Amit Majmudar, nuclear radiologist and former Poet Laureate of Ohio, spends some days discerning patterns in diagnostic scans of the human body, and other days patterning language into poems to represent the interior of the human experience. His most recent literary project is Godsong, an ambitious translation of the ancient Sanskrit poem and Hindu scripture, the Bhagavad Gita. Majmudar learned Sanskrit in order to craft the 700-verse translation, for which The New York Times called it "ravishing and faithful, marked by what Nabokov once called 'the precision of a poet and the imagination of a scientist.'" Join Majmudar as he zeroes in specifically on the 11th Session of Godsong—an extended vision of the Universal Form of the divine—in order to explore the function of "visions," both in poetry and in religious iconography.
This program is generously underwritten by Anita and Prabhakant Sinha.
Preorder your copy of Godsong: A Verse Translation of the Bhagavad-Gita, with Commentary through the CHF box office and save 20%.


text:
Get ready to start vaccinating by December 14, GPs told
Dec 4th 2020 5:10PM
GP surgeries have been told to be ready to start staffing Covid-19 vaccination centres by December 14.
In letter sent out across England's primary care networks, NHS England and NHS Improvement warned the "scale and complexity" of the immunisation programme would make it "one of the greatest challenges the NHS has ever faced".
It was signed by Dr Nikita Kanani, medical director for primary care at NHS England and NHS Improvement, and Ed Waller, director of primary care at the same institutions.
"It is crucial we start to activate local vaccination services to allow priority patient cohorts to start accessing the vaccine," it said.
The vaccination sites must be ready to administer 975 doses of the vaccine to priority patients within three-and-a-half days of delivery on December 14.
Speed is of the essence with the vaccine, as it is usually stored at -70C and will only remain stable at fridge temperatures of between 2C-8C for a limited period of time.
Ultra-low temperature freezers used to store the vaccine (Paul Reid/HSE)
There are 975 doses in each of the Pfizer/BioNTech vaccine packs, which has posed a logistical problem of how they can be broken up and distributed to other key sites such as care homes.
The first to receive the vaccine in the centres will be those aged 80 and over, as long as their other risk factors "clinical or otherwise" had been taken into account.
NHS England and NHS Improvement said the number of vaccination sites in each clinical commissioning group (CCG) area will vary according to the number of residents it has who are over 80.
CCGs have been asked to consider inequalities and deprivation – some of the biggest Covid-19 risk factors – when choosing the sites for their vaccination centres.
Each centre will also be supplied with the "IT equipment necessary to deliver the programme and a fridge", the letter said.
It added: "We will be writing to sites identified as part of wave 1 on Monday, setting out full details of vaccine supply dates, delivery of other consumables and equipment to the site, and the process for assuring readiness before delivery of vaccines."
It said staff at the vaccination sites would be provided with training, and be "given full support to mobilise within the timescale".
The staff at the first sites to open are due to get login details for the IT system set up to deliver the vaccination programme "as soon as possible".
CCGs were told they must "offer all possible assistance" to the vaccine sites as the programme gets under way, including helping them with logistics and setting up clinical waste arrangements.
They were also warned that while urgent care must be provided by GPs even when the vaccine is being delivered, the vaccination programme must be their top priority.
It urged all primary care providers to work together to maximise uptake and minimise wastage.
The letter added there is £150 million in funding is available to ensure their is ensure their is staff and resources for GPs to meet their usual obligations and deliver the vaccine programme.
It finished: "This is a hugely exciting moment for general practice and you who will be playing a key part in this important programme."
Meanwhile, the Medicines and Healthcare products Regulatory Agency (MHRA) said it has approved a solution to splitting vaccine packs and getting it into care homes.
Assemblers – a mixture of companies licensed to assemble packs and NHS hospitals with appropriate facilities – will break down the vaccines into smaller batches in special cold rooms at between 2C and 8C.
The process is extremely time sensitive, with assemblers having just 12 hours to thaw the vials, repack the medicines, label them and distribute them to the mobile teams delivering to care homes.
Despite elderly care home residents being the highest priority, they will not be first in the queue as manufacturers have to test and validate the assembly process and ensure all staff are thoroughly trained.
The MHRA said: "We have put in place arrangements to support delivery of the vaccine to top priority groups as safely and swiftly as possible.
"Regulatory approvals required to proceed with the splitting up of packs have been put in place.
"We are working with NHS and their assemblers to help support them to implement the processes and training they need to have in place to meet our conditions."
Dr June Raine, chief executive of the MHRA, told the BBC on Friday that she estimated the vaccine would begin to be delivered to care homes "within the next two weeks".

redpajama_set_name: RedPajamaCommonCrawl


Imagine your day that starts and ends something like this: No one waking us up in the morning after the alarm clock has failed to do its job. Not able to find that black dress or shirt which you wanted to wear today, no breakfast or lunch prepared, the room looks like a mess, and you don't remember where you kept your watch last night when you were too sleepy to keep it where it belonged. Now, this is just the start and when you have a bad start your entire day is already doomed. This is the condition of most unlucky ones who do not have that miracle angel called "mother" or are living away from their moms. Mother do a lot without making their children realise or expecting something back in return. Whatever we do, it is impossible to thank our mothers for all the sacrifices they make. The least we can do is celebrate special occasions like her birthday, Women's Day and most importantly Mother's Day. Mother's Day this year falls on the 12th of May 2019. IGP brings some of the most expressive and thoughtful Mother's Day gift ideas that feel like they are made especially for your mother dearest.
A gourmet hamper that says 'you are very special'- that's the perfect gift for any mother. This special Mother's Day gift idea has a rich chocolate cake bar along with walnut cookies in a stylish can. Both the treats come packed inside a wooden tray, which can be reused by your mother for storing her knick-knacks.
She is the epitome of strength, who went through the pain of giving birth to you and the challenges of raising a misbehavioured you. And undoubtedly, she has been your first true love. So get this beautiful hand-tied arrangement of blue orchids and white roses that stand as a tribute to your one in a million mother.
If you are looking for a more exciting gift than just a bouquet, we recommend this mix flower bouquet with assorted cupcakes. The tissue wrapped bouquet includes stunning and fragrant lilies, roses, and gerberas. There are 6 cupcakes in different flavours that are sure to delight your mother's taste buds.
If your mother has a green thumb, she would love to receive a plant to add to her growing garden. Your mother can decorate her living space with this bamboo plant and benefit with its good luck bearing quality. Along with the plant, this Mother's Day gift idea also has soothing and nourishing St.Ives body wash. There is also a good wishes card included in the hamper.
Surprise your mom this Mother's Day, with this awe-inspiring Mother's Day gift idea. The star of the hamper is a rose gold finish necklace, which is surely going to look pretty on your mummy. Along with it, there is a Lindt Sea Salt chocolate bar and a happy Mother's Day greeting card.
Bangles are the most picked jewellery piece for mothers. If your mother likes adorning herself with jewellery, choose these luxurious looking but budget-friendly bangles for her. The bangles are studded with different sized pearls that draw a stunning pattern across its round surface. Complimenting the white pearls are crystals that add a charming touch to the pair.
She has spent countless days and evenings thinking and cooking up the most delicious dishes for you. Now it's your turn to pamper her with a lip-smacking treat this Mother's Day. Our delicious and fresh pineapple cake is among the most loved Mother's Day gift idea. The half kg cake is creamy and soft and topped with cherries.
If she is a working mom, then you can consider this Mother's Day gift idea for her. This hamper includes desk accessories like stationary holder and mug with Like a Boss, Perfect Mother, etc. written on them. These will help her stay organized and fresh all through the day. She will finish her work on time always and come back home to you quickly.
For your lord Shiva and Parvati bhakt mom, nothing can be a better gift than this. This beautiful Shivalaya Tarkashi painting is 9 inches wide and long and can easily fit into the crampiest wall easily.


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text:
Koroshi Ai (Love of Kill)
Action Anime, Romance Anime [Genres]
Koroshi Ai (Love of Kill)Koroshi Ai (Love of Kill) [Anime Title]
Action, Romance, Josei
©2022 Fe/KADOKAWA/殺し愛製作委員会
At a certain workplace, two assassins—Chateau, a cool female bounty hunter, and Ryang-Ha, a powerful man of many mysteries—face off.
Chateau and Ryang-Ha should have become enemies after their initial fight, but for some reason, Ryang-Ha is intrigued by her and starts following her around. Chateau slowly comes around to Ryang-Ha and they form a cooperative partnership. However, there are organizations that are targeting Ryang-Ha, and Chateau becomes involved in the struggle. And what's more, their fight against the organizations is also linked to her past.
Why did Ryang-Ha get close to Chateau? What secrets are buried in her past?
The strange gears of fate begin to turn in this twisted assassin thriller weaved together by an unlikely pair of assassins.
Chateau Dankworth: Saori Onishi
Ritzland Support Company's newest bounty hunter recruit. She's a skilled agent who executes jobs in a cool and pragmatic fashion, but after meeting Ryang-Ha, her performance takes a hit. Her origins and past are shrouded in mystery, and as a child she was discovered at the site of a certain incident.
Ryang-Ha Song: Hiro Shimono
An assassin whose affiliation and true identity are unknown. He once dealt a huge blow to the underworld, resulting in a bounty being placed on his head. Now, several organizations are trying to kill him for the reward. Although he has a carefree attitude, he's incredibly strong and when it comes to his work, he executes jobs in a bold and level-headed manner. For some reason, he's taken a liking to Chateau.
Euripedes Ritzland : Kenyu Horiuchi
The representative of the Ritzland Support Trading Company, affectionately referred to as "The President." He is a talented manager who is capable of dealing with the underworld, but he has a hard time reigning Chateau in. Due to his size, intensity, and calm disposition, he is often seen as older than he really is, but he is actually 25.
Jim: Kohei Amasaki
An accountant at Ritzland Support Trading Company. He speaks with a unique intonation. Although he often appears slow-paced and absentminded, he's actually an excellent worker. His occasional harsh remarks sometimes cause problems for the people around him...the President in particular.
Hou: Tomoaki Maeno
A former member of the organization that Ryang-Ha used to belong to. He has a personal vendetta against Ryang-Ha due to their past relationship. Although he used to be an emotional young man, he has since become a cold-hearted and brutal avenger. He has a tattoo on his left cheek.
Donny: Hochu Otsuka
Full name: Donald Bachmann. He is the boss of an organization based in the port city of Barosela and is also known as "Trigger." He trains war orphans into capable subordinates. He sends an assassin to Ryang-Ha for a certain purpose.
Opening Song: "Midnight Dancer" by Toshiki Masuda
Ending Song: "Makoto Period" by Aika Kobayashi
"Koroshi Ai (Love of Kill)" (Manga)
Original Creator: Fe
Director: Hideaki Oba
Script: Ayumu Hisao
Series Composition: Ayumu Hisao
Character Design: Yoko Satou
Animation Director: Yoko Sato / Toshimitsu Kobayashi
Sound Director: Hajime Takakuwa
Music: Kei Yoshikawa
Koroshi Ai (Love of Kill) Post
View All Koroshi Ai (Love of Kill) Post


This brand new 3 bedroom home offers an open living/dining layout with an indoor utility/mud room and an outdoor covered patio. The kitchen features stainless steel appliances with a gas range, microwave, Quartz countertops, tile back splash, and breakfast bar. The home boasts a HERS score of 66, guaranteeing you low heating and cooling costs all year round. Scheduled to be move-in ready in March 2019.


the lowest price and save the most when you shop online for, please check the following offers page! 15 OFF, coupon Code: Thanksgivingcp. 15 OFF Coupon Code: veteransdaycp Veterans Day Sale: Get 15 off on all history books when using this coupon code at eBooks * Expiry Date: * This percent off coupon only applies to selected items. 15 OFF Coupon Code: cybermondayecp Cyber Monday Sale at m: Get 15 off on your favorite books using this promo code, valid 11/28 - 11/30 * Expiry Date: 11/30/2010 * This percent off coupon applies to your entire creative loafing coupons order. 10 OFF, coupon Code: LibraryLovercp, simply add the coupon code to the coupon box under the shopping cart, for a 10 off Reference category books. Regularcoupons reserves the right to deactivate any offers without prior notice. Internet coupons and discount links may expire anytime without prior notice. Get Coupon Code, expiration Date: Jan 29, 2018, yes. Folge diesem Link und erhalte mit dem Code einen 15 eBook Gutschein für Kalender 2018. Bring it home today.
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JERUSALEM (AP) — The Israeli military says it bombed Hamas military targets in the Gaza Strip, hours after an explosive device detonated near an Israeli army vehicle along the Gaza-Israel border.
The Palestinian health ministry in Gaza confirmed that a Palestinian was killed and another man was seriously wounded in the strike early on Thursday.
The bombing came hours after an Israeli airstrike targeted Hamas training sites in the central and southern Gaza Strip.
Tensions are rising on the Gaza-Israel border as Hamas, the Islamic militant group that controls the coastal territory, has mobilized people for unprecedented weekly protests along the borders since March 30.
Thursday's fatality raises the number of Palestinians killed by Israeli fire over the past two weeks to 33. At least 26 were killed during protests.


