text:
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Home BREAKING NEWS His Holiness the Dalai Lama Offers Prayers at Mahabodhi Temple in Bodh...
His Holiness the Dalai Lama Offers Prayers at Mahabodhi Temple in Bodh Gaya
By Shyamal Sinha,New Delhi
His Holiness the Dalai Lama offering prayers infront of the main Buddha statue in Mahabodhi Temple on second day of his arrival in Bodh Gaya, 29 December 2016.
HH second day of arrival in Bodh Gaya, the Dalai Lama today visited the Mahabodhi Temple, the seat of Lord Buddha's enlightenment to offer prayers at around 8:50 am.
In 1891, the Mahabodhi Society was founded with the aim of reclaiming Bodhgaya and the Mahabodhi temple from Hindu to Buddhist control. The 1949 Bodhgaya Act recognize the site as a Buddhist holy place.
In 2002, UNESCO declared the Mahabodhi Temple a World Heritage Site and recommended "the Indian authorities to develop an overall management plan to protect the values of the World Heritage site.
His Holiness was accompanied by Chair of the 34th Kalachakra Organising Committee, Kalon Ven Karma Gelek Yuthok of Department of Religion and Culture, Kalon Phagpa Tsering Labrang, Department of Security and other representatives of the Central Tibetan Administration.
Devotees from all around India and world lined up the streets leading to the holy site of Mahabodhi temple to greet His Holiness.
Earlier His Holiness gave a private audience to a group of Tibetan pilgrims from Tibet.
The Mahabodhi Mahavihara is revered as the holiest place of Buddhist pilgrimage in the world. It is among the 84,000 shrines built by King Ashoka in the 3rd century BC.
His Holiness the Dalai Lama visiting the Mahabodhi Temple, the seat of Buddha's enlightenment, in Bodh Gaya, 29 December 2016.
Immediately next to the Mahabodhi Temple is the Bodhi Tree, a descendent of the very tree under which the Buddha was enlightened, and the Jewel Walk, marking the place where the Buddha is said to have practiced walking meditation for seven days after his enlightenment.
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redpajama_set_name: RedPajamaCommonCrawl


This article is about a concept in Islamic jurisprudence. For the college admission tests in Saudi Arabia, see Qiyas Tests.
In Islamic jurisprudence, qiyās (Arabic: قياس) is the process of deductive analogy in which the teachings of the Hadith are compared and contrasted with those of the Qur'an, in order to apply a known injunction (nass) to a new circumstance and create a new injunction. Here the ruling of the Sunnah and the Qur'an may be used as a means to solve or provide a response to a new problem that may arise. This, however, is only the case providing that the set precedent or paradigm and the new problem that has come about will share operative causes (عِلّة, ʿillah). The ʿillah is the specific set of circumstances that trigger a certain law into action. An example of the use of qiyās is the case of the ban on selling or buying of goods after the last call for Friday prayers until the end of the prayer stated in the Quran 62:9. By analogy this prohibition is extended to other transactions and activities such as agricultural work and administration. Among Sunni Muslim in recent centuries Qiyas has been accepted as a fundamental source of Sharia law along with Ijmāʿ and secondary to the Qur'an, and the Sunnah.
Late and modern Sunni jurisprudence regards analogical reason as a fourth source of Islamic law, following the Qur'an, prophetic tradition and binding consensus. While Muslim scholarship in the later period traditionally claimed that analogy had existed in Islamic law since their religion's inception, modern scholarship generally points to Muslim scholar Abu Hanifa as the first to incorporate analogical reason as a source of law. Since its inception, analogical reason has been the subject of extensive study in regard to both its proper place in Islamic law and its proper application.
Among Sunni traditions, there is still a range of attitudes regarding the validity of analogy as a method of jurisprudence. Imam Bukhari, Ahmad bin Hanbal, and Dawud al-Zahiri for example, rejected the use of analogical reason outright, arguing that to rely on personal opinion in law-making would mean that each individual would ultimately form their own subjective conclusions. Bernard G. Weiss, one of today's foremost experts on Islamic law and philosophy, explains that while analogical reason was accepted as a fourth source of law by later generations, its validity was not a foregone conclusion among earlier Muslim jurists. Thus, while its status as a fourth source of law was accepted by the majority of later and modern Muslim jurists, this was not the case at the inception of Muslim jurisprudence as a field.
Some of them argued that qiyâs is contrary to reason. One argument given in this light was that: "Delving into this method is intellectually repugnant in its own right". Another argument was: "Islamic legal rulings are based on human well-being, and no one knows human well-being except the One who gave us the sacred law. Therefore, the only way we can know the sacred law is from the revelation." Other scholars said that qiyâs is not contrary to reason, but prohibited by the sacred law itself.
Imam Bukhari maintained a negative position towards qiyas, as he held views aligned with the Zahiris of his time. Scott Lucas states that Bukhari's rejection of qiyas was placed within the context of what Bukhari perceived as invalid techniques of ijtihad, which included religious innovation (bid'a), ra'y, and tamthil.
Lucas also points out common mistakes other scholars make when analyzing Bukhari's position on qiyas. The biggest source of confusion for scholars is the fact that, while rejecting qiyas, Bukhari accepts the idea of tashbih (comparison), which seems similar to analogy. However, this is not the case, as tashbih is a comparison used in explanation (such as a metaphor), whereas qiyas applies a specific legal ruling to another case.
If the Prophet was asked about something about which he had not received a revelation, he either said, 'I do not know' or did not reply until he received a revelation. he did not [reply] by means of ra'y or qiyas, due to the [Qur'anic] verse, "…in accordance to what God has shown you" (4:105).
On Ahmad's views, Christopher Melchert states "Ahmad and his fellow traditionalists of the ninth century expressly condemned the Hanafi exercise of qiyas…" When compared with Dawud al-Zahiri's intensely negative stance towards qiyas, Melchert also states "Ahmad ibn Hanbal could likewise be quoted, as we have seen, in total rejection of ra'y (opinion) and qiyas (analogy)."
Early support for the validity of analogical reason in jurisprudence came from Abu Hanifa and his student Abu Yusuf. Al-Shafi'i was a proponent of analogical reasoning as well, though his usage was less frequent than that of Abu Hanifa.
Acceptance of analogical reason gradually increased within the Muslim world. With the Malikite and Hanbalite schools eventually granting full acceptance as the Hanafites and Shafi'ites already had done, the overwhelming majority of Sunni jurists from the late period onward affirmed its validity. Japanese scholar of Islam Kojiro Nakamura defined the orthodox Sunni schools in regard to their eventual acceptance of analogy in descending order of that acceptance: Hanafis, Malikis, Shafi'is, Hanbalis and Zahiris. Much work was performed on the details of proper analogy, with major figures such as Al-Qastallani, Al-Baqillani, Al-Juwayni and al-Amidi from the Shafi'ite school and Ibn Abidin from the Hanafite school providing rules and guidelines still used to this day.
Sunni scholar Baghawi gave a commonly accepted definition of analogy in Islamic law: analogical reasoning is the knowledge by which one learns the method of deriving a ruling from the Quran and prophetic tradition. In this case, the above-mentioned ruling should not already be apparent in the Quran, prophetic tradition or consensus. If there is no derivation involved due to the explicitness of the ruling in the Quran and prophetic tradition, then such a person is not, by definition, a mujtahid.
The Imam (a.s.) said, "My father narrated from my great-great-great-great grandfather, the holy Prophet (s.a.) who said, 'Those who act on the basis of analogy will face their destruction and lead others to their destruction. Those who give fatwas without the knowledge of the abrogating and the abrogated, the clear text and that which requires interpretation, they will face destruction and lead others to their destruction."
Among the most notable Ismaili thinkers, Bu Ishaq Quhistani regarded the notion of subjective opinion (qiyās) as completely contradictory to the Islamic notion of tawhīd (unity) as it ultimately gave rise to a countless divergent conclusions, besides which those who exercised deductive analogy relied on little more than their imperfect individual intellects. According to Bu Ishaq, there must be a supreme intellect in every age, just as Muhammad was in his time. Without this, it would be impossible for any ordinary individual to attain knowledge of the Divine using mere speculation. The supreme intellect, he reasoned, could be none other than the Imam of the age.
Primarily being a school of theology and not jurisprudence, the Mu'tazila generally did not hold independent positions on such issues. The majority of the Mu'tazila, despite being a distinct sect from both Sunni and Shi'ite Islam, still preferred the juristic school of Abu Hanifa, with a minority following Al-Shafi'i's views. This resulted in the odd combination of being Mu'tazilite in creed but Sunni in jurisprudence, and consequently most of the Mu'tazila accepted analogical reason in addition to pure reason.
Mu'tazilite scholar Abu'l Husayn al-Basri, a major contributor to early Muslim jurisprudence, said that in order for a jurist to perform analogical reason, they must possess a thorough knowledge of the rules and procedures for which allows the application of revealed law to an unprecedented case, in addition to basic knowledge of the Qur'an and prophetic tradition.
Not all of the Mu'tazila followed Sunni jurisprudence. Al-Nazzam in particular denied the validity of analogical reason wholesale, preferring to rely on pure reason instead.
The Inquisition that took place in the middle of the 9th century, which was initiated by the caliph al-Ma'mun, ensured the persecution of many scholars who did not agree with the caliph's rationalistic views. The most famous of these persecuted scholars is Ahmad ibn Hanbal, who maintained his view that the Qur'an was not created, but eternal.
Christopher Melchert similarly argues that the Mihna demonstrated a relationship between the Hanafis of Baghdad, who were associated with the heavy use of qiyas, and the Mu'tazilites.
Before the Middle Ages there was a logical debate among Islamic logicians, philosophers and theologians over whether the term qiyas refers to analogical reasoning, inductive reasoning or categorical syllogism. Some Islamic scholars argued that qiyas refers to inductive reasoning, which Ibn Hazm (994-1064) disagreed with, arguing that qiyas does not refer to inductive reasoning, but refers to categorical syllogism in a real sense and analogical reasoning in a metaphorical sense. On the other hand, al-Ghazali (1058–1111) and Ibn Qudāmah al-Maqdīsī (1147-1223) argued that qiyas refers to analogical reasoning in a real sense and categorical syllogism in a metaphorical sense. Other Islamic scholars at the time, however, argued that the term qiyas refers to both analogical reasoning and categorical syllogism in a real sense.
^ "Usul Fiqh: THE RULE OF QIYAS: ITS MEANING, JUSTIFICATION, TYPES, SCOPE, APPLICATION, FEASIBILITY AND REFORM PROPOSALS". Islamic Jurisprudence - The Collection of articles for Islamic Jurisprudence II, LLM- Administration Of Islamic Law, International Islamic Universiti of Malaysia. session 2007/2008. February 10, 2008. Retrieved 8 September 2015.
^ Reuben Levy, Introduction to the Sociology of Islam, pg. 236-237. London: Williams and Norgate, 1931-1933.
^ Chiragh Ali, The Proposed Political, Legal and Social Reforms. Taken from Modernist Islam 1840-1940: A Sourcebook, pg. 280. Edited by Charles Kurzman. New York City: Oxford University Press, 2002.
^ a b c d Mansoor Moaddel, Islamic Modernism, Nationalism, and Fundamentalism: Episode and Discourse, pg. 32. Chicago: University of Chicago Press, 2005.
^ Keith Hodkinson, Muslim Family Law: A Sourcebook, pg. 39. Beckenham: Croom Helm Ltd., Provident House, 1984.
^ a b Understanding Islamic Law: From Classical to Contemporary, edited by Hisham Ramadan, pg. 18. Lanham, Maryland: Rowman & Littlefield, 2006.
^ Christopher Roederrer and Darrel Moellendorf [de], Jurisprudence, pg. 471. Lansdowne: Juta and Company Ltd., 2007.
^ Nicolas Aghnides, Islamic Theories of Finance, pg. 69. New Jersey: Gorgias Press LLC, 2005.
^ a b Chiragh Ali, pg. 281.
^ Lucas, Scott C. (2006). "The Legal Principles of Muhammad B. Ismāʿīl Al-Bukhārī and Their Relationship to Classical Salafi Islam". Islamic Law and Society. 13 (3): 310.
^ Lucas, Scott C. (2006). "The Legal Principles of Muhammad B. Ismāʿīl Al-Bukhārī and Their Relationship to Classical Salafi Islam". Islamic Law and Society. 13 (3): 307-309.
^ Ibn Hajr. Fath al-bari. 15:263.
^ Melchert, Christopher (1992). "The Formation of the Sunni Schools of Law, 9th-10th Centuries C.E.": 47.
^ Melchert, Christopher (1992). "The Formation of the Sunni Schools of Law, 9th-10th Centuries C.E.": 329.
^ al-Barbahari. Sharh K. al-Sunnah. 2:19.
^ Melchert, Christopher (1992). "The Formation of the Sunni Schools of Law, 9th-10th Centuries C.E.": 38.
^ Elhadj, Elie. The Islamic Shield: Arab Resistance to Democratic and Religious Reforms (Boca Raton: Brown Walker Press), 2007, p.48.
^ Shafique Virani. The Ismailis in the Middle Ages: A History of Survival, A Search for Salvation (New York: Oxford University Press), 2007, p.149.
^ Hallaq, Wael B. (2005). The Origins and Evolution of Islamic Law. p. 125.
^ Melchert, Christopher (1996). "Religious Policies of the Caliphs from al-Mutawakkil to al-Muqtadir, AH 232-295/AD 847-908". Islamic Law and Society. 3 (3): 324.
^ Wael Hallaq (1993), Ibn Taymiyya Against the Greek Logicians, p. 48. Oxford University Press, ISBN 0-19-824043-0.


