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130+ World Leaders, Nobel Laureates, Scientists Call for End to Vaccine Patents
Marking two years since the WHO's pandemic declaration, a letter urges rich countries to stop blocking a patent waiver.
Activists hold placards during a protest in favor of an agreement to waive patent protections for COVID-19 vaccines on November 30, 2021, in Geneva, Switzerland.
Fabrice Coffrini / AFP via Getty Images
Marking the two-year anniversary of the World Health Organization's official Covid-19 pandemic declaration, more than 130 current and former world leaders, Nobel laureates, scientists, and humanitarians published an open letter Friday imploring rich countries to finally end their obstruction of a proposed patent waiver and share key vaccine technology with the world.
"The European Union, the United Kingdom, and Switzerland continue to block the lifting of intellectual property rules which would enable the redistribution and scale-up of Covid-19 vaccines, test, and treatment manufacturing in the Global South," reads the new letter, which was signed by the ex-leaders of more than 40 countries — including Malawi's former president Joyce Banda — and prominent human rights advocates, from UNAIDS executive director Winnie Byanyima to former U.N. Secretary-General Ban Ki-moon.
Released ahead of major global demonstrations marking the grim pandemic anniversary, the letter argues that a patent waiver proposed by South Africa and India back in October 2020 would help fast-track production of coronavirus vaccines, which remain heavily concentrated in wealthy countries two years into the pandemic.
To date, the pharmaceutical firms that largely control global vaccine production have resisted calls to share their publicly funded technology, leaving poor nations to rely on charity for their supply of shots — an imbalance that has proved disastrous. Earlier this week, Moderna's billionaire CEO Stéphane Bancel said working with a key WHO technology transfer hub in South Africa that's attempting to make its own mRNA shot would not be a "good use of our time."
In their open letter on Friday, the campaigners argued that the current approach to global vaccinations — under which profit-seeking companies have the "the power to decide who lives and who dies" — is "immoral, entirely self-defeating, and also an ethical, economic, and epidemiological failure":
As Mask Mandates End, COVID Experts Fear It's Too Soon
As the masks come off at my daughter's school -- and around the country -- I'm worried about COVID's next move.
William Rivers Pitt ,
"We appeal to world leaders to end this strategy of counter-productive nationalism and of protecting pharmaceutical monopolies and to finally act with international solidarity," the letter reads. "Now is the time to renew the commitments made at the founding of the World Health Organization, where all states agreed to deliver "the highest attainable standard of health as a fundamental right of every human being."
The letter's signatories called on governments around the world to take five specific steps:
Urgently agree and implement a global roadmap to deliver the WHO goal of fully vaccinating 70% of people by mid-2022, and beyond this ensure sustained, timely, and equitable access worldwide to Covid-19 vaccines, treatments, tests, and other medical technologies, including next generations effective and safe Covid-19 vaccines and medical technologies.
Maximize the production of safe and effective vaccines and other Covid-19 products by suspending relevant intellectual property rules and ensuring the mandatory pooling of all Covid-19 related knowledge, data, and technologies so that any nation can produce or buy sufficient and affordable doses of vaccines, treatments, and tests.
Invest public funding now in a rapid and massive increase in vaccine manufacturing as well as research and development (R&D) capacity to build a global distributed network capable of and governed to deliver affordable vaccines as global public goods to all nations.
Make Covid-19 vaccines, treatments, and tests available to governments and institutions at a price as close to the true cost as possible, and provided free of charge to everyone, everywhere, and allocated according to need.
Scale up sustainable investment in public health systems to ensure that low- and middle-income country governments have adequate resources to get shots into arms and save lives. These investments will pay dividends in the global economy and help restore economic and development gains which the global Covid-19 pandemic has partially reversed.
"Every life lost now to vaccine apartheid is avoidable," the letter concludes. "Only a People's Vaccine — based on the principles of equity and solidarity — can protect all of humanity and create a fairer, safer, more prosperous world."
While countries across the globe have begun lifting coronavirus-related public health restrictions, Covid-19 continues to kill nearly 7,000 people on average each day, with poor countries bearing the brunt of the devastation. According to new research by the People's Vaccine Alliance, the coronavirus death toll is four times higher in lower-income countries than in rich nations.
Pharmaceutical companies, meanwhile, continue to rake in huge profits from vaccines that were developed with the support of massive infusions of public money. Oxfam International estimates that the U.S.-based corporation Pfizer is making more than $1 million in profit per hour from its vaccine, which relies on government-developed technology.
"Let us be clear: this pandemic is far from over in Africa and across the world," said Banda, who served as Malawi's president from 2012 to 2014. "We are seeing, with each day, thousands of avoidable deaths. We are seeing women and girls being disproportionately impacted by the pandemic, through lost educational opportunities, domestic violence, and economic hardship. We must recapture the spirit of solidarity to end the suffering and create a better future."
"That starts now," she continued, "with ending these callous pharmaceutical monopolies on Covid-19 vaccines, so Africa and the world can tackle this crisis and the next."
Two years since #COVID19 was declared a pandemic and billions have not received a first dose.
100+ countries are united in demanding equitable vaccine access – the UK opposes them.@BorisJohnson: side with people not big pharma.
#EndCOVIDMonopoliespic.twitter.com/IzUnn2IDc4
— STOPAIDS (@STOPAIDS) March 11, 2022
Just 13.7% of people in low-income countries have received at least one coronavirus vaccine dose, according to the latest figures from Our World in Data. The head of the WHO recently warned that nearly 90 countries are not on track to fully vaccinate 70% of their populations by the start of July 2022.
Millions of people have died of Covid-19 since South Africa and India introduced their patent waiver proposal at the World Trade Organization (WTO), where the governments of rich countries have stalled and undermined negotiations for more than a year.
While European countries — including Germany and the United Kingdom — have led opposition to the patent waiver, the United States has been accused of taking a passive stance at WTO talks despite endorsing a waiver in May. One former WHO official recently argued that rich countries should face prosecution at the International Criminal Court for perpetuating deadly vaccine inequities.
"Rich country leaders are protecting pharmaceutical monopolies on Covid-19 vaccines, diagnostics, and therapeutics over the health and lives of billions of people," Ban, who led the U.N. from 2007 to 2016, said in a statement Friday. "And we can only imagine how damaging a new profoundly lethal variant could be for everyone on the planet. That is why this is a historic test of multilateralism. It truly affects us all."
"If world leaders can't rise to the challenge of vaccine equity," Ban warned, "they diminish hope that they will rise to the existential challenge of tackling the climate crisis."
Jake Johnson is a staff writer for Common Dreams. Follow him on Twitter: @johnsonjakep.
COVID Shows It's Time to End the Pharmaceutical Industry
The only way out of this pandemic is a global prioritization of public health above profit.
Rob Wallace ,
Capitalism Is Preventing Bold Steps to Curb COVID as Cases Skyrocket
The president needs to get a handle on this at long last, and stop mollifying capitalists with exercises in futility.


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Analyst Update: Cepheid, Chipotle Mexican Grill, Twitter
Analysts adjusted their ratings on Cepheid (CPHD), Chipotle Mexican Grill, Inc. (CMG), and Twitter Inc (TWTR)
Mark Fightmaster
CPHD
Analysts are weighing in today on molecular diagnostic company Cepheid (NASDAQ:CPHD), restaurateur Chipotle Mexican Grill, Inc. (NYSE:CMG), and social media network Twitter Inc (NYSE:TWTR). Here's a quick roundup of today's brokerage notes on CPHD, CMG, and TWTR.
CPHD is 19.6% lower at $31.82 -- and just off a two-year low of $31.42 -- after the company cut its quarterly revenue expectations (subscription required). As such, no fewer than eight brokerages cut their price targets on Cepheid, including Canaccord Genuity (to $42 from $68), Goldman Sachs (to $30 from $53), Needham (to $47 from $70), and Craig Hallum (to $52 from $75). The stock is now more than 41% lower year-to-date, and sitting well below support at its 40-month moving average. CPHD's analyst configuration leaves the stock vulnerable to even more negative attention, as it sports 10 "strong buys" and two "buys" out of 16 total rankings.
CMG is 3.3% lower at $725.83 this afternoon, even though the restaurant chain saw RBC raise its price target to $825 from $740 and reiterate an "outperform" rating. Prior to today, Chipotle Mexican Grill, Inc. had been attempting to claw atop the $750-$760 region, home to its record high hit in early August. On the International Securities Exchange (ISE), Chicago Board Options Exchange (CBOE), and NASDAQ OMX PHLX (PHLX), it appears that bears have a bigger-than-usual appetite for CMG. The equity's 50-day put/call volume ratio of 1.32 is higher than 89% of the readings taken in the past year. Perhaps they're betting on an earnings miss for the company when it reports next Tuesday, Oct. 20.
Shares of social media giant TWTR have reversed early losses, up 0.9% at $29.29, after the company named former Alphabet Inc (NASDAQ:GOOGL) -- aka Google -- exec Omid Kordestani its executive chairman. Earlier, TWTR was reeling from a downgrade to "market perform" from "market outperform" at JMP Securities. Twitter Inc has been in the headlines of late, naming a new CEO and announcing mass layoffs. Yesterday, bulls took the reins in the option pits, and TWTR's 50-day call/put volume ratio on the ISE/CBOE/PHLX of 2.74 is ranks in the 81st percentile of the readings taken in the past 52 weeks. On the charts, however, TWTR has shed 18.1% in 2015, and is running into a wall in the form of its 20-week moving average.
For other stocks in analysts' crosshairs, read Analyst Upgrades: Intel Corporation, Chesapeake Energy Corporation, and Ford Motor Company and Analyst Downgrades: GoPro Inc, Transocean LTD, and JPMorgan Chase & Co.

redpajama_set_name: RedPajamaCommonCrawl


CLARKS SUMMIT, Pa. - Two first half goals was just enough as Clarks Summit (5-5-2, 1-3-1 CSAC) defeated Bryn Athyn (2-12, 0-5) 2-1 in men's soccer action on Saturday.
Hugh Hayner scored in the 7th minute, followed by a goal from Jan Rodriguez at 30:20 for the scoring for the Defenders. Caleb Holloman assisted on both goals.
Kevin Conroy scored for the Lions in the 57th minute, but, the Lions could not get the equalizer.
Shane McCurdy had one save on the afternoon.


