rationalize more power for itself, suffocate opposition, and integrate itself into every facet of life. Society is being remade into an environment where no one can opt out of their services.
Is there a darker agenda? That might be the wrong question. How much information about ourselves are we willing to give away in the name of safety, attention or convenience? And when does aggregating that information go too far? (One of the novel's most memorable fictional phone apps scans a target's Internet history to compile advice for dating them.) Do our actions have a right to be forgotten?
At what point do we categorically reject invasions of privacy for whatever reason? What practices will our future selves allow to become normalized? Because maybe the ultimate threat to our autonomy won't come from conscious oppressors, but the well-meaning, and blindly ambitious. Maybe the danger comes from those who build things without thinking of how they might be used.
In the Circle, the ideology of openness and the democratization of data slowly harrowingly gives way from "Sharing is caring." to "Everything that happens, will be known." and "Privacy is theft." The characters are eventually navigating the beginnings of a world where everything they do is accessible for judgement by the masses. And attempts to unplug are met with suspicion and hounding.
Dion Graham does terrific, seamless work with the narration. He's always tuned into the right emotions when reading prose, keeping the listening experience dynamic. His character voices are district and well-realized, including the two lead female characters, who have a lot of dialogue together.
At thirteen hours and forty minutes, once the Circle gets its hooks into you and the terrifying implications of this day after tomorrow technology are in the open, much like your phone, you might find it impossible to put down.
James Alexander lives in Durham.
This entry was posted in reviews and tagged dave eggers, dion graham, james alexander, the circle. Bookmark the permalink.


How do I sell my house Fishers residents often ask. Check out the answers to this question and more from Rocket Realty agents.
Whether it is how do I list, show, or sell my house Fishers residents like you are asking the right questions about real estate. If you are a first time seller, getting the answers to these questions is especially important as you navigate the real estate process. Even if you have bought and sold houses before, getting the updated facts and recommendations about listing, showing, and selling can go a long way in helping you to have the best possible real estate experience. At Rocket Realty, we are dedicated to helping Fishers homeowners like you have a stress free sale! One way we try to help out is by addressing some of your frequent questions and concerns.
How do I sell my house Fishers residents like you often ask. At Rocket Realty, we find that this question is usually getting at a different, though related question. Homeowners like you really want to know if there is a correct or a best way to sell their homes. This question is completely fair! We all want to find the best methods, especially when it comes to new things. However, it might not have the answer you think.
Really, there isn't just one right way to sell your home. There are a range of different methods, from hiring a full service agent to handle the whole process to listing your home FSBO and handling most of the work on your own. The best method, then, is going to be the one that fits your unique situation. Are you a busy person who really needs to focus on your new job or family? Hiring a full service agent will probably be the most satisfying experience. Do you have extra time, energy, and a dedication to getting the greatest possible savings? Jumping into the FSBO adventure may be the right choice for you.
If I Sell My House Fishers Still Requires Complicated Paperwork… Who Can Help?
Starting with one real estate option doesn't mean that you are stuck doing it the same way the whole time. If you choose to list and sell your home, you may find you share the same experience of many of our clients: after I sell my home Fishers requires complicated paperwork. Am I on my own with this?
Not at all! You can find great help from a local a la carte real estate agency like Rocket Realty. With an a la carte service, you pay only for the services that you need, so you can get help with that paperwork without signing on for a full set of services that you already have covered on your own.
What if I Find a Buyer After I've Hired A Full Service Agent?
The answer to this question is going to vary depending upon which agency you've hired. If you talk to almost any agency in Fishers, they may congratulate you for finding the buyer… but still charge you the full price. At Rocket Realty, though, you'll find a different story. If you are able to find your own buyer, instead of the agent, we offer you a reduction in commission! This is one offer you won't find with any other agency, Fishers or beyond!
What other questions do you have about selling your home? Let us know in the comment sections below! Or, give us a call today at 844-5ROCKET (844-576-2538).


text:
Asean morning news for July 19
Bulk carrier explosion, South China sea – Maritime Bulletin
Scientists hail stealth that may blind military radar systems - South China Morning Post
U.S. House votes to set aside impeachment resolution against Trump
South China Sea Standoff Likely Signals China's 'Push of Maritime Claims' Amid Vietnam Backdown
Encounters frequent in South China Sea
May 16, 2019 spratlys
The U.S. Navy's USS Blue in Singapore, on May 9, 2019.
The voice on the radio in the middle of the South China Sea follows a familiar script for Capt. Eric Anduze, who helms the USS Blue Ridge. It's China on the phone.
"They'll contact us and they'll go — 'U.S. government vessel, this is Chinese Navy vessel' number whatever — 'we will maintain 5 miles from you and escort you as you make your transit,"' Anduze said, describing the English-speaking voice from a rival Chinese warship.
The U.S. response is short: "Chinese vessel, this is government vessel 1 9, copy, out." From there on, silence, as the vessels of the world's rival powers steam onward together.
The ship-to-ship interactions are a regular potential flash point for the world's two biggest militaries in contested waters. In September, a Chinese destroyer sailed within a football field's distance of the USS Decatur in what the U.S. said was an "unsafe and unprofessional" maneuver. That hasn't deterred future sailings — the U.S. sent two guided-missile destroyers within 12 nautical miles of disputed islands earlier this month.
Based in Japan, the Blue Ridge is a frequent traveler through the South China Sea, which Beijing considers its waters against an international community increasingly concerned by its encroachment. The area is home to key shipping lanes and fisheries that have sparked disputes between China and its neighbors.
China has built up and mans artificial islands with buildings, communications towers, ports and runways to cement its claims in what many international observers say are military provocations. The U.S. says it travels the waters as if the expansive claims don't exist, running through shipping lanes as the Blue Ridge does, or on so-called freedom of navigation patrols near the disputed islands. Those latter sailings are viewed by China as aggravating.
The U.S. Navy allowed media outlets, including Bloomberg, an inside look at the sort of ship it's using to sail through the disputed waters. The oldest operational warship in the American Navy, the Blue Ridge is the flagship of the 7th Fleet, and docked in Singapore as part of a tour of Southeast Asian port cities.
The Blue Ridge is billed as one of the most technologically advanced ships in the world. It operates as a central information node for a fleet whose range stretches from the India-Pakistan border to the International Date Line in the Pacific Ocean. Through its several computers flow a database the Navy says gives it a "complete tactical picture of air, surface and subsurface contacts." That's what it does: it sails and it knows things.
The ship is a small floating town of more than 1,000 sailors at any given time. There are beds and cafeterias, fitness centers and a post office. A miniature hospital has sick beds and an operating room, along with a dentist who can fill a cavity or pull a tooth — unless the waves get too rough. Up on deck, sailors can jog on a makeshift track around the ship, at about seven laps to a mile.
Since February, Anduze said the Blue Ridge has been escorted by Chinese vessels about six times in an almost unremarkable and now routine manner.
In Washington, the view is that China uses "coercive tactics" including its naval and paramilitary vessels to enforce claims in the South China Sea, the Pentagon said last week in its annual report on China's military power. Those are targeted "in ways calculated to be below the threshold of provoking conflict," though have escalated into near-misses with U.S. warships.
Naval officials on the Blue Ridge declined to comment in detail on those more aggressive encounters, except to say nothing similar had happened with them as they passed through the South China Sea's shipping lanes.
A Section on 05/15/2019
Print Headline: Encounters frequent in South China Sea
Lively debate on interpreting laws at maritime conference, Singapore News & Top Stories
'Our presence has been consistent over decades' says U.S. Navy on South China Sea – Herald Sun
July 19, 2019 spratlys 0
Asean morning news Compiled by Stella-maris Ewudolu @stmarisewudolu Southeast Asia organized crime profits unprecendented: UN...
Post Views: 599 Bulk carrier suffered explosion at around in the morning Jul 17, no AIS...
Scientists hail stealth that may blind military radar systems – South China Morning Post
Scientists hail stealth that may blind military radar systems South China Morning Post Researchers at academy of...

redpajama_set_name: RedPajamaCommonCrawl


text:
Faculty and Staff Members Receive Awards
Three members of the Truett-McConnell College faculty and staff received recognition at the college's annual Honors Day held on April 23, 2009.
Vicki Steele
Vicki Steele, associate professor of biology, was the recipient of the Vulcan Materials Company Teaching Excellence Award. The award is presented each year to a Truett-McConnell faculty member who exhibits outstanding skills and abilities in classroom teaching and in related faculty work.
Vulcan Materials Company, based in Birmingham, AL, is the nation's largest producer of construction aggregates, including crushed stone, sand and gravel that are used in nearly all forms of construction. Vulcan is also a major producer of other construction materials including asphalt and ready-mixed concrete. The company employs approximately 11,000 people across the entire company. Vulcan was named to Fortune magazine's list of American's Most Admired Companies six times in the last ten years.
The Vulcan Materials Teaching Excellence Award is presented each year to a faculty member at Truett-McConnell College who exhibits outstanding skills and abilities in classroom teaching and in related faculty work. Mrs. Steels is an associate professor of biology, having been employed on the TMC faculty since 1990. During the 2008-2009 academic year, in addition to her teaching responsibilities Mrs. Steele has directed the college's Compliance Certification study for reaffirmation of the college's accreditation by the Southern Association of Colleges and Schools.
A lifelong resident of Toccoa, GA, Mrs. Steele and her husband, Roger, a retired public school teacher, are the parents of two adult sons and the grandparents of two grandsons. She holds a Master of Education degree from North Georgia College and State University and a Specialist in Education certificate from Clemson University.
In announcing the college's recipient for the 2009 Vulcan Materials award, Dr. Mike Simoneaux, vice president for academic services, said: "Vicki Steele exemplifies the qualities of an outstanding teacher. She is highly competent in teaching, she has the best interest of her students at heart, she performs any task she is asked to do with enthusiasm. I am pleased that her long and dedicated service to Truett-McConnell College and its students is being recognized in this award."
T. Michael Davis
Dr. T. Michael Davis, assistant professor of mathematics, received the 2009 Award for Faculty Excellence from Truett-McConnell College. This award is given each year to a teaching faculty member in recognition of superior skills in classroom teaching and related responsibilities.
He and his wife, Joy, live in Cleveland and are the parents of two adult children. Dr. Davis holds a bachelor's degree from North Georgia College and State University, a master's degree from New Orleans Baptist Theological Seminary, a Doctor of Ministry from Luther Rice University and a Ph.D. from Clemson University. His teaching responsibilities are in the disciplines of mathematics and physical science; in addition, he heads the college's Quality Enhancement Program and chairs the Division of Mathematics, Science and Business. He joined the TMC faculty in 2007.
Cindy Erbele
Cindy Erbele, administrative assistant to the Vice President for Academic Services, was awarded the 2009 Presidential Achievement Award by TMC President Dr. Emir F. Caner. Each year the award goes to a staff member whose dedication to the institution is inspirational and whose service to the college, its faculty and students is exemplary.
A resident of White County, Cindy Erbele is an outstanding example of these qualities and is a deserving recipient of the award. She is the mother of two children: Martin Erbele, 20, and Allison Erbele, 18. She was initially employed by Truett-McConnell College in 2004.