This is no ordinary Pizza Stone. This is the SteakStones Pizza Stone, designed to cook any Pizza to a crisp base and keep every last slice as hot and delicious as the first. Whilst it's perfect way to cook and serve Pizza, our Pizza Stone is also ideal as a fun way to serve and carve your Christmas Turkey or any other large cuts of meat, with the main joint and slices meat being kept lovely and hot, whilst the dry heat of the stone carries no concern of drying out the food. Or a chilled serving platter for your favourites Cheeseboard, Antipasti Platter or Sushi Selection. This is the perfect serving platter, marrying form and function in a simple, stylish design.
You can both cook and serve on The SteakStones Pizza Stone and to do so we recommend heating the stone in your oven for 10 mins before placing your Pizza on top in the oven. Once your Pizza is perfectly cooked, simply remove the stone with a pair of oven gloves and place carefully into the Bamboo Board. You can now easily and safely transfer the piping hot Pizza to your table to cut and serve and enjoy every slice as hot and delicious as first. Alternatively to chill your stone, simply place in the fridge or freezer (depending how cold you would like it to be) for around 30 mins. The stone can be heated to any temperature, but we don't recommend anything higher than 250c in order to ensure the Bamboo Board doesn't get scorched.
We've used this on our Barbecue to cook steaks on and it keeps them lovely and hot once we take it back to the table in the board. Really makes a difference in the English summer!
This pizza stone is beautiful – I actually cook my pizza on it in the oven and this makes them really crispy.
I bought this because it looked so lovely in the picture. It definitely lives up to my expectations although it is quite heavy.


The guys hanging the drywall (aka sheet rock, gypsum board, gyp board, wall board, etc.) are like a breath of fresh air around here. Yesterday there were 4 in their crew, and today there were 5.
They show up early, work hard, and get an amazing amount of work done in a day's time. They split up into teams of 2 or 3, with one (or two) measuring and fitting the cut pieces into place (often up on a ladder or scaffold), while the other one is down on the floor, cutting the pieces to fit.
Add to that the (attempted) gospel stylings of one of them, intermittently breaking out in between the dimension calling. Today it was predominantly "Swing Low, Sweet Chariot" intermixed with "I'll Fly Away".
Things are rolling along. After what seems like a couple of really slow weeks on the project, (although I have managed to say pretty busy,) it's getting really fun now.
I just realized that it's Tuesday, which means tonight is OList Happy Hour on Twitter. I've spent the day running all around the county (actually 2 counties, skillfully avoiding having to drive all the way to Knoxville just to have one piece of steel cut, but I digress) for tasks related to the house I'm building in Tennessee.
Last week I realized that one of the most active of the OLists has yet to be represented on the Happy Hour Menu, and that OList would be none other than OEvolve. OEvolve has a special place in my heart since it was the first of the OLists to which I subscribed. I also think it's the one from which I get the most practical advice.
It's the end of Day 42. That makes it the end of our 6th week since we left L.A. I feel like saying, "I can't believe it has been 6 weeks since Paul, Todd and I started out on our big adventure" except that, I actually can believe it.
And, at this point, I think I can begin to see the end point. I think we have about 2 more weeks, maybe 2 1/2 left here. Then it will be back to L.A., via a slightly more circuitous route than brought us here.
These past few days have seen a few minor, but fun, adventures.
It's Tuesday, which means tonight is OList Happy Hour on Twitter.
I got the idea for tonight's Happy Hour menu from a couple of very good, recent blog posts over at Kelly's blog, Reepicheep's Coracle. She had posted them to the OAcademics list, and they really jumped out at me as great subjects for further discussion.
Things are still moving along somewhat more slowly, and probably will be for the coming week. Later in the week they should be starting to hang drywall, which will be tremendously exciting.
Brunch is an open-topic social group chat. We've been doing these for a few weeks now, and they have consistently been a lot of fun. That hour really flies by quickly.
Brunch takes place from 9-10 a.m. Pacific, and 12-1 p.m. Eastern, on Twitter.
You can see the whole image gallery here.
There is also a little bit more to report, so maybe you'll get 2 posts today! I know that's rare, and I'm not promising. I'm just sayin.
Appetizer: For our appetizer, let's discuss blogging frequency. How often do you post on your blog? Do you blog on a schedule? Have you ever tried posting on specific topics at regular intervals? Or, are you a go-with-the-flow type of writer, simply putting up whatever topic is personally relevant to you? What can you see as advantages and disadvantages of each approach? For readers, how often do you visit your various blogs? Do you use an aggregator like Google Reader to organize your blog reading? What other blog reading/organizing tools are out there?
Drink Special: What tricks have you adopted to make your blogging and/or blog-reading more productive? What has worked and hasn't worked for you in this regard?
I'm writing this on Day 31, being Sunday, September 12th. I am going to make a greater effort, moving forward, to get these updates out earlier in the day. Today we're going to look at wood flooring up in Knoxville, and I may get some welding in later in the afternoon.
Today's post, however, is about yesterday's fun day of showing the house to some special guests, who drove from Atlanta and Nashville.
Well, people, I really have to apologize for not posting all week.
It seemed like little got done this week, although there were some important things that did get done. But, there were also things that I felt should have gotten done, but didn't, and not for any apparent (and meaningful) reason.
And, I have to be honest and say that I'm feeling a little burnt out. I've been away from my 'normal' life for about a month, and working nonstop on the house for over 3 weeks, without any real break, and it's beginning to wear on me, just a little bit.
There is some relief in sight, because we are getting to a point where we are close to drywall going on, and some other more mundane things, that will give me a break from the constant attention the project has demanded. I might even take the dogs and go away for a few days, just for a little change of scene.
What got done this week? Let's see.
Monday was a holiday, but the framing crew worked anyway. They worked on exterior siding and miscellaneous things inside. I believe the electricians were here also.
Tuesday should have been the day the roofer came to start on the 2 roof decks, but he decided to extend his camping trip a day. *SIGH* This was also the last day for the full-time framing crew, and since the roof wasn't on those roof deck areas, because the roofer was still camping, the framer peopleguys couldn't install the doors there.
Because the doors weren't on, I had to scramble when it rained to put some plastic sheeting up, for some protection on the openings. By myself. In the rain. But everything was ok in the end. Mostly.
I don't have a huge reservoir of confidence in these guys at this point. The product they are installing is a great roofing system, when installed correctly. I just hope they can get get the job done. I guarantee you that I will be blasting it with a garden hose the day after they finish. If this thing is going to leak, I want to know about it now, not next spring or summer.
The framers are peopleguys, and the roofers are clearly not.
Yesterday, the roofers came back and worked on the roof over the front door. They stayed late, and yet, somehow, still didn't finish it.
Also, the steel brackets for the timber beams showed up, having been powder coated.
You could scratch the finish off them with your finger nail. The worst powder coating job I have ever seen. But, since they are installed up high, and not where they can be reached and touched, we decided to install them anyway.
I can touch up the paint manually. If they were going to be installed outside in the weather, or inside where they would be handled (like the mezzanine railings) they would be rejected.
Ms. Client chimed in on those, and affirmed that in all her years at her old job, in automotive parts quality control, where she dealt with a lot of powder coating, she had never encountered powder coating that flaked off like that, and that that would have definitely been rejected.
We plan on putting the powder coating company on notice that if this happens again with the upcoming railings (which will be subject to handling and touching, as well as have some outdoor components) they will be rejected.
This morning, the electricians were back, finishing up a few things they needed to finish up. They left early, and Dave, the genius framing crew boss peopleguy, came over and installed the steel brackets. I helped, and it was a great way to end the week.
The real excitement this week is in what went on last night and will be happening here tomorrow.
Last night I went down to Atlanta and gave a talk on the house and my design work for the Atlanta Objectivist Society (aka ATLOS).
It was really fun, the audience was very attentive, and they asked great questions. It was also really fun that Jenn, aka Rational Jenn, one of the group's organizers, brought her son Ryan. To get the complete picture of how clever and brilliant this little guy is, you should just read her blog. Suffice it to say that if Jenn should ever fall short in her stated quest to take over the world, I have absolutely no doubt but that Ryan will carry on and finish the job for her. So it was a treat to have him at the lecture.
Tomorrow, the group is converging here, right here at the house construction site, for a picnic, tour, and pony rides. How great is that!?
And, there's even a bit of a special event on top of the special event. I get to host the first in-real-life meeting of two of my Objectivist blogging friends, Trey and Jenn. So, it will be a meeting of the minds, and I'm going to open a bottle of wine or two for the occasion!
Farmer Jo has been working on cleaning up her barn, and getting things ready for pony rides. It's actually a horse (named Lacy) that the kids will be riding, not a pony per se. I may even go for a ride myself!
Todd was trying to help out in the barn but kept getting distracted by Bonjo the Donkey in the next stall. That dog is utterly obsessed with the donkeys. Lucky for him, he keeps getting shocked by the electric fence, or he would end up getting kicked into the next county.
The herd and flock actually formed into rows all by themselves as the two groups of animals approached each other. It was pretty interesting to watch. I don't believe there was any bloodshead.
I can't wait until I don't have to look at that awful green foam ceiling any more!
Appetizer: Last Sunday, during OLists Brunch, the subject of Local Community Groups enjoyed quite a bit of discussion. There are tons of folks who participate regularly in established community groups, focused on Objectivism in some way. Activism, reading groups, and social events are all great ways of building a community of people interested in Objectivist ideas.
Those of us who are part of a group, how do you participate or contribute? For the leaders of these groups, what ideas do you have for expansion? How do you promote your group? If you have no such group in your area, and are interested in starting one, how do you begin?
Drink Special: Meeting new people who are new to Ayn Rand and Objectivism can be both exciting and frustrating at the same time. Practically everyone holds some degree of mixed premises, and the rise of religion and spirituality in the culture can make it challenging, if not difficult (or even impossible) to discuss philosophical ideas with people holding fundamental premises at odds with your own.
What are some of the best and worst conversations you've had, with people you didn't know well, about religion and/or philosophy? When you find yourself discussing a particular issue, were you able to agree or disagree effectively, while respecting the honesty and/or context of the other person?
Want to lurk and follow along with the discussion without joining Twitter? You can do that here.
Last time I checked in with y'all (actually, that's what they say in Atlanta. Around here they say y'uns. Sometimes I have no idea what people are even saying) was last Thursday evening. A lot got done Friday too, and the crew is coming back to work tomorrow also. They will be laboring on Labor Day. That's their choice. I'm happy about it. We have rain in the forecast and doors that still need installation.
Almost all of the windows are in, except for 3 that were special order items (just because they were slightly odd-sized) and won't be here for about 2 more weeks.
Today was another big big day on the construction site.
I interacted with the plumber today, for the first time. He seems competent, and is friendly and chatty like the locals in Middle-of-Nowhere, East Tennessee tend to be.


