text:
function symbolize(opt) {
	jsdoc = null;
	symbols = null;
	jsdoc = new JSDOC.JsDoc(opt);
	symbols = jsdoc.symbolGroup.getSymbols();
}

var testCases = [
	function() {
		symbolize({a:true, p:true, _: [SYS.pwd()+"test/prototype.js"]});
//print(Dumper.dump(symbols[0].get("methods")));

		is('symbols[0].get("name")', "Article", 'Function set to constructor prototype with inner constructor name is found.');
		is('symbols[0].get("methods")[0].get("name")', "init", 'The initializer method name of prototype function is correct.');
		is('symbols[0].get("properties")[0].get("name")', "title", 'A property set on the initializer "this"  is on the outer constructor.');
		is('symbols[3].get("name")', "counter", 'A static property set in the initializer has the name set correctly.');
		is('symbols[3].get("memberof")', "Article", 'A static property set in the initializer has the memberof set correctly.');
		is('symbols[3].get("isStatic")', true, 'A static property set in the initializer has isStatic set to true.');

	}
	,
	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/prototype_oblit.js"]});

		is('symbols[0].get("name")', "Article", 'Oblit set to constructor prototype name is found.');
		is('symbols[0].get("memberof")', "", 'The memberof of prototype oblit is correct.');
		is('symbols[0].get("methods")[0].get("name")', "getTitle", 'The nonstatic method name of prototype oblit is correct.');
		is('symbols[0].get("methods")[0].get("isStatic")', false, 'The isStatic of a nonstatic method of prototype oblit is correct.');
		is('symbols[0].get("methods")[1].get("name")', "getTitle", 'The static method name of prototype oblit is correct.');
		is('symbols[0].get("methods")[1].get("isStatic")', true, 'The isStatic of a static method of prototype oblit is correct.');
		is('symbols[1].get("alias")', "Article#getTitle", 'The alias of non-static method of prototype oblit is correct.');
		is('symbols[1].get("isa")', "FUNCTION", 'The isa of non-static method of prototype oblit is correct.');
		is('symbols[2].get("alias")', "Article.getTitle", 'The alias of a static method of prototype oblit is correct.');
		is('symbols[1].get("isa")', "FUNCTION", 'The isa of static method of prototype oblit is correct.');
	}
	,
	function() {
		symbolize({a:true, p:true, _: [SYS.pwd()+"test/prototype_oblit_constructor.js"]});

		is('symbols[0].get("name")', "Article", 'Oblit set to constructor prototype with inner constructor name is found.');
		is('symbols[0].get("methods")[0].get("name")', "init", 'The initializer method name of prototype oblit is correct.');
		is('symbols[0].get("properties")[0].get("name")', "pages", 'Property set by initializer method "this" is on the outer constructor.');
		is('symbols[1].get("name")', "Title", 'Name of the inner constructor name is found.');
		is('symbols[1].get("memberof")', "Article", 'The memberof of the inner constructor name is found.');
		is('symbols[1].get("isa")', "CONSTRUCTOR", 'The isa of the inner constructor name is constructor.');
		is('symbols[1].get("properties")[0].get("name")', "title", 'A property set on the inner constructor "this"  is on the inner constructor.');
	}
	,
	function() {
		symbolize({a:true, p:true, _: [SYS.pwd()+"test/inner.js"]});

		is('symbols[0].get("name")', "Outer", 'Outer constructor prototype name is found.');
		is('symbols[0].get("methods").length', 1, 'Inner function doesnt appear as a method of the outer.');
		is('symbols[0].get("methods")[0].get("alias")', "Outer#open", 'Outer constructors methods arent affected by inner function.');
		is('symbols[1].get("alias")', "Outer-Inner", 'Alias of inner function is found.');
		is('symbols[1].get("isa")', "CONSTRUCTOR", 'isa of inner function constructor is found.');
		is('symbols[1].get("memberof")', "Outer", 'The memberof of inner function is found.');
		is('symbols[1].get("name")', "Inner", 'The name of inner function is found.');
		is('symbols[2].get("name")', "name", 'A member of the inner function constructor, attached to "this" is found on inner.');
		is('symbols[2].get("memberof")', "Outer-Inner", 'The memberof of an inner function member is found.');
	}
	,
	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/prototype_nested.js"]});

		is('symbols[0].get("name")', "Word", 'Base constructor name is found.');
		is('symbols[0].get("methods")[0].get("name")', "reverse", 'Base constructor method is found.');
		is('symbols[0].get("methods").length', 1, 'Base constructor has only one method.');
		is('symbols[0].get("memberof")', "", 'Base constructor memberof is empty.');
		is('symbols[1].get("name")', "reverse", 'Member of constructor prototype name is found.');
		is('symbols[1].get("memberof")', "Word", 'Member of constructor prototype memberof is found.');
		is('symbols[1].get("methods")[0].get("name")', "utf8", 'Member of constructor prototype method name is found.');
		is('symbols[2].get("name")', "utf8", 'Static nested member name is found.');
		is('symbols[2].get("memberof")', "Word#reverse", 'Static nested member memberof is found.');
	}
	,
	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/namespace_nested.js"]});

		is('symbols[0].get("name")', "ns1", 'Base namespace name is found.');
		is('symbols[0].get("memberof")', "", 'Base namespace memberof is empty (its a constructor).');
		is('symbols[1].get("name")', "ns2", 'Nested namespace name is found.');
		is('symbols[1].get("alias")', "ns1.ns2", 'Nested namespace alias is found.');
		is('symbols[1].get("memberof")', "ns1", 'Nested namespace memberof is found.');
		is('symbols[2].get("name")', "Function1", 'Method of nested namespace name is found.');
		is('symbols[2].get("memberof")', "ns1.ns2", 'Constructor of nested namespace memberof is found.');
	}
	,
	function() {
		symbolize({a:true, p:true, _: [SYS.pwd()+"test/functions_nested.js"]});

		is('symbols[0].get("name")', "Zop", 'Any constructor name is found.');
		is('symbols[0].get("isa")', "CONSTRUCTOR", 'It isa constructor.');
		is('symbols[0].get("methods")[0].get("name")', "zap", 'Its method name, set later, is in methods array.');
		is('symbols[1].get("name")', "Foo", 'Containing constructor name is found.');
		is('symbols[1].get("methods")[0].get("name")', "methodOne", 'Its method name is found.');
		is('symbols[1].get("methods")[1].get("name")', "methodTwo", 'Its second method name is found.');
		is('symbols[2].get("alias")', "Foo#methodOne", 'A methods alias is found.');
		is('symbols[2].get("isStatic")', false, 'A methods is not static.');
		is('symbols[4].get("name")', "Bar", 'A function set inside another function is found.');
		is('symbols[4].get("isa")', "FUNCTION", 'It isa function.');
		is('symbols[6].get("name")', "inner", 'An inner functions name is found.');
		is('symbols[6].get("memberof")', "Foo", 'It is member of the outer function.');
		is('symbols[6].get("isInner")', true, 'It is an inner function.');
		is('symbols[6].get("alias")', "Foo-inner", 'The inner functions alias is found.');
	}
	,
	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/memberof_constructor.js"]});

		is('symbols[1].get("name")', "Tangent", 'Constructor set on prototype using @member has correct name.');
		is('symbols[1].get("memberof")', "Circle", 'Constructor set on prototype using @member has correct memberof.');
		is('symbols[1].get("alias")', "Circle#Tangent", 'Constructor set on prototype using @member has correct alias.');
		is('symbols[1].get("isa")', "CONSTRUCTOR", 'Constructor set on prototype using @member has correct isa.');
		is('symbols[1].get("isStatic")', false, 'Constructor set on prototype using @member is not static.');
		is('symbols[2].get("name")', "getDiameter", 'Method set on prototype using @member has correct name.');
		is('symbols[2].get("memberof")', "Circle#Tangent", 'Method set on prototype using @member has correct memberof.');
		is('symbols[2].get("alias")', "Circle#Tangent#getDiameter", 'Method set on prototype using @member has correct alias.');
		is('symbols[2].get("isa")', "FUNCTION", 'Method set on prototype using @member has correct isa.');
		is('symbols[2].get("isStatic")', false, 'Method set on prototype using @member is not static.');
	}
	,
	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/inherits.js"]});
//print(Dumper.dump(symbols));
		is('symbols[0].get("name")', "Layout", 'Constructor can be found.');
		is('symbols[0].get("methods")[0].get("name")', "init", 'Constructor method name can be found.');
		is('symbols[0].get("properties")[0].get("name")', "orientation", 'Constructor property name can be found.');
		is('symbols[4].get("methods")[0].get("name")', "reset", 'Second constructor method name can be found.');
		is('symbols[4].get("properties")[0].get("name")', "orientation", 'Second constructor inherited property name can be found in properties.');
		is('symbols[4].get("properties")[0].get("memberof")', "Page", 'Second constructor inherited property memberof can be found.');
		is('symbols[6].get("methods")[0].get("alias")', "ThreeColumnPage#init", 'Third constructor method can be found even though method with same name is inherited.');
		is('symbols[6].get("methods")[1].get("alias")', "ThreeColumnPage#reset", 'Inherited method can be found.');
		is('symbols[6].get("properties")[0].get("alias")', "ThreeColumnPage#orientation", 'Twice inherited method can be found.');

	}
	,
	function() {
		symbolize({a: true, _: [SYS.pwd()+"test/augments.js", SYS.pwd()+"test/augments2.js"]});

		is('symbols[4].get("augments")[0]', "Layout", 'An augmented class can be found.');
		is('symbols[4].get("methods")[0].get("alias")', "Page#reset", 'Method of augmenter can be found.');
		is('symbols[4].get("methods")[1].get("alias")', "Layout#init", 'Method from augmented can be found.');
		is('symbols[4].get("properties")[0].get("alias")', "Layout#orientation", 'Property from augmented can be found.');
		is('symbols[4].get("methods").length', 3, 'Methods of augmented class are included in methods array.');

		is('symbols[6].get("augments")[0]', "Page", 'The extends tag is a synonym for augments.');
		is('symbols[6].get("methods")[0].get("alias")', "ThreeColumnPage#init", 'Local method overrides augmented method of same name.');
		is('symbols[6].get("methods").length', 3, 'Local method count is right.');

		is('symbols[12].get("augments")[0]', "ThreeColumnPage", 'Can augment across file boundaries.');
		is('symbols[12].get("augments").length', 2, 'Multiple augments are supported.');
		is('symbols[12].get("inherits")[0].alias', "Junkmail#annoy", 'Inherited method with augments.');
		is('symbols[12].get("methods").length', 6, 'Methods of augmented class are included in methods array across files.');
		is('symbols[12].get("properties").length', 1, 'Properties of augmented class are included in properties array across files.');
	}
	,

	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/static_this.js"]});

		is('symbols[0].get("name")', "box.holder", 'Static namespace name can be found.');
		is('symbols[1].get("name")', "foo", 'Static namespace method name can be found.');
		is('symbols[1].get("isStatic")', true, 'Static namespace method is static.');

		is('symbols[2].get("name")', "counter", 'Instance namespace property name set on "this" can be found.');
		is('symbols[2].get("alias")', "box.holder#counter", 'Instance namespace property alias set on "this" can be found.');
		is('symbols[2].get("memberof")', "box.holder", 'Static namespace property memberof set on "this" can be found.');
	}
	,
	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/scope.js"]});

		is('symbols[0].get("name")', "Person", 'Class defined in scope comment is found.');
		is('symbols[0].get("methods")[0].get("name")', "initialize", 'Scoped instance method name can be found.');
		is('symbols[0].get("methods")[1].get("name")', "say", 'Second instance method can be found.');
		is('symbols[0].get("methods")[1].get("isStatic")', false, 'Instance method is known to be not static.');

		is('symbols[0].get("methods")[2].get("name")', "sing", 'Instance method name from second scope comment can be found.');
		is('symbols[4].get("name")', "getCount", 'Static method name from second scope comment can be found.');
		is('symbols[4].get("isStatic")', true, 'Static method from second scope comment is known to be static.');

		is('symbols[5].get("name")', "Unknown.isok", 'Static instance method from scope comment is kept.');
		is('symbols[6].get("name")', "_global_", 'Orphaned instance method from scope comment is discarded.');
	}
	,
	function() {
		symbolize({a:true, _: [SYS.pwd()+"test/param_inline.js"]});

		is('symbols[0].get("params")[0].type', "int", 'Inline param name is set.');
		is('symbols[0].get("params")[0].desc', "The number of columns.", 'Inline param desc is set from comment.');
		is('symbols[1].get("params")[0].name', "id", 'User defined param documentation takes precedence over parser defined.');
		is('symbols[1].get("params")[0].isOptional', true, 'Default for param is to not be optional.');
		is('symbols[1].get("params")[1].isOptional', false, 'Can mark a param as being optional.');
		is('symbols[1].get("params")[1].type', "number, string", 'Type of inline param doc can have multiple values.');
		is('symbols[2].get("params")[0].type', "", 'Type can be not defined for some params.');
		is('symbols[2].get("params")[2].type', "int", 'Type can be defined inline for only some params.');
		is('symbols[4].get("params").length', 0, 'Docomments inside function sig is ignored without a param.');
		is('symbols[5].get("params")[2].type', "zoppler", 'Doccomments type overrides inline type for param with same name.');
	}
	,
	function() {
		symbolize({a: true, _: [SYS.pwd()+"test/shared.js", SYS.pwd()+"test/shared2.js"]});

		is('symbols[1].get("name")', 'some', 'The name of a symbol in a shared section is found.');
		is('symbols[1].get("alias")', 'Array#some', 'The alias of a symbol in a shared section is found.');
		is('symbols[1].get("desc")', "Extension to builtin array.", 'A description can be shared.');
		is('symbols[2].get("desc")', "Extension to builtin array.\nChange every element of an array.", 'A shared description is appended.');
		is('symbols[3].get("desc")', "A first in, first out data structure.", 'A description is not shared when outside a shared section.');
		is('symbols[4].get("alias")', "Queue.rewind", 'Second shared tag can be started.');
		is('symbols[5].get("alias")', "_global_#startOver", 'Shared tag doesnt cross over files.');