text:
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Crews locate body of girl swept away by floodwaters in N.C.; boy still missing
Sean Noone
Posted: Sep 2, 2020 / 06:07 PM CDT | Updated: Sep 2, 2020 / 06:07 PM CDT
SMITHFIELD, N.C. (NewsNation Now) – Search teams have found the body of a young girl who was swept away by floodwaters with her brother in North Carolina late Monday.
Sheriff Steve Bizzell confirmed the body was found in a tributary of the Neuse River around 5 p.m.
"We are still searching for that little boy," Bizzell said.
The children, 5-year-old Alexa and 4-year-old Abraham Jr., were in their mother's car around 11:30 p.m. when it ended up in a flooded creek off Galilee Road in Smithfield.
"It's tough on the guys today, but at least we found the girl we will continue the search on the ground and by boat," Bizzell said. "Everyone's emotional, the family, deputies, I'm emotional. Anytime we have a case involving children — someone's child someone's loved one — brother and sister, I'm emotional, deputies are emotional. It's tough."
Heavy rains pounded the area in recent days – bringing the Neuse River to moderate flood stage.
The mother was able to call 911 after the car was swept off the road.
As a rescue team reached the car, they were able to pull the mother and child from the car but the boat capsized and the boy was swept out of a firefighter's arm, according to Bizzell.
The car was found later.
K-9s and helicopters were deployed to help in the search on Tuesday.
Bizzell says the search has been difficult and dangerous, and the thought of the children was heavy on the minds of the first responders.
This is a developing story. Refresh for updates.
NewsNation affiliate WNCN contributed to this report.
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text:
About EIFA
Opinion: Iraq and Syria—One Country, One War
By Abdulrahman Al-Rashed
Asharq al-Awsat - About a year ago, US President Barack Obama attempted to justify Washington's different approach on the crises in Iraq and Syria. While drones, military consultants, and arms have been sent to Iraq, only humanitarian aid—blankets, medicines, supplies—have been sent to Syria. Back then, Obama said his administration was committed to Iraq's security because the country has strategic value for the US. Regarding the US position on Syria he said meanwhile that it would remain firmly focused on the political and humanitarian fronts since there were no plans for any other kind of American involvement in the crisis.
Naturally, most governments speak about Iraq and Syria separately. But this divide between the two neighbors is based on old maps and is in reality no longer tangible on the ground. There are no borders, border guards, passports, or armies separating Iraq and Syria anymore. Many checkpoints have become mere stopovers for Islamic State of Iraq and Syria (ISIS) militants and other fighters crossing into and out of Iraq's western Anbar province. War and terrorism have united both countries. And, so, ISIS now seems justified in having named itself the "Islamic State of Iraq and Syria."
Back when Iraq and Syria's borders first began to "disappear," I wrote about how viewing the situation according to the old, traditional rules—border markers, flags, countries, religion—would result in a failure to understand the depth of the crisis. But the picture is now clearer. Current events in Iraq are an indispensable part of developments in Syria, and both countries' borders are now worn-out old lines jotted on paper at the offices of various foreign affairs ministries around the world. We are currently witnessing a crisis that unites the two countries, all the way from Syria's Bab Al-Hawa border crossing north of Turkey, to Jordan's Trebil on the border with Iraq, to Saudi Arabia's Arar crossing, just south of Iraq. ISIS militants now loom dangerously on the suburbs of both the Syrian capital Damascus and the Iraqi capital Baghdad.
If concerned countries wish to confront this crisis and stop ISIS's expansion, they must deal with Iraq and Syria as one country, because success and failure in one is now connected to the other. It is no longer relevant that the world, and particularly the United States, categorizes Iraq as an oil-rich country with strategic importance while at the same time categorizing Syria as a mere radish farm! We are before a Siamese twin facing one war against the same enemy.
All this means of course that depending on the Baghdad government will not achieve much. It also means that supporting Shi'ite militias in Iraq via the Shi'ite-dominated Popular Mobilization volunteer forces will deepen wounds and increase collective Sunni resentment toward Baghdad as well increase hostility against the US. Such policies will in the end make ISIS's pretensions to statehood and being the representative of the majority of Sunnis in both Syria and Iraq, an actual reality.
The remaining reasonable option, after the failure of the above alternatives, is to support opposition Sunni forces in Syria and Sunni tribal forces in Iraq to help them fight ISIS—which claims to be a Sunni group—and to stop using the Popular Mobilization forces and Shi'ite militias in Iraq—which are under the control of the Iranians and actually end up serving the aims of ISIS. It also goes without saying that concrete steps must be taken to resolve the tragedy which the Syrian people are currently experiencing. Syrian Sunnis, who make up around 80 percent of the country's population, cannot remain silent towards Assad's regime after his forces have now killed more than a quarter of a million Syrians. Iran and its ally Assad will not begin to accept a political solution that leads to healing the Sunni majority unless an aerial plan is developed to confront and prevent the Syrian regime's daily massacres. Assad's air force is currently allowed to fly and drop barrel bombs on civilians as it resumes its ethnic cleansing operation. Meanwhile the Syrian opposition is prohibited from obtaining access to advanced weapons that could help them defend themselves against Assad's air power. To rub salt in the wound, the international community refuses to impose a no-fly zone, which would put an end to this tragic situation.
In the absence of an adequate understanding and resolution of these crises, ISIS will no doubt expand and find itself more supporters and fighters—even more than the current 100,000 militants fighting alongside it in Iraq and Syria.
Read more: http://www.aawsat.net/…/opinion-iraq-and-syria-one-country-…
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text:
Armaan Malik, a popular YouTuber, is married to two women, Kritika Malik and Payal Malik.
And both are pregnant. YOUTUBER had shared his wives' pregnancy announcements.
Don't blink; this is not a movie, but a real-life family scenario.
Armaan Malik, a popular content producer with 1.5 million Instagram followers, has married two females and lives a lovely life with them.
The Youtuber shared a few photos of his wives, Kritika and Payal, showing off their baby bumps.
YOUTUBER ARMAAN MALIK TROLLED:
It was described as "cheap" by social media users.
"Dono pregnant," said another.
"Yh koun se ladkiya h jo hubby share kr leti h," said another.
Armaan married Payal Malik in 2011. They also have a child, Chirayu Malik, together.
In 2018, After six years of marriage Armaan married Kritika Malik, the best friend of Armaan's first wife
Geeta Devi, Manoj Bajpayee's mother, died at the age of 80.


text:
\section{Introduction}
Grand unified theories (GUTs) are an important class of extensions to standard model (SM) physics,
 with most theories attempting to unify the strong and electroweak interactions within
 the framework of a single, larger gauge symmetry $G$. The simple groups $SU(5)$~\cite{su5}
and $SO(10)$~\cite{so10},
which may be derived from a possible underlying $E_6$~\cite{e6},
have been the common groups of interest. $SU(5)$ is the smallest group with
complex representations
 that can accommodate the SM gauge structure, with the fermions placed in the
$\mathbf{1 \oplus \overline{5} \oplus 10}$
 representations. As both quarks and leptons are contained in both the
$\mathbf{\overline{5}}$ and the
 $\mathbf{10}$, gauge-mediated quark-lepton transformations exist, giving rise to
baryon number violation.
 Similarly, $SO(10)$ can house an entire generation of fermions, including
the right-handed neutrino, in a
 single $\mathbf{16}$. This enumeration of fermions provides great simplicity
but also places unrealistic bounds on
 proton stability once a unification scale is identified.

Additionally, these groups are plagued by a lack of
phenomenologically successful coupling constant unification.
As these unified theories
 are based upon a simple group $G$, a single gauge coupling constant describes
the strength of all
gauge interactions. The three coupling parameters of the low energy SM
field theory need
to separately evolve as a function of energy until concordance at some possible
 unification energy scale
results and we have only one effective coupling constant~\cite{gqw}.
Whether unification of the gauge coupling constants can be achieved represents
a crucial test for the feasibility of a GUT.
The running of the coupling constants in $SU(5)$ and $SO(10)$ theories fail to
satisfy this criterion unless appropriate new physics at an intermediate scale, such as
supersymmetry, is invoked.
This motivates the application of product groups
$G \otimes G \otimes ...$,
augmented by a discrete symmetry permuting the $G$ factors, as an alternative class of
unified theories. These models need not have gauge
boson mediated proton decay as the quarks and leptons are often in
separate representations, however even if so, proton instability
can still orginate through Higgs-fermion Yukawa interactions.

The smallest such group consistent with SM phenomenology at low energy is
trinification, based on $SU(3)_q \otimes SU(3)_L \otimes SU(3)_R$, which has shown
promise both within and independent of a supersymmetric context. It can also
be obtained from an $E_6$ theory and as a result has been studied
extensively \cite{trinification,Willenbrock}.

A natural extension to trinification is quartification.\footnote{See also Ref.~\cite{carone1}
for an extension of trinification to $3N$ $SU(3)$ factors, where $N$ is a positive integer.}
Quartification
was first proposed by Volkas and
Joshi~\cite{Ray} and then independently revisited by Babu, Ma and
Willenbrock~\cite{BMW}.
These theories represent
an implementation of the idea of quark-lepton universality at high energies
postulated by Foot and Lew~\cite{Foot}.
In the low energy world described by the SM, there are significant disparaties between the quarks and
leptons. They have different electric charges, and the quarks are confined by
colour interactions whereas the leptons are not. Extended models employing discrete quark-lepton
symmetry~\cite{Foot} allow quarks and leptons to become indistinguishable
above energy scales as low as a few $TeV$. To achieve this, one must introduce new degrees of
freedom for the leptons which
are embodied in a separate gauge group $SU(3)_{\ell}$. With this gauge
group supplementing the familiar quark
colour $SU(3)_q$ group, a discrete exchange symmetry between the quark and leptons can be imposed.
The quark-lepton indistinguishability afforded by this scheme does not require
gauge coupling constant unification.
Quartification is the simplest known way to extend such models to also
provide for full coupling constant unification.