It's taken us too long to start following the brilliant parody account Awful Fantasy on twitter. You don't have to be a diehard Tolkien fan or have the entire Wheel of Time series in hardcover to appreciate their dead-on stabs at genre fiction. Somebody give these people a Hugo award, or at the very least follow them on facebook and tumblr.


This investment of $500,776 in the NOIC will help support innovative entrepreneurs in their efforts to start, expand, or strengthen their business. In recent years, the demand for the centre's services has more than doubled, so today's investment will help meet this growing demand.
"Economic development is a priority throughout Northwestern Ontario, including First Nations throughout the riding. Ensuring community members have access to training and business development services is an important factor in securing the economic future of First Nations."


The lounge chair is ideal for using in the garden, patio, poolside, yard, so you can lie down on it and enjoy your leisure time.
6 Height position: 13.0", 23.5", 29.0", 32.5", 36.5" and 39.5"
Distance from seat to floor: 13.0"
Made of steel and rattan, the lounge chair has a sturdy and durable construction. The high back can adjust to 6 positions to fit your body, offering you with comfortable experience. Two wheels make the lounge chair easy to move. In addition, the thick cushion can add more comforts when you lie on it.
If you are looking for this lounge chair, do not hesitate to buy one!
Excellent value for a great chaise lounger!
I bought these for my daughter and my son-in-law for their new home! They said they were easy to assemble and they are so comfortable!! I am actually going to buy two more so when we visit my husband and I can lay down as well and lounge around the pool!
Great chair for the money. Bought 4 and putting them together took some time. The way they're machined it is a little awkward. Regardless, the cushions are comfortable, of good quality and the look great. One of the backs weave is off kilter, but with the cushion on top you don't really notice it.
This is wonderful!! It was not hard to put together - one person. The pad is as comfortable as a mattress! It feels and looks luxurious. A very good buy for the money.
The cushions dry really fast when they get wet in the rain! The only thing I don't like is when you sit on it you can feel the seam below the cushion of the two sections. I would still get them again and when I have time I'm going to post some photos I took that will help with putting them together as the horrible copy that comes with it has really bad pictures.
The chaise is comfortable to lie in. Cushion is soft and thick. Design is classic, great addition to my pool. I don't know how long the chaise will last. However, I think I will buy it again if this one damaged. For such price this one is satisfying.
LOVE IT! Height is what I needed not to low to the ground. Comfortable I may consider buying a second one!
Could not keep/open this one, but intend to re-order! This lounge is the right height, not too low! THANK You for this great design.
First, arrived soon as promised. Four days later I saw it in front door. Second, item was intact, exactly as advertised. Third, cushions feels good and comfortable. I love my yard even better than past. One more word, it is indeed WORTH a TRY.
When I opened the package, orderly arrangement smoothed the past days of fear away. It arrived with intact set. Instruction is easy to understand. I can assemble it effortlessly. No longer need to ask help from my neighbor. even got a sense of self-achievement with my practical ability. LOL.


By default, Ren'Py will show one one selection of text after another. When used with scene or show, images can be shown, hidden, or replaced. However, sometimes there is a need for an effect to better inform a player of movement between events or places. For this, transitions can be used.
Transitions work with other keywords like scene and show to use effects like fade, dissolve, or none to merge from one set of one or more interactions to another.
Open the "script.rpy" file for "The Question".
The first use of a transition occurs on line 18. It is paired with the line immediately above it, line 17.
The use of the keyword with pairs a transition with another keyword. In the case of line 17, the keyword fade is used with the keyword scene to fade into the use of the "bg lecturehall" image.
Note: The use of the keyword with groups one or more uses of the keywords scene and show together. It will work on all of them up until the next text, interaction, or transition is used.
The next use of a new transition occurs on lines 34 – 37.
The keyword dissolve is used to create an effect of the image "sylvie green normal" appearing slowly in parts.
Throughout "The Question", the transitions of fade and dissolve are used in the same way. The keyword fade is often used with the new establishing of a scene and the keyword dissolve with a character appearing in the same scene.
Later in "The Question", the use of the keyword dissolve is used to express the quick passage of time, as if it is "dissolving" between different points in time as the story moves quickly between events.
Note: Although not used in "The Question", the last major transition keyword, None, bypasses any timed transition effect and simply updates. It can be paired with other transitions to break up complex visual events such as using scene and multiple uses of show with their own transitions.


Terror Jr is an American pop trio consisting of Felix Snow, former The Cataracs band member David "Campa" Benjamin Singer-Vine, and a lead singer that goes by the alias Lisa. Their single "Come First" peaked at number 10 on the Billboard Spotify Velocity chart, number 54 in the Czech Republic, and number 59 in Slovakia. Their single "3 Strikes" was featured in a Kylie Jenner lipgloss commercial. The pop trio released a musical project named Bop City on October 21, 2016.
Members Felix Snow and David "Campa" Benjamin Singer-Vine met through Twitter in 2014 and first worked together on a project called Momma, losing touch shortly after. They reconnected the following year working with singer Kiiara on her single "Gold". They then started Terror Jr as a "social experiment", with Singer-Vine adding "It's a puzzle, and we're gonna give out pieces with each song." Terror Jr launched their first single "3 Strikes" in a 2016 Kylie Jenner lipgloss commercial.
In November 2016, Jenner was rumored to be the lead singer of Terror Jr. Although she is listed as a performer on the ASCAP profile for the song "3 Strikes", Jenner denied being a part of the group in a Snapchat post on November 9, 2016.


text:
Q: Moq.Mock - how to set up a method that takes an expression I am Mocking my repository interface and am not sure how to set up a method that takes an expression and returns an object? I am using Moq and NUnit.
Interface:
public interface IReadOnlyRepository : IDisposable
{
    IQueryable<T> All<T>() where T : class;
    T Single<T>(Expression<Func<T, bool>> expression) where T : class;
}

Test with IQueryable is already set up, but don't know how to set up the T Single:
private Moq.Mock<IReadOnlyRepository> _mockRepos;
private AdminController _controller;
[SetUp]
public void SetUp()
{
    var allPages = new List<Page>();
    for (var i = 0; i < 10; i++)
    {
        allPages.Add(new Page { Id = i, Title = "Page Title " + i, Slug = "Page-Title-" + i, Content = "Page " + i + " on page content." });
    }
    _mockRepos = new Moq.Mock<IReadOnlyRepository>();
    _mockRepos.Setup(x => x.All<Page>()).Returns(allPages.AsQueryable());
    //Not sure what to do here???
    _mockRepos.Setup(x => x.Single<Page>()
    //----
    _controller = new AdminController(_mockRepos.Object);
}

A: Have the .Returns call return the result of the expression against your allPages variable.
_mockRepos.Setup(x => x.Single<Page>(It.IsAny<Expression<Func<Page, bool>>>()))
    .Returns( (Expression<Func<Page, bool>> predicate) => allPages.Where(predicate) );

A: You can set it up like this:
_mockRepos.Setup(x => x.Single<Page>(It.IsAny<Expression<Func<Page, bool>>>()))//.Returns etc...;

However you are coming up against one of Moq's shortcomings.  You would want to put an actual expression there instead of using It.IsAny, but Moq doesn't support setting up methods that take expressions with specific expressions (it's a difficult feature to implement).  The difficulty comes from having to figure out whether two expressions are equivalent.
So in your test you can pass in any Expression<Func<Page,bool>> and it will pass back whatever you have setup the mock to return.  The value of the test is a little diluted.

A: I have found that It.Is<T> should be used in place of It.IsAny<T> for more accurate results.
Page expectedPage = new Page {Id = 12, Title = "Some Title"};
_mockRepos.Setup(x => x.Single<Page>(It.Is<Expression<Func<Page, bool>>>(u => u.Compile().Invoke(expectedPage))))
                       .Returns(() => expectedPage);

A: Using Moq's It.IsAny<> without a .CallBack forces you to write code that's not covered by your test. Instead, it allows any query/expression at all to pass through, rendering your mock basically useless from a unit testing perspective.
The solution: You either need to use a Callback to test the expression OR you need to constrain your mock better. Either way is messy and difficult. I've dealt with this issue for as long as I've been practicing TDD. I finally threw together a helper class to make this a lot more expressive and less messy. Here's an example of one possible end-result:
mockPeopleRepository
  .Setup(x => x.Find(ThatHas.AnExpressionFor<Person>()
    .ThatMatches(correctPerson)
    .And().ThatDoesNotMatch(deletedPerson)
    .Build()))
  .Returns(_expectedListOfPeople);

Here's the blog article that talks about it and gives the source code: http://awkwardcoder.com/2013/04/24/constraining-mocks-with-expression-arguments/

redpajama_set_name: RedPajamaStackExchange


text:
Assistant/Associate Professor of Clinical Pediatrics
Location: IU School of Medicine
Department: IUSM - Pediatrics
Campus: IU School of Medicine Indianapolis
Position Announcement:
The Ryan White Center for Pediatric Infectious Diseases & Global Health at Indiana University School of Medicine and Riley Hospital for Children in Indianapolis, Indiana, are seeking Assistant/Associate Professor. The position is on the clinician-educator track and will provide a highly competitive salary for the Division of Infectious Disease and Global Health.
Successful candidates will have protected time for academic (education and research) activities and will have access to state of the art equipment and facilities. Remaining effort will be dedicated to clinical service within Riley Hospital for Children.
Indiana University is an equal opportunity and affirmative action employer. All qualified applicants will receive consideration for employment without regard to race, color, ethnicity, religion, age, sex, sexual orientation or identity, national origin, disability status, or protected veteran status. This institution is also a provider of ADA services.
The successful candidate(s) will possess an MD degree, academic credentials appropriate for an Assistant/Associate level appointment, and eligibility for license in Indiana if applicable. It is preferred that candidates have clinical and research interests in global health education and/or immunocompromised patient populations.
Department Contact for Questions:
Dr. Chandy John
Director, Ryan White Center for Pediatric Infectious Disease and Global Health
Indiana University School of Medicine,
1044 W Walnut St,
Email: johnsore@iu.edu
OAA #: IUSM005052016
About Indiana University
Founded in 1820, Indiana University is the state's public university system, comprised of two core campuses—IU-Bloomington and IUPUI—and six regional campuses located throughout the state. Combined, the total student population is more than 114,000 undergraduate, graduate and professional school students. IU is one of the largest employers in the state, with over 20,000 faculty and staff. Indiana University is an equal employment and affirmative action employer and a provider of ADA services. Diversity and inclusion remains a core commitment to IU's mission to cultivate an educational environment that supports equal access, participation and representation on all of its campuses and that provides educational and career opportunities for all.
Professor (Tenure Track) or Clinical Professor (Non-Tenure Track) Storrs, Connecticut
University of Connecticut Today
Assistant, Associate or Full Professor (HS Clin, Clin X, Adjunct, In Residence) Pediatric Neurologist La Jolla, California
University of California San Diego 2 Days Ago
Assistant, Associate or Full Professor (HS Clin, Clin X, Adjunct, In-Residence) - Pediatric Surgery La Jolla, California


text:
What Season Is Made In Chelsea On? (Question)
Overview of the series
Originally aired
First aired
16 11 8 October 2018
17 13 25 March 2019
18 11 2 September 2019
The twentieth season of Made in Chelsea, a British structured-reality television program, began broadcasting on September 28, 2020, and finished on December 14, 2020, after twelve episodes had been broadcast. Following the cancellation of the previous season's production due to the COVID-19 epidemic, it was revealed that the actors will adhere to government regulations while filming the current season.
1 What season is Made in Chelsea up to?
6 Are Miles and Ruby together?
7 What order do I watch Made in Chelsea?
8 How many episodes of Made in Chelsea are there?
9 How many seasons is maid on Netflix?
What season is Made in Chelsea up to?
The premiere date for Series 22. The program is presently on its 22nd (!) season on E4, which is a record.
You might be interested: Liverpool Vs West Ham Which Channel? (TOP 5 Tips)
Are Miles and Ruby together?
What order do I watch Made in Chelsea?
1.1 Series 1 (2011)
1.8 NYC (2014)
1.4 Series
How many episodes of Made in Chelsea are there?
Our beloved Chelsea residents return for what will be the 22nd series of the E4 reality program. The SW3 socialites will be back in Chelsea for the next season, after previously erupting in a stunning Cotswolds house for the previous season.
How many seasons is maid on Netflix?
It is not presently planned that Maid will be renewed for a second season. "Maid" is a television show that is available on Netflix as a "limited series." And fans may be unhappy to find that this generally signifies that the series will be limited to a single installment.
Previous Post:Chelsea Vs Burnley Where To Watch? (TOP 5 Tips)
Next Post:How To Draw A Chelsea Badge? (Solved)