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angular.module('service.car')
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    var getCars = function () {
      return $http.get(addres+'/get-cars');
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    function Car() {
      this.id;
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      this.make;
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    return {
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      uploadPhotos: function(files, car) { return uploadCarPhotos(files, car) },
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redpajama_set_name: RedPajamaGithub


text:
Tv Tropes Franklin Pierce
by Tommy April 12, 2019
MRC is a diversified global media company with operations in filmed entertainment, television programming and original digital content. The company is the industry's largest independent film, TV.
invocation of the trope of "mysteries," the idea that beneath the surface of city life lay. Franklin Pierce;65 Melville could have stayed a sailor, or a pin-setter in a. media such as film or television (where the audience is both more easily.
Paul Verhoeven's gloriously OTT erotic neo-noir caused a sensation on its release. Sharon Stone's bisexual novelist is a controversial femme fatale for the ages, while Michael Douglas is the troubled homicide cop tasked with investigating the murder-by-icepick she may have committed. A pulsating and memorably sleazy blockbuster, packed with sex and violence.
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While it is true that some (including the RAE, the "Spanish Royal Academy") maintain that the etymology of gringo is uncertain, to say that "no one has ever put forth a definite origin story for 'gringo' " is untrue and does grave disservice to the memory of the great Hispanic linguist and etymologist Joan Corominas (1905-1997). Corominas (or Coromines in his native Catalan) was.
A guide to some of the best neo-noirs, the most political and stylish genre of film and TV, from "Jackie Brown" to "Gone Girl."
avoids the standard genre tropes and feels more like a Gaijin-in-Japan action movie than yet another superhero blockbuster. If you're feeling fatigued by tights, suits and capes, this may be the.
Woodrow Wilson United Nations Woodrow Wilson summary: Woodrow Wilson was the 28th president of the United States of America. He was born in Virginia in 1856. He was the son of a Reverend and traveled quite a bit as a child with his family. He attended college at what is now Princeton University, studied law at the University of.
What We Saw: In the demo we saw, the mysterious Aiden Pierce went into an art gallery. His presence there was a trap — he was trying to get an equally mysterious (and apparently wealthy) man named.
Mar 5, 2018. I believe TV Tropes refers to that one as a "Meaningful Echo," but I could be wrong. Fitz asks folks to look for Doctor Franklin's original notes.
You can point out all the tropes (quirky townies, plots that revolve entirely around snacks, minorities in anonymous supporting roles) and sniff in disapproval. You can fall asleep in the middle of.
He has since repudiated any such suggestion, but the incident proved how easily even well-informed Americans can stray into Lost Cause tropes—and how even astute. As soon as Congress passed, and.
This character sheet is for X-Men.It's under construction, but trust us, if it's ever complete, it's going to be HUGE. For the 2008 animated series Wolverine and the X-Men versions of the characters, see the related character page. For the film incarnations of these characters, see X-Men Film Series.
He died suddenly while travelling with former president Franklin Pierce (a friendship. Gatsby), pastoralism survives as a key trope but is incapable of resolution. He has written many television and film screenplays, including adaptations of.
For The Record A Documentary History Of America Coinciding with the film's UK release, Flickering Myth's Tom Beasley caught up with director Jack Bryan to discuss his. The previous record for black Oscar winners was set in 2016, when five black artists won: supporting actors Viola Davis for "Fences" and Mahershala Ali for "Moonlight;" documentary feature. Why Was Benjamin Franklin Famous Benjamin Franklin
"The progressives are at it again, and for a small fee you can join the primary proponent of this apostate religious movement to get answers," Tony Pierce, a pastor of First. Christian leaders such.
In the small coastal community of Rhode Island's South County, economic and social shifts have disrupted the lives of the irascible seaman Dick Pierce and the women (wife. With video game tropes.
What Caused The Declaration Of Independence To Be Written What Do The Points On The Statue Of Liberty Represent Higher Education Budget History Usa Delaware's three largest higher education institutions. While the bond bill is not the largest capital budget in state history, it does include one of the biggest infusions of cash from the. Subscribe. If you like what you're reading online, why
Questions about Clinton's health just won't go away — partly because they fit so neatly with longstanding sexist cultural tropes about women's weakness. fainting is hardly a female condition.
Writers and Editors, linking writers and editors to resources (including each other), markets, clients, and fans; maintained by Pat McNees, writer, personal and organizational historian, journalist, editor.
<http://tvtropes.org/pmwiki/pmwiki.php/Main/ForeignQueasine> Accessed. 48 See, for instance, Ted J. Kaptchuk, and Franklin G. Miller, "Placebo Effects in Medicine," New England Journal of. Medicine. Though beams of light pierce the.
AvaxHome mirrors list, the real AVAXHOME is here, all other sites are impostors!
It's not far from reality – at the end of the eighteenth century a famous automaton called The Turk travelled the world and won a number of chess matches against the rich and famous, including.
Sep 17, 2018. But there are tropes in the story that extend back to a bevy of other classic. Like Joan Crawford in the glossy 1940s melodrama Mildred Pierce (a movie. diner employee in a timeless cameo by Aretha Franklin in The Blues Brothers. of Jennifer Aniston's Rachel (another waitress!) in TV's long-running.
and "Franklin Delano!". In the Naruto censored/TV version dub, due to censoring they referred to Lee while in a drunken state as. It is possible that Pierce managed to get this word past the censors simply by virtue of using so many other.
"When the hitchhiker in The Texas Chain Saw Massacre slits open his hand for the thrill, the onlookers recoil in horror– all but Franklin, who seems fascinated. The flip side of destabilizing the.
The return address was 40 University Dr, Rindge, NH 03461— Franklin Pierce University. Lo and behold (because we like to keep tabs on things and are sneaky like that) we have discovered a website for.
He slams GVF's overused psychedelic tropes mixed with watered down pseudo-sexual imagery and refers to them merely as "proficient" musicians — which, in context of this review, is a glaring dig. After.
Pierce's Foundations of Science Fiction. and the field from which it derives and transforms its tropes, paradigms, historical resonances—but she does not seem to recognize that a conferral of a new.
Apr 18, 2012. you love" is one of the most popular dramatic tropes in the 20th century. made any easier by punching President Franklin Pierce in the jaw.
Tayler, Jackson, Ankrom, and Schlichting do their best to stir in a charming self-awareness and humor to these tropes we've seen so many times but they face a Sisyphean task. On a more human level,
Antisemitic canards are unfounded rumors or false allegations which are defamatory towards Judaism as a religion, or defamatory towards Jews as an ethnic or religious group. They often form part of broader theories of Jewish conspiracies.According to defense attorney Kenneth Stern, "Historically, Jews have not fared well around conspiracy theories.Such ideas fuel anti-Semitism.
This is a building, first opened in 1913, and fully restored in 1998, that helped keep President Franklin Delano Roosevelt's polio. The first female-led Marvel movie subverts superhero tropes in.
Rather, it mixed American nativist tropes with the growing fears of immigration and Islamization. leadership has now virtually all died out, figures like William L. Pierce of the National Alliance.
The reference site TV Tropes named this trope after a BSG quote, "They. Look Like Us Now.. Allaire, Franklin. "The Only Good Walker is a. Joan Crawford , Mildred Pierce, and Mommie Dearest," The Velvet Light. Trap, vol. 29, 1992, pp.
Elizabeth Burgin American Revolution He gave us Tillie Burgin ("one drill sergeant in the Army of Compassion. rumors about "Arab-looking" suspects generated nationwide hysteria and attacks on Arab-American citizens–the authorities and. Davy Crockett Facts Born. 8/17/1786. Died. 3/6/1836. Spouse. Polly Finley Elizabeth Patton. Davy Crocket Articles. Explore articles from the History Net archives about Davy Crockett Dec 14, 2014.
Szwast). My New Year's resolution: No cheesy tropes in the column. On to the totally novel content format. Let's take a look at how last year's trends may influence ECAC Hockey's future. Brown is 1-5.
Jeffrey Tobias "Jeff" Winger (Joel McHale), is a sardonic, quick-witted ex-lawyer attending Greendale Community College. He was born in Denver, Colorado to his parents William Winger and Doreen Winger née Fitzgerald. His father divorced his mother when he was very young and later had another child named Willy Jr. who is Jeff's half-brother.
2 days ago. Omar's "use of anti-Semitic tropes and prejudicial accusations about Israel's. At Republican rallies, bands struck up Franklin D. Roosevelt's. Herbert Hoover , Franklin Pierce, William Henry Harrison and James Buchanan. the rise of Donald Trump from celebrity TV show host to POTUS has put the.
This page lists the many characters that appear in the Marvel Cinematic Universe across all mediums. For an index of where specific characters appear, refer to the bottom of this page. For an index of the actors and actresses who have their own page on this wiki, see here.
The cross-examineress for the defense went very easy on him, letting him waffle through a less-than-compelling testimony.
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Brown, the former Massachusetts senator, appeared at Franklin Pierce University for what was originally slated as a Senate faceoff, one of several election debates co-sponsored by the Marlin Fitzwater.
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In 2010, I wrote a paper titled "Nobody Puts Baby in a Corner: Moving Beyond Compulsory Heterosexuality in 21st Century Media," focusing on the relationship between two characters on Glee, Santana.
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For The Record A Documentary History Of America


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You are at:Home»Posts Tagged "Senior Executive Service"
Browsing: Senior Executive Service
New financial reporting requirements now in force for Senior Executive Service members
By on	 July 3, 2012 Workplace
Today marks a milestone for thousands of Senior Executive Service members, general military officers and other top government officials: The onset of new reporting requirements for most stock purchases and other securities transactions worth more than $1,000. Up to now, such transactions had to be reported only once a year on the public financial disclosure statement known as Office of Government Ethics Form 278. Now, senior officials must file "transaction reports" within 30 to 45 days, under the recently passed STOCK Act. The new mandate applies to almost all 28,000 Form 278 filers, according to the ethics office, which issued…
Senior executives organization warns against online financial disclosure
By on	 June 19, 2012 Agency Management
As a key deadline draws nearer, the Senior Executives Association continues to press its case against Internet posting of top career federal officials' financial disclosure reports. That recently enacted requirement is set to kick in by the end of August for some 28,000 Senior Executive Service members, political appointees, and military general and flag officers. In a six-page paper released today, SEA urges a delay in implementation "for careful consideration" of the potential impact, followed by action to exclude career feds if the case against the new mandate "is as one-sided as we believe." Among the association's arguments: Online access…
Rep. Moran: Too many feds protecting their comfort zones
By on	 February 29, 2012 Staffing
Rep. Jim Moran, D-Va., is usually numbered among the federal workforce's best friends on Capitol Hill. But this morning, he evidently decided it was time for a little tough love. When feds get a new assignment, they "hire a consultant," Moran told participants at a Partnership for Public Service event. "They don't take it on themselves." "We've got too many people, even in managerial positions, who are protecting their comfort zone," he continued. "I'm really discouraged because these are good people that can do far more than they are attempting to accomplish. They're worth more than they really give themselves…
Federal Times on the air
By on	 September 16, 2010 HR Management
In case you missed my appearance last night on Capital Insider to discuss the government's difficulties recruiting qualified Senior Executive Service candidates, here you go: [HTML1]
Obama to SES: We need you
By on	 September 14, 2010 Agency Management, Government reform, OMB
President Obama is urging the Senior Executive Service to embrace his accountability agenda, according to a memo released today. "As the most senior managers in the federal government, you know how essential the work you and your colleagues do is to the nation," Obama told more than 7,000 SES members. "You also are aware what happens when your best efforts are thwarted by outdated technologies and outmoded ways of doing business. "You understand the consequences of accepting billions of dollars in waste as the cost of doing business and of allowing obsolete or under-performing programs to continue year after year."…


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Below is our selection of nylon hose suitable for use in hydraulic steering set ups. We offer hose from Vetus, available in a wide variety of lengths, as well as hose from our own Seaflow range of approved hose. This hose connects to the helm pump at one end and the steering ram at the other end.