The gauge group of these theories is $SU(3)^4$ with an anomaly-free fermion
assignment.
The symmetry breaking is accomplished with Higgs multiplets in a certain $36$-dimensional
representation of $SU(3)^4$, and in Refs.~\cite{Ray} and \cite{BMW}
the symmetry is broken down to $SU(3)_q \otimes SU(2)_L \otimes U(1)_Y \otimes SU(2)_{\ell}$.
In the low-energy limit,
there is an $SU(2)_\ell$ remnant of leptonic colour left unbroken. Each standard lepton has heavy exotic
partners, which following tradition we call ``liptons'', in an $SU(2)_\ell$ doublet.  If the liptons are heavier
than a few 100 GeV, then the existence of the new unbroken gauge symmetry $SU(2)_\ell$ is hidden,
though potentially to be found at Large Hadron Collider energies.

The original proposal of Ref.~\cite{Ray} succeeds only in partial unification~\cite{ps}, with
two independent gauge coupling
constants at the unification scale. The model of Babu et al.~\cite{BMW} is a variant
that achieves full unification, with
the running coupling constants meeting at around $4 \times 10^{11}\ GeV$
without invoking
supersymmetry. Vital to this result is the presence of the liptons at the $TeV$ scale,
to help ensure appropriate running for the coupling constants, assisted by a second
light Higgs doublet.
In non-quartified Foot-Lew type models~\cite{Foot}, $TeV$-scale liptons are a natural
possibility.  In quartification, however, the default situation sees the liptons
acquiring unification-scale masses.  This is precisely why the original paper~\cite{Ray}
proposed partial unification only.  By contrast, Babu et al.~\cite{BMW} impose an
additional discrete symmetry for the sole purpose of avoiding ultra-large lipton masses.
While it is of course technically natural, the additional discrete symmetry is in other
respects very much an afterthought. It is compounded by the
resolutions employed (non-renormalisable operators) to obtain realistic $TeV$-scale
lipton masses.  There are also issues with neutrino mass generation.
This scheme is reviewed more fully in Sec.~\ref{cha:BMW}.

The purpose of this paper is to show how to achieve full unification through quartification
without the imposition of the additional discrete symmetry. In fact,
we demonstrate that the unification scheme of Ref.~\cite{BMW} is not unique, and can be obtained
without restricting
Yukawa couplings simply by introducing intermediate stages in the symmetry breaking.
Importantly, we show that
heavy exotic liptons need not spoil unification.
In Sec.~\ref{cha:masses}
we describe the matter content of our models and state any assumptions
associated with the Higgs sector.
We outline in Sec.~\ref{cha:SU2l} all possible choices of symmetry
breaking which leave a residual
$SU(2)_{\ell}$, while in the subsequent subsection we break this symmetry
entirely~\cite{lbroken} and
resolve the neutrino mass issues.
We systematically solve the renormalisation-group equations for each,
showing the choices that give rise to successful
unification. In Sec.~\ref{cha:phenomenology} we comment on the low-energy
phenomenology in those scenarios
in which unification is possible. We conclude in Sec.~\ref{cha:conclusion}.

\section{The quartification model} \label{cha:BMW}

The quartification gauge group is
\begin{equation}
\centering
G_4 = SU(3)_q \otimes SU(3)_L \otimes SU(3)_{\ell} \otimes SU(3)_R.
\label{eqn:quart}
\end{equation}
A $Z_4$ symmetry which cyclicly permutes the gauge groups as per $q \rightarrow L \rightarrow
\ell \rightarrow R \rightarrow q$ is imposed, ensuring a single gauge coupling constant $g_4$.
The fermions are contained within a left-handed $\mathbf{36}$ of Eq.~\ref{eqn:quart},
\begin{eqnarray}
\mathbf{36} & = &\mathbf{(3,\overline{3},1,1)+(1,3,\overline{3},1,)+(1,1,3,\overline{3})+(\overline{3},1,1,3)}, \\
& \equiv & \qquad q \qquad \oplus \qquad \ell \qquad \oplus \qquad {\ell}^c \qquad \oplus \qquad q^c,
\end{eqnarray}
where $q (\ell)$ denotes the left-handed quarks (leptons) and $q^c ({\ell}^c )$ the left-handed
anti-quarks(leptons). Under $G_4$, these have the transformations
\begin{equation}
\centering
q \rightarrow U_q \, q \, U_L^\dagger, \qquad q \rightarrow U_R\,
q^c\, U_q^\dagger,\qquad \ell \rightarrow U_L\, \ell \,U_\ell^\dagger, \qquad
\ell^c \rightarrow U_\ell \, \ell^c \, U_R^\dagger,
\label{eqn:fermtransforms}
\end{equation}
where $U_{q, L, \ell, R} \in SU(3)_{q, L, \ell, R}$, and the
multiplets are represented by $3 \times 3 $ matrices:
\begin{eqnarray}\label{eqn:fermions}
\centering
q &\sim& (\mathbf{3,\overline{3},1,1})=
\left(\begin{array}{ccc}
d & u & h \\
d & u & h \\
d & u & h \end{array} \right),\qquad
q^c \sim  (\mathbf{\overline{3},1,1,3})=
\left(\begin{array}{ccc}
d^c & d^c & d^c \\
u^c & u^c & u^c \\
h^c & h^c & h^c \end{array} \right),
\\ \nonumber
\ell&\sim&(\mathbf{1,3,\overline{3},1})=
\left(\begin{array}{ccc}x_1&x_2&\nu\\
y_1&y_2&e\\
z_1&z_2&N \end{array} \right),\qquad
\ell^c \sim  (\mathbf{1,1,3,\overline{3}})=
\left(\begin{array}{ccc}x_1^c&y_1^c&z_1^c\\
x_2^c&y_2^c&z_2^c\\
\nu^c&e^c&N^c\end{array}\right).
\end{eqnarray}
Note the existence of exotic particles as a necessary ingredient in
the representations.
Defining the generator of electric charge $Q$ as
\begin{equation}
\centering
Q = I_{3L} - \frac{Y_L}{2} - \frac{Y_{\ell}}{2} + I_{3R} - \frac{Y_R}{2},
\end{equation}
we identify $N, N^c$ as neutral particles, $h(h^c)$ as charge $Q=-\frac{2}{3}(\frac{2}{3})$ exotic quarks
and the liptons $(x,y,z)$ to have charges $(\frac{1}{2},-\frac{1}{2},\frac{1}{2})$.

In Babu et al.'s model~\cite{BMW}, the Higgs fields are contained in two
different $\mathbf{36}$'s, and shall be denoted
\begin{eqnarray}
\centering
\Phi_a & \sim (\mathbf{1, \overline{3},1,3}), \qquad
\Phi_b \sim (\mathbf{3,1, \overline{3},1}), \qquad
\Phi_c \sim (\mathbf{1,3,1, \overline{3}}), \qquad
\Phi_d \sim (\mathbf{\overline{3},1,3,1}), \nonumber \\
 \Phi_\ell & \sim (\mathbf{1, 3, \overline{3},1}), \qquad
\Phi_{\ell^c} \sim (\mathbf{1, 1,3, \overline{3}}), \qquad
\Phi_{q^c} \sim (\mathbf{\overline{3},1,1,3}), \qquad
\Phi_{q} \sim (\mathbf{3, \overline{3},1,1})
\label{eqn:higgstrans}
\end{eqnarray}
in this paper (Ref.~\cite{BMW} used different subscripts).  Note that $\Phi_a \sim \Phi_c^{\dagger}$; this
effective replication of a Higgs multiplet is achieved as a natural consequence of the $Z_4$
symmetry.
These fields, which comprise two sets of multiplets that are closed under the $Z_4$,
are sufficient to break the quartification symmetry
and generate realistic fermion masses and mixings. The VEV pattern is given by
\begin{eqnarray}
\langle \Phi_{a} \rangle &\sim \langle \Phi_{c}^\dagger \rangle \sim \left(\begin{array}{ccc}
u&0&u\\
0&u&0\\
v&0&v\end{array}\right),\qquad
\langle \Phi_\ell \rangle \sim \left(\begin{array}{ccc}
0&0&u\\
0&0&0\\
0&0&v\end{array}\right),\qquad
\langle \Phi_{\ell^c} \rangle \sim \left(\begin{array}{ccc}
0&0&0\\
0&0&0\\
v&0&v\end{array}\right),\\
\langle \Phi_b \rangle &= \langle \Phi_d \rangle = \langle \Phi_{q^c} \rangle =
\langle \Phi_{q} \rangle =
\left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right),\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\end{eqnarray}
where the $u$'s and $v$'s are at the electroweak and unification scales, respectively.
This VEV structure induces the strong symmetry breaking
$SU(3)_q \otimes SU(3)_L \otimes SU(3)_{\ell} \otimes SU(3)_R \rightarrow
SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes U(1)_Y$
in a single step, and then instigates electroweak symmetry breaking.
To do this, a delicate
hierarchy within some of these multiplets must exist which necessitates unnatural fine-tuning. (This
unwelcome feature also occurs in trinification models and in the new quartification schemes
we propose below.)

The coupling to the fermions is described by the $Z_4$-invariant
Lagrangian\footnote{The notation $\ell \ell^c \Phi_a$ means $\overline{\ell}_R \ell_L \Phi_a$, etc.}
\begin{equation}\label{eqn:lag}
\centering
\mathcal{L} = Y_1 Tr \left( \ell \ell^c \Phi_a + \ell^c q^c \Phi_b + q q^c \Phi_c + q \ell \Phi_d \right)
+ Y_2 Tr \left( \ell \ell^c \Phi_c^{\dagger} + \ell^c q^c \Phi_d^{\dagger} + q q^c \Phi_a^{\dagger}
+ q \ell \Phi_b^{\dagger} \right) + H.c.
\end{equation}
The theory also admits couplings of the type $\lambda \ell \ell \Phi_\ell$ and cyclic permutations.
These terms, however, give GUT scale masses to the liptons
$x_1$, $x_2$, $y_1$, $y_2$, $x_1^c$, $x_2^c$, $y_1^c$ and $y_2^c$,
and, according to~\cite{BMW}, it is essential that these particles remain light for
gauge coupling constant unification. An additional
$Z_4^{'}$ symmetry defined by
\begin{equation}
\centering
(q, \ell, q^c, {\ell}^c) \rightarrow i ( q, \ell, q^c, {\ell}^c), \qquad
\Phi_{a-d} \rightarrow -\Phi_{a-d} \qquad \textrm{and}
\qquad \Phi_{\ell,\ell^c,q,q^c} \rightarrow \Phi_{\ell,\ell^c,q,q^c}
\label{eqn:z4'}
\end{equation}
is then imposed to forbid these terms,
reducing the Yukawa Lagrangian exclusively to Eq.~\ref{eqn:lag}.
As a consequence of this $Z_4'$, after symmetry breaking, the massless particle
spectrum contains the $x,x^c$ and $y,y^c$ liptons in
addition to the minimal SM particles. This massless spectrum, which contains two SM Higgs doublets,
is sufficient to unify the coupling constants within experimental error at approximately
$4 \times 10^{11} GeV$.