What's that sound? Oh, it's HOUDINI turning over in his grave!
You know, I rarely comment on your postings (because I nearly always agree and love what you do) but I could not sit back on this one. Way too much room for laughter. "I think they should just leave the chains on him"
"You don't need to go to church to be a Christian. If you go to Taco Bell, that doesn't make you a taco."
That's actually a pretty decent joke. Thought it was gonna finish with "...doesn't make you Mexican," but "...doesn't make you a taco" is just ridiculous. I chuckled.
Would have been funnier with chalupa. Chalupa is just a funny sounding word.
"You don't need to go to church to be a Christian. If you go to Taco Bell, that doesn't make you a chalupa."


text:
Q: Azure ServiceBus Eventhub, is the "offset" still available/durable when some of event data is expired? When I write some code to test the EventHub which is a newly released on azure service bus.
As there is very few article online and msdn also do not have rich documentation about the detail of event hub. So I hope someone could share your experience for my question.
For EventHub, we have following statement:

*

*we use "offset" to remember where we are when reading the event data from some partition

*the event data on the EventHub would be expired (automatically?) after some configurable time span
So my question is, can the offset still be available/durable when some of the event data is deleted as the result of expiration?
For example, we have following data on one of partition:
M1 | M2 | M3 | M4  ( oldest --> latest )

After my processing logic runs, let's say that I have processed M1 and M2, so the offset would be the start of M2(when use exclusive mode).
After some time, and if my service is down during that time. M1 is deleted as the result of expiration. so the partition would become:
M2 | M3 | M4 | M.... ( oldest -> latest )

In this case, when my server is restart again, is the offset i stored before is still be available to be used to read from M3?
We can also image this case on runtime when my consumer server is reading the event data on eventhub when some of the oldest event data is expired, does the offset still be available on runtime?
Thanks for any sharing of this question.

A: Based upon how various parts of the documentation is written I believe you will start from the beginning of the current stream as desired if your starting offset is no longer available. EventProcessorHost should follow similar restrictions. Since the sequence numbers are 64 bits, I would expect one of those to be able to serve as an offset within a partition since they monotonically increase without being recycled. The offset should have a similar property. So if EventHubs are designed in a reasonable fashion (ie like similar solutions), then the offsets within partitions can hold despite data expiration. But since I have not yet tested this myself, I will be very unhappy if it is not so, and I'd expect an Azure person to be able to give true confirmation.


text:
Norwalk LA Fitness reopens following closure from Health Department
Erin Kayata
Dec. 8, 2020 Updated: Dec. 8, 2020 1:30 p.m.
Patrons visit the LA Fitness Norwalk location to find that it has been closed Wednesday, December 2, 2020, at iPark in Norwalk, Conn. The Norwalk Health Department and Norwalk Police Department signed a closure order for LA Fitness today due to several repeated violations of State of Connecticut sector rules. Violations include patrons not wearing facemasks or other cloth face coverings, failure to maintain six feet of space between equipment and people, and failure to enforce the revised capacity limits.
Erik Trautmann / Hearst Connecticut Media
NORWALK — LA Fitness has reopened after being shut down due to violations of reopening rules.
According to city spokesperson Josh Morgan, the gym was allowed to reopen on Dec. 4 after submitting a reopening plan to the city.
"LA Fitness was very responsive and submitted a reopening plan on December 2 that addressed the concerns of the Health Department and Police Department," Morgan said.
The gym was originally ordered to close on Dec. 1 after health officials found members weren't wearing masks or adhering to social distancing while exercising. The facility also failed to enforce revised capacity limits, according to Norwalk Health Director Deanna D'Amore.
The inspection was conducted following complaints from patrons about the lack of adherance to reopening rules, officials said.
erin.kayata@hearstmediact.com
Reach Erin on
Erin Kayata is the city hall reporter for the Norwalk Hour. She previously covered education for the Stamford Advocate. Erin got her start covering Darien and New Canaan for the weekly papers there. Before that, she worked at the Boston Globe as part of their co-op program. Erin grew up in southeastern Massachusetts and graduated from Emerson College with a degree in journalism.


Proc Natl Acad Sci U S A. 2016 Aug 2;113(31):E4558-66. doi: 10.1073/pnas.1608319113. Epub 2016 Jul 18.
Loss of TRIM33 causes resistance to BET bromodomain inhibitors through MYC- and TGF-β-dependent mechanisms.
Shi X1, Mihaylova VT1, Kuruvilla L1, Chen F1, Viviano S1, Baldassarre M1, Sperandio D2, Martinez R2, Yue P2, Bates JG2, Breckenridge DG2, Schlessinger J3, Turk BE3, Calderwood DA4.
Department of Pharmacology, Yale University School of Medicine, New Haven, CT 06520; joseph.schlessinger@yale.edu ben.turk@yale.edu david.calderwood@yale.edu.
Department of Pharmacology, Yale University School of Medicine, New Haven, CT 06520; Department of Cell Biology, Yale University School of Medicine, New Haven, CT 06520 joseph.schlessinger@yale.edu ben.turk@yale.edu david.calderwood@yale.edu.
Bromodomain and extraterminal domain protein inhibitors (BETi) hold great promise as a novel class of cancer therapeutics. Because acquired resistance typically limits durable responses to targeted therapies, it is important to understand mechanisms by which tumor cells adapt to BETi. Here, through pooled shRNA screening of colorectal cancer cells, we identified tripartite motif-containing protein 33 (TRIM33) as a factor promoting sensitivity to BETi. We demonstrate that loss of TRIM33 reprograms cancer cells to a more resistant state through at least two mechanisms. TRIM33 silencing attenuates down-regulation of MYC in response to BETi. Moreover, loss of TRIM33 enhances TGF-β receptor expression and signaling, and blocking TGF-β receptor activity potentiates the antiproliferative effect of BETi. These results describe a mechanism for BETi resistance and suggest that combining inhibition of TGF-β signaling with BET bromodomain inhibition may offer new therapeutic benefits.
shRNA screening reveals TRIM33 as a regulator of BETi resistance in cancer cells. (A) Structures of the two different BETi used in this study, JQ1 and GS-626510. (B) KD values of GS-626510 for 40 bromodomains () were determined using BROMOscan (DiscoveRx). The dendrogram Image was generated using TREEspot Software Tool and reprinted with permission from KINOMEscan DiscoveRx Corporation, © DISCOVERX CORPORATION 2010. (C) Dose-dependent inhibition of RKO cell proliferation by JQ1 and GS-626510 in a 5-d assay. Relative viable cell number was determined by CellTiter Glo assay. (D) GS-626510 and JQ1 both down-regulate MYC protein levels. RKO cells were treated with increasing concentrations of BETi for 3 h, and MYC levels in whole cell lysates were assessed by immunoblotting. Actin was used as a loading control. (E) Scheme of shRNA screening procedure. Cells infected by the pooled shRNA library were propagated through eight doublings in presence of either DMSO vehicle control or different concentrations of JQ1 or GS-626510. Genomic DNA was extracted from the T0 (reference) and T4 conditions for determination of proviral shRNA abundance. (F) Top 10 enriched target genes revealed by RIGER analysis in each condition. (G) Multiple individual TRIM33 shRNAs are enriched in BETi-treated, but not in DMSO control treated, conditions. Log2 fold change (T4/T0) of each shRNA is plotted from the most depleted to the most enriched. Each red line represents a single shRNA targeting TRIM33.
Proc Natl Acad Sci U S A. 2016 Aug 2;113(31):E4558-E4566.
(A) Gene expression changes induced by JQ1 and GS-626510 in shCTRL cells are highly correlated. The log2 [fold-change (BETi/DMSO)] of all gene expression following 3-h treatment with 1 µM JQ1 or 0.3 µM GS-626510 was fitted to a line (Microsoft Excel). Two replicate experiments result in an R2 of 0.93 and 0.92, respectively. (B) The pooled shRNA screen described in was repeated and carried out to T5 (10 population doublings). The top 10 enriched genes for each condition, as assessed by RIGER analysis, are listed. TRIM33 was among the top three ranked genes in all four BETi conditions but not in the DMSO condition.
Loss of TRIM33 confers resistance to BETi. (A, Top) Schematic of TRIM33 domain organization and positions of two pairs of RT-PCR primers. (Middle) TRIM33 mRNA levels determined by RT-PCR in shCTRL cell line and cell lines expressing four different TRIM33-targeting shRNAs. (Bottom) TRIM33 protein levels in these cell lines. (B) shCTRL or shTRIM33 cells were seeded in a six-well plate (3 × 105 cells per well) in the presence of DMSO, 100 nM JQ1, or 50 nM GS-626510 and cumulative cell numbers were assessed every 3 d for up to 15 d. (C) Growth inhibition assay. shCTRL and shTRIM33 cells were cultured with different concentrations of JQ1 or GS-626510 for 120 h and relative cell numbers were determined using CellTiter Glo. (D) IC50 values (mean ± SEM) were calculated from five independently performed growth inhibition assays using shCTRL and shTRIM33 cells. P values are based on paired t test. (E) Cells (2 × 104 shCTRL or shTRIM33) were plated in six-well plates, treated with DMSO, 100 nM JQ1, or 50 nM GS-626510 for 2 wk, and then stained with Crystal violet. The Crystal violet staining was quantified at 590-nm absorbance. (F) shCTRL or shTRIM33 cells were transduced with either an empty vector control or TRIM33-expressing lentivirus and cell growth was assessed as in E. (G) TRIM33 expression levels in cells from F were assessed by immunoblotting.
(A and B) The effect of TRIM33 depletion on JQ1 or GS-626510 sensitivity in a panel of cancer cell lines. (A) The IC50 values for each cell line expressing either shCTRL or shTRIM33 was derived from three independent growth inhibition assays and the mean ± SEM of the fold-change in IC50 (shTRIM33/shCTRL) was calculated (*P < 0.05, paired t test). (B) IC50 values (mean ± SEM) for each cell line. (C) Cell proliferation assay of cell lines expressing two independent shRNAs (B5 and A12) targeting TRIM33. Cells were cultured in 100 nM JQ1, or 50 nM GS-626510 for two weeks and then stained with Crystal violet. (D) Crystal violet quantification measured at 590-nm absorbance corresponding to .
(A) RT-PCR quantification of mRNA levels of 15 selected genes whose expression was changed with shTRIM33 or BETi treatment. Error bars represent SD (n = 3). (B) Normalized RNAseq reads of the 15 genes above from two replicate experiments. Error bars represents the SD between the two replicates. (C) Immunoblot showing that BRD4 protein level is not changed by TRIM33 knockdown.
RNAseq analysis of vehicle or BETi-treated shCTRL or shTRIM33 cells. Waterfall plots show gene-expression changes induced by 3-h treatment of shCTRL RKO cells with 1 µM JQ1 (A) or 0.3 µM GS-626510 (B). MYC (red) is down-regulated by both JQ1 and GS-626510. (C) Top 10 sequence motifs enriched in promoter regions of genes down-regulated >twofold by JQ1 and GS-626510 in shCTRL cells were determined by GSEA (Broad Institute). (D) Gene-expression changes induced by shTRIM33 in RKO cells.
TRIM33 modulates MYC sensitivity to BETi. (A) Normalized RNAseq reads of MYC mRNA from two replicate experiments before and after JQ1 or GS626510 treatment. (B) RT-PCR quantification of MYC mRNA in shCTRL, shTRIM33, and shTRIM33 rescued (shTRIM33RES) cells, either untreated or treated with BETi for 3 h. (C) Cells treated similarly to the cells in B were analyzed for MYC protein. (D) MYC protein level in control or MYC overexpressing cells before and after BETi treatment for 3 h. (E) Crystal violet staining of control or MYC overexpressing cells growing with DMSO, JQ1 or GS-626510 for 2 wk. (F) Cumulative cell growth of control or MYC-overexpressing cells over 15 d.
(A and B) ChIP at MYC gene promoter region. Red line threshold indicates IgG control level. (A) TRIM33 ChIP PCR using four different primer pairs (#2, #14, #15, and #16) in the MYC promoter region. (B) BRD4 ChIP PCR using the same set of primers as in A. (C) shCTRL or shTRIM33-transduced RKO and SK-CO-1 cells were treated with 1 μM of JQ1 or 0.3 μM of GS626510 for 24 h. MYC levels were determined by immunoblotting and ERK1/2 was used as a loading control. (D and E) ChIP at the TβRII gene promoter region. Red line threshold indicates IgG control level. (D) TRIM33 ChIP PCR using two different pair of primers (#4 and #3) amplifying the TβRII gene promoter region (E) BRD4 ChIP PCR, using the same set of primers as in D, amplifying the TβRII gene promoter region.
Inhibition of TGF-β signaling potentiates the antiproliferative effects of BETi. (A) TGF-β1 ligand-stimulated phosphorylation of SMAD2 is potentiated in shTRIM33 cells. shCTRL or shTRIM33 RKO cells were treated with increasing doses of TGF-β1 for 25 min (Left) or with 2 ng/mL TGF-β1 for various times (Right); cells were lysed and immunoblotted for pSMAD2, total SMAD2, and TRIM33. (B) shCTRL or shTRIM33 cells were untreated or treated with 100 pM of TGF-β1 for 25 min and SMAD4 was immunoprecipitated. Coprecipitating pSMAD2 was assessed by immunoblotting. (C) shCTRL or shTRIM33 cells were infected with lentivirus encoding shCTRL or one of two hairpins targeting SMAD4 (shSMAD4-3 or shSMAD4-4). Cells were untreated or treated with 100 pM of TGF-β1 for 25 min. SMAD4, pSMAD2, and total SMAD2 levels were assessed by immunoblotting. (D) TGF-β receptor II (TβRII) mRNA from RNAseq in shCTRL and shTRIM33 cells. (E–G) Inhibition of TGF-β pathway by silencing TβRII increases the magnitude of cell growth inhibition by BETi. (E) RT-PCR quantification of TβRII mRNA levels in shCTRL and shTRIM33 cells expressing control (shCTRL) or two different TβRII-targeting shRNAs (shTβRII-3 and shTβRII-4). (F) Cells from E were stimulated with 100 pM of TGF-β1 for 25 min and pSMAD2 levels assessed by immunoblotting. (G) shCTRL cells (Left) or shTRIM33 cells (Right) expressing control and TβRII-targeting shRNAs were cultured for 2 wk with DMSO or different concentrations of BETi (as indicated) and then stained with Crystal violet. (H–J) The TβRI inhibitor LY2157299 potentiates BETi-mediated inhibition of cell proliferation. (H) shTRIM33 cells were pretreated with increasing doses of LY2157299 and then exposed to 100 pM TGF-β1 for 25 min. Immunoblotting shows dose-dependent inhibition of pSMAD2 by LY2157299. (I) shCTRL and two shTRIM33 knockdown cell lines were cultured in the presence of JQ1 or GS-626510, with or without LY2157299 for 2 wk and stained with Crystal violet. (J) shCTRL or shTRIM33 cells were treated with 1 µM JQ1 or 0.3 µM GS-626510 with or without 5 µM LY2157299 overnight, and MYC protein levels were assessed by immunoblotting.
(A) GSEA shows that the down-regulation of TGF-β and MYC signatures by JQ1 was significantly decreased in shTRIM33 in comparison with shCTRL cells. NES, normalized enrichment score. (B and C) Crystal violet quantification measured at 590-nm absorbance corresponding to , respectively. (D–F) Overexpression of TβRII is not sufficient to induce resistance to BETi. (D) pLentiCMV-EV or pLentiCMV-TβRII transduced stable cell lines were treated with increasing doses of TGF-β1 for 25 min and pSMAD2 levels assessed by immunoblotting. (E) Cells from D were cultured in the presence of DMSO, 100 nM JQ1, or 50 nM GS-626510 for 2 wk and stained with Crystal violet. (F) pLentiCMV-EV or pLentiCMV-TβRII stable cell lines were treated with 1 µM JQ1 or 0.3 µM GS-626510 for overnight either in the presence or in the absence of 100 pM TGF-β1. MYC levels were determined by immunoblotting and actin was used as a loading control.