	}
	,
	function() {
		symbolize({a: true, _: [SYS.pwd()+"test/config.js"]});
		is('symbols[0].get("params")[0].name', 'person', 'The name of a param is found.');
		is('symbols[0].get("params")[1].name', 'person.name', 'The name of a param set with a dot name is found.');
		is('symbols[0].get("params")[2].name', 'person.age', 'The name of a param set with a dot name is found.');
		is('symbols[0].get("params")[4].name', 'connection', 'The name of a param after config is found.');

		is('symbols[1].get("params")[0].name', 'persons', 'Another name of a param is found.');
		is('symbols[1].get("params")[1].name', 'persons.Father', 'The name of a param+config is found.');
		is('symbols[1].get("params")[2].name', 'persons.Mother', 'The name of a second param+config is found.');
		is('symbols[1].get("params")[3].name', 'persons.Children', 'The name of a third param+config is found.');

	}
];

//// run and print results
print(testrun(testCases));


WELCOME to the 2019 school year, which has begun with the hustle and bustle of new staff and new students.
The term has started with a fantastic sense of calm and great student attendance. Students are interacting with each other and staff with respect and confidence.
New students are received well by students across the school.
The buddy system is working well with students who volunteered to be a buddy to the preps receiving their first "high 5" for the year.
Prep students have spent the first few weeks familiarising themselves with the school surrounds, staff and other students and have settled in well under the guidance of Mrs Walker and Mrs Ruth.
We also welcome Miss Loren McInnenery to the school who will be teaching the Year 1/2 class this year.
Loren has joined us from Greenslopes School in Brisbane and is thoroughly enjoying the first few weeks in a small school.
She is a great addition to our school.
Loren is excited to be at Wallangarra School and is looking forward to being part of such a wonderful community.
Wallangarra School now has four full-time teachers and a part-time teacher who works four days a week.
BILLIE Wilson won second place in the school art at the Stanthorpe Show last weekend. Against fierce competition, she won a ribbon, certificate and money with her Mother Nature poster.


text:
Reps accept Emefiele's apology, CBN gov allays fears
Home/Celebrity Gossip/The Plot to Kill My Mother movie cast revealed
The Plot to Kill My Mother movie cast revealed
Lifetime has an astonishing assortment of TV films, similar to its most recent, The Plot to Kill My Mom. With the Christmas season behind us, Lifetime is presently zeroing in on additional serious movies like this one. The Plot to Kill My Mom will debut on Sunday, January 22, 2023, on Lifetime. The charming reason for the film is supported by some fascinating projecting decisions, as Romy Weltman and Milton Barnes.
The film will follow a young lady who unwittingly experienced childhood in government witness security, finding it a lot further down the road, prompting tumult. The outline for the film, as delivered by Lifetime, peruses:
"A young lady who unconsciously experienced childhood in government witness security reels after her mom's homicide, driving her to address all that she believes is valid. She chooses to leave the program and find the executioner before he eliminates once more, yet recovering a day to day existence she never realized won't be simple."
The movie is coordinated by Simone Stock with a content from Lisa Satisfaction.
Romy Weltman will play the hero in The Plot to Kill My Mom. As it is an extremely overbearing job, Weltman should place in a great deal to squeeze into this character's perspective.
Romy Weltman is maybe most popular for her part in Behind the stage. She assumes the part of Katie Dunn in the Jennifer Pertsch and Lara Azzopardi show. A great many people know her from Specialist Loathsome's Outsider Dissection business, which was a seriously famous one of now is the right time.
Born in Canada, Weltman purportedly prepared under a confidential acting mentor prior to joining the business. She is the most youthful of three kids in her loved ones. One of her most important jobs was in The Returned, an exemplary thriller from 2013.
Not much is been aware of the job Milton Barnes will play in the forthcoming Lifetime film, yet fans can trust that it's a big one. The entertainer was born and raised in Canada. He shot to popularity with his part in Death By Fellowship.
Barnes is supposedly a traditionally prepared entertainer with experience in both theater and screen. He even began his profession as a theater entertainer.
He followed it up with other well known jobs in probably the most famous movies of the 2010s, including Comprehensive recollection (2012), Pacific Edge (2013), and Station Eleven (2021).
Other cast individuals from The Plot to Kill My Mom Unfortunately, the organization has uncovered almost no about the cast of the impending Lifetime spine chiller. Lifetime has uncovered just two additional cast individuals in front of the film's debut on Sunday.
Samantha Earthy colored will assume the part of Allie, about which there is little thought. The entertainer is known for her parts in Holdout (2021), Y: The Last Man (2021), and Hostage (2013).
Arwen Humphreys, an essayist, and maker, is the main other cast part uncovered by Lifetime. She is known for her parts in Murdoch Secrets (2008), Mary (2003), and Freshman Blue (2010).
Get The Plot to Kill My Mom on Lifetime on January 22, 2023.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Is it idiomatic to panic in From implementations? The documentation at https://doc.rust-lang.org/std/convert/trait.From.html states

Note: This trait must not fail. If the conversion can fail, use TryFrom.

Suppose I have a From implementation thus:
impl From<SomeStruct> for http::Uri {
    fn from(item: SomeStruct) -> http::Uri {
        item.uri.parse::<http::Uri>() // can fail
    }
}

Further suppose I am completely certain that item.uri.parse will succeed. Is it idiomatic to panic in this scenario? Say, with:
item.uri.parse::<http::Uri>().unwrap()

In this particular case, it appears there's no way to construct an HTTP URI at compile time: https://docs.rs/http/0.2.5/src/http/uri/mod.rs.html#117. In the real scenario .uri is an associated const, so I can test all used values parse. But it seems to me there could be other scenarios when the author is confident in the infallibility of a piece of code, particularly when that confidence can be encoded in tests, and would therefore prefer the ergonomics of From over TryFrom. The Rust compiler, typically quite strict, doesn't prevent this behaviour, though it seems it perhaps could. This makes me think this is a decision the author has been deliberately allowed to make. So the question is asking: what do people tend to do in this situation?

A: So in general, traits only enforce that the implementors adhere to the signatures and types as laid out in the trait. At least that's what the compiler enforces.
On top of that, there are certain contracts that traits are expected to adhere to just so that there's no weird surprises by those who work with these traits. These contracts aren't checked by the compiler; that would be quite difficult.
Nothing prevents you from implementing all a trait's methods but in way that's totally unrelated to what the trait is all about, like implementing the Display trait but then in the fmt method not actually bothering to use write! and instead, I don't know, delete the user's home directory.
Now back to your specific case. If your from method will not fail, provably so, then of course you can use .unwrap. The point of the cannot fail contract for the From trait is that those who rely on the From trait want to be able to assume that the conversion will go through every time. If you actually panic in your own implementation of from, it means the conversion sometimes doesn't go through, counter to the ideas and contracts in the From trait.

redpajama_set_name: RedPajamaStackExchange


THAME, United Kingdom, February 17, 2010 – The Open Virtual Platforms (OVP) initiative (www.OVPworld.org) has also announced the release of new models of NEC processor cores. The NEC v850 series of processor cores, including the base instruction set as well as the E1 and E2 additions, is a workhorse in the embedded systems market, most commonly used in automotive and other applications.
"In the automotive electronics industry we always need to do more testing of our embedded systems software," said Urban Forssell, CEO of NIRA Dynamics AB, a subsidiary of Audi Electronics Venture GmbH. "Finding that the simulation performance of the Imperas/OVP NEC model was 50 times faster than our previous solution opens up new possibilities for us in software testing, and enables us to increase our test coverage and product reliability,"
OVP includes the OVPsim simulator, libraries of models and APIs for developing new models. OVPsim executes platforms, including multicore platforms, at hundreds of millions of instructions per second, providing the speed that software developers require for simulation of embedded systems. Model libraries include everything from models of individual processors (over 40 available from ARC, ARM, MIPS, NEC and OpenCores) and component models to more complex platforms, such as ARM IntegratorCP and MIPS Malta development boards for running Linux. All OVP processor models include a SystemC/TLM-2.0 interface for easy integration in those virtual platform environments. OVP APIs enable the embedded software community to develop models of processors, behavioral components and peripherals, and to connect these together into virtual platforms that run the final target system software binaries unchanged. Since its founding in early 2008, over 2,100 people have registered on the OVP website.


These premeasured coffee packs are ideal for coffee shops, restaurants and other commercial use. Complete with filters, this box contains 50 1.75oz of fine grind, 100% premium Arabica coffee. Each pack makes a 64oz pot of gourmet coffee. Choose between our Breakfast Roast (Light), Carolina Classic (Medium), Cerrado Gold (Full City Roast) and French Roast (Dark).


Stick this striking black-and-white sticker anywhere you want to add a little Arizona pride! This Arizona sticker features a bold chevron design. Each sticker measures 3" high by 2.5" wide, and is printed on heavy-duty, waterproof vinyl with a matte finish. Plus, sticker orders always ship free!


Deep clean in half the time!
Large animal groomer... cleans deeper in half the time. It can move 135 cubic ft. of air per min. for fast, high-powered grooming. The Pro-3AG, Vac-N-Blo does all the work and cuts grooming time in half. As a vacuum, its super suction gets down deep to remove parasites, dirt, dandruff, and loose hair.
As a vacuum, its super suction gets down deep to remove parasites, dirt, dandruff, and loose hair.
As a blower, it dries coats evenly and completely. Leaves hair smooth and lustrous. Removes 95% of the water from an animals coat. Can even be used to blow dust off animals.
Portable and lightweight, can be used anywhere. Works great on large dogs, horses, donkeys, cows, even llamas love it. Adjustable shoulder strap frees hands for easy use. Excellent for cleaning debris from barn stables, grain drills, tractors, feed mills, etc. Built for continuous heavy-duty operation.
Weighs 9.5 lbs. Comes complete with cloth bag, shoulder strap, 2-6' flex hoses with air flow control, air concentrator, air flare, groomer rake, 3 piece brush comb set, 3 disposable bags, 2 filters.
- Triple filtration. Paper bag, permanent cloth and filter.
- 100" of water lift.
- Grounded power supply cord.
- Two 6.5 foot flexible hoses with air flow control.
- Air concentrator, air flare, and groomer rake.
- 3 piece comb set (grooming, curry comb and pin comb brushes).


text:
\section{Introduction \label{intro}}

Flavour symmetries, both discrete and continuous, have been extensively employed in the literature as a way to solve the puzzling questions associated with flavour. Examples of well-known discrete symmetries applied to flavour are $S_3$, $A_4$, $S_4$ and $A_5$.
Recently, these same symmetries are used in flavour models as modular symmetries $\Gamma_2\simeq S_3$~\cite{Kobayashi:2018vbk, Kobayashi:2018wkl},
$\Gamma_3\simeq A_4$~\cite{Feruglio:2017spp, Criado:2018thu, Kobayashi:2018scp, Okada:2018yrn, Kobayashi:2018wkl, Novichkov:2018yse, Ding:2019zxk, Zhang:2019ngf, Wang:2019xbo},
$\Gamma_4\simeq S_4$~\cite{Penedo:2018nmg, Novichkov:2018ovf, deMedeirosVarzielas:2019cyj, King:2019vhv, Wang:2019ovr,Wang:2020dbp}, and
$\Gamma_5\simeq A_5$~\cite{Novichkov:2018nkm, Ding:2019xna}.

Models using multiple $S_4$ modular symmetries can be found at \cite{deMedeirosVarzielas:2019cyj, King:2019vhv}. In this framework of multiple modular symmetries, each one with its own modulus field, it is possible to use one for the charged leptons, one or two for the neutrinos, while also breaking from these multiple symmetry groups to a single symmetry group at low energy, thus effectively explaining the use of multiple moduli together with a single modular symmetry group \cite{deMedeirosVarzielas:2019cyj}.

Before the mixing angles were observed experimentally with more precision, a commonly used mixing texture for the PMNS matrix was the tri-bimaximal mixing (TBM):
\begin{equation}
U_{TBM}=\begin{pmatrix}
\sqrt{\frac{2}{3}} & \sqrt{\frac{1}{3}} & 0\\
-\sqrt{\frac{1}{6}} & \sqrt{\frac{1}{3}} & -\sqrt{\frac{1}{2}}\\
-\sqrt{\frac{1}{6}} & \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{2}}
\end{pmatrix}.
\label{TBM matrix}
\end{equation}
This ansatz, ruled out since the measurement of non-zero $\theta_{13}$ mixing angle, remains an appealing leading order solution with no free parameters. Mixing schemes such as Tri-Maximal 1 (TM1) and Tri-Maximal 2 (TM2) preserve respectively the 1st and 2nd columns of tri-bimaximal mixing \cite{Albright:2010ap}, and remain viable.

\cite{deMedeirosVarzielas:2019cyj} presents a general mechanism of employing multiple modular symmetries to construct a high energy theory which is then broken to a low energy model with a single modular symmetry when their modulus fields gain different vacuum expectation values (VEVs) at fixed points of the modular symmetry (stabilisers). The preserved residual symmetries then lead to the realization of different mass textures in the charged lepton and neutrino sectors in modular flavour models without flavons.
These modular symmetries are thus able to generate all masses and mixing parameters for the leptons, using a much smaller set of free parameters.
As an example, a $S_4$ flavour model featuring TM1 mixing \cite{Varzielas:2012pa} is constructed in an elegant manner from multiple $S_4$ modular symmetries \cite{deMedeirosVarzielas:2019cyj}.

