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Dear President Obama, it's not too late for diplomacy with Iran
December 20, 2011 Jasmin Ramsey 3 Comments
Fifteen top analysts and former U.S. officials have signed a letter by the National Iranian American Council and the Arms Control Association urging President Obama to pursue diplomatic engagement with Iran before it's too late.
Toward a Diplomatic Solution on the Iranian Nuclear Issue
Dear Mr. President,
We write to convey our strongly held view that it is vitally important for U.S. and international security to reinvigorate direct diplomatic engagement with Iran to seek a resolution to the current standoff.
We applaud your Administration's success in uniting the international community in its response to Iran's nuclear and missile programs and its human rights violations, and we welcome recent statements by Secretary of State Hillary Rodham Clinton and National Security Advisor Thomas Donilon that your Administration remains committed to a diplomatic resolution with Iran.
But in order to capitalize on the positive steps your Administration has achieved, we believe that persistence and creativity in pursuing engagement with Iran are essential.
We recognize that achieving an agreement with Iran to fulfill its international obligations and provide full transparency over its nuclear activities is no easy task. The disappointing conclusion to the last P5+1 meeting with Iran in Istanbul showed that Tehran needs to be much more serious about a negotiated solution.
The lack of diplomatic progress with Iran has led to calls from some to take the diplomatic option off the table. We are concerned such a step would limit the U.S. ability to prevent a nuclear-armed Iran, limit the international community's ability to effectively address Iran's human rights situation, reduce the international pressure on Iran due to skepticism over U.S. motives, and increase the likelihood of disastrous military confrontation.
We believe the United States and the P5+1 must continue to highlight the framework of step-by-step confidence-building measures required to ease sanctions and end Tehran's diplomatic isolation. It must also be persuasively conveyed that sanctions are not a permanent reality but instead are dependent on Iranian behavior.
A near-term goal should be to test Iran's recent publicly stated offer to halt uranium enrichment to 20% levels if it could have access to fuel for its Tehran Research Reactor. A stockpile of 20% would allow Iran to shorten its time frame to produce weapons, if it chose to do so. We should forgo no opportunities to reduce that risk. A verifiable pledge from Iran not to produce enriched uranium above normal fuel grade and an agreement to export the 20%-enriched uranium it has produced would reduce the proliferation risk, while an arrangement to provide fuel for the reactor would be a humanitarian gesture benefiting the Iranian people.
While such an agreement would not resolve the many areas of concern we have with Iran—including not just its nuclear program, but its human rights situation and role in the region—it would be a critical first step towards achieving a resolution to these critical issues.
The targeted sanctions your Administration has obtained are helping to slow Iran's nuclear and missile programs and can help leverage concessions on the Iranian side. But winning those concessions requires the renewal of effective negotiations.
We offer these observations with no illusions that diplomacy with Iran will yield quick fixes. Iran must be willing to respond to reasonable proposals to clarify the IAEA's outstanding questions about its activities and provide confidence that its nuclear program is not being used for weapons purposes. We also recognize that there are elements in Iran who may be seeking to undermine progress toward a resolution of the nuclear issue.
The United States needs to reinvigorate the diplomatic initiative at this critical juncture. International pressure on Iran is now at an all time high. Iran's nuclear program is struggling to overcome technical problems. The time available must be used to convince Iran's current and future leaders they stand to gain more from forgoing nuclear weapons than from any decision to build them.
We are eager to assist in any way we can with this matter and look forward to your Administration's response.
Barry Blechman, co-founder of the Henry L. Stimson Center
Sir Richard Dalton, former Ambassador of the United Kingdom to the Islamic Republic of Iran
Charles Ferguson, President, Federation of American Scientists
Lt. General (USA, Ret.) Robert G. Gard, Jr.
Daryl G. Kimball, Executive Director, Arms Control Association
Lawrence Korb, former U.S. Assistant Secretary of Defense, and Senior Felow, Center for American Progress,
Amb. John Limbert, former U.S. Deputy Assistant Secretary of State for Iran
Amb. Francois Nicoullaud, former Ambassador of France to the Islamic Republic of Iran
Trita Parsi, President, National Iranian American Council
Bruno Pellaud, former Deputy Director General of the International Atomic Energy Agency
Ambassador Tom Pickering, former U.S. Under Secretary of State for Political Affairs, Ambassador to the United Nations, Russia, India, Israel, Nigeria, Jordan and El Salvador
Paul Pillar, former National Intelligence Officer for the Near East and South Asia; currently Director of Graduate Studies, Center for Peace and Security Studies
Gary Sick, former member of the National Security Council staff under Presidents Ford, Carter, and Reagan; currently Senior Research Scholar at Columbia University's Middle East Institute
Ambassador Roberto Toscano, former Ambassador of Italy to the Islamic Republic of Iran
James Walsh, Research Associate, Security Studies Program, Massachusetts Institute of Technology
Please address replies to: 1313 L Street NW, Suite 130, Washington, D.C. 20005. For more information or to contact the signatories of the letter, contact ACA at 202-463-8270.
The Arms Control Association (ACA) is an independent, membership-based organization dedicated to providing authoritative information and practical policy solutions to address the dangers posed by the world's most dangerous weapons. To contribute to our work, please visit:http://www.armscontrol.org/supportaca/
Message, US Foreign Policy ACA, Arms Control Association, Arms Control Association letter to obama, Iran, Iran nuclear, iran nuclear program, Iran threat, Iran War, Obama, P5+1, President Obama
Jasmin Ramsey
Jasmin Ramsey is an Iranian-born journalist based in Washington, DC.
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The most important steps to prevent Iran from becoming nuclear are:
1-Put pressure on Israel to disarm.(Real pressure including sanctions.)
2-Give Iran concrete promises to not invade it by US and/or Israel.
3-Put serious pressure on Pakistan to disarm it. Pakistan is even a more serious danger to Iran in the long run.
4-Do not meddle with Iran's affairs and accept it as a regional power (as Turkey and Israel)
delia ruhe
Futile. How much are these signatories contributing to Obama's re-election campaign? Not much, I'll bet.
William Rice
It is depressing to see history repeating itself again. 100 Senators are lined up with the Israeli Lobby and the Military,Industrial , Congressional Complex. Millions protested before the Iraq war in vain. Does anybody think this will be different? Nobody even talks about the disaster that is Libya. Another illegal war! Now the same neocon warmongerers are spreading their war propaganda across the Main Stream Media and the lazy incurious sheep will believe what they are told. Ron Paul 2012 because he is the only one running willing to tell the truth. They don't hate us for our freedoms,they hate us because we keep bombing them.