text:
Sen. Josh Hawley claims without evidence that liberals are attacking masculinity Hawley is calling for a "revival of ... manhood in America." Kristin Kobes Du Mez, a Calvin University professor and the author of Jesus and John Wayne, explains how masculinity is a political issue.
< Sen. Josh Hawley claims masculinity is under attack. This historian disagrees
Updated November 11, 202112:39 PM ET Originally published November 11, 20215:06 AM ET
Sen. Josh Hawley claims masculinity is under attack. This historian disagrees
Embed <iframe src="https://www.npr.org/player/embed/1054615028/1054615029" width="100%" height="290" frameborder="0" scrolling="no" title="NPR embedded audio player">
Senator Josh Hawley says he's defending men. The Missouri Republican spoke last week at the National Conservatism Conference. He attacked the political left, as many Republicans do, and alleged they are targeting masculinity.
JOSH HAWLEY: This is an effort that the left has been at for years now, and they have had alarming success. American men are working less. They're getting married in fewer numbers. They're fathering fewer children. They're suffering more anxiety and depression. They're engaging in more substance abuse.
INSKEEP: In a TV interview with Axios, Hawley said he wants to make this a signature political issue. We heard a critique of Hawley's speech from Kristin Kobes Du Mez. She is the author of an acclaimed book on Christian nationalism called "Jesus And John Wayne." It argues that white evangelicals embraced an idea of men drawn more from Western movies than from the Bible. She teaches at Calvin University, which is a Christian school in Michigan. When she read Hawley's speech, she said something was missing.
KRISTIN KOBES DU MEZ: It's never entirely clear how he defines masculinity, even though he's quite certain that masculinity is under attack, and the left is trying to do away with real men. He uses words like courage, independence and assertiveness. He is calling on conservative men to step up to their roles as providers and protectors - protectors of faith, family and nation and to protect what he calls our culture.
INSKEEP: I want to ask how this compares to what's really going on in society because he seems to have keyed off of a Wall Street Journal article that interviewed a lot of men, and it's not the only article I've seen that's played on this theme. The September 6 article had the headline "A Generation Of American Men Give Up On College." And then there's a quote from a young man, I just feel lost. And they cite a real statistic that of college students right now, they're almost 60% women. Would you agree that something is going on or even going wrong there?
DU MEZ: I think that there are many challenges that the younger generation is facing right now, women and men. But there are a lot of assumptions that Hawley's making that the problems are caused by some sort of destruction of manhood or destruction of masculinity when we could look at, what are the expectations of masculinity that might be inappropriate, that might be outmoded that are perhaps exacerbating this crisis?
INSKEEP: I want to hear more of Senator Hawley's speech. Let's listen.
HAWLEY: It's hard to accept that the pathologies gripping so many American men are good for American society. I'd argue just the opposite. Now, this is not to say that American women aren't central to this story - far from it. American women have shaped our culture every bit as much as men, and their virtues are every bit as necessary to the success of our republic.
DU MEZ: He's drawing on these notions of gender difference that women and men are created by God in very distinct ways, often kind of pitted as opposites. So men are to be courageous and independent and assertive or aggressive, whereas women are made by God to be dependent, to be submissive, to be delicate. Men are protectors. Women are designed to be protected. And so these - this vision of gender difference really runs through conservative Christianity and through American conservatism more generally.
INSKEEP: I guess we should be clear. We're not asserting exactly what Senator Hawley's religious belief is. But you're telling me this is a very widespread religious belief, and this is how a lot of people would read that speech.
DU MEZ: Exactly. It would resonate powerfully with conservative evangelicals in particular and with conservatives more broadly.
INSKEEP: I'm obliged to note that Senator Hawley made enormous news on January 6, 2021, when he held up a fist to protesters outside the Capitol who were soon inside the Capitol in an attack on democracy. And Senator Hawley voted to object to the election in which Donald Trump was defeated. How does Donald Trump and his story match up with these traditional masculine virtues?
DU MEZ: Well, it's a bit of a leap. Donald Trump isn't generally seen as a particularly virtuous man, but here we have to understand that these traditional masculine virtues are in the service of white Christian nationalism, really. And that's clear that for Hawley, he's really using this militant language. And that militancy does sanction violence, and that would resonate with many among his base. We have survey data that shows the majority of white evangelicals believe the election was stolen. And of those, 39% believe that violence may be necessary to save the country.
INSKEEP: You know, if Senator Hawley were with us here - and I should note we've invited him on the program; he is welcome - if he were here, I wonder if he would take issue with a word that you've used. You said white Christian nationalism. Senator Hawley might say, I didn't say anything about race here. And in fact, he can point to his speech in which he refers to dads at Southwood High School in Shreveport, La. These are, if I'm not mistaken, largely African American fathers who were concerned about school violence and went into the school to sort of patrol the hallways. Why do you use the word white when you're talking about this Christian nationalism?
DU MEZ: Yeah. With this calling on men to defend our shared culture, in his words, he really does seem to be tapping into a distinctive notion of who real Americans are. And those are Americans who share his conservative values, not just around gender, but arguably also around what this country is supposed to be, what this country is supposed to look like. I think we do need to understand how these words that he's using resonate in particular ways with his base.
INSKEEP: One other question - we're in this period of really intense identity politics, as you know. And there are a lot of social narratives and political narratives in which white men are explicitly made out to be the bad guys. What do you think white men should make of that?
DU MEZ: I can understand that, and I think it could be frustrating in some cases. If you're looking to strengthen fathers, there are many ways that, you know, the left is actually working to do so, such as paid paternity leave and broader family leave policies - and that there can be ways to find common ground here rather than pitting half of America against the other half. And I think that white men actually have a really critical role to play in that respect.
INSKEEP: Kristin Kobes Du Mez, it's always a pleasure talking with you. Thank you so much.
DU MEZ: Thank you.


text:
\section{Introduction}

There are many indications for physics beyond the standard model (SM)
\cite{bsm}. One among
them is the existence of non-zero neutrino masses \cite{neurev}.
Some of the indications
for new physics can be sucessfully explained in supersymmtric models
\cite{susy}.
For this reason, neutrino masses have been addressed in supersymmetry.
In a neutrino mass model, there is a possibility for
lepton flavor violation (LFV) \cite{lfvrev},
for which there is no direct evidence.
Experiments have put upper bounds on the branching ratios of these
LFV processes \cite{meg,lfvboun,pdg}.
Due to Glashow-Iliopoulos-Maiani cancellation mechanism, these processes
are highly suppressed in
the SM and the above mentioned upper bounds are obviously satisfied in it.
However, a signal for any LFV process with an appreciable branching ratio
gives a confirmation for new physics.

In this work, we study LFV processes of the form $\ell_i\to\ell_j\gamma$
in a supersymmetrized model for neutrino masses \cite{ma}. Here,
$\ell_i,i=1,2,3$, are charged leptons.
The above mentioned model arises after supersymmetrizing the
inert Higgs doublet model \cite{scoto,IH}. The inert Higgs doublet model
\cite{scoto} offers explanation for neutrino masses and dark matter.
In this model \cite{scoto}, dark matter is stable due to
an exact $Z_2$ symmetry
and the neutrinos acquire masses at 1-loop level.
This model has been extensively studied and some recent works on this
can be seen in \cite{IHrec}. Supersymmetrizing this model could bring new
features and it is done in \cite{ma}. In the supersymmetrization of the
inert Higgs doublet model \cite{ma}, the discreet symmetry is extended to
$Z_2\times Z_2^\prime$.
In this
model, dark matter can be multi-partite \cite{multi} due to the
presence of $R$-parity
and the $Z_2^\prime$ symmetry. Some variations of this model
are also presented in \cite{refs,st}.
In the model of \cite{ma}, gauge coupling unification
is possible by embedding it in a supersymmetric SU(5) structure \cite{su5}.
The origin of the discrete symmetry $Z_2\times Z_2^\prime$, which is
described above,
is also explained by realizing it as a residual symmetry from a U(1)
gauged symmetry \cite{u1ma}.

In this work we consider the model of \cite{ma} and present the expression
for neutrino masses, which arises from two 1-loop diagrams. We will
demonstrate that neutrino masses are tiny in this model if either the
neutrino Yukawa couplings are suppressed
or some certain soft parameters of the scalar potential are
fine-tuned. We consider the later case, in which the neutrino Yukawa couplings
can be ${\cal O}(1)$, and they can drive LFV processes such as
$\mu\to e\gamma$. In our work we assume flavor diagonal in the slepton
mass matricies as well as in the $A$-terms of sleptons. Hence, in our model,
lepton flavor violation is happening due to non-diagonal
Yukawa couplings. Under the above mentioned scenario, we compute
branching ratio for the decays $\ell_i\to\ell_j\gamma$.
Among these decays, we show that $\mu\to e\gamma$ can give stringent
constraints on model parameters, especially on right-handed neutrino mass.
Early calculations on $\mu\to e\gamma$ in a lepton number violating
supersymmetric model can be seen in \cite{ihl}.