St. Etienne play host to Guingamp in what is set to be an engaging encounter. Everything points to a close tie and both teams settling for a point would not be a bad result for either to kick off the new season.
The home side enjoyed fantastic form in the second half of last season, thanks to the appointment of Coach Jean-Louis Gasset. In that period, Gasset refused to lose to the likes of Lyon, Marseille, and PSG and concluded the season with a 5-0 thumping of Lille. St. Etienne looks poised to continue from where they left and the lack of European action this term means they can focus on the league.
Guingamp finished their pre-season preparations without defeat, providing great momentum going into this fixture. Stand-offs against high-profile opponents in the later stages of last season also show that Guingamp can be stubborn on a good day.
The away side welcome PSG on matchday two, so collecting at least a point in this one is a must. Although Les Verts won this fixture last time out, we can expect Guingamp to resist a defeat this time around.
St. Etienne have lost just once in all their last eleven games (W8, D2).
Guingamp has never won away at St. Etienne in all previous eight attempts.


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Richard Thompson (Cascade, 7 juni 1985) is een sprinter uit Trinidad en Tobago. Hij is recordhouder van zijn land op de 100 m en mederecordhouder op de 4 x 100 m. Hij nam driemaal deel aan de Olympische Spelen en won hierbij in totaal twee medailles, aanvankelijk een zilveren en een bronzen. In 2015 werd de bronzen medaille van 2012 alsnog omgezet in een zilveren na de diskwalificatie van de Amerikaanse ploeg als gevolg van een geconstateerde dopingovertreding van een van zijn teamleden. Vervolgens gebeurde in 2017 iets vergelijkbaars met de in 2008 behaalde zilveren medaille. Die werd omgezet in een gouden, nadat de winnende ploeg van Jamaica was gediskwalificeerd om precies dezelfde reden als de Amerikaanse in 2015.

Loopbaan
Thompson verliet in 2004 zijn geboorteland en schreef zich in voor de Louisiana State University. In 2007 nam hij deel aan de wereldkampioenschappen in Osaka, maar werd hierbij in de kwartfinale van de 100 m uitgeschakeld met een tijd van 10,44 s.

In 2008 won hij bij de NCAA-kampioenschappen een gouden medaille op de 60 m (indoor) en 100 m (outdoor). Op de Olympische Spelen van 2008 in Peking veroverde hij een zilveren medaille op de 100 m. Met een tijd van 9,89 eindigde hij achter de Jamaicaan Usain Bolt, die het wereldrecord verbeterde tot 9,69 en voor de Amerikaan Walter Dix, die het brons won in een persoonlijk record van 9,91. Samen met zijn landgenoten Keston Bledman, Aaron Armstrong en Marc Burns nam hij vervolgens deel aan de 4 x 100 m estafette. Met een tijd van 38,26 kwalificeerden ze zich voor de finale. Armstrong werd hierbij vervangen door Emmanuel Callender en in de finale werden ze tweede in 38,06 achter het Jamaicaanse estafetteteam, dat met onder andere Usain Bolt in de gelederen de wedstrijd won in een wereldrecordtijd van 37,10. In 2017 werd Jamaica gediskwalificeerd vanwege een positieve test van Nesta Carter. Waardoor Burns en zijn teamgenoten negen jaar na afloop van de Spelen tot olympisch kampioen werden uitgeroepen.

Vier jaar later, op de Olympische Spelen van Londen in 2012, deed de estafetteploeg van Trinidad en Tobago op de 4 x 100 m een nieuwe poging om de Jamaicaanse concurrenten te kloppen. In dezelfde samenstelling als tijdens de olympische finale van Peking in 2008 kwam de Trinidadiaanse ploeg in de finale ditmaal tot 38,12. Daarmee moesten zij niet alleen Jamaica opnieuw voorlaten, dat ook nu weer in wereldrecordtijd, 36,84, naar het goud snelde, maar deze keer was ook de Amerikaanse ploeg hen in 37,04 te snel af.Enkele jaren later, in 2015, kreeg het viertal echter het zilver alsnog in de schoot geworpen: als gevolg van de geconstateerde overtreding van het dopingreglement door de Amerikaan Tyson Gay en de hieruit voortkomende schorsing plus diskwalificatie van al diens prestaties vanaf 15 juli 2012, had het IOC het Amerikaanse USOC in mei opgedragen om de zilveren medailles die de Amerikaanse ploeg met Tyson Gay in de gelederen tijdens de Spelen van Londen op de 4 x 100 m estafette had veroverd, terug te vorderen van de betreffende atleten. Dit betekende dat bronzenmedaillewinnaar Trinidad en Tobago opschoof naar de tweede plaats en alsnog met het zilver werd beloond.

Titels
Olympisch kampioen 4 x 100 m - 2008
Centraal Amerikaans en Caribisch kampioen 4 x 100 m - 2008
NCAA-kampioen 100 m - 2008
NCAA-indoorkampioen 60 m - 2008

Persoonlijke records
Outdoor

Indoor

Palmares

100 m
Kampioenschappen
 2007:  NACAC - 10,32 s
 2008:  OS - 9,89 s
 2009: 5e WK - 9,93 s
 2012: 7e OS - 9,98 s
 2013: 5e in ½ fin. WK - 10,19 s (in serie 10,14 s)

Golden League-podiumplekken
2008:  Weltklasse Zürich – 10,09 s
2009:  Meeting Areva – 10,04 s

Diamond League-podiumplekken
2010:  Bislett Games – 9,90 s
2010:  Adidas Grand Prix – 9,89 s
2010:  DN Galan – 10,10 s
2010:  London Grand Prix – 10,05 s
2014:  Bislett Games – 10,02 s
2014:  Meeting Areva – 10,08 s

4 x 100 m
2007:  NACAC - 39,92 s
2008:  Centraal Amerikaanse en Caribische kamp. - 38,54 s
2008:  OS - 38,06 s (na DQ Jamaica)
2009:  WK - 37,62 s (NR)
2011: 6e WK - 39,01 s
2012:  OS - 38,12 s (na DQ Verenigde Staten)(in serie 38,10 s)
2013: 7e WK - 38,57 s
2014:  IAAF World Relays - 38,04 s
2014:  Gemenebestspelen - 38,10 s
2014:  IAAF Continental Cup - 37,97 s
2015: 7e IAAF World Relays - 38,92 s
2016: DSQ OS (in serie: 37,96 s)

Atleet uit Trinidad en Tobago
Olympisch kampioen uit Trinidad en Tobago
Sprinter (atletiek)

redpajama_set_name: RedPajamaWikipedia


Welcome to our CaringBridge website. We are using it to keep family and friends updated in one place. Twenty-four hours ago Curt began his medical journey (well, I shouldn't say 24 hours, as it has been years in the making), and Curt's triple bypass surgery was a success. He currently remains in Cardiac ICU, but we are hopeful he will break out later today from all the bells and whistles that make him nervous. He has remained stable and even sat up in a chair for a bit last night. His pain is now well controlled and it no longer feels like his brother John is sitting on his chest. They harvested arteries out of his arm and leg and he does still complain of numbness in his arm, but this should pass. All in all, he has been a trooper during the first 24 hours and I expect nothing but continued improvement. Curt and I have been blessed with amazing children, parents, brothers, sisters, friends, and co-workers who without their support would have made this journey so much tougher. We appreciate your support and words of hope and encouragement. Thank you for visiting.
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text:
Design Architecture
Step Inside This Contemporary Masterpiece in the Desert
by NANCY ERDMANN
Architect Dale Gardon takes on a new challenge: designing a home of his own.
Scottsdale architect Dale Gardon, AIA, has a lot to be proud of. For almost two decades, he has been designing large-scale master-planned communities, commercial office buildings, private golf clubhouses and multi-million-dollar custom residences as the principal of his own namesake firm, Dale Gardon Design. But a few years back, when it came time for the renowned architect to come up with a plan for his own home, he had to take a step back and rethink his design aesthetic.
Having previously lived in a traditional-style home with his wife, Mary, and their two sons, he decided to go in a completely different direction. "This house was conceived to be a very comfortable contemporary. My wife and I dislike very austere contemporary, so I wanted to have clean modern lines, but maintain a warmth by selecting wood tones to contrast with crisp white walls," he says.
"This house was conceived to be a very comfortable contemporary. My wife and I dislike very austere contemporary, so I wanted to have clean modern lines, but maintain a warmth by selecting wood tones to contrast with crisp white walls."
The house is located in the exclusive DC Ranch community, just two lots over from their first home. Surrounded by the breathtaking tapestry of desert vegetation with sweeping mountain vistas, it's a great place to raise a family. "It's where we live, work and play," says Gardon, who was the lead land planner and architect for the community which was developed more than 20 years ago. "I can't imagine living anywhere else to afford such convenience combined with phenomenal natural and man-made beauty."
It took more than five years to plan, design and construct the residence, but when it was complete, an extraordinary modern masterpiece emerged. Constructed of integrally colored stucco, environmentally-friendly insulated panels, and aluminum siding that resembles wood, the sleek, light-filled residence looks perfectly at home in the desert.
Designed for plenty of indoor/outdoor interplay and a dynamic use of natural light, the house makes the most of what Gardon describes as a "left-over corner of the community." Most notably, he sited the residence to perfectly align with three of the valley's most distinctive mountains—Pinnacle Peak, Camelback Mountain and McDowell Mountain. The grounds, landscaped with a natural desert palette, blend into the neighborhood and adjacent natural areas.
Understandably, Gardon wanted something unique. "I really wanted the house to be distinctive. I wanted it to radiate with the warmth of wood, yet at the same time use aluminum cladding that is highly durable to our extreme heat," says the architect, who is LEED accredited. "The soft-gray stucco we chose is balanced by the crisp green accents of natural desert foliage."
"I really wanted the house to be distinctive. I wanted it to radiate with the warmth of wood, yet at the same time use aluminum cladding that is highly durable to our extreme heat."
What stands out most about the open-concept, two-story residence is that it was designed upside-down, with the main living spaces on top and the secondary spaces on the bottom. Awash in natural light from clerestory and wall-to-wall windows, the second floor is home to the kitchen, living room, master suite, and office. "This afforded us the most spectacular views of the entire valley on a daily basis," he points out. Three guest bedrooms, two baths, and a rec room make up the lower half.
Designing a home of one's own comes with its challenges. "The difficulty is that you want to expose yourself to all the possibilities of the new and intriguing products and materials that are out there to define what you are capable of within your own budget," he says. "You are forever redefining your home to incorporate all those newfound elements."
The kitchen, where all the major appliances are concealed behind wood cabinet doors, is a perfect example. "There is a window between the countertop and some upper cabinets, affording a view to the rear yard and summer sunsets," says Gardon. To enhance the area, black mirror was used on either side of the window as a backsplash to conceal appliance garages, and at the same time the mirror reflects the mountain backdrop from the opposite side of the room.
Designed in several hues, the kitchen cabinets integrate white, gray and truffle-brown-color wood with reconstituted Zebrano wood. Concealed under-cabinet lighting illuminates counter surfaces for task lighting, and offers a soft glow in the evening. And nearby, a cozy breakfast nook surrounded on three sides by glass has an atrium quality.
To keep the mood light and bright, the couple filled their interiors with soft, contemporary furnishings and finishes in a palette of white, gray, brown and blue. Flooring, too, is easy on the eye, with the use of white oak wood with a gray stain, natural concrete and porcelain tile. Sprinkled throughout are photographs taken by their sons during their travels, to remind them of the journeys the boys have taken.
Outdoor living is equally important to the Gardons, and there are plenty of options for enjoying it on their acre property. For entertaining, an upper terrace resembles an alfresco bistro with its stylish modern furnishings, centerpiece fire table, and stainless steel barbecue with a sleek dining table with room for eight.
Down below and off the game room is the pool terrace. Designed with a play shelf at one end where lounge chairs can be set into the water, and a sheer-descent water wall at the other, the pool is sited perfectly for capturing the mountain scenery. The water feature also acts as a privacy screen from street views into the yard. For playtime, poles can be set in the pool coping for setting up a volleyball net. At night, full-spectrum LED pool lights glow in colors ranging from white to blue to green.
A private dog run, an outdoor shower that Gardon says he uses daily except in the coldest of days, a shaded hammock and an artificial turf area for playing lawn games keep the fun going.
"Seeing the panoramic view every day is like living in a penthouse, yet with all the privacy and amenities of a beautiful master-planned community," enthuses Gardon. "You become so much more aware of your surroundings, and it is a daily treat to watch the sunrise and sunset from your own home."
Dale Gardon Design
Salcito Custom Homes
Bill Nassikas' Golden Touch Takes Hospitality To A New Level
Perfecting Elegant, Modern Minimalism House Design
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text:
\section{Introduction}

