text:
Middle East District celebrates Quarles, welcomes Kinsman
Col. Patrick Kinsman assumed command of the U.S. Army Corps of Engineers, Middle East District, in a ceremony at the District's headquarters in Winchester, Va. Dec. 8.
Huntington District's Medley discovers ERDC University opportunities
The lure of hands-on involvement in research and lab work attracted Huntington District Geologist
Lab hosts Norwegian Polar Institute researcher
Engineer Research and Development Center's Cold Regions Research and Engineering Laboratory Polar
Finding a pathway to natural resource management opportunities
How does a college student find a career with the government? In 2010, an Executive Order was signed by President Obama to implement a program titled Pathways, this authorizes the U.S. Army Corps of Engineers, Kansas City District to hire students two different ways. The first authority allows the hiring of temporary park rangers and maintenance workers to fulfill a seasonal work program at lake projects. The second authority allows students to enter the Natural Resource Management or Maintenance Training programs in which students are provided employment while they are trained for these specialized careers.
The lure of hands-on involvement in research and lab work attracted Huntington District Geologist Erica Medley to apply for ERDC University, and both she and her district leaders were delighted when Medley joined the inaugural class to begin project work in August.


text:
New Video: Kanye West "All Day/I Feel Like That"
Posted by Andy B. under Videos
Close to a full year after Kanye West debuted the song at the BRIT Awards in London, the Steve McQueen-directed video for "All Day" has finally arrived online. The one-take production, which was initially screened during Paris Fashion Week last year, finds Ye bouncing around an abandoned building. Sweaty and out of breath, he then slouches on the floor and performs an unreleased song called "I Feel Like That."
Still no word on when the "final version" of The Life of Pablo will be dropping, but it's unlikely "All Day" will appear on the album.
UPDATE: Watch the video below…
[Nah Right]
Related: The Messy Rollout—and Aftermath—of Kanye West's 'The Life of Pablo'
Monday, Feb 22nd, 2016 at 9:27 am

redpajama_set_name: RedPajamaCommonCrawl


How citizens of Marshall Islands apply a Vietnam visa in Taiwan?
This section explains how citizens of Marshall Islands who currently travel or do businesses in Taiwan want to get a Visa to visit Vietnam.
If you hold passport of Marshall Islands, you have two options to apply for Vietnam visa in Taiwan.


I woke up in a bad mood this morning. Something was gnawing at me. So I sat down to write, as I do most mornings, and just put it all down. I'm not joking when I say I felt better in ten minutes. It doesn't always happen so fast, but I can tell you that writing it down makes me feel better nine times out of ten.
This is an outlet more people need to explore. Don't worry about writing well. Just set aside ten or twenty minutes and don't stop writing. You'll get to the bottom of what's bothering you real quick. Writing this morning made me realize that the problem was I was being too hard on myself. But I wouldn't have got there if I hadn't written. Not even close. I would've stewed in it all day but now I get to move on.
I remember being grateful when I woke up because I knew writing would make me feel better. Too many people walk around with their feelings untouched and unexamined. Ignoring how you feel only makes it worse, and we all know that. Writing can help you pin down what's really going on. Ten minutes. That's all you need. Just don't stop. Don't censor yourself.


Arsenal defeated Cardiff City on Tuesday so let's take a look at the three lessons learnt from this game which has put them level on points with Chelsea.
Arsenal began on a bright note as they opened the scoring through usual suspect Pierre-Emerick Aubameyang. This gave the Gunners the leverage and mental assertion to control proceedings in the contest.
Alexandre Lacazette then later doubled the lead of the gunners but they however conceded one goal in the last minutes of the game because of poor defending from the North Londoners.
With the 2-1 win against Cardiff City, we take a look at the possible lessons as observed from the match.
Arsenal picked up the valuable three points at home but their inability to keep a clean sheet was a let down to Unai Emery's side at home.
Arsenal's two goals lead was cut as Bernd Leno was well beaten in between the sticks as Cardiff aimed to cause a scare like Burnley did at Old Trafford against Manchester United.
There is need for the club manager to tighten his sides defence while also establishing balance and minimum equilibrium in the team as Arsenal keep themselves in contention for the major positives this season.
The Gunners inability to keep a clean sheet was made possible as the visitors scored to half the deficit at 2-1 away from home.
This showed they have something to prove as Unai Emery needed to be very mindful to seal the win for Arsenal in the latter stages of the game.
That moment was necessary as it produced an interesting game, all through the 90 minute duration.
If you are an Arsenal fan, you will be delighted at what the table standings is showing at the moment. This is because Arsenal has leveled up with London rivals Chelsea on points.
This will motivate the gunners to steer up a grand finale when they take on the wounded Manchester City side who lost 2-1 at St. James Park to Newcastle United despite taking an early lead in the fixture.


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Alternatively, you can use these helpful tools to access them on the web.
Use the KCL CADalog search engine to find our specs.


text:
Q: Why does scipy.signal.correlate2d() use such an inefficient algorithm? A few days ago, I started a script to utilize scipy.signal.correlate2d to compute the 2-D autocorrelation of a 4096x4096 image.  The exact call is
zauto = signal.correlate2d(image, image, mode='full', boundary='wrap')

Three days later, it's still running with no end in sight.  I eventually realized it must be doing a brute-force element-by-element convolution, a procedure that goes with N^2, thus 4096^4 = 281 trillion multiplies and adds.
In the meantime, I eventually figured out that I can get the desired result by taking the 2-D FFT of the image, converting it to a 2-D power spectrum, and then taking the inverse FFT;  i.e.,
image -= np.mean(image)            # remove constant bias
zfft = np.fft.fft2(image)          # take 2-D FFT (complex)
zpower = zfft*np.conjug(zfft)      # convert to power spectrum
zauto = np.real(np.ifft2(zpower))  # take inverse FFT
zauto /= zauto[0,0]                # normalize

The above lines take under one minute to complete.
My question:  why doesn't scipy.signal.correlate2d include at least the option to use the much more efficient algorithm when feasible rather than letting the user discover the hard way that it simply can't be used on larger images?