In this paper, we will use two $A_4$ modular symmetries to build models that lead to TM2 mixing due to two independent modulus stabilisers, similarly to the use of multiple $S_4$ modular symmetries in \cite{King:2019vhv,deMedeirosVarzielas:2019cyj}. We note that \cite{Novichkov:2018yse} employs a single $A_4$ modular symmetry and two moduli in a model leading to TM2 mixing, where neutrino mass arises through the effective Weinberg operator. In the models constructed here, we use the type I seesaw mechanism and justify the presence of two distinct moduli by starting with two $A_4$ symmetries which are subsequently broken to the diagonal subgroup.

In Section \ref{mms} we briefly review the framework of multiple modular symmetries. In Section \ref{sec:A4res} we describe the $A_4$ modular symmetry and respective stabilisers associated to residual symmetries. In Section \ref{sec:models} we present two possible models for TM2 mixing. We conclude in Section \ref{sec:conclusion}.

\section{Flavour modular symmetries \label{mms}}

This section provides the definition of modular group and modular forms, and some fundamental aspects of constructing a realistic model with single and multiple modular symmetries, as in \cite{deMedeirosVarzielas:2019cyj}.

\subsection{Modular group and modular forms}

The modular group $\overline{\Gamma}$ is the group of linear fractional transformations $\gamma$ that act on the complex modulus $\tau$, for $\tau$ in the upper-half complex plane, i.e. $Im(\tau)>0$:
\begin{equation}
\gamma:\tau\rightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d},
\end{equation}
where $a,b,c,d$ are integers and satisfy $ad-bc=1$.

It is convenient to use $2 \times 2$ matrices to represent the elements of $\overline{\Gamma}$ as
\begin{equation}
\overline{\Gamma}=\left\{ \begin{pmatrix}
a&b\\
c&d
\end{pmatrix}/\{\pm 1\},~a,b,c,d\in \mathbb{Z}, ~ ad-bc=1 \right\}.
\label{modular group matrices}
\end{equation}

Note that, since $\gamma$ and $-\gamma$ are the same modular transformation, the  group $\overline{\Gamma}$ is isomorphic to $PSL(2,\mathbb{Z}) = SL(2,\mathbb{Z})/\mathbb{Z}_2$, where $SL(2,\mathbb{Z})$ is the group of $2 \times 2$ matrices with integer entries and determinant one.

The modular group has two generators, $S_{\tau}$ and $T_{\tau}$, which satisfy $S_{\tau}^2=(S_{\tau}T_{\tau})^3=1$. One possible choice for these generators is the following:
\begin{equation}
S_{\tau}:\tau\rightarrow-\frac{1}{\tau},~T_{\tau}:\tau\rightarrow\tau+1
\end{equation}
and their corresponding representations are
\begin{equation}
S_{\tau}=\begin{pmatrix}
0&1\\
-1&0
\end{pmatrix},~
T_{\tau}=\begin{pmatrix}
1&1\\
0&1
\end{pmatrix}.
\label{generators representation 2times2}
\end{equation}

It is possible to define subgroups $\overline{\Gamma}(N)$ of $\overline{\Gamma}$ modding out the entries of the representation matrices:
\begin{equation}
\overline{\Gamma}(N)=\left\{ \begin{pmatrix}
a&b\\
c&d
\end{pmatrix}\in PSL(2,\mathbb{Z}),~
\begin{pmatrix}
a&b\\
c&d
\end{pmatrix}=
\begin{pmatrix}
1&0\\
0&1
\end{pmatrix} ~(mod~\text{N})\right\}.
\end{equation}

Although the groups $\overline{\Gamma}(N)$ are discrete but infinite, the quotient groups $\Gamma_{N}=\overline{\Gamma}/\overline{\Gamma}(N)$ are finite, thus being called finite modular groups. For $N \leq 5$, these groups are isomorphic to well-known groups:
$\Gamma_2 \simeq S_3$,
$\Gamma_3 \simeq A_4$,
$\Gamma_4 \simeq S_4$,
$\Gamma_5 \simeq A_5$.
These finite modular groups can be obtained by imposing an additional condition, $T_{\tau}^N=1$, which implies that $\tau=\tau+N$.

Modular forms of weight $2k$ and level $N$ are holomorphic functions of $\tau$ that transform
under $\overline{\Gamma}(N)$ in the following way:
\begin{equation}
f(\gamma\tau)=(c\tau+d)^{2k}f(\tau),~\gamma=
\begin{pmatrix}
a&b\\c&d
\end{pmatrix}
\in \overline{\Gamma}(N),
\end{equation}
where $k$ is a non-negative integer and $N$ is natural (we are only interested in even weights).
These modular forms are invariant under $\overline{\Gamma}(N)$,
up to the factor $(c\tau + d)^{2k}$, but they transform under the quotient group
$\Gamma_N$.

Modular forms of weight $2k$ and level $N$ span a linear space of finite dimension $\mathcal{M}_{2k}(\overline{\Gamma}(N))$. Here we are interested in the $A_4$ modular group, so it is useful to know that the dimension of the linear space of modular forms of weight $2k$ and level $3$ is $2k+1$.
It is possible to choose a basis in $\mathcal{M}_{2k}(\overline{\Gamma}(N))$ such
that the transformation of the modular forms under $\Gamma_N$ is described by a unitary representation $\rho$ of $\Gamma_N$:
\begin{equation}
f_i(\gamma\tau)=(c\tau+d)^{2k}\rho(\gamma)_{ij}f_j(\tau),~
\gamma=
\begin{pmatrix}
a&b\\c&d
\end{pmatrix}
\in\Gamma_N.
\end{equation}

\subsection{Models with a single modular symmetry}

Considering an N = 1 supersymmetric model invariant under a finite modular symmetry, the action in general takes the form
\begin{equation}
S=\int
d^4x d^2\theta d^2\overline{\theta}~
K(\phi_i,\overline{\phi}_i;\tau,\overline{\tau})+\left(
\int d^4xd^2\theta~W(\phi_i;\tau)+h.c.\right).
\end{equation}
Under $\Gamma_N$ the K\"ahler potential $K$ transforms at most by a K\"ahler transformation and the superpotential $W$ stays invariant:
\begin{align}
K(\phi_i,\overline{\phi}_i;\tau,\overline{\tau}) & \rightarrow
K(\phi_i,\overline{\phi}_i;\tau,\overline{\tau})+f(\phi_i;\tau)+\overline{f}(\overline{\phi}_i; \overline{\tau})\\
W(\phi_i;\tau) & \rightarrow W(\phi_i;\tau).
\end{align}

The superpotential is in general a function of the modulus $\tau$ and superfields $\phi_i$ and can be expanded as:
\begin{equation}
W(\phi_i;\tau)=\sum_n \sum_{\{i_1,...,i_n\}} \sum_{I_Y} ~
(Y_{I_Y}\phi_{i_1}...\phi_{i_n})_{\mathbf{1}}.
\end{equation}

We want the superpotential to be invariant under $\Gamma_N$. This is possible if we assume the couplings $Y_{I_Y}$ to be multiplet modular forms, and the superfields $\phi_i$ transforming as
\begin{align}
\phi_i(\tau) & \rightarrow \phi_i(\gamma\tau)=(c\tau+d)^{-2k_i}\rho_{I_i}(\gamma)\phi_i(\tau)\\
Y_{I_Y}(\tau) & \rightarrow
Y_{I_Y}(\gamma\tau)=(c\tau+d)^{2k_Y}\rho_{I_Y}(\gamma)Y_{I_Y}(\tau),
\end{align}
where $-2k_i$ is the modular weight of $\phi_i$, $I_i$ is the representation of $\phi_i$, $2k_Y$ is the modular weight of $Y_{I_Y}$, $I_Y$ is the representation of $Y_{I_Y}$ and $\rho_{I_i}(\gamma)$ and $\rho_{I_Y}(\gamma)$ are the unitary representation matrices of $\gamma$ with $\gamma\in\Gamma_N$.
For the superpotential to be invariant as wanted, the sum of the weights needs to equal zero, i.e. $k_Y=k_{i_1}+\ldots+k_{i_n}$, and the multiplication of the representations $\rho_{I_Y}\times\rho_{i_1}\times\ldots\times\rho_{i_n}$ has to contain an invariant singlet.

\subsection{Models with multiple modular symmetries}

It is also possible to consider a theory that has multiple modular symmetries, based on a series of M modular groups
$\overline{\Gamma}^1$,
$\overline{\Gamma}^2$, $\ldots$, $\overline{\Gamma}^M$, where the modulus field for each symmetry $\overline{\Gamma}^J$, $J=1,\ldots,M$, is denoted as $\tau_J$. The associated modular transformations take the form:
\begin{equation}
\gamma_J:\tau_J\rightarrow\gamma_J\tau_J=
\frac{a_J\tau_J+b_J}{c_J\tau_J+d_J}.
\end{equation}

A series of finite modular groups $\Gamma_{N_J}^J$ for $J = 1,\ldots,M$ can be obtained by modding out
an integer $N_J$ as done for only one modular group in the previous subsection and taking the quotient finite groups. Take into account that $N_J$ does not need to be identical to $N_{J'}$ for $J \neq J'$.

Consider again an N = 1 supersymmetric model invariant under multiple modular symmetries; the action is
extended to the form
\begin{equation}
S=\int
d^4x d^2\theta d^2\overline{\theta}~
K(\phi_i,\overline{\phi}_i;\tau_1,\ldots,\tau_M,\overline{\tau}_1,\ldots,\overline{\tau}_M)+\left(
\int d^4xd^2\theta~W(\phi_i;\tau_1,\ldots,\tau_M)+h.c.\right).
\end{equation}

In this case, the superpotential expansion in powers of the superfields takes the form
\begin{equation}
W(\phi_i;\tau_1,\ldots,\tau_M)=\sum_n \sum_{\{i_1,\ldots,i_n\}} \sum_{(I_{Y,1},\ldots,I_{Y,M})}~
(Y_{(I_{Y,1},\ldots,I_{Y,M})}\phi_{i_1}\ldots\phi_{i_n})_{\mathbf{1}}
\end{equation}
and under any finite modular transformation $\gamma_1,\ldots,\gamma_M$ in $\Gamma^1_{N_1}\times \Gamma^2_{N_2} \times \ldots \times \Gamma^M_{N_M}$ the superfields and multiplet couplings transform as
\begin{align}
\phi_i(\tau_1,\ldots,\tau_M) & \rightarrow \phi_i(\gamma_1\tau_1,\ldots,\gamma_M\tau_M)\nonumber\\
=&\prod_{J=1,\ldots,M} (c_J\tau_J+d_J)^{-2k_{i,J}}
\bigotimes_{J=1,\ldots,M}\rho_{I_{i,J}}(\gamma_J)~\phi_i(\tau_1,\ldots,\tau_M)\\
Y_{(I_{Y,1},\ldots,I_{Y,M})}(\tau_1,\ldots,\tau_M) & \rightarrow
Y_{(I_{Y,1},\ldots,I_{Y,M})}(\gamma_1\tau_1,\ldots,\gamma_M\tau_M)\nonumber\\
=&\prod_{J=1,\ldots,M}
(c_J\tau_J+d_J)^
{2k_{Y,J}}
\bigotimes_{J=1,\ldots,M}\rho_{I_{Y,J}}(\gamma_J)~Y_{(I_{Y,1},\ldots,I_{Y,M})}(\tau_1,\ldots,\tau_M).
\label{transform superfields and couplings for multi fields}
\end{align}

As discussed previously, for the superpotential to be invariant, $k_{Y,J}=k_{i_{1,J}}+\ldots+k_{i_{n,J}}$, and $\rho_{I_{Y,J}}\times\rho_{i_{1,J}}\times\ldots\times\rho_{i_{n,J}}$ must contain an invariant singlet, for $J=1,\ldots,M$.

\section{Modular \texorpdfstring{$A_4$}{A4} symmetry and residual symmetries \label{sec:A4res}}

The models present in this paper use two $A_4$ modular symmetries, so the following subsection will deal with the $A_4$ symmetry group in particular. The stabilisers which are included here apply for the specific case of $A_4$ modular symmetries and, as well as the stabilisers for the modular groups from $N=2$ to 5, can be found in \cite{deMedeirosVarzielas:2020kji} (we note also the stabilizers or fixed points for $N=3,4$ were presented in \cite{Ding:2019gof}). The directions at the stabilisers can also be found in \cite{Novichkov:2018yse}, although the factors for the modular forms were corrected here.

