text:
actor ranjith wife
Notably, its lead actor Mammootty received no payment to act in the film as the script impressed him so much. Towards the end of 1997, Ranjith penned Aaraam Thampuran, which went to become one of the biggest hits of his career. The critic K. Vijayan of New Straits Times praised Ranjith's acting citing that, despite his big build, impresses in his debut performance. Ranjith has celebrated the total number of 47 birthdays till date. Also learn how He earned most of networth at the age of 56 years old? The year 2000 began with the release of Narasimham, directed by Shaji Kailas. Ranjith was born in Balussery in Kozhikode, Kerala. 47 years, 11 months, The film, directed by K. S. Rajkumar, was produced by R. K. Selvamani and he acted alongside Rahman, Sivaranjani and Ramya Krishnan. In 2004 Ranjith teamed up with Mammooty for Black, which was a complete commercial entertainer targeting the fans of Mammootty. In 1997, Ranjith teamed up again with Shaji Kailas for Asuravamsham, starring Manoj K Jayan and Biju Menon; the film yielded an average commercial response. In 2007, Ranjith teamed up with Joshiy for the big budget film Nasrani starring Mammootty. In 2010 Ranjith wrote the story for the film Penpattanam, directed by V. M. Vinu. His real name is Shivaji Rao. Ranjith Balakrishnan (born 5 September 1964) is a veteran Indian film director, screenwriter, producer, and actor who works in Malayalam cinema.Ranjith made his directorial debut in 2001 with Ravanaprabhu, a sequel to Devaasuram.His films Thirakkatha and Indian Rupee won the National Film Award for Best Feature Film in Malayalam in 2008 and 2011 respectively. [2] He appeared in two episodes of The Office, as Vikram, a telemarketer who worked with Michael, and was briefly hired for The Michael Scott Paper Company. He made his film debut in Basu Chatterjee's Khatta Meetha, following which he played prominent parts in Bollywood comedy classics such as Basu Chatterjee's Baton Baton Mein (1979) and Hrishikesh Mukherjee's Khubsoorat (1980). Get more info like birthplace, age, birth sign, biography, family, relation & latest news etc. The movie was able to bring out the best in the actors. The different segments in the movie were conceived and connected based on the concept of Yatra, or journey. His father is from a Gujjar background, while his mother, Pearl Padamsee, was of partial Jewish descent on her mother's side, but remained Christian during her life. Although the film was well-appreciated by critics, it was a commercial failure. In the start of 1998, Ranjith played as an antagonist in Maru Malarchi, the film was released to critical acclaim and he won the Tamil Nadu State Film Award for Best Villain for his role. His next film with Mammootty, Kadal Kadannoru Mathukkutty, underperformed at the box office. 20 days old age Ranjith will turn 48 on 15 November, 2020. His next directorial feature was the musical-comedy Rock & Roll, starring Mohanlal. His stepfather, Alyque Padamsee, was a theatre actor and director who also headed an advertising company in Mumbai. Then came Kaikudanna Nilavu, in 1998, directed by Kamal, starring Jayaram, yet another average grosser. Directed by Shaji Kailas, the film broke several box office collection records and received praise from critics. Sasi, Devasuram was both critically and commercially acclaimed; Mohanlal's performance was lauded and it opened a new genre of feudal stories in Malayalam cinema. In 1988, Ranjith made his formal entry into the film world by writing the story for Orkkapurathu, an adventure film directed by Kamal, with Mohanlal in the lead role. He net worth has been growing significantly in 2018-19. This was his last script for Shaji Kailas and with this film Ranjith became the most successful scriptwriter of the time. His next venture was an offbeat film Kaiyoppu. He was married to actress Priya Raman in 1999 and divorced in 2014. In 2011, Ranjith scripted and directed a critically acclaimed satirical film Indian Rupee starring Prithviraj, which was well received by the critics and also a commercial success. Yet again in 2000, together with Shaji Kailas, Ranjith repeated history: his Onam release Valliettan starring Mammooty was a commercial success. In 2001, Ranjith made his directorial debut with Ravanaprabhu, the sequel to Devasuram. Ranjith produced this film along with his friend, actor Siddique. According to Ranjith, Maya Mayuram is one of his best films and its failure affected him greatly. His next film, Spirit, primarily focused on the increasing habits of alcoholism in Kerala. In the same year Ranjith scripted Ammakilikood, directed by Padmakumar with Prithviraj in the lead role, which also got the cold shoulder from the public. Njaan, based on the novel K T N Kottoor: Ezhuthum Jeevithavum by T P Rajeevan and featuring Dulquer Salmaan in the lead role, was released on 19 September 2014. Discover Ranjith's Biography, Age, Height, Physical Stats, Dating/Affairs, Family and career updates. After completing a course in acting, he made his acting debut in 1993 with Pon Vilangu. In the mid of 1998, he acted in the blockbuster Natpukkaga and he played the villain role in Dharma. "[8], His mother, Pearl Padamsee, was a well-known theatre personality, drama teacher and actress on stage and film. He said, "Mohanlal fans expected too much. Film director, screenwriter, producer, actor, Agnivesh Balakrishnan, Ashutosh Balakrishnan. Starring Mohanlal in the lead role, the film was critically acclaimed and a box office success. For his role as Rocky in Deepa Mehta's 2002 film Bollywood/Hollywood, he was nominated for Best Performance by an Actor in a Supporting Role at the 23rd Genie Awards[3][4] His other most noted role was in Last Holiday (2006), starring Queen Latifah. 1993 was a huge turning point in Ranjith's career: Devasuram, his film based on the life of his father-figure, Mullassery Rajagopal, turned out to be a huge box office hit. In 1992, he wrote Neelagiri for I. V. Sasi, which underperformed at the box office. He graduated from the School of Drama and Fine Arts, Thrissur, in 1985. I don't think it was a bad film. In 1987, he made his debut as a writer in Oru Maymasa Pulariyil, produced by Alex I. Kadavil and directed by V. R. Gopinath. He is a member of famous Film director with the age 56 years old group. This film, written by Ranjith, was a super hit at the box office. His films Thirakkatha and Indian Rupee won the National Film Award for Best Feature Film in Malayalam in 2008 and 2011 respectively. This film was also a turning point in the career of Mohanlal, and with it Ranjith gained a reputation as a commercially viable scriptwriter. Starring Anoop Menon and Priyamani, the movie featured Prithviraj and Samvrutha Sunil in important supporting roles. Ranjith Balakrishnan (born 5 September 1964) is a veteran Indian film director, screenwriter, producer, and actor who works in Malayalam cinema. View popular celebrities life details, birth signs and real ages. With Mohanlal playing the lead role, this film became one of the biggest hits ever in the history of Malayalam cinema at the time. Both children are in the custody of his mother. In 1999, Ranjith and Shaji Kailas produced the film Ustaad, directed by Sibi Malayil, starring Mohanlal. At 56 years old, Ranjith height not available right now. He died on 15 April 2020. In 2015, his next film, Loham, starring Mohanlal, became a superhit at the box office. In 1996, he appeared in Minor Mappillai and Avathara Purushan. After PMK formed an ally with AIADMK for the 2019 general elections, he switched from PMK to AAMK (Amma Makkal Munnetra Kazhagam's). Fans come in with pre-conceived notions on how the actor will be in the movie and the kind of story it will be, hence they could not grasp the poetic element in the film.". We recommend you to check the complete list of Famous People born on 5 September. He wrote the screenplay and acted in Sam & Me (directed by Deepa Mehta), which won an honorable mention at Cannes in 1991. Ranjith's income source is mostly from being a successful Film director. After the divorce, he remarried to actress Ragasudha in 2014, but the couple starts living separately after some time. Learn How rich is He in this year and how He spends money? Discover today's celebrity birthdays and explore famous people who share your birthday. In the same year he came up with another unique creation, Kerala Cafe — a fusion of ten different short films by ten directors. He is from Indian. Sharing a picture of him with the baby, Jishin wrote: "Happy first birthday to Ranjith's Iza mol". He then produced and scripted for G. S. Vijayan's Mammootty-starring Bavuttiyude Namathil in 2012, which was a flop at the box office. Ranjith mentioned in an interview that several directors passed on the opportunity to make Maya Mayuram, and it was Mohanlal who convinced Sibi Malayil to get involved. See the analysis by days count and bar graph. He is an Indian Film Actor, Film Director & Politician who is known for his work in Tamil cinema as well as the Malayalam film industry. He had one older sister named Rohini (c. 1951 – 26 September 1961), who died from nephritis. Also learn how He earned most of networth at the age of 56 years old? Ranjith believes that it was not a bad film. During the late 1980s, Ranjith tried his hand at skits and television serials for Doordarshan.. Ranjith's entry into Mollywood was accidental, he was inspired by his friend, the late film producer and director Alex I. Kadavil, in whose residence he had stayed for his debut in the film industry. The immediate success of Orkkapurathu brought Ranjith several opportunities to work in films. Ranjith (Indian, Film Actor) was born on 15-11-1972. At the end of 1998, he acted in supporting role in Desiya Geetham and Pudhumai Pithan while he played one of the leads in the low-budget film Cheran Chozhan Pandian. In 2006, he scripted and directed Prajapathi, starring Mammootty, was also blasted by critics, and was another disastrous movie. Only 10 days, 19 hours, 29 minutes has left for his next birthday. Ranjit Chowdhry (19 September 1955 – 15 April 2020) was an Indian character actor, known for his roles in television, movies, and theatre.
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Talented teacher Ashling Murphy was 'an inspiration to us all' says Durrow national school principal
Candles were burning at Durrow National School on its first day without First Class teacher Ashling Murphy
Gearoid Keegan
THE principal of the school where murdered Offaly teacher Ashling Murphy worked has paid a heartfelt tribute to his former staff member.
James Hogan, principal of Durrow National School, said the 23-year-old was a beautiful, gifted and talented lady.
"She was a talented teacher, extraordinary sportsperson, traditional Irish music enthusiast, very active and fit, an inspiration to all," said Mr Hogan.
"She gave 110% to everything she put her hand to. And the kids and their education was her priority.
She taught a class of 28 boys and girls and those children were today dealing with the sudden loss of a teacher they loved.
"Today was very much about talking, being open and saying it's ok to be sad and have feelings, and say we're all hurting and remember the good things about Ms Murphy," said the principal
Man arrested in connection with Ashling Murphy murder released from detention and is 'no longer a suspect'
"They all did little notes and lovehearts, poems and prayers and Mass cards and left them in memory boxes."
He said Ashling's death was devastating for the school and the whole community and he paid tribute to the psychologists who came to provide support.
"The road ahead is going to be tough but I can reassure everyone that the school and the community will always have support services her to cater for the kids, the families, and ourselves, the staff. Because when this new reality dies down it's going to be tough and hard for everyone."
Ashling Murphy began work in Durrow as a substitute teacher in March 2021 and was then appointed to First Class as a permanent teacher.
Mr Hogan said his thoughts were now with her family and he expressed his deepest sympathy.
Woman remanded in custody for failing to leave crime scene in the midlands
"I know them at a personal level too outside of the school through traditional music. I grew up playing music and her family are musical and I know them really well. Words can't describe how to express sympathy to them."
Mary Daly, chair of the school board of management spoke of the terrible shock and sadness she had felt.
"The loss of any life is tragic but to know a girl after getting her college finished and starting work here, it was just devastation for the whole community and the wider community and worldwide," said Ms Daly.
"She was the light in shining armour here. That was the feeling from everybody and the little pupils, to see the way they came up so dignified, and to see them go home this evening. It was heartbreaking."
Fr Joe Gallagher, parish priest of Tullamore, joined the school staff for a period of reflection this afternoon and said that was important "to have that few moments to pray and reflect and strengthen the bonds and support one another and understand how valuable we are to one another and what we can do for one another".
Fr Gallagher added: "When somebody is pulled out like that all of a sudden it is then we realise how important we are to each other and how we can do so much. It's something that touches everybody very deeply."
Book of condolence to open at Longford County Council for murdered Offaly teacher Ashling Murphy
Members of the public will be able to relay their sympathy to the family of murdered primary school teacher Ashling Murphy tomorrow when a book of condolence is opened tomorrow by Longford County Council.
He said the feeling when he heard the news of the tragedy was "surreal".
"It was unbelievable, it was just a total, total shock, and you think this could just not be true. For it to happen in your own town and a place we all know and walk. You meet so many people down there.
"And to think that she was coming home from school and going out to get a bit of air and exercise and then for that to happen. The whole town is feeling it and it has touched so many people all over the country and we've got messages from so many places, all expressing their sympathy and support and offering their prayers."