A: Congratulations, you've rediscovered the convolution theorem!
Joking aside, scipy does give you the option of doing convolution in either the signal or Fourier domain, just not with the explicit 2D method you've selected. The signal.correlate function will handle N-dimensional convolution, and will either try to select a good method for you (the method="auto" argument), or you can force it to use the one you want.
But note that the Fourier-method is not always the most efficient. While the FFT-based convolution does have better asymptotic complexity than the direct method, the constant factors are larger. There is a significant amount of book-keeping and setup required to perform the FFTs involved, while the direct method is dead-simple. This is the reason for the "auto" option and the choose_conv_method function: the "best" method depends on the sizes of the inputs.
As to why? I can't really answer that, other than to say that the library authors might have felt that most people using these tools would be aware of the trade-offs. The scipy.signal module provides a number of similar methods, and several parts of the documentation make clear the fact that FFT-based convolution is not a panacea.

redpajama_set_name: RedPajamaStackExchange


text:
In Defense of Being Your Own Gatekeeper
If you're still feeling squeamish about self-publishing, consider this: the Bowker book data company recently reported that the number of self-published books in the U.S. reached 391,000 in 2012. While it may not sound like a lot, this statistic marked a 59% increase over 2011 and, in a more dramatic turn, exceeded the number of traditionally published hard copy books for that same year (only 301,642).
While the numbers speak loud and clear about a cultural shift in book publishing and buying, it does not necessarily sound a death knell for big publishers – it simply means writers are faced with more options regarding how to go about getting their work out to readers. By the same toll, readers are now faced with the same choice, from the other side of the lens. Will they take self-published books as seriously as those published the customary way?
In a March 2014 issue of The Beacon, I was asked to comment on this topic for their cover article "A New Era for Self Publishing" (which cited the statistics above). The article explores the self-publishing movement especially as it relates to older writers, who are quickly catching onto the new trend and enjoying the departure from traditional publishing houses, thanks in part to the fact that the cost to self-publish has dropped significantly in the past decade.
The article also included comments from two Columbia-area writers, psychologist and poet Pamela Armstrong and mystery novelist Peter Pollak, as well as two of my clients, novelist Patty Sroka (who writes romantic suspense under the pen name P.J. O'Dwyer) and Jeanne Ketley, who recently put out a homeowners' guide to homeowner's associations. All of these writers have made the decision, some after varying degrees of success with traditional publishers, to independently publish their work.
The upside to putting your own book out is, as the article points out, that writers can collect 80 to 90 percent of their sales revenue, a stark comparison to the 10 percent publishing houses typically offer. The downside is that you'll also have to independently take care of your own marketing, PR, book design, editing – all of which a publishing house will manage for you, in the event that they read your book proposal, accept it, and agree to publish the work, a process that means waiting in the hands of gatekeepers whose job it is to decide your fate.
Allowing writers the satisfaction of publishing on their own terms in their twilight years gives them the power to get their words out there when many have already been through what Sroka called (and many others have echoed) the "emotional" process of traditional publishing. But you certainly don't have to be a veteran of the publishing battle to take advantage of the opportunity to be the gatekeeper of your own work.
The Beacon Newspapers is "a family-owned business dedicated to providing well-written, useful information of interest to people 50 and over and their families. The company was founded in 1989 by publisher Stuart Rosenthal and his wife, Judy, the associate publisher."
Free monthly print editions of The Beacon cover the Greater Washington (D.C.) area, the Greater Baltimore (Maryland) area and Howard County (Maryland).
Many thanks to Robert Friedman for inviting me to interview and to the folks at The Beacon for their great work.
Check here for more of my posts on self-publishing!
Filed Under: Latest News, Publishing Business, Self-Publishing Tagged With: client success, indie publishing, publishing, self-publishing, writing business
Client Success Story: Author Finds Missing Pages From the Parent Handbook
For many parents, the mountain of self-help style books on the art and science of child rearing can seem overwhelming. Frustration sets in, and mom and dad typically resign themselves to "winging it." Unlike most parenting resources, however, The Missing Pages of the Parent Handbook taps into collective wisdom to encourage and educate parents who are feeling overloaded.
Rather than induce anxiety or hyperactive planning, the book humanizes and personalizes the trials and tribulations of parenthood by incorporating wisdom from many different parenting stories. The topics covered in The Missing Pages are variously heavy or light, and range from dealing with teachable moments gone wrong to bullying to postpartum depression. Both expectant parents as well as parents of grown children will find insightful, supportive information inside.
Taking Control of the Content Through Self-Publishing
Once the manuscript was complete, Christina Brockett began to research her options for bringing it to the market. She was faced with spending a lot of time querying agents in the hope it would be chosen, or she could go the self-publishing route. The ability to maintain control over the content, cover, and final outcome appealed to her, so she chose to self-publish.
Professional Help for Professional Results
Christina says, "I could not have created the book I did without help from Ambitious Enterprises. Ally's developmental editing insight was invaluable and the copyediting services provided were outstanding." And the results speak for themselves: Since its release in May, The Missing Pages of the Parent Handbook has received many great reviews and was selected by Independent Publisher as a Notable New Release in August. Most recently, it was selected as a Finalist in the Parenting & Family category of the 2013 USA Best Book Awards, sponsored by USA Book News. Christina has also spoken on numerous radio shows nationally, and appeared on Al Jazeera America TV as well, to discuss parenting and aspects of the book.
But Christina isn't one to rest on her laurels. "While the success so far is exciting, I'm always thinking about other ways to promote the book and what I could be doing better."
Before The Missing Pages, Christina wrote mostly for pleasure, usually short works of fiction or personal essays. Researching, writing, and revising a full-length book opened a whole new world to her. She was amazing to watch as her creation morphed. She quickly realized that the actual writing and development of the book was one thing; however, the marketing of the book was another. "Despite what I thought I knew," she says, "I wasn't fully prepared for the amount of work that goes into the post-release marketing of a book."
"There are so many different avenues to pursue when marketing a book, it honestly can be overwhelming," she continues. "As a novice, I really didn't know which ones would yield the best results and made some mistakes along the way. For instance, my book is geared towards parents so I tried to capitalize on the Mother's Day holiday and planned my launch very close to it. Unfortunately some elements of the cover and interior design work were delayed. While the book was ready for the launch, it was not ready in time for me to pursue a lot of avenues for pre-publication marketing opportunities. If I were to do it again, I probably would have delayed the launch. Given that I can't go back and change the approach I took pre-publication, I closely monitor my previously developed plan and am constantly evaluating what is and is not working and adjust accordingly.
The other thing to take into consideration is the amount of time you need to allocate to marketing efforts. Although I had developed a plan, I had no way of understanding how much time I would spend on each marketing task."
Invest In Your Success
Christina recommends that anyone looking to publish a book, especially to self-publish a book, makes sure to align themselves with true professionals. "The market is full of books that could be great, but aren't because the reader is distracted by poor editing or they can't get past the book cover," she cautions. "Investing in a good editing team and book designer is critical to the success of any book. These are investments you really can't skimp on."
Christina Brockett is a working mother, speaker, writer and mom blogger. Her work has appeared in the Frederick News Post, Frederick Magazine's City and County Guide, the Torpedo Factory Art Center and Getsparked.org. Christina has appeared on Al Jazeera America and numerous radio shows. She lives outside of Washington, D.C. with her husband, two children and dog. To learn more about Christina Brockett, please visit her website at www.christinabrockett.com or her blog at www.momevolving.com.
Print and eBook forms of the book are available for purchase at Amazon, BarnesandNoble.com, and through iTunes.
Filed Under: Book Marketing, Latest News, Self-Publishing Tagged With: book marketing, client success, editing, indie publishing, marketing, marketing platform, publishing advice, self-publishing
Client Success Story: Award-Winning Author Battles Corruption in the Foster Care System
When Sherrie Clark and her husband decided to become foster parents so they could eventually adopt they expected a few challenges, but they never anticipated the deceit, manipulation, corruption, secrecy, and conflict they would encounter. This Floridian couple found themselves entangled in a true-life horror story battling a powerful secret society where, on a daily basis, children end up victims of personal agendas, a financial bottom line, and apathy. "My family's desire to adopt a child led us down a path into an underground world that very few know exist," Clark reveals.