\subsection{Modular \texorpdfstring{$A_4$}{A4} symmetry and modular forms of level 3}

The group $A_4$ is the group of even permutations of 4 objects and has 12 elements. It is generated by two operators $S_{\tau}$ and $T_{\tau}$ obeying
\begin{equation}
S_{\tau}^2=(S_{\tau}T_{\tau})^3=T_{\tau}^3=1.
\end{equation}

This group has three singlets and one triplet as its irreducible representations and the multiplication rules and other properties can be found in Appendix \ref{app:A4}.
In the so-called complex basis (basis where $T_{\tau}$ is diagonal), the $A_4$ generators in the triplet representation are
\begin{equation}
\rho_{\textbf{3}}(S_{\tau})=\frac{1}{3}
\begin{pmatrix}
-1&2&2\\
2&-1&2\\
2&2&-1
\end{pmatrix}~~\text{and}~~
\rho_{\textbf{3}}(T_{\tau})=
\begin{pmatrix}
1&0&0\\
0&\omega&0\\
0&0&\omega^2
\end{pmatrix}.~~\omega=e^{i2\pi/3}.
\label{triplet representations}
\end{equation}

The flavour models we build here employ $A_4$ as a modular symmetry group and the Yukawa couplings are going to be modular forms.
The three linearly independent weight 2 modular forms of level 3 $Y_{1,2,3}^{(2)}$ form a triplet of $A_4$ and can be expressed in terms of the Dedekind eta functions (see Appendix \ref{modular forms Dedekind}).
The modular forms of higher weight can be generated starting from these modular forms of weight 2.
For example, the five linearly independent weight 4 modular forms decompose into a triplet \textbf{3} and two singlets \textbf{1} and \textbf{1$'$}.
Using the weight 2 modular forms, one obtains the weight 4 modular forms:
\begin{equation}
Y_{\textbf{3}}^{(4)}
=\frac{2}{3}
\begin{pmatrix}
(Y_1^{(2)})^2-Y_2^{(2)}Y_3^{(2)}\\
(Y_3^{(2)})^2-Y_1^{(2)}Y_2^{(2)}\\
(Y_2^{(2)})^2-Y_1^{(2)}Y_3^{(2)}
\end{pmatrix}
\end{equation}
and
\begin{equation}
Y_{\textbf{1}}^{(4)}=
(Y_1^{(2)})^2+2Y_2^{(2)}Y_3^{(2)},~~
Y_{\textbf{1}'}^{(4)}=
(Y_3^{(2)})^2+2Y_1^{(2)}Y_2^{(2)}.
\end{equation}
The singlet \textbf{1$''$} vanishes because $Y_\textbf{3}^{(2)}(\tau)$ satisfy the constraint
\begin{equation}
Y_{\textbf{1}''}^{(4)}=(Y_2^{(2)})^2+2Y_1^{(2)}Y_3^{(2)}=0.
\end{equation}

Furthermore, the modular forms of weight 6, whose linear space has dimension 7 and decomposes into 2 triplets and 1 singlet, are \cite{Feruglio:2017spp}:
\begin{align}
Y_{{\textbf{3}_1}}^{(6)}&=
\begin{pmatrix}
(Y_1^{(2)})^3+2Y_1^{(2)}Y_2^{(2)}Y_3^{(2)}\\
(Y_1^{(2)})^2Y_2^{(2)}+2(Y_2^{(2)})^2Y_3^{(2)}\\
(Y_1^{(2)})^2Y_3^{(2)}+2(Y_3^{(2)})^2Y_2^{(2)}
\end{pmatrix}\\
Y_{{\textbf{3}_2}}^{(6)}&=
\begin{pmatrix}
(Y_3^{(2)})^3+2Y_1^{(2)}Y_2^{(2)}Y_3^{(2)}\\
(Y_3^{(2)})^2Y_1^{(2)}+2(Y_1^{(2)})^2Y_2^{(2)}\\
(Y_3^{(2)})^2Y_2^{(2)}+2(Y_2^{(2)})^2Y_1^{(2)}
\end{pmatrix}
\end{align}
and
\begin{equation}
Y_{\textbf{1}}^{(6)}=
(Y_1^{(2)})^3+(Y_2^{(2)})^3+(Y_3^{(2)})^3-3Y_1^{(2)}Y_2^{(2)}Y_3^{(2)},
\end{equation}
and again the other triplet that we are able to construct vanishes:
\begin{equation}
Y_{{\textbf{3}_3}}^{(6)}=
\begin{pmatrix}
(Y_2^{(2)})^3+2Y_1^{(2)}Y_2^{(2)}Y_3^{(2)}\\
(Y_2^{(2)})^2Y_3^{(2)}+2(Y_3^{(2)})^2Y_1^{(2)}\\
(Y_2^{(2)})^2Y_1^{(2)}+2(Y_1^{(2)})^2Y_3^{(2)}
\end{pmatrix}=0.
\end{equation}
For the second model, the following weight 8 modular forms will also be useful: an invariant triplet:
\begin{equation}
Y_{\textbf{1}}^{(8)}=(Y_1^{(2)})^4+4(Y_1^{(2)})^2Y_2^{(2)}Y_3^{(2)} + 4(Y_2^{(2)})^2(Y_3^{(2)})^2
\label{eq:mf singlet 8}
\end{equation}
and one of the triplets:
\begin{align}
Y_{{\textbf{3}}}^{(8)}=
\begin{pmatrix}
(Y_1^{(2)})^4+(Y_1^{(2)})^2Y_2^{(2)}Y_3^{(2)} - 2(Y_2^{(2)})^2(Y_3^{(2)})^2\\
(Y_1^{(2)})^2(Y_3^{(2)})^2 - (Y_1^{(2)})^3Y_2^{(2)} - 2 Y_1^{(2)}(Y_2^{(2)})^2Y_3^{(2)} + 2 Y_2^{(2)}(Y_3^{(2)})^3\\
(Y_1^{(2)})^2(Y_2^{(2)})^2 - (Y_1^{(2)})^3Y_3^{(2)} - 2 Y_1^{(2)}Y_2^{(2)}(Y_3^{(2)})^2 + 2 (Y_2^{(2)})^3Y_3^{(2)}
\end{pmatrix}.
\label{eq:mf triplet 8}
\end{align}

\subsection{Stabilisers and residual symmetries of modular \texorpdfstring{$A_4$}{A4}}

Stabilisers of the symmetry play a crucial role in residual symmetries. Given an element $\gamma$ in the modular group $A_4$, a stabiliser of $\gamma$ corresponds to a fixed point in the upper half complex plane that transforms as $\gamma\tau_{\gamma}=\tau_{\gamma}$. Once the modular field acquires a VEV at this special point,
$\langle\tau\rangle=\tau_{\gamma}$, the modular symmetry is broken but an Abelian residual modular symmetry generated by $\gamma$ is preserved. Obviously, acting $\gamma$ on the modular form at its stabiliser leaves the modular form invariant:
\begin{equation}
\gamma:Y_I(\tau_{\gamma})\rightarrow Y_I(\gamma\tau_{\gamma})=
Y_I(\tau_{\gamma}),
\end{equation}
which implies that
\begin{equation}
\rho_I(\gamma)Y_I(\tau_{\gamma})=
(c\tau_{\gamma}+d)^{-2k}Y_I(\tau_{\gamma}).
\end{equation}

This means that, at the stabiliser, the modular form is an eigenvector of the representation matrix $\rho_\textbf{3}(\gamma)$ for the given stabiliser that corresponds to the eigenvalue $(c\tau_{\gamma}+d)^{-2k}$, and thus the directions of the modular forms at the stabilisers can be easily determined.
Furthermore, since the representation matrix is unitary, $|c\tau_{\gamma}+d|=1$.

The stabilisers for the $A_4$ modular group are shown in TABLE \ref{table:stabilisers} \cite{deMedeirosVarzielas:2020kji}.
\begin{table}[ht]
\centering
\begin{tabular}{c|c}
\toprule
$\gamma$ & $\tau_{\gamma}$\\ [0.5ex]
\colrule
$T_{\tau}, T_{\tau}^2$ & $i\infty,~ \frac{3}{2}+\frac{i}{2\sqrt{3}}$ \\
$S_{\tau}T_{\tau}, T_{\tau}^2S_{\tau}$ & $1,~ -\frac{1}{2}+\frac{i\sqrt{3}}{2}$ \\
$T_{\tau}S_{\tau}T_{\tau}, S_{\tau}T_{\tau}S_{\tau}$ & $0,~ \frac{3}{2}+\frac{i\sqrt{3}}{2}$ \\
$T_{\tau}S_{\tau}, S_{\tau}T_{\tau}^2$ & $-1,~ \frac{1}{2}+\frac{i\sqrt{3}}{2}$ \\
$T_{\tau}^2S_{\tau}T_{\tau}$ & $-1+i,~ \frac{1}{2}+\frac{i}{2}$ \\
$S_{\tau}$ & $i,~ \frac{3}{2}+\frac{i}{2}$ \\
$T_{\tau}S_{\tau}T_{\tau}^2$ & $1+i,~ -\frac{1}{2}+\frac{i}{2}$ \\ [1ex]
\botrule
\end{tabular}
\caption{Stabilisers for the $A_4$ elements \cite{deMedeirosVarzielas:2020kji}.}
\label{table:stabilisers}
\end{table}

For the transformations $S_{\tau}$, $T_{\tau}$, $S_{\tau}T_{\tau}$ and $T_{\tau}S_{\tau}$, the coefficients $(c\tau_{\gamma}+d)^{-2k}$ are
\begin{equation}
(c \tau_{\gamma} + d)^{-2k} =
\left\{
\begin{array}{ll}
      (-1)^{k} & \gamma = S_{\tau}, \tau_{S_{\tau 1}} = i \text{ or } \tau_{S_{\tau 2}} = \frac{3}{2} + \frac{i}{2} \\
      1 & \gamma = T_{\tau}, \tau_{T_{\tau 1}} = i\infty \\
      \omega^{2k} & \gamma = T_{\tau}, \tau_{T_{\tau 2}} = \frac{3}{2} + \frac{i}{2\sqrt{3}} \\
      \omega^{2k} & \gamma = S_{\tau}T_{\tau}, \tau_{S_{\tau}T_{\tau}} = \omega \\
      \omega^{2k} & \gamma = T_{\tau}S_{\tau}, \tau_{T_{\tau}S_{\tau}} = -\omega^2 \\
\end{array}
\right. .
\label{stabilisers eigenvalues}
\end{equation}

The directions of the modular forms of weight $2k=2$, $4$ and $6$ for the stabilisers of these four elements are shown in TABLE \ref{table:directions}. Additionally, we include the factors for each modular form (\cite{Novichkov:2018yse} presents the directions for the modular forms of weight 2 and 4, although the factors have been corrected here for the weight 4 triplets). These factors are written in function of $Y$, which is defined in general as the first component $Y_1^{(2)}$ of $Y_\textbf{3}^{(2)}$, except for $\tau_{T_{\tau 2}}=\frac{3}{2}+\frac{i}{2\sqrt{3}}$, when we define it as the third component $Y_3^{(2)}$ of that triplet since the first component happens to vanish. For $Y$, the definitions for the weight 2 modular forms present in Appendix \ref{modular forms Dedekind} were used. The values the modular form singlets of weight 4 and 6 take at the stabilisers are also included.

The other two stabilisers for $S_{\tau}T_{\tau}$ and $T_{\tau}S_{\tau}$ were not considered since the modular forms approach infinity for these two values of the modulus field.
Notice that the two stabilisers of $S_\tau$ and $T_\tau$ stabilise these modular transformations but for different, although equivalent, representations in terms of $2\times2$ matrices, and thus for different $c$ and $d$. This explains the different eigenvalues obtained for the two stabilisers of $T_\tau$. However, for $S_\tau$ the eigenvalues happen to be the same for both stabilisers. Nevertheless, the directions for these two stabilisers of $S_\tau$ are indeed different. In this case the difference comes from the existence of two eigenvectors for the same eigenvalue, eigenvectors that are introduced in the example that follows.

\begin{table}[t!]
\centering
\resizebox{.98\hsize}{!}{
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\toprule
\multirow{2}{*}{$\tau_\gamma$} & weight 2 & \multicolumn{3}{c|}{weight 4} & \multicolumn{3}{c|}{weight 6} & \multirow{2}{*}{$Y$}\\
\cline{2-8}
& \textbf{3} & \textbf{3} & \textbf{1} & \textbf{1$'$} & \textbf{3}$_1$ & \textbf{3}$_2$ & \textbf{1} & \\
\colrule
\multicolumn{1}{l|}{$\tau_{S_{\tau}T_{\tau}}=-\frac{1}{2}+\frac{i\sqrt{3}}{2}$} & $Y\begin{pmatrix}1\\\omega\\-\frac{1}{2}\omega^2\end{pmatrix}$ & $Y^2\begin{pmatrix}1\\-\frac{1}{2}\omega\\\omega^2\end{pmatrix}$ & 0 & $\frac{9}{4}\omega Y^2$ & $0$ & $-\frac{9}{8} Y^3 \begin{pmatrix}1\\-2\omega\\-2\omega^2\end{pmatrix}$ & $\frac{27}{8}Y^3$ & $0.94867\ldots$ \\
\multicolumn{1}{l|}{$\tau_{T_{\tau}S_{\tau}}=\frac{1}{2}+\frac{i\sqrt{3}}{2}$} & $Y\begin{pmatrix}1\\\omega^2\\-\frac{1}{2}\omega\end{pmatrix}$ & $Y^2\begin{pmatrix}1\\-\frac{1}{2}\omega^2\\\omega\end{pmatrix}$ & 0 & $\frac{9}{4}\omega^2Y^2$ & 0 & $-\frac{9}{8} Y^3 \begin{pmatrix}1\\-2\omega^2\\-2\omega\end{pmatrix}$ & $\frac{27}{8}Y^3$ & $0.94867\ldots$ \\
\multicolumn{1}{l|}{$\tau_{S_{\tau 1}}=i$} & $Y\begin{pmatrix}1\\1-\sqrt{3}\\-2+\sqrt{3}\end{pmatrix}$ & $(4-2\sqrt{3})Y^2\begin{pmatrix}1\\1\\1\end{pmatrix}$ & $(6\sqrt{3}-9)Y^2$ & $-(6\sqrt{3}-9)Y^2$ & $(6\sqrt{3}-9)Y^3\begin{pmatrix}1\\1-\sqrt{3}\\-2+\sqrt{3}\end{pmatrix}$ & $(21\sqrt{3}-36)Y^3\begin{pmatrix}1\\-2-\sqrt{3}\\1+\sqrt{3}\end{pmatrix}$ & $0$ &$1.02253\ldots$ \\
\multicolumn{1}{l|}{$\tau_{S_{\tau 2}}=\frac{3}{2}+\frac{i}{2}$} & $Y\begin{pmatrix}1\\1+\sqrt{3}\\-2-\sqrt{3}\end{pmatrix}$ & $(4+2\sqrt{3})Y^2\begin{pmatrix}1\\1\\1\end{pmatrix}$ & $-(6\sqrt{3}+9)Y^2$ & $(6\sqrt{3}+9)Y^2$ & $-(6\sqrt{3}+9)Y^3\begin{pmatrix}1\\1+\sqrt{3}\\-2-\sqrt{3}\end{pmatrix}$ & $-(21\sqrt{3}+36)Y^3\begin{pmatrix}1\\-2+\sqrt{3}\\1-\sqrt{3}\end{pmatrix}$ & $0$ & $0.54798\ldots$ \\
\multicolumn{1}{l|}{$\tau_{T_{\tau 1}}=i\infty$} & $Y\begin{pmatrix}1\\0\\0\end{pmatrix}$ & $\frac{2}{3} Y^2\begin{pmatrix}1\\0\\0\end{pmatrix}$ & $Y^2$ & $0$ & $Y^3\begin{pmatrix}1\\0\\0\end{pmatrix}$ & $0$ &$Y^3$ & $1$ \\
\multicolumn{1}{l|}{$\tau_{T_{\tau 2}}=\frac{3}{2}+\frac{i}{2\sqrt{3}}$} & $Y\begin{pmatrix}0\\0\\1\end{pmatrix}$ & $\frac{2}{3} Y^2\begin{pmatrix}0\\1\\0\end{pmatrix}$ & 0 & $Y^2$ & $0$ & $Y^3\begin{pmatrix}1\\0\\0\end{pmatrix}$ & $Y^3$ & $-4.26903\ldots$ \\
\botrule
\end{tabular}}}
\caption{Directions for the modular forms of weight 2, 4 and 6 of level 3 for 4 $A_4$ elements.}
\label{table:directions}
\end{table}