If the discrete symmetry of Eq.~\ref{eqn:z4'} was not imposed, then the intersection of the three
SM coupling constants would not occur. A less nice aspect of this scheme is that
there is no natural origin for this symmetry -- why, on theoretical grounds, should
certain Yukawa terms be forbidden and not others? This unattractive feature is exacerbated by
the need to introduce non-renormalisable terms of the form $\epsilon_{jkl}\epsilon_{mnp} \ell^{jm}
\ell^{kn} ( \Phi_a^{\dagger} \Phi_{\ell^c}^{\dagger} ) ^{lp}$
and $\epsilon_{jkl}\epsilon_{mnp} (\ell^c)^{jm}
(\ell^c)^{kn} ( \Phi_{\ell}^{\dagger} \Phi_c )^{lp}$. These terms must exist to give $TeV$ scale
masses to the liptons, otherwise their mass terms would be indistinguishable from the
ordinary leptons at the electroweak level.  Although the proton decay mediated by these terms is predicted
at a realistic rate, from a model-building point of view they are a little ad hoc.
Babu et al.\ then introduce another ingredient
to resolve the issue of neutrino mass. In the bare model, neutrinos naturally acquire
Dirac masses of the same order as the charged fermions.
This is circumvented by the addition of a Higgs singlet whose
coupling with the right-handed neutrino induces the seesaw mechanism.
We comment later on how neutrinos naturally acquire light masses without a
Higgs singlet when $SU(2)_{\ell}$ is broken.  All in all, the very pleasing gauge unification
property of
this scheme is partially spoilt by the additional discrete symmetry required,
the non-renormalisable operators, and the extra Higgs singlet.

\section{mass spectrum}\label{cha:masses}
\subsection{$SU(2)_{\ell}$ unbroken}

The matter content and mass thresholds of a unified theory govern the running of the coupling constants.
It is thus important to elucidate at what energy scales the various particles gain mass and how
they contribute to the renormalisation-group equations.
We employ the same fermion and Higgs multiplet assignments as Ref.~\cite{BMW}, summarised
by Eqs.\ \ref{eqn:fermtransforms}-\ref{eqn:fermions} and \ref{eqn:higgstrans}.

Given that we are aiming for as natural a quartification model as possible, one needs to be aware of the
most obvious approach in determining the Higgs VEV structure short of
performing a minimisation analysis of a Higgs potential.
First, those Higgs fields of Eq.~\ref{eqn:higgstrans} which are not singlets under quark
colour necessarily can not acquire VEVs, and we also naturally assume
them to have mass of unification scale always.
Thus the fields $\Phi_b$, $\Phi_d$, $\Phi_q$ and $\Phi_{q^c}$ have no influence on the renormalisation-group
equations and can be ignored for now. For the remaining fields our policy is the following:
We first choose a symmetry breaking cascade. At a given
stage in the symmetry breaking chain, those components that can acquire a VEV
consistent with the symmetry breakdown pattern do so at that scale,
and that the corresponding Higgs masses are also at that same scale.

With this in mind, consider the breaking
\begin{equation}\label{eqn:G4toSMSUl}
\centering
SU(3)_q \otimes SU(3)_L \otimes SU(3)_{\ell} \otimes SU(3)_R \stackrel{v}{\longrightarrow}
SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes U(1)_Y \stackrel{u}{\longrightarrow}
SU(3)_q \otimes SU(2)_{\ell} \otimes U(1)_Q.
\end{equation}
The VEV pattern which achieves this is
\begin{equation}\label{eqn:SUlVEV}
\langle \Phi_{\ell} \rangle = \left(\begin{array}{ccc}
0&0&u_{\ell}\\
0&0&0\\
0&0&v_{\ell}\end{array}\right)\qquad
\langle \Phi_{\ell^c} \rangle = \left(\begin{array}{ccc}
0&0&0\\
0&0&0\\
v_{\ell^{c}_{1}}&0&v_{\ell^{c}_{2}}\end{array}\right)\qquad
\langle \Phi_{a} \rangle = \langle \Phi_c^{\dagger} \rangle =
\left(\begin{array}{ccc}u_{a1}&0&u_{a2}\\
0&u_{a3}&0\\
v_{a1}&0&v_{a2}\end{array}\right).
\end{equation}
(If we were to introduce intermediate steps in the breaking, then the $v$'s would be of
different orders.) It is unfortunate that this VEV structure has
an intra-multiplet hierarchy, with entries of both order $u$ and $v$ contained
in a single multiplet.
We shall accept this as we think it would be more unnatural for the Higgs fields
to get smaller VEVs than the
symmetry breaking scheme requires. Solving this hierarchy problem would require additional Higgs fields (not necessarily
in the same representations as those above), which would necessarily imply a larger Higgs potential
with a greater number of arbitrary parameters,
result in only a partially unified theory as in Ref.~\cite{Ray}, or
require a completely different symmetry breaking mechanism, such as
the employment of inhomogeneous scalar field configurations~\cite{clash}
or orbifold symmetry breaking in a brane-world setting~\cite{kawamura}(see Refs.~\cite{Alison}
and \cite{carone2} for applications of the former and latter, respectively, to trinification models).

To give all Yukawa couplings even grounding, we remove the discrete symmetry employed in Ref.~\cite{BMW},
leaving four independent Yukawa interactions which can endow the fermions
with mass. These are
\begin{eqnarray}\label{eqn:higgscouplings}
\centering
\lambda_q Tr[q^c \, q \, \Phi_{a} ] &
\lambda_{\ell}  Tr [ \ell \, \ell^{c} \, \Phi_{c} ],\\
\lambda_{L} \epsilon^{jkl} \epsilon^{mnp}\ell^{jm} \ell^{kn} (\Phi_{\ell}^{\dagger})^{lp},&
\lambda_R \epsilon^{jkl} \epsilon^{mnp}(\ell^c)^{jm} (\ell^c)^{kn} (\Phi_{\ell^c})^{lp},
\end{eqnarray}
giving the quark mass term
\begin{equation}\label{eqn:quarkmass}
\mathcal{L}_{\rm quark\;mass}=
\lambda_q \left(\begin{array}{cc}d&h\end{array}\right)
\left(\begin{array}{cc} u_{a 1}& u_{a 2}\\ v_{a 1} & v_{a 2}\end{array}\right)
\left(\begin{array}{c} d^c \\ h^c \end{array}\right)
+ ( \lambda_q u_{a 3} ) u  u^c + H.c.
\end{equation}
The up quarks acquire electroweak scale Dirac masses, while the $d$ and $h$ quarks
are mixed. Upon diagonalisation of this mass matrix, we have only one $Q = -1/3$
quark per family with electroweak scale mass to be identified as the down quark, and
the exotic quark gains a GUT scale mass. Note that mixing between
the $h$ and $d$ quarks is suppressed by $u/v$.

The mass terms of the leptons are solely of Dirac nature. The $Q=-1$ charged leptons gain
masses of electroweak order and do not mix with any other states. The liptons
$x_1$, $x_2$, $y_1^c$, $y_2^c$, $z_1$ and $z_2$ have charge $Q=+1/2$ and pair up with
the charge $-1/2$ liptons $x^c_1$, $x^c_2$, $y_1$, $y_2$, $z^c_1$ and $z^c_2$,
to acquire GUT scale masses. The electrically neutral leptons $\nu$, $N$, $N^c$ and $\nu^c$ also
only have Dirac mass terms. One sector is of GUT scale, identified as heavy neutral
leptons, and the other, the ordinary neutrinos, is of electroweak scale. We see that we
encounter the same problem as did Babu et al.\ with respect to obtaining a light neutrino mass.

In summary, all SM particles including the Dirac neutrinos have electroweak scale masses,
and all exotics have
GUT scale masses. If the symmetry breaking occurs via intermediate scales,
then the masses of the exotic particles will be at
the unification or one of these intermediate scales.

In determining the running of the gauge coupling constants, we must also
know the full structure of the light Higgs spectrum at each stage of symmetry
breaking. The VEV structure above neither provides enough information to define all the
masses of the Higgs' components nor how many SM doublets there are. One is forced to make an
assumption to deal with this, and, again, we adopt as natural a one as possible.
The assumption chosen involves looking at the branchings, and
particularly at what scale components branch away from those components that acquire VEVs.
If a component gains a VEV, then the SM multiplet in which it is contained is taken to get a mass at the
same scale. In the case where $SU(2)_{\ell}$ remains unbroken, there are SM multiplets which
have no VEVs but are embedded within a quartification multiplet that does. We assume that
these gain mass at the scale of the largest VEV in the quartification multiplet. For example, the VEV
\begin{equation}
\langle \Phi_{\ell} \rangle = \left( \begin{array}{ccc}
0&0&u\\
0&0&0 \\
0&0&v \end{array} \right)
\end{equation}
implies that the components $(\Phi_{\ell})^1_3,(\Phi_{\ell})^2_3$ have masses of order
$u$, while the remaining components all have mass of order $v$.
This gives us seven candidate light Higgs doublets at the SM level: one from $\Phi_\ell$
and three each from $\Phi_a$ and $\Phi_c$.  The Higgs doublet multiplicity has
a beneficial effect on the achievement of gauge coupling constant unification~\cite{Willenbrock}.
Although the Higgs sector of our models has been burdened with these assumptions,
we have avoided the introduction of an additional discrete symmetry.