The heart-wrenching journey that Clark and her family experienced inspired her to write. "I wanted to tell my story and blow the whistle on what this 'secret society' is doing to victimize children," Clark explains. In her award-winning memoir Small Voices Silenced: The Secret Society of Sacrificed Children, Clark exposes some of the most horrid practices among foster-care agencies. While the book includes all the components of a juicy fiction novel, that this is a true story delivers an extra emotional punch that can shake readers to their core.
Why Self-Publishing?
Clark decided to self-publish Small Voices Silenced instead of seeking an agent and publishing contract. "I believe that my book has an important message that needs to get out to the public without any more delays," she explains. "I worked on this book for so long, and I knew that it would add more time to write the query and book proposal, submit it to publishers/agents, and have them evaluate it. And if it was picked up by a traditional publisher, it probably wouldn't hit the marketplace for at least another year after that." Self-publishing allowed Clark to get her message out faster.
Additionally, Clark felt her personal story was too special to hand over to a publisher. "I put so much time, effort, and sacrifice into writing it…sending it to publishers felt like giving birth to a baby and then signing away my rights to someone else. Being a surrogate was not for me, so self-publishing was the best way to go."
Self-publishing brings its own challenges, but Clark spent a lot of time researching her options. Having studied just about every aspect of the publishing industry, she knew she needed a strong editor to help her effectively craft her story in a way that would appeal to readers. "My writing experience is as a journalist, and I realized I was writing a memoir without changing my "journalistic" mind-set and writing habits," she describes. "Hiring Ambitious Enterprises for a developmental edit of my manuscript was one of the best decisions I've ever made. I credit Ally for helping me transition and cross over from the journalistic to the creative."
From Rough Draft to Award-Winning Book
"When I first brought my book to Ally, my manuscript was very raw," Clark says. "Ally showed me my weaknesses and my strengths, what to do to improve my story and how to do it, and she did so in a way that clicked in my journalistic brain. I realized I do have a creative part inside of me, and I needed to tap into it and connect to it when writing this story. Without Ally, I truly believe it wouldn't be the book it is today."
Since rewriting her manuscript, Clark has received some very notable feedback. Small Voices Silenced has won the iUniverse Editor's Choice Award, which honors excellence in writing, and the prestigious Rising Star Award. "Only those who have won the Editor's Choice Award are invited to apply for the Rising Star Award, and only a few applicants are selected, so I am honored to receive this award," Clark says. Beyond the awards, Clark already sees her book taking on a life of its own. "I have received some strong feedback from those who have read my book, even strangers outside of my family and friends who read it from a critiquing perspective. All of it is encouraging and surreal and very motivating to keep moving forward with writing more books."
As Clark pursues future writing projects, she will be able to apply what she's learned about the self-publishing process while promoting Small Voices Silenced. "I wish I'd known that there would be much more time, effort, and sacrifice needed to bring the book into the marketplace after writing the words 'The End'," she says. "And it doesn't stop there. You can write the best book in the world, but if no one knows about it, then you're just going to have a book that you've written. Marketing your book is an ongoing process, and you have to believe in what you wrote. If you don't, the process can be quite daunting, especially if you're not a salesperson."
The challenges, though, make the rewards worth that much more. "This is an endless road, but that also means the possibilities are endless," she says. For authors currently entrenched in the writing and publication process, Clark urges persistence. "Be determined. Keep going despite the obstacles and challenges that can and will crop up to keep you from writing that book inside of you. Push past them and keep going. And keep learning about your craft. The results will be well worth the time invested."
You can find Small Voices Sacrificed at iUniverse, Amazon.com, and BN.com.
Filed Under: Book Marketing, Latest News, Self-Publishing Tagged With: client success, indie publishing, publishing, self-publish, self-publishing, Sherrie Clark, Small Voices Silenced
Client Success Story: Fluid Explores Good and Evil Using New Digital Format
How does a writer successfully craft a cohesive, engaging story that offers readers over 500 different storyline possibilities where they decide, by their own impulses, whether it ends happily ever after or not? The answer: With ingenuity, a special flair for storytelling, and a little help from an ambitious editor.
Author Travis Sentell went beyond the ordinary romance novel when he wrote Fluid, a digital novel that uses today's ebook technology to completely transform the "choose-your-own-adventure" reading experience. It's truly a unique approach to crafting a story because readers have the ultimate control over how the characters interact, and how the story eventually ends. "In Fluid, readers nudge the thoughts and actions of key characters, resulting in over 500 different pathways through a sprawling literary romance novel. You have only yourself to blame as the world burns or rejoices," Sentell explains.
Fluid is both the large story of a cosmic battle between God and Satan, and the small story of two lost teenagers, Chastity and Austin, yearning for connection in a greedy, manipulative world. As often as fate throws them together, life after reincarnated life, so too does fate rip them apart, time after time. When they once again meet face-to-face in Dr. Abramson's office, it's time for the doctor himself to play a role in deciding whether they will ultimately be together or live apart forever. As God and the Devil watch and plot, conspiring to destroy or create true love, the clock ticks and nothing short of the end of all Creation is at stake.
Right Editor + Right Publisher = Awesome
"It is a difficult book to edit, with many divergent story lines, possibilities and endings," Sentell acknowledges. But I'm always up for a good challenge and an exciting project, so I was in. "Early on in the project," Sentell says, "Ambitious Enterprises helped to craft and shape the central narrative of a fairly sprawling and experimental novel. Ally was responsive, intelligent, and firm with her feedback, going above and beyond the call of duty to do what was best for the book. In the end, she helped me deliver a fantastic final product to the marketplace."
Sentell knew from the start that finding the right outlet for this unique project would be difficult. With the help of his agent, he discovered Coliloquy, a publisher he knew would be able to fulfill his vision for Fluid. "Coliloquy is a new, digital-media publisher out of Silicon Valley. They were doing new things, and Fluid is a new type of book, and it seemed to be a natural fit. The book couldn't really exist in any other form, and no other publisher had the available technology to execute it properly," describes Sentell.
Some Things I Wish I Knew
Coliloquy and Fluid launched about the same time, resulting in a great flurry of press for Sentell. "Reviewers and journalists were intrigued by the format and interested by the content, so the phone calls came fast and furiously. I received a boatload of positive reviews on Amazon, and was featured in the magazine The Daily with a very positive review."
Now, as he speaks to book clubs around the country, he's really enjoying meeting his fans. But despite the early and immediate media attention, Sentell wishes he would have had a more developed, more complete advertising strategy prior to the launch of the book. "It's a complicated, busy media world out there, and it's very easy for books to get lost in the shuffle," he explains. [I urge you to check out these fascinating promo videos in which Sentell challenges strangers to help him experiment with free will and predestination. I bet you can't figure out how he performs these magical tricks! If you crave more after viewing these—and I think you will—you can find more videos on Sentell's own site here.]
Luckily, Sentell was able to grasp the opportunities uncovered by the media whirlwind and wasn't afraid to pursue new outlets for his digital adventure. "Fluid comes out on iPad and iPhone soon," he reveals, "so I'm preparing to advertise and hopefully to introduce myself to a whole new group of fans!"
If only Sentell had had a crystal ball to see what the final product was going to look like—he says he would have hired Ambitious Enterprises sooner and saved himself some time. "I took a lot of wrong turns trying to get the narrative crafted," he admits. Yet, the experience he gained through the writing, editing, and publishing process has allowed him to grow as a writer and extract new stories from unexpected places as he travels the country promoting Fluid.
Advice for Hopeful Authors
Sentell encourages hopeful authors to stick to a schedule and write every day. "Like everything else, it's a job and you should treat it as such," he says. He also recommends writing a steady stream of short fiction and submitting it for publication, as well as making efforts to find your local writing community. "There are tons of reading and storytelling events springing up across the country – find your local groups and go meet other writers. It will definitely help your drive, creativity and discipline." His best advice for developing ideas is to have as many adventures as possible and to always carry a notebook as you figure out where you want to go next. "Time moves quickly, so make sure you're sending time on projects that matter to you, stories and characters that you love!" he recommends. "Also, don't forget to make tons of mistakes. They make for the best stories."
Filed Under: Book Marketing, Get Published, Latest News Tagged With: client success, Coliloquy, ebooks, editing, Fluid, marketing, marketing platform, nurturing creativity, publishing, publishing advice, Travis Sentell