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Why Facebook And Mark Zuckerberg Went All In On Live Video
Facebook threw the company behind a massive push to add a slew of new features to its Live video as it takes on Snapchat, Periscope, and traditional television itself. Get ready to broadcast.
by Mat Honan
BuzzFeed News San Francisco Bureau Chief
Posted on April 6, 2016, 12:32 pm
Michelle Rial / BuzzFeed News
It was February, and Facebook CEO Mark Zuckerberg was just back from paternity leave, sitting in on his first M team meeting since returning. M team meetings are where Facebook global management comes together — managers from engineering, product, sales, you name it — to talk about what's going on with the company. And on that day, Fidji Simo and Maher Saba, who respectively run product and engineering for Facebook's media projects, spoke about the company's video efforts, Live in particular.
Facebook launched Live in August of last year. It's a dead simple feature: Hit a button to live-stream video. Initially, Live was available only to celebrities and other verified Facebook users, like journalists. But in December, the company rolled it out to everyone. By February, it was clear Live was gaining impressive momentum — and quickly, too. Facebook had been betting on Live for a long time, but it was at this meeting that the company's vision for it really crystalized.
Photo illustration by BuzzFeed News; Photo: Getty images
"When we roll things out occasionally you have these feelings like you're looking at the beginning of something special, and we all had that feeling," Chris Cox, Facebook's chief product officer, told BuzzFeed News.
Simo described a similar pop of revelation. "There was this a-ha moment for everyone," she said. "Zuck was like, 'Wait, if this thing is really taking off, why don't we focus a lot more resources on it?'"
Facebook has been barreling hard into video. But for the most part, its offerings haven't looked that distinct from what you might see on YouTube or other video platforms. Live, on the other hand, was something new. Something different. And it was working.
"The big decision we made was to shift a lot of our video efforts to focus on Live, because it is this emerging new format; not the kind of videos that have been online for the past five or ten years," Zuckerberg said in an interview with BuzzFeed News. When the meeting was over, he fired off an email. A long one. And with that, it was on.
"For an engineer, when you read an email that long and detailed you immediately start treating it like a spec," Saba explained. He and Simo broke out into two major parts: product work (the consumer-facing stuff we all see) and infrastructure (things like computing power, network issues, and storage). It was a challenge, but one the pair were certain they could meet. They just needed more bodies. "I read Zuck's email and thought of Jaws," said Saba. "We need a bigger boat."
The original Live team was composed of only a dozen or so people. But the vision laid out for the product at that February meeting would require more than 100 engineers to build. "The meeting was on a Thursday, and on Monday, Saba and I were standing in front of 150 engineers," said Simo. Cox, Saba, and Simo began holding a series of meetings to rally people around the project. They recruited engineers straight out of Facebook boot camp, pitching them by showing off what they were building, often using Live video itself to make their case. "At Facebook, we use Facebook to build Facebook," Saba explained.
And so for much of February, and the entire month of March, Live video became Facebook's monomania.
"The whole company used to run this way," said Cox, who has been with Facebook since 2005. "Ten or eleven years ago, we were all working on one thing at a time. People still want to be a part of that ... When everybody is on the floor all working on the same thing, there's an extra focus and energy that comes with that."
They had a small mountain of problems to solve. The list of new features to be added to Live was daunting — from streaming reactions across the video screen in real time, to adding video filters, to creating a map of live broadcasts, to allowing the capability to broadcast just to certain groups, even building what amount to channel guides that show which of your friends or pages you follow are live now. There is even a new Live video button right on the home row of the app itself. And the technical hurdles to implementing them were numerous.
Consider just the challenge of streaming from hundreds of thousands of phones all at once. Yes, you have to make sure you have the network and storage capacity to do that. But what's much harder is video conversion. Facebook builds for the world. And the world runs a whole hell of a lot of Android phones. Unlike iPhones, which you can reliably count on to have the same hardware from one device to another, Android handsets come in an array of different chipsets, with different video encoding and decoding capabilities. Facebook had to ensure that it could process live video regardless of source, and then transmit it in a way that could be viewed no matter the destination. And it had to do that instantly. That requires a massive amount of computational power and engineering acumen. But it is also very much what Facebook is designed to do.
"We built this big technology platform so we can go and support whatever the most personal and emotional and raw and visceral ways people want to communicate are as time goes on," said Zuckerberg.
The payoff had enormous potential. Live solves a lot of problems for Facebook. It gives people an easy way to create video content that doesn't require scripting or much production. Which in turn creates more content for Facebook. Live also helps the company tap into real-time events, an area where it's struggled compared to Twitter. It helps Facebook create its own celebrities; there are already proto–Live stars emerging. (Facebook is also working with media companies to make sure there's plenty of slick stuff as well. BuzzFeed is among the partners to whom Facebook is advancing cash to encourage the production of Live videos, as Re/code first reported.) And if you look way down the road, you can just begin to see a version of the future in which Facebook has emerged as the place to watch the kind of live events you might now tune into a local or cable station to see. But most immediately, Live was a solid and novel win for Facebook video. And for Facebook, video is increasingly important.
"We're entering this new golden age of video," Zuckerberg told BuzzFeed News. "I wouldn't be surprised if you fast-forward five years and most of the content that people see on Facebook and are sharing on a day-to-day basis is video."
While Live began as a tool for celebrities, after opening up to the public, it was getting traction from the very demographics you want when you're thinking five years out.
"The thing that was really surprising for us was that it wasn't just for public figures," Zuckerberg said. "It was a new, raw way that people wanted to share on a day-to-day basis. We're seeing this especially with young people and teens."
Indeed, one recent trend in social media has been a move away from highly produced content, particularly video. As Alex Kantrowitz reported for BuzzFeed News last year, what's hot now in social is the raw, unfiltered window into the lives of others. A host of apps that launched in 2015 and 2016 — Periscope, Meerkat, Peach, and Beme among them — try to nail this experience. This is precisely what Snapchat is so good at, and why it has become such a threat to Facebook. And it's clearly something that's been on Zuckerberg's mind as well.
"People look at live video and they think this is a lot of pressure because it's live; it takes a lot of courage to go live and put yourself out there. But what we're finding is the opposite," Zuckerberg said in a phone interview the day before the Live relaunch. "A lot of the biggest innovations have been things that take some of the pressure out of posting a photo or video."
"Because it's live, there is no way it can be curated," he said. "And because of that it frees people up to be themselves. It's live; it can't possibly be perfectly planned out ahead of time. Somewhat counterintuitively, it's a great medium for sharing raw and visceral content."
And in fact, a lot of what Facebook built in the new version of the app seems to be designed to enhance that raw feeling, that sensation that you're not seeing something scripted or performative. It's why the company spent so much effort on latency — lag time, in other words — so that everyone watching and commenting on a video can do so at precisely the same time, and the broadcaster can see and act on those comments as they happen. It's why it will now let people broadcast Live just to specific events or groups, because doing so frees people up to be more real than they might otherwise be with a wider, less specific audience. The idea is to get people to act like they normally would. But, you know, on video.
I saw this firsthand at the company's headquarters in Menlo Park, when Simo cracked open her laptop to show me one of Live's new features, a map that displays where everyone in the world is going live. Circles show how big, or small, a Live broadcaster's audience is. At my request, Simo clicked on a small one in Africa to reveal...two guys in Angola staring languidly into the camera. They seemed to be just hanging out. It was oddly compelling.
We checked out another new feature, one that lets you see who is Live right now. These might be people or pages you follow, but you can also follow topics, like, say, sports. There's also a new search function to help with discovery. And Facebook will offer suggestions, so there will always be something to check out.
Simo follows Hillary Clinton. And, as luck would have it, the Democratic presidential candidate was broadcasting to Facebook Live from the Apollo Theater at the time of our meeting. As Clinton spoke, a steady stream of thumbs-up Facebook "like" icons flowed across the bottom of the screen. When she mentioned taking on the gun lobby, a "wow" face appeared, followed by a scowling "angry" face. These are yet another new feature of Live. Previously, viewers could only react to video in a static way — you gave one thumbs-up or angry face or whatever. That was, effectively, your rating for the entire video. The new version lets you react as the video plays, and streams these animated reactions, floating them across the bottom of the video as they roll in. If a reaction comes from someone you follow, you will see their profile picture as well.
What's more, if you were to go back and watch the video later, you would see that reaction as it happened. If someone adds a thumbs-up or a comment, say, two minutes in, it will show up at the two-minute mark when the video is played back later. It's sort of like a DVR for commentary and reactions.
If this all sounds a bit like Twitter's live-streaming video app Periscope, which also has a map of people who are live, and also lets you react to videos in real time with icons (hearts that float up vertically), and also will play back those reactions and comments as they appeared in the video, well, yes, there are many similarities. When I asked Zuckerberg about this and what sets Live apart from Periscope or Meerkat or any number of other video apps with, er, ~similar~ features, he didn't mince words: audience.
"If you're a public figure, the audience is unprecedented," Zuckerberg said. "If you're Jimmy Fallon and want to go live, it's not really worth your time unless you can reach an audience that's similar to what you can reach on TV." And while those numbers matter in aggregate for people like Fallon, they may even matter more on an individual level for regular people. "If you're a person that just wants to share with your friends, it helps to have your friends there."
Take that Hillary Clinton speech from Harlem. More than 240,000 people watched it, many live, and many more after the fact. Those sorts of numbers, if not routine, are at least not completely unusual. That's been especially true since Facebook prioritized Live in its News Feed. A rush of publishers (BuzzFeed included) have barreled in to experiment and build audiences. Facebook already enjoys massive network effects, which means that you don't have to build an audience for Live. It's already there.
Meanwhile, Facebook is already creating its own Live celebrities and even genres. There's Liz Cook, a tattoo artist who goes Live with inkings and has amassed more than 1 million followers. And Esther the Wonder Pig, who is, well, a highly successful pig. On Live, weird seems to work. (One of the most popular Live videos BuzzFeed has produced was a close-up of a guy popping bubble wrap during a GOP debate. Nearly 750,000 people watched it.)
And it seems clear where Facebook will eventually go with this: right at traditional broadcast. The company already made a play to win the rights to stream the NFL on Thursday nights (Twitter won these). It has reportedly been in discussions with celebrities about paying them to make Live video. And if someone can generate the same kind of audiences on Facebook Live as they can on a cable channel (and they can), why bother investing in much more expensive television broadcasts?
When I asked Zuckerberg about this, he sidestepped the question. "I think you would just do both," he offered. "The point isn't that you have to make a choice, it's that we're giving people new tools."
That may be true for now, but it doesn't seem like the kind of thing that will hold over the long term. A broadcaster's choices may be infinite, but your attention is not. And while Live may not signal the end of live television as we know it, Cox is correct: It definitely feels like the beginning of something.
Apr. 06, 2016, at 13:18 PM
Updated to add disclosure based on Re/code's initial reporting that BuzzFeed is a paid media partner, which BuzzFeed News has now confirmed.
Mat Honan
Mat Honan is the San Francisco bureau chief for BuzzFeed News.
Contact Mat Honan at mat.honan@buzzfeed.com.

redpajama_set_name: RedPajamaCommonCrawl


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Updated 1/16/2019 4:02 PM
Feder: Ann Dwyer named editor of Crain's Chicago Business
Courtesy of WTTW/Chicago TonightAnn Dwyer has been promoted to editor of Crain's Chicago Business, becoming the first woman to lead the editorial staff in the publication's 41-year history.
Robert Feder
Ann Dwyer has been promoted to editor of Crain's Chicago Business, becoming the first woman to lead the editorial staff in the publication's 41-year history.
Dwyer, 54, who most recently was managing editor, joined Crain's in 1995 as a copy editor and later moved to the news desk. She succeeds Michael Arndt, who resigned in November after six years as editor.
"Ann is an outstanding, creative editor who comes into this role at the right time in the paper's evolution," Jim Kirk, publisher and executive editor of Crain's, said in announcing the appointment Wednesday.
• Read the full report at robertfeder.com.
Feder, Robert


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Stewart brings innovative interdisciplinary approach to new Central Piedmont Community College student complex
Construction of the new Central Campus Library and Student Success Center will begin in August
CHARLOTTE, N.C. – Stewart, an interdisciplinary design, engineering and planning firm, has played a pivotal role in the design of Central Piedmont Community College's new Central Campus Library and Student Success Center. The firm has provided structural engineering and landscape architecture services for the project, helping to lay a strong foundation for the 173,000 square-foot complex that will welcome visitors and expand learning opportunities for students and faculty.
Construction of the new Central Campus Library and Student Success Center will begin in August and will replace the former Terrell Building, Hagemeyer Learning Resource Center and Pease Auditorium that were recently demolished for the new project. Stewart's unique interdisciplinary approach has allowed for innovative designs and enhanced features within the complex. The new library and Student Success Center will include advanced technology amenities that offer flexible and modern collaboration and study spaces for students. The center will host student organizations and services such as financial aid, testing, career, and transfer programs and advising, as well as specialized amenities including a dining area, student commons and multi-purpose spaces.
Libraries are intended to draw people in and encourage learning and interaction, and Stewart's structural engineering group helped intentionally design the new library to do so. The building will be welcoming to Central Piedmont Community College visitors with its new 450-seat Pease Auditorium, and will also include instructional classrooms, a private reading room, more than 25 collaboration rooms and more. The new complex will also contain several outdoor terraces.
"This building is filling a need on campus by creating a space that will be student-centered, focused on welcoming spaces, enhanced services, and socially friendly environments – all to enhance opportunities for student success," said Vicki Saville, associate vice president of facilities and construction at Central Piedmont Community College. "We're committed to providing collaborative learning opportunities and are thankful for Stewart's role in the design and development process of the new complex."
In collaboration with Rodgers Builders and R.J. Leeper Construction, and architectural firm Morris-Berg Architects in association with Moody Nolan, Stewart is helping to bring the new complex to life. Construction is expected to be completed in December 2021.
"Our growth in the Charlotte community really mirrors the growth of Central Piedmont Community College and we're honored to have partnered with an institution that shares our passion for innovation," said Lance Williams, associate vice president and manager of structural engineering at Stewart. "We're excited to have the opportunity to demonstrate our interdisciplinary strength and help bring innovative designs to the campus that will help drive student engagement and involvement."
Stewart's impact can be found across Central Piedmont Community College's main and suburban campuses, with several projects currently underway. The firm provided structural engineering services for the new North Classroom Building on Central Campus that will be completed in December 2019, as well as a new classroom building on the corner of Charlottetown and 5th Street that will be completed in July 2020. Stewart also provided structural engineering services for a new library on the Merancas Campus in Huntersville, and is providing structural engineering, civil engineering, and landscape architecture services for an Advanced Technology Building on the Harper Campus in Charlotte.
To learn more about Stewart, please visit www.stewartinc.com.
Central Piedmont Community College'
Lance Williams
Vicki Saville
dbrigham


Steve Gerdes has focused his practice during most of his career on the tax aspects of state and local government obligations. His practice has covered the full range of tax-exempt obligations appearing in the marketplace, including governmental new money, commercial paper and working capital financings, and current and advance refundings. These transactions also include exempt facility bond financings such as seaport and airport financings, student loan financings, solid waste and sewage financings, single- and multi-family housing financings, as well as professional sports facility and development district financings.
His clients have included cities, counties, school districts, universities and other types of political subdivisions within the State of Texas, investment banking firms active in Texas, and other law firms requiring the services of special tax counsel.


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Q: Checking for a limit and finding the sum of a series 1) Find the sum of the series $\sum_{n = 1}^\infty (\frac{i}{2})^{2n}$
2) Find the sum of the series $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+......$

How do i go about finding the sum of the two series?
As I understand it, if the partial sums tend to a limit then that limit is the sum of the series. And if not then it has no sum?
Given that I guess that 2) equal 2 but I don't know how a proper argument could be made for that result.