redpajama_set_name: RedPajamaCommonCrawl


text:
package org.aesh.command.man;

import org.aesh.terminal.utils.ANSI;
import org.aesh.terminal.utils.Config;
import org.aesh.command.man.parser.ManSection;
import org.junit.Test;

import java.util.ArrayList;
import java.util.List;

import static org.junit.Assert.assertEquals;

/**
 * @author Ståle W. Pedersen
 */
public class ManSectionTest {

    @Test
    public void testManSection() {
        List<String> input = new ArrayList<String>();
        input.add("OPTIONS");
        input.add("-------");
        input.add("*-a, --attribute*='ATTRIBUTE'::");
        input.add("  Define or delete document attribute. ");
        input.add(Config.getLineSeparator());
        input.add("*-b, --backend*='BACKEND'::");
        input.add("  Define or delete document attribute.");

        ManSection section = new ManSection().parseSection(input, 80);
        assertEquals("OPTIONS",section.getName());

        assertEquals(ANSI.BOLD+"OPTIONS"+ ANSI.DEFAULT_TEXT+
                Config.getLineSeparator()+
                "  "+ ANSI.BOLD+
                "-a, --attribute"+
                ANSI.DEFAULT_TEXT+
                "="+ ANSI.UNDERLINE+
                "ATTRIBUTE"+
                ANSI.DEFAULT_TEXT+ Config.getLineSeparator()+
                "    Define or delete document attribute. "+ Config.getLineSeparator()+
                " "+ Config.getLineSeparator()+
                "  "+ ANSI.BOLD+
                "-b, --backend"+
                ANSI.DEFAULT_TEXT+
                "="+ ANSI.UNDERLINE+
                "BACKEND"+
                ANSI.DEFAULT_TEXT+ Config.getLineSeparator()+
                "    Define or delete document attribute. "+ Config.getLineSeparator()+
                " "+ Config.getLineSeparator(),
                section.printToTerminal());
    }
}

redpajama_set_name: RedPajamaGithub


Enjoy a Hot Summer Nights Music Festival Chamber Recital at the Broadway Christian Church. Missouri Symphony Orchestra musicians perform in small groups with a wide range of repertoire from the baroque to the very newest works. While enjoying the superb acoustics of the church, the audience has a unique opportunity to get close to the musicians and meet with them after the concert.


Wisconsin's season ends with loss to Michigan State in the Big Ten quarterfinals, but Bucky didn't go down without a major fight to the end.
Wisconsin will not go on to the NCAA tournament, thanks to a 63-60 loss to the Michigan State Spartans in the Big Ten tournament quarterfinals.
It would've taken the Badgers winning the Big Ten tournament championship, but officially the streak of 19-straight NCAA tournament appearances is over. A last-second desperation three-point attempt by Brad Davison came up short and Wisconsin fell to No. 1 seed Michigan State 63-60 in the quarterfinals of the Big Ten tournament.
Everyone knew it would happen at some point, but few saw the Badgers tournament streak ending the way it did on the season or in this game.
Once again, the Spartans found a way to pull a game out late against Wisconsin, just like it did four days ago and it has over the last five straight in this series.
Ethan Happ had 22 points, while Michigan State star Miles Bridges had 20 in the winning effort. However, it was Cassius Winston that did the Badgers in for the second time in less than a week. The sophomore guard poured in 17 points, but unlike last time was just 1 of 7 from beyond the arc.
Wisconsin shot 45.6 percent from the field, but ultimately couldn't overcome a huge 41-27 deficit in rebounds. Michigan State held a 16-10 advantage on the offensive glass as well.
The game was within a possession or two for the vast majority of the day, but a 7-0 run by Michigan State in the second half to give the Spartans a 58-51 lead seemed too steep a hill to climb. Happ delivered though, putting the team on his back and getting UW to a bucket at 58-56 with 3:41 to play.
Michigan State's Matt McQuaid had other ideas though, giving the Spartans a 5-point cushion on a step-back three pointer of his own for a 63-58 lead with 1:21 to play.
Wisconsin came up empty on the possession before that make by MSU as well, and it was all Michigan State's control at that point.
However, Davison wasn't going to let his Badgers go down without a major fight. The freshman guard was given a clear path to the basket and made a layup with 28 seconds left to push the game to 63-60. Miles Bridges, normally a good free throw shooter, was fouled on the inbounds and missed the front-end of a one-and-one opportunity.
UW got the ball past mid-court, called a timeout and it was advantage Michigan State out of the time out. The Spartans played terrific perimeter defense and it left a horrific look for Davison with the game clock running out. His effort fell way short and with it the season ended.
For the first time since the 1997-98 season, the Wisconsin Badgers will be watching the NCAA tournament from the comfort of their homes for its entirety.
We all knew that we were witnessing one of the most special players in Wisconsin Badgers history with Ethan Happ's senior season.
But, on Friday night he was rewarded for that special season by winning the Kareem Abdul-Jabbar award. It's been given to the nation's best center since 2015.
Happ is the second Badgers big man to win the award, following the very first award winner in Frank Kaminsky.
This season, Happ finished 8th in the country with 21 double-doubles on the year, which tied the school record held previously by Joe Franklin in the 1967-68 season.
He also finished this year averaging 17.3 points, 10.1 rebounds and 4.5 assists. All three of those were career highs as well.
With 2,130 points, 1,217 rebounds and 423 assists, Happ is one of six players in NCAA history to accumulate at least 2,000 points, 1,000 rebounds and 400 assists. He joins Oscar Robertson, Larry Bird, Danny Ferry, Stacey Augmon and John Konchar.
Previous winners of the Kareem Abdul-Jabbar Award were Kaminsky (2015), Utah's Jakob Poeltl (2016), Gonzaga's Przemek Karnowski (2017) and Seton Hall's Angel Delgado (2018).


Hey guys! It's been a while. I've been busy and enjoying the summer vacation last May that's why I haven't updated the blog since.
So today I am showing you the last miniature bedroom kit that I received from GearBest. I re-designed the bedroom with Nendoroid Snow Miku 2017 in my mind.
I like adding a personal touch to each dollhouse kits that I receive from the company, so make sure to watch the video below to see the changes I made!
Whenever I think of Nendoroid Snow Miku Twinkle Ver, I think of moon and stars and so I made tiny decors for the room using sintra board and glitters.
Good thing about having so many dollhouse kits, I can just borrow different miniature pieces from the old one and use it to the new ones.
The tent makes the entire more visually appealing, however, it takes a lot of space so sometimes, I remove it especially when I am taking pictures of my figures.
I love that the room is big enough to fit my Nendoroid Snow Miku inside it.
Instead of the tent, you can place Snow Miku's table for a breakfast treat.
Yukine seems to find his perfect spot in the room. The tent is his to play.
Overall, the size of the bedroom kit is recommended for Nendoroid. However, the bed seems a little small for them. For photography purposes, I think this room is a great backdrop for your toy/figure.
Ah, this tiny room is so cute (。♥‿♥。) and even has a light feature.
It is sweet how much work and love you've put into the room assembling and decoration. When looking at the pics it was totally worth the effort.
The tent is a great addition to the setting.
I always like the dollhouse to be different from the others. Thank you so much for checking my blog. I'm glad you also loved the music.


Sapa Easy Trekking is great tour for all guests with interesting sight seeing and highland scape. Trekking and visiting ethnic minorities are main parts in this tour. Let's explore and experience with our trip.
Visiting Muong Hoa valley and many villages in Sapa: Y Linh Ho, Lao Chai, Ta Van, Cat Cat, and Sin Chai – the village located at the foot of Fansipan Mountain.
Giving you a worth chance to learn about the local house architecture which is full of their daily life styles and exclusive custom.
We will pick you up at your hotel in Hanoi at 7.30 pm and transfer you to Hanoi train station for the overnight train to Lao Cai province. Experience the romantic overnight train travel.
Arrive at Lao Cai's train station at 6.00 am. We will take 1-hour car ride uphill to the beautiful town of Sapa. The ride gives you a glimpse of the stunning vistas and impressive rice terraces. Upon arrival in Sapa town you have your breakfast and freshen up before the trip. Starting with 6km of walking down to the south east of Sapa along the side of Muong Hoa valley to Y Linh Ho village of the Black H'mong. And then continuing with 3km of trekking down to Lao Chai village of the H'mong people. Lunch at a local house, and then walk through terraced rice fields to Ta Van village of the Giay minority people. It gives you a worth chance for you to learn about the local house architecture which is full of their daily life styles and exclusive custom. Enjoy the landscape on the way. Get back to Sapa town by jeep or motorbike.
A walk down to the South-west of Sapa (around 3 km) to visit Cat Cat village of the Black H'mong. Relax at the foot of the stunning waterfall and the vestige of a hydroelectric power station that was constructed by the French during the colonial time in Vietnam. Enjoy picnic lunch by the side of the river, and then next 3 km to Sin Chai, the village located at the foot of Fansipan Mountain – "the roof of Vietnam and Indochina". From here you can enjoy huge stunning landscapes before climbing up on the way by a jeep to return to Sapa. In late afternoon we will transfer you back to Lao Cai station to take the overnight train to Hanoi. You will be arriving in Hanoi at 5.30 am next day.
Pick-up and transfer by air-conditioned vehicle as specified in the program.
Travel in an appropriate A/C vehicle and/or a jeep with an experienced safe driver.