text:
An amazing, genre-breaking story from Grant Morrison, one of the most original and inventive writers in comics today.Legendary Comics proudly presents Annihilator, an original graphic novel odyssey from the subversive mind of Grant Morrison. This 6-issue series is a reality-bending sci-fi adventure like no other, brought to life with stunning artwork from Frazer Irving (Batman and Robin, Judge Dredd, Necronauts). Washed-up Hollywood screenwriter Ray Spass is caught in a downward spiral of broken relationships, wild parties and self-destruction. Out of luck and out of chances, he's one failed script away from fading into obscurity. Little does he know he's about to write the story of his life.As his imagination runs rampant, Ray must join forces with his own fictional character Max Nomax on a reality-bending race to stop the entire universe from imploding... without blowing his own mind in the process. • New York Times Bestselling Author • Written by Grant Morrison (Animal Man, Batman, All Star Superman, Fantastic Four) • This is the compilation of the 6 issue series • Featuring interiors and covers by Frazier Irving (Batman and Robin, Uncanny X-Men) • Includes bonus materials on the making of the comic "Morrison is one of the most adventurous and commercially successful comic-book writers of the past 25 years, retrieving superheroic and science-fictional dispatches from the fringes of consciousness, splattering his chaotic visions onto the page." -- Rolling Stone "I think [Annihilator is] the best stuff I've done yet." -- Frazer Irving
Do you want ot get the Annihilator?
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Federigo Gonzaga succeeded his father, Francesco II, as Marquis of Mantua in 1519, and, in 1530, he was conferred the title of duke by Emperor Charles V. In 1531, he married Maria Palaeologo, daughter of the Marquis of Monferrato, which gained Mantua that territory in 1536. Federigo was an able military commander. In 1521, as Captain of the Church, he was involved in the siege of Pavia and, in 1529, was appointed captain to the Imperial troops in Italy. Federigo was also a major patron, his passion for the arts inherited from his mother, Isabella d'Este, who was an avid art collector. As a child, Federigo was kept hostage at the Vatican in the court of Julius II, along with his father whose political inclinations did not accord with those of the pope. This coincided with the time when Michelangelo, Raphael, and Donato Bramante were working at the Vatican, which would have piqued Federigo's interest in art. Federigo was the patron of Correggio who rendered for him mythologies depicting the loves of Jupiter, including Danaë (c. 1531; Rome, Galleria Borghese), Ganymede, and Io (both early 1530s; Vienna, Kunsthistorisches Museum). Giulio Romano built for him the Palazzo del Tè (1527-34) and frescoed the Sala dei Giganti with his greatest painted master-piece, the Fall of the Giants (1530-1532).