In the model of \cite{ma}, apart from $\mu\to e\gamma$ there can also
be an LFV decay of $\mu\to 3e$. In a Type-II seesaw mechanism for neutrino
masses, the decay $\mu\to 3e$ can take place at tree level, due to the
presence of triplet Higgs boson. In our model
\cite{ma}, there are no triplet Higgses, hence the decay $\mu\to 3e$ will
take place at loop level. The current experimental upper limit on
$Br(\mu\to 3e)$ is $1\times 10^{-12}$ \cite{mu3e}, which is about two times
larger than that of $Br(\mu\to e\gamma)$. So we can expect
$Br(\mu\to e\gamma)$ to put
somewhat tighter constraints on model parameters than that due to
$Br(\mu\to 3e)$. Hence, in this work we focus on the computation of
$Br(\mu\to e\gamma)$. It may happen that $Br(\mu\to 3e)$ and
$Br(\mu\to e\gamma)$ may put some additional constraints on model parameters,
but we study these in a separate work.

This paper is organized as follows. In the next section, we describe the
model of \cite{ma}. In section 3, we present the expressions for neutrino
masses and branching ratios for the decays $\ell_i\to\ell_j\gamma$.
In section 4, we give
neumerical results on neutrino masses and $\mu\to e\gamma$. We conclude in
section 5.

\section{The model}

The model of Ref.\cite{ma} is an extension of minimal supersymmetric standard
model (MSSM).
The additional superfields of this model are as follows:
(i) three right-handed neutrino fields, $\hat{N}_i$, $i=1,2,3$, (ii)
two electroweak doublets $\hat{\eta}_1=(\hat{\eta}_1^0,\hat{\eta}_1^-)$,
$\hat{\eta}_2=(\hat{\eta}_2^+,\hat{\eta}_2^0)$, (iii)
a singlet field $\hat{\chi}$. Under the electroweak gauge group
SU(2)$_L\times$U(1)$_Y$, the charges of these additional superfields are
given in Table 1.
\begin{table}[!h]
\begin{center}
\begin{tabular}{c|cccc}\hline\hline
Field & $\hat{N}_i$ & $\hat{\eta}_1$ & $\hat{\eta}_2$ & $\hat{\chi}$ \\\hline
SU(2)$_L\times$U(1)$_Y$ & (1,0) & (2,-1/2) & (2,1/2) & (1,0) \\\hline\hline
\end{tabular}
\end{center}
\caption{Charge assignments of additional superfields of the model
under the electroweak gauge group.}
\end{table}
The model of Ref.\cite{ma} contains discrete symmetry $Z_2\times Z_2^\prime$,
under which all the quark and Higgs superfields can be taken to be even.
The leptons and the additional fields described above are charged
non-trivially under this discrete symmetry \cite{ma}. The purpouse
of this symmetry is to disallow the Yukawa term $\hat{L}_i\hat{H}_u\hat{N}_j$
in the superpotential of the model, and as a result the neutrino
remains massless at tree level. Here,
$\hat{L}_i=(\hat{\nu}_i,\hat{\ell}_i)$, $i=1,2,3$, are
the lepton doublet superfields. The singlet charged lepton superfield
is represented by $\hat{E}^c_i$, $i=1,2,3$.
We denote up- and down-type Higgs superfields
as $\hat{H}_u$ and $\hat{H}_d$ respectively.

The superpotential of our model consisting of electroweak fields can be
written as \cite{ma}
\begin{eqnarray}
W&=& (Y_E)_{ij}\hat{L}_i\hat{H}_d\hat{E}^c_j+
(Y_\nu)_{ij}\hat{L}_i\hat{\eta}_2\hat{N}_j+\lambda_1\hat{H}_d\hat{\eta}_2
\hat{\chi}+\lambda_2\hat{H}_u\hat{\eta}_1\hat{\chi}+
\nonumber \\
&&\mu \hat{H}_u\hat{H}_d+\mu_\eta\hat{\eta}_2
\hat{\eta}_1+\frac{1}{2}\mu_\chi\hat{\chi}\hat{\chi}+\frac{1}{2}M_{ij}
\hat{N}_i\hat{N}_j
\label{eq:W}
\end{eqnarray}
Here, there is a summation over indices $i,j$ which run from 1 to 3.
The first and second terms in the above equation are Yukawa terms
for charged leptons and neutrinos, respectively. But,
as described before, $\hat{\eta}_2$ is odd under the discrete symmetry
of the model and hence the scalar component of it does not acquire vacuum
expectation value (vev) \cite{ma}. So neutrinos are still massless at
tree level.
Apart from the superpotential of Eq. (\ref{eq:W}), we should consider
the scalar potential. The relavant terms in the scalar potential
are given below.
\begin{eqnarray}
V &=& (m^2_L)_{ij}\tilde{L}_i^\dagger\tilde{L}_j+m^2_{\eta_1}\eta_1^\dagger
\eta_1+m^2_{\eta_2}\eta_2^\dagger\eta_2+m^2_\chi\chi^*\chi+
(m^2_N)_{ij}\tilde{N}^*_i\tilde{N}_j+
\nonumber \\
&& \left[(AY_\nu)_{ij}\tilde{L}_i\eta_2\tilde{N}_j+(A\lambda)_1H_d\eta_2\chi
+(A\lambda)_2H_u\eta_1\chi \right.
\nonumber \\
&&\left. +b_\eta\eta_2\eta_1+\frac{1}{2}b_\chi\chi\chi
+\frac{1}{2}(b_M)_{ij}\tilde{N}_i\tilde{N}_j+{\rm c.c.}\right].
\label{E:spot}
\end{eqnarray}

As we have explained before that our motivation is to study LFV processes
in the above described model. The LFV processes can be driven by charged
sleptons. For instance, the off-diagonal elements of soft parameters,
$(m^2_L)_{ij}$, can drive LFV processes. Similarly, we can write soft mass
terms for singlet charged sleptons, $\tilde{E}_i,i=1,2,3$, in the
scalar potential. Also, there can exist
$A$-terms connecting $\tilde{L}_i$ and $\tilde{E}_j$. The off-diagonal
terms of the above mentioned soft terms can drive LFV processes, which actually
exist in MSSM. Since our model \cite{ma} is an extension of MSSM, we
are interested in LFV processes generated by the additional fields
of this model. Hence, we assume that the off-diagonal terms of the
soft terms, which are described above, are zero.

For simplicity, we assume that the parameters of the superpotential and
scalar potential of our model are real. Then, by an orthogonal transformation
among the neutrino superfields, $\hat{N}_i$, we can make the
the following parameters to be diagonal, which are given below.
\begin{equation}
M_{ij}=M_i\delta_{ij},\quad (m_N^2)_{ij}=(m_N^2)_i\delta_{ij},\quad
(b_M)_{ij}=(b_M)_i\delta_{ij}
\end{equation}
By going to an appropriate basis of $\hat{L}_i$ and $\hat{E}_j$, we can
get the Yukawa couplings for charged leptons to be diagonal.
After doing this, we are left with no freedom and hence the neutrino
Yukawa couplings, $(Y_\nu)_{ij}$, can be non-diagonal. These
non-diagonal Yukawa couplings can drive LFV processes such as
$\ell_i\to\ell_j\gamma$.
These LFV processes are driven at the 1-loop level, which we describe
in the next section. As explained before, neutrinos also acquire masses
at 1-loop level in this model \cite{ma}. To calculate these loop diagrams
we need to know the mass eigenstates of the scalar and fermionic partners
of the fields shown in Table 1, since these fields enter into the
loop processes. Expressions for these mass eigenstates are given in
Ref.\cite{akot}. However, our notations and conventions are different from
that of Ref.\cite{akot}. Hence, for the sake of completeness we present
them below.

The charged components of $\hat{\eta}_1,\hat{\eta}_2$ can be
fermionic and scalar, which can be written as
$(\tilde{\eta}_1^-,\tilde{\eta}_2^+)$ and $(\eta_1^-,\eta_2^+)$,
respectively. The two charged fermions
represent chargino-type fields, whose mass is $\mu_\eta$.  Whereas,
the charged scalars, in the basis $\Phi_+^{\rm T}=(\eta_2^+,\eta_1^{-*})$,
will have a mass matrix which is given below.
\begin{equation}
{\cal L}\ni -\Phi_+^\dagger
\left(\begin{array}{cc}
\mu_\eta^2+m_{\eta_2}^2+\frac{g^2-g^{\prime 2}}{4}v^2\cos(2\beta) & b_\eta \\
b_\eta & \mu_\eta^2+m_{\eta_1}^2-\frac{g^2-g^{\prime 2}}{4}v^2\cos(2\beta)
\end{array}\right)\Phi_+
\end{equation}
Here, $g,g^\prime$ are the gauge couplings of SU(2)$_L$ and U(1)$_Y$,
respectively. $\beta$ is defined as $\tan\beta =\frac{v_2}{v_1}=
\frac{\langle H_u^0\rangle}{\langle H_d^0\rangle}$ and $v^2=v_1^2+v_2^2$.
We can diagonalize the above mass matrix by taking $\Phi_+$ as
\begin{eqnarray}
\Phi_+=\left(\begin{array}{cc}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{array}\right)
\left(\begin{array}{c}
\eta^+_{m2}\\\eta^+_{m1}\end{array}\right),
\quad
\tan 2\theta = \frac{2b_\eta}{m^2_{\eta_2}-m^2_{\eta_1}+(g^2-g^{\prime 2})
v^2\cos(2\beta)/2}
\end{eqnarray}
Here, $\eta^+_{m1}$ and $\eta^+_{m2}$ are mass eigenstates of the
charged scalar fields and we denote their mass eigenvalues by
$m_{1+}$ and $m_{2+}$, respectively.

The neutral fermionic and scalar components of
$\hat{\eta}_1,\hat{\eta}_2,\hat{\chi}$ can be written as
$\Psi^{\rm T}=(\tilde{\eta}_1^0,\tilde{\eta}_2^0,\tilde{\chi})$
and $\Phi_0^{\rm T}=(\eta_1^0,\eta_2^0,\chi)$, respectively.
The neutral fermionic fields will have a mixing mass matrix,
which is given below.
\begin{equation}
{\cal L} \ni -\frac{1}{2}\Psi^{\rm T}M_\eta\Psi,
\quad
M_\eta = \left(\begin{array}{ccc}
0 & -\mu_\eta & -\lambda_2v_2 \\
-\mu_\eta & 0 & \lambda_1v_1 \\
-\lambda_2v_2 & \lambda_1v_1 & \mu_\chi \end{array}\right)
\end{equation}
The above mixing matrix can be diagonalized by an orthogonal matrix as
\begin{equation}
U_\eta^{\rm T}M_\eta U_\eta = {\rm diag}(m_{\tilde{\eta}_1},m_{\tilde{\eta}_2},
m_{\tilde{\eta}_3})
\end{equation}