text:
info@boilingpointmedia.com | (405) 652-9033
Oklahoma City Film Rebate 2022
by Emily Taylor | Jan 30, 2023 | Blog, Film
Rumor has it that Oklahoma City is the next Hollywood –"Hollywood on the Prairie". Here at Boiling Point Media, we knew that Oklahoma City has a lot of potential for the film industry, that's why we have been providing post production services, visual effects and...
Continuous Scroll On Google Now On Desktop
by Emily Taylor | Dec 22, 2022 | marketing
Now introducing Google's continuous scrolling feature now available on desktop! The continuous scroll was first introduced on mobile searches only back in October of 2021. It's no surprise that a desktop version is available now too. You might be asking yourself...
What E-E-A-T In SEO Is & Why It's Important
E-E-A-T in search engine optimization (SEO) is an acronym that stands for experience, expertise, authoritativeness, and trustworthiness. These elements are what Google uses to determine whether or not a site is credible enough to have an effective result that will...
What is Position Zero & Why is it Important?
by Emily Taylor | Nov 30, 2022 | Blog, marketing
Achieving position zero is a goal every SEO Agency should have. Search Engine Optimization (SEO) experts will tell you that position zero is SEO gold because of what it signifies. But what exactly is position zero, where did it come from, how is it determined, and why...
Things To Consider When Choosing A Marketing Agency
by Emily Taylor | Nov 11, 2022 | Blog
Choosing an advertising agency, aka a marketing agency, is oftentimes either what makes or breaks your business. Alright, that might sound a little dramatic, but the marketing agency you decide to put your trust in is crucial to promoting your business. Creative and...
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text:
She says / He says (A–B)
She says / He says (C–F)
She says / He says (G-L)
She says / He says (M–R)
She says / He says (S–Z)
Cultureguru's Weblog
Of food, technology, movies, music, and travel—or whatever else strikes my fancy
March 9, 2019 by cultureguru	1 Comment
Not a particularly significant birthday this year, so I wasn't thinking too much about it. However, some months ago, when looking to pick a date to go see Sting's The Last Ship in Toronto, I figured why not pick my birthday weekend.
Then events got built around that. I took the Friday off (to do a whole lot of nothing special—but still better than a work day). And I noticed that the KW Comedy Festival was having their opening gala the Thursday night before, so I got tickets.
I don't think it was as strong as last year. My favorite act of the first half was Arthur Simeon, originally from Uganda but now living in Toronto. In the second half it was Emily Galati, the only woman featured, along with the headliner, Sean Majunder. The rest of the comedians were a bunch of white guys. And to be fair, one of them, host Derek Seguin, provided the evening's most hilarious bit, in his description of the challenges of man-scaping.
But overall, it was some absurdist comedy, which is not really my thing, and a lot of jokes about their kids, or about why they don't have kids—maybe one of the few safe subjects for white guys to joke about these days? But not as effective, for me, as Simeon, Galati, and Majunder's takes on politics, social media, racism, and sexism. Tricky time to be funny, I guess, but the event would have benefited from more diversity than it had.
Everywhere you go, always take the weather
When we booked our bus to Toronto, we discovered that the Greyhound schedule isn't as good as it used to be. Not as many buses, and they all have more stops. (This is just annoying. It's not as though the train service is any better on Saturdays.) There was one bus that would have gotten us there around 10:50, which would have been ideal, but it would have taken three hours. So we went with the one that scheduled to arrive around 11:30, because it was only supposed to take two hours.
I didn't think the forecasted 2 cm of snow would really affect it, but I was wrong. For one, I think it was somewhat more snow than that. Regardless, it slowed down all the traffic. We clearly weren't going to make our 12:15 lunch reservation, so I texted my sister about that, and suggested that she could order for us, and we'd aim to arrive by 1:00.
Off the bus, we had trouble finding a cab, so we called an Uber, and initially had trouble finding them, too, but we did connect. Only to find that they had the wrong Holiday Inn listed as the destination, which I needed to change in the app. Which was not as easy to do as one would hope. By the time I finally got it to work (Jean's suggestion to turn off wifi was key), we were there!
Fortunately, hotel check-in went smoothly, and calling a second Uber to take us to lunch was drama-free. We ate the O&B Canteen at the TIFF Bell Lightbox. A bit pricey for what you got ($19 for a burger?), but everything was very good.
I didn't know a whole lot about The Last Ship other than that Sting wrote the music (it was not one of those musicals built around an artist's famous songs), inspired by the ship building heritage of his home town. But it was really good! Strong cast, great singers, and a very moving story. In the first half, I was kind of with the capitalists ("Be realistic! The ship building industry is dying!") and identifying with the characters who felt they just had to leave the town to secure their futures elsewhere. But by the end of the second half, I was totally with the workers.
We had an hour after the play before our dinner reservation, which gave us time to walk back to the hotel, and stay there briefly before walking to dinner at Buca Yorkville.
That was a fine meal. We started with three kinds of house-cured fish, which were small taste sensations. We added in a nice rosemary foccacia that was served with the most amazing olive oil. As a main, I had chestnut-stuffed ravioli with porcini, that was just fantastic, and a side of Swiss chard.
Jean had the day's special of uni spaghetti, also good, but not quite as good as the ravioli.
The wine with that was the waiter's suggestion of an Italian Riesling, which did work well.
For dessert, Jean went with the waiter's suggestion of the affogato using decaf espresso, and it really was delicious (they make their own ice cream). I also enjoyed the cranberry millefeuilles that I had.
Apart from the candle on the dessert plate, as my birthday bonus I got a takeout of fresh pasta with little containers of olive oil and pepper and little containers of cheese. And instructions on how to cook this into a meal for two. This I did this past Thursday, and it was very nice.
The whole experience somewhat reminded of New York dining: Impeccable service, fantastic food, but no dawdling. One course arrived promptly after another, and we were done by 8:00. Probably because they needed the table for someone else.
After that rather packed Saturday, it was nice not to have anything planned ahead for Sunday, other than our bus back. We had breakfast at Cora, and decided it was better than the Cora we'd tried previously (forget where, but not the one in KW). We then decided to visit the ROM, as they were featuring this year's winners of the Wildlife Photography contest. That exhibit was terrific, again. The work to get some of those shots!
We then visited the "Treasures of the Earth" exhibit, that I don't recall having been to before. It featured some beautiful minerals, gems, meteorites, and rocks, and had a section on Canadian mining, in which my home town featured prominently.
Gold from Northern Ontario mines
Since Richmond Station is very difficult to get dinner reservations at, but recently started opening Sundays, we thought we'd try to just go there and see if we could get in for a late lunch. It worked! We got a table.
To start, with shared the duck liver pate—creamy and rich. Then I had the lamb forestiere cavatelli, while Jean had duck two ways. We had a half-glass of sparkling to start with that, then a glass of red each. We were left too full for dessert.
All that was left was to gather our luggage back at the hotel, then get to the station. There was a bit of Uber drama here too, that I won't get into. But we made it to the station in plenty of time, and that bus was not delayed.
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text:
\section{Introduction}
\label{int}

\vspace*{-2mm}
Most of the recent progress in the understanding of strong interaction physics
at large momentum transfer has been due to the comparison between precise
experimental data and very accurate QCD calculations to higher perturbative
orders [\ref{QCDrev}].

These higher-order computations have been carried out over a period of
about fifteen years, often long after the accuracy of experimental data
has made them necessary, because of the difficulties in setting up a
general and straightforward calculational procedure. The physical origin of
these difficulties is in the necessity of factorizing the long- and
short-distance components of the scattering processes
and is reflected in the perturbative
calculation by the presence of divergences.

In general, when evaluating higher-order QCD cross sections,
one has to consider
real-emission contributions and virtual-loop corrections and one has to deal
with different kind of singularities. The customary {\em ultraviolet\/}
singularities, present in the virtual contributions, are removed by
renormalization. The low-momentum ({\em soft\/}) and small-angle
({\em collinear\/}) regions instead produce singularities both in the real and
in the virtual contributions. In order to handle these divergences, the
observable one is interested in has to be properly defined. It has to be a
{\em jet quantity}, that is, a hadronic observable that turns out to be
infrared safe and either collinear safe or collinear factorizable: its actual
value
has to be independent of the number of soft and collinear particles in the
final state (see Sect.~\ref{siee} for a formal definition).
In the case of jet
quantities, the coherent sum over different (real and virtual) soft and
collinear partonic configurations in the final state leads to the cancellation
of soft singularities. The left-over collinear singularities are then
factorized into the process independent structure and fragmentation functions
of partons (parton distributions), leading to predictable scaling violations.
As a result, jet cross sections are finite (calculable) at the partonic level
order by order in perturbation theory. All the dependence on long-distance
physics is either included in the parton distributions or in non-perturbative
corrections that are suppressed by inverse powers of the (large)
transferred momentum $Q$ that controls the scattering process.

Because of this pattern of singularities,
QCD calculations of jet cross sections beyond leading order (LO)
are very
involved. Owing to the complicated phase space for multi-parton
configurations, analytic calculations are in practice impossible for
all but the simplest quantities, but the use of numerical methods is
far from trivial because soft and collinear singularities present in the
intermediate steps have first to be regularized. This is usually done by
analytic continuation to a number of space-time dimensions $d=4-2\epsilon$
different from four, which greatly complicates the Lorentz algebra in the
evaluation of the matrix elements and prevents a straightforward implementation
of numerical
integration techniques. Despite these difficulties, efficient computational
techniques have been set up, at least to next-to-leading order (NLO),
during the last few years.

There are, broadly speaking, two types of algorithm used for NLO
calculations: one based on the phase-space {\em slicing\/} method and the
other
based on the {\em subtraction\/} method\footnote{We refer the reader to the
Introduction of Ref.~[\ref{KS}] for an elementary description of the basic
difference between the two methods.}. The main difference between these
algorithms and the standard procedures of analytic calculations is that only
a minimal part of the full calculation is treated analytically, namely only
those contributions giving rise to the singularities. Moreover, for any given
process, these contributions are computed in a manner that is independent of
the particular jet observable considered. Once every singular term has been
isolated and the cancellation/factorization of divergences achieved, one can
perform the remaining part of the calculation in four space-time dimensions.
Although, when possible, one still has the freedom of completing
the calculation analytically, at this point the use of numerical integration
techniques (typically, Monte Carlo methods) is certainly more convenient.
First of all, the numerical approach allows one to calculate any number and
any type of observable simultaneously by simply histogramming the appropriate
quantities, rather than having to make a separate analytic calculation for each
observable. Furthermore, using the numerical approach, it is easy to
implement different experimental conditions, for example, detector acceptances
and experimental cuts. In other words, the phase-space slicing and subtraction
algorithms provide the basis for setting up a general-purpose Monte
Carlo program for carrying out arbitrary NLO QCD calculations in a given
process.

Both the slicing [\ref{KL}] and the
subtraction [\ref{ERT}] methods were first used in the context of NLO
calculations of three-jet cross sections in $e^+e^-$\ annihilation. Then they
have been applied to other cross sections adapting the method each time
to the particular process. Only recently has it
become clear that both algorithms are generalizable in a process-independent
manner. The key observation is that the singular parts of the QCD matrix
elements for real emission can be singled out in a general way by using
the factorization properties of soft and collinear radiation [\ref{BCM}].

At present, a general slicing algorithm is available for
calculating NLO cross sections for {\em any\/} number of jets both in
lepton [\ref{GG}] and hadron [\ref{GGK}] collisions. To our knowledge,
fragmentation processes have been considered only in the particular case of
direct-photon production [\ref{BOO}].
The complete generalization of this method to include fragmentation functions
and heavy flavours is still missing.

As for the subtraction algorithm, a general NLO formalism has been set up
for computing three-jet observables in $e^+e^-$\ annihilation [\ref{ERT},\ref{KN}]
and cross sections up to two final-state jets [\ref{KS},\ref{EKS}]
in hadron collisions\footnote{An extension of the
method for three-jet cross sections has been recently presented
[\ref{Frix}].}. Also the treatment of massive partons has been
considered in the particular case of heavy-quark correlations in hadron
collisions [\ref{MNR}].

In this paper, we present the basic idea to set up a {\em completely general\/}
version of the  subtraction algorithm. This generality is obtained by fully
exploiting the factorization properties of soft and collinear emission and,
thus, deriving new improved factorization formulae, called {\em dipole
factorization formulae}. They allow us to introduce a set of universal
counter-terms that can be used for {\em any\/} NLO QCD calculation.

For the purpose of illustration, in this short contribution we describe the
implementation of our general method to the calculation of jet cross sections
in processes with no initial-state hadrons, typically $e^+e^-$\ annihilation.
Full details of the method and its application to all the other hard-scattering
processes will appear elsewhere [\ref{CS}].

We first recall the main features of the subtraction method in
Section~\ref{gen}. Then in Section~\ref{dff} we present our dipole
factorization formulae, for the case in which there are no incoming QCD
partons. These allow us to calculate cross-sections for an arbitrary number of
jets in $e^+e^-$\ annihilation, which we do in Section~\ref{siee}.  After briefly
recapping the resulting formulae in Section~\ref{finfor}, we illustrate them
\enlargethispage*{1mm}
with a simple example in Section~\ref{3jet}, $e^+e^-\to3$ jets.  Finally in
Section~\ref{outl} we give a summary and outlook.

\section{The subtraction procedure}
\label{gen}

Suppose we want to compute a jet cross section $\sigma$ to NLO, namely
\begin{equation}
\label{sig}
\sigma = \sigma^{LO} + \sigma^{NLO} \;.
\end{equation}
Here the LO cross section $\sigma^{LO}$ is obtained by integrating the
exclusive cross section $d\sigma^{B}$ in the Born approximation over the phase
space for the corresponding jet quantity. Suppose also that this LO calculation
involves $m$ partons in the final state. Thus, we write
\begin{equation}
\label{sLO}
\sigma^{LO} = \int_m d\sigma^{B} \;,
\end{equation}
where, in general, all the quantities (QCD matrix elements and
phase space) are evaluated in $d=4-2\epsilon$ space-time dimensions. However, by
definition, at this LO the phase space integration in Eq.~(\ref{sLO}) is finite
so that the whole calculation can be carried out (analytically or numerically)
in four dimensions.

Now we go to NLO\@. We have to consider the exclusive cross section
$d\sigma^{R}$
with $m+1$ partons in the final-state and the one-loop correction $d\sigma^{V}$
to the process with $m$ partons in the final state:
\begin{equation}
\label{sNLO}
\sigma^{NLO} \equiv \int d\sigma^{NLO} =
 \int_{m+1} d\sigma^{R} + \int_{m} d\sigma^{V} \;.
\end{equation}

The two integrals on the right-hand side of Eq.~(\ref{sNLO}) are separately
divergent if $d=4$, although their sum is finite. Therefore, before any
numerical calculation can be attempted, the separate pieces have to be
regularized. Using dimensional regularization, the divergences (arising out
of the integration) are replaced
by double (soft and collinear) poles $1/\epsilon^2$ and single (soft, collinear
or ultraviolet) poles $1/\epsilon$. Suppose that one has already carried out the
renormalization procedure in $d\sigma^{V}$ so that all its ultraviolet poles
have been removed.