\subsection{$SU(2)_{\ell}$ broken}

The Higgs fields of Eq.~\ref{eqn:higgstrans} also have the capacity to break the
leptonic colour symmetry completely, leaving no residual $SU(2)_\ell$ gauge group unbroken~\cite{lbroken}.
Consider the breaking cascade
\begin{equation}\label{eqn:quarttosm}
\centering
SU(3)_q \otimes SU(3)_L \otimes SU(3)_{\ell} \otimes SU(3)_R \stackrel{v}{\longrightarrow}
SU(3)_q \otimes SU(2)_L \otimes U(1)_Y \stackrel{u}{\longrightarrow}
SU(3)_q \otimes U(1)_Q.
\end{equation}
With this symmetry breaking pattern, the electric charge generator is given by
\begin{equation}\label{eqn:Q2}
\centering
Q=I_{3L}-\frac{Y_L}{2}+I_{3\ell}-\frac{Y_\ell}{2}+I_{3R}-\frac{Y_R}{2}.
\end{equation}
Notice that the three spontaneously broken $SU(3)$ factors contribute in a
symmetric manner to the electric charge generator.
This breakdown pattern is accomplished by Higgs fields obtaining VEVs of the form
\begin{equation}\label{eqn:VEV2}
\centering
\langle \Phi_{\ell} \rangle = \left(\begin{array}{ccc}
u_{\ell 1}&0&u_{\ell 2}\\
0&u_{\ell 3}&0\\
v_{\ell 1}&0&v_{\ell 2}\end{array}\right), \qquad
\langle \Phi_{\ell^c} \rangle =
\left(\begin{array}{ccc}
v_{\ell^c 1}&0&v_{\ell^c 2}\\
0&v_{\ell^c 3}&0\\
v_{\ell^c 4}&0&v_{\ell^c 5}\end{array}\right), \qquad
\langle \Phi_{a} \rangle = \langle \Phi_{c}^{\dagger} \rangle = \left(\begin{array}{ccc}
u_{a1}&0&u_{a2}\\
0&u_{a3}&0\\
v_{a1}&0&v_{a2}\end{array}\right),
\end{equation}
where all Higgs components that can acquire a VEV at a given scale do so.
The Higgs mass spectrum here is ``derived'' in a more obvious fashion than before. All members of SM
multiplets  which get a VEV, acquire masses at that scale.
This leaves nine light, left-handed Higgs doublets
at the standard model level, three each from $\Phi_\ell$, $\Phi_a$ and $\Phi_c$.

As before, Eq.~\ref{eqn:higgscouplings} describes the Yukawa couplings.
Breaking leptonic colour completely has no impact on the quarks
as they are singlets under this gauge group: the quark masses are identical
irrespective of whether or not $SU(2)_{\ell}$ is broken.
The leptons, however, possess leptonic colour and their electric charges are altered
due to the different electric charge generator of Eq.~\ref{eqn:Q2}, and their mass terms are greatly influenced by
the different VEV pattern of Eq.~\ref{eqn:VEV2}.  The components that were previously
half-integrally charged liptons now acquire integral charges $Q = \pm 1,0$, so they
are no longer liptons but are instead charged and neutral heavy leptons.

The leptons with a charge of $+1$ are the $e^c$, $y_1^c$, $z_2$ and $x_2$
components of Eq.~\ref{eqn:fermions}. They mix and form Dirac mass terms
with the charge $-1$ lepton components $e$, $y_1$, $z_2^c$ and $x_2^c$,
in the manner
\begin{equation}\label{leptonmass2}
\left(\begin{array}{cccc} e& y_1 & z_2^c & x_2^c\end{array}\right)
\left(\begin{array}{cccc}
u_{a 3} & 0 & -u_{\ell 1} & v_{\ell 1} \\
0 & u_{a 3} & u_{\ell 2} & -v_{\ell 2} \\
-v_{\ell^c 1} & v_{\ell^c 4} & v_{a 2} & u_{a 2} \\
v_{\ell^c 2} & -v_{\ell^c 5} & v_{a 1} & u_{a 1}
\end{array}\right)
\left(\begin{array}{c}
e^c\\y_1^c\\z_2\\x_2
\end{array}\right) + H.c..
\end{equation}
There are three
Dirac mass eigenvalues (per family) of GUT scale, and one eigenvalue (per family)
of electroweak scale corresponding to the $e$, $\mu$ and $\tau$ masses.

The leptonic components $N$, $N^c$, $\nu$, $\nu^c$, $x_1$, $x_1^c$, $y_2$, $y_2^c$, $z_1$
and $z_1^c$ are all neutral. Unlike the former case these ten leptons gain
Majorana masses, as per
\begin{equation}
\left(\begin{array}{cccccccccc}\!N&\!N^c&\!\nu&\!\nu^c&\!x_1&\!x_1^c&\!y_2&\!y_2^c&\!z_1&\!z_1^c\end{array}\!\right)
\left(\begin{array}{cccccccccc}
0  & v_{a2} & 0  & v_{a 1}& u_{L3} & 0 & u_{L 1} & 0 & 0 & 0\\
v_{a2} & 0  & u_{a 2} & 0 & 0 & v_{\ell^c 3} & 0 & v_{\ell^c 1} & 0 & 0 \\
0 & u_{a 2} & 0 &u_{a 1} & 0 & 0 &-u_{\ell 1} & 0 & -u_{L 3} & 0 \\
v_{a 1} & 0 & u_{a1} & 0 & 0 & 0 & 0& -v_{\ell^c 2} & 0 & -v_{\ell^c 3} \\
u_{\ell 3} & 0 & 0 & 0 & 0& u_{a 1}  &v_{\ell 2} & 0&0&0 \\
0 & v_{\ell^c 3} & 0 & 0 &u_{a 1} & 0 & 0 & v_{\ell^c 5} & 0 & 0 \\
u_{\ell 1} & 0 & -v_{\ell 1} & 0 & v_{\ell 2} & 0 & 0 & u_{a 3} & -u_{\ell
2} & 0 \\
0 & v_{\ell^c 1} & 0 & -v_{\ell^c 2} & 0 & v_{\ell^c 5} & u_{a 3} & 0 & 0
& 0 \\
0 & 0 & -u_{\ell 3} & 0 & 0 & 0 & -u_{\ell 2} & 0 & 0 & v_{a 2} \\
0 & 0 & 0 & -v_{\ell^c 3}& 0 & 0 & 0 & 0 &v_{a 2} & 0
\end{array}\!\right)\nonumber
\left(\begin{array}{c}N \\ N^c \\ \nu \\ \nu^c \\ x_1 \\ x_1^c \\ y_2 \\ y_2^c \\ z_1 \\ z_1^c\end{array}\right).
\label{eq:10x10}
\end{equation}
Nine of the resulting mass eigenvalues are of the order of the GUT scale, and the tenth
has is a small mass of the order of $\frac{u^2}{v}$, which is precisely the
mass scale that would result from a regular seesaw mechanism~\cite{seesaw}.
This particle
displays the correct weak coupling with the electron to be identified as
the neutrino, and all interactions
involving the light leptons with the heavy leptons are very suppressed.
When intermediate scales are introduced, some of the order $v$ entries decrease in size
and thus some of the large eigenvalues also decrease.  One anticipates that this
raises the value of the smallest eigenvalue.

In summary, the VEV patterns of Eq.~\ref{eqn:VEV2} through the Yukawa coupling
terms provide large masses to exotic fermions, electroweak-scale masses for standard
charged fermions, and a see-saw suppressed masses for the neutrinos.

\section{Renormalisation-group equations}

\subsection{$SU(2)_{\ell}$ unbroken}\label{cha:SU2l}

We begin by analysing the renormalisation-group equations for the schemes featuring
a remnant of the leptonic colour symmetry at low energy.
There is no physical reason why the symmetry breaking has to directly proceed via
Eq.~\ref{eqn:G4toSMSUl}. In fact in this case, without the restrictions
proposed in Ref.~\cite{BMW} imposed on the Yukawa sector,
the gauge coupling constants only come within the vicinity of intersecting if
the Higgs sector is enlarged significantly.

An alternative to the one-step scenario
is the introduction of intermediate symmetry breaking scales.
There are four independent symmetry breaking routes from
$G_4 \rightarrow SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes U(1)_Y$
which can be achieved with our Higgs sector. They are labelled as per:
\begin{eqnarray}
\centering
1. \: \: \: G_4 \: \: \: \stackrel{v}{\rightarrow} & SU(3)_q & \otimes SU(3)_L \otimes SU(2)_{\ell} \otimes SU(2)_R
\otimes U(1)_{X_1} \\ \nonumber
\stackrel{w}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R
\otimes U(1)_{X_2} \\ \nonumber
 \stackrel{x}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell}
\otimes U(1)_{Y} \label{eqn:cascade1a}\\
2. \: \: \: G_4 \: \: \: \stackrel{v}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(3)_R
\otimes U(1)_{X_1} \\ \nonumber
\stackrel{w}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R
\otimes U(1)_{X_2} \\ \nonumber
 \stackrel{x}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell}
\otimes U(1)_{Y}\\
3.\: \: \:  G_4 \: \: \: \stackrel{v}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(3)_{\ell} \otimes SU(2)_R
\otimes U(1)_{X_1} \\ \nonumber
\stackrel{w}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R
\otimes U(1)_{X_2} \\ \nonumber
 \stackrel{x}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell}
\otimes U(1)_{Y}\\
4.\: \: \: G_4 \: \: \: \stackrel{v}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(3)_{\ell} \otimes SU(2)_R
\otimes U(1)_{X_1} \\ \nonumber
\stackrel{w}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(3)_{\ell}
\otimes U(1)_{X_2} \\ \nonumber
 \stackrel{x}{\rightarrow} & SU(3)_q & \otimes SU(2)_L \otimes SU(2)_{\ell}
\otimes U(1)_{Y}. \label{eqn:cascade4a}
\end{eqnarray}
For each cascade, the symmetry breaking is achieved by the VEVs of Eq.~\ref{eqn:SUlVEV}
where the energy of the $v_i$ VEV entries are not uniform. Each of these cascades is of
course followed by electroweak symmetry breakdown.
As we now have several scales, the masses of the exotic fermions and Higgs bosons
will stagger with energy and
they will have varying contributions to the renormalisation-group equations.
The exact nature of the VEV entries and at which scale the fermions gain masses are
detailed in Table \ref{tab:masses}. Notice that the $x_1,x_2,y_1, y_2$ particles which had to
be at the $TeV$ scale in Ref.~\cite{BMW}, can only potentially be light in the case of
cascade four.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Cascade & Energy scale of VEVs & Masses at $v$  & Masses at $w$  & Masses at $x$\\
\hline
1 & $ v_{R_2} \sim v,$ & $x_1^c, x_2^c, y_1^c,y_2^c$ & $h, h^c, z_1,z_1^c z_2, z_2^c,$ & none  \\
  & $v_{a2} \sim w, \; v_L \sim w,$ & &$N,N^c,x_1, x_2, y_1,y_2$ & \\
  & $v_{R_1} \sim x, \; v_{a1} \sim x$ & & & \\
\hline
2 & $v_L \sim v, $ & $x_1, x_2, y_1, y_2$ & $h, h^c, z_1,z_1^c z_2, z_2^c,$  & none \\
  & $v_{R_2} \sim w,\; v_{a2} \sim w, $ & &$N,N^c,x_1^c, x_2^c, y_1^c,y_2^c$ & \\
  & $ v_{R_1} \sim x, \; v_{a1} \sim x$ & & & \\
\hline
3 & $v_{a2} \sim v,$ & $h, h^c, z_1, z_2, N,$ & $x_1, x_2, y_1, y_2,$ & none \\
  & $v_L \sim w,\;  v_{R2} \sim w, $ & $z_1^c, z_2^c,N^c$ & $ x_1^c, x_2^c, y_1^c,y_2^c$ & \\
  & $v_{a1} \sim x, \; v_{R1} \sim x$ & & & \\
\hline
4 & $v_{a2} \sim v$,  & $h, h^c, z_1, z_2, N,$ & none & $x_1, x_2, y_1, y_2,$\\
  & $v_{a1} \sim w$  & $z_1^c, z_2^c,N^c$ & & $ x_1^c, x_2^c, y_1^c,y_2^c$ \\
  & $v_L \sim x, \; v_{R2} \sim x$ & & & \\
\hline
\end{tabular}
\end{center}
\caption{The energy scale of the VEV entries of Eq.~\ref{eqn:SUlVEV}
with $v \geq w \geq x$ , and the enumeration of the
fermion masses for the four cascades of Eqs.~\ref{eqn:cascade1a}-\ref{eqn:cascade4a} where
the quartification gauge symmetry is broken down to $G_{SM} \otimes SU(2)_{\ell}$
in stages. Notice that
cascade four is the only symmetry breaking route that will potentially permit light
liptons.}
\label{tab:masses}
\end{table}