A: As regards 1), since $i^2=-1$, then
$$\left(\frac{i}{2}\right)^{2n}=\left(\left(\frac{i}{2}\right)^2\right)^{n}=\left(\frac{i^2}{2^2}\right)^{n}=\left(\frac{-1}{4}\right)^{n}.$$
Hence the partial sum of the first $N$ terms is
$$S_N=\sum_{n = 1}^{N} \left(\frac{i}{2}\right)^{2n}=
\sum_{n = 1}^{N}\left(\frac{-1}{4}\right)^{n}
=\frac{-1}{4}\sum_{n = 1}^{N}\left(\frac{-1}{4}\right)^{n-1}\\
=\frac{-1}{4}\sum_{n = 0}^{N-1}\left(\frac{-1}{4}\right)^{n}
=\frac{-1}{4}\cdot \frac{1-(-1/4)^{N}}{1-(-1/4)}=
\frac{-1+(-1/4)^{N}}{5}$$
where we used a basic property of the geometric series.
Is the series convergent?
What is the limit of the partial sum $S_N$ as $N\to\infty$?
For 2), as remarked by Frank Lu, note that $1+\dots+n=\frac{n(n+1)}{2}$ and the partial sum is
$$S_N=\sum_{n=1}^N\frac{2}{n(n+1)}=2\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{n+1}\right)=2\sum_{n=1}^N\frac{1}{n}-2\sum_{n=1}^N\frac{1}{n+1}\\=2\sum_{n=1}^N\frac{1}{n}-2\sum_{m=2}^{N+1}\frac{1}{m}=2-\frac{2}{N+1}$$
We may now conclude that the limit of the partial sum $S_N$ as $N\to\infty$ is $2$.

redpajama_set_name: RedPajamaStackExchange


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NEHires
Continuance is granted in attempted murder case
New generation joins local business
Home / Infant's death inspires local woman
Infant's death inspires local woman
Cedar County News
HARTINGTON — Sarah Gustafson was not going to let adversity keep her from doing bigger and better things for more people.
She was a foster parent to her 29-year-old niece's premature son.
In October 2018, the infant died in her arms at the age of three months.
Kheondrae was born at approximately just shy of his eighth month and would go home with Gustafson in Hartington.
"The State of Nebraska was going to take him away because he was born with drugs in his system and other problems," Gustafson said.
Kheondrae's death was ruled as Sudden Unknown Infant Death Syndrome (SUIDE) and Gustafson wanted his death to be more than just a passing footnote.
It needed to have more meaning, she said.
That's why she decided to make a difference by creating K's Closet.
"I started what is called a foster care closet and I run it out of our house," Gustafson said. "We take donations of any kind for all different ages and sizes – even adult size. When foster families receive kids or have them as they get older, they can come and shop for free."
Gustafson said that even non-fostering parents can come to her in a pinch.
"We have helped a family in Norfolk where the mom is a single parent and found out she has uterine cancer and we were able to help her out over Christmas," Gustafson said. "All together, we have helped about 15 different families since this started."
Chelsey Pena has known Gustafson for many years and reconnected through Facebook.
There, Pena said she posted on her wall how her nephew has been born premature at 25 weeks-plus. The infant stayed at the hospital for the next four months while Pena tried to get everything together to help the child move forward.
"Sarah had a bunch of stuff and she used what she had from Kheondrae and we have clothes, diapers and bottles," Pena said. "She basically helped us start our journey when we brought him home on New Year's Eve."
HNS Girls Basketball - Homer
Names are now being sought for Veterans' Memorial
102 W. Main
Hartington NE 68739
Cedar County News Copyright © 2020


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

A future high-energy electron-positron collider will allow a very
precise measurement of the
top-antitop production cross section near threshold. From the
threshold scan several standard model parameters, like top-quark mass,
width, and Yukawa coupling, can be extracted with high precision. It
was found in several
studies~\cite{Martinez:2002st,Seidel:2013sqa,Horiguchi:2013wra} that
the top mass can be determined with an uncertainty well below
$100$~MeV. Contrary to direct reconstructions at hadron colliders,
there is no ambiguity in relating the result to a precisely defined
mass parameter. To achieve this level of accuracy requires
precise theoretical predictions for the threshold cross section. The
challenge is that conventional perturbation expansions in the strong
coupling $\alpha_s$ fail for threshold production, since it involves
multiple scales. In terms of the mass $m_t$ and velocity $v$ of the
top-quark these are the hard scale $m_t$, the soft scale $m_tv$, and
the ultrasoft scale $m_tv^2$. At the ultrasoft scale, the colour-Coulomb
force is non-perturbatively strong. In terms of Feynman diagrams,
this implies that when the velocity is of order of the
strong coupling, Coulomb singularities of the form $(\alpha_s/v)^n$
have to be summed to all orders. This can be achieved by successively
integrating out the hard and soft scale leading to the non-relativistic
QCD (NRQCD)~\cite{Thacker:1990bm,Lepage:1992tx,Bodwin:1994jh} and potential
NRQCD
(PNRQCD)~\cite{Pineda:1997bj,Pineda:1997ie,Beneke:1998jj,Beneke:1999qg,Brambilla:1999xf} effective field theories, respectively. Within this framework,
described in detail for top-quark pair production near threshold
in \cite{Beneke:2013PartI}, the dominant contribution from
the S-wave correlation function has been computed at
next-to-next-to-next-to-leading order
(NNNLO)~\cite{Beneke:2005hg,Beneke:2008ec,Beneke:2008cr,Beneke:2013PartII}.

The axial-vector coupling of the top quark to the $Z$ boson gives
rise to a P-wave contribution to the top-pair production cross section.
In this work we compute the corresponding P-wave PNRQCD correlation
function. Being suppressed by $v^2$ relative to the S-wave it contributes
only starting from NNLO. The complete NNNLO calculation of the threshold
correction therefore requires a NLO calculation of the P-wave
correlation function, which we perform here in dimensional regularization.
Some results for the P-wave Green function were
already obtained
in~\cite{Bigi:1991mi,Penin:1998ik,Penin:1998mx,Kuhn:1999hw}, but
none of these computations were performed in dimensional
regularization. Dimensional regularization is, however, required
for the following reason: The imaginary part of the P-wave Green function,
which is relevant for the cross section, is
divergent already at leading order in the non-relativistic expansion,
if the finite width of the top quark is included. This divergence and
the resulting scheme dependence cancel only when non-resonant
corrections to the process $e^+ e^-\to W^+ W^-b\bar b$ are added.
The separation of resonant and non-resonant contributions can be
performed consistently in unstable-particle effective field
theory~\cite{Beneke:2003xh, Beneke:2004km}, in which the non-resonant
terms appear as a hard region. The corresponding diagrams are
computed as usual in dimensional regularization as has
already been done in~\cite{Beneke:2010mp,Jantzen:2013gpa}. Consistency then
requires that the non-relativistic, resonant part is also computed
in dimensional regularization. We emphasize that to determine the
top-quark mass precisely it is necessary to compute
the process $e^+ e^-\to W^+ W^-b\bar b$ including non-resonant terms,
since the top mass is ultimately determined from the rise of the
cross section near threshold, where non-resonant effects are
important~\cite{Beneke:2010mp}.

The outline of the paper is as follows.
In Sec.~\ref{sec:theory} we give a brief overview of the framework
of the calculation. A detailed discussion can be found
in~\cite{Beneke:2013PartI,Beneke:2013PartII}, however, we repeat some
of the formulas given there in order to make the present paper
self-contained. The P-wave Green function up to NLO is computed in
Sec.~\ref{sec:computation} with some technical details relegated to
the appendix. In Sec.~\ref{sec:crosssection} a numerical
analysis of the P-wave contribution to top-pair production at
threshold is presented. The absolute size of the P-wave contribution
is rather small due to the small axial couplings of the top quark.
We therefore also  discuss the P-wave dominated (s)top threshold
production processes $\gamma\gamma\rightarrow t\bar{t}$ with different photon
helicities and $e^+e^-\rightarrow\tilde{t}\tilde{\bar{t}}$ in
Sec.~\ref{sec:photon} and Sec.~\ref{sec:susy}, respectively, where
P-wave production is the dominant production mechanism. We
conclude in Sec.~\ref{sec:conclusion}.

\section{Effective theory setup\label{sec:theory}}

The production of a top pair in $e^+e^-$ annihilation is mediated by
photons and $Z$ bosons. While the coupling of photons to fermions is
purely vector-like the $Z$ boson couples to vector currents and
axial-vector currents with the respective strengths
\begin{equation}
  v_f=\frac{T_3^f-2e_f\sin^2\theta_w}{2\sin\theta_w\cos\theta_w},
  \hspace{1cm}
  a_f=\frac{T_3^f}{2\sin\theta_w\cos\theta_w},
  \label{eq:Zffcoupling}
\end{equation}
where $\theta_w$ is the Weinberg angle, $e_f$ the electric charge and
$T_3^f$ the third component of the weak isospin of the fermion
$f$. For the vector current $j_\mu^{(v)}=\bar{t}\gamma_\mu t$ and the
axial-vector current $j_\mu^{(a)}=\bar{t}\gamma_\mu\gamma_5 t$ we
define the two-point functions
\begin{equation}
  \begin{aligned}
    \Pi_{\mu\nu}^{(X)}(q^2) =& \hspace*{0.15cm}
    i\int d^dx\,e^{iq\cdot x}\Braket{0|T\left[j_\mu^{(X)}(x)j_\nu^{(X)}(0)\right]|0}\\
    =& \hspace*{0.15cm}
(q_\mu q_\nu-q^2g_{\mu\nu})\,\Pi^{(X)}(q^2)+q_\mu q_\nu\Pi_L^{(X)}(q^2).
  \end{aligned}
  \label{eq:twopointfct}
\end{equation}
We denote by $R=\sigma_{X\bar{t}t}/\sigma_0$ the inclusive $\bar{t}t$
production cross section $\sigma_{X\bar{t}t}=\sigma(e^+e^-\rightarrow
t\bar{t}X)$ normalized to the high-energy limit of the $\mu^+\mu^-$
production cross section
$\sigma_0=4\pi\alpha_{\text{em}}^2/(3q^2)$. It can be related to the
imaginary part of the two-point functions by the optical theorem
\begin{equation}
  \begin{aligned}
    R = 12\pi\text{ Im} \bigg[&
    e_t^2\Pi^{(v)}(q^2)-\frac{2q^2}{q^2-M_Z^2}v_ev_te_t\Pi^{(v)}(q^2)\\
    &+\left(\frac{q^2}{q^2-M_Z^2}\right)^2(v_e^2+a_e^2)(v_t^2\Pi^{(v)}(q^2)+a_t^2\Pi^{(a)}(q^2))
    \bigg].
  \end{aligned}
  \label{eq:Rratio}
\end{equation}

Near the production threshold $s\equiv q^2\approx4m_t^2$ the usual
perturbation theory in $\alpha_s$ breaks down, because the velocity
$v$ of the top quark is of the same order as the strong coupling and
contributions that scale as $(\alpha_s/v)^k$ have to be summed to
all orders. Instead, in non-relativistic perturbation theory, one expands
in $\alpha_s$ and $v$ around the non-perturbative solution
that sums these terms. Explicitly, the re-organized expansion takes the form
\begin{eqnarray}
  \begin{aligned}
    R \sim v \sum\limits_k \left(\frac{\alpha_s}{v}\right)^k
    \{& 1(\text{LO}); \alpha_s,v(\text{NLO});
    \alpha_s^2,\alpha_sv,v^2(\text{NNLO}); \\
    &\alpha_s^3,\alpha_s^2v,\alpha_sv^2,v^3(\text{NNNLO});\dots\}.
  \end{aligned}
  \label{eq:Rexpansion}
\end{eqnarray}