The general idea of the subtraction method for writing a general-purpose
Monte Carlo program is to use the identity
\begin{equation}
\label{dsNLO}
d\sigma^{NLO} = \left[ d\sigma^{R} - d\sigma^{A}  \right]
+  d\sigma^{A} +  d\sigma^{V} \;\;,
\end{equation}
where $d\sigma^{A}$ is a proper approximation of $d\sigma^{R}$ such as to have
the same {\em pointwise\/} singular behaviour (in $d$ dimensions) as
$d\sigma^{R}$ itself.
Thus, $d\sigma^{A}$ acts as a {\em local\/} counterterm for
$d\sigma^{R}$ and, introducing the phase space integration,
\begin{equation}
\label{sNLO1}
\sigma^{NLO} = \int_{m+1} \left[ d\sigma^{R} - d\sigma^{A}  \right]
+  \int_{m+1} d\sigma^{A} +  \int_m d\sigma^{V} \;\;,
\end{equation}
one can safely perform the limit $\epsilon \rightarrow 0$ under the integral sign in the
first
term on the right-hand side of Eq.~(\ref{sNLO1}). Hence, this first term can be
integrated numerically in four dimensions.

All the singularities are now associated to the last two terms on the
right-hand side of Eq.~(\ref{sNLO1}). If one is able to carry out analytically
the integration of $d\sigma^{A}$ over the one-parton subspace leading to the
$\epsilon$ poles, one can combine these poles with those in $d\sigma^{V}$, thus
cancelling all the divergences, performing the limit $\epsilon \rightarrow 0$ and
carrying out numerically the remaining integration over the $m$-parton phase
space. The final structure of the calculation is as follows
\begin{equation}
\label{sNLO2}
\sigma^{NLO} =
\int_{m+1} \left[ d\sigma^{R}_{\epsilon=0} - d\sigma^{A}_{\epsilon=0} \right]
+  \int_m \left[ d\sigma^{V} +  \int_1 d\sigma^{A} \right]_{\epsilon=0} \;\;,
\end{equation}
and can be easily implemented in a \lq partonic Monte Carlo' program, which
generates appropriately weighted partonic events with $m+1$ final-state partons
and events with $m$ partons.

The key for the subtraction procedure to work is obviously
the actual form of $d\sigma^{A}$.
One needs to find an expression for $d\sigma^{A}$ that fulfils the following
properties: $i)$~for any given process, $d\sigma^{A}$ has to be obtained in a
way
that is independent of the particular jet observable considered; $ii)$~it has
to exactly match the singular behaviour of $d\sigma^{R}$ in $d$ dimensions;
$iii)$~its form has to be
particularly convenient for Monte Carlo integration techniques;
$iv)$~it has to be exactly integrable analytically in $d$
dimensions over the single-parton subspaces leading to soft and collinear
divergences.

In Ref.~[\ref{ERT}], a suitable expression for $d\sigma^{A}$ for the process
$e^+e^-$$\rightarrow 3$~jets was obtained by starting from the explicit expression (in
$d$ dimensions) of the corresponding $d\sigma^{R}$ and by performing extensive
partial fractioning of the $3+1$-parton matrix elements, so that each divergent
piece could be extracted. This is an extremely laborious and ungeneralizable
task, in the sense that having done it for $e^+e^-$$\rightarrow 3$~jets does not help
us to do this for, say, $e^+e^-$$\rightarrow 4$~jets or for any other process.

In Ref.~[\ref{KS}], the general properties of soft and collinear emission were
first used (in the context of the subtraction method) to construct
$d\sigma^{A}$, for one- and two-jet production in hadron collisions, in a way
that is independent of the detailed form of the corresponding $d\sigma^{R}$.

The central proposal of our version of the subtraction method is that one can
give a recipe for constructing $d\sigma^{A}$ that is completely {\em process
independent\/} (and not simply independent of the jet observable).
Starting from our physical knowledge of how the $m+1$-parton matrix elements
behave in the soft and collinear limits that produce the divergences, we
introduce improved factorization formulae, called
dipole formulae (see Sect.~\ref{dff}), which
allow us to obtain in a straightforward way (see Sect.~\ref{siee}) a
counter-term $d\sigma^A$
satisfying all the properties listed above.

\section{Dipole factorization formulae}
\label{dff}

\noindent {\it Notation}
\vspace{0.1cm}

In general we use dimensional regularization in $d=4-2\epsilon$
space-time dimensions and consider $d-2$ helicity states for gluons
and 2 helicity states for massless quarks. This defines the usual
dimensional-regularization scheme. Other dimensional-regularization
prescriptions can be used, as well [\ref{CS}].

The dimensional-regularization scale, which appears in the calculation
of the matrix elements, is denoted by $\mu$.
In the perturbative calculation of physical cross sections,
after having combined the renormalized matrix elements,
the dependence on $\mu$ exactly cancels and is replaced by the dependence on
the renormalization scale $\mu_R$. Therefore, in order to avoid a cumbersome
notation, we set $\mu=\mu_R$.

Throughout the paper, $\as$ stands for $\as(\mu)$,
the NLO QCD running coupling evaluated at the renormalization scale $\mu$.
The actual value of the QCD coupling $\as(\mu)$ depends on the
renormalization scheme used to subtract the ultraviolet divergences
from the (bare) one-loop matrix element (or, equivalently, from $d\sigma^V$
in Eq.~(\ref{sNLO})).

The $d$-dimensional phase space, which involves the integration over the
momenta $\{p_1, ...,$ $p_m\}$ of $m$ final-state partons, will be denoted as
follows
\begin{equation}
\label{psm}
\left[ \;\prod_{l=1}^{m} \frac{d^dp_l}{(2\pi)^{d-1}}
\,\delta_+(p_l^2) \right] \;(2\pi)^{d} \,\delta^{(d)}(p_1 + ... + p_m - Q)
\equiv d\phi_m(p_1, ...,p_m;Q) \;\;.
\end{equation}

In the case of processes without initial-state
QCD partons ($e^+e^-$-type processes),
the ({\em tree-level\/}) matrix element with $m$ QCD partons in the final state
has the following general structure (non-QCD partons, namely
$\gamma^*, Z^0, W^{\pm}, \ldots$, carrying
a total incoming momentum $Q_\mu$, are always understood)
\begin{equation}
{\cal M}_m^{c_1, ...,c_m; s_1, ...,s_m}(p_1, ...,p_m)
\end{equation}
where $\{c_1, ...,c_m\}$, $\{s_1, ...,s_m\}$ and
$\{p_1,..,p_m\}$ are respectively colour indices
($a=1, ...,$ \mbox{$N_c^2-1$} different colours for each gluon,
$\alpha=1, ..,N_c$ different colours for each quark or antiquark),
spin indices ($\mu=1,...,d$ for gluons, $s=1,2$ for massless fermions)
and momenta.

It is useful to introduce a basis
$\{ \ket{c_1,...,c_m} \otimes \ket{s_1,...,s_m} \}$
in colour + helicity space in such a way that
\begin{equation}
{\cal M}_m^{c_1, ...,c_m; s_1, ...,s_m}(p_1, ...,p_m) \equiv
\Bigl( \bra{c_1,...,c_m} \otimes \bra{s_1,...,s_m} \Bigr) \mket{1, ..., m} \;.
\end{equation}
Thus $\mket{1, ..., m}$ is a vector in colour + helicity space.
According to this notation, the matrix element squared (summed
over final-state colours and spins) $\M{}{m}$ can be written as
\begin{equation}
\M{}{m} = {}_m\!\!<{1, ..., m}|{1, ..., m}>_m \;.
\end{equation}

As for the colour structure\footnote{Within our formalism, there is no need
to consider the decomposition of the matrix elements into colour subamplitudes,
as in [\ref{GG},\ref{MP}].}, it is convenient to associate a colour charge
${\bf T}_i$ with the emission of a gluon from each parton $i$. We thus define
the square of colour-correlated tree-amplitudes as follows
\begin{eqnarray}
\label{colam}
|{\cal{M}}_{m}^{i,k}|^2 &\equiv&
{}_m\!\!<{1, ..., m}| \,{\bom T}_i \cdot {\bom T}_k \,|{1, ..., m}>_m
\nonumber \\
&=&
\left[ {\cal M}_m^{a_1.. b_i ... b_k ... a_m}(p_1,...,p_m) \right]^*
\; T_{b_ia_i}^c \, T_{b_ka_k}^c
\; {\cal M}_m^{a_1.. a_i ... a_k ... a_m}(p_1,...,p_m) \;,
\end{eqnarray}
where $T_{c b}^a \equiv i f_{cab}$ (colour-charge matrix
in the adjoint representation)  if the emitting particle $i$
is a gluon and $T_{\alpha \beta}^a \equiv t^a_{\alpha \beta}$
(colour-charge matrix in the fundamental representation)
if the emitting particle $i$ is a quark (in the case of an emitting
antiquark $T_{\alpha \beta}^a \equiv {\bar t}^a_{\alpha \beta}
= - t^a_{\beta \alpha }$). It is straightforward to check that
the colour-charge algebra is:
\begin{equation}
{\bom T}_i \cdot {\bom T}_j ={\bom T}_j \cdot {\bom T}_i \;\;\;\;{\rm if}
\;\;i \neq j; \;\;\;\;\;\;{\bom T}_i^2= C_i,
\end{equation}
where $C_i$ is the Casimir operator, that is,
$C_i=C_A=N_c$ if $i$ is a gluon and $C_i=C_F=(N_c^2-1)/2N_c$ if $i$ is a quark
or antiquark.

In this notation, each vector $\mket{1, ..., m}$ is a colour singlet, so
colour conservation is simply
\begin{equation}
\sum_{i=1}^m {\bom T}_i \mket{1, ..., m} = 0.
\end{equation}

\vspace{0.3cm}
\noindent {\it Dipole formulae}
\vspace{0.1cm}

The real contribution $d\sigma^R$ to the NLO cross section in Eq.~(\ref{sNLO})
is proportional to the tree-level matrix element ${\cal M}_{m+1}$ for
producing $m+1$ partons in the final state.
The dependence of $|{\cal M}_{m+1}|^2$
on the momentum $p_j$ of a final-state parton $j$ is singular in two different
phase-space regions: when the momentum $p_j$ vanishes ({\em soft\/}
region) and/or when it becomes parallel to the momentum $p_i$ of
another parton in ${\cal M}_{m+1}$ ({\em collinear\/} region).
This singular behaviour of $|{\cal M}_{m+1}|^2$ is well-known [\ref{BCM},\ref{AP}]
and universal.
Indeed, in the soft and
collinear limits, ${\cal M}_{m+1}$ is essentially factorizable
with respect to ${\cal M}_{m}$, the tree-level amplitude with $m$ partons,
and the singular factor only depends
on the momenta and quantum numbers of the QCD partons in ${\cal M}_{m}$.

We thus introduce the following  dipole factorization formula:
\begin{equation} \label{ff}
\!\!|{\cal M}_{m+1}(p_1, ...,p_{m+1})|^2 =
\obra{1, ..., m+1} \oket{1, ..., m+1} = \sum_{k \neq i,j}
{\cal D}_{ij,k}(p_1, ...,p_{m+1}) + \dots
\end{equation}
where $\dots$ stands for terms that are not singular in the
limit $p_i \cdot p_j \rightarrow 0$ (i.e.\ when $i$ and $j$ become collinear or
when either $i$ or $j$ is soft) and the dipole contribution
${\cal D}_{ij,k}$ is given by
\begin{eqnarray} \label{dipff}
\!\!\!&{\cal D}_{ij,k}&\!\!\!\!(p_1, ...,p_{m+1}) =
\frac{-1}{2 p_i \cdot p_j}  \\
&&\cdot \;\;
\mbra{1, .., {\widetilde {ij}},.., {\widetilde k},.., m+1}
\,\frac{{\bom T}_k \cdot {\bom T}_{ij}}{{\bom T}_{ij}^2} \; {\bom V}_{ij,k} \,
\mket{1, .., {\widetilde {ij}},.., {\widetilde k},.., m+1} \;.
\nonumber
\end{eqnarray}

The $m$-parton matrix element on the right-hand side of Eq.~(\ref{dipff}) is
obtained from the original $m+1$-parton matrix element by replacing $a)$ the
partons $i$ and $j$ with a single parton ${\widetilde {ij}}$ (which plays the
role of {\em emitter\/})
and $b)$ the parton $k$ with the parton $\widetilde k$
(which plays the role of {\em spectator\/}).
All of the quantum numbers (colour, flavour) except momenta are assigned as
follows. The spectator
parton $\widetilde k$ has the same quantum numbers as $k$. The quantum numbers
of the emitter parton ${\widetilde {ij}}$ are obtained according to their
conservation in the collinear splitting process ${\widetilde {ij}} \rightarrow i + j$
(i.e.\ \lq anything + gluon' gives \lq anything' and \lq quark + antiquark'
gives \lq gluon').