For $S_{\tau}:\tau\rightarrow-1/\tau$, and using the stabiliser $\tau_{S_{\tau 1}}=i$, which stabilisers the modular transformation represented by the $2\times2$ matrix in (\ref{generators representation 2times2}), the expression for the modular form at the stabiliser is
\begin{equation}
\rho_{\textbf{3}}(S_{\tau})Y_{\textbf{3}}^{(2k)}(\tau_{S_{\tau 1}})
=(-\tau_{S_{\tau 1}})^{-2k}Y_{\textbf{3}}^{(2k)}(\tau_{S_{\tau 1}})=
(-i)^{-2k}Y_{\textbf{3}}^{(2k)}(\tau_{S_{\tau 1}})=
(-1)^{k}Y_{\textbf{3}}^{(2k)}(\tau_{S_{\tau 1}}),
\end{equation}
and thus we obtain its directions from the eigenvectors of the representation matrix for S ((\ref{triplet representations})) corresponding to the eigenvalue in the previous equation:
\begin{align}
Y_{\textbf{3}}^{(2k)}(\tau_{S_{\tau 1}}) & = y_{\tau_{S_{\tau 1}},1}^{(2k)}
\begin{pmatrix}
1\\ 1 \\ -2
\end{pmatrix} +
y_{\tau_{S_{\tau 1}},2}^{(2k)}
\begin{pmatrix}
0\\ -1 \\ 1
\end{pmatrix}, ~k=1~(mod~2)\\
Y_{\textbf{3}}^{(2k)}(\tau_{S_{\tau 1}}) & = y_{\tau_{S_{\tau 1}}}^{(2k)}
\begin{pmatrix}
1 \\ 1 \\ 1
\end{pmatrix}, ~k=2~(mod~2).
\end{align}

For the lowest weights that appear in TABLE \ref{table:directions}, $y_{\tau_{S_{\tau 1}},1}^{(2)}=Y$ and $y_{\tau_{S_{\tau 1}},2}^{(2)}=\sqrt{3} Y$, and from the definitions of the modular forms of higher weight in terms of those of weight 2 we have that $y_{\tau_{S_{\tau 1}}}^{(4)}=(4-2\sqrt{3}) Y^2$ and the factors for the two triplets with weight 6 are obtained similarly.

\section{Models with two modular \texorpdfstring{$A_4$}{A4} symmetries \label{sec:models}}

Models that consider the symmetry breaking from multiple modular symmetries groups to a single symmetry group at low energy have already been constructed using $S_4$ symmetry groups in order to obtain TM1 mixing \cite{deMedeirosVarzielas:2019cyj,King:2019vhv}. In this section, we construct two models that consider two $A_4$ modular symmetries in order to obtain TM2 mixing, using different weights for the fields and modular forms in each model. Although the current experimental evidence excludes TBM mixing, TM2 remains a viable and appealing scheme for lepton mixing.
At high energies, the model is based in two modular symmetries, $A_4^l$ and $A_4^{\nu}$, with modulus fields denoted by $\tau_l$ and $\tau_{\nu}$, respectively. After the modulus fields acquire different VEVs, different mass textures are realised in the charged lepton and neutrino sectors.

\subsection{\texorpdfstring{$A_4^l \times A_4^{\nu} \rightarrow A_4$ breaking}{A4l x A4nu breaking to A4} \label{sec:symmetry breaking}}

First, we start by discussing how the symmetry breaking from two independent $A_4^l \times A_4^{\nu}$ to a single $A_4$ is achieved (this mechanism is shared by both models).
The superfields considered for these models are $L$, which is a doublet of SU(2)$_L$ containing the left-handed leptons and a triplet under $A_4^l$, $\nu^c$, which is a triplet under $A_4^{\nu}$ containing the conjugate of the right-handed neutrino fields added to the Standard Model, and $H_u$, an additional Higgs doublet as required in Supersymmetric models. A bi-triplet $\Phi$, which is a triplet under both $A_4^l$ and $A_4^\nu$, is introduced. $Y_{\nu}$ represents the Yukawa couplings that in the case of modular symmetries should be modular forms. One model considers a weight zero modular form (i.e. a modular field independent constant), and the other a singlet and a triplet under $A_4^{\nu}$.

We consider that neutrinos get their mass through the type 1 see-saw mechanism and the term from the superpotential that gives rise to a Dirac mass matrix is $\frac{1}{\Lambda} L \Phi Y_{\nu} \nu^c H_u$. This term is an effective term that can arise from renormalizable interactions of the fields shown with heavy messengers (not shown explicitly - a possibility for the messenger is an electroweak neutral field).
Considering the multiplication rules for two triplets to get a trivial singlet, the term $\frac{1}{\Lambda} L \Phi Y_{\nu} \nu^c H_u$ can be explicitly expanded as:
\begin{equation}
\frac{1}{\Lambda}(L_1,L_2,L_3)P_{23}
\begin{pmatrix}
\Phi_{11}&\Phi_{12}&\Phi_{13}\\
\Phi_{21}&\Phi_{22}&\Phi_{23}\\
\Phi_{31}&\Phi_{32}&\Phi_{33}
\end{pmatrix}
P_{23}Y_{\nu}(\tau_{\nu})\otimes
\begin{pmatrix}
\nu^c_1\\\nu^c_2\\\nu^c_3
\end{pmatrix}H_u,
\end{equation}
where $Y_{\nu}\otimes\nu^c$ is the product between $Y_{\nu}$ and $\nu^c$ that gives a triplet of $A_4^{\nu}$, and $P_{23}$ is the matrix that permutes the second and third columns/rows:
\begin{equation}
P_{23}=\begin{pmatrix}
1&0&0\\
0&0&1\\
0&1&0
\end{pmatrix}.
\end{equation}

If $\Phi$ acquires the VEV $\langle\Phi\rangle = v_{\Phi}P_{23}$ (see Appendix \ref{vac_aligns}), the symmetry $A_4^l\times A_4^{\nu}$ is broken but given that the same transformation $\gamma$ can be performed in $A_4^l$ and $A_4^{\nu}$ simultaneously, there is still a single modular symmetry $A_4$ that is conserved (the diagonal subgroup). Under this symmetry, a modular transformation takes the form
\begin{equation}
\gamma : (\tau_l,\tau_{\nu}) \rightarrow (\gamma \tau_l,\gamma\tau_{\nu}) = \left( \frac{a\tau_l+b}{c\tau_l+d}, \frac{a\tau_{\nu}+b}{c\tau_{\nu}+d} \right), \gamma \in A_4.
\end{equation}
The term $\frac{1}{\Lambda}L\Phi Y_{\nu} \nu^c H_u$ gets the form $\frac{v_{\Phi}}{\Lambda} (L Y_{\nu}\nu^c)_\textbf{1} H_u$, which implies a Dirac matrix term for the neutrinos when the Higgs $H_u$ acquires a VEV.

\subsection{Model 1}

The first model we consider is a model were the Yukawa coupling $Y_{\nu}$ is simply a constant.
The transformation properties of fields, Yukawa couplings and masses for this model are in TABLE \ref{transformationp_model1}.

\begin{table}[b]
\centering
\begin{tabular}[t]{c||c|c|c|c|c}
\toprule
Fields & $SU(2)$ & $A_4^l$ & $A_4^{\nu}$ & $2k_l$ & $2k_{\nu}$ \\ \colrule
$L$ & \textbf{2} & \textbf{3} & \textbf{1} & $0$ & $-2$ \\
$e^c$ & \textbf{1} & \textbf{1} & \textbf{1} & $+6$ & $+2$ \\
$\mu^c$ & \textbf{1} & \textbf{1$''$} & \textbf{1} & $+6$ & $+2$ \\
$\tau^c$ & \textbf{1} & \textbf{1$'$} &  \textbf{1} & $+6$ & $+2$ \\
$\nu^c$ & \textbf{1} & \textbf{1} & \textbf{3} & $0$ & $+2$ \\
$H_{u,d}$ & \textbf{2} & \textbf{1} & \textbf{1} & $0$ & $0$ \\
$\Phi$ & \textbf{1} & \textbf{3} & \textbf{3} & $0$ & $0$ \\ \botrule
\end{tabular}~~
\begin{tabular}[t]{c||c|c|c|c}
\toprule
Yukawas/Masses & $A_4^l$ & $A_4^{\nu}$ & $2k_l$ & $2k_{\nu}$ \\ \colrule
$Y_l$ & \textbf{3} & \textbf{1} & $+6$ & $0$ \\
$Y_{\nu}$ & \textbf{1} & \textbf{1} & $0$ & $0$ \\
$M_\textbf{1}$ & \textbf{1} & \textbf{1} & $0$ & $+4$ \\
$M_\textbf{1$'$}$ & \textbf{1} & \textbf{1$'$} & $0$ & $+4$ \\
$M_\textbf{3}$ & \textbf{1} & \textbf{3} & $0$ & $+4$ \\ \botrule
\end{tabular}
\caption{Transformation properties of fields, Yukawa couplings and masses for the right-handed neutrinos for model 1.}
\label{transformationp_model1}
\end{table}

The Yukawa coefficients for the charged leptons are a modular form which transforms as a triplet of $A_4^l$ with weight $2k_l=+6$, whereas $Y_{\nu}$ is simply a modulus independent constant, a modular form of weight $0$. For the right-handed neutrino masses we consider three modular forms transforming under $A_4^{\nu}$ as a trivial singlet \textbf{1} $M_{\textbf{1}}$, as a singlet \textbf{1$'$} $M_{\textbf{1$'$}}$ and as a triplet \textbf{3} $M_\textbf{3}$, all with weights $2k_{\nu}=+4$. The weights were chosen in such a way that the modular forms acquire the desired directions as we show below.

The right-handed electron, muon and tau fields are respectively singlets \textbf{1}, \textbf{1$'$} and \textbf{1$''$} of $A_4^l$ and trivial singlets \textbf{1} of $A_4^{\nu}$, with weights $2k_l=+6$ and $2k_{\nu}=+2$. The lepton doublets $L$ are arranged as a triplet of $A_4^l$ and a singlet of $A_4^{\nu}$, with weights $2k_l=0$ and $2k_{\nu}=-2$. In this model, the three right-handed neutrinos introduced form a triplet of $A_4^{\nu}$ with weight $2k_{\nu}=+2$.
These are the correct choices for the weights such that the modular forms and fields in each term sum up to zero since the weight for the fields is not $2k$, which are the values that were introduced in this section, but $-2k$ instead (recall the transformation relations for the modular forms and the superfields, (\ref{transform superfields and couplings for multi fields}), and how the signs of the exponents where the weights enter differ).

Note that, in spite of the charged leptons only having non-trivial singlet transformations under $A_4^l$ and the right-handed neutrinos only under $A_4^\nu$ (which justifies the nomenclature used), the respective weights introduce non-trivial transformations under both modular symmetries for these fields.

With the fields assigned in this manner, the superpotential for this model, which can be separated into one part containing the mass terms for the charged leptons and the other the neutrino mass terms, has the following form:
\begin{align}
w&=w_e+w_{\nu},\\
w_e &= \left( \alpha LY_l(\tau_l)e^c + \beta LY_l(\tau_l) \mu^c + \gamma LY_l(\tau_l) \tau^c \right) H_d ,\\
w_{\nu} &= \frac{Y_{\nu}}{\Lambda} L\Phi \nu^c H_u +
\frac{1}{2}M_\textbf{1}(\tau_{\nu})(\nu^c \nu^c)_\textbf{1} +
\frac{1}{2}M_\textbf{1$'$}(\tau_{\nu})(\nu^c \nu^c)_\textbf{1$''$} +
\frac{1}{2}M_\textbf{3}(\tau_{\nu})(\nu^c \nu^c)_\textbf{3}.
\end{align}

The bi-triplet $\Phi$ will then acquire a VEV as presented in Section \ref{sec:symmetry breaking}, and the two modular symmetries are broken to a single $A_4$, getting for $w_\nu$ (the $w_e$ terms remain exactly the same):
\begin{align}
w_{\nu} &= y_D(L\nu^c)_\textbf{1}H_u +
\frac{1}{2}M_\textbf{1}(\tau_{\nu})(\nu^c \nu^c)_\textbf{1} +
\frac{1}{2}M_\textbf{1$'$}(\tau_{\nu})(\nu^c \nu^c)_\textbf{1$''$} +
\frac{1}{2}M_\textbf{3}(\tau_{\nu})(\nu^c \nu^c)_\textbf{3},
\end{align}
where $y_D=Y_{\nu}v_{\Phi}/\Lambda$.