The computation of the vector current spectral function
$\Pi^{(v)}(q^2)$ to NNNLO is summarized in~\cite{Beneke:2013PartI,Beneke:2013PartII}. This
work focuses on the axial-vector current spectral function
$\Pi^{(a)}(q^2)$. In a first step the hard modes are integrated
out. Matching of the axial-vector current to NRQCD yields an
expansion
\begin{equation}
 j^{(a)k}=\frac{c_a}{2m_t}\psi^\dagger\left[\sigma^k,(-i)\boldsymbol{\sigma}\cdot\mathbf{D}\right]\chi+\dots,
 \label{eq:axialvectorcurrent}
\end{equation}
where bold face letters and latin indices refer to $d-1$ dimensional vectors,
and $d=4-2\epsilon$ is the space-time dimension.
The hard matching coefficient $c_a$ is given by~\cite{Kniehl:2006qw}\footnote{
The $\mathcal{O}(\epsilon)$ term will be needed later on and was not given
in~\cite{Kniehl:2006qw}. For $\gamma_5$ the naive anti-commuting (NDR) scheme
is employed.}
\begin{equation}
 c_a=1-4 C_F\,\frac{\alpha_s}{4\pi} \left[1-\epsilon\ln\frac{m_t^2}{\mu^2}
+\mathcal{O}(\epsilon^2)\right] +\mathcal{O}(\alpha_s^2).
  \label{eq:ca}
\end{equation}
We observe that~\eqref{eq:axialvectorcurrent} contains a covariant
derivative. Thus the axial-vector current spectral function is
suppressed by the well-known P-wave factor of $v^2$ compared to the S-wave
and contributes to $R$ only starting from NNLO. The two-point function
takes the form
\begin{equation}
\label{eq:PiaNRQCD}
  \begin{aligned}
   \Pi^{(a)}(q^2) =&  \hspace*{0.15cm}\frac{1}{(d-1)q^2}\,\Pi_{ii}^{(a)}(q^2)\\
    =&  \hspace*{0.15cm}\frac{N_c c_a^2}{8m_t^4}\frac{i}{2N_c(d-1)}
    \int d^dx\,e^{iEx^0}
    \Braket{0|T\left([\psi^\dagger\Gamma^i\chi]^\dagger(x)
        [\psi^\dagger\Gamma^i\chi](0)\right)|0}_{\text{NRQCD}}+\dots,
 \end{aligned}
\end{equation}
where
$\Gamma^k=(-i)\left[\sigma^k,\boldsymbol{\sigma}\cdot\mathbf{D}\right]$. A
further matching to potential NRQCD integrates out the soft modes and
the potential gluons and light quarks. To the required order the
PNRQCD Lagrangian reads
\begin{equation}
 \begin{aligned}
  \mathcal{L}_{\text{PNRQCD}}=&\psi^\dagger\left(i\partial_0+g_sA_0(t,\mathbf{0})+\frac{\boldsymbol{\partial}^2}{2m}\right)\psi+\chi^\dagger\left(i\partial_0+g_sA_0(t,\mathbf{0})-\frac{\boldsymbol{\partial}^2}{2m}\right)\chi\\
  &+\int d^{d-1}\mathbf{r}\left[\psi_a^\dagger\psi_b\right](x+\mathbf{r})V_{ab;cd}(\mathbf{r},\partial)\left[\chi_c^\dagger\chi_d\right](x),
 \end{aligned}
\label{eq:LPNRQCD}
\end{equation}
where $\psi$ ($\chi$) denotes the potential quark (antiquark) field. The
coupling to the ultrasoft gluon field
$A_0(t,\mathbf0)$ can be removed by a field redefinition,
which, in general, modifies the external current~\cite{Beneke:2010da},
but cancels for the colour-singlet currents relevant to production
through photons and $Z$ bosons.
Henceforth, ultrasoft gluons can be ignored, since they contribute only at higher
order.
Since the top pair is produced in a colour-singlet state, only
the colour-singlet projection
\begin{equation}
 V(\mathbf{p},\mathbf{p}')=\frac{1}{N_c}\delta_{bc}\delta_{da}V_{ab;cd}(\mathbf{p},\mathbf{p}')
\end{equation}
of the general potential is required. To the considered order it
consists purely of the Coulomb potential
\begin{equation}
 V(\mathbf{p},\mathbf{p}')=-\frac{4\pi C_F\alpha_s}{\mathbf{q}^2}\left[\mathcal{V}_C^{(0)}+\frac{\alpha_s}{4\pi}\mathcal{V}_C^{(1)}+\mathcal{O}(\alpha_s^2)\right]+\dots,
 \label{eq:VCoulomb}
\end{equation}
where the $d$ dimensional coefficients of the LO and NLO Coulomb potential are
\begin{align}
 \mathcal{V}_C^{(0)}=&\hspace*{0.15cm}1,\\
 \mathcal{V}_C^{(1)}=&\hspace*{0.15cm}\left[\left(\frac{\mu^2}{\mathbf{q}^2}\right)^\epsilon-1\right]\frac{\beta_0}{\epsilon}+\left(\frac{\mu^2}{\mathbf{q}^2}\right)^\epsilon a_1(\epsilon),
 \label{eq:VCoulombcoefficients}
\end{align}
with
\begin{equation}
 a_1(\epsilon)=\Big(C_A[11-8\epsilon]-4T_Fn_f\Big)\frac{e^{\gamma_E\epsilon}\Gamma(1-\epsilon)\Gamma(2-\epsilon)\Gamma(\epsilon)}{(3-2\epsilon)\Gamma(2-2\epsilon)}-\frac{\beta_0}{\epsilon}.
\end{equation}
They are required up to order $\epsilon$ for reasons that will become
clear later. The LO Coulomb potential contributes to the same order as
the leading kinetic terms in the PNRQCD Lagrangian and thus has to be treated
non-perturbatively, while the NLO correction $ \mathcal{V}_C^{(1)}$ is a
perturbation. Thus the LO Lagrangian describes the propagation of
quark-antiquark pairs, where ladder diagrams with exchange of an
arbitrary number of potential gluons between the quark-antiquark pair have been
resummed. The quark-antiquark pair propagator
$\tilde{G}_0(\mathbf{p},\mathbf{p}';E)$ satisfies the $d$-dimensional
Lippmann-Schwinger equation
\begin{equation}
 \begin{aligned}
  &\left(\frac{\mathbf{p}^2}{m_t}-E\right)
\tilde{G}_0(\mathbf{p},\mathbf{p}';E)-\tilde{\mu}^{2\epsilon}
\int\frac{d^{d-1}\mathbf{k}}{(2\pi)^{d-1}}
\frac{4\pi C_F\alpha_s}{\mathbf{k}^2}\,
\tilde{G}_0(\mathbf{p}-\mathbf{k},\mathbf{p}';E)\\
  & = (2\pi)^{d-1}\delta^{(d-1)}(\mathbf{p}-\mathbf{p}'),
 \end{aligned}
 \label{eq:LippmannSchwinger}
\end{equation}
where $E=\sqrt{s}-2m_t$ denotes the energy relative to the production
threshold, $m_t$ the top-quark pole mass,
and the scale $\tilde\mu=\mu \,[e^{\gamma_E}/(4\pi)]^{1/2}$ has
been chosen such that the minimal subtraction of $1/\epsilon$
poles corresponds to
the $\overline{\text{MS}}$ rather than the MS scheme. Here and in the
following the tilde is used to indicate that the Green function is
given in momentum space. Its Fourier transform
\begin{equation}
 G_0(\mathbf{r},\mathbf{r}';E)=
\int\frac{d^{d-1}\mathbf{p}}{(2\pi)^{d-1}}\int\frac{d^{d-1}\mathbf{p}'}
{(2\pi)^{d-1}}\,e^{i\mathbf{p}\cdot\mathbf{r}}e^{-i\mathbf{p}'\cdot\mathbf{r}'}
\tilde{G}_0(\mathbf{p},\mathbf{p}';E)
\end{equation}
is the solution to the Schr\"odinger equation
\begin{equation}
 \left(-\frac{\nabla_{(r)}^2}{m_t}-\frac{C_F\alpha_s}{r}-E\right)
G_0(\mathbf{r},\mathbf{r}';E)=\delta^{(3)}(\mathbf{r}-\mathbf{r}')
 \label{eq:Schroedinger}
\end{equation}
in four dimensions. An expression for general $d$ is not available,
but for $d=4$ several representations are
known~\cite{Wichmann:1961,Schwinger:1964zzb,Voloshin:1985bd,Voloshin:1979uv}. We
find it convenient to use the integral representation
from~\cite{Wichmann:1961},
\begin{equation}
  \begin{aligned}
    G_0(\mathbf{r},\mathbf{r}';E) =&\hspace*{0.15cm}
    \sum\limits_{l=0}^\infty\,
    (2 l+1)\, P_l\!\left(\frac{\mathbf{r}\cdot\mathbf{r}'}{rr'}\right)
    \,\frac{m_t p}{2\pi} \frac{(2pr)^{l}(2 pr')^l}
    {\Gamma(l+1+\lambda)\Gamma(l+1-\lambda)}\,
    \\
    &\times\int\limits_0^1dt\int\limits_1^\infty ds \left[s(1-t)\right]^{l+\lambda}
    \left[t(s-1)\right]^{l-\lambda}\exp\left\{-p\,[r'(1-2t)+r(2s-1)]\right\},
  \end{aligned}
  \label{eq:LOGFintegralrep}
\end{equation}
with $r'<r$ and $p=\sqrt{-m_tE}$, and $P_l(z)$ the Legendre polynomials.
We also introduced the variable
\begin{equation}
\lambda=\frac{C_F\alpha_s}{2\sqrt{-\frac{E}{m_t}}}.
\end{equation}

The Feynman rules required for higher-order computations have been
derived in~\cite{Beneke:2013PartI}. As also discussed there
the soft matching of the two-point
function is trivial and hence
\begin{equation}
  \Pi^{(a)}(q^2) = \frac{N_c c_a^2}{2m_t^4}\,\frac{d-2}{d-1}\, G^P(E),
 \label{eq:PiaPNRQCD}
\end{equation}
where now
\begin{equation}
 \begin{aligned}
  G^P(E)=&\hspace*{0.15cm}\frac{i}{8N_c(d-2)}\int d^dx\,
e^{iEx^0}\Braket{0|T\left([\psi^\dagger\Gamma^i\chi]^\dagger(x)[\psi^\dagger\Gamma^i\chi](0)\right)|0}_{\text{PNRQCD}}\\
  =&\hspace*{0.15cm}\frac{i}{2N_c}\int d^dx\,
e^{iEx^0}\Braket{0|T\left([\chi^\dagger i D^i\psi](x)[\psi^\dagger i D^i\chi](0)\right)|0}_{\text{PNRQCD}}
 \end{aligned}
\label{eq:GP}
\end{equation}
is the P-wave Green function at the origin.
In passing to the second equation, we used that due to the
spin-independence of the Coulomb potential~\eqref{eq:VCoulomb} the
$(d-1)$-dimensional spin algebra in~\eqref{eq:GP} can be
evaluated once and for all.
In PNRQCD perturbation theory up to NLO, the P-wave correlation
function reads
\begin{eqnarray}
    G^P(E) & = & G_0^P(E)+\delta_1G^P(E)+\dots \nonumber \\
    &=& \tilde{\mu}^{4\epsilon}
    \int\frac{d^{d-1}\mathbf{p}}{(2\pi)^{d-1}}
    \int\frac{d^{d-1}\mathbf{p}'}{(2\pi)^{d-1}}\,
    \mathbf{p}\cdot\mathbf{p}' \times
    \bigg[G_0(\mathbf{p},\mathbf{p}';E) \nonumber \\
    && + \,\tilde{\mu}^{4\epsilon}
    \int\frac{d^{d-1}\mathbf{p}_1}{(2\pi)^{d-1}}
    \int\frac{d^{d-1}\mathbf{p}_2}{(2\pi)^{d-1}}\,
    G_0(\mathbf{p},\mathbf{p}_1;E)\,
    i\left(-\frac{\alpha_s^2C_F}{(\mathbf{p}_1-\mathbf{p}_2)^2}\mathcal{V}_C^{(1)}\right)
    iG_0(\mathbf{p}_2,\mathbf{p}';E)\nonumber  \\
    && + \,\dots\bigg],
 \label{eq:GFPNRQCDexpansion}
\end{eqnarray}
where we have inserted the NLO Coulomb potential. As yet we have
neglected the sizable width of the top quark
$\Gamma_t = 1.33$~GeV. For the leading-order S-wave contribution the
width effect is accounted for by the replacement $E\rightarrow E+i\Gamma_t$ in the
spectral function~\cite{Fadin:1987wz,Fadin:1988fn}. We follow this
prescription to {\em define} the pure QCD calculation of the
pair-production cross section and therefore assume henceforth that
$E$ can take complex values. Due to
the $\mathbf{p}\cdot\mathbf{p}'$ factor
already the leading order result for the P-wave
contains ultraviolet divergences of the form $E/\epsilon$. Following
the above description yields poles proportional to $\Gamma_t/\epsilon$
in the imaginary part, and a scale-dependence related to these poles.
To make this explicit, we distinguish between the scale $\mu_r$ at
which we evaluate the running coupling $\alpha_s=\alpha_s(\mu_r)$ and
the scale $\mu_w$ that arises from the finite-width divergences.
While the residual $\mu_r$ scale-dependence must always be of
higher-order, the
divergences and the dependence on the scale $\mu_w$ have to cancel
against non-resonant electroweak corrections involving a
$Wb$-loop correction to the off-shell top propagator~\cite{Beneke:2008cr}.
So far only the $1/\epsilon$ pole corresponding to the LO P-wave correlation
function is known~\cite{Jantzen:2013gpa,Hoang:2004tg,Hoang:2010gu}. The finite
term is required to cancel the corresponding scheme dependence.
Since the finite term as well as the NNNLO non-resonant terms related
to the NLO   P-wave correlation
function are presently unknown, we keep the dependence on $\mu_w$
(and the associated poles) explicit in our analytical result.