Motivated by the goal to obtain predictions that are as precise as possible
for quantities entering benchmark high-energy processes, there have been
significant recent developments of computational techniques and methodologies
in higher-order perturbative calculations for Quantum Chromodynamics (QCD)\@.
As a result,
computations of the basic renormalization group functions of QCD~%
\cite{%
Gross:1973id,%
Politzer:1973fx,%
Caswell:1974gg,%
Jones:1974mm,%
Egorian:1978zx,%
Tarrach:1980up,%
Tarasov:1980au,%
Tarasov:1982gk,%
Larin:1993tp,%
vanRitbergen:1997va,%
Chetyrkin:1997dh,%
Vermaseren:1997fq,%
Chetyrkin:2004mf,%
Czakon:2004bu%
}
have recently reached the five-loop level~%
\cite{%
Baikov:2014qja,%
Baikov:2016tgj,%
Luthe:2016ima,%
Luthe:2016xec,%
Herzog:2017ohr,%
Luthe:2017ttc,%
Baikov:2017ujl%
}.
In the modified minimal subtraction ({\ensuremath{\overline{\text{MS}}}}{}) scheme~%
\cite{tHooft:1973mfk,Bardeen:1978yd},
the standard renormalization scheme for perturbative QCD which is closely
related to dimensional regularization~%
\cite{Bollini:1972ui,tHooft:1972tcz},
these functions are obtained by computing single poles in corresponding
Green's functions.
Since only the poles are required, the above four- and five-loop results were
first obtained using the method of infrared rearrangement~%
\cite{%
Vladimirov:1979zm,%
Chetyrkin:1982nn,%
Chetyrkin:1984xa,%
Chetyrkin:2017ppe,%
Herzog:2017bjx,%
Chetyrkin:1996sr,%
Chetyrkin:1996ez,%
Misiak:1994zw,%
Chetyrkin:1997fm%
}
which simplifies computations without changing the ultraviolet singular
structure, but modifies the finite parts.

\newpage

The main aim of this work is to provide the self-energies and a set of
vertices with one vanishing external momentum for massless QCD at the
four-loop level.
The unrenormalized results are exact in terms of $\varepsilon =(D-4)/2$, where
$D$ is the space-time dimension, and four-loop master integrals~%
\cite{Baikov:2010hf,Lee:2011jt}.
The renormalized $D\!=\!4$ results are given in the {\ensuremath{\overline{\text{MS}}}}{} scheme.
The computation has been performed for a general gauge group and in
an arbitrary covariant linear gauge, by using the {\sc Forcer}{} program~%
\cite{Ueda:2016sxw,Ueda:2016yjm,FORCER}
for massless four-loop propagator-type integrals.
For the vertices, setting one of the momenta to zero effectively reduces
vertex integrals to propagator-type integrals. In QCD this does not create
IR divergences, which means the poles do not change.
At the three-loop level, similar computations were performed in ref.~%
\cite{Chetyrkin:2000dq}, but with an expansion in $\varepsilon$.
In addition, studies of QCD vertices in perturbation theory for various
configurations include refs.~%
\cite{%
Celmaster:1979km,%
Ball:1980ax,%
Pascual:1980yu,%
Braaten:1981dv,%
Davydychev:1996pb,%
Davydychev:2001uj,%
Binger:2006sj,%
Davydychev:1997vh,%
Davydychev:1998aw,%
Davydychev:2000rt,%
Chetyrkin:2000fd,%
Gracey:2011vw,%
Gracey:2014mpa%
}.

We compute all QCD vertices in a general linear covariant gauge, with the
exception of the four-gluon vertex for which there are at least three
difficulties:
first, two momenta have to be nullified before the diagrams become
propagator-like.
Second, the number of diagrams is large at four loops.
Third, the colour structure for a generic group is no longer an overall factor,
but will be term dependent. Additionally, the renormalization constant
is completely determined through the Slavnov-Taylor identities~%
\cite{Taylor:1971ff,Slavnov:1972fg}, which means the quartic gluon
vertex does not provide extra information in this context.

A direct application of our results is to compute conversion factors from
the {\ensuremath{\overline{\text{MS}}}}{} scheme to momentum subtraction schemes, see, e.g.,~refs.~%
\cite{Celmaster:1979km,Braaten:1981dv}, for renormalization group functions.
Unlike the {\ensuremath{\overline{\text{MS}}}}{} scheme, momentum subtraction schemes are defined in
a regularization-independent way.
In these schemes, the field renormalizations are performed such that finite
radiative corrections on propagators are absorbed as well as divergences and
hence they coincide with their tree-level values at the renormalization point.
Then one of (or an arbitrary linear combination of) the vertex functions
is normalized to its tree-level value and the other vertices are fixed via the
Slavnov-Taylor identities.
Common choices for the subtraction point of the vertex are a symmetric point
(referred as MOM{} schemes) and an asymmetric point where one of the momenta
is nullified, sometimes referred as {\ensuremath{\widetilde{\text{MOM}}}}{} schemes.
The latter choice corresponds to our result for the vertex functions.
Indeed, ref.~\cite{Chetyrkin:2000dq} derived four-loop beta functions in
four particular {\ensuremath{\widetilde{\text{MOM}}}}{} schemes from that in the {\ensuremath{\overline{\text{MS}}}}{} scheme
by computing conversion factors via finite parts of two- and three-point
functions in the {\ensuremath{\overline{\text{MS}}}}{} scheme.

As an example application, we provide the five-loop beta function in the
minimal momentum subtraction (MiniMOM{}) scheme introduced in ref.~%
\cite{vonSmekal:2009ae}, thus extending previous results
\cite{vonSmekal:2009ae,Gracey:2013sca} by one order in the coupling constant.
This scheme, see the preceeding references for a detailed discussion, is
more convenient than {\ensuremath{\overline{\text{MS}}}}{} for extending analyses of the strong coupling
constant and its scale dependence into the non-perturbative regime, e.g.,
via lattice QCD; for a recent analysis see ref.~\cite{Ayala:2017tco}.
In the perturbative regime the MiniMOM{} scheme provides an alternative to
{\ensuremath{\overline{\text{MS}}}}{} for studying the behaviour and truncation uncertainty of the
perturbation series for benchmark quantities such as the $R$-ratio in
$e^+e^-$ annihilation and the Higgs-boson decay to gluons, see refs.~%
\cite{Gracey:2014pba,Kataev:2015yha,Zeng:2015gha}.

The remainder of this article is organized as follows:
In section~\ref{sect:prelim} we specify our notations for the self-energies
and vertex functions and recall their renormalization.
In~section~\ref{sect:compt} we address technical details and checks of the
computation.
Due to their length, the results for the renormalized self-energies and vertex
functions for a generic gauge group and a general linear covariant gauge are
deferred to appendix \ref{sec:results}\@.
In section~\ref{sect:betaMM} the five-loop beta function is presented in the
MiniMOM{} scheme for QCD in the Landau gauge; the general result and the
corresponding {\ensuremath{\overline{\text{MS}}}}{}-to-MiniMOM{} conversion factor for the coupling
constant are provided in appendix \ref{sec:mmresults}\@.
In section~\ref{sect:conclusion} we summarize and give a brief outlook.

Our result can be obtained in a digital form as ancillary files of this article
on the preprint server \url{https://arXiv.org}.
They are also available from the authors upon request.
The files contain the bare results for the self-energies and vertices in terms
of master integrals with coefficients that are exact for any dimension $D$,
as well as the results in the {\ensuremath{\overline{\text{MS}}}}{} scheme for $D=4$. The notations in these
files can be found in appendix~\ref{sec:notationfile}\@.