The neutral scalar fields of $\Phi_0$ can be written as
\begin{equation}
\Phi_0=\frac{1}{\sqrt{2}}\Phi_R+\frac{i}{\sqrt{2}}\Phi_I=
\frac{1}{\sqrt{2}}\left(\begin{array}{c}
\eta^0_{1R}\\\eta^0_{2R}\\\chi_R
\end{array}\right)+\frac{i}{\sqrt{2}}
\left(\begin{array}{c}
\eta^0_{1I}\\\eta^0_{2I}\\\chi_I
\end{array}\right)
\end{equation}
The mixing matrix among these fields can be written as
\begin{equation}
{\cal L}\ni -\frac{1}{2}\Phi_R^{\rm T}m_{\eta_R}^2\Phi_R
-\frac{1}{2}\Phi_I^{\rm T}m_{\eta_I}^2\Phi_I
\end{equation}
Here, the mixing matrices $m_{\eta_R}^2,m_{\eta_I}^2$ can be obtained from the
following matrix
\begin{eqnarray}
m_\eta^2(\epsilon)&=&\left(\begin{array}{ccc}
m_{11}^2 & m^2_{12} & m^2_{13} \\
m^2_{12} & m^2_{22} & m^2_{23} \\
m^2_{13} & m^2_{23} & m^2_{33}
\end{array}\right),
\quad
m_{11}^2= \mu_\eta^2+m_{\eta_1}^2+\lambda_2^2v_2^2+\frac{g^2+g^{\prime 2}}
{4}v^2\cos(2\beta),
\nonumber \\
m_{22}^2&=& \mu_\eta^2+m_{\eta_2}^2+\lambda_1^2v_1^2-\frac{g^2+g^{\prime 2}}
{4}v^2\cos(2\beta),
\quad
m_{33}^2= \mu_\chi^2+m_\chi^2+\lambda_1^2v_1^2+\lambda_2^2v_2^2+\epsilon
b_\chi ,
\nonumber \\
m_{12}^2&=& -\lambda_1\lambda_2v_1v_2-\epsilon b_\eta ,
\quad
m_{13}^2= -\lambda_1v_1\mu_\eta -\lambda_2v_2\mu_\chi -\epsilon
[(A\lambda)_2v_2-\mu \lambda_2v_1]
\nonumber \\
m_{23}^2&=& \lambda_1v_1\mu_\chi +\lambda_2v_2\mu_\eta +\epsilon
[(A\lambda)_1v_1-\mu \lambda_1v_2]
\label{Eq:m2eta}
\end{eqnarray}
Here, $\epsilon$ can take $+1$ or $-1$. We have $m_{\eta_R}^2=m_\eta^2(+1)$ and
$m_{\eta_I}^2=m_\eta^2(-1)$. These two mixing mass matrices can be
diagonalized by
orthogonal matrices $U_R$ and $U_I$, which are defined below.
\begin{equation}
U_R^{\rm T}m_{\eta_R}^2U_R={\rm diag}(m^2_{\eta_{R1}},m^2_{\eta_{R2}},
m^2_{\eta_{R3}}),
\quad
U_I^{\rm T}m_{\eta_I}^2U_I={\rm diag}(m^2_{\eta_{I1}},m^2_{\eta_{I2}},
m^2_{\eta_{I3}})
\end{equation}

At last, the fermionic and scalar components of right-handed neutrino
superfields, $\hat{N}_i$, can be donted by $N_i$ and $\tilde{N}_i$,
respectively. The fermionic components have masses $M_i$. The scalar
components can be decomposed into mass eigenstates as
\begin{equation}
\tilde{N}_i=\frac{1}{\sqrt{2}}\left(\tilde{N}_{Ri}+i\tilde{N}_{Ii}\right)
\label{Eq:sneu}
\end{equation}
The mass-squares of $\tilde{N}_{Ri}$ and $\tilde{N}_{Ii}$, respectively, are
given below.
\begin{equation}
m_{Ri}^2=M_i^2+(m_{N}^2)_i+(b_{M})_i,\quad
m_{Ii}^2=M_i^2+(m_{N}^2)_i-(b_{M})_i
\label{Eq:m2RI}
\end{equation}

\section{Neutrino masses and LFV processes}

As described before that in the model of Ref.\cite{ma} neutrinos are
massless at tree level due to the presence of the discrete symmetry
$Z_2\times Z_2^\prime$. However, in this model
neutrinos acquire masses at 1-loop level, whose diagrams are shown
in Figure 1 \cite{ma}.
\begin{figure}[!h]
\begin{center}

\includegraphics[width=6in]{radmas.eps}

\end{center}
\caption{Radiative masses for neutrinos.}
\end{figure}
After computing these 1-loop diagrams, we have found the following
mass matrix for neutrinos.
\begin{eqnarray}
\left(m_\nu\right)_{ij}&=&\sum_{k,l=1}^3\frac{(Y_\nu)_{ik}(Y_\nu)_{jk}}
{16\pi^2}M_k
\left[[U_R(2,l)]^2\frac{m_{\eta_{Rl}}^2}{m_{\eta_{Rl}}^2-M_k^2}\ln
\frac{m_{\eta_{Rl}}^2}{M_k^2}-
[U_I(2,l)]^2\frac{m_{\eta_{Il}}^2}{m_{\eta_{Il}}^2-M_k^2}\ln
\frac{m_{\eta_{Il}}^2}{M_k^2}\right]
\nonumber \\
&&+\sum_{k,l=1}^3\frac{(Y_\nu)_{ik}(Y_\nu)_{jk}}{16\pi^2}[U_\eta(2,l)]^2
m_{\tilde{\eta}_l}
\left[\frac{m_{Rk}^2}{m_{Rk}^2-m_{\tilde{\eta}_l}^2}\ln\frac{m_{Rk}^2}
{m_{\tilde{\eta}_l}^2}-
\frac{m_{Ik}^2}{m_{Ik}^2-m_{\tilde{\eta}_l}^2}\ln\frac{m_{Ik}^2}
{m_{\tilde{\eta}_l}^2}\right]
\label{Eq:mnu}
\end{eqnarray}
It is to be noticed that the first and second lines of the above
equation arises from the left- and right-handed diagrams of Figure 1.

In our work we assume supersymmetry breaking to be around 1 TeV. Hence,
we can take all the supersymmetric (SUSY) particle masses to be around few
hundred GeV. With this assumption, we can estimate the
neutrino Yukawa couplings by requiring that the neutrino mass scale to be
around 0.1 eV \cite{neurev}. With this requirement, we have found that
$(Y_\nu)_{ij}\sim 10^{-5}$.
Here there are six different Yukawa couplings, which need to be suppressed
to ${\cal O}(10^{-5})$. This could be one possibility in this model
in order to explain the
correct magnitude for neutrino masses. However, in this case, since
the Yukawa couplings are suppressed, LFV processes such as
$\ell_i\to\ell_j\gamma$ would also be suppressed. These LFV processes
will be searched in future experiments \cite{fulfv}, hence
it is worth to consider the case where these processes can have substancial
contribution in this model. In otherwords, we have to look for a parameter
region where we can have $(Y_\nu)_{ij}\sim{\cal O}(1)$.

From Eq. (\ref{Eq:mnu}), it can observed that each diagram of Figure 1
contribute positive and negitive quantities to the neutrino mass
matrix. Without suppressing Yukawa couplings, by fine-tuning
the masses of SUSY particles, we may achieve
partial cancellation between the positive and negative contributions
of Eq. (\ref{Eq:mnu}) and endup with tiny masses for neutrinos. To
demonstrate this explicitly, using Eq. (\ref{Eq:m2RI}), we can notice
that in the limit $(b_M)_i\to 0$ we get $m_{Ri}^2- m_{Ii}^2\to 0$,
and hence the second line of Eq. (\ref{Eq:mnu}) would give tiny contribution.
The first line of Eq. (\ref{Eq:mnu}) can give very small value in the
following limiting process: $U_R(2,l)-U_I(2,l)\to 0$ and
$m_{\eta_{Rl}}-m_{\eta_{Il}}\to 0$. To achieve this limiting process
we have to make sure that the elements of the matrices $m_{\eta_R}^2$ and
$m_{\eta_I}^2$ are close to each other. From the discussion around Eq.
(\ref{Eq:m2eta}), we can observe that the elements of $m_{\eta_R}^2$ and
$m_{\eta_I}^2$ can differ by quantities which are proportional to $\epsilon$.
These quantities depend on the following parameters:
$b_\chi$, $b_\eta$, $(A\lambda)_1$ and $(A\lambda)_2$. By taking the
following limit:
$(A\lambda)_{1}-\lambda_{1}\mu v_2/v_1\to 0$,
$(A\lambda)_{2}-\lambda_{2}\mu v_1/v_2\to 0$, $b_\eta\to 0$, $b_\chi\to 0$,
we can get tiny contribution from the first line of Eq. (\ref{Eq:mnu}).
To sum up the above discussion, without suppressing the neutrino Yukawa
couplings we can fine-tune the below seven paramters, in order to
get very small neutrino masses in this model.
\begin{equation}
(b_M)_i,i=1,2,3,\quad b_\eta,\quad b_\chi,\quad (A\lambda)_1,\quad
(A\lambda)_2
\label{Eq:para}
\end{equation}
Apparently, the above parameters are SUSY breaking soft parameters
of the scalar potential of this model. A study of neutrino masses depending
on SUSY breaking soft parameters can be seen in \cite{af}.

In the previous paragraph we have argued that Majorana masses for neutrinos
are vanishingly small when we fine tune certain soft parameters of the model.
We can understand these features from symmetry arguments.
For instance, when lepton number
is conserved, neutrinos cannot have Majorana masses. For lepton number,
we can propose a group U(1)$_L$, under which the following fields are assigned
the corresponding charges and the rest of the superfields are singlets.
\begin{equation}
\hat{L}_i\mapsto +1,\quad\hat{E}^c_i\mapsto -1,\quad\hat{N}_i\mapsto -1
\end{equation}
With the above mentioned charges, we can see that the last term in Eqs.
(\ref{eq:W}) and (\ref{E:spot}) are forbidden. In fact, in the limit
$M_i\to 0$ and $(b_M)_i\to 0$, the two diagrams of Figure 1 give zero masses
to neutrinos. Hence, in order to get Majorana masses for neutrinos, we have
softly broken the lepton number symmetry. Now, even if we have $M_i\ne 0$,
we have described in the previous paragraph that the left-handed diagram of
Figure 1 can still give vanishingly small masses by fine tuning some
soft parameters.
This suggests that apart from U(1)$_L$ there can exist some additional
symmetries. Suppose we set $(A\lambda)_{1}v_1-\lambda_{1}\mu v_2=0$,
$(A\lambda)_{2}v_2-\lambda_{2}\mu v_1=0$. Then, as argued previously that
the left-handed diagram of Figure 1 gives zero neutrino masses for $b_\eta\to 0$
and $b_\chi\to 0$, even if $M_i\ne 0$. This case can be understood by
proposing additional symmetry U(1)$_\eta$, under which the following fields
have non-trivial charges and the rest of the fields are singlets.
\begin{equation}
\hat{L}_i\mapsto +1,\quad\hat{E}^c_i\mapsto -1,\quad\hat{\eta}_1\mapsto -1,
\quad\hat{\eta}_2\mapsto -1,\quad\hat{\chi}\mapsto +1
\end{equation}
Using the above charges, we can notice that $\mu_\eta$-, $\mu_\chi$-terms
in Eq. (\ref{eq:W}) and $b_\eta$-, $b_\chi$-terms in Eq. (\ref{E:spot}) are
forbidden. Thus, the additional symmetry U(1)$_\eta$ can forbid the Majorana
masses for neutrinos in the left-handed diagram of Figure 1. Finally, one may
ask how the relations $(A\lambda)_{1}v_1-\lambda_{1}\mu v_2=0$,
$(A\lambda)_{2}v_2-\lambda_{2}\mu v_1=0$ can be satisfied. In these two
relations, SUSY breaking soft masses are related to SUSY conserving mass
$\mu$. These relations may be achieved my proposing certain symmetries
in the mechansim
for SUSY breaking, which is beyond the reach of our present work.

Previously, we have motivated a parameter region where
the neutrino Yukawa couplings can be ${\cal O}(1)$. For these values
of neutrino Yukawa couplings, LFV processes such as $\ell_i\to\ell_j\gamma$
can have substantical contribution in our model, and worth to compute them.
The Feynman diagrams for $\ell_i\to\ell_j\gamma$ are given in Figure 2.
\begin{figure}[!h]
\begin{center}

\includegraphics[width=6in]{lfv.eps}

\end{center}
\caption{Lepton flavor violating decays of the form $\ell_i\to\ell_j\gamma$.}
\end{figure}

The general form of the amplitude for $\ell_i\to\ell_j\gamma$ is as follows.
\begin{equation}
{\cal M}=e\epsilon^*_\mu(q)\bar{u}_j(p-q)\left[A_L^{(ij)}\frac{1-\gamma_5}{2}
+A_R^{(ij)}\frac{1+\gamma_5}{2}\right]i\sigma^{\mu\nu}q_\nu u_i(p)
\label{Eq:amp}
\end{equation}
It is to be noted that in the above equation, there is no summation over
the indices $i,j$. The quantities $A_{L,R}^{(ij)}$ of the above
equation can be found from the 1-loop diagrams of Figure 2, which we
have given below.
\begin{eqnarray}
A_L^{(ij)}&=&A^{(ij)}m_j,\quad A_R^{(ij)}=A^{(ij)}m_i,
\nonumber \\
A^{(ij)}&=&\sum_{k=1}^3\frac{(Y_\nu)_{ik}(Y_\nu)_{jk}}{16\pi^2}
\left\{\frac{1}{4\mu_\eta^2}\left[f_2(x_{Rk})+f_2(x_{Ik})\right]
-\left[\cos^2\theta\frac{f_2(x_{k2})}{2m^2_{2+}}+\sin^2\theta\frac{f_2(x_{k1})}
{2m^2_{1+}}\right]\right\},
\nonumber \\
x_{Rk}&=&\frac{m_{Rk}^2}{\mu_\eta^2},\quad x_{Ik}=\frac{m_{Ik}^2}{\mu_\eta^2},
\quad x_{k2}=\frac{M_k^2}{m^2_{2+}},\quad x_{k1}=\frac{M_k^2}{m^2_{1+}},
\nonumber \\
f_2(x)&=&\frac{1}{(1-x)^4}\left[\frac{1}{6}-x+\frac{1}{2}x^2+\frac{1}{3}
x^3-x^2\ln(x)\right].
\end{eqnarray}
From the above expressions, we can notice that in the curly brackets of
$A^{(ij)}$, the first two and the last two terms are arising from the
left- and right-handed diagrams of Figure 2, respectively. Moreover,
there is a relative minus sign in the contribution from these two diagrams.