\section{Computation of the P-wave Green
  function\label{sec:computation}}

\subsection{Leading order\label{sec:LO}}

Simple power counting shows that ladder diagrams with up to four loops
are ultraviolet divergent, as compared to two loops for the S-wave, due to the
additional factor $\mathbf{p}\cdot\mathbf{p}'$
in~\eqref{eq:GFPNRQCDexpansion}. These diagrams, the sum of which is
denoted by $G_0^{P(\leq3ex)}$, therefore have to be computed in $d$
dimensions. We have used FIRE~\cite{Smirnov:2008iw,Smirnov:2013dia} to
perform the reduction to a small set of master integrals. Results for
these master integrals are available in the
literature~\cite{Schroder:2005va} and are in agreement with special
cases of the more general calculation for the NLO contribution presented below.
We obtain
\begin{equation}
 G_0^{P(\leq3ex)}(E)=\frac{m_t^4C_F^3\alpha_s^3}{32\pi\lambda^3}
+4\pi C_F\alpha_s\left[I_{\text{P}}^{00}[1]+I_{\text{P}}^{10}[1]+I_{\text{P}}^{20}[1]\right],
\end{equation}
where we have given the contribution from the one-loop, zero-gluon exchange
diagram explicitly. It exhibits the characteristic
$1/\lambda^3 \sim E^{3/2} \sim v^3$ threshold behaviour of P-wave
production. The higher-loop integrals $I_{\text{P}}^{(n-1)0}[1]$,
corresponding to diagrams with $n$ gluon exchanges, are given in
App.~\ref{sec:diagrams}. (The notation is explained in more detail in
the context of the NLO calculation after~\eqref{eq:IPa}.)
The remaining part $G_0^{P(\geq4ex)}$ is finite and can be
calculated in $d=4$ dimensions. We perform this calculation
in position space. Eq.~(\ref{eq:GFPNRQCDexpansion}) implies
\begin{equation}
\begin{aligned}
 G_0^{P(\geq4ex)}(E)&\hspace*{0.15cm}
=\mathop{\text{lim}}\limits_{\mathbf{x},\mathbf{y}\ra0}\Braket{\nabla_x\cdot\nabla_{y}\; G_0^{(\geq4ex)}(\mathbf{x},\mathbf{y};E)}\\
 &\hspace*{0.15cm}
=\mathop{\text{lim}}\limits_{\mathbf{x},\mathbf{y}\ra0}\frac{1}{(4\pi)^2}\int d\Omega_{\mathbf{x}}\int d\Omega_{\mathbf{y}}\left[\nabla_x\cdot\nabla_{y}\; G_0^{(\geq4ex)}(\mathbf{x},\mathbf{y};E)\right],
\end{aligned}
 \label{eq:G0Pmorethan4ex}
\end{equation}
where $\Braket{\dots}$ denotes the angular mean of the respective
expression. This projects out the P-wave, $l=1$, component of the Green
function $G_0(\mathbf{r},\mathbf{r}';E)$.
The expression~\eqref{eq:G0Pmorethan4ex} can be computed
using the representation~\eqref{eq:LOGFintegralrep} with appropriate
subtractions for the parts with less than three gluon exchanges. The
sum $G_0^{P(\leq3ex)}+G_0^{P(\geq4ex)}$ gives the expression for the
correlation function in dimensional regularization:
\begin{equation}
  \begin{aligned}
    G_0^P(E) = \frac{m_t^4C_F^3\alpha_s^3}{32\pi\lambda^3}
    \bigg[& 1 -
      \left( \frac{1}{2\epsilon} + 2L_\lambda^w +4 \right) \lambda
      - 3\lambda^2
      + \left( \frac{1}{4\epsilon} + 2L_\lambda^w + \frac{7}{2}
      \right) \lambda^3 \\
      & + 2(\lambda-\lambda^3)\hat{\psi}(2-\lambda) \bigg],
  \end{aligned}
  \label{eq:GFMSbar}
\end{equation}
where
\begin{equation}
\hat{\psi}(z)=\gamma_E+\psi(z),
\hspace{1cm}
L_\lambda^a=\ln\left(\frac{\lambda\mu_a}{m_t\alpha_sC_F}\right)
=-\frac{1}{2}\ln\left(\frac{-4m_tE}{\mu_a^2}\right),
\end{equation}
with $a\in \{r,w\}$, $\gamma_E$ the Euler-Mascheroni constant, and $\psi(z)$
the logarithmic derivative of the Gamma function.
In (\ref{eq:GFMSbar}) subtracting the
$1/\epsilon$ poles gives the result in the $\overline{\text{MS}}$ scheme.
Note that due to the overall $1/\lambda^3$ factor only the term
proportional to $\lambda$ in square brackets results in a
finite-width divergence. We have checked that this divergence in the
imaginary part of the LO Green function~\eqref{eq:GFMSbar} agrees
with the result given
in~\cite{Jantzen:2013gpa,Hoang:2001mm}. Neglecting the width of the
top, the imaginary part is finite, and reads
\begin{eqnarray}
 \text{Im}\!\left[G_0^P(E)\right] &=&
\left(\frac{m_tC_F\alpha_s}{2}\right)^{\!5}
\sum\limits_{n=2}^\infty\frac{n^2-1}{n^5}\,\delta(E-E_n)
\nonumber\\
&& +\,\frac{m_t^4}{4\pi}\left(\frac{E}{m_t}
+\frac{C_F^2\alpha_s^2}{4}\right)
\frac{\pi C_F\alpha_s}{1-e^{-C_F\alpha_s\pi/v}}\,\theta(E),
 \label{eq:IMLO}
\end{eqnarray}
where $v\equiv \sqrt{E/m_t}$, and
$E_n=-(m_tC_F^2\alpha_s^2)/(4n^2)$ with $n\geq2$ are the $l=1$
bound state energies. Eq.~(\ref{eq:IMLO}) agrees with~\cite{Bigi:1991mi}.

\subsection{Next-to-leading order\label{sec:NLO}}

Analogous to the S-wave computation \cite{Beneke:2013PartII},
we define the single-insertion function
\begin{equation}
 I_{\text{P}}[x+u]=
 \int\left[\prod\limits_{i=1}^4\frac{d^{d-1}\mathbf{p}_i}{(2\pi)^{d-1}}\right]
 \mathbf{p}_1\cdot\mathbf{p}_4\;
 \tilde{G}_0(\mathbf{p}_1,\mathbf{p}_2;E)\frac{1}{(\mathbf{q}_{23}^2)^x}
 \left(\frac{\mu^2}{\mathbf{q}_{23}^2}\right)^{\!u}
 \tilde{G}_0(\mathbf{p}_3,\mathbf{p}_4;E),
\label{eq:IP}
\end{equation}
where $\mathbf{q}_{ij}=\mathbf{p}_i-\mathbf{p}_j$.
In terms of this the NLO correction to the Green function contained in
(\ref{eq:GFPNRQCDexpansion}) is given by
\begin{equation}
 \delta_1G^P(E)=C_F\alpha_s^2
 \left[\frac{\beta_0}{\epsilon}\Big(I_{\text{P}}[1+\epsilon]-
 I_{\text{P}}[1]\Big)+a_1(\epsilon)I_{\text{P}}[1+\epsilon]\right].
\label{eq:delta1G}
\end{equation}
As for the LO Green function divergences only occur in diagrams with
up to four loops. We therefore split the NLO correction into a
divergent (a) and a finite part (b) as indicated in
Fig.~\ref{fig:SingleCoulombInsertionab}. Since the top quark width
$\Gamma_t$ cannot be neglected, the imaginary part of $(a)$ contains
divergences of the type $\Gamma_t/\epsilon$ arising from poles of the
form $E/\epsilon$. Thus, contrary to the S-wave case, where no such
divergences are present in the computation of corrections from the
Coulomb potential, the NLO Coulomb
potential~\eqref{eq:VCoulombcoefficients} cannot be expanded in
$\epsilon$ prior to integration
in the computation of the loop integrals of part $(a)$.
However, the momentum integrals of part $(b)$ are
finite and the potential can be expanded before integration here.
\begin{figure}[t]
\centering
 \includegraphics[width=0.85\textwidth]{fig_SingleCoulombInsertionab.eps}
 \caption{We split the NLO correction to the Green function into a
   divergent part (a) that contains all diagrams with up to four loops
   and the finite remainder (b).\label{fig:SingleCoulombInsertionab}}
\end{figure}