There is really a wealth information and facts on the internet about gambling and betting. How to start a gambling website, what exactly do people bet on, can be a betting exchange better compared to a sports book or casino website, or do users prefer betting online associated with betting not online reputation? What is lacking in many of these posts isn't the information offer you on starting a gambling website, but rather exactly the place to begin. So Where would you start, you ask? and how do you start? Would you require funding when thinking of starting ones own gambling net site? Are there legal pitfalls in running an online sports book or casino gambling world-wide-web?
Asking these questions is simply tip from the iceberg in knowing exactly what you are growing yourself into in running a betting exchange or several other gambling websites, but knowing resolutions will raise your chance or odds in running a successful online gambling site. You discover them ? do having? You can search and search around the online world and gather all information (highly recommended) on starting your own casino, using a proper betting exchange, or on providing a fair online sports book atmosphere to your users, but knowing information on how to work with this information is key in running your set of clothing.
Before we're into utilizing the information you gather properly, you may want to visit some for these places if you have not already done so: gambling forums, casino pages, sports website, news pages on gambling, pages on the "internet gambling regulation act" and what will affect your online website, laws governing gambling and paying close focus to the dates the country each specific information pertains to and the like.
By congratulations, you must be turning over this is to utmost. But ask yourself this: are you aware what each gambler want, do there's more will these use website is over your next Joe web pages. Simply put you need recognize the stuff that will drive advertisers, sponsors to website is (funding) and most importantly what will drive users to your website. You can have topic . house on the block but if not one is aware of your house or you, well you receive the vision.
If an individual visited the suggested headings listed above and educate yourself a extra on gambling and the laws governing online sports book, casinos, poker rooms and betting exchanges however wish to carry on then keep reading my friend, you have gotten in the first obstacle. Now that we have you thinking beyond your box this will better understanding on what's in store for you lets tackle the first and most important question you would first need answered "Is gambling legal".
Now this is the most important because in the event you thinking of conducting or running a gambling website in a country, city, state, province, town, to it could be and its not legal. Like monopoly you'll be fined or worst you'll need go in order to jail. Sorry boys and girls but we must make benefit clear here being company doesn't mean a thing to brand new and the authorities. You will need to register a business or obtain a business license (not simillar to a gambling license), you'll need hosting and you will probably need a bank account to support the money generated from your sports book, betting exchange, poker or casino webpage. So finding out if gambling is legal where are going to running your business is extremely notable.
To give you a better picture of things 9/10 trying to run a gambling website have had to opt to offshore business. Why you ask? Well that obvious needless to say. In places like Costa rica there are laws within position for gambling, but there aren't any law against online gambling neither is there a need gambling license (more on gambling license later), a person will again have to do your research, if you are prepared about having a sports book or the other gambling websites, http://104.156.233.15 wait for it to gather information. Totally free Tip #1 – Speak to a Lawyer (a reputable one if you can) they'll shed a lot of light on your situation and point you in proper direction.


text:
Home / POLITICS / IMF Bailout: NPP Gov't Has The Most Revenue Sources In The History Of Our Country – Henry Osei Akoto
IMF Bailout: NPP Gov't Has The Most Revenue Sources In The History Of Our Country – Henry Osei Akoto
Adwenpa-Hene July 5, 2022 POLITICS 234 Views
The 2020 Parliamentary candidate for the National Democratic Congress (NDC) in the Oforikrom Constituency and National Organizer Hopeful, Henry Osei Akoto, believes the Nana Addo led government has had a lot of revenue sources and allocations and wonders where they went.
Ghana has officially contacted the International Monetary Fund to seek financial support.
In a statement, President Akufo-Addo, said he had authorized Finance Minister Ken Ofori-Atta to commence formal engagements with the International Monetary Fund (IMF), inviting the Fund to support an economic program put together by the Government of Ghana.
The statement added that this followed a telephone conversation between the President and the IMF Managing Director, Miss Kristalina Georgieva, conveying Ghana's decision to engage with the Fund.
See also Asiedu Nketia Runs From Campaign Posters
On CTV's Political Talk Show, National Organizer Hopeful reiterated his position on the country's request for an IMF bailout.
"There are no useful projects or improvements. What exactly is going on? Finally, we have been dragged to the IMF by the same party that claims the country is rich in resources. The same party claims it will never use the IMF. The same party saw it as a failure to go to the IMF," he said.
Mr Osei Akoto has stated that the President, Nana Addo Dankwa Akuffo-Addo, has a way with words and appears to love the country while pursuing his own interests.
See also World Bank Leads Review Of Free SHS, Agenda 111 And Others
"When listening to that man, one would be thoroughly convinced that he's willing to lay down his life for the country, but that's not the case. They are people who continue to look after their own interests and have no better plan to help the country get out of this crisis" he added.
Mr Osei Akoto pressed the government on where all the money from the Covid-19 fund, our oil refineries, and a few others had gone.
"When John Dramani Mahama left office, he left three oil fields for the country. Jubilee Oil Field, Saltpond Oil Field, and Tweneboa Enyenra Ntomme Oil Field produce 170,000 barrels of oil per day together. Over 2 billion dollars have also been obtained from the Covid-19 fund. I'd like to ask the president and his administration what happened to all that revenue," he concluded.
See also IMF Officials Expected In Ghana On July 6 To Assess Bailout Request
Source: Daniel Frimpong
IMF Bailout 2022-07-05
Tags IMF Bailout
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There is a great need for priests from all Eastern Spiritual Traditions in the west. Hindu temples and societies also need such services for their members. To this end, Hindu University, a seminary for Hindus serving in the community at large, has developed a certification course in Hindu Priesthood. This certification requires 4 courses which can be completed in 24 months. The credits earned (12 credits) can be transferred to masters' program which requires 30 credits.
All the courses in this certification are taught online using our Litmos Course Management System which provides a full virtual class experience. All courses are practical and require class participation and online submission of video assignments. Tradition specific training will be provided under the guidance of the qualified instructors in traditional systems. The specific requirements of the training would be decided by the instructors depending upon the student's abilities.
The major Hindu saṃskāras as per the select traditions (Sampradāyas) are discussed in theory and practice.
The major Hindu festivals and pūjās as per the select traditions (Sampradāyas) discussed in theory and practice.
Major Hindu Rituals in different traditions (Sampradāyas) as per the select specializations are discussed in depth of theory and practice to bring out the similarities and differences in various sects of Hindu traditions.
Saṃskāras, Festivals and Pūjās, Rituals in different traditions (Sampradāyas), in their aspects of practical observance in the contemporary society. This is an integration of the theory and the practical components of the entire course for societal application.
*The courses will be offered based on sufficient number of student registered.