text:
\section{Charge conjugation symmetry}
\label{sec:appB}

The charge conjugation symmetry transformation takes the same form for the fields
at vanishing flow-time, at positive flow time and the Lagrange multipliers.
Here we indicate generically the fermion field as $\phi = \left\{\psi(0,x),\chi(t,x),\lambda(t,x)\right\}$
and the corresponding $\overline{\phi}$.
For charge conjugation we have
\be
\mathcal{C} \colon
\begin{cases}
   V(t,x;\mu) \longrightarrow  V(t,x;\mu)^*, \\
   \phi    \longrightarrow
                C^{-1}\overline{\phi}^T,\\
   \overline{\phi} \longrightarrow
               - \phi^T C ,
\end{cases}
\label{eq:chargeconj}
\ee
where $V(t,x;\mu)$ is the gauge link at positive flow-time and $C$ satisfies
\be
- \gamma_\mu^T = C \gamma_\mu  C^{-1} , \qquad \gamma_5 = C \gamma_5  C^{-1}.
\ee

\section{Derivation of eq.~\eqref{eq:delta_chiral1}}
\label{sec:appC}
In this appendix we derive eq.~\eqref{eq:delta_chiral1} that we rewrite here for convenience
\bea
\frac{1}{\mcZ_{\chi,\lambda}}\int\mcD\left[\chi,\chibar \right] \mcD\left[\lambda,\lambdabar \right]
\mcO \left[\lambdabar(s,x) \Gamma^a(s,x) \right.  &+&  \left.
\overline{\Gamma}^a(s,x)\lambda(s,x) \right] \exp\left\{-S_{F,fl} \right\} = \nonumber \\
&=& \int_s^{\infty} dt~\int d^Dz~\left[\overline{\Gamma}^a(s,x) \overline{J}(t,z;s,x) \frac{\delta \mcO}{\delta \chibar(t,z)}
\right. \nonumber \\
&+&\left. \Gamma^a(s,x)J(t,z;s,x)\frac{\delta \mcO}{\delta \chi(t,z)}\right]_0\,.
\label{eq:delta_chiral_app}
\eea
where
\be
\mcZ_{\chi,\lambda} = \int\mcD\left[\chi,\chibar \right] \mcD\left[\lambda,\lambdabar \right] \exp\left\{-S_{F,fl} \right\}\,,
\ee
and the index $0$ indicates that the r.h.s of eq.~\eqref{eq:delta_chiral_app} has to be evaluated
with fields $\chi$ and $\chibar$ solutions of the GF equations.

The proof is based on the observation that, given the form
of the bulk action, an operator linear in $\lambda$ or $\lambdabar$,
as in the l.h.s of eq.~\eqref{eq:delta_chiral_app}, can be generated by a suitable change of variable.
The non-singlet nature of the transformation parametrized by the functions $\Gamma^a$ and $\overline{\Gamma}^a$
guarantees that the Jacobian of this change of variable has unit determinant.

We start considering the kernel $K(t,z;s,x)$ defined as solution of
\be
\begin{cases}
\left(\partial_t - \Delta_z\right) K(t,z;s,x) = 0 & t >s \\
\lim_{t \rightarrow s^+}K(t,z;s,x) = \delta(z-x) & \\
K(t,z;s,x) = 0 & t<s\,. \\
\end{cases}
\ee
The Jacobians $J$ and $\overline{J}$ defined in eq.~\eqref{eq:jaco} satisfy the same equations.
This can be checked applying the gradient flow operator to $J$ and $\overline{J}$ and recalling that in the Jacobians the fields
$\chi$ and $\chibar$ are solutions of the gradient flow equations.

We now consider the expectation value of a generic operator $\mcO$ defined at positive flow-time
and function of the fermion or gauge fields. We assume that $\mcO$ is independent of the Lagrange multipliers.
In the functional integral that defines the expectation value we perform
the following change of variable
\be
\delta \chi(t,z) = i \int_0^tds~\int d^4 x \Gamma^a(s,x)K(t,z;s,x)\alpha^a(s,x)\,,
\label{eq:delta_chi_app}
\ee
\be
\delta \chibar(t,z) = i \int_0^tds~\int d^4 x \overline{\Gamma}^a(s,x)K(t,z;s,x)^{\dagger}\alpha^a(s,x)\,.
\label{eq:delta_chibar_app}
\ee
As we will show below the Jacobian of this change of variable is one, thus the variation
of the expectation value of $\mcO$ should vanish. Similarly to what happens when deriving
Ward identities this implies that $\langle \delta \mcO \rangle = \langle O \delta S\rangle$.