The momenta of the emitter and the spectator are defined as follows
\begin{equation} \label{pk}
{\widetilde p}_{ij}^\mu =
p_i^\mu  + p_j^\mu  - \frac{y_{ij,k}}{1-y_{ij,k}} \,p_k^\mu \;\;,
\;\;\;\;{\widetilde p}_k^\mu = \frac{1}{1-y_{ij,k}} \,p_k^\mu \;\;,
\end{equation}
where the dimensionless variable $y_{ij,k}$ is given by
\begin{equation} \label{yijk}
y_{ij,k} = \frac{p_ip_j}{p_ip_j+p_jp_k+p_kp_i} \;.
\end{equation}

In the bra-ket on the right-hand side of Eq.~(\ref{dipff}),
${\bom T}_{ij}$ and ${\bom T}_k$ are the colour charges of the emitter
and the spectator and ${\bom V}_{ij,k}$ are matrices in the helicity space
of the emitter. These matrices, which depend on $y_{ij,k}$ and
on the kinematic variables ${\tilde z}_i, {\tilde z}_j$:
\begin{eqnarray}
\label{zitil}
{\tilde z}_i
= \frac{p_ip_k}{p_jp_k+p_ip_k} =
\frac{p_i{\widetilde p}_k}{{\widetilde p}_{ij}{\widetilde p}_k}
\;, \;\;\;
{\tilde z}_j = \frac{p_jp_k}{p_jp_k+p_ip_k}=
\frac{p_j{\widetilde p}_k}{{\widetilde p}_{ij}{\widetilde p}_k}=
1-{\tilde z}_i  \;,
\end{eqnarray}
are universal factors related to the $d$-dimensional Altarelli-Parisi
splitting functions [\ref{AP}].
For fermion + gluon splitting we have
($s$ and $s'$ are the spin indices of the fermion ${\widetilde {ij}}$
in $\bra{..,{\widetilde {ij}},..}$ and $\ket{..,{\widetilde {ij}},..}$
respectively)
\begin{eqnarray}
\label{vqgk}
\bra{s}
{\bom V}_{q_ig_j,k}({\tilde z}_i;y_{ij,k}) \ket{s'}
&=& 8\pi \mu^{2\epsilon} \as\; C_F\,
\left[ \frac{2}{1-{\tilde z}_i(1-y_{ij,k})}
- \frac{}{} (1+{\tilde z}_i)
-\epsilon (1-{\tilde z}_i)\right] \;\delta_{ss'} \nonumber \\
&\equiv&  V_{q_ig_j,k} \;\delta_{ss'} \;.
\end{eqnarray}
For quark + antiquark and gluon + gluon splitting we have
($\mu$ and $\nu$ are the spin indices of the gluon ${\widetilde {ij}}$
in $\bra{..,{\widetilde {ij}},..}$ and $\ket{..,{\widetilde {ij}},..}$
respectively)
\begin{equation} \label{vqbqk}
\bra{\mu}
{\bom V}_{q_i{\bar q}_j,k}({\tilde z}_i)
\ket{\nu}
= 8\pi \mu^{2\epsilon} \as \;
T_R \;
\left[ -g^{\mu \nu} - \frac{2}{p_ip_j} \,
( {\tilde z}_i p_i^{\mu} - {\tilde z}_j p_j^{\mu} )
\,( {\tilde z}_i p_i^{\nu} - {\tilde z}_j p_j^{\nu} ) \,\right]
\equiv  V_{q_i{\bar q}_j,k}^{\mu \nu} \,,
\end{equation}
\begin{eqnarray} \label{vggk}
&\!&\!\!\!\!\!\!\!\!\!\!\bra{\mu}{\bom V}_{g_ig_j,k}({\tilde z}_i;y_{ij,k})
\ket{\nu} = 16\pi
\mu^{2\epsilon} \as \;C_A\,
\left[ -g^{\mu \nu} \left(
\frac{1}{1-{\tilde z}_i(1-y_{ij,k})} \right. \right.\nonumber \\
&\!&\!\!\!\!\!\!\!\!\!\!+ \left. \left.
\frac{1}{1-{\tilde z}_j(1-y_{ij,k})} - 2 \right)
+ (1-\epsilon) \frac{1}{p_ip_j} \,
( {\tilde z}_i p_i^{\mu} - {\tilde z}_j p_j^{\mu} )
\,( {\tilde z}_i p_i^{\nu} - {\tilde z}_j p_j^{\nu} ) \,\right]
\equiv  V_{g_ig_j,k}^{\mu \nu} \;.
\end{eqnarray}

The factorization formula in Eq.~(\ref{ff}) has a dipole structure with respect
to the {\em colour\/} and {\em spin\/} indices of the factorized partons.
As shown in Ref.~[\ref{CS}], Eqs.~(\ref{ff},\ref{dipff}) coincide with
the soft-gluon [\ref{BCM}] and Altarelli-Parisi [\ref{AP}] factorization
formulae respectively in the soft and collinear limits. However,
Eqs.~(\ref{ff},\ref{dipff}) are completely well-defined also outside these
limiting regions of the phase space. Indeed, in the factorized $m$-parton
matrix element both the emitter ${\widetilde {ij}}$ and the spectator
${\widetilde k}$ are on-shell
$({\widetilde p}_{ij}^2 = {\widetilde p}_k^2 = 0)$ and, performing the
replacement $\{i,j,k\} \rightarrow \{{\widetilde {ij}},{\widetilde k}\}$, momentum
conservation is implemented exactly:
\begin{equation} \label{momcon}
p_i^\mu  + p_j^\mu + p_k^\mu = {\widetilde p}_{ij}^\mu +
{\widetilde p}_k^\mu \;\;.
\end{equation}
The importance of these kinematical features is twofold. Firstly, momentum
conservation leads to a smooth interpolation between the soft and collinear
limits, thus avoiding double counting of overlapping soft and collinear
singularities. Secondly, the definition (\ref{pk}) of the dipole
momenta allows us to factorize exactly the $m+1$-parton phase space into an
$m$-parton subspace times a single-parton contribution (see
Eqs.~(\ref{psfac},\ref{dpi})).
The first property allows us to construct a counter-term $d\sigma^A$ that
produces a pointwise cancellation of the singularities of $d\sigma^R$ as in
Eq.~(\ref{sNLO1}). The second property makes this counter-term fully integrable
analytically over the subspace leading to soft and collinear divergences.

\section{The calculation of jet cross sections}
\label{siee}

The dipole formulae form the basis for our general algorithm for NLO jet
calculations, as we describe below.  First we define the leading order cross
section and give a formal definition of the requirements a jet definition must
fulfil.  Then we introduce the subtraction term, which cancels all
singularities of the real matrix element, and show how it can be integrated in
$d$ dimensions to cancel the singularities of the virtual matrix element.

\vspace{0.3cm}
\noindent {\it Leading order and jet definition}
\vspace{0.1cm}

The Born-level cross section in Eq.~(\ref{sLO}) has the following expression
\begin{equation}
\label{dsmb}
d\sigma^{B} = {\cal N}_{in} \sum_{\{ m \} } \, d\phi_m(p_1, ...,p_m;Q)
\;\frac{1}{S_{\{ m \} }} \;|{\cal{M}}_{m}(p_1, ...,p_m) |^2
\;F_J^{(m)}(p_1, ...,p_m) \;\;,
\end{equation}
where ${\cal N}_{in}$ includes all the factors that are QCD independent,
$\sum_{\{ m \} }$ denotes the sum over all the configurations with $m$
partons, $d\phi_m$ is the partonic phase space in Eq.~(\ref{psm}),
$S_{\{ m \} }$ is the Bose symmetry factor for identical partons in the final
state and
${\cal{M}}_{m}$ is the tree-level matrix element.

The function $F_J^{(m)}(p_1, ...,p_m)$ defines the jet observable in terms of
the momenta of the
$m$ final-state partons. In general, $F_J$ may contain $\theta$-functions
(thus, Eq.~(\ref{dsmb}) defines precisely a cross section), $\delta$-functions
(Eq.~(\ref{dsmb}) defines a differential cross section), numerical and
kinematical factors (Eq.~(\ref{dsmb}) refers to an inclusive observable), or
any combination of these. The essential property of $F_J^{(m)}$ is that the
jet observable we are interested in has to be infrared
and collinear safe. That is, it has to be
experimentally (theoretically) defined in such a
way that its actual value is independent of the number of soft and
collinear hadrons (partons) produced in the final state. In particular, this
value has to be the same in a given $m$-parton configuration and in all
$m+1$-parton configurations that are kinematically degenerate with it
(i.e.\ which are obtained from the $m$-parton configuration by adding a soft
parton or replacing a parton with a pair of collinear partons carrying the same
total momentum). These properties can be simply restated in a formal way, as
follows
\begin{equation}
\label{fjsoft}
F_J^{(n+1)}(p_1,..,p_j,..,p_{n+1} ) \rightarrow
F_J^{(n)}(p_1,...,p_{n+1} )  \;\;\;\;\;\ {\rm if} \;\;p_j \rightarrow 0 \;\;,
\end{equation}
\begin{equation}
\label{fjcoll}
F_J^{(n+1)}(p_1,..,p_i,..,p_j,..,p_{n+1} ) \rightarrow
F_J^{(n)}(p_1,..,p_i+p_j,..,p_{n+1} ) \;\;\;\;\;\
{\rm if} \;\;p_i \parallel p_j \;,
\end{equation}
\begin{equation}
\label{fjLO}
F_J^{(m)}(p_1,...,p_m ) \rightarrow 0 \;\;\;\;\;\ {\rm if} \;\;p_i \cdot p_j
\rightarrow 0 \;\;.
\end{equation}
Equations (\ref{fjsoft}) and (\ref{fjcoll}) respectively guarantee that the
jet observable is infrared and collinear safe for {\em any\/} number $n$ of
final-state partons, i.e.\ to {\em any\/} order in QCD perturbation theory.
Equation (\ref{fjLO}) defines the LO cross section, that is, it
ensures that the Born-level cross section $d\sigma^{B}$ in Eq.~(\ref{dsmb})
is well-defined (i.e.\ finite after integration) in $d=4$ dimensions.

\vspace{0.3cm}
\noindent {\it Next-to-leading order: the subtraction term}
\vspace{0.1cm}

The real contribution $d\sigma^{R}$ to the NLO cross section in
Eq.~(\ref{sNLO})
has the same expression as $d\sigma^{B}$ in Eq.~(\ref{dsmb}),
apart from the replacement $m \rightarrow m + 1$.
In particular, the $m$-parton matrix element ${\cal M}_m$ is replaced by
${\cal M}_{m+1}$. Therefore
an explicit and general form for the local counter-term $d\sigma^{A}$ in
Eq.~(\ref{dsNLO}) is provided
by the dipole factorization formula (\ref{ff}):
\begin{eqnarray}
\label{dsadef}
d\sigma^{A} &=& {\cal N}_{in} \sum_{\{ m+1 \} } \,
d\phi_{m+1}(p_1, ...,p_{m+1};Q)
\; \frac{1}{S_{\{ m+1 \} }} \nonumber \\
&\cdot& \sum_{\mathrm{pairs}\atop i,j}
\;\sum_{k\not=i,j} {\cal D}_{ij,k}(p_1, ...,p_{m+1}) \;
F_J^{(m)}(p_1, .. {\widetilde p}_{ij}, {\widetilde p}_k, ..,p_{m+1}) \;\;.
\end{eqnarray}
Here ${\cal D}_{ij,k}(p_1, ...,p_{m+1})$ is the dipole contribution
in Eq.~(\ref{dipff})
and $F_J^{(m)}(p_1, .. {\widetilde p}_{ij}, {\widetilde p}_k, ..,p_{m+1})$
is the jet function for the corresponding $m$-parton state
$\{ p_1, .. {\widetilde p}_{ij}, {\widetilde p}_k, ..,p_{m+1} \}$.
Note that this is completely independent of $p_i,$ which is how we are able to
integrate $d\sigma^A$ analytically over the phase-space of $i$ without any
information about the form of $F_J,$ as we perform below.