\section{Preliminaries}
\label{sect:prelim}

We first summarize the notations for self-energies and vertex
functions with one vanishing momentum presented in this article.
In most cases we follow the conventions in ref.~\cite{Chetyrkin:2000dq}.%
\footnote{%
  We note that these conventions may be different from the ones commonly used
  in the literature. In fact, the Feynman rules in
  {\sc Forcer}{} are different as well, and hence we occasionally had to convert
  intermediate results from one convention to the other and back.
}

\subsection{Self energies}

The gluon, ghost and quark self-energies (figure~\ref{fig:self-energy})
are of the form
\begin{align}
\label{firstdef}
  \Pi^{ab}_{\mu\nu}(q)
    &\:=\: - \delta^{ab} (q^2 g_{\mu\nu} - q_\mu q_\nu) \Pi(q^2) , \\[1mm]
  \tilde{\Pi}^{ab}(q)
  &\:=\: \delta^{ab} q^2 \tilde{\Pi}(q^2) , \\[1mm]
  \Sigma^{ij}(q)
  &\:=\: \delta^{ij} \slashed{q} \Sigma_V(q^2) .
\end{align}
The colour indices are understood such that $a$ and $b$ are for the adjoint
representation of the gauge group, $i$ and $j$ for the representation to which
the quark belongs.
The form factors $\Pi(q^2)$, $\tilde{\Pi}(q^2)$ and $\Sigma_V(q^2)$ can
easily be extracted from contributions of the corresponding one-particle
irreducible diagrams by applying projection operators~\cite{Chetyrkin:2000dq}
(the same holds for the vertex functions discussed below).
They are related to the full gluon, ghost and quark propagators as follows:
\begin{align}
  D^{ab}_{\mu\nu}(q)
  &\:=\: \frac{\delta^{ab}}{-q^2} \biggl[
     \Bigl( - g_{\mu\nu} + \frac{q_\mu q_\nu}{q^2} \Bigr)\,\frac{1}{1+\Pi(q^2)}
     - \xi \,\frac{q_\mu q_\nu}{q^2}
     \biggr] , \\
  \Delta^{ab}(q)
  &\:=\: \frac{\delta^{ab}}{-q^2} \,\frac{1}{1+\tilde{\Pi}(q^2)} \:, \\
  S^{ij}(q)
  &\:=\: \frac{\delta^{ij}}{-q^2} \,\frac{\slashed{q}}{1+\Sigma_V(q^2)} \:.
\end{align}
Here the Landau gauge corresponds to $\xi=0$, and the Feynman gauge to $\xi=1$.
 We note that this convention differs from that in the widely used {\sc Form}{}
version \cite{Larin:1991fz} of the {\sc Mincer}{} program~\cite{Gorishnii:1989gt}
for three-loop self-energies, where the symbol {\tt xi} represents $1 - \xi$.

\begin{figure}
  \centering
  \setlength{\tabcolsep}{15pt}
  \begin{tabular}{ccc}
    \begin{axopicture}(106,46)(-53,-23)%
      \Gluon(-50,0)(-15,0){3}{4}%
      \Gluon(15,0)(50,0){3}{4}%
      \GCirc(0,0){15}{0.82}%
      \SetArrowInset{0}%
      \SetArrowAspect{1}%
      \SetColor{Blue}%
      \LongArrow(42,10)(30,10)%
      \Text(36,18){$q$}%
      \SetColor{Red}%
      \Text(-47,-12){$a,\mu$}%
      \Text(47,-12){$b,\nu$}%
    \end{axopicture}
    &
    \begin{axopicture}(106,46)(-53,-23)%
      \DashArrowLine(50,0)(15,0){3}%
      \DashArrowLine(-15,0)(-50,0){3}%
      \GCirc(0,0){15}{0.82}%
      \SetArrowInset{0}%
      \SetArrowAspect{1}%
      \SetColor{Blue}%
      \LongArrow(42,10)(30,10)%
      \Text(36,18){$q$}%
      \SetColor{Red}%
      \Text(-47,-12){$a$}%
      \Text(47,-12){$b$}%
    \end{axopicture}
    &
    \begin{axopicture}(106,46)(-53,-23)%
      \ArrowLine(50,0)(15,0)%
      \ArrowLine(-15,0)(-50,0)%
      \GCirc(0,0){15}{0.82}%
      \SetArrowInset{0}%
      \SetArrowAspect{1}%
      \SetColor{Blue}%
      \LongArrow(42,10)(30,10)%
      \Text(36,18){$q$}%
      \SetColor{Red}%
      \Text(-47,-12){$i$}%
      \Text(47,-12){$j$}%
    \end{axopicture}
    \\ (a) & (b) & (c)
  \end{tabular}
  \caption{%
    The gluon, ghost and quark self-energies $\Pi^{ab}_{\mu\nu}(q)$ (a),
    $\tilde{\Pi}^{ab}(q)$ (b) and $\Sigma^{ij}(q)$ (c).
  }
  \label{fig:self-energy}
\end{figure}

\subsection{Triple-gluon vertex}

\begin{figure}[b]
  \vspace{0.4cm
  \centering
  \begin{axopicture}(106,100)(-53,-40)%
    \Gluon(0,50)(0,15){3}{4}%
    \Gluon(-43,-25)(-13,-8){3}{4}%
    \Gluon(43,-25)(13,-8){3}{4}%
    \GCirc(0,0){15}{0.82}%
    \SetArrowInset{0}%
    \SetArrowAspect{1}%
    \SetColor{Blue}%
    \LongArrow(10,42)(10,30)%
    \LongArrow(-40.4,-12)(-30,-6)%
    \LongArrow(30.4,-31)(20,-25)%
    \Text(18,37){$q$}%
    \Text(-43,1){$-q$}%
    \Text(22,-38){$0$}%
    \SetColor{Red}%
    \Text(0,58){$a,\mu$}%
    \Text(-46,-34){$b,\nu$}%
    \Text(46,-34){$c,\rho$}%
  \end{axopicture}
  \caption{The triple-gluon vertex with one vanishing momentum,
           $\Gamma^{abc}_{\mu\nu\rho}(q,-q,0)$.}
  \label{fig:ggg}
\end{figure}

Without loss of generality, one can set the momentum of the third gluon to
zero, as depicted in figure~\ref{fig:ggg}.
Then the triple-gluon vertex can be written in the following form:
\begin{equation}
  \Gamma^{abc}_{\mu\nu\rho}(q,-q,0)
  \,=\,
  - i g f^{abc} \biggl[
    (2 g_{\mu\nu} q_\rho - g_{\mu\rho} q_\nu - g_{\rho\nu} q_\mu)\, T_1(q^2)
    - \Bigl( g_{\mu\nu} - \frac{q_\mu q_\nu}{q^2} \Bigr)\, q_\rho T_2(q^2)
  \biggr ] ,
  \label{eq:3g}
\end{equation}
where $g$ is the coupling constant and $f^{abc}$ are the structure constants of
the gauge group.
The first term in the square bracket corresponds to the tree-level vertex
while the second term arises from radiative corrections,
i.e., at the tree-level the form factors $T_{1,2}(q^2)$ read
\begin{equation}
  T_1(q^2) \bigl|_\text{tree} \,=\, 1 , \qquad
  T_2(q^2) \bigl|_\text{tree} \,=\, 0 .
\end{equation}

Because of Furry's theorem~\cite{Furry:1937zz} and the fact that we have no
  colour-neutral particles, symmetric invariants with an odd number of
  indices cannot occur for internal fermion lines. Neither can such
  invariants occur for the adjoint representation. Hence, if we project out a
  $d^{abc}$ structure, we would get a scalar invariant with an odd number of
  $f$ tensors, and such a combination must be zero. This has been checked
  explicitly to the equivalent of six-loop vertices in
  ref~\cite{vanRitbergen:1998pn}.
Due to the bosonic property of gluons, the totally antisymmetric colour factor
$f^{abc}$ leads to antisymmetric Lorentz structure as in eq.~\eqref{eq:3g}.
  One could consider another Lorentz structure,
  \begin{equation}
    - i g f^{abc} q_\mu q_\nu q_\rho T_3(q^2) .
    \label{eq:T3}
  \end{equation}
  However, a Slavnov-Taylor identity requires $T_3(q^2)$ to vanish.

\subsection{Ghost-gluon vertex}

\begin{figure}[t]
  \centering
  \setlength{\tabcolsep}{20pt}
  \begin{tabular}{cc}
    \begin{axopicture}(106,100)(-53,-40)%
      \Gluon(0,50)(0,15){3}{4}%
      \DashArrowLine(-13,-8)(-43,-25){3}%
      \DashArrowLine(43,-25)(13,-8){3}%
      \GCirc(0,0){15}{0.82}%
      \SetArrowInset{0}%
      \SetArrowAspect{1}%
      \SetColor{Blue}%
      \LongArrow(10,42)(10,30)%
      \LongArrow(-40.4,-12)(-30,-6)%
      \LongArrow(40.4,-12)(30,-6)%
      \Text(18,37){$q$}%
      \Text(-43,1){$-q$}%
      \Text(43,1){$0$}%
      \SetColor{Red}%
      \Text(0,58){$c,\mu$}%
      \Text(-46,-34){$a$}%
      \Text(46,-34){$b$}%
    \end{axopicture}
    &
    \begin{axopicture}(106,100)(-53,-40)%
      \Gluon(0,50)(0,15){3}{4}%
      \DashArrowLine(-13,-8)(-43,-25){3}%
      \DashArrowLine(43,-25)(13,-8){3}%
      \GCirc(0,0){15}{0.82}%
      \SetArrowInset{0}%
      \SetArrowAspect{1}%
      \SetColor{Blue}%
      \LongArrow(10,42)(10,30)%
      \LongArrow(-40.4,-12)(-30,-6)%
      \LongArrow(40.4,-12)(30,-6)%
      \Text(18,37){$0$}%
      \Text(-43,1){$-q$}%
      \Text(43,1){$q$}%
      \SetColor{Red}%
      \Text(0,58){$c,\mu$}%
      \Text(-46,-34){$a$}%
      \Text(46,-34){$b$}%
    \end{axopicture}
    \\ (a) & (b)
  \end{tabular}
  \caption{%
    The ghost-gluon vertex:
    (a) $\tilde{\Gamma}^{abc}_\mu(-q,0;q)$ with the vanishing incoming ghost
    momentum and (b) $\tilde{\Gamma}^{abc}_\mu(-q,q;0)$ with the vanishing gluon
    momentum.
  }
  \label{fig:hhg}
\end{figure}