The variation of the action according to eq.~(\ref{eq:delta_chi_app},\ref{eq:delta_chibar_app}) is given by
\be
i \frac{\delta S}{\alpha^a(t,y)} = - \left[\lambdabar(t,y)\Gamma^a(t,y) + \overline{\Gamma}^a(t,y) \lambda(t,y)\right]\,.
\ee
If we consider now that the kernels $K$ and $K^\dagger$ coincide with the Jacobians in eq.~\eqref{eq:jaco}
computing the variation of the operators $\left\langle \delta \mcO \right\rangle$
we obtain the desired formula~\eqref{eq:delta_chiral_app}.

We are left to show that the Jacobian of the transformation~(\ref{eq:delta_chi_app}), (\ref{eq:delta_chibar_app})
has determinant equal to one.
From the definition~(\ref{eq:delta_chi_app},\ref{eq:delta_chibar_app})
and given that we consider a transformation with infinitesimal $\alpha^a$
this is a result of the non-singlet nature of the transformation,
i.e. of the fact the matrices $\Gamma^a$ and $\overline{\Gamma}^a$ are traceless.

\section{Concluding remarks}
\label{sec:conclu}

The gradient flow is becoming an important tool to probe
in interesting ways the non-perturbative dynamic of QCD, one aspect being
the non-perturbative renormalization of local fields. The example of the chiral condensate
shows that mixing with lower-dimensional operators can be avoided if we
relate the quark condensate defined at positive and zero flow-time.
A key role for this relation to hold is played by chiral Ward identities.

In this work we have shown that it is possible to use local chiral variations
to derive the corresponding Ward identities with operators located at positive flow-time
reproducing the identities obtained by L\"uscher~\cite{Luscher:2013cpa}.
While this alternative derivation is interesting on its own, it is also useful when considering more complicated
operators. In fact Ward identities can be used to give non-perturbative definitions
to the coefficients appearing in small flow-time expansions.

We have then considered the gradient flow in combination with twisted mass fermions.
We have shown that the property of automatic O($a$) improvement for Wilson twisted mass
fermions at maximal twist is still valid at positive flow-time.
Using the vector Ward identity derived for twisted mass fermions we find a definition
for the chiral condensate that not only is multiplicatively renormalizable but
also automatic O($a$) improved.

\section*{Acknowledgments}

I wish to thank Martin L\"uscher and Agostino Patella for enlightening
discussions on the gradient flow. I want to thank the CERN Theory division,
where part of this work has been done, for their kindness and hospitality.
I also thank Tom Luu for reading a first version of this work and
for constant encouragement.

\section{Introduction}
\label{sec:intro}
The gradient flow (GF), for gauge~\cite{Luscher:2010iy}
and fermion~\cite{Luscher:2013cpa} fields, used in combination with a lattice regulator,
can probe the non-perturbative dynamics of QCD
in advantageous manners.
It is defined by a differential equation that gauge and fermion
fields satisfy as a function of the space-time coordinates $x$ and of a new
scale, the flow-time $t$.
In a way the flow equations modify the short-distance behavior of local fields and
can be considered as a continuous form of stout smearing~\cite{Morningstar:2003gk}.

Smearing has been a useful tool for lattice gauge theories
since many years~\cite{Albanese:1987ds,Gusken:1989qx,Morningstar:2003gk}.
It was never clear though whether smeared operators had a continuum limit
or not. The only exception was the discussion of the continuum limit for smeared Wilson
loops in a paper by Narayanan and Neuberger~\cite{Narayanan:2006rf}.

The works by L\"uscher and Weisz~\cite{Luscher:2010iy,Luscher:2011bx,Luscher:2013cpa}
give us a complete understanding of the continuum limit
of observables at non-vanishing flow-time. This allows us to use the GF
to define observables that otherwise would be difficult
to compute with standard methods.
For example the GF has been used to give operational definitions to the
fundamental parameters of QCD as the strong coupling~\cite{Luscher:2010iy,Fodor:2012td,Fritzsch:2013je}
or other quantities, as the chiral condensate~\cite{Luscher:2013cpa}
and the energy-momentum tensor~\cite{Suzuki:2013gza,DelDebbio:2013zaa}.
The GF can also be used to define new relative ways to set the scale in
lattice QCD calculations~\cite{Luscher:2010iy,Borsanyi:2012zs} or
to define a renormalization scheme for fermionic operators~\cite{Monahan:2013lwa}.

To study correlation functions involving fields
at positive flow-time it is, in some cases, advantageous to consider a suitable extra-dimensional local field theory
where the flow-time $t$ is the extra-dimensional coordinate and
QCD is located at the $t=0$ boundary.
The theory contains the fundamental fields and appropriate Lagrange multipliers
to guarantee that the fundamental fields, once the Lagrange
multipliers are integrated out, satisfy the proper GF equations at positive flow-time.

In this work we derive non-singlet chiral Ward identities
considering the transformation properties of the action and of local operators under a local
chiral variation. We derive axial and vector Ward identities when the local
variation is performed at the $t=0$ boundary
and bulk Ward identities that defines $4+1$ conserved axial and vector currents.
To obtain the Ward identities we use two different perspectives.

The first one takes into account the dependence of the GF equations solutions
on the initial conditions. A local chiral variation of the initial conditions
propagates into a variation of local fields located in the bulk.
Using the Lagrange multipliers it is possible to obtain a universal
formula that relates the chiral variation located at the boundaries
with a multilocal correlation function that depends exclusively on the
type of symmetry transformation. This generates additional terms to the standard Ward identities.
The same idea has been investigated in~\cite{DelDebbio:2013zaa}
in the context of space-time symmetries.