We can check that the definition (\ref{dsadef}) makes the difference
$( d\sigma^{R} - d\sigma^{A} )$ integrable in $d=4$ dimensions. Its
explicit expression is
\begin{eqnarray}
\label{dsra}
d\sigma^{R} - d\sigma^{A} &=& {\cal N}_{in} \sum_{\{ m+1 \} } \,
d\phi_{m+1}(p_1, ...,p_{m+1};Q)
\; \frac{1}{S_{\{ m+1 \} }}  \nonumber \\
&\cdot& \left\{ \frac{}{} |{\cal{M}}_{m+1}(p_1, ...,p_{m+1}) |^2
\;F_J^{(m+1)}(p_1, ...,p_{m+1}) \frac{}{} \right. \nonumber \\
&-& \left.
\sum_{\mathrm{pairs}\atop i,j}
\;\sum_{k\not=i,j} \right. \left. \frac{}{}
 {\cal D}_{ij,k}(p_1, ...,p_{m+1}) \;
F_J^{(m)}(p_1, .. {\widetilde p}_{ij}, {\widetilde p}_k, ..,p_{m+1}) \right\}
\;\;.
\end{eqnarray}
Each term in the curly bracket is separately singular in the soft and
collinear regions. However, as stated in Sect.~\ref{dff}, in each of these
regions both the matrix element ${\cal{M}}_{m+1}$ and the phase space
for the $m+1$-parton configuration behave as the corresponding dipole
contribution and dipole phase space:
\begin{eqnarray}
|{\cal{M}}_{m+1}(p_1, ...,p_{m+1}) |^2 &\rightarrow& {\cal D}_{ij,k}(p_1, ...,p_{m+1})
\;\;, \\
\{ p_1, .. p_i,..p_j,..p_k, ..,p_{m+1} \} &\rightarrow&
\{ p_1, .. {\widetilde p}_{ij}, {\widetilde p}_k, ..,p_{m+1} \} \;\;.
\end{eqnarray}
Thus, because of Eqs.~(\ref{fjsoft}) and (\ref{fjcoll}), the singularities
of the first term in the curly bracket are cancelled by similar singularities
due to the second term. On the other hand, each dipole ${\cal D}_{ij,k}$ in
Eq.~(\ref{dipff}) has no other singularities but those due to the $m$-parton
matrix element $\mket{1, .., {\widetilde {ij}},.., {\widetilde k},.., m+1}$.
Because of Eq.~(\ref{fjLO}), these singularities are screened (regularized)
by the jet
function $F_J^{(m)}(p_1, .. {\widetilde p}_{ij}, {\widetilde p}_k, ..,p_{m+1})$
in the curly bracket of Eq.~(\ref{dsra}).

Note that this cancellation mechanism is completely independent of the actual
form of the jet defining function but it is essential that $d\sigma^R$ and
$d\sigma^A$ are proportional to $F_J^{(m+1)}$ and $F_J^{(m)}$ respectively.
Nonetheless, because the terms on the
left- and  right-hand sides of Eq.(\ref{ff}) depend on the same kinematic
variables, $\{ p_1, ...,p_{m+1} \},$
both $d\sigma^R$ and $d\sigma^A$ live on the same $m+1$-parton
phase space. Thus the numerical integration (in $d=4$
dimensions) of Eq.~(\ref{dsra}) via Monte Carlo techniques is straightforward.
One
simply generates an $m+1$-parton configuration and gives it a positive
($+ \,|{\cal{M}}_{m+1}|^2$) or negative ($- \,{\cal D}_{ij,k}$) weight.
The role
of the two different jet functions $F_J^{(m+1)}$ and $F_J^{(m)}$
is that of binning these weighted events into different bins of the jet
observable. Any time that the generated configuration approaches a singular
region, these two bins coincide and the cancellation of the large  positive
and negative weights takes place.

Note, also, that the helicity dependence
of the splitting kernels ${\bom V}_{ij,k}$ in Eqs.~(\ref{vqbqk},\ref{vggk})
is essential if $d\sigma^A$ is to act as a local counterterm  that makes
$[ d\sigma^R - d\sigma^A ]$ integrable in four dimensions.  Indeed, the parton
azimuthal correlations due to this dependence are not only essential in the
most general case when $F_J$ explicitly depends on them, but even when it does
not\footnote{In this case the evaluation of
$\int_{m+1} d\sigma^R$ in four dimensions usually involves double angular
integrals of the type $\int_{-1}^{+1} d\cos\theta \int_{0}^{2\pi} d\varphi
\;\cos\varphi/(1-\cos\theta),$ where $\varphi$ is the azimuthal angle. These
integrals are mathematically ill-defined. If their numerical integration is
attempted, one can obtain any answer whatsoever, depending on the detail of the
integration procedure.  Performing the integral analytically before going to 4
dimensions, one obtains $\int_{-1}^{+1} d\cos\theta \int_{0}^{2\pi} d\varphi
\;\cos\varphi/(1-\cos\theta)\sin^{-2\epsilon}\theta\sin^{-2\epsilon}\varphi=0$.}.

\vspace{0.3cm}
\noindent {\it Next-to-leading order: integral of the subtraction term}
\vspace{0.1cm}

Having discussed the four-dimensional integrability of
$( d\sigma^{R} - d\sigma^{A} )$, the only other step we have to consider
is the $d$-dimensional analytical integrability of $d\sigma^{A}$ over the
one-parton subspace leading to soft and collinear divergences.
In this respect, our dipole formalism is particularly efficient and simple.
The definition (\ref{pk}) of the dipole momenta
allows us to {\em exactly\/} factorize the phase space of the partons $i,j,k$
into the dipole phase space times a single-parton contribution, as follows
\begin{equation} \label{psfac}
d\phi_{m+1}(p_1,..,p_i,p_j,p_k,..,p_{m+1};Q) =
d\phi_m(p_1,..,{\widetilde p}_{ij},
{\widetilde p}_k, ..,p_{m+1};Q) \;\left[ dp_i({\widetilde p}_{ij},
{\widetilde p}_k) \right] \;\;,
\end{equation}
where
\begin{eqnarray}
\label{dpi}
\left[ dp_i({\widetilde p}_{ij},{\widetilde p}_k) \right]
= \frac{d^{d}p_i}{(2\pi)^{d-1}} \,\delta_+(p_i^2) \; \Theta(1- {\tilde z}_i)
\,\Theta(1-y_{ij,k}) \;
\frac{(1-y_{ij,k})^{d-3}}{1- {\tilde z}_i} \;,
\end{eqnarray}
and the kinematic variables $y_{ij,k}$ and ${\tilde z}_i$ are defined in
Eqs.~(\ref{yijk},\ref{zitil}).

Inserting Eq.~(\ref{psfac}) and the explicit expression (\ref{dipff}) for
${\cal D}_{ij,k}$ into Eq.~(\ref{dsadef}), we can factorize completely the
$p_i$ dependence and carry out the integration over the phase space region
(\ref{dpi}) with the $m$ parton momenta
$\{ p_1,..,{\widetilde p}_{ij},{\widetilde p}_k, ..,p_{m+1} \}$ kept fixed.
Remarkably, this integration can be exactly performed in closed analytical form
in any number of space-time dimensions.
As shown in detail in Ref.~[\ref{CS}], after integration the spin
correlations between ${\bom V}_{ij,k}$ and
$\mket{1, .., {\widetilde {ij}},.., {\widetilde k},.., m+1}$ vanish and only
colour correlations survive.
The final result for $\int_{m+1} d\sigma^A$ can be written
in terms of an $m$-parton integral of the LO
(colour-correlated) matrix element times a factor [\ref{CS}]:
\begin{eqnarray}
\label{intdsma}
\int_{m+1} d\sigma^{A} = \int_m \left[ \int_1 d\sigma^A \right] &=&
\int_m {\cal N}_{in} \sum_{\{ m \} } \, d\phi_m(p_1, ...,p_m;Q)
\;\frac{1}{S_{\{ m \} }} \nonumber \\
&\cdot&
{}_m\!\!<{1, ..., m}| \;{\bom I}(\epsilon) \;|{1, ..., m}>_m \;
F_J^{(m)}(p_1, ...,p_m) \;\;.
\end{eqnarray}
Comparing Eqs.~(\ref{intdsma}) and (\ref{dsmb}), we see that the integration
of $d\sigma^A$ over the one-parton subspace that produces soft and collinear
singularities leads to an expression completely analogous to $d\sigma^B$.
One should simply replace the matrix element squared
$\M{}{m} =$ \mbox{${}_m\!\!<{1, ..., m}|{1, ..., m}>_m$} in $d\sigma^B$
with
\begin{equation}
\label{insop}
{}_m\!\!<{1, ..., m}| \;{\bom I}(\epsilon) \;|{1, ..., m}>_m \;,
\end{equation}
where ${\bom I}(\epsilon)$ is an insertion operator that depends on the colour
charges
and momenta of the $m$ final-state partons.
Its explicit expression is [\ref{CS}]:
\begin{eqnarray}
\label{iee}
{\bom I}(p_1,...,p_m;\epsilon) = -
\frac{\as}{2\pi}
\frac{1}{\Gamma(1-\epsilon)} \sum_i \;\frac{1}{{\bom T}_{i}^2} \;{\cal V}_i(\epsilon)
\; \sum_{k \neq i} {\bom T}_i \cdot {\bom T}_k
\; \left( \frac{4\pi \mu^2}{2 p_i\cdot p_k} \right)^{\epsilon}
 \;\;,
\end{eqnarray}
where the singular factors ${\cal V}_i(\epsilon)$
have the following $\epsilon$-expansion\footnote{Their exact expressions in
any number $d=4-2\epsilon$ of dimensions are given in [\ref{CS}].}
\begin{eqnarray}
\label{calvexp}
{\cal V}_{i}(\epsilon) = {\bom T}_{i}^2 \left( \frac{1}{\epsilon^2} -
\frac{\pi^2}{3} \right) + \gamma_i \;\frac{1}{\epsilon}
+ \gamma_i + K_i + {\cal O}(\epsilon) \;\;,
\end{eqnarray}
with ($T_R=1/2$ and $N_f$ is the number of flavours)
\begin{equation}
\label{gadef}
\gamma_{i=q,{\bar q}} = \mbox{$\frac{3}{2}$}\, C_F \;,
\;\;\;\;
\gamma_{i=g} = \mbox{$\frac{11}{6}$} \, C_A - \mbox{$\frac{2}{3}$}
T_R N_f \;,
\end{equation}
\begin{eqnarray}
\label{kcondef}
K_{i=q,{\bar q}} = \left( \mbox{$\frac{7}{2}$} - \mbox{$\frac{\pi^2}{6}$}
\right) C_F \;,
\;\;\;\;
K_{i=g} = \left( \mbox{$\frac{67}{18}$} - \mbox{$\frac{\pi^2}{6}$} \right) C_A
- \mbox{$\frac{10}{9}$} T_R N_f \;.
\end{eqnarray}

In the calculation of the NLO cross section (\ref{sNLO2}), Eq.~(\ref{intdsma})
has to be combined with the virtual contribution, whose expression
in terms of the
(renormalized) one-loop matrix element is the following
\begin{equation}
\label{dsv1}
d\sigma^{V} = {\cal N}_{in} \sum_{\{ m \} } \, d\phi_m(p_1, ...,p_m;Q)
\;\frac{1}{S_{\{ m \} }} \;|{\cal{M}}_{m}(p_1, ...,p_m) |^2_{(1-loop)}
\;F_J^{(m)}(p_1, ...,p_m) \;\;.
\end{equation}
As discussed in Ref.~[\ref{CS}], the addition of these two contributions
correctly produces the cancellation of all the $\epsilon$-poles, thus leading
to a finite NLO cross section.

\section{Final formulae}
\label{finfor}

The final results of the application of our algorithm to the calculation
of jet cross sections with no hadron in the initial state are summarized
below.

The full QCD cross section in Eq.~(\ref{sig}) contains a LO and a NLO
component.
Assuming that the LO calculation involves $m$ final-state partons, the LO
cross section is given by
\begin{equation}
\label{LOeefin}
\sigma^{LO} = \int_m d\sigma^{B} = \int d\Phi^{(m)}
\;| {\cal M}_{m}(p_1, ...,p_m)|^2 \;F_J^{(m)}(p_1, ...,p_m) \;\;,
\end{equation}
where ${\cal M}_{m}$ is the tree-level QCD matrix element for producing $m$ partons
in the final state and the function $F_J^{(m)}$ defines the particular jet
observable we are interested in (see Eqs.~(\ref{fjsoft}-\ref{fjLO})
for the general
properties that $F_J^{(m)}$ has to fulfil). The factor $d\Phi^{(m)}$ collects
all the relevant phase space factors, i.e.\ all the remaining terms
on the right-hand side of Eq.~(\ref{dsmb}). The whole calculation (phase space
integration and evaluation of the matrix element) can be carried out in
four space-time dimensions.