To deal with part (b), we start with the computation of the
complete $I_{\text{P}}[1+u]$ in position space and later perform the
necessary subtractions to obtain $I_{\text{P}}^{(b)}[1+u]$:
\begin{eqnarray}
 I_{\text{P}}[1+u] &=&
    \mathop{\text{lim}}\limits_{\mathbf{x},\mathbf{y}\ra0}
    \Braket{ \left(\nabla_x\cdot\nabla_y\right)
      \int d^{d-1}\mathbf{r}\,G_0(\mathbf{x},\mathbf{r};E)
      \frac{\mu^{2u}\left(\mathbf{r}^2\right)^{u-\frac12}}
      {4\pi\Gamma(1+2u)\cos(\pi u)}
      G_0(\mathbf{r},\mathbf{y};E)} \nonumber \\
    && \hspace*{-2cm} = \,\mathop{\text{lim}}\limits_{\mathbf{x},\mathbf{y}\ra0}
    \frac{9m_t^2(2p)^6\mu^{2u}}
    {(4\pi)^3\Gamma(1+2u)\cos(\pi
      u)\,\Gamma(2+\lambda)^2\Gamma(2-\lambda)^2}\nonumber \\
    && \hspace*{-1.8cm} \times \,\int\limits_0^1dt_1\int\limits_0^1dt_2
    \left[(1-t_1)(1-t_2)\right]^{1+\lambda}
    \left[t_1t_2\right]^{1-\lambda}
    \int\limits_1^\infty ds_1\int\limits_1^\infty ds_2
    \left[s_1s_2\right]^{1+\lambda}\left[(s_1-1)(s_2-1)\right]^{1-\lambda}
\nonumber \\
    && \hspace*{-1.8cm} \times \, \Braket{ \left(\nabla_x\cdot\nabla_y\right)
      \int d^{d-1}\mathbf{r} \left(\mathbf{x}\cdot\mathbf{r}\right)
      \left(\mathbf{y}\cdot\mathbf{r}\right)
      \left(\mathbf{r}^2\right)^{u-\frac12}
      e^{-p\left[x(1-2t_1)+y(1-2t_2)+2r(s_1+s_2-1)\right]}}.
 \end{eqnarray}
We proceed by first solving the integral over $\mathbf{r}$:
\begin{equation}
  \int d^{d-1}\mathbf{r}\ r^ir^j\,r^{2u-1} e^{-2pr(s_1+s_2-1)}=
  \frac{\delta^{ij}}{d-1}\,
  \frac{2\pi^{(d-1)/2} \Gamma(d+2u)}{\Gamma((d-1)/2) }
   \left[2p(s_1+s_2-1)\right]^{-d-2u}.
\end{equation}
Taking the derivatives and the limit $\mathbf{x},\mathbf{y}\ra0$, we
can perform the integrations over $t_1$ and $t_2$ and obtain
\begin{equation}
  \begin{aligned}
    I_{\text{P}}[1+u] = &\hspace*{0.15cm}
    \frac{m_t^2(2p)^{6-d-2u}\mu^{2u}}{4(4\pi)^3\Gamma(1+2u)\cos(\pi u)}
    \frac{2\pi^{(d-1)/2}\Gamma(d+2u)}{\Gamma((d-1)/2)} \\
    & \times \int\limits_1^\infty ds_1\int\limits_1^\infty ds_2\,
    \frac{\left[s_1s_2\right]^{1+\lambda}
      \left[(s_1-1)(s_2-1)\right]^{1-\lambda}}{(s_1+s_2-1)^{d+2u}}.
  \end{aligned}
\end{equation}
After the necessary subtraction the remaining part $(b)$ is finite and
can be computed in $d=4$ dimensions. We obtain
\begin{equation}
 I_{\text{P}}^{(b)}[1+u]=\frac{m_t^2p^2}{(4\pi)^2}
\left(-\frac{\mu^2}{4m_tE}\right)^uj_{\text{P}}(u),
 \label{eq:Ib}
\end{equation}
where
\begin{eqnarray}
 j_{\text{P}}(u)&=&\frac{\Gamma(4+2u)}{\Gamma(1+2u)\cos(\pi u)}
\int\limits_0^\infty dt_1\int\limits_0^\infty dt_2\frac{t_1t_2(1+t_1)(1+t_2)}{(1+t_1+t_2)^{4+2u}}\Bigg[\left(\frac{(1+t_1)(1+t_2)}{t_1t_2}\right)^\lambda
\nonumber\\
&&-\,1-\lambda\log\left(\frac{(1+t_1)(1+t_2)}{t_1t_2}\right)
-\frac{\lambda^2}{2}\log^2\left(\frac{(1+t_1)(1+t_2)}{t_1t_2}\right)\Bigg].
 \label{eq:ju}
\end{eqnarray}
Here the last three terms are the first three terms in the expansion
in $\alpha_s$, which corresponds to up to two gluon exchanges to the
left or right of the NLO Coulomb potential insertion. This is precisely
part (a), hence the above, subtracted expression is part (b) as desired.
A strategy to solve this kind of integral is presented
in~\cite{Beneke:2013PartII}. We obtain
\begin{eqnarray}
 j_{\text{P}}(0) &=&
-1+\left(\frac{\pi^2}{3}-2\right)\lambda+3\zeta (3)\lambda ^2+\left(1-3\lambda^2\right)\hat{\psi}(2-\lambda)+\left(\lambda^3-\lambda\right)\psi_1(2-\lambda),
\nonumber\\[-0.4cm]
&& \label{eq:j0}
\\
  j'_{\text{P}}(0)&=&\frac{\pi ^2}{6}-\frac{50}{9}+\left[-4+\frac{2 \pi ^2}{3}-4 \zeta (3)\right]\lambda+\left[\frac{34}{3}+\frac{\pi ^2}{6}-\frac{\pi ^4}{180}+6 \zeta (3)\right]\lambda ^2 \nonumber\\
  &&+\,\left[4+6\lambda-10\lambda^2\right]\hat{\psi}(2-\lambda)+\left(3 \lambda ^2-1\right)\left[\hat{\psi}(2-\lambda)^2-3\psi_1(2-\lambda)\right] \nonumber\\
  &&+\,\left(\lambda^3-\lambda\right)\left[\left(\frac{22}{3}-2\hat{\psi}(2-\lambda)\right)\psi_1(2-\lambda)+\psi_2(2-\lambda)\right] \nonumber\\
  &&+\,\frac{3}{2(\lambda -2)}\, _4F_3(1,1,4,4;5,5,3-\lambda;1).
 \label{eq:jprime0}
\end{eqnarray}
Here $\psi_n(z)$ is the $n$th derivative of the psi function.
We provide some useful formulas for the evaluation of the generalized
hypergeometric function $_4F_3$ in App.~\ref{sec:pFq}. In terms of
$j_{\text{P}}(u)$ and $j'_{\text{P}}(u)$ from~\eqref{eq:j0}
and~\eqref{eq:jprime0} we can write part $(b)$ of~\eqref{eq:delta1G} as
\begin{equation}
 \begin{aligned}
 \delta_1G^{P(b)}(E)=&\hspace*{0.15cm}
C_F\alpha_s^2\left[\beta_0\frac{d}{du}I_{\text{P}}^{(b)}[1+u]\Big|_{u=0}
+a_1(0)I_{\text{P}}^{(b)}[1]\right]\\
 =&\hspace*{0.15cm}
\frac{m_t^4\alpha_s^4C_F^3}{64\pi^2\lambda^2}
\left[\beta_0\left(2j_{\text{P}}(0)L_\lambda+j_{\text{P}}'(0)\right)
+a_1(0)j_{\text{P}}(0)\right],
 \end{aligned}
 \label{eq:delta1GPb}
\end{equation}
where we have expanded the NLO Coulomb potential
(\ref{eq:VCoulombcoefficients}) and written the
logarithm of $\mathbf{q}^2$ as a derivative at zero $u$.

The divergent part $(a)$ is given by
\begin{equation}
 I_{\text{P}}^{(a)}[1+u]=I_{\text{P}}^{00}[1+u]+2I_{\text{P}}^{10}[1+u]
+2I_{\text{P}}^{20}[1+u]+I_{\text{P}}^{11}[1+u],
 \label{eq:IPa}
\end{equation}
where the $I_{\text{P}}^{nm}[1+u]$ denotes the contribution to the
single insertion function from the diagram with $n$ potential gluon
exchanges to the left and $m$ to the right of the potential
insertion. Part $(a)$ then takes the form of~\eqref{eq:delta1G}
with $I_P$ replaced by~\eqref{eq:IPa}, and requires the calculation
of some four-loop diagrams in dimensional regularization.
The results for the  $I_{\text{P}}^{nm}[1+u]$ needed are given in
App.~\ref{sec:diagrams}. The complete NLO correction to the Green function
in dimensional regularization given by the sum of parts $(a)$ and $(b)$ reads:
\begin{eqnarray}
  \lefteqn{\delta_1G^P(E)=-\frac{m_t^4\alpha_s^4C_F^3}{64\pi^2\lambda^2}}
  && \nonumber \\
  && \times \Bigg\{ \beta_0 \Bigg[
  \left(-\frac{1}{12 \epsilon ^2}+\frac{59}{9}+\frac{5 \pi ^2}{72}
    +4L_\lambda^r+2L_\lambda^rL_\lambda^w-\left(L_\lambda^w\right)^2\right)
  +\left(9+6L_\lambda^r\right)\lambda \nonumber \\
  && \phantom{\times\{ \beta_0}
  +\left(\frac{3}{40 \epsilon ^2}+\frac{1}{20 \epsilon }
    -\frac{344}{15}-\frac{\pi ^2}{8}
    -\frac{21}{2}L_\lambda^r+\frac{1}{2}L_\lambda^w-6L_\lambda^wL_\lambda^r
    +3\left(L_\lambda^w\right)^2\right)\lambda ^2 \nonumber \\
  && \phantom{\times\{ \beta_0}
  +\left[-4-6
    \lambda+10\lambda^2+2(3\lambda^2-1)L_\lambda^r\right]\hat\psi(2-\lambda)
  \nonumber \\
  && \phantom{\times\{ \beta_0}
  +\left(\lambda -\lambda ^3\right)
  \left[\psi_1(2-\lambda)\left(\frac{22}{3}+2L_\lambda^r-2\hat\psi(2-\lambda)\right)
    +\psi_2(2-\lambda )\right] \nonumber \\
  && \phantom{\times\{ \beta_0}
  +\left(3 \lambda ^2-1\right) \left[3 \psi_1(2-\lambda)-\hat\psi(2-\lambda)^2\right]
  +\frac{3}{4-2 \lambda } \, _4F_3(1,1,4,4;5,5,3-\lambda ;1)\Bigg]
  \nonumber \\
  && \phantom{\times}
  + a_1(\epsilon) \bigg[
  \frac{1}{6\epsilon} + 2+L_\lambda^w + 3\lambda
  - \left(\frac{3}{10 \epsilon} + \frac{26}{5} + 3L_\lambda^w \right) \lambda^2
  + \left(3 \lambda ^2 - 1\right) \hat\psi(2-\lambda) \nonumber \\
  && \phantom{\times + a_1(\epsilon)}
  +\left(\lambda -\lambda ^3\right) \psi_1(2-\lambda )\bigg]\Bigg\}.
\label{eq:delta1GP}
\end{eqnarray}
The dependence on the two scales $\mu_r$ and $\mu_w$ has been obtained
with the procedure described in App.~\ref{sec:diagrams}. Alternatively,
the dependence on $\mu_r$ can be obtained using one-loop running
of $\alpha_s$ in the LO Green function. The remaining logarithms of
$\mu$ must then be assigned to $\mu_w$. We note that the dependence on
the scale $\mu_w$ is polynomial in $E$ and cancels in the
imaginary part for $\Gamma_t=0$,
which provides a useful consistency check.