Since the tree-level vertex is proportional to the outgoing ghost momentum,
nullifying this momentum gives identically zero in perturbation theory.
Therefore, we only have two possibilities to set one of the external momenta to zero.
One is the incoming ghost momentum and the other is the gluon momentum
(figure~\ref{fig:hhg}):
\begin{align}
  \tilde{\Gamma}_\mu^{abc}(-q,0;q)
    &\:=\: - i g f^{abc} q_\mu \tilde{\Gamma}_h(q^2) , \\[1mm]
  \tilde{\Gamma}_\mu^{abc}(-q,q;0)
    &\:=\: - i g f^{abc} q_\mu \tilde{\Gamma}_g(q^2) .
\end{align}
The subscript `$h$' of $\tilde{\Gamma}_h(q^2)$ indicates the function with
vanishing incoming ghost momentum, whereas `$g$' of $\tilde{\Gamma}_g(q^2)$
denotes the vanishing gluon momentum.
These functions are equal to one at the tree-level,
\begin{equation}
  \tilde{\Gamma}_h(q^2) \bigl|_\text{tree} \,=\,
  \tilde{\Gamma}_g(q^2) \bigl|_\text{tree} \,=\, 1.
\end{equation}

\subsection{Quark-gluon vertex}

\begin{figure}[b]
  \vspace{0.4cm
  \centering
  \setlength{\tabcolsep}{20pt}
  \begin{tabular}{cc}
    \begin{axopicture}(106,100)(-53,-40)%
      \Gluon(0,50)(0,15){3}{4}%
      \ArrowLine(-13,-8)(-43,-25)%
      \ArrowLine(43,-25)(13,-8)%
      \GCirc(0,0){15}{0.82}%
      \SetArrowInset{0}%
      \SetArrowAspect{1}%
      \SetColor{Blue}%
      \LongArrow(10,42)(10,30)%
      \LongArrow(-40.4,-12)(-30,-6)%
      \LongArrow(40.4,-12)(30,-6)%
      \Text(18,37){$q$}%
      \Text(-43,1){$-q$}%
      \Text(43,1){$0$}%
      \SetColor{Red}%
      \Text(0,58){$a,\mu$}%
      \Text(-46,-34){$a$}%
      \Text(46,-34){$b$}%
    \end{axopicture}
    &
    \begin{axopicture}(106,100)(-53,-40)%
      \Gluon(0,50)(0,15){3}{4}%
      \ArrowLine(-13,-8)(-43,-25)%
      \ArrowLine(43,-25)(13,-8)%
      \GCirc(0,0){15}{0.82}%
      \SetArrowInset{0}%
      \SetArrowAspect{1}%
      \SetColor{Blue}%
      \LongArrow(10,42)(10,30)%
      \LongArrow(-40.4,-12)(-30,-6)%
      \LongArrow(40.4,-12)(30,-6)%
      \Text(18,37){$0$}%
      \Text(-43,1){$-q$}%
      \Text(43,1){$q$}%
      \SetColor{Red}%
      \Text(0,58){$a,\mu$}%
      \Text(-46,-34){$i$}%
      \Text(46,-34){$j$}%
    \end{axopicture}
    \\ (a) & (b)
  \end{tabular}
  \caption{%
    The quark-gluon vertex:
    (a) $\Lambda^a_{\mu,ij}(-q,0;q)$ with the vanishing incoming quark momentum
    and (b) $\Lambda^a_{\mu,ij}(-q,q;0)$ with the vanishing gluon momentum.
  }
  \label{fig:qqg}
\end{figure}

We consider the case of a vanishing incoming quark momentum and
the case of a vanishing gluon momentum (figure~\ref{fig:qqg}).
Nullifying the outgoing quark momentum gives the same result as nullifying
the incoming quark momentum.
Then the vertex can be written as
\begin{align}
  \Lambda^a_{\mu,ij}(-q,0;q)
    &\:=\: g T^a_{ij} \Bigl[
      \gamma_\mu \Lambda_q(q^2)
      + \gamma^\nu \biggl( g_{\mu\nu} - \frac{q_\mu q_\nu}{q^2} \biggr)
        \Lambda_q^T(q^2)
    \Bigr] , \\
  \Lambda^a_{\mu,ij}(-q,q;0)
    &\:=\: g T^a_{ij} \Bigl[
      \gamma_\mu \Lambda_g(q^2)
      + \gamma^\nu \biggl( g_{\mu\nu} - \frac{q_\mu q_\nu}{q^2} \biggr)
        \Lambda_g^T(q^2)
    \Bigr] .
\end{align}
$T_{ij}$ is the generator of the representation for the quark.
The subscript `$q$' indicates the functions with vanishing incoming quark
momentum and `$g$' indicates those with vanishing gluon momentum.
At the tree-level we have
\begin{align}
  \Lambda_q(q^2) \bigl|_\text{tree} =
  \Lambda_g(q^2) \bigl|_\text{tree} &\,=\, 1 , \\[1mm]
  \Lambda_q^T(q^2) \bigl|_\text{tree} =
  \Lambda_g^T(q^2) \bigl|_\text{tree} &\,=\, 0 .
\label{lastdef}
\end{align}

\subsection{Renormalization}

In a generic renormalization scheme $R$, the respective renormalizations of
the gluon, ghost and quark fields can be written as
\begin{align}
  (A^B)^a_\mu &\:=\: \sqrt{Z_3^R} \: (A^R)^a_\mu , \\
  (\eta^B)^a  &\:=\: \sqrt{\tilde{Z}_3^R} \: (\eta^R)^a , \\
  \psi^B_{if} &\:=\: \sqrt{Z_2^R} \: \psi^R_{if} .
\end{align}
The superscript `$B$' indicates a bare quantity and `$R$' a renormalized
one.
For the coupling constant,
we define $a = \alpha_{\rm s} / (4\pi) = g^2 / (16\pi^2)$.
Then $a$ and the gauge parameter $\xi$ are renormalized in dimensional
regularization ($D=4-2\varepsilon$) as follows:
\begin{align}
  a^B   &\:=\: \mu^{2\varepsilon} Z_a^R a^R , \\[1mm]
  \xi^B &\:=\: Z_3^R \xi^R .
\end{align}
Here $\mu$ is the 't~Hooft mass scale.
We have used the fact that the gauge parameter is also renormalized by
the gluon field renormalization constant, $Z_\xi^R = Z_3^R$.
The renormalization of the self-energies and vertex functions is performed as
\begin{align}
  1 + \Pi^R &\:=\: Z_3^R (1 + \Pi^B) , \\[1mm]
  1 + \tilde{\Pi}^R &\:=\: \tilde{Z}_3^R (1 + \tilde{\Pi}^B) , \\[1mm]
  1 + \Sigma_V^R &\:=\: Z_2^R (1 + \Sigma_V^B) ,
\end{align}
and
\begin{alignat}{3}
  T_i^R &\:=\: Z_1^R T_i^B ,
    &&
    &\qquad i &= 1, 2, \\
  \tilde{\Gamma}_i^R &= \tilde{Z}_1^R \tilde{\Gamma}_i^B ,
    &&
    &\qquad i &= h, g, \\
  \Lambda_i^R &\:=\: \bar{Z}_1^R \Lambda_i^B , &\qquad
  \Lambda_i^{T,R} &\:=\: \bar{Z}_1^R \Lambda_i^{T,B} ,
    &\qquad i &= q, g ,
\end{alignat}
where the vertex renormalization constants are related to the field and coupling
renormalization constants via the Slavnov-Taylor identities by
\begin{equation}
  \sqrt{Z_a^R Z_3^R}
    \,=\, \frac{Z_1^R}{Z_3^R}
    \,=\, \frac{\tilde{Z}_1^R}{\tilde{Z}_3^R}
    \,=\, \frac{\bar{Z}_1^R}{Z_2^R} \: .
\end{equation}

In {\ensuremath{\text{MS}}}{}-like schemes, the renormalization constants contain only pole terms
with respect to $\varepsilon$ and thus take the form
\begin{equation}
  Z_i^{\ensuremath{\text{MS}}}
    \:=\: 1
    + \sum_{l=1}^\infty a^l \sum_{n=1}^l \frac{Z_i^{{\ensuremath{\text{MS}}},(l,n)}}{\varepsilon^n} \: .
\end{equation}
The coefficients $Z_i^{{\ensuremath{\text{MS}}},(l,n)}$ are determined order by order in such a way
that any renormalized Green's function becomes finite.
The choice of the subtraction point defines the specific ({\ensuremath{\text{MS}}}{}-like)
renormalization scheme; the beta function is independent of this choice.

\section{Computations and checks}
\label{sect:compt}

The results in this article are obtained by direct computation using the
{\sc Forcer}{} package~\cite{Ueda:2016sxw,Ueda:2016yjm,FORCER}, written in {\sc Form}{}~%
\cite{Vermaseren:2000nd,Tentyukov:2007mu,Kuipers:2012rf}. {\sc Forcer}{} is a
four-loop equivalent of {\sc Mincer}{}~\cite{Gorishnii:1989gt,Larin:1991fz}:
for each topology a parametric reduction to simpler topologies is provided.
Most topologies can be reduced using direct integration of one-loop insertions,
by the triangle rule~\cite{Chetyrkin:1981qh,Tkachov:1981wb} or by the diamond rule~%
\cite{Ruijl:2015aca}. If these substructures are absent, a manually designed
reduction scheme is required.
Through parametric integration-by-parts (IBP) identities~\cite%
{Chetyrkin:1981qh,Tkachov:1981wb},
similar to the triangle rule, such reductions are found in a computer-assisted
manner.

Great care has to be taken to prevent term blow-up from rewriting momenta to a
new basis for the transitions between topologies. In {\sc Mincer}{} this was done
manually, but in {\sc Forcer}{} a basis is found that minimizes the terms that are
created automatically.
The performance of the {\sc Forcer}{} program has been demonstrated by computing the
four-loop beta function in ten minutes in the Feynman gauge and in eight hours
with all powers of the gauge parameter on a modern 24-core machine~%
\cite{Ueda:2016sxw,Ueda:2016yjm}.

The diagrams have been generated using Qgraf~\cite{QGRAF}.
Subsequently, we filter propagator insertions. These insertions
are known lower-loop propagators, and can be factorized out of the diagram,
see ref.~\cite{Herzog:2016qas}.
Next, the topology is mapped to a built-in {\sc Forcer}{} topology, after nullifying
a leg for the vertices, and the colour factor is determined using the {\sc Form}{}
program of ref.~\cite{vanRitbergen:1998pn}.
To extract the form factors defined above, a generalization of the projection
operators in ref.~\cite{Chetyrkin:2000dq} to a generic gauge group is used.
Then the Feynman rules are applied. The remaining Lorentz-scalar integrals
(which include loop-momenta in numerators) are computed by the {\sc Forcer}{} program.

The computation time varied between an hour and a week, on a single computer.
The easy cases, such as the ghost propagator and quark propagator took an hour.
The gluon propagators and ghost-ghost-gluon vertex and quark-quark-gluon vertex
took about eight hours per configuration. The triple gluon vertex was the
hardest case and took a week per configuration on a single machine with 24
cores. Had we chosen to compute with an expansion in $\varepsilon$, the computations
would have been much faster.