ANNAPOLIS – In a presentation on Wednesday to the Maryland Commission on Innovation and Excellence ("Kirwan"), race equity expert Dr. Ivory Toldson delivered his highly anticipated second presentation along with a call to action to the 25 Commission members to stand up for Black and Brown children in Maryland. The ACLU of Maryland (ACLU) joins Dr. Toldson in urging the Commission to include explicit policies, strategies, and staffing allocations to address racial inequities.
To evaluate the "Kirwan" Commission working group documents, Dr. Toldson used a methodological strategy, School Equity Model: Equity to Excellence, to provide guidance on state-level educational equity using an equity framework, which involves three core tenets: (1) good data; (2) thoughtful analysis; and (3) compassionate understanding.
Explicit acknowledgement of racial disparities in access to education and policies.
Ways to serve diverse populations of students that address their unique cultural needs.
Acknowledgement that teachers who are already successful in our classrooms across cultures are not just smart but are compassionate, empathetic, deliver culturally meaningful content, which are teacher qualities that must be sought and encouraged.
Cultural competency training is necessary for educators of all races.
Cultural biases may be present in early education assessments.
Maryland's new funding formula will impact an entire generation of students. The Commission's charge is to make recommendations for improving education in Maryland through funding, policies, and allocating resources. Since Black and Brown students make up more than half of Maryland's public school enrollment—and Maryland's future rests on how well it serves these students—the state's "Kirwan" Commission should center race equity as the group finalizes their recommendations, which are expected by the end of the year.
State Superintendent Dr. Karen Salmon affirmed a long-term commitment to race equity and that she wants to continue to work with Dr. Toldson to ensure race equity is integrated throughout recommendations.
The Constitution of Maryland establishes the mandate for a "thorough and efficient" free public education for all children in the state. However, the ACLU is deeply concerned that Black and Latinx students are not being adequately and equitably served by Maryland's education system. The Education Trust recently released a new set of analyses that examine racial inequities in academic achievement, funding, access to high-quality teachers and leaders, and more. The data make a clear and compelling case that race equity must be a centerpiece of our deliberations about policy changes to Maryland's educational system.
In addition to providing his own observations and recommendations, Dr. Toldson lifted up those from stakeholders, such as ACLU of Maryland, and pointed to the gaps between feedback and the initial Working Group drafts. He strongly encouraged the Commission to incorporate stakeholder recommendations.
"While poverty is a critical proxy for meeting the needs of underserved children, that factor should be combined with focused strategies that address racial inequities when the state decides how best to target resources," said Bebe Verdery, ACLU of Maryland's Education Director. The ACLU is advocating for a full and equitable funding formula that addresses racial equity throughout the formula and via other reforms to erase racial inequities in achievement.
The ACLU believes by incorporating Dr. Toldson's recommendations, combined with those of The Education Trust, ACLU, and other groups who are working extensively on race equity issues in education across Maryland, the "Kirwan" Commission can recommend reforms that will begin to eliminate these racial inequities.
Dr. Ivory Toldson is a national expert on educating students of color and building channels for post-secondary success. President and CEO of The Quality Education for Minorities (QEM) Network and professor of counseling psychology at Howard University, Dr. Toldson is also the former head of the White House Initiative on Historically Black Colleges and Universities.


Discover DH Lawrence country on the edge of Nottingham and follow in the footsteps of the county's most controversial literary hero.
From 'Lady Chatterley's Lover' to 'Women in Love', DH Lawrence was a man ahead of his time, known for pushing the boundaries of the day when it came to the risque.
Now, a hundred years later, his literary genius is widely recognized and his working class origins in Eastwood, Nottinghamshire, can easily be explored today.
Visit the house where DH Lawrence was born, the DH Lawrence Birthplace Museum, and find out about his family history and how the local area influenced his writing. Follow the Blue Line Trail in his footsteps and see firsthand many of the places which feature in his work. Stay nearby in a friendly B&B, and make the most of a literary inspired break in Nottinghamshire.
If you're a cheese connoisseur then a trip to Colston Bassett is a must when you are in Nottinghamshire.
Soak up the festive atmosphere at a Christmas fair, while staying in a luxury apartment, and sipping cuppas with cute kitties.
Visit the new Harley Gallery, a permanent home for the Portland Collection on the Welbeck Estate.


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

Search for lepton flavor violation (LFV) is an important probe of
the possible physics beyond the Standard Model (SM).
At present LFV is an established fact, since it has been already
observed in neutrino oscillations, and therefore it is natural to expect
that LFV is also going to manifest itself in the sector of charged leptons.

A search strategy for LFV should consider those  processes
which have the best prospect for discovery, both from the viewpoint of
of their possible experimental identification
and from theoretical limitations on the corresponding rates.
The latter should incorporate the study of model independent
relations between different processes, some of which are already strongly
limited by experimental data.

The three-body purely leptonic decays of $\mu$ and $\tau$ are among
the most stringently constrained LFV processes, with the following current
limits on their branching ratios~\cite{PDG}
\begin{eqnarray}
\label{eq:ExpLim-mu}
&&{\rm Br}(\mu^{-} \to e^{-} e^{+} e^{-}) < 1.0 \times 10^{-12}\,,\\
\label{eq:ExpLim-tau-e}
&&{\rm Br}(\tau^{-} \to  e^{-} e^{+} e^{-}) < 2.7 \times 10^{-8}\,,\\
\label{eq:ExpLim-tau-mu}
&&{\rm Br}(\tau^{-} \to  \mu^{-} e^{+} e^{-}) < 1.8 \times 10^{-8}\,,\\
\label{eq:ExpLim-mu-e-2g}
&&{\rm Br}(\mu^{-} \to  e^{-} \gamma\gamma) \hspace{12pt} < 7.2 \times 10^{-11}\,.
\end{eqnarray}

\begin{figure}[htb]
\vspace*{-3cm}
\includegraphics[scale=0.45]{ThreeBody.pdf}
\noindent
\mbox{}\\[-35mm]
\caption{Three-body LFV decays:  (a) $\tau(\mu) \to 3 \ell$ and
(b) $\tau(\mu) \to \ell \, \gamma\gamma$.
\label{fig:tau3ell}}
\end{figure}
The purpose of the present paper is to relate the three-body lepton and
lepton-photon decays of $\mu$ and $\tau$ (see Fig.~\ref{fig:tau3ell})
to the two-body LFV decays of neutral vector bosons and pseudoscalar mesons,
and to give upper limits for these two-body branching ratios in a model
independent way.
We also study the LFV dimension-six quark-lepton effective
operators underlying these processes and derive limits on their scales
from the limits~(\ref{eq:ExpLim-mu})-(\ref{eq:ExpLim-tau-mu}).

There already exist in the literature similar studies of limits on
the two-body LFV decays of vector mesons/bosons
$V = \rho^0, \omega, \phi, J/\psi, \Upsilon, Z^0 \to \mu^{\pm}e^{\mp}$,
which use the constraint given in~(\ref{eq:ExpLim-mu})  and
unitarity-inspired arguments~\cite{Nussinov:2000nm}.

The idea of using effective quark-lepton and
hadron-lepton Lagrangians for studying
LFV processes (lepton-flavor changing decays, lepton-flavor
conversion, double beta decay) have been proposed and developed in
Refs.~\cite{Vergados:1985pq}-\cite{Kuno:1999jp}
and further used in a series of papers
(see, e.g., Refs.~\cite{Faessler:2004jt}-\cite{Davidson:2018kud}).
In particular, in Refs.~\cite{Faessler:1996ph}, the on-mass-shell
matching condition between the quark-level effective Lagrangian
and the effective hadronic-level (e.g., nucleon) Lagrangian was proposed,
which sets the relations between the couplings at the quark level to those
at the hadronic level.
In a series of papers~\cite{Faessler:2004jt}-\cite{Gonzalez:2013rea},
 $\mu^--e^-$ conversion in nuclei was studied
in the framework of an effective
Lagrangian approach, without referring to any specific realization of
the physics beyond the SM responsible for LFV.
Limits on various LFV couplings of vector and scalar mesons
to the $\mu-e$ current were derived from the existing experimental data on
$\mu^{-}-e^{-}$ conversion in nuclei.
Here, we extend the application of these techniques, in order to extract
limits on two-body LFV decays of vector and pseudoscalar mesons by searching
for LFV three-lepton decays of tau leptons and muons.

The paper is organized as follows.
In Sec.~II, we introduce the relevant effective quark-lepton and
meson-lepton LFV operators, without referring to specific mechanisms of LFV.
In Sec.~III, we derive the relations between three-body lepton LFV
decays and two-body LFV meson decays, which is done by taking into
account the contribution of neutral vector and pseudoscalar mesons in the
three-body lepton LFV process. With these relations, we set the limits on
the two-body LFV meson decays. In Sec.~IV, we derive the relations between
branching ratios of two-body LFV decays of the same quark content and examine
the limits on the effective quark-lepton operators from purely leptonic
processes. Section V contains our summary and conclusions.

\section{Effective Quark-Lepton and Meson-Lepton LFV operators}
\label{sec:Effective Lepto-quark and Lepto-Meson LFV operators}