Among the various decays of the form $\ell_i\to\ell_j\gamma$, the upper bound
on the branching ratio of $\mu\to e\gamma$ is found to be stringent
\cite{meg}. Moreover, we have $Br(\mu\to e\bar{\nu}_e\nu_\mu)\approx 100\%$.
Using this and neglecting the electron mass, the branching ratio of
$\mu\to e\gamma$ is found to be
\begin{eqnarray}
Br(\mu\to e\gamma)&=&\frac{3\alpha}{16\pi G_F^2}
\left|\sum_{k=1}^3(Y_\nu)_{1k}(Y_\nu)_{2k}\times \right.
\nonumber \\
&& \left.
\left\{\frac{1}{4\mu_\eta^2}\left[f_2(x_{Rk})+f_2(x_{Ik})\right]
-\left[\cos^2\theta\frac{f_2(x_{k2})}{2m^2_{2+}}+\sin^2\theta\frac{f_2(x_{k1})}
{2m^2_{1+}}\right]\right\}\right|^2
\label{Eq:br}
\end{eqnarray}
Here, $\alpha =\frac{e^2}{4\pi}$ and $G_F$ is the Fermi constant.

Here we compare our work with that of Ref.\cite{st}. The model in
\cite{st} is similar to that of \cite{ma}. But, in \cite{st}, a theory
at a high scale with an anomalous U(1)$_X$ symmetry is assumed.
The U(1)$_X$ symmetry breaks into $Z_2$ symmetry at a low scale.
Due to these differences,
there exists three 1-loop diagrams for neutrinos in \cite{st}, whereas
only two diagrams generate neutrino masses in \cite{ma}. The diagrams
for the LFV processes of $\ell_i\to\ell_j\gamma$ in \cite{st} is
similar to the diagrams given in this paper (see Figure 2).
But the expression for
$Br(\mu\to e\gamma)$, which is given in Eq. (\ref{Eq:br}), is found to
be different from that in \cite{st}. We hope that these differences might
have arised since the model in \cite{st} has different origin
from that of \cite{ma}.

Although the main motivation of this paper is to study the correlation
between neutrino masses and $Br(\mu\to e\gamma)$, below we mention about
muon $g-2$ in our model. It is known that the theoretical \cite{thg-2}
and experimental \cite{exg-2}
values of muon $g-2$ differ by about $3\sigma$ deviation. However,
there are hadronic uncertainities to muon $g-2$, which need to be
improved \cite{thg-2}. Hence, the above mentioned result is still an indication
for new physics signal. In our model \cite{ma}, muon $g-2$ get contributions
from MSSM fields \cite{mssmg-2} as well as from additional fields, which are
shown in Table 1. The contribution from MSSM fields can fit the $3\sigma$
discrepancy of muon $g-2$\footnote{In Ref.\cite{rsh}, the discrepancy
in muon $g-2$ is fitted in a supersymmetric model, where the contribution is
actually from the MSSM fields.}. Hence, in our model \cite{ma}, it is
interesting to know how large would be the contribution from the
additional fields of this model. The contribution from these additional fields
can be found from the amplitude of Eq. (\ref{Eq:amp}), which is given below.
\begin{equation}
\Delta a_\mu =\frac{m^2_\mu}{16\pi^2}\sum_{k=1}^3\left[(Y_\nu)_{2k}\right]^2
\left\{\frac{1}{2\mu_\eta^2}\left[f_2(x_{Rk})+f_2(x_{Ik})\right]
-\left[\cos^2\theta\frac{f_2(x_{k2})}{m^2_{2+}}+\sin^2\theta\frac{f_2(x_{k1})}
{m^2_{1+}}\right]\right\}
\label{Eq:damu}
\end{equation}
Here, $m_\mu$ is mass of the muon.

\section{Analysis and results}

As described in section 1 that our motivation is to study the correlation
between neutrino masses and $Br(\mu\to e\gamma)$. We have given expression
for neutrino masses in Eq. (\ref{Eq:mnu}). We have explained in the
previous section that to explain neutrino mass scale of 0.1 eV,
we can make neutrino Yukawa couplings to be about ${\cal O}(1)$, but we need to
fine-tune certain SUSY breaking soft parameters which are given in
Eq. (\ref{Eq:para}).
We consider this case, since for unsuppressed neutrino Yukawa couplings,
$Br(\mu\to e\gamma)$ can have maximum values. As mentioned before,
experiments have put the following upper bound: $Br(\mu\to e\gamma)<5.7\times
10^{-13}$ \cite{meg}.
Hence, for the above mentioned parameter space, where neutrino
Yukawa couplings are unsuppressed, we compute
$Br(\mu\to e\gamma)$ by fitting neutrino masses. We check if the computed
values for $Br(\mu\to e\gamma)$ satisfy the experimental bound \cite{meg}.

Before we compute $Br(\mu\to e\gamma)$,
we first need to ensure that the neutrino Yukawa couplings can be unsuppressed
in our model. We can calculate these Yukawa couplings from
Eq. (\ref{Eq:mnu}) by fitting to the neutrino oscillation data. The neutrino
mass matrix of Eq. (\ref{Eq:mnu}) is related to neutrino mass eigenvalues
through the following relation.
\begin{equation}
m_\nu=U_{\rm PMNS}^*{\rm diag}(m_1,m_2,m_3)U_{\rm PMNS}^\dagger.
\label{Eq:neumax}
\end{equation}
Here, $m_{1,2,3}$ are the mass eigenvalues of neutrinos and $U_{\rm PMNS}$
is the Pontecorvo-Maki-Nakagawa-Sakata matrix. The matrix $U_{\rm PMNS}$
depends on three mixing angles ($\theta_{12},\theta_{23},\theta_{13}$)
and Dirac CP-violating phase, $\delta_{\rm CP}$. In the above equation
there is a possibility of Majarona phases, which we have taken to be
zero, for simplicity. We have parametrized $U_{\rm PMNS}$ in terms of
mixing angles and $\delta_{\rm CP}$ as it is given in \cite{pdg}.

By fitting to various neutrino oscillation data, we haven known solar
and atmospheric neutrino mass-square differences and also about the
neutrino mixing angles \cite{neuos}. In the case of normal hierarchy (NH) of
neutrino masses, we have taken the mass-square differences as
\begin{equation}
\Delta m_{21}^2=m_2^2-m_1^2=7.6\times 10^{-5}~{\rm eV}^2,\quad
|\Delta m_{31}^2|=|m_3^2-m_1^2|=2.48\times 10^{-3}~{\rm eV}^2
\end{equation}
In the case of inverted hierarchy (IH) of neutrino masses, the value of
$\Delta m_{21}^2$ remains the same as mentioned above, but,
$|\Delta m_{31}^2|=2.38\times 10^{-3}~{\rm eV}^2$.
In this work, the neutrino mixing angles and CP-violating phase are
chosen to be
\begin{equation}
\sin\theta_{12}=\frac{1}{\sqrt{3}},\quad \sin\theta_{23}=\frac{1}{\sqrt{2}},
\quad \sin\theta_{13}=0.15,\quad \delta_{\rm CP}=0
\end{equation}
The above mentioned neutrino mass-square differences, mixing angles
and CP-violating phase are consistent with the fitted values in \cite{neuos}.
From the mass-square differences, we can estimate neutrino mass eigenvalues
which are given below for the cases of NH and IH, respectively.
\begin{eqnarray}
&& m_1=0,\quad m_2=\sqrt{\Delta m_{21}^2},\quad m_3=\sqrt{|\Delta m_{31}^2|}
\\
&& m_3=0,\quad m_1=\sqrt{|\Delta m_{31}^2|},\quad m_2=\sqrt{\Delta m_{21}^2
+m_1^2}
\end{eqnarray}

In the previous paragraph, we have mentioned neumerical values of
neutrino mass eigenvalues, mixing angles and CP-violating phase. By
plugging these values in Eq. (\ref{Eq:neumax}), we can compute the
elements of the matrix $m_\nu$, which are related to neutrino Yukawa
couplings and SUSY parameters through Eq. (\ref{Eq:mnu}).
Using Eq. (\ref{Eq:mnu}), we can calculate neutrino Yukawa couplings,
in order to satisfy neutrino oscillation data. This calculation procedure
would become simplified if we assume degenerate masses for right-handed
neutrinos and right-handed sneutrinos. For $i=1,2,3$, we assume the following:
\begin{equation}
M_i=M,\quad (m_N^2)_i=m_N^2,\quad (b_M)_i=b_M
\end{equation}
Under the above assumption, all the three right-handed neutrinos have mass
$M$. The corresponding sneutrinos have real and imaginary components
(see Eq. (\ref{Eq:sneu})), whose masses would be
\begin{equation}
m_R^2=M^2+m_N^2+b_M,\quad m_I^2=M^2+m_N^2-b_M
\end{equation}
Under the above mentioned assumption, the neutrino mass matrix of Eq.
(\ref{Eq:mnu}) will be simplified to
\begin{eqnarray}
(m_\nu)_{ij}&=&\frac{S_{ij}}{16\pi^2}\sum_{l=1}^3\left\{
M\left[[U_R(2,l)]^2\frac{m_{\eta_{Rl}}^2}{m_{\eta_{Rl}}^2-M^2}\ln
\frac{m_{\eta_{Rl}}^2}{M^2}-
[U_I(2,l)]^2\frac{m_{\eta_{Il}}^2}{m_{\eta_{Il}}^2-M^2}\ln
\frac{m_{\eta_{Il}}^2}{M^2}\right]+ \right.
\nonumber \\
&&\left. [U_\eta(2,l)]^2m_{\tilde{\eta}_l}
\left[\frac{m_{R}^2}{m_{R}^2-m_{\tilde{\eta}_l}^2}\ln\frac{m_{R}^2}
{m_{\tilde{\eta}_l}^2}-
\frac{m_{I}^2}{m_{I}^2-m_{\tilde{\eta}_l}^2}\ln\frac{m_{I}^2}
{m_{\tilde{\eta}_l}^2}\right]\right\},
\label{Eq:simmnu}
\\
S_{ij}&=&\sum_{k=1}^3(Y_\nu)_{ik}(Y_\nu)_{jk}
\label{Eq:S}
\end{eqnarray}
The elements $S_{ij}$ are expressed quadratic in neutrino Yukawa couplings.
From the above relation we can see that for certain values of SUSY parameters,
$S_{ij}$ can be calculated from $(m_\nu)_{ij}$.
Using the above mentioned assumption of
degenerate masses for right-handed neutrinos and right-handed
sneutrinos, we can see
that Eqs. (\ref{Eq:br}) $\&$ (\ref{Eq:damu}) would give us
$Br(\mu\to e\gamma)\propto S_{21}^2$ and $\Delta a_\mu\propto S_{22}$.