\pagebreak

\noindent
We have checked our setup and results in various ways:
\begin{itemize}
  \item The longitudinal component of the gluon self-energy
    $\delta^{ab} q_\mu q_\nu \Pi_L(q^2)$ was shown to be zero by an explicit
    calculation at the four-loop level.
  \item The form factor $T_3(q^2)$ of the triple-gluon vertex in
    eq.~\eqref{eq:T3} was computed and indeed vanished at the four-loop level.
  \item All the self-energies and vertex functions computed in this work were
    compared up to three loops with those in ref.~\cite{Chetyrkin:2000dq}.
    Note that the finite parts of the vertex-function results in
    ref.~\cite{Chetyrkin:2000dq} are only correct for SU($N$) gauge groups,
    since the presence of quartic Casimir operators was not taken into account
    in the reconstruction of the general case. This fact was also noted in
    ref.~\cite{Luthe:2017ttc}.
  \item The four-loop renormalization constants and anomalous dimensions for
    the case of SU($N$) and a general linear covariant gauge were provided in
    ancillary files \cite{Kostja:2004anc} of ref.~\cite{Chetyrkin:2004mf}.
    Directly after {\sc Forcer}{} was completed, we established agreement with
    those results. For a generic group our results are in agreement with
    ancillary files of ref.~\cite{Luthe:2017ttc}.
  \item We remark that the ghost-gluon vertex is unrenormalized
    $\tilde{Z}_1^{\ensuremath{\overline{\text{MS}}}} = \tilde{Z}_3^{\ensuremath{\overline{\text{MS}}}} \sqrt{Z_a^{\ensuremath{\overline{\text{MS}}}} Z_3^{\ensuremath{\overline{\text{MS}}}}} = 1$
    in the Landau gauge.
    Moreover, our results confirm that the vertex has no radiative corrections
    when the incoming ghost momentum is nullified
    (i.e., $\tilde{\Gamma}_h^{\ensuremath{\overline{\text{MS}}}} = 1$) in the Landau gauge up to four loops.
\end{itemize}

The renormalized results for all self-energies and vertex functions are rather
lengthy and are thus given in appendix~\ref{sec:results}\@.
The unrenormalized results in terms of master integrals, and the values of the
master integrals in $4-2 \varepsilon$ dimensions are available online on
\texttt{arxiv.org}, as ancillary files to the paper.
A description of the files is given in appendix~\ref{sec:notationfile}\@.

\section{Five-loop Landau-gauge QCD beta function in the MiniMOM{} scheme}
\label{sect:betaMM}

In the MiniMOM{} scheme~\cite{vonSmekal:2009ae},
the self-energies are completely absorbed into the field renormalization
constants at the subtraction point $q^2 = - \mu^2$:
\begin{align}
  1 + \Pi^\text{MM}(-\mu^2)
    &\:=\: Z_3^\text{MM} \Bigl[ 1 + \Pi^B(-\mu^2) \Bigr] \:=\: 1 , \\
  1 + \tilde{\Pi}^\text{MM}(-\mu^2)
    &\:=\: \tilde{Z}_3^\text{MM} \Bigl[ 1 + \tilde{\Pi}^B(-\mu^2) \Bigr] \:=\: 1 , \\
  1 + \Sigma_V^\text{MM}(-\mu^2)
    &\:=\: Z_2^\text{MM} \Bigl[ 1 + \Sigma_V^B(-\mu^2) \Bigr] \:=\: 1 .
\end{align}
Here the superscript `\text{MM}{}' indicates a quantity in the MiniMOM{} scheme.
In addition, motivated by the non-renormalization of the ghost-gluon
vertex in the Landau gauge~\cite{Taylor:1971ff}, the vertex renormalization
constant for this vertex is chosen the same as that in {\ensuremath{\overline{\text{MS}}}}{},
\begin{equation}
  \tilde{Z}_1^\text{MM} \:=\: \tilde{Z}_1^{\ensuremath{\overline{\text{MS}}}} ,
\end{equation}
which is equal to one in the Landau gauge.

The above renormalization conditions lead to the following relations for
the coupling constant and gauge parameter in the two schemes:
\begin{align}
  a^\text{MM}(\mu^2) &\:=\: a^{\ensuremath{\overline{\text{MS}}}}(\mu^2)
  \:\frac{1}{\bigl[1+\Pi^{\ensuremath{\overline{\text{MS}}}}(-\mu^2)\bigr]
           \bigl[1+\tilde{\Pi}^{\ensuremath{\overline{\text{MS}}}}(-\mu^2)\bigr]^2} \: , \label{aMM} \\[1mm]
  \xi^{\ensuremath{\overline{\text{MS}}}}(\mu^2) &\:=\: \xi^\text{MM}(\mu^2) \:\frac{1}{1+\Pi^{\ensuremath{\overline{\text{MS}}}}(-\mu^2)} \: .
  \label{xiMM}
\end{align}
Eq.~\eqref{aMM} allows one to convert a value of $\alpha_s^{\ensuremath{\overline{\text{MS}}}}$ to
$\alpha_s^\text{MM}$.  For example, $\alpha_s^{\ensuremath{\overline{\text{MS}}}}(M_Z^2) = 0.118$ leads to
$\alpha_s^\text{MM}(M_Z^2) = 1.096 \, \alpha_s^{\ensuremath{\overline{\text{MS}}}}(M_Z^2)$ for QCD in the Landau
gauge with $n_f=5$ quark flavours.  The general expansion of eq.~\eqref{aMM}
is given in appendix~\ref{sec:mmresults}.

The scale dependence of the coupling constant in eq.~\eqref{aMM} in this scheme
is given by
\begin{equation}
  \beta^\text{MM}
  \:=\: \mu^2 \frac{d a^\text{MM}}{d \mu^2}
  \:=\: \frac{\partial a^\text{MM}}{\partial a^{\ensuremath{\overline{\text{MS}}}}} \beta^{\ensuremath{\overline{\text{MS}}}}
  \:+\: \frac{\partial a^\text{MM}}{\partial \xi^{\ensuremath{\overline{\text{MS}}}}} \gamma_3^{\ensuremath{\overline{\text{MS}}}} \xi^{\ensuremath{\overline{\text{MS}}}} ,
  \label{betaMM}
\end{equation}
where we have used the beta function and gluon field anomalous dimension
in {\ensuremath{\overline{\text{MS}}}}{},
\begin{align}
  \beta^{\ensuremath{\overline{\text{MS}}}} &\:=\: \mu^2 \frac{d a^{\ensuremath{\overline{\text{MS}}}}}{d \mu^2} \: , \\[1mm]
  \gamma_3^{\ensuremath{\overline{\text{MS}}}} \xi^{\ensuremath{\overline{\text{MS}}}} &\:=\: \mu^2 \frac{d \xi^{\ensuremath{\overline{\text{MS}}}}}{d \mu^2} \: .
\end{align}
Note that the right-hand side of eq.~\eqref{aMM}, and hence that of
eq.~\eqref{betaMM}, is naturally given in terms of $a^{\ensuremath{\overline{\text{MS}}}}$ and $\xi^{\ensuremath{\overline{\text{MS}}}}$.
One has to convert them into $a^\text{MM}$ and $\xi^\text{MM}$ by inverting the series
of eq.~\eqref{aMM} and by using eq.~\eqref{xiMM}.%
\footnote{%
  In ref.~\cite{vonSmekal:2009ae}, the results are presented in $\xi^{\ensuremath{\overline{\text{MS}}}}{}$
  instead of $\xi^\text{MM}$.
  On the other hand, in ref.~\cite{Gracey:2013sca} the conversion
  from $\xi^{\ensuremath{\overline{\text{MS}}}}$ to $\xi^\text{MM}$ was performed.
  The results become the same in the Landau gauge $\xi^{\ensuremath{\overline{\text{MS}}}} = \xi^\text{MM} = 0$.
  The same is true for the ``{\ensuremath{\widetilde{\text{MOM}}}}{}h'' scheme of
  ref.~\cite{Chetyrkin:2000dq}.
}

Having results for the four-loop self-energies in the {\ensuremath{\overline{\text{MS}}}}{} scheme at hand,
one can obtain the five-loop beta function in the MiniMOM{} scheme
from the five-loop beta function \cite{Baikov:2016tgj,Herzog:2017ohr} and the
four-loop gluon field anomalous dimension in the {\ensuremath{\overline{\text{MS}}}}{} scheme.
The result for SU(3) in the Landau gauge ($\xi^\text{MM}=0$) reads
\begin{equation}
  \beta^{\,\text{MM}}
    \:=\: - \sum_{l=0}^4 \bigl(a^\text{MM}\bigr)^{l+2} \beta_l^{\,\text{MM}}
    \:+\: \mathcal{O}\Bigl(\bigl(a^\text{MM}\bigr)^7\Bigr) ,
\end{equation}
with
{
  \renewcommand{\colorfactor}[1]{#1}
  \def#1{\beta_{#1}^{\,\text{MM}}}
  \def#1{form2tex/betaLandauQCDMM#1.inc}
  \def\exprlabel#1{betaMMcoeffs#1}

  \label{betaMMcoeffs}
}
The result for a generic group and in an arbitrary covariant linear gauge can
be found in appendix~\ref{sec:mmresults}; it agrees with the result given in
ref.~\cite{Gracey:2013sca} up to four loops.
As is well known, the first coefficient $\beta_0^{\,\text{MM}}$ is scheme independent.
The next coefficient $\beta_1^{\,\text{MM}}$ has a gauge dependence and the universal
value is obtained only in the Landau gauge.
The last coefficient $\beta_4^{\,\text{MM}}$ is the new result.
In the {\ensuremath{\overline{\text{MS}}}}{} scheme, some of higher values of the zeta function (e.g.,
$\zeta_3^2$, $\zeta_6$ and $\zeta_7$ at five loops) do not occur, for a
discussion of this issue see refs.~%
\cite{Baikov:2010hf,Baikov:2012zm,Baikov:2017ujl}.
In contrast, one cannot expect their absence in the MiniMOM{} scheme.
Indeed eq.~\eqref{res:betaMMcoeffs} includes terms with $\zeta_3^2$ and
$\zeta_7$, and for $\xi^\text{MM} \neq 0$ also $\zeta_6$ occurs.