Let us assume generic LFV sources, leading to $\tau\rightarrow \mu(e) e e$
and $\mu\rightarrow 3e$ decays, in the form of effective operators as the
low-energy limit of a renormalizable ``fundamental'' LFV theory
at a scale $\Lambda$. The leading-order operators have been
proposed in Refs.~\cite{Vergados:1985pq}-\cite{Kuno:1999jp}.
The set of these operators can be written as
\begin{eqnarray}\label{eq:EffOperLept-1}
\mbox{4-lepton:}&&\mathcal{L}_{4\ell}
= \frac{1}{\Lambda^2} \,
\sum\limits_{(IJ)} \,
C^{\Gamma_I\Gamma_J}_{\ell_{1}\ell_{2}} \,
[\, \bar{\ell}_{1} \Gamma_I \ell_{2} \, ] \cdot
[\,\bar{e}   \Gamma_J e\, ]  + {\rm H.c.}
\\
\label{eq:MagneticOper}
\mbox{Magnetic:}&&\mathcal{L}_{M} =
\frac{1}{\Lambda}
\bigg( \,
\bar{C}^{T}_{\ell_{1}\ell_{2}}  [\, \bar\ell_{1}  \sigma_{\mu\nu} \ell_{2}  \, ]
+
 \bar{C}^{T5}_{\ell_{1}\ell_{2}}  [\,  \bar\ell_{1}
\sigma_{\mu\nu} \gamma_{5}\ell_{2}  \, ]  \,
 \bigg) \, F^{\mu\nu}\, + \, {\rm H.c.}\,,
\\
\label{eq:l-q-operators}
\mbox{Quark-Lepton:}&&\mathcal{L}_{\ell q} =
\frac{1}{\Lambda^2} \,
\sum\limits_{(IJ)} \,
C^{\Gamma_I\Gamma_J}_{if,\ell_{1}\ell_{2}} \,
 [\,  \bar{\ell}_{1} \Gamma_I \ell_{2}  \, ]  \cdot
 [\,  \bar{q}_{f}    \Gamma_J q_{i}  \, ]  + {\rm H.c.}
\nonumber\\
&=&\frac{1}{\Lambda^2}
\, \bigg(
\
C^{SS}_{if,\ell_{1}\ell_{2}} \,
[\bar{\ell}_{1}  \ell_{2} ] \cdot  [ \bar{q}_{f}    q_{i}]
\ +\
C^{PS}_{if,\ell_{1}\ell_{2}} \,
[\bar{\ell}_{1}  \gamma^5 \ell_{2} ] \cdot  [ \bar{q}_{f}    q_{i}]
\ +\
C^{SP}_{if,\ell_{1}\ell_{2}} \,
 [\,  \bar{\ell}_{1} \ell_{2}  \, ]  \cdot  [\,   \bar{q}_{f} \gamma_5  q_{i}  \, ]
\nonumber
\\
&& \hspace{-6pt}+\
 C^{PP}_{if,\ell_{1}\ell_{2}} \,
 [\,  \bar{\ell}_{1} \gamma^5 \ell_{2}  \, ]  \cdot   [\, \bar{q}_{f} \gamma_5  q_{i}  \, ]
\ +\
C^{VV}_{if,\ell_{1}\ell_{2}} \,
 [\,  \bar{\ell}_{1} \gamma^\mu \ell_{2}  \, ] \cdot
 [\,  \bar{q}_{f} \gamma_\mu q_{i}  \, ]
\  +\
 C^{AV}_{if,\ell_{1}\ell_{2}} \,
  [\,  \bar{\ell}_{1} \gamma^\mu \gamma^5 \ell_{2}  \, ]
\cdot
 [\,  \bar{q}_{f} \gamma_\mu q_{i}  \, ]
\nonumber
\\
&& \hspace{-6pt}+\
C^{VA}_{if,\ell_{1}\ell_{2}} \,
 [\,  \bar{\ell}_{1} \gamma^\mu \ell_{2}  \, ]
\cdot
 [\,  \bar{q}_{f} \gamma_\mu\gamma_5 q_{i}
 \, ]
\  +\
 C^{AA}_{if,\ell_{1}\ell_{2}} \,
  [\,  \bar{\ell}_{1} \gamma^\mu\gamma^5\, \ell_{2}  \, ]  \cdot
 [\,  \bar{q}_{f} \gamma_\mu\gamma_5 q_{i}  \, ]
\nonumber\\
&& \hspace{-6pt}+\
C^{TT}_{if,\ell_{1}\ell_{2}} \,
 [\,  \bar{\ell}_{1} \sigma^{\mu\nu} \ell_{2}  \, ]  \cdot
 [\,  \bar{q}_{f} \sigma_{\mu\nu} q_{i}  \, ]
\ \bigg)
 + {\rm H.c.} \,,
\label{eq:T-term-1}
\end{eqnarray}
where $\ell= \mu, e$ and
$F^{\mu\nu} = \partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$
is the electromagnetic field tensor.
In (\ref{eq:EffOperLept-1}) and (\ref{eq:l-q-operators}),
we use $I,J= S, P, V, A, T$ and
$\Gamma_{I,J} = 1, \gamma_{5}, \gamma_{\mu}, \gamma_{\mu}\gamma_{5},
\sigma_{\mu\nu}$, so that
the summation runs over $(IJ) = (SS), (PS), (SP), (PP),
(AV), (VV), (VA), (AA), (TT)$. In Eq.~(\ref{eq:T-term-1}),
we displayed the terms in the sum explicitly.
After specifying all possible Lorentz structures
in Eqs.~(\ref{eq:EffOperLept-1}) and
(\ref{eq:l-q-operators}), we used the identity
$
\bar{a} \sigma^{\mu\nu}\gamma_{5} b \cdot
\bar{c} \sigma_{\mu\nu}\gamma_5 d =\bar{a} \sigma^{\mu\nu} b \cdot
\bar{c} \sigma_{\mu\nu} d.
$
Here, we denoted the LFV scale by $\Lambda$.

The operators~(\ref{eq:EffOperLept-1}) and~(\ref{eq:MagneticOper}) lead to
tree-level contributions to $\tau\rightarrow \ell e e$, while
the dipole-type  operator~(\ref{eq:MagneticOper}) directly contributes to
$\tau\rightarrow \ell \gamma$.
Limits on the scales of these operators are readily extracted from
data~\cite{PDG} and can be found in the literature (see, for instance,
Ref.~\cite{Lims-4-lept-M}).
The quark-lepton LFV operators~(\ref{eq:l-q-operators})
have been studied by many authors, which consider the two-body decays
$\tau\rightarrow \ell M$, $M\rightarrow \ell_{1} \ell_{2}$,
deep inelastic conversion
$\tau(\mu) q \rightarrow \ell q$~\cite{Gninenko:2018num}
as well as nuclear $\mu-e$- conversion (for a recent review see, for instance,
Ref.~\cite{Nucl-MuE}). The existing data on the rates of these processes
allowed extraction of rather stringent limits on the scale of
the corresponding operators~(\ref{eq:l-q-operators}), which also
contribute to leptonic LFV decays of
mesons $M\rightarrow \ell_{1} \ell_{2}$ at tree level.
At one-loop level they contribute
to purely leptonic LFV processes $\tau^{-}\rightarrow \mu^{-} e^{+} e^{-}$
and $\mu^{-}\rightarrow e^{-} e^{+} e^{-}$.
However, quark-hadron duality \cite{Nussinov:2000nm} relates
these loop contributions, taking into account nonperturbative QCD effects,
with the sum over the tree-level contributions (Fig.~\ref{fig-duality}) of all
the intermediate meson states with the allowed quantum numbers.

Therefore, effectively the operators in Eq.~(\ref{eq:l-q-operators})
trigger tree-level contributions to
$\ell_{1}\rightarrow \ell_{2} \, e^{+} e^{-}$
via intermediate meson states. The relevant meson-lepton vertices
involving vector $V$, axial $A$, pseudoscalar $P$, and scalar $S$ mesons
with quantum numbers $J^{PC} = 1^{--}$, $1^{++}$ ($1^{+-}$), $0^{-+}$,
and $0^{++}$, respectively, are
\begin{eqnarray}
\label{eq:Meson-Lepton-LFV-Vert}
\mathcal{L}_{\ell M} &=&
V_{\mu} \left(g^{(V)}_{V\ell_{1}\ell_{2}} \,
[\bar{\ell}_{1} \gamma^{\mu} \ell_{2}]+
g_{V\ell_{1}\ell_{2}}^{(A)} \,
[\bar{\ell}_{1}
\gamma^{\mu}\gamma_{5} \ell_{2} ] \right) +
A_{\mu} \left(g^{(V)}_{A\ell_{1}\ell_{2}} \,
[ \bar{\ell}_{1} \gamma^{\mu} \ell_{2} ]+
g_{A\ell_{1}\ell_{2}}^{(A)} \,
[ \bar{\ell}_{1}
\gamma^{\mu}\gamma_{5} \ell_{2} ] \right)
\nonumber\\
&+&
\frac{g^{(T)}_{V\ell_1\ell_2}}{M_V} \,
F^{V}_{\mu\nu}\, \left[\bar{\ell}_{1} \sigma^{\mu\nu} \ell_{2}\right]
+\frac{g^{(T)}_{A\ell_1\ell_2}}{M_A} \,
F^{A}_{\mu\nu}\, \left[\bar{\ell}_{1} \sigma^{\mu\nu} \gamma^5 \ell_{2}\right]
\nonumber\\
&+&S\, \left(g^{(S)}_{S\ell_{1}\ell_{2}} \,
[ \bar{\ell}_{1}  \ell_{2} ]
+
g_{S\ell_{1}\ell_{2}}^{(P)} \,
[ \bar{\ell}_{1} \gamma_{5} \ell_{2}] \right) +
P\, \left( i g^{(S)}_{P\ell_{1}\ell_{2}} \, [ \bar{\ell}_{1}   \ell_{2} ] +
i g_{P\ell_{1}\ell_{2}}^{(P)} \, [\bar{\ell}_{1} \gamma_{5} \ell_{2} ]
\right)
\nonumber\\
&+&
\frac{\partial_\mu P}{M_P} \,
\left(g^{(V)}_{P\ell_1\ell_2} [ \bar{\ell}_{1} \gamma^\mu \ell_{2} ] +
g^{(A)}_{P\ell_1\ell_2}  [ \bar{\ell}_{1} \gamma^\mu\gamma^5 \ell_{2} ] \right)
\,+\, {\rm H.c.}
\end{eqnarray}
\begin{center}
\begin{figure}[htb]
\vspace{-2.6cm}
\includegraphics[scale=0.35,angle=0]{Duality.pdf}
\vspace{-2.4cm}
\caption{
Quark-lepton contact interaction contribution to
$l_{1}\rightarrow l_{2} l_{3} l_{4}$ via meson exchange
according to quark-hadron duality.}
\label{fig-duality}
\end{figure}
\end{center}
Here we introduced the notation
$F^{M}_{\mu\nu} = \partial_{\mu}M_{\nu}- \partial_{\nu}M_{\mu}$
(with $M=V ,A$) for the field tensors of the vector and axial mesons,
respectively.
Obviously, the lightest mesons dominate in the diagram in
Figs.~\ref{fig:tau3ell}(a) and 1(b),
because the contributions of meson resonances to the three-body LFV
decays scale as $1/M^4$, where $M$ is the mass of intermediate meson.

In the next sections, we shall use the effective hadronic-level
Lagrangian of Eq.~(\ref{eq:Meson-Lepton-LFV-Vert}) in order to constrain
the quark-lepton operators of Eq.~(\ref{eq:l-q-operators}),
using the bounds given in
Eqs.~(\ref{eq:ExpLim-mu})-(\ref{eq:ExpLim-tau-mu}). This is done by
applying an appropriate matching condition at the hadronization scale.
In this way, we shall constrain the vector and tensor operators related to
the corresponding vector boson contribution
$(V = \rho^0, \omega, \phi, J/\psi, \Upsilon, Z^0)$
to the processes $\tau(\mu) \to 3 \ell$, and also constrain
the pseudoscalar and scalar operators from the contribution of
the pseudoscalar $(P = \pi^0, \eta, \eta', \eta_c(1S))$ and
scalar meson states $(S = f_{0}(500), f_{0}(980), a_{0}(980), \chi_{c0}(1P))$
to the processes $\tau(\mu) \to \ell \gamma\gamma$.
Expressions for the LFV two-body decay widths of different meson states
are shown in Appendix~\ref{App:B}.

Let us recall a key point of the present study: non-perturbative
QCD effects leading to the formation of the $M$ meson bound states
in the intermediate state of $\ell_{1}\rightarrow \ell_{2}\, e^{+}e^{-}$
are taken into account according
to the quark-hadron duality, via two parameters: the meson masses $M_{M}$ and
their leptonic decay constants $f_{M}$.
Numerical values of these parameters are known either from direct
experimental measurements, from lattice simulations or some reliable
models. The list of these parameters are given in Appendix \ref{App:A}.
We shall study these meson exchange  mechanisms in the next sections.