Defining the fine-structure constants as $\alpha_q$, $\alpha_L$, $\alpha_{\ell}$ and
$\alpha_Y$, respectively, for quark colour, weak $SU(2)_L$, leptonic colour and hypercharge, the
one-loop renormalisation-group equations which describe their evolution have the form
\begin{equation}
\centering
\frac{1}{\alpha_i (M_1)} = \frac{1}{\alpha_i (M_2)} + \frac{b_i}{2 \pi} \ln{ \left( \frac{M_1}{M_2} \right)}.
\end{equation}
$i=q,L,\ell,Y$, $M_{1,2}$ denote two mass scales of our theory, and the $b$ factor is given by
\begin{equation}
\centering
 b = - \frac{11}{3} T({\rm gauge \:\: bosons}) + \frac{2}{3} T ({\rm Weyl\:\: fermions})
+ \frac{1}{3} T({\rm complex \:\: scalars}).
\end{equation}
The $T$'s are group theoretical properties which depend on the gauge group representations
and are defined
by the generators $\lambda^a$ in the representation $R$ as per
\begin{equation}
\centering
Tr \left( \lambda^a \, \lambda^b \right) = T_R \delta^{ab}.
\end{equation}
At each stage of the symmetry breaking, $b$ harbours all knowledge of particles with masses
lighter than that particular scale. Our labelling scheme for these factors is best illustrated
by an example: $b_{q_1}$
refers to the cumulative effect of fields which possess quark colour between $v$ and $w$;
while $b_{q_2}$ is concerned with the energy range $x \leftrightarrow w$; and $b_{q_3}$ the range
$M_{EW} \leftrightarrow x$.  The $b$-factors for the $\ell$, $L$ and $R$ sectors are denoted
similarly, with $q$ replaced by the appropriate subscript.
The quantity $b_{u_j}, \, j=1,2,3$ shall denote the running of the $U(1)$ coupling and takes into account
the normalisation of the generators defining $X_1, X_2$ and $Y$. The generator of hypercharge is
taken to be the conventional embedding and is given by
\begin{equation}
\centering
Y = I_{3R} + \frac{2}{\sqrt{3}} \left( \lambda_{L8} + \lambda_{\ell 8} + \lambda_{R8} \right),
\end{equation}
where the $\lambda$'s are the usual Gell-Mann generators.

For all cascades, the running of the $\alpha_q , \alpha_L$ and $\alpha_{\ell}$ constants have the
same generic form given by
\begin{equation}
\centering
\frac{1}{\alpha_i (v)} = \frac{1}{\alpha_i (M_{EW})} - \frac{b_{i_1}}{2 \pi} \ln{ \left( \frac{v}{M_{EW}} \right)}
+  \frac{b_{i_1}-b_{i_2}}{2 \pi} \ln{ \left( \frac{w}{M_{EW}} \right)}
+ \frac{b_{i_2}-b_{i_3}}{2 \pi} \ln{\left( \frac{x}{M_{EW}} \right)}.
\end{equation}
This arises as the $SU(2)_{L,\ell}$ groups have the same coupling constants
as $SU(3)_{L,\ell}$ because of
the way in which these subgroups are embedded within their parent $SU(3)$. At each scale, the
coupling constants are analysed and we can determine a relationship between the
the different energy scales $v,w,x$ and the values of the fine-structure constants at
the electroweak scale $M_{EW}$.
The evolutions of the $U(1)$ coupling constants are different for each cascade
as they depend on the specific linear combinations of generators defining $X_1$ and $X_2$,
and so the forms of the renormalisation group equations
depend on the symmetry breaking pattern. The relationship between the
$U(1)$ electroweak-scale fine structure constant and
the electroweak-scale coupling constants for $SU(2)_\ell$ and $SU(2)_L$ is
\begin{eqnarray}
\centering
\frac{1}{\alpha_Y} & = & \frac{1}{3 \alpha_{\ell}} + \frac{5}{3 \alpha_L}
+ \frac{3 (b_{u_1} + b_{R_1})-b_{\ell_1}-4 b_{L_1}}{6 \pi} \ln{\left( \frac{v}{M_{EW}}\right)} \nonumber\\
& + & \frac{3 (b_{u_2} + b_{R_2}- b_{u_1} -  b_{R_1}) + b_{\ell_1}- b_{\ell_2} + 4 b_{L_1}-
5 b_{L_2}}{6 \pi} \ln{\left( \frac{w}{M_{EW}}\right)} \\ \nonumber
& + & \frac{3 ( b_{u_3} -  b_{u_2} - b_{R_2}) + b_{\ell_2}- b_{\ell_3}+ 5 ( b_{L_2}-
b_{L_3})}{6 \pi} \ln{\left(\frac{x}{M_{EW}}\right)}
\end{eqnarray}
for cascade one, but takes a different form for the other cascades.
The experimental values for the fine-structure constants at $M_{EW}$ are~\cite{PDG}
\begin{equation}
\centering
\alpha_q = 0.1172, \qquad \alpha_L = 0.0338, \qquad \alpha_Y = 0.0102,
\end{equation}
where remember we have absorbed the normalisation into the $b_{u_j}$'s.

In this section we shall focus on the renormalisation-group equation analysis for cascade one and present only a
 summary of results for the other symmetry breaking routes.
The full details of their equations are relegated to appendix \ref{cha:appendix1}.
After the first stage of symmetry breaking, the particles $x_1^c$, $x_2^c$, $y_1^c$ and $y_2^c$
gain mass and the light Higgs spectrum is
\begin{eqnarray}
\centering
& \Phi_{\ell}& \sim \left( \mathbf{1,3,2,1} \right) \left( \frac{1}{3} \right) \oplus
\left( \mathbf{1,3,1,1} \right) \left(-\frac{2}{3}\right), \qquad \qquad \qquad
\Phi_{\ell^c} \sim \left( \mathbf{1,1,1,2} \right) (1), \\
& \Phi_a & \sim  \left( \mathbf{1,3,1,2} \right) \left(\frac{1}{3}\right) \oplus
\left( \mathbf{1,3,1,1} \right) \left(-\frac{2}{3}\right), \qquad \Phi_c \sim \Phi_a^{\dagger}.
\end{eqnarray}
The remaining exotic fermions gain masses at the $w$ scale and the
light Higgs spectrum is
\begin{eqnarray}
\centering
& \Phi_{\ell}& \sim \left( \mathbf{1,2,1,1} \right) (-1), \qquad
\Phi_{\ell^c} \sim \left( \mathbf{1,1,1,2} \right) (1), \\
& \Phi_a & \sim  \left( \mathbf{1,2,1,2} \right) (0) \oplus
\left( \mathbf{1,2,1,1} \right) (-1) \oplus
\left( \mathbf{1,1,1,2} \right) (1), \qquad \Phi_c \sim \Phi_a^{\dagger}.
\end{eqnarray}
After the final breaking particles of the minimal SM and $\nu^c$ are massless and
\begin{eqnarray}
\centering
& \Phi_{\ell}& \sim \left( \mathbf{1,2,1} \right) (-1), \\
& \Phi_a & \sim  \left( \mathbf{1,2,1} \right) (-1) \oplus
\left( \mathbf{1,2,1} \right) (1) \oplus
\left( \mathbf{1,2,1} \right) (-1), \qquad \Phi_c \sim \Phi_a^{\dagger},
\end{eqnarray}
giving us seven SM Higgs doublets.

Defining $N_H$ to be the multiplicity of the Higgs fields in
Eq.~\ref{eqn:higgstrans}\footnote{We take the same multiplicity $N_H$ for
{\it{both}} of the $\mathbf{36}$'s representing our Higgs fields. This
can obviously be generalised, but we shall have no need to do this
because we shall focus only on the simplest possible
schemes in which the multiplicity of each representation is precisely $N_H=1$.
Happily, it turns out $N_H > 1$ is not required.}
and summing over three generations of fermions, this spectrum of particles
defines the values of the $b$ quantities as
\begin{eqnarray}
\begin{array}{ccccc}
b_{q_1}  =  -5,&  b_{L_1} = - 5 + \frac{3 N_H}{2},&
b_{{\ell}_1} = - \frac{10}{3} + \frac{N_H}{2}, &
b_{R_1} = -\frac{10}{3} + \frac{7 N_H}{6}, & b_{u_1} = 4 + \frac{2 N_H}{3},\\
b_{q_2} =  -7, & b_{L_2} = - \frac{10}{3} + \frac{7 N_H}{6}, &
b_{{\ell}_2} = -\frac{22}{3},&  b_{R_2}= -\frac{10}{3}+\frac{7 N_H}{6}, &
b_{u_2} = \frac{8}{3} + N_H, \\
b_{q_3} =  b_{q_2}, & b_{L_3}=b_{L_2}, & b_{{\ell}_3} = b_{{\ell}_2}, & &
b_{u_3}=\frac{20}{3}+\frac{7 N_H}{6}.\end{array}
\end{eqnarray}

Substituting these numbers into the renormalisation-group equations, we have
\begin{eqnarray}
\centering
\frac{1}{\alpha_q (v)} & =& \frac{1}{\alpha_q} + \frac{5}{2 \pi} \ln{ \left( \frac{v}{M_{EW}} \right) }
+ \frac{1}{\pi} \ln{ \left( \frac{w}{M_{EW}} \right) }, \\
\frac{1}{\alpha_L (v)} & = &\frac{1}{\alpha_L} + \frac{10-3 N_H}{4 \pi} \ln{ \left( \frac{v}{M_{EW}} \right) }
+ \frac{N_H - 5 }{6 \pi} \ln{ \left( \frac{w}{M_{EW}} \right) }, \\
\frac{1}{\alpha_{\ell} (v)} & =& \frac{1}{\alpha_{\ell}} + \frac{20-3 N_H}{12 \pi}
\ln{ \left( \frac{v}{M_{EW}} \right) }
+ \frac{8 + N_H }{4 \pi} \ln{ \left( \frac{w}{M_{EW}} \right) }, \\
\frac{1}{\alpha_Y} & = & \frac{1}{3 \alpha_{\ell}} + \frac{5}{3 \alpha_L}
+ \frac{76-3 N_H}{18 \pi} \ln{ \left( \frac{v}{M_{EW}} \right) }
+ \frac{5 N_H-10 }{18 \pi} \ln{ \left( \frac{w}{M_{EW}} \right) }
+ \frac{22-3 N_H }{6 \pi} \ln{ \left( \frac{x}{M_{EW}} \right) }.
\end{eqnarray}
With these inputs, we have a large degree of freedom in the unification of the couplings, with
the simplest scheme being that in which $N_H=1$.