The flavour structure after \texorpdfstring{$A_4$}{A4} symmetry breaking now follows.
We assume that the charged lepton modular field $\tau_l$ acquires the VEV $\langle\tau_l\rangle=\tau_T=\frac{3}{2}+\frac{i}{2\sqrt{3}}$, which is a stabiliser of $T_\tau$.
At this stabiliser, a residual modular $Z_3^T$ symmetry is preserved in the charged lepton sector. This implies that the modular form $Y_l$, which has weight $+6$, has the direction
\begin{equation}
Y_l(\tau_l) \propto \begin{pmatrix} 1\\0\\0 \end{pmatrix}
\end{equation}
This direction leads to a diagonal charged lepton mass matrix when the Higgs field $H_d$ acquires a VEV $\langle H_d \rangle = (0,v_d+h(x))$:
\begin{equation}
m_e=v_d\begin{pmatrix}
\alpha & 0 & 0 \\
0 & \beta & 0 \\
0 & 0 & \gamma
\end{pmatrix}.
\label{mass matrix charged leptons}
\end{equation}
The masses for the charged leptons can be reproduced by adjusting the parameters $\alpha$, $\beta$ and $\gamma$. These constants were redefined to include the constant associated with $Y_l(\tau_l)$.

For the other modular field $\tau_{\nu}$, since we want to obtain the trimaximal mixing TM2,
which preserves the second column of the tri-bimaximal mixing matrix, the modular form $M_\textbf{3}$ should acquire the direction
\begin{equation}
M_\textbf{3}(\tau_\nu) \propto
\begin{pmatrix}
1\\1\\1
\end{pmatrix},
\end{equation}
which occurs for the VEV $\langle \tau_{\nu} \rangle = \tau_S = i$, and thus it should have an even $k$, as happens for $2k_\nu=+4$.
In this case, a residual modular $Z_2^S$ symmetry is preserved in the neutrino sector.
This implies the following structure for the right-handed neutrino mass matrix:
\begin{equation}
M_R = c_\textbf{1} \begin{pmatrix} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{pmatrix} +
c_\textbf{1$'$} \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{pmatrix} +
\frac{c_\textbf{3}}{3} \begin{pmatrix} 2&-1&-1 \\ -1&2&-1 \\-1&-1&2
\end{pmatrix},
\end{equation}
where $c_{\textbf{1}},c_{\textbf{1$'$}},c_{\textbf{3}}$ are complex constants associated with the respective modular form.
The Dirac mass matrix that relates the right-handed and active neutrinos after the Higgs field $H_u$ acquires a VEV $\langle H_u \rangle = (0,v_u+h(x))$ is simply
\begin{equation}
M_D=y_D v_u P_{23}.
\end{equation}
Consequently, the active neutrino mass matrix for the seesaw mechanism gets the form
\begin{equation}
M_{\nu}= - M_D M_R^{-1} M_D^T = - y_D^2 v_u^2 P_{23} M_R^{-1} P_{23}.
\end{equation}

We want now to diagonalize $M_{\nu}$, such that $U^T M_{\nu} U = M_{\nu_d} = \text{diag}(m_1,m_2,m_3)$, where $m_i$ are the neutrino masses and $U$ is an unitary matrix. It is also true that $U^T M_{\nu} U = - U^T M_D M_R^{-1} M_D^T U = M_{\nu_d}$. So $M_D^T U$ also diagonalizes the matrix $M_R^{-1}$ and thus $V = M_D^{\dagger} U^*$ diagonalizes $M_R$ such that $V^T M_R V  = M_{R_d} = \text{diag}(M_1,M_2,M_3)$ where $M_i= - \frac{y_D^2 v_u^2}{m_i}$. Conversely, $U = M_D^* V^*$ when $V$ diagonalizes $M_R$.

In the present model, when we apply the tri-bimaximal matrix in (\ref{TBM matrix})
to the heavy neutrino mass matrix we obtain:
\begin{equation}
U_{TBM}^T M_R U_{TBM} = \begin{pmatrix}
a & 0 & c\\
0 & \frac{a - b}{2} + \sqrt{3}c & 0 \\
c & 0 & b
\end{pmatrix},
\label{eq:UMRU1}
\end{equation}
where $a = c_\textbf{3} + c_\textbf{1} - \frac{1}{2}c_\textbf{1$'$}$, $b = c_\textbf{3} - c_\textbf{1} + \frac{1}{2} c_\textbf{1$'$}$ and $c = \frac{\sqrt{3}}{2}c_\textbf{1$'$}$.
This matrix has only an element on the second row and second column and four elements on the corners that form a $2\times2$ symmetric matrix and so can be put into block diagonal form by permuting the first and second columns and rows.
Thus, the full matrix can be fully diagonalized adding a matrix $V_r$ that introduces a rotation among the first and third columns.
This rotation preserves the second column so $M_R$ is diagonalized by a TM2 mixing matrix, since this mixing matrix can be written as the product of the TBM mixing matrix and a rotation on the first and third columns.
For the present model, $M_D$ is only a permutation, so we have that, being $V = U_{TBM} V_r$ the matrix that diagonalizes $M_R$, the matrix that diagonalizes $M_{\nu}$ is $U = P_{23} U_{TBM} V_r$, which can also be written as $U_{TBM} U_r$, where $U_r$ is a rotation between the first and third columns. Using the parametrization
\begin{equation}
U_r = \begin{pmatrix}
\cos\theta e^{i \alpha_1} & 0 & \sin\theta e^{-i \alpha_2} \\
0 & e^{i \alpha_3} & 0 \\
-\sin\theta e^{i \alpha_2} & 0 & \cos\theta e^{- i \alpha_1}
\end{pmatrix},
\end{equation}
which implies that
\begin{equation}
V_r = \begin{pmatrix}
\cos\theta e^{- i \alpha_1} & 0 & \sin\theta e^{i \alpha_2}\\
0 & e^{- i \alpha_3} & 0 \\
\sin\theta e^{- i \alpha_2} & 0 & -\cos\theta e^{i \alpha_1}
\end{pmatrix},
\label{UTM2_for_Mnu}
\end{equation}
we are then able to diagonalize both $M_{\nu}$ and $M_R$. HEre, $\theta$ is the angle that governs the rotation and the three $\alpha_i$ are introduced such that $M_i$ are purely real values.

It is also possible to start from the diagonal matrix $M_{R_d}$ and get $U_{TBM}^T M_R U_{TBM}$. We have that
\begin{equation}
V_r^* M_{R_d} V_r^{\dagger} =
\begin{pmatrix}
M_1 \cos^2\theta e^{2 i \alpha_1} + M_3 \sin^2\theta e^{-2 i \alpha_2} & 0 & \frac{1}{2} (M_1 e^{i (\alpha_1+\alpha_2)} - M_3 e^{-i(\alpha_1+\alpha_2)})\sin2\theta \\
0 & M_2 e^{2i\alpha_3} & 0\\
\frac{1}{2} (M_1 e^{i (\alpha_1+\alpha_2)} - M_3 e^{-i(\alpha_1+\alpha_2)})\sin2\theta & 0 & M_1 \sin^2\theta e^{2 i \alpha_2} + M_3 \cos^2\theta e^{-2 i \alpha_1}
\end{pmatrix}
\label{eq:VMRdV1},
\end{equation}
and comparing with (\ref{eq:UMRU1}) we obtain that
$\alpha_2=\frac{1}{2} \arg\left(\frac{a - b}{2} + \sqrt{3}c\right)$ and, more importantly, we get a mass sum rule for $M_i$ that can also be expressed in terms of the active neutrino masses $m_i$:
\begin{align}
\frac{1}{m_2} & = -\frac{1}{y_D^2 v_u^2} \Big| \frac{a-b}{2} + \sqrt{3} c \Big| \nonumber \\
& = \Big| \frac{1}{2m_1} \left( e^{2i\alpha_1}\cos^2\theta-e^{2i\alpha_2}\sin^2\theta+\sqrt{3} e^{i(\alpha_1+\alpha_2)}\sin2\theta \right)  \nonumber\\
& ~~~~~ + \frac{1}{2m_3} \left( -e^{-2i\alpha_1}\cos^2\theta+e^{-2i\alpha_2}\sin^2\theta-\sqrt{3} e^{-i(\alpha_1+\alpha_2)}\sin2\theta \right) \Big|.
\label{sum_rule_model1}
\end{align}

The angles and phases from the standard parametrization of the PMNS matrix in \cite{Zyla:2020zbs} can be expressed in terms of the model parameters $\theta$, $\alpha_1$ and $\alpha_2$ using the expressions between the parameters and the PMNS matrix elements (these expressions are equivalent to the ones in \cite{Novichkov:2018yse})
\begin{align}
\label{sintheta13_relation} \sin^2\theta_{13} &= |U_{e3}|^2  = \frac{2\sin^2\theta}{3} \\
\label{sintheta12_relation} \sin^2\theta_{12} &= \frac{|U_{e2}|^2}{1-|U_{e3}|^2} = \frac{1}{3-2\sin^2\theta} \\
\label{sintheta23_relation} \sin^2\theta_{23} &= \frac{|U_{\mu3}|^2}{1-|U_{e3}|^2} = \frac{1}{2} + \frac{\sqrt{3}}{2} \frac{ \sin 2\theta}{2+\cos 2\theta}\cos (\alpha_1-\alpha_2) \\
\delta &= - \arg \left( \frac{ U_{e3} U_{\tau 1} U_{e1}^* U_{\tau 3}^*}{\cos\theta_{12} \sin\theta_{13} \cos^2\theta_{13} \cos\theta_{23}}
+ \cos\theta_{12} \sin\theta_{13} \cos\theta_{23} \right) \nonumber \\
\label{delta_relation} & = \arg \left(\left(e^{i (\alpha_1-\alpha_2)} \sin^2\theta - 3 e^{- i (\alpha_1-\alpha_2)} \cos^2\theta \right) \sin2\theta \right).
\end{align}

Using the $3\sigma$ C.L. range of $\sin\theta_{13}$ for NO(IO), $0.02034(0.02053)\rightarrow0.02430(0.02436)$ \cite{Esteban:2020cvm}, we obtain the allowed range for $\sin\theta$:
\begin{equation}
0.1747(0.1755)\lesssim|\sin\theta|\lesssim0.1909(0.1912),
\end{equation}
which implies also ranges for the other mixing angles  (using that $-1\leq\cos(\alpha_1-\alpha_2)\leq1$):
\begin{align}
0.3403 (0.3403) \lesssim \sin^2\theta_{12} \lesssim 0.3416 (0.3417) \\
0.3891 (0.3890) \lesssim \sin^2\theta_{23} \lesssim 0.6109 (0.6110).
\end{align}
The $1\sigma$ region is within the interval found for $\sin^2\theta_{23}$, which overlaps with the $3\sigma$ region for this parameter, with our result extending below 0.407(0.411) for NO(IO) and not reaching its upper limit. The range of allowed values for $\sin\theta_{12}$ is near the upper allowed limit, which is a characteristic feature of the TM2 mixing, since the lowest value allowed for $\sin \theta_{12}^2$ is $1/3$ as can be seen from (\ref{sintheta12_relation}).

The sum rule (\ref{sum_rule_model1}) and (\ref{sintheta13_relation}-\ref{delta_relation}) provide us relations between the observables and the parameters of the TM2 mixing, and hence provide what is needed to do a numerical minimization recurring to the $\chi^2$ function:
\begin{equation}
\chi^2 = \sum_i \left(\frac{P_i(\{x\})-BF_i}{\sigma_i}\right)^2,
\end{equation}
where $P_i$ are the values provided by the considered model, $BF$ the best fit value from NuFit \cite{Esteban:2020cvm} and $\sigma_i$ is also provided by NuFit, when averaging the upper and lower $\sigma$ provided.
For the fitting, the three mixing angles, the atmospheric and solar neutrino squared mass differences and the Dirac neutrino CP violation phase were considered.