\section{Relations between three- and two-body LFV decays}
\label{sec:Relations between Tree- and two-body LFV decays}

Here we derive unitarity-inspired relations between the three-body
lepton decays and the two-body vector, scalar, and pseudoscalar
meson decays.
Unitarity implies the contribution of all intermediate meson states to
$\tau,\mu\rightarrow\ell e e, \ell\,  \gamma \gamma$.
Following Ref.~\cite{Nussinov:2000nm}, we retain as a good approximation
only the lightest mesons,
so that their contributions are described by the meson exchange diagrams
in Figs.~\ref{fig:tau3ell}, with the LFV vertices given
by the Lagrangian~(\ref{eq:Meson-Lepton-LFV-Vert}).
We shall not consider flavored mesons, because their decay
rates \mbox{$M\rightarrow e^{+}e^{-}, \gamma\gamma$}, which enter
in the above-mentioned relations, are GIM-suppressed, and this does
not allow us to derive significant limits for their LFV decays.

\subsection{Vector mesons}
\label{sec:Vector meson LFV decays}

Let us consider the vector mesons
$V = \rho^0, \omega, \phi, J/\psi, \Upsilon, Z^0$.
Our goal is to analyze their contribution to $\mu,\tau\rightarrow 3\ell$
decays. For the case of
$\mu^{-}\rightarrow e^{-} e^{+}e^{-}$ and vector
mesons, this was done in Ref.~\cite{Nussinov:2000nm}.

Neglecting the final lepton masses, for the muon decay rates we have
\begin{eqnarray}\label{eq:mu-3e-1}
\Gamma(\mu^{-}\to e^{-} e^{+}e^{-}) &=&
\kappa \frac{\tilde{g}_{V\mu e}^{2}
\tilde{g}_{Vee}^{2}}{M_{V}^{4}}\\
\label{eq:mu-3e-2}
\Gamma(\mu^{-}\to e^{-} \bar\nu_{e} \nu_{\mu}) &=&
\Gamma (\mu \to All) = \kappa \frac{g_{W}^{4}}{M_{W}^{4}}\,,
\end{eqnarray}
where $M_V$ is the vector meson/boson mass,
$\kappa = M_\mu^5/(384 \pi^3)$
is a kinematic-spin factor common to all decay modes involving
vector mesons in the intermediate state, while
$g_{W}$ and $M_{W}$ are the electroweak coupling and the $W$ boson mass, respectively
(here $g_W$ is normalized so that the Fermi coupling is
$G_F/\sqrt{2} =  g_W^2 /(2 M_W^2)$).
By definition
$\tilde{g}_{V\mu e}^{2} = |g^{(V)}_{V\mu e}|^{2}+|g^{(A)}_{V\mu e}|^{2}$.
Then one finds for the LFV branching ratio
\begin{eqnarray}\label{eq:BrMu3E}
{\rm Br}(\mu^{-}\to e^{-} e^{+}e^{-}) &=&
\frac{\tilde{g}_{V \mu e}^2
\tilde{g}_{Vee}^2}{M_V^4} \
\frac{M_W^4}{g_W^4}\,.
\end{eqnarray}
Formulae for the meson two-body decay rates are given
in Appendix~\ref{App:B}. Neglecting the
final lepton masses they can be written as
\begin{eqnarray}
\label{eq:V2LL-1}
& &\Gamma(V \to e^+e^-) = a\,
\tilde{g}_{Vee}^2 \, M_V
\,, \\
\label{eq:V2LL-2}
& &\Gamma(V \to \mu^\pm e^\mp) = a\,
\tilde{g}_{V \mu e}^2 \, M_V
\,, \\
\label{eq:V2LL-3}
& &\Gamma(W \to e\bar\nu_e) = a\,  g_W^2 \, M_W\, , \qquad
\end{eqnarray}
where $a = 1/(12\pi)$ is a kinematic factor common to all these processes.

The branching ratio of Eq.~\eqref{eq:BrMu3E} can then be written
in terms of the two-body decay rates as:
\begin{eqnarray}\label{eq:NussinovFivRes-1}
{\rm Br}(\mu \to 3 e) &=&
\frac{\Gamma(V \to \mu  e) \,
\Gamma(V \to e^+e^-)}{\Gamma(W \to e \bar\nu_e)^2} \,
\biggl(\frac{M_W}{M_V}\biggr)^6 \,.
\end{eqnarray}

For the case of $\tau^{-}\rightarrow e^{-}(\mu^{-}) e^{+}e^{-}$
there are two main differences with respect to the muon decays:
(i) due to the large mass of the $\tau$ lepton,
there are some on-mass-shell meson contributions to this process;
(ii) the $\tau$ decay width is not purely an electroweak quantity, i.e.
\mbox{$\Gamma(\tau \to All) \neq \Gamma (\tau \to \ell \bar\nu \nu)$},
since it contains hadronic channels. The latter suffer from considerable
theoretical uncertainties.
However, the tau decay width is an experimentally well measured observable~\cite{PDG}.
Combining the above formulae~(\ref{eq:mu-3e-1}), (\ref{eq:mu-3e-2})
and~(\ref{eq:V2LL-1})-(\ref{eq:V2LL-3}) with the corresponding replacements,
for $M_{V}> M_{\tau}$ we find:
\begin{eqnarray}\label{eq:TauLEE-1}
{\rm Br}(\tau^{-}\rightarrow \ell^{-} e^{+}e^{-}) &=&
 \frac{\Gamma(V \to \tau \ell) \,
\Gamma(V \to e^+e^-)}{\Gamma(W \to e \bar\nu_e)^2} \,
 \frac{\Gamma(\mu^{-}\to e^{-} \bar\nu_{e} \nu_{\mu})}{\Gamma(\tau \to All)}
  \left(\frac{M_W}{M_V}\right)^6
\left(\frac{M_{\tau}}{M_{\mu}}\right)^5\,,
\end{eqnarray}
and for $M_{V}<M_{\tau}$
\begin{eqnarray}\label{eq:TauLEE-2}
{\rm Br}(\tau^{-}\rightarrow \ell^{-} e^{+}e^{-}) &=&
{\rm Br}(\tau \to V \ell)\,  {\rm Br}(V \to e^{+}e^{-})\,
\end{eqnarray}
where $\ell = \mu, e$.
The latter case is not interesting for our analysis,
which is related to constraints on $\tau \to V \ell$.

\subsection{Unflavored pseudoscalar and scalar mesons.}
\label{sec:Pseudoscalar meson qq LFV decays}

The unflavored pseudoscalar and scalar mesons contribute to
$\mu^{-} \to e^{-} \gamma \gamma$ and $\tau^{-}\to \ell^{-} \gamma\gamma$,
according to the diagram in Fig.~\ref{fig:tau3ell}(b),
with the LFV vertex $P(S) \bar\ell_{1}\ell_{2}$
given in Eq.~(\ref{eq:Meson-Lepton-LFV-Vert}), and
\begin{eqnarray}
\label{eq:Vert-PGammaGamma}
{\cal L}_{P\gamma\gamma} &=& \frac{e^2}{4} \, g_{P\gamma\gamma} \,
P \,
F_{\mu\nu}\,  \varepsilon^{\mu\nu\alpha\beta}\,   F_{\alpha\beta}\,, \\
\label{eq:Vert-SGammaGamma}
{\cal L}_{S\gamma\gamma} &=& \frac{e^2}{4} \, g_{S\gamma\gamma} \,
S\, F_{\mu\nu} \, F^{\mu\nu},
\end{eqnarray}
where
$F_{\mu\nu} = \partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu}$
is the electromagnetic stress tensor, $\varepsilon^{\mu\nu\alpha\beta}$
is the Levi-Cevita tensor,
and $g_{I\gamma\gamma}$ ($I=P, S$) are
the effective couplings of the $I \to \gamma\gamma$
decay widths: \begin{eqnarray}
\Gamma(I \to \gamma\gamma) = \frac{\pi\alpha^2}{4} \,
g_{I\gamma\gamma}^2 \, M_I^3\,,
\end{eqnarray}
where $\alpha \simeq 1/137.036$ is the fine structure constant.
In the case of $\pi^0$, the coupling
$g_{\pi\gamma\gamma}$ is related to the pion decay constant
$F_\pi \simeq 92.4$ MeV as
\begin{eqnarray}
g_{\pi\gamma\gamma} = \frac{1}{4 \pi^2 F_\pi}\,.
\end{eqnarray}
The pion contribution to the decay $\mu \to e \gamma\gamma$ was
discussed in Ref.~\cite{Nussinov:2000nm}.
Extending this analysis to include other scalar and
pseudoscalar mesons we can write
\begin{eqnarray}\label{eq:MU-EGG-1}
{\rm Br}(\mu^{-}\rightarrow e^{-} \gamma\gamma) &\approx&
\frac{\Gamma(I \to \mu  e) \,
\Gamma(I \to \gamma\gamma)}{\Gamma^2(W \to e \bar\nu_e)} \,
\left(\frac{M_W}{M_I}\right)^6 \left(\frac{M_{\mu}}{2 M_{I}}\right)^{4}.
\end{eqnarray}
For the $\tau$ lepton decay we find, in analogy  to Eq.~(\ref{eq:TauLEE-1}),
and for $M_{I}>M_{\tau}$:
\begin{eqnarray}\label{eq:Tau-EGG-1}
{\rm Br}(\tau^{-}\rightarrow \ell^{-} \gamma\gamma) &\approx&
\frac{\Gamma(I \to \tau \ell) \,
\Gamma(I \to \gamma\gamma)}{\Gamma^2(W \to e \bar\nu_e)} \,
\frac{\Gamma(\mu^{-}\to e^{-} \bar\nu_{e} \nu_{\mu})}{\Gamma(\tau \to All)}
\left(\frac{M_W}{M_I}\right)^6   \left(\frac{M_{\tau}}{M_{\mu}}\right)^5
  \left(\frac{M_{\tau}}{2 M_{I}}\right)^{4}\,.
\end{eqnarray}
The case $M_I < M_{\tau}$ with on-mass-shell mesons is not interesting
for our analysis.