According to the subtraction formula, Eq.~(\ref{sNLO2}),
the NLO cross section
is split into two terms with $m+1$-parton and  $m$-parton kinematics,
respectively. The contribution with $m+1$-parton kinematics is the following
\begin{eqnarray}
\label{m1eefin}
\int_{m+1} \left[ d\sigma^R_{\epsilon=0} - d\sigma^A_{\epsilon=0} \right]&=&
\int d\Phi^{(m+1)}  \left\{ \frac{}{}
\;| {\cal M}_{m+1}(p_1, ...,p_{m+1})|^2 \;F_J^{(m+1)}(p_1, ...,p_{m+1}) \right. \\
&&\left. - \sum_{\mathrm{pairs}\atop i,j}
\;\sum_{k\not=i,j} {\cal D}_{ij,k}(p_1, ...,p_{m+1}) \;
F_J^{(m)}(p_1, .. {\widetilde p}_{ij}, {\widetilde p}_k, ..,p_{m+1}) \right.
\left. \frac{}{} \right\}
\;\;, \nonumber
\end{eqnarray}
where the term in the curly bracket is exactly the same as that in
Eq.~(\ref{dsra}): ${\cal M}_{m+1}$ is the tree-level matrix element,
${\cal D}_{ij,k}$ is the dipole factor in Eq.~(\ref{dipff}) and $F_J^{(m)}$
is the
jet defining function for the corresponding $m$-parton state (note, again,
the difference between the two jet functions $F_J^{(m+1)}$ and $F_J^{(m)}$
in the curly bracket). In spite of their original $d$-dimensional
definition, at this stage the full calculation is carried out in four
dimensions.

The NLO contribution with $m$-parton kinematics is given by
\begin{eqnarray}
\label{meefin}
&&\!\!\!\!\!\!\!\int_{m} \left[ d\sigma^{V}  + \int_1 d\sigma^{A}
 \right]_{\epsilon=0}  \\
&&\!\!\!\!\!\!\!
=  \int d\Phi^{(m)} \left\{ | {\cal M}_{m}(p_1, ...,p_{m})|^2_{(1-loop)}
+ {}_m\!\!<{1, ..., m}| \;{\bom I}(\epsilon) \;|{1, ..., m}>_m \right\}_{\epsilon=0}
 F_J^{(m)}(p_1, ...,p_{m}) \;\;.\nonumber
\end{eqnarray}
The first term in the curly bracket is the one-loop {\em renormalized\/} matrix
element squared for producing $m$ final-state partons. The second term is
obtained by inserting the colour-charge operator of Eq.~(\ref{iee}) into the
tree-level matrix element for producing $m$ partons as in Eq.~(\ref{insop}).
These two terms have to be first evaluated in $d=4-2\epsilon$
dimensions. Then one has to carry out their expansion in $\epsilon$-poles (the
expansion for the singular factors ${\cal V}_i(\epsilon)$ is given in
Eq.~(\ref{calvexp})),
cancel analytically (by trivial addition) the poles and perform the limit
$\epsilon \rightarrow 0$. At this point the phase-space integration is carried out in four
space-time dimensions.

\section{\boldmath$e^+e^-$$\rightarrow 3$ jets}
\label{3jet}

In this Section we consider the simplest non-trivial application of our
algorithm, namely the calculation of three-jet observables in $e^+e^-$\ annihilation.
Thus our formalism can be directly compared with that in Ref.~[\ref{ERT}].

The LO partonic process to be considered is
$e^+ e^- \rightarrow q(p_1) + {\bar q}(p_2)  + g (p_3)$. The corresponding tree-level
matrix element is denoted by ${\cal M}_3(p_1,p_2,p_3)$. We use customary
notation for the kinematic variables:
$Q^2$ is the square of the centre-of-mass energy, $y_{ij}=2p_i\cdot p_j/Q^2$
and $x_i=2p_i\cdot Q/Q^2$.

At NLO, two different real-emission subprocesses contribute:
a) $e^+ e^- \rightarrow q(p_1) + {\bar q}(p_2)  + g (p_3) + g(p_4)$;
b) $e^+ e^- \rightarrow q(p_1) + {\bar q}(p_2)  + q(p_3) + {\bar q}(p_4)$. In addition
one has to compute the one-loop correction to the LO process.

The calculation of the subtracted cross section (\ref{m1eefin}) for the
subprocess
a) involves the evaluation of the following dipole contributions:
${\cal D}_{13,2}, {\cal D}_{13,4}, {\cal D}_{14,2}, {\cal D}_{14,3},
{\cal D}_{23,1}, {\cal D}_{23,4},
{\cal D}_{24,1},$ ${\cal D}_{24,3}, {\cal D}_{34,1}, {\cal D}_{34,2}$.
The associated colour
algebra can be easily performed in closed form because the several colour
projections of the three-parton matrix element fully
factorize\footnote{If the LO matrix element involves two or three partons,
the colour algebra can always be carried out in closed (factorized) form.}.
Thus we do not need to compute any colour-correlated tree amplitudes
and we find
\begin{eqnarray}
\label{dipa}
{\cal D}_{13,2}(p_1,p_2,p_3,p_4) &=& \frac{1}{2p_1p_3}
\left(1- \frac{C_A}{2C_F}\right)
V_{q_1g_3,2} \;|{\cal M}_3({\widetilde p_{13}},{\widetilde p_2},p_4)|^2 \;\;,
\nonumber \\
{\cal D}_{13,4}(p_1,p_2,p_3,p_4) &=& \frac{1}{2p_1p_3} \,\frac{C_A}{2C_F} \;
V_{q_1g_3,4} \;|{\cal M}_3({\widetilde p_{13}},p_2,{\widetilde p_4})|^2 \;\;, \\
{\cal D}_{34,1}(p_1,p_2,p_3,p_4) &=& \frac{1}{2p_3p_4} \,\frac{1}{2} \;
V_{g_3g_4,1}^{\mu \nu}
\;{\cal T}_{\mu \nu}({\widetilde p_{1}},p_2,{\widetilde p_{34}}) \;\;.
\nonumber
\end{eqnarray}
The dipole contributions ${\cal D}_{23,1}, {\cal D}_{23,4},
{\cal D}_{34,2}$ are obtained respectively
from ${\cal D}_{13,2},{\cal D}_{13,4}, {\cal D}_{34,1} $ by the replacement
$p_1 \leftrightarrow p_2$. Analogously, one can obtain
${\cal D}_{14,2}$ and ${\cal D}_{14,3}$ respectively from
${\cal D}_{13,2}$ and ${\cal D}_{13,4}$ by the replacement
$p_3 \leftrightarrow p_4$, and ${\cal D}_{24,1}$ and
${\cal D}_{24,3}$ respectively from ${\cal D}_{13,2}$ and
${\cal D}_{13,4}$ by the replacement $p_1 \leftrightarrow p_2, \;
p_3 \leftrightarrow p_4$.

In the case of the subprocess b) we have to consider the following
dipole contributions:
${\cal D}_{12,3}, {\cal D}_{12,4}, {\cal D}_{14,2}, {\cal D}_{14,3},
{\cal D}_{23,1}, {\cal D}_{23,4}, {\cal D}_{34,1}, {\cal D}_{34,2}$.
Performing the colour algebra we get
\begin{equation}
\label{dipb}
{\cal D}_{34,1}(p_1,p_2,p_3,p_4) = \frac{1}{2p_3p_4} \frac{1}{2} \;
V_{q_3{\bar q}_4,1}^{\mu \nu}
{\cal T}_{\mu \nu}({\widetilde p_{1}},p_2,{\widetilde p_{34}}) \;,
\end{equation}
and all the other dipoles are obtained by the corresponding permutation of
the parton momenta.

The splitting functions $V_{ij,k}$ of Eqs.~(\ref{dipa},\ref{dipb}) are given
in Eqs.~(\ref{vqgk}-\ref{vggk}). The tensor ${\cal T}_{\mu \nu}$ is
the squared amplitude for the LO process $\mathrm{e^+e^-\rightarrow q\bar{q}g}$
not summed over the gluon polarizations ($\mu$ and $\nu$ are the gluon spin
indices and $-g^{\mu \nu} {\cal T}_{\mu \nu} = |{\cal M}_3|^2 $).
This can be easily calculated. In the case of jet observables averaged over
the directions of the incoming leptons (un-oriented events) we find
(in $d=4$ dimensions)
\begin{equation}
{\cal T}^{\mu \nu}(p_1,p_2,p_3) = - \frac{1}{x_1^2 + x_2^2}
\; |{\cal M}_3(p_1,p_2,p_3)|^2 \;T^{\mu \nu} \;,
\end{equation}
where
\begin{eqnarray}
  T^{\mu\nu} &=&
    +2                            \frac{p_1^\mu p_2^\nu}{Q^2}
    +2                            \frac{p_2^\mu p_1^\nu}{Q^2}
    -2\frac{1-x_1}{1-x_2}         \frac{p_1^\mu p_1^\nu}{Q^2}
    -2\frac{1-x_2}{1-x_1}         \frac{p_2^\mu p_2^\nu}{Q^2}
\nonumber\\&&
    -\frac{1-x_1-x_2+x_2^2}{1-x_2}\left[
                                  \frac{p_1^\mu   p_3^\nu}{Q^2}
    +                             \frac{p_3^\mu p_1^\nu}{Q^2}
                                  \right]
    -\frac{1-x_2-x_1+x_1^2}{1-x_1}\left[
                                  \frac{p_2^\mu p_3^\nu}{Q^2}
    +                             \frac{p_3^\mu p_2^\nu}{Q^2}
                                  \right]
\nonumber\\&&
    +\left(1+\smfrac{1}{2} x_1^2+\smfrac{1}{2} x_2^2-x_1-x_2\right)g^{\mu\nu} \;.
\end{eqnarray}

To complete the NLO calculation we also need the virtual cross section.
In the case of un-oriented events, we take the one-loop matrix element
in the ${\overline {\rm MS}}$ renormalization scheme from Ref.~[\ref{ERT}]
(we use slightly different notation):
\begin{eqnarray}
\label{ert1loop}
&&|{\cal M}_3(p_1,p_2,p_3)|^2_{(1-loop)} =
|{\cal M}_3(p_1,p_2,p_3)|^2 \; \frac{\as}{2\pi}\frac{1}{\Gamma(1-\epsilon)}
\left(\frac{4\pi\mu^2}{Q^2}\right)^\epsilon
\nonumber \\
&&\cdot
\left\{ \,-\frac{1}{\epsilon^2} \left[ (2C_F -C_A) y_{12}^{-\epsilon} +
C_A \left(y_{13}^{-\epsilon} + y_{23}^{-\epsilon} \right) \right]
-\frac{1}{\epsilon} \left( 3C_F + \frac{11}{6}C_A -\frac{2}{3}T_R N_f \right)
\right. \nonumber \\
&&+ \left. \frac{\pi^2}{2}(2C_F+C_A) - 8 C_F \right\}
+ \frac{\as}{2\pi}\left[ F(y_{12},y_{13},y_{23}) +{\cal O}(\epsilon) \right] \;\;,
\end{eqnarray}
where $F(y_{12},y_{13},y_{23})$ is defined in Eq.~(2.21) of Ref.~[\ref{ERT}].