The second is to consider fermion fields at positive flow-time
simply as integration variables of the extra-dimensional theory.
We extend the $SU(N_f)_L \times SU(N_f)_R$ chiral symmetry transformations
for the fermion fields and the corresponding Lagrange multipliers at positive flow-time.
It becomes clear then that the QCD chiral Ward identities get modified by boundary terms.
These boundary terms are identical to the additional terms obtained with the previous method.
This second approach provides a general framework to determine Ward identities
in the bulk and at the boundary of the $4+1$ theory
that can be easily extended to any operator one is interested in.
As a check we compare the chiral Ward identities with the identities obtained by L\"uscher
in~\cite{Luscher:2013cpa} finding perfect agreement.

Chiral symmetry transformations for the $(4+1)$-dimensional theory have several applications.
In this work, after extending the GF formalism for twisted mass fermions,
we prove automatic O($a$) improvement for correlators computed with Wilson twisted mass fermions at maximal twist
at non-zero flow-time.
As a by-product we show that the newly proposed way to determine
the chiral condensate with standard Wilson fermions~\cite{Luscher:2013cpa}
can be easily generalized to Wilson twisted mass
with the additional advantage that there is no need
to determine any improvement coefficient.

\section{Gradient flow}
\label{sec:gf}
In this section, to fix the notation, we give a short introduction to
the GF for gauge and fermions fields~\cite{Luscher:2010iy,Luscher:2013cpa}.
We consider the $SU(N)$ gauge potential, $G_{\mu}(x)$
and the quark and anti-quark fields $\psi(x)$ and $\psibar(x)$.
The Dirac, color and flavor indices, unless specifically needed,
are suppressed for simplicity.

The Yang-Mills gradient flow~\cite{Luscher:2010iy} of gauge fields as a function
of the flow-time $t$ is defined as follows
\be
\partial_t B_{\mu}=D_{\nu,t}G_{\nu\mu}\,,
\label{eq:GF_gauge}
\ee
where
\be
G_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}+[B_{\mu},B_{\nu}]\,,
\qquad
D_{\mu,t}=\partial_{\mu}+[B_{\mu},\,\cdot\;]\,,
\ee
and the initial condition on the flow-time-dependent field $B_{\mu}(t,x)$
at $t=0$ is  given by the fundamental gauge field.
The flow-time $t$ has a time-squared dimension,
but in the following, with a slight abuse of notation,
we will refer to it as a single extra-dimension.

The flow for the quark fields considered in ref.~\cite{Luscher:2013cpa}
is given by
\be
\partial_t\chi_t=\Delta\chi_t \qquad \partial_t\chibar_t=\chibar_t\overleftarrow{\Delta}\,,
\label{eq:flow_ferm}
\ee
\be
\Delta=D_{\mu,t}D_{\mu,t} \qquad D_{\mu,t}=\partial_{\mu}+B_{\mu}\,,
\label{eq:Dmu_def}
\ee
\be
\overleftarrow{\Delta}=\overleftarrow{D}_{\mu,t}\overleftarrow{D}_{\mu,t}
\qquad \overleftarrow{D}_{\mu,t}=\overleftarrow{\partial}_{\mu}-B_{\mu}\,,
\label{eq:Dmu_def}
\ee
which, together with the initial conditions
\be
\left.\chi_t\right|_{t=0}=\psi \qquad \left.\chibar_t\right|_{t=0}=\psibar\,,
\label{eq:bc}
\ee
define the time-dependent quark and antiquark
fields $\chi_t(x)$ and $\chibar_t(x)$.
In the following we will indicate with $\chi_t(x)$ and $\chibar_t(x)$
the solutions of the GF equations and with $\chi(t,x)$ and $\chibar(t,x)$
generic fermion fields functions of $(t,x)$.
We note that the GF equations for the fermion fields contain the gauge fields evolved with
eq.~\eqref{eq:GF_gauge}.

Solving the GF equations at tree-level it becomes clear the smoothing
effect
\be
\chi_t(x) = \int d^4y~K(t;x-y) \psi(y)\,,\qquad K(t;x) = \frac{\e^{-\frac{x^2}{4t}}}{\left(4\pi t\right)^{2}}\,,
\ee
with smearing radius of $\sqrt{8t}$.

Using the solutions of the flow equations we can construct
operators located at positive flow-time and consider correlation functions between them.
Correlation functions of operators located at positive flow-time can be shown to be
equivalent to correlation functions defined in a local field theory in $4+1$ dimensions,
the extra dimension being the flow-time~\cite{ZinnJustin:1986eq,ZinnJustin:1987ux,Luscher:2010iy,Luscher:2013cpa}.
The fields content of the $4+1$ field theory consists of
the gauge and fermion fields at vanishing flow-time, at positive flow-time
and the corresponding Lagrange multipliers.
The Lagrange multipliers are needed to make sure that
once they are integrated over the gauge and fermion fields
satisfy the appropriate flow equations.
The local extra-dimensional theory is a useful theoretical tool to investigate
correlators involving operators at positive flow-time. For example
the renormalizability of the pure-gauge gradient flow has been established~\cite{Luscher:2011bx}
to all orders of perturbation theory.

In this work we consider only the fermionic sector
of the theory and details about the gauge sector, the ghost sector and the gauge fixing terms
can be found in refs.~\cite{Luscher:2010iy,Luscher:2013cpa}.
The fermionic part of the action reads
\be
S_{\rm F} = S_{\rm F,QCD} + S_{\rm F,fl}\,,
\ee
where $S_{\rm F,QCD}$ is the usual fermionic QCD action
including the quark and gauge fields at zero flow-time
\be
S_{\rm F,QCD} = \int d^4x~\psibar(x)\left[ \gamma_\mu D_\mu + M\right]\psi(x)\,,
\ee
where $M$ is the mass matrix of the theory and $D_\mu = \partial_\mu + G_\mu$.
The bulk action $S_{\rm F,fl}$ of the $(4+1)$-dimensional theory including the fields at non-zero flow-time
is given by
\be
S_{\rm F,fl} =\int_0^{\infty}\rmd t\int\rmd^4x
  \left[\lambdabar(t,x)
         \left(\partial_t-\Delta\right)\chi(t,x) +\chibar(t,x)
  \left(\overleftarrow{\partial}_t-\overleftarrow{\Delta}\right)
  \lambda(t,x)\right]\,,
\label{eq:action_bulk}
\ee
where $\lambda(t,x)$ and $\lambdabar(t,x)$ are the Lagrange multiplier
for the fermion fields and have energy-dimension of $\left(4+1\right)/2$.
The relative sign between the two terms in eq.~\eqref{eq:action_bulk}
is dictated by charge conjugation symmetry defined in app.~\ref{sec:appB}.
Integrating out the Lagrange multipliers one identifies correlation functions of fields
evolved with their flow equations to correlation functions in
a local field theory in $4+1$ dimensions, the extra dimension being the flow-time.