In our numerical analysis, we use the central values of the decay widths
of pseudoscalar and scalar mesons quoted from the Particle Data
Group~\cite{PDG}:
\begin{eqnarray}
& &
\Gamma(\pi^0   \to \gamma\gamma)  = 7.64   \ {\rm eV}\,,
\quad \hspace{26pt}
\Gamma(\eta    \to \gamma\gamma)  = 0.52   \ {\rm keV}\,,
\nonumber\\
& &
\Gamma(\eta'   \to \gamma\gamma)  = 4.35   \ {\rm keV}\,,
\quad \hspace{24pt}
\Gamma(\eta_c  \to \gamma\gamma)  = 5.02   \ {\rm keV}\,,
\nonumber\\
& &
\Gamma(f_0(500)\to \gamma\gamma)  = 2.05   \ {\rm keV}\,,
\quad
\Gamma(f_0(980)\to \gamma\gamma)  = 0.31   \ {\rm keV}\,,
\nonumber\\
& &
\Gamma(a_0(980)\to \gamma\gamma)  = 0.30   \ {\rm keV}\,,
\quad
\Gamma(\chi_{c0}(1P)\to \gamma\gamma)  = 2.20 \ {\rm keV}\,.
\end{eqnarray}

Note that up to now there are no experimental constraints on
$\tau \to \ell \gamma\gamma$ decay rates.
Therefore, in the present paper we present
only theoretical formula~(\ref{eq:Tau-EGG-1}) relating three-body LFV decay
of $\tau$ with two-body LFV decays $P(S) \to \tau \ell$, which could be
useful in future searches of these processes.

\subsection{Limits on two-body LFV meson decays}
\label{sec:Limits on two-body LFV meson decays}

From Eqs.~(\ref{eq:NussinovFivRes-1}), (\ref{eq:TauLEE-1}),
(\ref{eq:MU-EGG-1}) and (\ref{eq:Tau-EGG-1}),
we deduce upper limits for the branching ratios of the two-body LFV decays
$M(Z) \to \ell_1 \ell_2$ of neutral vector and pseudoscalar mesons and
$Z$-boson, using the existing data
(\ref{eq:ExpLim-mu})-(\ref{eq:ExpLim-tau-mu}) for
three-body LFV decays $\tau(\mu) \to 3 \ell$.
We present our results in the second column of
Table~\ref{tab:two-body-LFV} and compare them with the limits
derived from the study of lepton conversion~\cite{Nussinov:2000nm}
and available experimental data~\cite{PDG}.

In the case of the $\pi^0$ and $J/\psi$ contributions,
we also show in parenthesis our results for the constraints which
take into account the $Q^2$-dependence of the meson propagator
and the form factor $\tilde{g}_{M\ell_{1}\ell_{2}}(Q^{2})$, when this last
effect is significant. For other meson contributions the effect of
the $Q^{2}$-dependence is negligible. A detailed discussion and
estimation of this effect is presented in Appendix~\ref{app:Q2dep}.

\begin{table}[ht]
\begin{center}
\caption{Upper limits for the branching ratios of two-body LFV decays
of neutral vector, pseudoscalar, and scalar mesons, and $Z$-boson,
extracted from the bound on the indicated three-body $\mu$ and $\tau$ decays.
``EO-improved'' are limits obtained from Eq.~(\ref{eq:M1-M2-1})
relating different LFV processes with the same underlying
effective operators (EO).}
\label{tab:two-body-LFV}
\def.9{.9}
\begin{tabular}{|c|c|c|c|c|}
\hline
Mode & \multicolumn{2}{|c|}{Our results} & Existing Limits & Data \\
\hline
\multicolumn{3}{|c|}{from $\mu^- \to e^- \gamma\gamma$ process}& &\\
\cline{1-3}
$\pi^0   \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{
$5.8 \times 10^{-11}$ ($3.2 \times 10^{-11}$)
}
& $10^{-10}$~\cite{Nussinov:2000nm}
& $3.8 \times 10^{-10}$~\cite{PDG} \\
$\eta   \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{$6.2 \times 10^{-9}$}
& $10^{-8}$~\cite{Nussinov:2000nm}
& $3.0 \times 10^{-6}$~\cite{PDG} \\
$\eta'   \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{$1.3 \times 10^{-9}$}
& $4.7 \times 10^{-4}$~\cite{PDG} & \\
$\eta_c   \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{$5.9 \times 10^{-7}$}
& $1.57 \times 10^{-4}$~\cite{PDG} & \\
$f_0(500)   \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{$1.6 \times 10^{-15}$}
& &\\
$f_0(980)   \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{$1.0 \times 10^{-10}$}
& &\\
$a_0(980)   \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{$6.2 \times 10^{-11}$}
& &\\
$\chi_{c0}(1P)  \to \mu^\pm e^\mp$
& \multicolumn{2}{|c|}{$1.5 \times 10^{-5}$}
& &\\
\hline\hline
Mode & Our results & EO-improved& Existing Limits & Data \\
\hline
\multicolumn{3}{|c|}{from $\mu^- \to e^- e^+ e^-$ process} &&\\
\cline{1-3}
$\rho^0   \to \mu^\pm e^\mp$  & $5.8 \times 10^{-21}$ & &
$3.5 \times 10^{-24}$~\cite{Gutsche:2011bi}
& \\
$\omega   \to \mu^\pm e^\mp$  & $6.8 \times 10^{-20}$
& $9.1 \times 10^{-21}$ &
$6.2 \times 10^{-27}$~\cite{Gutsche:2011bi}
& \\
$\phi     \to \mu^\pm e^\mp$  & $1.6 \times 10^{-19}$
& $1.1 \times 10^{-19}$ &
$4 \times 10^{-17}$~\cite{Nussinov:2000nm};
$(1.1 \times 10^{-25} - 5.6
\times 10^{-22})$~\cite{Gutsche:2011bi}
&
$2.0 \times 10^{-6}$~\cite{PDG} \\
$J/\psi   \to \mu^\pm e^\mp$  & $2.9 \times 10^{-17}$
& $2.6 \times 10^{-18}$ &
$4 \times 10^{-13}$~\cite{Nussinov:2000nm};
$5.4 \times 10^{-14}$~\cite{Gutsche:2011bi}
&
$1.6 \times 10^{-7}$~\cite{PDG} \\
$\Upsilon \to \mu^\pm e^\mp$  & $1.0 \times 10^{-13}$
&$2.5 \times 10^{-16}$ &
$2 \times 10^{-9}$~\cite{Nussinov:2000nm};
$1.1 \times 10^{-6}$~\cite{Gutsche:2011bi}
& \\
$Z^0 \to \mu^\pm e^\mp$  & $1.3 \times 10^{-12}$ & &
$5 \times 10^{-13}$~\cite{Nussinov:2000nm};
$4.2 \times 10^{-6}$~\cite{Gutsche:2011bi}
&
$7.5 \times 10^{-7}$~\cite{PDG} \\
\hline
\multicolumn{3}{|c|}{ from $\tau^- \to e^- e^+ e^-$ process}&& \\
\cline{1-3}
$J/\psi   \to \tau^\pm e^\mp$ &
$4.5 \times 10^{-12}$ ($2.8 \times 10^{-12}$)
& &
$6 \times 10^{-7}$~\cite{Nussinov:2000nm}  &
$8.3 \times 10^{-6}$~\cite{PDG} \\
$\Upsilon \to \tau^\pm e^\mp$ & $1.6 \times 10^{-8}$  & $7.3 \times 10^{-10}$ &
$1 \times 10^{-2}$~\cite{Nussinov:2000nm}  & \\
$Z^0      \to \tau^\pm e^\mp$ & $1.9 \times 10^{-7}$  & &
$3 \times 10^{-6}$~\cite{Nussinov:2000nm}  &
$9.8 \times 10^{-6}$~\cite{PDG} \\
\hline
\multicolumn{3}{|c|}{from $\tau^- \to \mu^- e^+ e^-$ process}&& \\
\cline{1-3}
$J/\psi   \to \tau^\pm \mu^\mp$ &
$3.0 \times 10^{-12}$ ($1.9 \times 10^{-12}$)
& &  &
$2.0 \times 10^{-6}$~\cite{PDG} \\
$\Upsilon \to \tau^\pm \mu^\mp$ & $1.0 \times 10^{-8}$
& $4.9 \times 10^{-10}$ & No limits&
$6.0 \times 10^{-6}$~\cite{PDG} \\
$Z^0      \to \tau^\pm \mu^\mp$ & $1.3 \times 10^{-7}$  & &  &
$1.2 \times 10^{-5}$~\cite{PDG} \\
\hline
\end{tabular}
\end{center}
\end{table}

One can see from Table~\ref{tab:two-body-LFV} that in most cases
we get more stringent constraints on
the branching ratios of the two-body LFV decays.
In particular, our limits are $3-4$ orders of magnitude better
than the existing ones for $J/\Psi, \Upsilon \rightarrow \mu e$,
while for $J/\Psi, \Upsilon \rightarrow \tau e$ the improvement is
5 orders of magnitude.
To the best of our knowledge, in the
literature there are no phenomenological limits
for $J/\Psi, \Upsilon, Z \rightarrow \tau \mu$, and
our limits are significantly more stringent than the existing
experimental bounds~\cite{PDG}.
In Table~\ref{tab:two-body-LFV},
we also displayed for completeness the LFV decays of
$f_{0},a_{0},\chi_{c0} \rightarrow \mu e$, which are unrealistic
for experimental observations. We recall that these mesonic states,
together with other mesons, are needed for the implementation of
the quark-hadron duality and the derivation of the limits on
the quark-lepton operators (\ref{eq:T-term-1}).

\section{Quark-lepton effective operators in LFV decays of $\mu, \tau$}
\label{sec:Lepto-quark effective LFV operators in LFV decays MU-TAU}