In our model, there are plenty of SUSY parameters, and we need to fix
some of them to simplify our analysis.
In our analysis, we have chosen the following SUSY parameters as follows.
\begin{eqnarray}
&&\mu_\chi=600~{\rm GeV},\quad m_{\eta_1}=400~{\rm GeV},\quad
m_{\eta_2}=500~{\rm GeV},\quad m_\chi=600~{\rm GeV},
\nonumber \\
&&m_N=700~{\rm GeV},\quad \lambda_1=0.5,\quad \lambda_2=0.6,\quad \tan\beta=10
\end{eqnarray}
We have varied the parameters $\mu_\eta$ and $M$, freely. In the previous
section, we have explained that we need to fine-tune the parameters of
Eq. (\ref{Eq:para}) in order to get small neutrino masses. Among these
parameters, we take $(A\lambda)_1=\lambda_1\mu v_2/v_1$ and
$(A\lambda)_2=\lambda_2\mu v_1/v_2$. The other parameters of
Eq. (\ref{Eq:para}), without loss of generality, are taken to be degenerate,
which are given below.
\begin{equation}
b_M=b_\eta=b_\chi=b_{\rm susy}
\end{equation}

We have explained before that we have assumed degenerate masses for
right-handed neutrinos and right-handed sneutrinos.
Under this assumption, the information of neutrino Yukawa couplings is
contained in the quantities $S_{ij}$. Hence, it is worth to plot these
quantities to know about neutrino Yukawa couplings.
In Figure 3, for the case of NH,
we have plotted $S_{21}$ and $S_{22}$ versus
right-handed neutrino mass, $M$, for $\mu_\eta=$ 1 TeV.
\begin{figure}[!h]
\begin{center}

\includegraphics[height=2.5in,width=2.5in]{coupl.eps}
\includegraphics[height=2.5in,width=2.5in]{couph.eps}

\end{center}
\caption{The quantities $S_{21}$, $S_{22}$ are plotted against
right-handed neutrino mass for $\mu_\eta=$ 1 TeV, in the case of NH.
In the left-
and right-handed plots, $b_{\rm susy}$ is taken to be
$(3\times 10^{-2})^2$ GeV$^2$ and $(7\times 10^{-2})^2$ GeV$^2$, respectively.}
\end{figure}
The plots of Figure 3 indicate that $S_{22}$ and $S_{21}$ are around
${\cal O}(1)$. Since these quantities are sum of squares of neutrino Yukawa
couplings (see, Eq. (\ref{Eq:S})), we can expect that
the neutrino Yukawa couplings should be in the range of ${\cal O}(1)$.
We have not plotted the values of $S_{11}$, $S_{31}$, etc in Figure 3,
but we have found that these will also be around ${\cal O}(1)$. We have
plotted $S_{21}$ and $S_{22}$ in Figure 3, since these two determine
$Br(\mu\to e\gamma)$ and $\Delta a_\mu$.

From the plots of Figure 3, we can notice that the values of $S_{22}$ are
higher than that of $S_{21}$. This fact follows from Eq. (\ref{Eq:simmnu}),
where we can see that $S_{ij}$ are proportional to $(m_\nu)_{ij}$,
which are determined by neutrino oscillation parameters. In the case of
NH, we have seen that $(m_\nu)_{22}$ is greater than $(m_\nu)_{21}$ by a
factor of 3.4, hence $S_{22}$ is always found to be larger than $S_{21}$.
It is clear from the plots of Figure 3 that by increasing $b_{\rm susy}$,
$S_{21}$ and $S_{22}$ would decrease. Again, this feature can be understood
from Eq. (\ref{Eq:simmnu}). As explained in the previous section, the
square brackets of Eq. (\ref{Eq:simmnu}) would tend to zero in the limit
$b_{\rm susy}\to 0$. So for large value of $b_{\rm susy}$ there will be
less partial cancellation in the square brackets, and hence $S_{21}$ and
$S_{22}$ would decrease. In both the plots of Figure 3, it is found that
the values of
$S_{21}$ and $S_{22}$ initially decreases with $M$, goes to a minima and then
increases. The shape of these curves can be understood by applying the
approximation of $\frac{b_{\rm susy}}{M^2}\ll 1$ in Eq. (\ref{Eq:simmnu}).
In the limit $b_{\rm susy}\to 0$, we can take
\begin{equation}
m_{\eta_{Rl}}^2=m^2_{\eta_l}(1+\delta_{Rl}),\quad
m_{\eta_{Il}}^2=m^2_{\eta_l}(1+\delta_{Il}),\quad
U_R(2,l)\approx U_I(2,l)=U_0(2,l)
\end{equation}
Here, $\delta_{Rl},\delta_{Il}\ll 1$.
Using the above mentioned approximations in Eq. (\ref{Eq:simmnu}), we get
\begin{eqnarray}
(m_\nu)_{ij}&=&\frac{S_{ij}}{16\pi^2}\sum_{l=1}^3\left\{[U_0(2,l)]^2
(\delta_{Rl}-\delta_{Il})M\frac{m^2_{\eta_l}}{m^2_{\eta_l}-M^2}
\left[1-\frac{M^2}{m^2_{\eta_l}-M^2}\ln\frac{m^2_{\eta_l}}{M^2}\right]+\right.
\nonumber \\
&& \left. [U_\eta(2,l)]^2m_{\tilde{\eta}_l}\frac{2b_{\rm susy}}
{M^2+m_N^2-m^2_{\tilde{\eta}_l}}\left[1-\frac{m^2_{\tilde{\eta}_l}}
{M^2+m_N^2-m^2_{\tilde{\eta}_l}}\ln\frac{M^2+m_N^2}{m^2_{\tilde{\eta}_l}}
\right]\right\}
\end{eqnarray}
In the summation of the above equation, the first and second lines arise
due to left- and right-handed diagrams of Figure 1. From the above equation,
we can understand that the contribution from the first line increases,
reaches a maximum, and then decreases with $M$. Whereas, the contribution
from the second line of the above equation decreases monotonically with $M$.
It is this functional dependence on $M$ that determine the shape of the
lines in Figure 3.
Physically, in the limit $b_{\rm susy}\to 0$, the above description
suggests that the right-handed diagram of Figure 1 is significant only for
very low values of $M$. For other values of $M$, the left-handed diagram
of Figure 1 gives dominant contribution to neutrino masses. One
remark about the plots in Figure 3 is that we have fixed $\mu_\eta=$ 1 TeV
in these figures. We have varied $\mu_\eta$ from 500 GeV to 1.5 TeV and
have found that the plots in Figure 3 would change quantitatively, but
qualitative features would remain same. Also, the plots in Figure 3
are for the case of NH. Again, these plots can change quantitatively,
if not qualitatively, for the case of IH. For this reason, below we
present our results on $Br(\mu\to e\gamma)$ and muon $g-2$ for the
case of NH only.

As described before that our motivation is to compute $Br(\mu\to e\gamma)$
in the model of \cite{ma}. In Figure 3 we have shown that the neutrino
Yukawa couplings in this model can be ${\cal O}(1)$, and for these values
of Yukawa couplings, $Br(\mu\to e\gamma)$ is unsuppressed. In the parameter
space where neutrino Yukawa couplings are unsuppressed, we have
plotted $Br(\mu\to e\gamma)$ as a function of right-handed neutrino mass.
These plots are shown in Figure 4, where we have also varied $\mu_\eta$
from 500 GeV to 1.5 TeV.
\begin{figure}[!h]
\begin{center}

\includegraphics[height=2.5in,width=2.5in]{brl.eps}
\includegraphics[height=2.5in,width=2.5in]{brh.eps}

\end{center}
\caption{$Br(\mu\to e\gamma)$ is plotted against
right-handed neutrino mass for different values of $\mu_\eta$. In the left-
and right-handed plots, $b_{\rm susy}$ has been taken as
$(3\times 10^{-2})^2$ GeV$^2$ and $(7\times 10^{-2})^2$ GeV$^2$, respectively.
The horizontal line indicates the current upper bound on
$Br(\mu\to e\gamma)$.}
\end{figure}
The horizontal line in these plots indicate the current upper bound of
$Br(\mu\to e\gamma)<5.7\times 10^{-13}$. This upper bound would impose
lower bound on the right-handed neutrino mass, as can be seen in the
plots of Figure 4. In the left-handed plot of Figure 4, for $\mu_\eta=$
500 GeV, the right-handed neutrino mass is allowed to be between
about 650 to 950 GeV. In the same plot, for $\mu_\eta=$ 1 or 1.5 TeV,
the right-handed neutrino mass has a lower bound of about 1 TeV.
In the right-handed plot of Figure 4, the lower bound on right-handed
neutrino mass is within 500 GeV, even for a low value of $\mu_\eta=$
500 GeV.

The lower bounds on the right-handed neutrino mass, $M$, are severe in
the left-handed plot of Figure 4. The reason is that for low value of
$b_{\rm susy}$, $S_{21}$ would be high, and hence $Br(\mu\to e\gamma)$ would
be large.
From Figure 4, we can observe that $Br(\mu\to e\gamma)$ initially
decreases with $M$, goes to a minimum and then increases.
For instance, in the left-handed plot of Figure 4, for
$\mu_\eta=$ 500 GeV, $Br(\mu\to e\gamma)$ goes to a minimum around $M=$
750 GeV, and then it will have a local maxima around $M=$ 1.5 TeV. The
reason for $Br(\mu\to e\gamma)$ to initially decrease with $M$ is due
to the fact that the decay $\mu\to e\gamma$ is driven by right-handed
neutrinos and right-handed sneutrinos, as given in Figure 2.
The masses of right-handed
neutrinos and right-handed sneutrinos are proportional to $M$, and hence
$Br(\mu\to e\gamma)$ would be suppressed with increasing $M$. After that,
at a certain value of $M$, $Br(\mu\to e\gamma)$ would tend to become zero. The
reason for this is that the sum of the two diagrams of Figure 2 gives
a relative minus sign to the contribution of $Br(\mu\to e\gamma)$,
which is given in Eq. (\ref{Eq:br}). Hence, for a particular value of
$M$, the contributions from both the two diagrams of Figure 2 cancel out
and give a minimum for $Br(\mu\to e\gamma)$. Also, $Br(\mu\to e\gamma)$
can go to zero asymptotically when $M\to\infty$, since in this limit
the masses of right-handed neutrinos and right-handed sneutrinos would become
infinitely large and suppress $Br(\mu\to e\gamma)$. Hence,
$Br(\mu\to e\gamma)$ has two zeros on the $M$-axis. As $Br(\mu\to e\gamma)$
is a continous function of $M$ and is always a positive quantity,
it is having a local maxima between the two zeros on the $M$-axis.

In the previous section we have described about muon $g-2$. In Eq.
(\ref{Eq:damu}), we have given the contribution due to additional
fields (see Table 1) of our model to the muon $g-2$. Apart from this
contribution, MSSM fields of our model also contribute to muon $g-2$
\cite{mssmg-2}, and it is known that this contribution fits the
$3\sigma$ discrepancy of muon $g-2$. Hence, it is interesting to know
if the additional contribution of Eq. (\ref{Eq:damu}) could
be as large as that of MSSM contribution to muon $g-2$. In Figure 5,
we have plotted the contribution of Eq. (\ref{Eq:damu}). In the plots
of Figure 5, we have chosen the parameter region such that the neutrino
oscillation data is fitted.
\begin{figure}[!h]
\begin{center}

\includegraphics[height=2.5in,width=2.5in]{damul.eps}
\includegraphics[height=2.5in,width=2.5in]{damuh.eps}

\end{center}
\caption{$\Delta a_\mu$ is plotted against
right-handed neutrino mass for different values of $\mu_\eta$. In the left-
and right-handed plots, $b_{\rm susy}$ has been taken as
$(3\times 10^{-2})^2$ GeV$^2$ and $(7\times 10^{-2})^2$ GeV$^2$, respectively.}
\end{figure}
From the plots of Figure 5, we can see that for low values of $M$,
$\Delta a_\mu$ can be negative and it becomes positive after certain large
value of $M$. From these plots we can notice that the overall magnitude of
$\Delta a_\mu$ is not more than about $10^{-12}$. This contribution is atleast
two orders smaller than the estimated discrepancy of muon $g-2$, which is
$(29\pm 9)\times 10^{-10}$ \cite{thg-2}. From this we can conclude that
the additional contribution to muon $g-2$ in our model, {\it i.e.}
Eq. (\ref{Eq:damu}), is insignificant compared to the MSSM contribution
to muon $g-2$.

\section{Conclusions}

We have worked in a supersymmetric model where neutrino masses arise at
1-loop level \cite{ma}. We have computed these loop diagrams and obtained
expressions for neutrino masses. We have identified a parameter region of
this model, where the neutrino osicllation data can be fitted without
the need of suppressing the neutrino Yukawa couplings. In our parameter
region, the SUSY breaking soft parameters such as $b_M$, $b_\eta$,
$b_\chi$, $(A\lambda)_1$,
$(A\lambda)_2$ need to be fine-tuned. In this parameter region, branching
fraction of $\mu\to e\gamma$ can be unsuppressed, and hence, we have computed
$Br(\mu\to e\gamma)$. We have shown that the current upper bound on
$Br(\mu\to e\gamma)$ can put lower bounds on the mass of right-handed
neutrino field. Depending on the parameteric choice, we have found that this
lower bound can be about 1 TeV. We have also computed the contribution to muon
$g-2$ arising from additional fields of this model, which are given in
Table 1. We have shown that, in the region where neutrino oscillation
data is fitted, the above mentioned contribution is two orders smaller than
the discrepancy in muon $g-2$.

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