\subsection{Pole resummation\label{sec:poles}}

At negative energies the P-wave Green function contains poles
corresponding to quark-antiquark bound states with angular momentum
$l=1$. In~\eqref{eq:delta1GP} they appear as poles of
polygamma and hypergeometric functions for positive integer
$\lambda\geq2$. Near these bound states the exact Green function has
the form
\begin{equation}
 G^P(E)\mathop{=}\limits^{E\rightarrow E_n^P}\frac{|\psi'_n(0)|^2}
{E_n^P-E-i\epsilon}+\text{regular},
 \label{eq:GFpolesexact}
\end{equation}
where $E_n^P$ is the energy of the $n$th P-wave bound state and
$\psi'_n(0)$ the derivative of the corresponding wave function at the
origin. Both take on a perturbative expansion in the strong coupling
constant
\begin{equation}
 E_n^P=E_n^{P(0)}\left(1+\frac{\alpha_s}{4\pi}\,e_1^P
+\mathcal{O}(\alpha_s^2)\right),
\hspace{0.5cm}|\psi'_n(0)|^2=|\psi_n'^{(0)}(0)|^2
\left(1+\frac{\alpha_s}{4\pi}f_1^P+\mathcal{O}(\alpha_s^2)\right).
\end{equation}
The NLO Green function expanded in $\alpha_s$ therefore takes the form
\begin{eqnarray}
    G^P(E)&\mathop{=}\limits^{E\rightarrow E_n^{P(0)}}&
    \frac{|\psi_n'^{(0)}(0)|^2}{E_n^{P(0)}-E-i\epsilon}
    +\frac{\alpha_s}{4\pi}
    \left(\frac{f_1^P|\psi_n'^{(0)}(0)|^2}{E_n^{P(0)}-E-i\epsilon}
      -\frac{e_1^P|\psi_n'^{(0)}(0)|^2E_n^{P(0)}}{\left(E_n^{P(0)}-E-i\epsilon\right)^2}\right)\nonumber \\
    &&+\,\mathcal{O}(\alpha_s^2)+\text{regular}.
  \label{eq:GFpolesexpanded}
\end{eqnarray}
The singular terms near $E_n^{P(0)}$ can be resummed into a single
pole to all orders by
subtracting~\eqref{eq:GFpolesexpanded} from the Green function and
adding~\eqref{eq:GFpolesexact} with the energies and derivatives of
the wave function in NLO approximation \cite{Beneke:1999qg}.
The leading-order expressions
can be read off from the imaginary part~\eqref{eq:IMLO} of the LO
result:
\begin{equation}
 E_n^{P(0)}=-\frac{m_tC_F^2\alpha_s^2}{4n^2},\hspace{1cm}
|\psi_n'^{(0)}(0)|^2=\frac1\pi\left(\frac{m_tC_F\alpha_s}{2}\right)^{\!
  5}\,
\frac{n^2-1}{n^5}.
\end{equation}
To obtain the NLO corrections we expand~\eqref{eq:delta1GP} for
$\lambda$ near positive integer $n$. We find
\begin{eqnarray}
  \lefteqn{\delta_1G^P(E)\,
    \mathop{=}\limits^{\lambda\rightarrow n}\,
    \frac{m_t^4 \alpha_s^4C_F^3}{4(4\pi)^2}} && \nonumber \\
  && \times\Bigg\{
  \frac{n^2-1}{n^2(n-\lambda )}
  \left[2\beta_0\left(2L_n^r+4+\frac{3}{n-1}-\frac{4n^2}{n^2-1}\hat\psi(n+2)
      -2n\psi_1(n+2)\right)+2a_1\right] \nonumber \\
  && \phantom{\times}
  +\frac{n^2-1}{n(n-\lambda)^2}\left[2\beta_0(L_n^r+\hat\psi(n+2))+a_1\right]
  +\text{regular}\Bigg\}.
\end{eqnarray}
With this it is straightforward to obtain:
\begin{align}
 e_1^P=&2a_1+4\beta_0\left[L_n^r+\hat{\psi}(n+2)\right],\\
 f_1^P=&5a_1+2\beta_0\left[5L_n^r+4+\frac{3}{n-1}-\frac{n^2+3}{n^2-1}\hat{\psi}(n+2)-2n\psi_1(2+n)\right],
\end{align}
where $L_n^r=\ln\left(n\mu_r/(m_tC_F\alpha_s)\right).$ We have checked
that this agrees with the results
of~\cite{Penin:1998ik,Penin:1998mx}.

\section{P-wave top-pair production cross
  section\label{sec:crosssection}}

In this section we discuss the phenomenological aspects of the P-wave
contribution to the top-pair production cross section near
threshold. All expressions so far have employed the pole mass
definition of the top quark. Since the pole mass suffers from an
infrared renormalon
ambiguity~\cite{Beneke:1994sw,Bigi:1994em,Beneke:1994rs}, which is
not present in the top-pair cross section itself, we
show results using the PS mass definition~\cite{Beneke:1998rk},
which eliminates this spurious infrared sensitivity. This
has been implemented in the PS Shift (PSS) and PS Insertion (PSI)
schemes~\cite{Beneke:2013PartII}. Denoting by $\delta m_t$
the difference between the pole mass and the PS mass,
the former is defined by
\begin{equation}
  G^{\text{PSS}}(\sqrt{s},m_t^{\text{PS}}) =
  G^{\text{pole}}(\sqrt{s}, m_t^{\text{PS}} + \delta m_t),
\end{equation}
where the value of $\delta m_t$ is order-dependent. That is, in LO,
we only use the leading-order expression $\delta m_t\propto \mu_f\alpha_s$,
whereas in NLO,  $\delta m_t$ includes the $\mu_f\alpha_s^2$ term that contains
the $a_1$ correction to the Coulomb potential~\cite{Beneke:1998rk}.\footnote{
However, when our result is combined with the NNNLO S-wave
contribution, $\delta m_t$ should be used at NNNLO as well.} The
PSI scheme is obtained by re-expanding the right-hand
side in $\alpha_s$. We find, however, that the difference between the
two schemes is very small and would not be visible
in the figures below. We therefore only show the results in the
PSS scheme. For the numerics we adopt the parameter values
$\alpha_s(M_Z) = 0.1184$, $m_t^{\text{PS}}
\equiv m_t^{\text{PS}}(\mu_f=20\,\mbox{GeV}) = 171\,\mbox{GeV}$,
which corresponds to a LO (NLO) pole mass of $m_t = 172.025
\,(172.433)\,$GeV, the top-quark width $\Gamma_t = 1.33\,$GeV and the
Weinberg
angle $\sin^2 \theta_w = 0.23$. The strong coupling is evolved to the
scale $\mu_r$ in the four-loop approximation.

Since $G^P(E)$ is divergent there exists an ambiguity in (\ref{eq:PiaPNRQCD})
whether $G^P(E)$ or $\Pi^{(a)}(q^2)$ should be minimally subtracted,
since the factor that relates both depends on $d$.
This ambiguity is only resolved when the finite term of the non-resonant
contribution is computed. Since the non-resonant calculation does not
refer to any kind of non-relativistic approximation it is natural to
define the non-resonant and resonant contribution by minimal subtraction
of $\Pi^{(a)}(q^2)$. This will be assumed in the following. To be
precise, we write the expansion of the hard coefficient (\ref{eq:ca})
in the form $c_a = 1+ [c_a^{(1)} +\epsilon c_a^{(1\epsilon)}]+
\mathcal{O}(\alpha_s^2)$, and the LO and NLO
Green functions (\ref{eq:GFMSbar}), (\ref{eq:delta1GP}), respectively, as
\begin{equation}
G_0^P(E) = \frac{1}{\epsilon}
\,G_{0, \rm div}+ G_{0,\overline{\mbox{\tiny MS}}}^P,
\qquad
\delta_1 G^P(E) =
\frac{1}{\epsilon^2} \,\delta_1 G_{\rm div2}+
\frac{1}{\epsilon} \,\delta_1 G_{\rm div1} +
 \delta_1 G_{\overline{\mbox{\tiny MS}}}^P.
\end{equation}
Then the expansion of $\Pi^{(a)}(q^2)$ to NLO in minimal subtraction
is given by
$\Pi^{(a)}(q^2) = \Pi_0^{(a)}(q^2) + \delta_1\Pi^{(a)}(q^2)$ with
\begin{eqnarray}
\text{Im}\,[\Pi_0^{(a)}(q^2)] &=& \frac{N_c}{2m_t^4}\,
\frac{2}{3}\, \text{Im}\,[G_{0,\overline{\mbox{\tiny MS}}}^P(E)] -
\frac{2}{9}\, \frac{N_c}{32\pi}\frac{\alpha_s C_F\Gamma_t}{m_t},
\label{PiMSLO}\\
\text{Im}\,[\delta_1 \Pi^{(a)}(q^2)] &=&
\frac{N_c}{2m_t^4}\,
\frac{2}{3}\, \text{Im}\,[\delta_1 G_{\overline{\mbox{\tiny MS}}}^P(E)] +
2 c_a^{(1)} \,\text{Im}\,[\Pi_0^{(a)}(q^2)]
\nonumber\\
&&  + 2 c_a^{(1\epsilon)} \,\frac{2}{3}\,
\frac{N_c}{32\pi}\frac{\alpha_s C_F\Gamma_t}{m_t}
+ \left(\frac{\beta_0}{81} - \frac{a_1(0)}{27}\right)
 \frac{N_c}{32\pi^2} \frac{\alpha^2_s C_F\Gamma_t}{m_t},
\label{PiMSNLO}
\end{eqnarray}
where the constant terms proportional to $\Gamma_t/m_t$
arise from the divergent
parts of $G^P(E)$ multiplying the order $\epsilon$ and $\epsilon^2$ terms
of the factor $(d-2)/(d-1)\times c_a^2$, which
relates  $G^P(E)$ to $\Pi^{(a)}(q^2)$.\footnote{The scale $\mu$
in the logarithm in $c_a^{(1\epsilon)}$, see (\ref{eq:ca}), should
be identified with $\mu_w$.} Note that we assume here that $\Gamma_t$
takes its numerical, four-dimensional, physical value, while for
an analytic combination with the (yet unknown) non-resonant
cross section, one eventually needs to use the analytic, $d$ dimensional,
leading-order expression for the top-quark width. The constant terms
in (\ref{PiMSLO}) and (\ref{PiMSNLO}) shift the
cross section only by a tiny amount, so that the scheme-dependence related
to the resonant-nonresonant separation that
can be resolved only once the non-resonant cross section is fully known,
is not relevant for the following discussion.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.7\textwidth]{fig_RaPSS.eps}
  \caption{The P-wave contribution to the R ratio in the PSS scheme
   for $m_t^{\text{PS}}=171$~GeV, $\Gamma_t=1.33$~GeV,
   $\mu_r=80$~GeV, and $\mu_w=m_t^{\text{PS}}$. The dashed and solid
   lines denote the LO and NLO contributions, respectively. The shaded
   regions show the respective scale uncertainties for a variation of
   $\mu_r$ in the region $[50\text{ GeV},m_t^{\text{PS}}]$. The dark-shaded
   band for the small NLO scale variation is hardly visible. The
   dotted line denotes the Born-level result.\label{fig:Ra}}
\end{figure}

In Fig.~\ref{fig:Ra} we show the LO and NLO P-wave contributions to the
$R$ ratio in the PSS scheme, employing the expressions (\ref{PiMSLO}),
(\ref{PiMSNLO}),
and including pole resummation up to the $n=6$ bound-state pole.
The overall size relative to the S-wave is below $1\%$
in the threshold region, because in addition to the $v^2$ suppression
the ratio of the couplings $(v_e^2+a_e^2)a_t^2/e_t^2\approx0.28$ is
small.\footnote{The possibility to extract the P-wave contribution
  using different beam polarizations was discussed in~\cite{Kuhn:1999hw}.}
In order to determine the theoretical
uncertainty we vary the renormalization scale $\mu_r$ in the range
$[50\text{GeV},m_t^{\text{PS}}]$, while keeping $\mu_w=m_t^{\text{PS}}$
fixed. The latter scale is chosen of order of the hard scale
in order to capture the logarithmically enhanced contribution of
the unknown non-resonant part. We observe that the dependence on
$\mu_r$ is much reduced at NLO, which implies
that perturbation theory works very well for the P-wave contribution.
A comparison with the Born level result (dotted curve) shows a
sizable difference and thus the importance of the Coulomb resummation,
which enhances the cross section. Note that unlike the S-wave case,
the strong Coulomb attraction does not lead to a peak structure in
the energy-dependence of the cross section, since the residue of
the lowest $n=2$ P-wave bound state is already too small to be visible given
the large top-quark width.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.45\textwidth]{fig_murdependenceabovethreshold.eps}
  \includegraphics[width=0.45\textwidth]{fig_muwdependenceabovethreshold.eps}\\[1.5ex]
  \includegraphics[width=0.45\textwidth]{fig_murdependenceatthreshold.eps}
  \includegraphics[width=0.45\textwidth]{fig_muwdependenceatthreshold.eps}\\[1ex]
  \includegraphics[width=0.45\textwidth]{fig_murdependencebelowthreshold.eps}
  \includegraphics[width=0.45\textwidth]{fig_muwdependencebelowthreshold.eps}
  \caption{\label{fig::scales}The scale dependence of the P-wave
    contribution to the R ratio. The left and right plots show the
    dependence on $\mu_r$ for fixed $\mu_w$ and vice versa. The
    dependence is shown above (upper plots), close to (middle plots), and
    below  threshold (lower plots) corresponding to
   $\sqrt{s} = 347.733, 344.866, 342\,$GeV, respectively. In the left plots
the finite width scale is fixed to $\mu_w = m_t^{\text{PS}}$. In the
right plots the coupling-renormalization scale is fixed to $\mu_r =
80\,$GeV.}
\end{figure}

A comparison of the dependence on the two scales is shown in
Fig.~\ref{fig::scales} for three values of the energy (above, at and
below threshold from top to bottom). We again see
that the dependence on the renormalization scale $\mu_r$ is strongly reduced at
NLO. This is not the case for the dependence on $\mu_w$,
which remains almost the same as at LO. This is expected, since
the finite-width scale dependence does not cancel by performing higher-order
QCD calculations. Rather, it has to cancel only when the non-resonant
corrections are added to the result. The fact that the finite-width
scale dependence is dominant at NLO shows the importance of this
cancellation. In particular, the lower-right plot shows that the
finite-width scale dependence changes the cross section by a large
factor below threshold, precisely where the non-resonant contributions are
expected to be important.