\subsection{Indirect contribution to $\ell_{1}\rightarrow \ell_{2} e e$}

Here we examine the limits on the effective quark-lepton
operators~(\ref{eq:l-q-operators}) from the purely leptonic
processes \mbox{$\tau^{-}\rightarrow \mu^{-}(e^{-}) e^{+} e^{-}$},
$\mu^{-}\rightarrow e^{-} e^{+} e^{-}$ or
$\tau^{-}\rightarrow \mu^{-}(e^{-}) \gamma\gamma$,
$\mu^{-}\rightarrow e^{-} \gamma\gamma$.
The operators (\ref{eq:l-q-operators}) contribute to
$\tau\rightarrow \ell e e$
at one-loop level.
However, as we discussed
in Sec.~\ref{sec:Effective Lepto-quark and Lepto-Meson LFV operators},
quark-hadron duality identifies these loop contributions with the tree-level
contribution of the mesons states with the corresponding quantum numbers,
as shown in  Fig.~\ref{fig:tau3ell}.
In order to constrain the  quark-lepton operators~(\ref{eq:l-q-operators}),
we match them to the corresponding meson-lepton operators
in Eq.~(\ref{eq:Meson-Lepton-LFV-Vert}), using the on-mass-shell matching
condition~\cite{Faessler:2004jt,Faessler:2005hx}:
\begin{equation}\label{match1}
\langle \ell_{1}^+ \, \ell_{2}^-|{\cal L}_{eff}^{lq}|M\rangle \approx
\langle \ell_{1}^+ \, \ell_{2}^-|{\cal L}_{eff}^{lM}|M \rangle ,
\end{equation}
where $M$ are the corresponding  mass-shell meson states.
This equation can be solved using the well-known quark current meson
matrix elements shown in Appendix~\ref{App:A},
and we find relations between the quark-lepton scaled Wilson coefficients,
$\mathcal{C}/\Lambda^{2}$ in Eq.~(\ref{eq:l-q-operators}),
and the meson-lepton couplings, $g_{M}$,
from~(\ref{eq:Meson-Lepton-LFV-Vert}),
which are shown in Appendix~\ref{App:C}.
Using these relations in the decay rate formulas
for $\Gamma (M\rightarrow l_{1}l_{2})$
from Appendix A and substituting them into Eqs.~(\ref{eq:NussinovFivRes-1}),
(\ref{eq:TauLEE-1}), (\ref{eq:MU-EGG-1}) and (\ref{eq:Tau-EGG-1}),
we set upper limits on the coefficients $\mathcal{C}/\Lambda^{2}$ of
the effective operators~(\ref{eq:l-q-operators}) from the experimental data on
$\tau(\mu) \to 3 \ell$.
There are several operators contributing simultaneously to each of these
processes, and therefore the data impose upper limits on linear combinations
of the corresponding Wilson coefficients shown in Appendix~\ref{App:E}.
In practice, it is useful to have individual upper limits for these
coefficients under certain reasonable assumptions. In the literature,
it is conventional to assume that there is no strong cancellation
between terms of different origin in the amplitudes and therefore extract
limits on each term as if it was present alone.
We apply this ``one-at-a-time'' approach
to Eqs.~(\ref{eq:gVVAV-Lambda2}-\ref{eq:gSSPS-Lambda2}).
The corresponding results are displayed in
Table~\ref{Table-2} in the form of  lower limits
on the individual mass scales, $\Lambda^{ij}_{\mu e}$,
of the operators in Eq.~(\ref{eq:l-q-operators}).
In the conventional definition (see, for instance,
Ref.~\cite{Gonzalez:2013rea}), these scales are
related to our notation as
\begin{eqnarray}\label{Lambda-LFV}
|C^{XY}_{a}| \, \left(\frac{1{\rm GeV}}{\Lambda}\right)^2 =
4\pi \left(\frac{1{\rm GeV}}{\Lambda^{XY}_{a}}\right)^2
\end{eqnarray}
with $a=0, 3, s, c,b, t$ and $z= hV, rS$, where $h = A, V$ and $r = P, S$
as defined before.

\begin{table}

\vspace*{.4cm}

\begin{center}
\begin{tabular}{|c|c|c||c|c|c||c|c|c|}
\hline
&&&&&&&&    \\[-3mm]
$\Lambda_{\ell_{1}\ell_{2}}$ & Our
& Existing &$\Lambda_{\ell_{1}\ell_{2}}$
& Our & Existing & $\Lambda_{\ell_{1}\ell_{2}}$
& Our &Existing  \\
&limits &limits &&limits &limits&&limits  &limits\\
&[TeV] &[TeV] &&[TeV] &[TeV]&&[TeV]  &[TeV]\\[3mm]
\hline
&& &&&&&&   \\[-3mm]
$\Lambda_{\mu e}^{(3) VV, AV}$ & 86 &   $10^{3}$
&$\Lambda_{\mu e}^{(3) PP,SP} $
& 8.0&none&$\Lambda_{\tau e}^{(c) VV,AV} $&13&none  \\[1mm]
\hline
&& &&&&&&   \\[-3mm]
$\Lambda_{\mu e}^{(3) AA, VA} $ & 7.1 & none &$\Lambda_{\mu e}^{(s) SS, PS} $
& none&
$3\times 10^{3}$ &$\Lambda_{\tau e}^{(b) VV, AV} $&7&none \\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(0) VV, AV} $ & 89 & $4.7 \times 10^{3}$
&$\Lambda_{\mu e}^{(s) PP, SP} $ & 1.3& none
&$\Lambda_{\mu e}^{(c) TT} $&19&none  \\[1mm]
\hline
&&   &&&&&& \\[-3mm]
$\Lambda_{\mu e}^{(0) AA, VA} $ & 2.4 & none
&$\Lambda_{\mu e}^{(c) SS, PS} $ & none
&   950 & $\Lambda_{\mu e}^{(b) TT} $&8.4&none \\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(s) VV,AV} $ & 134  &  $770$
& $\Lambda_{\mu e}^{(b) SS, PS} $
& none&  540 & $\Lambda_{\tau\mu}^{(c) VV, AV} $ &14.5&none  \\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(s) AA, VA} $ & 0.6 & none & $\Lambda_{\mu e}^{(t) SS, PS} $
&none&  90 & $\Lambda_{\tau\mu}^{(b) VV, AV} $&7.7&none \\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(c) VV, AV} $ & 300& 54   & $\Lambda_{\mu e}^{(3) TT} $
&103&none&$\Lambda_{\tau\mu}^{(c) TT} $ &19&none\\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(b) VV, AV} $& 138& 3   & $\Lambda_{\mu e}^{(0) TT} $
&107&none& $\Lambda_{\tau\mu}^{(b) TT} $&9.1&none\\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(3) SS, PS} $ & 0.5 &
$1.8 \times 10^{3}$  &$\Lambda_{\mu e}^{(s) TT} $&160&none&&&  \\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(0) SS, PS} $ & 0.6& $6.8 \times 10^{3} $
& $\Lambda_{\mu e}^{(c) TT} $&355&none&&& \\[1mm]
\hline
&&  &&&&&&  \\[-3mm]
$\Lambda_{\mu e}^{(0) PP, SP} $ & 6.0& none   &$\Lambda_{\mu e}^{(b) TT} $
&164&none&&&\\[1mm]
\hline
\end{tabular}
\end{center}
\caption{
Lower limits on the individual mass scales, $\Lambda_{\ell_{1}\ell_{2}}$,
of the effective operators (\ref{eq:l-q-operators}).
``Existing limits'' are taken from Ref.~\cite{Faessler:2004jt}.
All the limits  are derived assuming that only one operator contributes to
$\tau\rightarrow \mu(e) e e$,  $\mu\rightarrow 3e$
at a time.
}
\label{Table-2}
\end{table}

\subsection{Relations between LFV decays of different mesons}
\label{sec:Relations between LFV decays of different mesons}

Notice that the operators in Eq.~(\ref{eq:l-q-operators}), either individually
or in certain linear combination of them, underly LFV leptonic decay modes
of all the mesons with the same quark content and  $J^{PC}$.

Using the decay rate formulae, the meson matrix elements and the expressions
for the LFV meson couplings from Appendices~\ref{App:A}, \ref{App:B}
and \ref{App:C}, we find, in the limit of massless
final leptons, the following approximate relation
between the branching ratios of different mesons $\mathcal{M} =V,P$:
\begin{eqnarray}
\label{eq:M1-M2-1}
&&{\rm Br}(\mathcal{M}_{a}\rightarrow \ell_{1} \ell_{2}) \approx
\left(\frac{f_{{a}}}{f_{{b}}}\right)^{2}
\left(\frac{M_{{a}}}{M_{{b}}}\right)^{5}
\frac{\Gamma(\mathcal{M}_{b}\rightarrow All)}{\Gamma(\mathcal{M}_{a}
\rightarrow All)}\cdot
{\rm Br}(\mathcal{M}_{b}\rightarrow \ell_{1} \ell_{2}).
\end{eqnarray}
Using this relation and the upper limits in Table~\ref{tab:two-body-LFV}
on the branching ratios for one particular meson, we can set limits for
the other ones.
These ``cross-limits'', shown in the column ``EO improved''
of Table~\ref{tab:two-body-LFV},
are in some cases significantly more stringent than the limits
derived directly from the contribution of
the corresponding meson to $\tau\rightarrow \ell e e$.

\section{Summary}

We derived unitarity-inspired bounds on the two-body LFV
decays of unflavored neutral vector and pseudoscalar mesons as well
as of the $Z$-boson, from the experimental bounds
on the leptonic LFV decays
$\tau(\mu) \to  \ell e^{+}e^{-}, \ell \gamma\gamma$.
Many of our limits are better than those existing to date in the literature.
We also derived still nonexistent in the literature theoretical limits for
$J/\Psi, \Upsilon, Z \rightarrow \tau \mu$, which are significantly more
stringent that the experimental bounds.
Using the fact that the LFV decays of
the mesons with the same quark content and  $J^{PC}$
originate from the same linear combination of  quark-lepton operators,
Eqs.~(\ref{eq:l-q-operators}), we derived improved limits on the decay rate
of one meson from the more stringent limit of the decay rate of another meson.
In some cases, this improvement approaches 3 orders of magnitude.

We analyzed the contribution of
quark-lepton operators~(\ref{eq:l-q-operators}) to purely leptonic processes
$\tau(\mu) \to  \ell e^{+}e^{-}, \ell \gamma\gamma$, on the basis of
the quark-hadron duality, which takes into account these contributions
as coming from intermediate meson states.
In this approach, the nonperturbative QCD effects in the quark loops
are effectively considered by the meson masses and their
leptonic decay constants  In order to realize this approach,
we matched at the hadronization scale the quark-lepton and meson-lepton
effective Lagrangians and derived relations between the quark- and
meson-level effective LFV couplings. With this at hand, we extracted lower
limits on the individual scales of many LFV operators
from~(\ref{eq:l-q-operators}),
which are shown in Table~\ref{Table-2}. The limits for the scales of the
tensor, axial-vector, pseudoscalar operators, as well as
for $(\bar{q} \Gamma q)(\bar{e} \Gamma \tau)$,
$(\bar{q} \Gamma q)(\bar{\mu} \Gamma \tau)$
are new, nonexisting in the literature.
These limits can be useful for LFV phenomenology, allowing model independent
predictions for the LFV processes induced by the generic set of
quark-lepton operators (\ref{eq:l-q-operators}).

\begin{acknowledgments}

This work was supported
by the Carl Zeiss Foundation under Project ``Kepler Center f\"ur Astro- und
Teilchenphysik: Hochsensitive Nachweistechnik zur Erforschung des
unsichtbaren Universums (Gz: 0653-2.8/581/2),''
by Fondecyt (Chile) Grants No. 1150792, No. 1170171, No. 1180232
and by CONICYT (Chile) Ring ACT1406, PIA/Basal FB0821,
by the Russian Federation program ``Nauka'' (Contract No. 0.1764.GZB.2017),
by the Tomsk State University Competitiveness Improvement Program under
Grant No. 8.1.07.2018, and by the Tomsk Polytechnic University Competitiveness
Enhancement Program (Grant No. VIU-FTI-72/2017).

\end{acknowledgments}

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