The fit parameters obtained for normal ordering (NO) and inverted ordering (IO) of neutrino masses can be found in TABLE \ref{fitresults model1}. The best fit values lie inside the 1$\sigma$ range for all the observables except $\theta_{12}$, for both orderings near the upper limit of the 3$\sigma$ range, and $\delta$ for IO. Nonetheless, all the observables are within their $3\sigma$ intervals. The best-fit occurs for normal ordering of neutrino masses with a $\chi^2=1.57$.
\begin{table}[b!]
\centering
{\renewcommand{\arraystretch}{1.3}
\begin{tabular}{| c | c | c c c c c c c |}
\cline{1-9}
\multirow{4}{*}{NO} & \multirow{2}{*}{Para.} & & $\chi^2$/d.o.f. & $\theta$ & $\alpha_1$ & $\alpha_2$ & $m_1$ & $m_3$ \\
\cline{3-9}
& & & 1.57 & 10.51\degree & -67.60\degree & -24.26\degree & 0.0141 eV & 0.0521 eV \\
\cline{2-9}
& \multirow{2}{*}{Obs.} & $\theta_{12}$ & $\theta_{23}$ & $\theta_{13}$ & $\delta$ & $\Delta m_{21}^2$ & $\Delta m_{31}^2$ & $m_{\beta\beta}$\\
\cline{3-9}
& & 35.72\degree & 49.4\degree & 8.56\degree & 224\degree & 7.42$\times 10^{-5}$eV$^2$ & 2.514$\times 10^{-3}$eV$^2$ & 0.0131 eV\\
\cline{1-9}
\multirow{4}{*}{IO} &\multirow{2}{*}{Para.} & & $\chi^2$/d.o.f. & $\theta$ & $\alpha_1$ & $\alpha_2$ & $m_1$ & $m_3$ \\
\cline{3-9}
& & & 2.04 & 10.56\degree & -95.56\degree & -38.93\degree & 0.0546 eV & 0.0236 eV \\
\cline{2-9}
& \multirow{2}{*}{Obs.} & $\theta_{12}$ & $\theta_{23}$ & $\theta_{13}$ & $\delta$ & $\Delta m_{21}^2$ & $\Delta m_{32}^2$ & $m_{\beta\beta}$\\
\cline{3-9}
& & 35.73\degree & 48.4\degree & 8.61\degree & 237\degree & 7.42$\times 10^{-5}$eV$^2$ & -2.496$\times 10^{-3}$eV$^2$ & 0.0174 eV \\
\cline{1-9}
\end{tabular}}
\caption{Parameters (Para.) and observables (Obs.) for the best fit point for normal and inverted ordering for model 1.}
\label{fitresults model1}
\end{table}

It is also possible to obtain the expected $m_{\beta\beta}$ for neutrinoless beta decay using the formula
\begin{equation}
m_{\beta\beta} = |(M_{\nu})_{(1,1)}| = y_D^2 v_u^2 |(M_{R}^{-1})_{(1,1)}|.
\end{equation}
Doing a numerical computation, the allowed regions of $m_{\text{lightest}}$ vs $m_{\beta\beta}$ of FIG.\ref{fig:mee model1} (for NO, $m_{\text{lightest}} = m_1$ and for IO, $m_{\text{lightest}} = m_3$) were obtained, using again as constraints the data from \cite{Esteban:2020cvm}.
In both figures it is also shown the current upper limit provided by KamLAND-Zen, $m_{\beta\beta} < 61-165$ meV \cite{KamLAND-Zen:2016pfg}. Results from PLANCK 2018 constrain the sum of neutrino masses, although different constrains can be obtained depending on the data considered (for more details, see \cite{Aghanim:2018eyx}). In the figures are plotted two shadowed regions, a very disfavoured region $\sum m_i > 0.60$ eV (considering the limit 95\%C.L.,Planck lensing$+$BAO$+\theta_{MC}$) and a disfavoured region $\sum m_i>$ 0.12 eV (considering the limit 95\%C.L.,Planck TT,TE,EE$+$lowE$+$lensing$+$BAO$+\theta_{MC}$). These constraints on $\sum m_i$ can be expressed as constraints on $m_{\text{lightest}}$ using the best fit value for the squared mass differences: $m_{\text{lightest}}>0.198$ eV and $m_{\text{lightest}}>0.030$ eV for NO and $m_{\text{lightest}}>0.196$ eV and $m_{\text{lightest}}>0.016$ eV for IO, for the very disfavoured and the disfavoured regions respectively. We conclude then that both fits in TABLE \ref{fitresults model1} are in the disfavoured region.
\begin{figure}[t!]
     \centering
     \begin{subfigure}[t]{0.49\textwidth}
         \centering
         \includegraphics[width=\textwidth]{TM2_2A4_NO_model1.png}
		 \caption{Normal Ordering}
		 \label{fig:mee model1 NO}
     \end{subfigure}%
     ~
     \begin{subfigure}[t]{0.49\textwidth}
         \centering
         \includegraphics[width=\textwidth]{TM2_2A4_IO_model1.png}
         \caption{Inverted Ordering}
		 \label{fig:mee model1 IO}
     \end{subfigure}
        \caption{Predictions of $m_{\text{lightest}}$ vs $m_{\beta\beta}$ for both ordering of neutrino masses compatible with $1\sigma$ (dark-red, NO only, except $\theta_{12}$) and $3\sigma$ data from \cite{Esteban:2020cvm}. In both figures there were also included the current upper limit from KamLAND-Zen $m_{\beta\beta} < 61-165$ meV \cite{KamLAND-Zen:2016pfg} and cosmological constraints from PLANCK 2018 (disfavoured region 0.12 eV $ <\sum m_i <$ 0.60 eV and very disfavoured region $\sum m_i > 0.60$ eV) \cite{Aghanim:2018eyx}.}
        \label{fig:mee model1}
\end{figure}

For NO, there are some points compatible with the $1\sigma$ ranges of the observables other than $\theta_{12}$ (which is, as already said, always near the upper $3\sigma$ limit). These points were plotted with a darker red color.
For IO, at least one of the other observables is incompatible with its $1\sigma$ region, hence only the $3\sigma$ compatible points are shown for IO. Both mass orderings have points outside the disfavoured region, although the non-disfavoured region for IO is smaller.
The minimum values considering the $3\sigma$ ranges are
\begin{align}
(m_{\text{lightest}})_{\text{min}}^{\text{NO}}\approx  0.002\text{ eV}~~&~~(m_{\beta\beta})_{\text{min}}^{\text{NO}}\approx 0.003 \text{ eV}\nonumber
\\
(m_{\text{lightest}})_{\text{min}}^{\text{IO}}\approx 0.011\text{ eV}~~&~~(m_{\beta\beta})_{\text{min}}^{\text{IO}}\approx 0.015 \text{ eV}.
\end{align}

For this model, NO is hence the preferred mass ordering, although this means smaller values of $m_{\beta\beta}$, which are harder to access experimentally.

\subsection{Model 2}

It should be noted that it is possible to use other weights for the modular forms and still obtain a model with TM2 mixing.
One example is a model where we substitute the fixed constant $Y_{\nu}$ of the previous model by a triplet modular form under $A_4^\nu$.
However, in this case, the obtained matrix for the interactions between left and right-handed neutrinos would have a null eigenvalue associated with the eigenvector we want to preserve in TM2, which would lead to a massless effective neutrino corresponding to the second column of the PMNS. Therefore we need to further introduce an additional term for the model to be viable, or equivalently we should somehow keep the term from the previous model but taking into account that the weights in each term must always sum to zero.

\begin{table}[ht]
\centering
\begin{tabular}[t]{c||c|c|c|c|c}
\toprule
Fields & $SU(2)$ & $A_4^l$ & $A_4^{\nu}$ & $2k_l$ & $2k_{\nu}$ \\ \colrule
$L$ & \textbf{2} & \textbf{3} & \textbf{1} & $0$ & $0$ \\
$e^c$ & \textbf{1} & \textbf{1} & \textbf{1} & $+6$ & $0$ \\
$\mu^c$ & \textbf{1} & \textbf{1$''$} & \textbf{1} & $+6$ & $0$ \\
$\tau^c$ & \textbf{1} & \textbf{1$'$} &  \textbf{1} & $+6$ & $0$ \\
$\nu^c$ & \textbf{1} & \textbf{1} & \textbf{3} & $0$ & $+4$ \\
$H_{u,d}$ & \textbf{2} & \textbf{1} & \textbf{1} & $0$ & $0$ \\
$\Phi$ & \textbf{1} & \textbf{3} & \textbf{3} & $0$ & $0$ \\ \botrule
\end{tabular}~~
\begin{tabular}[t]{c||c|c|c|c}
\toprule
Yukawas/Masses & $A_4^l$ & $A_4^{\nu}$ & $2k_l$ & $2k_{\nu}$ \\ \colrule
$Y_l$ & \textbf{3} & \textbf{1} & $+6$ & $0$ \\
$Y_{\nu}$ & \textbf{1} & \textbf{1} & $0$ & $+4$ \\
$Y'_{\nu}$ & \textbf{1} & \textbf{3} & $0$ & $+4$ \\
$M_\textbf{1}$ & \textbf{1} & \textbf{1} & $0$ & $+8$ \\
$M_\textbf{3}$ & \textbf{1} & \textbf{3} & $0$ & $+8$ \\ \botrule
\end{tabular}
\caption{Transformation properties of fields, Yukawa couplings and masses for the right-handed neutrinos for model 2.}
\label{transformationp_model2}
\end{table}

The transformation properties of fields, Yukawa couplings and masses are in TABLE \ref{transformationp_model2}.
The weights for the $A_4^l$ symmetry remain the same as in the previous model given that the content of that symmetry was not changed, only of $A_4^\nu$, and as such just the $A_4^\nu$ weights will be introduced below.

The Yukawa coefficients $Y_l(\tau_l)$ remain the same. However, $Y_{\nu}(\tau_\nu)$ is now a singlet under both symmetries with weights $2k_l=0,2k_\nu=+4$, and $Y'_{\nu}(\tau_\nu)$, a triplet under $A_4^\nu$ with weight $2k_\nu=+4$, is introduced. It is then possible to assign a null value to $2k_{\nu}$ for the right-handed charged leptons $e^c,\mu^c,\tau^c$ and the lepton doublets $L$, which means that, conversely to what happened for model 1, no factors dependent on $\tau_\nu$ appear in the transformation relations for these superfields. That is to say that the leptons only transform under $A_4^l$ and the right-handed neutrinos only under $A_4^{\nu}$, with no addition of $(c\tau_\nu+d)^{-2k_\nu}$ factors for the right-handed leptons and a similar factor for $L$ as happened for model 1.

However, it should be pointed out that, even for model 1, it is possible to substitute the modular form of weight 0 by a singlet modular form of weight 4, thus changing the transformation properties under $A_4^\nu$, given that different weights are now attributed. In fact, if we consider model 2 without adding the triplet $Y'_\nu$, using the same weights, we obtain then the same mass matrices and mixing scheme as in model 1, although now no $\tau_l$ dependent factor would appear in $Y_\nu$ and the transformation relations for the leptons would look simpler too.

As already said, for the right-handed charged leptons $e^c,\mu^c,\tau^c$ and the lepton doublets $L$, the weights are now $2k_{\nu}=0$ and consequently, for the three right-handed neutrinos $\nu^c$, the weight is $2k_{\nu}=+4$ instead.
This implies that the right-handed neutrino masses $M_\textbf{1}$ and $M_\textbf{3}$ will have now $2k_{\nu}=+8$.
As before, these weights are chosen in such a way that the modular forms acquire the desired directions as we will see.

Again, neutrino masses can be generated through the type 1 seesaw mechanism, and, with the choices previously described, the superpotential has the following form:
\begin{align}
w&=w_e+w_{\nu},\\
w_e &= \left( \alpha LY_l(\tau_l)e^c + \beta LY_l(\tau_l) \mu^c + \gamma LY_l(\tau_l) \tau^c \right) H_d ,\\
w_{\nu} &= \frac{1}{\Lambda} L\Phi Y_{\nu}(\tau_\nu) \nu^c H_u + \frac{1}{\Lambda} L\Phi Y'_{\nu}(\tau_{\nu})\nu^c H_u +
\frac{1}{2}M_\textbf{1}(\tau_{\nu})(\nu^c \nu^c)_\textbf{1} +
\frac{1}{2}M_\textbf{3}(\tau_{\nu})(\nu^c \nu^c)_\textbf{3}.
\end{align}

The bi-triplet $\Phi$ will then acquire a VEV as before and the two modular symmetries are broken to a single one. As seen previously, $w_\nu$ gets the form:
\begin{align}
w_{\nu} &= \frac{v_\Phi}{\Lambda} (L Y_{\nu}(\tau_\nu) \nu^c + L Y'_{\nu}(\tau_{\nu}) \nu^c)_\textbf{1}H_u +
\frac{1}{2}M_\textbf{1}(\tau_{\nu})(\nu^c \nu^c)_\textbf{1} +
\frac{1}{2}M_\textbf{3}(\tau_{\nu})(\nu^c \nu^c)_\textbf{3},
\end{align}
and $w_e$ does not change.

The flavour structure after \texorpdfstring{$A_4$}{A4} breaking is now going to be covered.
We assume that the modular field $\tau_l$ acquires the VEV $\langle\tau_l\rangle=\tau_T=\frac{3}{2}+\frac{i}{2\sqrt{3}}$, stabiliser of $T_\tau$, as in the previous model.
A residual modular $Z_3^T$ symmetry is preserved in the charged lepton sector and, when the Higgs field $H_d$ acquires a VEV, the $Y_l$ direction leads to a diagonal charged lepton mass matrix as in (\ref{mass matrix charged leptons}), and the masses for the charged leptons can be reproduced by adjusting the parameters $\alpha$, $\beta$ and $\gamma$ as before.

For the other modular field $\tau_{\nu}$, a residual $Z_2^S$ symmetry is conserved, given that, as seen previously, the modulus should acquire the VEV $\langle \tau_{\nu} \rangle = \tau_S = i$ for $M_\textbf{3}$ to have the direction $(1,1,1)$, and thus $M_\textbf{3}$ must have an even $k_\nu$, as happens for $2k_\nu=+8$. Note that $(1,1,1)$ is precisely the direction of the weight 8 triplet in Eq.(\ref{eq:mf triplet 8}) and that there is indeed a non vanishing singlet at this stabiliser for the weight 8 singlet in Eq.(\ref{eq:mf singlet 8}), which accounts for the existence of $M_\textbf{1}$ .
This implies the following structure for the right-handed neutrino mass matrix:
\begin{equation}
M_R = c_\textbf{1} \begin{pmatrix} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{pmatrix} +
\frac{c_\textbf{3}}{3} \begin{pmatrix} 2&-1&-1 \\ -1&2&-1 \\ -1&-1&2 \end{pmatrix},
\end{equation}
where $c_{\textbf{1}},c_{\textbf{3}}$ are complex constants associated with the respective modular form.