\section{Top-pair production in photon collisions\label{sec:photon}}

The photon collider option via the back-scattering
method~\cite{Ginzburg:1982yr} was studied in the Technical Design
Report for TESLA~\cite{Badelek:2001xb} and is also considered at the
ILC (see Sec.~12.6 of~\cite{Adolphsen:2013kya}). We discuss here only
$\gamma\gamma\to t\bar t$ collisions
with opposite photon helicities, where the top pair is
produced in a P-wave state. We define the inclusive normalized cross section
$R_\gamma^{+-}=\sigma_{\gamma\gamma\rightarrow t\bar{t}X}^{+-}/\sigma_0$. Near
the production threshold the cross section takes the
form~\cite{Penin:1998ik,Penin:1998mx}
\begin{equation}
 R_\gamma^{+-}=\frac{32\pi N_ce_t^4}{m_t^4}\,
C_h^{+-}(\alpha_s)\,
\text{Im}\left[G^P(E)\right],
\end{equation}
with the hard matching coefficient
\begin{equation}
 C_h^{+-}(\alpha_s)=1-16C_F\frac{\alpha_s}{4\pi}.
\end{equation}
In the absence of a $d$ dimensional calculation of $C_h^{+-}$ we define
here the R ratio, to which non-resonant contributions
should eventually be added, by minimal subtraction of the P-wave
Green function $G^P(E)$.

We show results within the PSS scheme in Fig.~\ref{fig:R_pm}. The P-wave
cross section is an order of magnitude larger than in $e^+e^-$
collisions, due the different size of the electroweak couplings and
numerical prefactors. It can
be observed independently of the S-wave by adjusting the beam
polarizations. Since the $\gamma\gamma$ induced cross section differs
from the $e^+e^-$ one only by the hard-matching coefficient and
overall electroweak couplings, we observe essentially the same
features as in the previous section. Most importantly, the theoretical
uncertainty of the QCD contributions as measured by the residual
$\mu_r$ dependence is greatly reduced at NLO as seen from the (hardly visible)
width of the dark-shaded band in Fig.~\ref{fig:R_pm}.
\begin{figure}[t]
  \centering
  \includegraphics[width=0.7\textwidth]{fig_RgammaPSS.eps}
  \caption{$R_\gamma^{+-}=\sigma_{\gamma\gamma\rightarrow
      X\bar{t}t}^{+-}/\sigma_0$ at LO (dashed) and NLO (solid) in
   the PSS scheme. The same coding as in Fig.~\ref{fig:Ra} is
   adopted.\label{fig:R_pm}}
\end{figure}

We have compared our result for the Green function with a previous result
from~\cite{Penin:1998ik,Penin:1998mx} and found agreement of the
scheme-independent terms at LO. The scheme-dependent finite-width
terms have been fixed in these papers by matching the non-relativistic,
resonant computation to the full theory diagram with an off-shell
top-quark self-energy. This procedure accounts for part of the
non-resonant contributions, but does not eliminate the need for a
complete calculation of the $\gamma\gamma \to W^+ W^- b\bar b$
process with opposite photon helicities
in the vicinity of $\sqrt{s}\approx 2 m_t$ to achieve
parametric LO accuracy.\footnote{The corresponding calculations
for $e^+ e^-\to t\bar {t}$ ~\cite{Beneke:2010mp,Penin:2011gg} suggest,
however, that the contributions
from the off-shell self-energy diagrams might be numerically the most
important.}
Refs.~\cite{Penin:1998ik,Penin:1998mx} also
present a calculation of the NLO P-wave Green function, but contrary
to the closed expression (\ref{eq:delta1GP}) in dimensional regularization,
the result is given in a sum representation that makes the comparison
of even the scheme-independent terms difficult. We find, however,
that we are able to
reproduce the plots in~\cite{Penin:1998ik,Penin:1998mx} to a good
approximation, if we choose the scale of the hard matching coefficient
and $\mu_w$ at the hard scale.

\section{Stop-pair production at $e^+e^-$ and hadron
  colliders\label{sec:susy}}

The production of pairs of the scalar supersymmetric partner of the
top quark is of interest at both hadron and $e^+ e^-$ colliders.
The threshold cross section of stop-antistop pairs in $e^+e^-$ collisions
can be described in a fashion analogous to the previous
sections~\cite{Bigi:1991mi}. Since
the coupling of squarks to photons and $Z$ bosons contains a derivative,
stops are produced in a P wave. We focus here solely on production of
the lighter mass eigenstate $\tilde{t}\equiv\tilde{t}_1=
\tilde{t}_L\cos\theta_{\tilde{t}}+\tilde{t}_R\sin\theta_{\tilde{t}}$. The
ratio
$R_{\tilde{t}\tilde{\bar{t}}}=\sigma_{\tilde{t}\tilde{\bar{t}}X}/\sigma_0$
is given by
\begin{equation}
 R_{\tilde{t}\tilde{\bar{t}}}=12\pi\,\Bigg(
e_{\tilde{t}}^2-\frac{2q^2}{q^2-M_Z^2}v_ez_{\tilde{t}}
e_{\tilde{t}}+\left(\frac{q^2}{q^2-M_Z^2}\right)^{\!2}\!
(v_e^2+a_e^2)z_{\tilde{t}}^2\Bigg)\text{ Im}\left[\Pi^{(\partial)}(q^2)\right],
 \label{eq:Rratiostop}
\end{equation}
where $z_{\tilde{t}}$ is the coupling constant for the
$Z\tilde{t}\tilde{\bar{t}}$ vertex, which depends on the mixing
angle $\theta_{\tilde{t}}$, and $\Pi^{(\partial)}(q^2)$ is the
two-point function of the derivative current, which
matches to the NRQCD current
\begin{equation}
 j^{(\partial)k}=\frac{1}{m_{\tilde t}}\, \psi^*\,i{\partial}^k\chi^*.
\end{equation}
Here $\psi$ denotes the stop and $\chi$ the anti-stop field in the
non-relativistic normalization
$\psi\sim\chi\sim m_{\tilde{t}}^{3/2}$. Including
the hard matching coefficient, the two-point function takes the form
\begin{equation}
 \begin{aligned}
  \Pi^{(\partial)}(q^2)=\frac{N_c}{4(d-1)m_{\tilde t}^4}\, c_{\partial}^2\, G^P(E),
 \end{aligned}
 \label{eq:PipPNRQCD}
\end{equation}
where
\begin{equation}
 c_{\partial}=1-4C_F\,\frac{\alpha_s}{4\pi}+\mathcal{O}(\alpha_s^2)
\end{equation}
is the hard matching coefficient of the current
$j^{(\partial)k}$~\cite{Hoang:2005dk} and
\begin{equation}
 G^P(E)=\frac{i}{N_c}\int d^dx\,e^{iEx^0}
\Braket{0|T\left([\chi i{\partial}^k\psi](x)[\psi^* i{\partial}^k\chi^*](0)
\right)|0}_{\text{PNRQCD}}
\end{equation}
the P-wave Green function.
It is straightforward to evaluate this expression.
Since (\ref{eq:PipPNRQCD}) differs from the corresponding
expression for top-antitop production only by the prefactor, we observe
the same qualitative features. Most importantly, the theoretical
uncertainty of the QCD contributions as measured by the residual
$\mu_r$ dependence is greatly reduced at NLO as compared to the LO
calculation~\cite{Bigi:1991mi}.

In the following we comment on the relevance of the P-wave for the
production of stop-antistop pairs at hadron colliders. In quark-antiquark
annihilation the t-channel  gluino exchange
diagram that is dominant for pair production of
light-flavour squarks is strongly suppressed for stops due to the
negligible top parton distribution function in the proton.
Thus, in quark-antiquark annihilation the s-channel,
which produces stop pairs in a P-wave and
colour-octet state, is the dominant contribution. Production of stop-antistop
pairs in gluon-fusion can be described as for light-flavour
squarks. In~\cite{Beneke:2010da} a formalism for resummation of
soft and Coulomb corrections in S-wave production of pairs of heavy
coloured particles was derived at the next-to-next-to-leading logarithmic
approximation (NNLL). This was generalized to stop
pair production at NLL in~\cite{Falgari:2012hx,Falgari:2012sq}.
The arguments presented
there suggest that the factorization formula also holds at NNLL. The
NLO P-wave Green function derived in this work constitutes the
dominant part of the potential function $J^{R_\alpha}$ accounting for
NNLL terms beginning with $\alpha_s^2/\beta$. To obtain the colour-octet
Green function one only has to make the replacement $-C_F\rightarrow
D_8=1/(2N_c)$ in (\ref{eq:VCoulombcoefficients}). Additionally,
non-Coulomb potentials yield
terms beginning with $\alpha_s^2\log\beta$~\cite{Beneke:2013opa},
which also have to be included at NNLL order.

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

We have computed the P-wave Green function in dimensional
regularization up to NLO. We further confirmed results for the NLO
correction to energy levels and wave functions at the origin of
P-wave bound states. The NLO correction reduces the renormalization
scale uncertainty considerably. We have discussed the P-wave
contribution to three different pair production
processes. The NLO P-wave contribution to the top-quark pair production cross
section near threshold is part of the complete NNNLO result.
The P-wave production cross section turns out to be small relative
to the dominant S-wave, below 1\%. It is included in the forth-coming
NNNLO result for the $e^+e^-\rightarrow t\bar{t}X$ cross section~\cite{Beneke:2014}.
The photon collider option further offers the possibility to produce
tops in a pure P-wave and with a larger cross section.

In $e^+e^-$ collisions squark-antisquark pairs are also produced in a
P-wave. We have given the necessary formulas for the NLO cross
section. If squarks that are sufficiently light for production at a
future linear collider should be found, this will allow precision
studies including a precise mass determination. The NLO P-wave Green
function is also an important ingredient in the NNLL prediction of
stop-antistop production in hadron collisions.

\noindent
\subsubsection*{Acknowledgement}
This work has been supported by the DFG
Sonder\-forschungs\-bereich/Transregio~9 ``Com\-pu\-ter\-gest\"utzte
Theoretische Teil\-chen\-physik'', the Gottfried Wilhelm Leibniz
programme of the Deutsche Forschungsgemeinschaft (DFG),
and the DFG cluster of excellence ``Origin and Structure of the Universe.''

redpajama_set_name: RedPajamaArXiv


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