Given that we have no particles gaining mass at the final stage of breaking, the scale $x$
affects only the evolution of the
hypercharge fine-structure constant.
As a result, this scale can be as low as a few $TeV$ and as large as $6 \times 10^7\ GeV$
without spoiling the unification. Taking
$x \equiv x_{min} \sim 1\ TeV$, unification occurs for
\begin{equation}
\centering
w \sim 2.7 \times 10^{12}\ GeV, \qquad v \sim 1.2 \times 10^{17}\ GeV.
\end{equation}
At this unification scale, the value of the fine-structure constant for our unified theory is
$\alpha_{G_4}^{-1} \sim 43.85$, giving $\alpha_{\ell} \sim 0.0912$ at the electroweak scale,
which has a value between the weak $SU(2)_L$ and the strong couplings.
If we allow $x$ to increase
in energy, then the GUT scale decreases and the $w$ scale increases until $v = w \sim 7.5 \times 10^{13}\ GeV$
at $x_{max} \sim 6.4 \times 10^7\ GeV$. This unification scheme gives $\alpha_{G_4}^{-1} \sim 39.02$
and $\alpha_{\ell} \sim 0.1408$.
It is interesting to note here that the scale $w$ cannot be low for unification purposes, and
subsequently all our exotic fermions will be heavy, leaving only the $SU(2)_{\ell}$ gauge bosons
and additional Higgs fields as light particles foreign to the standard model.

Unification of the gauge coupling constants can also be obtained for the other three breaking patterns
with the range of possible, consistent scales summarised in Table \ref{tab:su2unbroken}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Cascade & $x$ & $w$ & $v$ & $\alpha_{G_4}^{-1}$ & $\alpha_{\ell}$ \\
\hline
1  & $x_{min}\sim 1 TeV$ & $2.7 \times 10^{12} GeV$ & $ 1.2 \times 10^{17} GeV$ & 43.85 & 0.0912 \\
  & $x_{max}\sim 6.4 \times 10^7 GeV$ & $7.5 \times 10^{13} GeV$ & $ 7.5 \times 10^{13} GeV$ & 39.02 & 0.1408 \\
\hline
2  & $x_{min} \sim 6.5 \times 10^5 GeV$ & $6.5 \times 10^{5} GeV$ & $3.9 \times 10^{19} GeV$ & 43.55 & 0.0526 \\
   & $x_{max} \sim 6.5 \times 10^{7} GeV$ & $7.4 \times 10^{13} GeV$ & $7.4 \times 10^{13} GeV$ & 39.01  & 0.1407 \\
 \hline
3  & $x_{min} \sim 6.3 \times 10^7 GeV$ & $ 7.7 \times 10^{13} GeV$ & $ 7.7 \times 10^{13} GeV$ & 39.02 & 0.1412 \\
   & $x_{max} \sim 4.9 \times 10^{10} GeV$ & $ 4.9 \times 10^{10} GeV$ & $ 7 \times 10^{12} GeV$ & 36.35  & 0.1210 \\
\hline
4  & $x_{min} \sim 6.2 \times 10^8 GeV$ & $1.7 \times 10^{12} GeV$ & $1.7 \times 10^{12} GeV $ & 34.77 & 0.111 \\
   & $x_{max} \sim 4.8 \times 10^{10} GeV$ & $4.8 \times 10^{10} GeV$ & $7 \times 10^{12} GeV$ & 36.35  & 0.1208 \\
\hline
\end{tabular}
\end{center}
\caption{Range of energy scales of symmetry breaking that yield unification of the gauge coupling constants.
There is only one scenerio which allows for a $TeV$ level breaking scale, while the scales
offered by the other choices are quite similar to each other.}
\label{tab:su2unbroken}
\end{table}
Unlike the first case, all the intermediate symmetry breaking scales for the other patterns must be high.
The lowest the final intermediate breaking scale can be is
about $6 \times 10^5\ GeV$ in option two, which is significantly
higher than the electroweak scale. Furthermore, this choice requires the unification scale
to be larger than the Planck scale, which is unacceptable. So, realistically,
in this scheme we would have to consider higher values of $x$ so as to lower the unification scale.
As a result, the theories prescribed by cascades two, three and four will
contain very heavy exotic particles that do not lie within reach of future colliders.
Note also that the value of the fine-structure constant for leptonic colour $SU(2)_\ell$
at the electroweak scale is generally always
larger than that describing quark colour.

Recall that the fourth symmetry breaking option is the only route that
returns exotic particles with masses of order $x$. However, the value of $x$ can
only be pushed down in energy to $6 \times 10^8\ GeV$ if unification is to be preserved.
Consequently, the possible existence of new low-energy phenomenology suggested by the presence of exotic
fermions at this last stage of breaking is denied by the demands placed on the energy scale
by the unification of the gauge coupling constants.

We shall comment further on the phenomenology of our models in Sec.~\ref{cha:phenomenology}

\subsection{$SU(2)_{\ell}$ broken} \label{cha:RGE-SU(2)broken}

By demanding that the leptonic colour symmetry is broken entirely, the number of symmetry
breaking routes from the quartification gauge group to the SM broadens to eight independent cascades.
These are labelled as per:
\begin{eqnarray}
\centering
1. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(3)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_2} \nonumber \\
& \stackrel{x}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes U(1)_{X_3} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y} \\
2. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(3)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_2} \nonumber \\
& \stackrel{x}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_R \otimes U(1)_{X_3} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y} \\
3. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(3)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(3)_L \otimes U(1)_{X_2} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y} \\
4. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(2)_L \otimes SU(3)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_2} \nonumber \\
& \stackrel{x}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes U(1)_{X_3} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y} \\
5. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(2)_L \otimes SU(3)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_2} \nonumber \\
& \stackrel{x}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_R \otimes U(1)_{X_3} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y} \\
6. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(2)_L \otimes SU(3)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(3)_{\ell} \otimes U(1)_{X_2} \nonumber \\
& \stackrel{x}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes U(1)_{X_3} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y} \\
7. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(3)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_2} \nonumber \\
& \stackrel{x}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes U(1)_{X_3} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y} \\
8. \;\; G_4 & \stackrel{v}{\rightarrow} &
SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(3)_R \otimes U(1)_{X_1} \nonumber \\
& \stackrel{w}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_2} \nonumber \\
& \stackrel{x}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes SU(2)_R \otimes U(1)_{X_3} \nonumber \\
& \stackrel{y}{\rightarrow} & SU(3)_q \otimes SU(2)_L \otimes U(1)_{Y},
\end{eqnarray}
where the generator of hypercharge now has the form
\begin{equation}
\centering
Y=I_{3R}+I_{3 \ell} +\frac{2}{\sqrt{3}} \left( \lambda_{L8} + \lambda_{\ell 8} +\lambda_{R8} \right).
\end{equation}
Of these eight choices, there are seven which can deliver unification
of the gauge coupling constants.
Cascade three does not offer a viable model assuming a minimal Higgs sector is used, so we eliminate it from
further consideration.
In all models, the light
particle spectrum consists of the standard model
particles and nine candidate SM Higgs doublets.
The exotic fermions gain masses either of order $v$ or $w$ in all cases except for cascade six.
The VEV structure of cascade six endows the particles
$x_1$,  $x_2$,  $y_1$,  $y_2$,  $x_1^c$,  $x_2^c$,  $y_1^c$ and $y_2^c$ with masses at the $x$ scale,
which at first sight could potentially result in lighter masses than the other scenarios.

For all cascades, the full details of the particle spectra, including the Higgs VEV patterns
instigating the breaking,
and the analysis of the renormalisation-group equations, are
contained in appendix~\ref{cha:appendix2}. Table~\ref{tab:unifscales} provides
a summary of results for the possible ranges of
energy scales which give unification of the gauge coupling constants for $N_H=1$.

Only a subset of these seven symmetry breaking schemes allow for a
flexible range of unification and intermediate breaking scales.
The final breaking to the SM gauge group can be as low as a $TeV$ for six of these seven options,
with cascades one, two, seven and eight demanding that this scale be precisely of $TeV$ order.
In fact, $y \equiv y_{max} \sim 7.1 \times 10^2\ GeV$ is the highest scale at which this breaking
occurs for these options. This choice of $y$-scale offers only two intermediate scales with unification
requiring $x=y$ and $w=v \sim 1.3 \times 10^{13}\ GeV$. The symmetry breaking patterns
for these four cascades thus
become equivalent, reducing to
$G_4 \rightarrow SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)
\rightarrow SU(3)_q \otimes SU(2)_L \otimes U(1)_Y$.

Unification can still be achieved in options four and five if the $y$ scale is as high as $\sim 10^6\ GeV$.
Furthermore, once at the $SU(3)_q \otimes SU(2)_L \otimes SU(2)_{\ell} \otimes SU(2)_R \otimes U(1)$
level, the choice of which $SU(2)_{\ell,R}$ factor to break first has no significant influence on
the outcome of the unification and intermediate scales.
It turns out that for all viable $y$ values, $x$ must be very close to $y$.
When the final breaking occurs at $y_{min} \sim 1\ TeV$, then
the unification scale is of order $10^{13}\ GeV$, whereas at $y_{max} \sim 10^6\ GeV$, the unification
scale is at a lower energy, of order $10^{11}\ GeV$, which could potentially be more dangerous with
respect to proton decay. Nevertheless, for all these cascades, the highest the unification
scale can be is $\sim 10^{13}\ GeV$ which is much lower than the GUT energies possible when
$SU(2)_{\ell}$ remains unbroken at low energy.