The term $(L Y'_{\nu}(\tau_{\nu}) \nu^c)_\textbf{1}$ will have two contributions, one when we multiply two of the triplets to get a symmetric triplet and then obtain a singlet by multiplying with the third triplet, and the other when we consider in the first step the antisymmetric contribution instead. In order to obtain the terms that give rise to the Dirac mass matrix, to these two terms the $Y_\nu$ term is added, all with different multiplicative constants, and thus having $g_1 Y_{\nu}(\tau_\nu) (L \nu^c)_\textbf{1} + g_2((L Y'_{\nu}(\tau_{\nu}))_{\textbf{3}_S} \nu^c)_\textbf{1}+g_3((L Y'_{\nu}(\tau_{\nu}))_{\textbf{3}_A} \nu^c)_\textbf{1}$ where the complex constants $g_i$ already account for the constants associated with the modular forms.

The VEV $\langle\tau_l\rangle=\tau_T=\frac{3}{2}+\frac{i}{2\sqrt{3}}$ the modulus field acquires will also imply a direction $(1,1,1)$ for $Y'_\nu(\tau_\nu)$.
The Dirac mass matrix that relates the right-handed and active neutrinos after the Higgs field $H_u$ acquires a VEV $\langle H_u \rangle = (0,v_u+h(x))$ will then be
\begin{equation}
M_D= y_D v_u g_1
\left(
\begin{pmatrix}
1&0&0 \\ 0&0&1 \\0&1&0
\end{pmatrix}
+ \frac{h_2}{3}\begin{pmatrix} 2&-1&-1 \\ -1&2&-1 \\-1&-1&2
\end{pmatrix}
+\frac{h_3}{2} \begin{pmatrix}
0&1&-1 \\ -1&0&1 \\1&-1&0
\end{pmatrix}
\right),
\end{equation}
where $h_i=g_i/g_1$ ($g_1$ was chosen given that, for the model to hold, it can not be taken to zero, as stated in the beginning of this section).
Again, the active neutrino mass matrix for the seesaw mechanism is obtained through the following formula:
\begin{equation}
M_{\nu}=-M_DM_R^{-1}M_D^T.
\end{equation}
The expressions for the entries of the neutrino mass matrix are much more complicated in this case since the Dirac mass matrix is not a permutation matrix as before. Nevertheless, the seesaw mass matrix is diagonalized by the TM2 mixing matrix and again, the PMNS matrix corresponds to the TM2 mixing.

Doing a similar derivation to what was done for the first model, the mass sum rule for this model was derived.
There are two solutions when solving $a$, $b$ and $c$ in terms of $c_\textbf{1}$, $c_\textbf{3}$ and $h_2$ when changing variables in $U_{TBM}^T M_\nu U_{TBM}$ as was done for the previous model in (\ref{eq:UMRU1}).
There are then two possible ways of relating $a$, $b$ and $c$ with $c_\textbf{1}$, $c_\textbf{3}$ and $h_2$, which implies that there are two different expressions for the entry (2,2) in $U_{TBM}^T M_\nu U_{TBM}$ and thus a different mass sum rule is obtained for each solution, and, consequently, different expressions for $m_{\beta\beta}$ too.
In spite of this, the expressions for each solution differ only in that $f_\text{first}(m_1,m_3,\theta,\alpha_1,\alpha_2,h_3)=f_\text{second}(m_1,m_3,\theta,\alpha_1\pm\pi/2,\alpha_2-\pi/2,h_3)$, where $f$ stands for either $m_2$ or $m_{\beta\beta}$.
Given that the expressions for $m_2$ and $m_{\beta\beta}$ only differ in the values for the complex phases $\alpha_i$ by a $\pm\pi/2$ shift, and also that, from (\ref{sintheta23_relation}) and (\ref{delta_relation}), this shift does not change the values taken by the observables $\theta_{23}$ and $\delta$, we conclude that, scanning over the phases $\alpha_i$, we obtain in the end the same results for both model 2 solutions.
Thus, since the results for the fits and the allowed regions for $m_\text{lightest}$ vs $m_{\beta\beta}$ only differ by a shift in the $\alpha_i$ values, we only study one of the solutions below.

The best fit values, shown in
TABLE \ref{fitresults_model2}
were obtained.
For NO, all the observables except $\theta_{12}$ are compatible with their $1\sigma$ ranges.
For IO, $\delta$ is outside its $1\sigma$ region, as happened for model 1, even though all of the observables are still within their $3\sigma$ ranges.
As for model 1, NO provides the best fit, with $\chi^2=1.57$, which is the same value found for model 1. This is not surprising given the contribution to the $\chi^2$ is coming not from the masses, but from the mixing angles, and both models give TM2 mixing.
\begin{table}[hb]
\centering
\begin{tabular}{| c | c | c c c c c c c |}
\cline{1-9}
\multirow{4}{*}{NO} & \multirow{2}{*}{Para.} & $\chi^2$/d.o.f. & $\theta$ & $\alpha_1$ & $\alpha_2$ & $h_3$ & $m_1$ & $m_3$ \\
\cline{3-9}
& & 1.57 & -10.51\degree & -54.88\degree & 168.45\degree & $-0.401+0.890i$ & 0.0164 eV & 0.0528 eV \\
\cline{2-9}
& \multirow{2}{*}{Obs.} & $\theta_{12}$ & $\theta_{23}$ & $\theta_{13}$ & $\delta$ & $\Delta m_{21}^2$ & $\Delta m_{31}^2$ & $m_{\beta\beta}$\\
\cline{3-9}
& & 35.72\degree & 49.4\degree & 8.56\degree & 224\degree & 7.42$\times 10^{-5}$eV$^2$ & 2.514$\times 10^{-3}$eV$^2$ & 0.0144 eV \\
\cline{1-9}
\multirow{4}{*}{IO} & \multirow{2}{*}{Para.} & $\chi^2$/d.o.f. & $\theta$ & $\alpha_1$ & $\alpha_2$ & $h_3$ & $m_1$ & $m_3$ \\
\cline{3-9}
& & 2.03 & 10.56\degree & 34.19\degree & 89.80\degree & $0.628-0.407i$ & 0.0521 eV & 0.0171 eV\\
\cline{2-9}
& \multirow{2}{*}{Obs.} & $\theta_{12}$ & $\theta_{23}$ & $\theta_{13}$ & $\delta$ & $\Delta m_{21}^2$ & $\Delta m_{32}^2$ & $m_{\beta\beta}$\\
\cline{3-9}
& & 35.73\degree & 48.5\degree & 8.61\degree & 236\degree & 7.42$\times 10^{-5}$eV$^2$ & -2.497$\times 10^{-3}$eV$^2$ & 0.0418 eV\\
\cline{1-9}
\end{tabular}
\caption{Parameters (Para.) and observables (Obs.) for the best fit point for normal and inverted ordering for model 2.}
\label{fitresults_model2}
\end{table}
\begin{figure}[ht]
     \centering
     \begin{subfigure}[t]{0.49\textwidth}
         \centering
         \includegraphics[width=\textwidth]{TM2_2A4_NO_model2_sol1.png}
         \caption{Normal Ordering}
		 \label{fig:mee model2 sol1 NO}
     \end{subfigure}%
     ~
     \begin{subfigure}[t]{0.49\textwidth}
         \centering
         \includegraphics[width=\textwidth]{TM2_2A4_IO_model2_sol1.png}
         \caption{Inverted Ordering}
		 \label{fig:mee model2 sol1 IO}
     \end{subfigure}
    \caption{Predictions of $m_{\text{lightest}}$ vs $m_{\beta\beta}$ for both orderings of neutrino masses for model 2 compatible with $1\sigma$ (dark-red, NO only, except $\theta_{12}$) and $3\sigma$ data from \cite{Esteban:2020cvm}. In both figures were also included the current upper limit from KamLAND-Zen amd PLANCK 2018 as in FIG.\ref{fig:mee model1}}
    \label{fig:mee model2}
\end{figure}

Furthermore, the allowed regions of $m_{\text{lightest}}$ vs $m_{\beta\beta}$, shown in FIG.\ref{fig:mee model2} (as before, for NO, $m_{\text{lightest}} = m_1$ and for IO, $m_{\text{lightest}} = m_3$) were obtained. The experimental constrains that were already discussed in the previous model are also included.
As before, there are some points (with a darker red color) in the NO figure compatible with the $1\sigma$ ranges for the observables other than $\theta_{12}$, even though for IO again no $1\sigma$ compatible points are shown, given that all points are outside the $1\sigma$ range for at least one of the other observables, as occurred for the best fit values. However, these $1\sigma$ points for NO do not form a characteristic structure as happened for model 1 but are dispersed within the other points that have at least an observable other than $\theta_{12}$ outside its $1\sigma$ region.
Then it is inferred that, similarly to what happened for model 1, NO is once more the preferred mass ordering.

However, this second model is much less predictive than the first one, since $m_\text{lightest}$ covers all orders of magnitude and almost all the available region for $m_\text{lightest}$ vs $m_{\beta\beta}$ and, more importantly, the minimum value for $m_{\beta\beta}$ also approaches zero.

\section{Conclusion \label{sec:conclusion}}

We employed the framework of multiple modular symmetries to build models with minimal field content that are able to reproduce viable Tri-Maximal 2 mixing. We describe how the multiple $A_4$ modular symmetries can break to a single group and show possible assignments of fields and weights under the modular symmetries leading to the desired mixing scheme. Two explicit models were built (with different weights) and shown to be predictive and to reproduce the observed mixing angles with good fits. Neutrinoless double beta decay is expected, with the inverted ordering possibility almost entirely disfavored by cosmological observations and less compatible with the $1\sigma$ best fit intervals for the experimental observables than the normal ordering of neutrino masses.

\section*{Acknowledgments}
IdMV acknowledges funding from Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia (FCT) through the contract IF/00816/2015 and was supported in part by FCT through projects CFTP-FCT Unit 777 (UID/FIS/00777/2019), PTDC/FIS-PAR/29436/2017, CERN/FIS-PAR/0004/2019 and CERN/FIS-PAR/0008/2019 which are partially funded through POCTI (FEDER), COMPETE, QREN and EU.
JL acknowledges funding from Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia (FCT) through project CERN/FIS-PAR/0008/2019.

redpajama_set_name: RedPajamaArXiv


text:
By Victor Hugo & Charles E. Wilbour
Genre: Classiques
Les Miserables is the great epic masterpiece of the mid-nineteenth century. Begun in 1845, the year Louis Philippe conferred a peerage and a lifetime seat in the Senate upon Victor Hugo, it was completed when the author was living in exile in the Channel Islands. Les Miserables is a product as well as a document of the political, social, and religious upheaval that followed the Napoleonic Wars and Europe's great democratic revolutions. The story is centered on Jean Valjean, a peasant who enters the novel a hardened criminal after nineteen years spent in prison for stealing a loaf of bread for the starving children of his sister. The path of Valjean's last twenty-five years, leading from the French provinces to the battlefield of Waterloo and the ramparts of Paris during the Uprising of 1832, introduces us to secret societies of revolutionaries and the vast world of the French lower classes. Jean Valjean's flight from the police agent Javert--the prototype of over a hundred years of fictional detectives--culminates in one of the most famous scenes in all literature, the chase through the sewers of Paris. Les Miserables sold out its large first printing in twenty-four hours and has remained enormously popular. This edition is the classic English translation of Hugo's friend Charles Wilbour, which appeared the same year the novel was published in France.
More by Victor Hugo & Charles E. Wilbour
50 Masterpieces you have to read before you die - Volume 1 (Beelzebub Classics)
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Les misérables Tome I
Victor Hugo & Isabel Florence Hapgood
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Les misérables Tome III
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Victor Hugo, Lee Fahnestock, Norman MacAfee & Chris Bohjalian
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Ulysses S. Grant, Theodore Roosevelt, Benvenuto Cellini, Horatio Alger, Jr, Charles Darwin, Jacob Abbott, Frank Lewis Dyer, Victor Hugo, Lytton Strachey, William M. Thayer, W.H.G. Kingston, John G. Nicolay, John T. Morse Jr., Edward Gibbon, Rabindranath Tagore, Oliver Wendell Holmes, John Stuart Mill, Albert Keim, Louis Lumet, Anthony Trollope & John Rae
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100 Books You Must Read Before You Die -
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La fin de Satan
Victor Hugo & Monica Kulling


She is a violist born in Spokane, Washington, who studied at Boston and Indiana Universities.
Since beginning her career as an orchestral and chamber musician she has performed with several orchestras and ensembles such as the Aire Born Strings , the Peaceful Valley Chamber Players, the Ronen Chamber Ensemble, the Milwaukee Symphony Orchestra and Suzuki and Friends, was principal violist for the San Diego Symphony and is currently a member of the Indianapolis Symphony Orchestra.
Working in committees she has been the Chair of the Indianapolis Symphony Orchestra, representing it at the International Conference of Symphony Musicians and she is currently on the American Federation of Musician's Executive Board of Local 3.
Veering away from classical music she has appeared on may recordings made by the comedy disc jockeys Bob & Tom who appear in their daily The Bob & Tom Show broadcast in Indianapolis.
Recognised for her contributions to music and education she won the Patch Leadership award in 1997 and takes part as an Artist in School and a "Side by Side" section coach in the educational programs set up by the Indianapolis Symphony Orchestra.
Here she is playing viola with members of the Indianapolis Symphony Orchestra at the Chatterbox Jazz Club..


text:
Country stars honor Las Vegas shooting victims at CMA show
By Mesfin Fekadu The Associated Press
November 8, 2017 - 8:50 pm November 8, 2017 - 8:56 pm
Eric Church, Keith Urban, Darius Rucker and Lady Antebellum opened the 2017 Country Music Association Awards with a touching performance honoring the 58 people killed at a country music festival on the Strip last month.


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