The explicit evaluation of the insertion operator ${\bom I}(\epsilon)$ in
Eqs.~(\ref{insop},\ref{iee}) gives:
\begin{eqnarray}
\label{ertinsop}
&&{}_3\!\!<{1,2,3}| \;{\bom I}(\epsilon) \;|{1,2,3}>_3 =
|{\cal M}_3(p_1,p_2,p_3)|^2 \; \frac{\as}{2\pi}\frac{1}{\Gamma(1-\epsilon)}
\left(\frac{4\pi\mu^2}{Q^2}\right)^\epsilon
\nonumber \\
&&
\left\{ \,\frac{1}{\epsilon^2} \left[ (2C_F -C_A) y_{12}^{-\epsilon} +
C_A \left(y_{13}^{-\epsilon} + y_{23}^{-\epsilon} \right) \right]
+\frac{1}{\epsilon} \left( 2\gamma_q + \gamma_g \right) \right.
\nonumber \\
&&- \left. \gamma_q \frac{1}{C_F} \left[ (2C_F - C_A) \ln y_{12} + \frac{1}{2}
C_A \ln (y_{13}y_{23}) \right] - \frac{1}{2} \gamma_g \ln (y_{13}y_{23})
\right. \nonumber \\
&&\left. -\frac{\pi^2}{3}(2C_F+C_A) + 2(\gamma_q + K_q) + \gamma_g + K_g
+{\cal O}(\epsilon) \right\} \;\;.
\end{eqnarray}
Combining the one-loop  matrix element (\ref{ert1loop}) with the result
(\ref{ertinsop}) according to Eq.~(\ref{meefin}) and using the explicit
expressions (\ref{gadef},\ref{kcondef}) for $\gamma_i$ and $K_i$,
all the pole terms cancel.
Note that as well as the pole terms, the closely related $\pi^2$ and
$\ln^2$ terms cancel:
\begin{eqnarray}
&&|{\cal M}_3(p_1,p_2,p_3)|^2_{(1-loop)} +
{}_3\!\!<{1,2,3}| \;{\bom I}(\epsilon) \;|{1,2,3}>_3 = |{\cal M}_3(p_1,p_2,p_3)|^2
\nonumber \\
&&\cdot \;\frac{\as}{2\pi}
\left [ - \frac{3}{2} (2C_F - C_A) \ln y_{12}
-\frac{1}{3} (5C_A - T_R N_f) \ln (y_{13}y_{23}) \right. \nonumber \\
&&\left. + 2 C_F +\frac{50}{9} C_A
- \frac{16}{9} T_R N_f \right]
+ \frac{\as}{2\pi}\left[ F(y_{12},y_{13},y_{23}) +{\cal O}(\epsilon) \right]] \;\;.
\end{eqnarray}

We have implemented these results as a working Monte Carlo program\footnote{The
  program can be obtained from
  \verb+http://surya11.cern.ch/users/seymour/nlo/+.} and the results are in
good agreement with Ref.~[\ref{KN}] for all distributions shown there.  As an
example we show the NLO coefficients for the thrust and $C$-parameter
distributions in Fig.~\ref{thefig}.
\begin{figure}
  \centerline{\epsfig{figure=nlolett_thrust.ps,height=6cm}\hfill
              \epsfig{figure=nlolett_cparam.ps,height=6cm}}
  \caption[]{Coefficient of $(\as/2\pi)^2$ for the thrust and $C$-parameter
    distributions. The dotted histograms show the size of the statistical
    errors.}
  \label{thefig}
\end{figure}
We find that in general, the numerical convergence is similar to the program of
Ref.~[\ref{KN}], except close to the two-jet region in which ours becomes
progressively better. More details of the Monte Carlo program and the
generalization to oriented three-jet events will be presented elsewhere.

\section{Summary and outlook}
\label{outl}

In this letter we have presented the basic idea to set up a completely
general algorithm for calculating jet cross sections in NLO QCD\@. By general
we mean that the algorithm applies to {\em any\/} jet observable in a given
scattering process as well as to {\em any\/} hard-scattering process. The
algorithm overcomes all the analytical difficulties related to the treatment
of soft and collinear divergences in the perturbative expansion. The output
of the algorithm is given in terms of effective matrix elements (the
contributions in the curly bracket of Eqs.~(\ref{m1eefin},\ref{meefin}))
with built-in
cancellation of soft and collinear singularities. These effective matrix
elements can be numerically or analytically (whenever possible) integrated
over the available phase space to compute the actual value of the NLO cross
section. If the numerical approach is chosen, Monte Carlo integration
techniques can be easily implemented to provide a general-purpose Monte
Carlo program for carrying out NLO QCD calculations in any given process.

Starting from the very general idea of the subtraction method, we have
discussed how one can automatically construct a pointwise and integrable
counter-term $d\sigma^A$ for the real contribution $d\sigma^R$ to the  NLO
cross section. This counter-term is independent of the details of the process
under study and can be handled once and for all to define the effective
matrix elements mentioned above. We have shown how this procedure works in
practice for processes with no hadron in the initial state (typically,
$e^+e^-$\ annihilation) and any number of jets in the final state. In this case, the
final result is given by the master formulae in
Eqs.~(\ref{m1eefin},\ref{meefin}). We
have provided explicit expressions for both the universal dipole factors
${\cal D}_{ij,k}$ and $\bom I$. Having
these factors at our disposal, the only other ingredients necessary for the
full NLO calculation, are the following
\begin{itemize}
\addtolength{\itemsep}{-4pt}
\item
a set
of independent colour projections\footnote{Actually, if the total number of
QCD partons involved in the LO matrix element is less than or equal to three,
one simply needs its incoherent sum over the colours (see Sect.~\ref{3jet}).}
of the matrix element
squared at the Born level, summed over parton polarizations, in $d$
dimensions;
\item
the one-loop matrix element in $d$ dimensions;
\item
an additional projection of the Born level matrix element over the helicity of
each external gluon in four dimensions;
\item
the tree-level NLO matrix element in four dimensions.
\end{itemize}

\noindent These few ingredients are sufficient for writing, in a
straightforward way,
a general-purpose NLO Monte Carlo algorithm. Note in particular
that there is no need to extract a proper counter-term $d\sigma^{A}$
starting from a
cumbersome expression for $d\sigma^{R}$ in $d$ dimensions. The NLO matrix
element
contributing to $d\sigma^{R}$ can be evaluated directly in four space-time
dimensions thus leading to an extreme simplification of the Lorentz algebra.

The key point of our method for constructing the counter-term $d\sigma^A$
is the dipole formalism described in Sect.~\ref{dff}. Starting from the
universal
behaviour of the QCD matrix elements in the soft and collinear regions, we have
introduced improved factorization formulae based on a dipole structure with
respect to the momenta, colours and helicities of the QCD partons. Our dipole
formulae correctly match the well-known singularities of the QCD scattering
amplitudes in the soft and collinear limits. Moreover, these limits are
approached smoothly, thus avoiding double counting of overlapping soft and
collinear divergences. This smooth transition is possible because our dipole
formalism is explicitly Lorentz covariant and the dipole formulae exactly
fulfil momentum conservation.

In the present paper, these main features of the dipole formulae have been used
in the context of NLO computations of jet cross sections with no initial-state
hadron. For lepton-hadron and hadron-hadron collisions, perturbative
QCD calculations face additional difficulties related to the factorization of
initial-state collinear singularities. Likewise, to calculate cross sections
for processes in which a final state hadron is identified, the equivalent
final-state collinear singularities must be dealt with. In a companion paper
[\ref{CS}], we show that the dipole formalism overcomes these difficulties in
a simple and general manner. Thus we can explicitly provide a set of universal
counter-terms or, equivalently, effective (non-singular) matrix elements
that can be straightforwardly used for {\em any\/} NLO QCD calculation.

\subsection*{Acknowledgements}
One of us (SC) would like to thank the CERN TH Division for hospitality and
partial support during the course of this work.

\subsection*{References}
\begin{enumerate}

\item \label{QCDrev}
S.\ Catani, in Proc.\ of the Int.\ Europhysics Conf.\ on High Energy Physics,
HEP 93,
eds.\ J.\ Carr and M.\ Perrottet (Editions
Frontieres, Gif-sur-Yvette, 1994), p.~771; B.R.\ Webber,
in Proc.\ of the 27th Int.\ Conf.\ on High Energy Physics,
eds. P.J.\ Bussey and I.G.\ Knowles (Institute of Physics,
Philadelphia, 1995), p.~213.

\item \label{KS}
Z.\ Kunszt and D.E.\ Soper, \pr{46}{192}{92}.

\item \label{KL}
K.\ Fabricius, G.\ Kramer, G.\ Schierholz and I.\ Schmitt, \zp{11}{315}{81};
G.\ Kramer and B.\ Lampe, Fortschr. Phys. 37 (1989) 161.

\item \label{ERT}
R.K.\ Ellis, D.A.\ Ross and A.E.\ Terrano,
\np{178}{421}{81}.

\item  \label{BCM}
See, for instance: A.\ Bassetto, M.\ Ciafaloni and G.\ Marchesini,
\prep{100}{201}{83}; Yu.L.\ Dokshitzer, V.A.\ Khoze, A.H.\ Mueller and
S.I.\ Troyan, {\em Basics of Perturbative QCD}, Editions Frontieres,
Paris, 1991.

\item \label{GG}
W.T.\ Giele and E.W.N.\ Glover, \pr{46}{1980}{92}.

\item \label{GGK}
W.T.\ Giele, E.W.N.\ Glover and D.A.\ Kosower, \np{403}{633}{93}.

\item \label{BOO}
H.\ Baer, J.\ Ohnemus and J.F.\ Owens, \pr{42}{61}{90}; B.\ Bailey,
J.F.\ Owens and J.\ Ohnemus, \pr{46}{2018}{92}.

\item  \label{KN}
Z.\ Kunszt and P.\ Nason, in \lq Z Physics at LEP 1', CERN 89-08, vol.~1,
p.~373.

\item \label{EKS}
S.D.\ Ellis, Z.\ Kunszt and D.E.\ Soper, \pr{40}{2188}{89}, \prl{69}{1496}{92}.

\item \label{Frix}
S.\ Frixione, Z.\ Kunszt and A.\ Singer, preprint SLAC-PUB-95-7073,
hep-ph/9512328.

\item \label{MNR}
M.L.\ Manga\-no, P.\ Nason and G.\ Ridolfi, \np{373}{295}{92}.

\item \label{CS}
S.\ Catani and M.H.\ Seymour, preprint CERN-TH/96-29, to appear.

\item \label{MP}
M.L.\ Mangano and S.J.\ Parke, \prep{200}{301}{91}.

\item \label{AP}
G.\ Altarelli and G.\ Parisi, \np{126}{298}{77}.

\end{enumerate}

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Q: How to call multiple Activity through single intent(from widget)? I'm looking for an answer to the question where on an event i want to open two activity parallely where one of them will be in background and another in foreground but when the foreground activity stops the background activity should fire up.
Is it possible? I would like to know the trick
@downvoters kindly comment if you find it useless
Update
Here is the code that I have tried:
 Intent intent = new Intent(context, ActivityTwo.class);
        Intent home= new Intent(context,MainActivity.class);
        Intent[] intents={intent,home};
        PendingIntent pendingIntent = PendingIntent.getActivities(context,0,intents,0);

But this is opening only the MainActivity class.
where i want it to open the Activitytwo class in the foreground and mainActivity in the background.
SOLVED

For those who wish to know how to open multiple activities parallel from widget
  OK here is my problem

The string array it should be
Intent[] intents={home,intent};

instead of
Intent[] intents={intent,home};

Reference

A: Its possible with startActivities(Context, Intent[], bundle). Only Argument here to be explained is  Intent[], which is array of intents of all the activities you need to fire.
Only one Activity (not sure the first or last intent in array, you can experiment on this) , will stay in foreground and remaining will stay in background. When you press back, they will sequentially keep firing until launcher activity.
From docs:

public static boolean startActivities (Context context, Intent[] intents, Bundle options)

Start a set of activities as a synthesized task stack, if able.
In API level 11 (Android 3.0/Honeycomb) the recommended conventions
  for app navigation using the back key changed. The back key's behavior
  is local to the current task and does not capture navigation across
  different tasks. Navigating across tasks and easily reaching the
  previous task is accomplished through the "recents" UI, accessible
  through the software-provided Recents key on the navigation or system
  bar. On devices with the older hardware button configuration the
  recents UI can be accessed with a long press on the Home key.
When crossing from one task stack to another post-Android 3.0, the
  application should synthesize a back stack/history for the new task so
  that the user may navigate out of the new task and back to the
  Launcher by repeated presses of the back key. Back key presses should
  not navigate across task stacks.
startActivities provides a mechanism for constructing a synthetic task
  stack of multiple activities. If the underlying API is not available
  on the system this method will return false.
Parameters context    Start activities using this activity as the
  starting context intents  Array of intents defining the activities that
  will be started. The element length-1 will correspond to the top
  activity on the resulting task stack. options Additional options for
  how the Activity should be started. See {@link
  android.content.Context#startActivity(Intent, Bundle) Returns true if
  the underlying API was available and the call was successful, false
  otherwise

UPDATE

The element length-1 will correspond to the top activity on the resulting task stack.

so this resolves my confusion too!
UPDATE
Intent intent = new Intent(context, ActivityTwo.class);
        Intent home= new Intent(context,MainActivity.class);
        Intent[] intents={home,intent};
        startActivities(context, intents, null);

A: Intent intent = new Intent(SplashActivity.this, FirstActivity.class);
startActivityForResult(intent, 10);

@Override
    protected void onActivityResult(int requestCode, int resultCode, Intent data) {
        // TODO Auto-generated method stub
        super.onActivityResult(requestCode, resultCode, data);

        Intent intent = new Intent(SplashActivity.this, SecondActivity.class);
startActivity(intent);
}

in this example second activity will opened after first activity closed.

A: 1) overwrite onStop() on activity1 and to start activity2.

redpajama_set_name: RedPajamaStackExchange


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