Cascade six affords the most tantalising spectrum of masses for the exotic fermions,
with masses of order $x$ resulting from the breaking
$G_{SM} \otimes SU(3)_{\ell} \rightarrow G_{SM} \otimes SU(2)_{\ell}$.
This scale can be as low as $\sim 10^4\ GeV$ with unification
occurring for
\begin{equation}
\centering
y_{min} \sim 1\ TeV, \qquad x \sim 8.8 \times 10^3\ GeV, \qquad w=v\equiv v_{min} \sim 3.6 \times 10^{10}\ GeV.
\end{equation}
A $10\ TeV$ scale for some of the exotic fermion masses provides hope for possible discovery
at the LHC.  However, this choice also requires the unification
scale $v \sim 10^{10}\ GeV$, which may be low enough to be troubling with respect to
proton decay. If we allow $x$ to increase, then we obtain
\begin{equation}
\centering
y_{min} \sim 1\ TeV, \qquad x = w \sim 4.2 \times 10^7\ GeV, \qquad v\equiv v_{max} \sim 3.8 \times 10^{11}\ GeV.
\end{equation}
There is flexibility in $y$; it can be pushed up to
\begin{equation}
y_{max} =x=w \sim 1.2 \times 10^6 GeV, \qquad v\sim 1.4 \times 10^{11}GeV, \qquad \alpha^{-1}_{G_4} = 32.02.
\end{equation}
With this choice, it becomes equivalent to the upper bound of unification for cascades four and five,
with the symmetry breaking now described by
$G_4 \rightarrow SU(3)_q \otimes SU(2)_L \otimes SU(3)_{\ell} \otimes SU(2)_R \otimes U(1)_{X_1} \rightarrow G_{SM}$.
Since the $x$-scale is now at about $10^6\ GeV$, we see that while this cascade is consistent
with exotic fermion masses of about $10\ TeV$, they can also be significantly higher without
spoiling unification.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
 & $y$ & $x$ & $w$ & $v$ & $\alpha_{G_4}^{-1}$ \\
\hline
1 and 2 &  $y_{max} \sim 7.1 \times 10^2 GeV$ & $7.1 \times 10^2 GeV$ &
$1.3 \times 10^{13} GeV$  & $1.3 \times 10^{13} GeV$& 37.05\\
\hline
4 and 5 & $y_{min} \sim 1 TeV$ & $1TeV$ & $6.2 \times 10^{12} GeV$ &$v_{max} \sim 1.1 \times 10^{13} GeV$ & 36.82 \\
  &    & $4.2 \times 10^7 GeV$ & $4.2 \times 10^7 GeV$ &$v_{max} \sim 3.8 \times 10^{11} GeV$ & 33.11\\
& $y_{max} \sim 1.2 \times 10^6 GeV$ & $1.2 \times 10^6 GeV$  &$1.2 \times 10^6 GeV$ &
$1.4 \times 10^{11} GeV$ & 32.02 \\
\hline
6 & $y_{min} \sim 1 TeV $ & $8.8 \times 10^{3} GeV$ &$3.6 \times 10^{10} GeV$ &
$v_{min} \sim 3.6 \times 10^{10} GeV$ & 30.48 \\
  &    & $4.2 \times 10^7 GeV$ & $4.2 \times 10^7 GeV$ &$v_{max} \sim 3.8 \times 10^{11} GeV$ & 33.11\\
 & $y_{max} \sim 1.2 \times 10^{6} GeV$ & $1.2 \times 10^{6} GeV$ &$1.2 \times 10^{6} GeV$ &
$1.4 \times 10^{11} GeV$ & 32.02 \\
\hline
7 and 8 & $y_{max} \sim 7.1 \times 10^2 GeV$&$ 7.1 \times 10^2 GeV$ & $1.3 \times 10^{13} GeV$ &
$1.3 \times 10^{13} GeV$ & 37.05\\
\hline
\end{tabular}
\end{center}
\caption{The range of energies for the symmetry breaking scales that will
consistently give unification of the gauge coupling constants, when $N_H=1$.
Four of the schemes become equivalent if unification is to be demanded, and the last
stage of symmetry breaking has to occur below a $TeV$. The other three choices allow
for a range in the intermediate scales while still preserving unification.
When $y_{max}$ chosen for cascades four, five and six, they also become equivalent. }\label{tab:unifscales}
\end{table}

\section{phenomenology}\label{cha:phenomenology}

We now round out our discussion of the phenomenological consequences of the various schemes above.
It is beyond the scope of this paper to provide a rigorous quantitative analysis of
phenomenological bounds for all of these models, so our remarks shall be qualitative and
our analysis necessarily incomplete.

We first deal with the four cascades featuring a remnant $SU(2)_\ell$.  All of them feature
seven electroweak Higgs doublets.
We emphasise that this multiplicity is not due to duplication of the fundamental Higgs multiplets.
Rather, they are an integral part of the minimal Higgs sector required for quartification.
Obviously, questions about Higgs-induced flavour-changing neutral processes arise.  Without a
detailed analysis we can only make the simple remark that some of the doublets will have to
acquire $TeV$ scale masses or have somewhat small Yukawa coupling constants.  It is
certainly interesting, though, that multiple Higgs doublets are a generic prediction of
quartification models.

Cascade one is the only
one that allows an intermediate scale as low as a $TeV$.  This scale is a right-handed weak-isospin
breaking scale, so the immediate phenomenological consequences are right-handed $W$ bosons
and a corresponding $Z'$ at the $TeV$ level.  Since this scale can be raised above $10^4\ TeV$
without spoiling unification, it is clear that it can be made phenomenologically acceptable.
However, it has no necessary new physics at LHC energies, apart from the
multi-Higgs doublet feature it shares with the other quartification schemes.
Recall from the earlier discussion that it also has no $TeV$-scale exotic fermions.
The unification scale is in the range $10^{14-17}\ GeV$, so we would guess that it
is safe from too-rapid Higgs-induced proton decay.

Cascades two, three and four all have high $x$-scales, so their only characteristic $TeV$-level
feature is the seven Higgs doublets.  The unification scales lie in the range
$7 \times 10^{12-13}\ GeV$, which should be safe from a proton-decay point of view.

We now turn to the schemes having no leptonic colour remnant symmetry.  As noted
earlier, cascade three is unsuccessful and hence discarded.  All the cascades feature
nine electroweak Higgs doublets.

The requirement of
unification makes cascades one, two, seven and eight identical, with the lowest
breaking scale being at about $700\ GeV$.  This scheme is possibly ruled out,
because it results in quite light right-handed $W$-bosons and other light gauge
particles including $Z'$ states.  The $700\ GeV$ scale follows from the central values for
the electroweak-scale gauge coupling constants; it can be pushed up to the $TeV$ range
by varying these values within the experimental error range.
All the exotic fermions, however, are quite heavy,
gaining masses at the $w$-scale which is about $10^{13}\ GeV$.  If detailed
study were to show it is not yet falsified, then it would be an interesting
situation in regards to possible discovery of new gauge bosons below $1\ TeV$.
The unification scale of $10^{13}\ GeV$ may be sufficient to suppress proton decay.

Cascades four and five can feature, respectively, $SU(2)_\ell$ or $SU(2)_R$ gauge bosons at
the $TeV$-level, although they need not.  It would be interesting to study
their GUT-scale proton-decay phenomenology, as a lower $y$-value implies a higher
GUT-scale, as summarised in Table~\ref{tab:unifscales}.  Conceivably, the suppression
of Higgs-induced proton decay might
favour new $TeV$-scale physics for these cascades.

As already noted, cascade six is unique in that it can have new fermions
at the relatively low scale of about $10\ TeV$. If so, this would be
correlated with a low breaking scale for $SU(2)_\ell$ and hence the presence
of exotic gauge bosons coupling leptons to exotic leptons.  There is also
extended neutral current phenomenology.  The danger for cascade six is the
low range for the unification scale, which is not allowed to be much
higher than $10^{11}\ GeV$.

In summary, there is obviously a wealth of phenomenology to be
explored within these schemes, both at the $TeV$-scale and at the GUT-scale.
The overall impression is that the schemes that totally break leptonic colour
are more constrained, either from $TeV$-scale considerations or from the GUT
regime or both.  This makes them more exciting, more easily tested; some are
possibly already ruled out.  Note that we have not yet
attempted a systematic study of the Higgs-induced proton decay question,
so our concerns about some of the lower unification scales are generic rather
than specific.

We have also not yet attempted a study of the Higgs potential and the minimisation
conditions.  It almost goes without saying that all proposed quartification
schemes suffer from the gauge hierarchy problem.  In our opinion, however, the
overall framework has considerable appeal, despite this
standard defect common to all non-supersymmetric GUTs.

Another interesting topic for future work is neutrino mass generation for the
totally-broken leptonic colour scenarios, to understand the effect of the intermediate
scales on the see-saw suppression given by Eq.~\ref{eq:10x10}.

\section{conclusion}\label{cha:conclusion}

Quartification schemes offer an alternative route to grand unification.  They
are conceptually rather appealing, with the fundamental fermion and Higgs
multiplets taking relatively simple and elegant forms.  As we have shown
in this paper, there are a variety of symmetry breaking cascades consistent
with successful gauge coupling constant unification.  None of them require
supersymmetry, though all of them of course require intermediate scales.
The non-trivial result is that appropriate intermediate scales are a natural
possibility.  Our results add to the important observation of Babu, Ma and
Willenbrock \cite{BMW} that complete unification is possible in quartification models,
rather than having to settle for the partial unification originally proposed
by Joshi and Volkas \cite{Ray}.

The various schemes have different phenomenological consequences, though
all have the existence of several electroweak Higgs doublets as a feature.
This multiplicity is not due to a replication of fundamental Higgs multiplets,
but is rather an inherent feature of the minimal Higgs sector required
for quartification. Depending on the scheme, rich phenomenology at LHC energies
such as additional gauge bosons and fermions is possible and in some cases required.
In addition, the models may have Higgs-induced proton decay,
though detailed analyses of this and the new physics at the $TeV$-scale
have yet to be carried out.

\acknowledgments{This work was supported by the Australian Research Council,
the Commonwealth of Australia, and the University of Melbourne.}

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