The numerical values of the above beta function for three to five quark
flavours are
\begin{equation}
\label{betaMMnum}
  \begin{alignedat}{2}
    \tilde{\beta}^\text{MM}(n_f=3) \:=\:
       1
       &+ 0.5658842421  \alpha_{\rm s}^\text{MM}
      &&+ 0.9419859046 (\alpha_{\rm s}^\text{MM})^2 \\
       &+ 2.304494526  (\alpha_{\rm s}^\text{MM})^3
      &&+ 6.647485913  (\alpha_{\rm s}^\text{MM})^4 , \\
    \tilde{\beta}^\text{MM}(n_f=4) \:=\:
       1
       &+ 0.4901972247  \alpha_{\rm s}^\text{MM}
      &&+ 0.6452147391 (\alpha_{\rm s}^\text{MM})^2 \\
       &+ 1.638457168  (\alpha_{\rm s}^\text{MM})^3
      &&+ 3.466865543  (\alpha_{\rm s}^\text{MM})^4 , \\
    \tilde{\beta}^\text{MM}(n_f=5) \:=\:
       1
       &+ 0.4013472477  \alpha_{\rm s}^\text{MM}
      &&+ 0.3288519562 (\alpha_{\rm s}^\text{MM})^2 \\
       &+ 1.026892491  (\alpha_{\rm s}^\text{MM})^3
      &&+ 0.8417657296 (\alpha_{\rm s}^\text{MM})^4 ,
  \end{alignedat}
\end{equation}
where $\tilde{\beta} \equiv \beta(a)/(-\beta_0 a^2)$ has been re-expanded
in powers of $\alpha_{\rm s} = 4 \pi a$.
These values may be compared with those in the {\ensuremath{\overline{\text{MS}}}}{} scheme
\cite{Baikov:2016tgj,Herzog:2017ohr} reading
\begin{equation}
\label{betaMSnum}
  \begin{alignedat}{2}
    \tilde{\beta}^{\ensuremath{\overline{\text{MS}}}}(n_f=3) \:=\:
       1
       &+ 0.5658842421  \alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}}
      &&+ 0.4530135791 (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^2 \\
       &+ 0.6769674420 (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^3
      &&+ 0.5809276379 (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^4 , \\
    \tilde{\beta}^{\ensuremath{\overline{\text{MS}}}}(n_f=4) \:=\:
       1
       &+ 0.4901972247  \alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}}
      &&+ 0.3087903795 (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^2 \\
       &+ 0.4859007965 (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^3
      &&+ 0.2806008338 (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^4 , \\
    \tilde{\beta}^{\ensuremath{\overline{\text{MS}}}}(n_f=5) \:=\:
       1
       &+ 0.4013472477   \alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}}
      &&+ 0.1494273313  (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^2 \\
       &+ 0.3172233974  (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^3
      &&+ 0.08092104151 (\alpha_{\rm s}^{\ensuremath{\overline{\text{MS}}}})^4 .
  \end{alignedat}
\end{equation}
Obviously, the MiniMOM{} coefficients in eqs.~(\ref{betaMMnum})
are (much) larger than their {\ensuremath{\overline{\text{MS}}}}{} counterparts in
eqs.~(\ref{betaMSnum}) starting from the
second order; moreover, they exhibit a definite growth with the
order that is absent in the {\ensuremath{\overline{\text{MS}}}}{} case.
One may expect that this behaviour, and the larger value of
$\alpha_{\rm s}^{\,\rm MM}$, is more than compensated by smaller expansion
coefficients for observables, leading to a better overall convergence in
MOM{}-like schemes.
However, this issue has been studied up to four loops in some detail for the
$R$-ratio in electron-positron annihilation, without arriving at such a
clear-cut conclusion~\cite{Gracey:2014pba}.

\section{Summary and outlook}
\label{sect:conclusion}

We have computed the four-loop three-particle vertices with one vanishing
momentum and the four-loop self-energies for QCD-like theories in a manner
that is as general as presently possible.
Our results, explicitly presented in the appendix of this article for the
$D\!=\!4$ renormalized quantities in the {\ensuremath{\overline{\text{MS}}}}{} scheme, and available
online in terms of their more lengthy bare counterparts, should be useful
for precision studies of QCD-like theories
  with any simple compact gauge group,
  in any linear covariant gauge,
  for any MS-like or {\ensuremath{\widetilde{\text{MOM}}}}{}-like renormalization scheme
    (see also ref.~\cite{Chetyrkin:2000dq}),
  and in any number of space-time dimensions $D$.
The latter requires replacing the master integrals, which is the only
component that is not known exactly in~$D$.

As an example application, we have determined the five-loop beta function in
the \mbox{MiniMOM{}} scheme of ref.~\cite{vonSmekal:2009ae}, i.e., we have
extended the result of refs.~\cite{vonSmekal:2009ae,Gracey:2013sca} by one
order in the coupling constant $\alpha_{\rm s}$.
This function appears to have a higher-order structure quite different from
that in the {\ensuremath{\overline{\text{MS}}}}{} scheme, thus inviting further studies especially
for the physical case of QCD in four dimensions.

Our computations have been made possible by the construction of {\sc Forcer}{}
\cite{Ueda:2016sxw,Ueda:2016yjm,FORCER}, a four-loop generalization of the
well-known {\sc Mincer}{} program \cite{Gorishnii:1989gt,Larin:1991fz} for the
parametric reduction of three-loop massless self-energy integrals.
We have verified that, except for an issue in the transition from SU($N$) to a
general gauge group that was also noted in ref.~\cite{Luthe:2017ttc}, our
renormalized self-energies and vertices up to three loops agree with the
results of ref.~\cite{Chetyrkin:2000dq}.
Furthermore, we have verified to four loops that the ghost-ghost-gluon vertex
is unrenormalized in the Landau gauge, i.e., its anomalous dimension is
proportional to $\xi$ and that the gluon propagator is transverse.
Earlier, we had checked the four-loop beta function and the four-loop
SU($N$)~\cite{Kostja:2004anc} renormalization constants and anomalous dimensions
of ref.~\cite{Chetyrkin:2004mf}.

Performing a similar calculation at five loops is a formidable challenge:
so far most methods used for computations at five loops use infrared
rearrangement, which modifies the finite terms.
A direct computation, as has been performed in this article, would require a
five-loop {\sc Forcer}{} equivalent. This is hard for at least two reasons: the number of
topologies that need a manually designed step-by-step IBP reduction is larger
than 200, and the number of parameters increases from 14 at four loops to 20
at five loops. Additionally, given the size of the step from three to four
loops, the required computer time could be an~issue.

\acknowledgments

We would like to thank J.A.~Gracey for useful discussions.
This work has been supported by the {\it European Research Council}$\,$ (ERC)
Advanced Grant 320651, {\sc HEPGAME}, and the UK {\it Science \& Technology
Facilities Council}$\,$ (STFC) under grant number ST/L000431/1.
The figures were made with {\tt Axodraw2}~\cite{Collins:2016aya}.

redpajama_set_name: RedPajamaArXiv


text:
Variety Plus Icon Read Next: China Orders Entertainment and Tech Firms to Stay in Their Lane
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CBS Sets July Launch, $11 Monthly Fee For Showtime OTT Service
By Cynthia Littleton
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CBS Corp. has made it official, announcing the launch of its broadband-only Showtime service in early July with Apple as its first partner.
CBS has set a price tag of $10.99 a month for the stand-alone offering, to be available to users of iPhone, iPad, iPod touch and Apple TV. The launch date coincides with the July 12 season premieres of dramas "Ray Donovan" and "Masters of Sex."
Other distribution partners are expected to sign on shortly.
"Going over-the-top means Showtime will be much more accessible to tens of millions of potential new subscribers," said CBS Corp. president-CEO Leslie Moonves. "Across CBS, we are constantly finding new ways to monetize our programming by capitalizing on opportunities presented by technology. This works best when you have outstanding premium content – like we do at Showtime – and when you have a terrific partner like Apple – which continues to innovate and build upon its loyal customer base."
Added Showtime chairman Matt Blank: "At Showtime, we're excited to offer consumers the opportunity to subscribe in new ways, by simply having an internet connection," said Blank. "Our intent is to make Showtime available to viewers via every manner possible, giving them an enormous amount of choice in accessing our programming."
Eddy Cue, Apple's senior vice president of Internet software and services, said the goal is to make subscribing to Showtime "as simple as buying an app."
"We're excited to offer Apple customers Showtime's live channels and extensive on demand library of great TV shows and movies, without a cable or satellite subscription," he said.
CBS moved aggressively into the OTT arena last October with the launch of CBS All Access, a $6 per month service that offers access to the Eye's vast library of shows, all episodes of current CBS series and the ability to live stream affiliate stations signals in CBS O&O markets and markets where CBS affiliates offer the service.
Showtime's OTT service follows on the heels of HBO's launch of HBO Now in April for $15 a month. HBO Now also launched with Apple, on the Apple TV device, as an exclusive distribution partner for a brief window, before signing Cablevision Systems and Google Play, which will offer the service on Android and Chromecast devices. HBO execs have said the volume of subscribers to date has exceeded expectations but it has not divulged any numbers.
CBS is in the midst of an expansion push for Showtime. In January, the Eye set its first deal to launch the Showtime brand outside the U.S. in a pact with Canada's Bell Media.
CBS Corp.,


text:
Home/Others/Latest developments/Dozens of Terrorists Killed in Damascus Countryside
Dozens of Terrorists Killed in Damascus Countryside
syriatr March 5, 2014
heavy clashes in the Damascus countryside on Tuesday.
The army units killed 20 terrorists in Al-Sahl area in the suburbs of Yabroud near the capital, Damascus.
Meantime, the army's artillery unit pounded several gathering centers of the terrorists in Khan Al-Sheikh in Damascus countryside and also repelled an attempt by the armed rebels who were trying to penetrate into the same area from the neighboring Douma.
On Sunday, the Syrian army launched massive ground operations to capture Yabroud.
Reports from the battlefield said the army units were advancing towards Yabroud in different fronts. The army seized back control over the strategic Al-Kuwaiti hill in the same area on Saturday.
The surrounding Rima farms and Al-Sahl town were also under the Syrian army's air and artillery attacks, FNA dispatches said.
Military sources told FNA that the Syrian army killed tens of militants and injured dozens more in fierce clashes with the terrorists after they reinforced their positions in Al-Kuwaiti hill and other surrounding areas of Yabroud.
On Saturday morning, the army units captured Al-Sahl town which was considered an important base for the militants.
The army had also earlier won back control over two strategic hills near the town of Yabroud.
Yabroud has been under the control of the foreign-backed militants over the past months. It was the last major town in Qalamoun that the Syrian army is fighting to retake.
The town is located near the Lebanese border and explosives used in fatal bomb attacks in Lebanon were reportedly smuggled from the militant-held town.
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Syrian Army Shocks Terrorists by Retaking Control of 2 Important Regions in Aleppo
Tens of Terrorists Killed, Wounded in Fresh Infighting in Northern Syria