If we consider a perturbative expansion, like for example in dimensional regularization,
a gauge fixing term should be added to the action.
In the following we will consider Ward identities (WIs)
involving gauge invariant operators, thus all possible additional
contributions coming from the gauge-fixing terms vanishes
and we can safely discard them. Additionally we will always have in mind
a non-perturbative definition of the theory using a lattice regulator
where gauge-fixing terms are not needed.

\section{Chiral symmetry}
\label{sec:chiral}

A global $SU_L(N_f) \times SU_R(N_f)$ chiral symmetry transformation
\be
\begin{cases}
\psi(x) \rightarrow  {\rm exp}\left\{i\left(\alpha_V^a \frac{T^a}{2} + \alpha_A^a \frac{T^a}{2}\gamma_5\right)\right\} \psi(x) \vspace{0.5cm}\\
\psibar(x) \rightarrow  \psibar(x) {\rm exp}\left\{i\left(-\alpha_V^a \frac{T^a}{2} + \alpha_A^a \frac{T^a}{2}\gamma_5\right)\right\}
\label{eq:chiral}
\end{cases}
\ee
leaves the massless QCD continuum action invariant.
$T^a/2$ are the generators of the SU($N_f$) flavor group in the fundamental
representation (e.g. Pauli matrices for $N_f = 2$, Gell-Mann matrices for $N_f = 3$
with $a=1 \ldots N_f^2-1$) and they satisfy
\be
\Tr\left[T^a T^b\right] = 2 \delta^{ab}\,, \quad
\left[\frac{T^a}{2},\frac{T^b}{2}\right] = i f^{abc} \frac{T^c}{2}\,, \quad
\left\{\frac{T^a}{2},\frac{T^b}{2}\right\} = i d^{abc} \frac{T^c}{2} + \frac{\delta^{ab}}{2} \mathbbm{1}\,,
\ee
where $f^{abc}$ are the SU($N_f$) structure constants, $d^{abc}$ are symmetric coefficients.
Using standard methods considering an infinitesimal
local chiral transformation
\be
\begin{cases}
\delta \psi(x) = \left[i \alpha_V^a(x) \frac{T^a}{2} + i \alpha_A^a(x) \frac{T^a}{2} \gamma_5\right]\psi(x)\,, \\
\delta \psibar(x) = \psibar(x)\left[-i \alpha_V^a(x) \frac{T^a}{2} + i \alpha_A^a(x) \frac{T^a}{2} \gamma_5\right]\,, \\
\end{cases}
\label{eq:deltapsi}
\ee
the corresponding Noether currents, the so-called axial and vector currents,
can be determined
\be
A_\mu^a(x) = \psibar(x)\gamma_\mu\gamma_5\frac{T^a}{2}\psi(x)\,, \qquad
V_\mu^a(x) = \psibar(x)\gamma_\mu\frac{T^a}{2}\psi(x)\,,
\ee
and the variations of the action reads
\be
i\frac{\delta S_{\rm QCD}}{\delta \alpha_A^a(x)} = \partial_\mu A^a_\mu(x) - \psibar(x) \gamma_5 \left\{\frac{T^a}{2},M\right\}\psi(x)\,,
\ee
\be
i\frac{\delta S_{\rm QCD}}{\delta \alpha_V^a(x)} = \partial_\mu V^a_\mu(x) + \psibar(x) \left[\frac{T^a}{2},M\right]\psi(x) \,.
\ee
For a generic observable $\mathcal{O}$ the variation under the transformations~\eqref{eq:deltapsi} is
\be
i\frac{\delta \mcO }{\delta \alpha_A^a(x)} =
-\left\{\psibar(x)\left[\gamma_5 \frac{T^a}{2}\right] \frac{\delta \mcO}{\delta \psibar(x)} -
\frac{\delta \mcO}{\delta \psi(x)}\left[\gamma_5 \frac{T^a}{2}\right] \psi(x)\right\} \,.
\ee
\be
i\frac{\delta \mcO }{\delta \alpha_V^a(x)} =
\left\{\psibar(x)\left[\frac{T^a}{2}\right] \frac{\delta \mcO}{\delta \psibar(x)} +
\frac{\delta \mcO}{\delta \psi(x)}\left[\frac{T^a}{2}\right] \psi(x)\right\} \,.
\ee

\subsection{Operators at positive flow-times}

In this section we want to derive non-singlet chiral WIs when the probe operator $\mcO$
depends on quark and gauge fields solutions of the GF equations~(\ref{eq:GF_gauge},\ref{eq:flow_ferm}).
We derive the variations of the action and the probe operator in the classical theory.
We then consider an interacting theory where a sensible regulator,
as the lattice or dimensional regularization, is used.
Once the local fields are properly renormalized and the regulator is removed we expect
the same WIs to be valid for the renormalized fields~\cite{Bochicchio:1985xa}.

To determine the form of the chiral WIs we need to understand how the
operator $\mcO$ transforms under a local chiral transformation.
A local chiral transformation of
fermion fields at vanishing flow-time modifies the initial conditions
of the flow-equations.
Consequently the fermion fields solutions of the flow equations
at positive flow-time are altered. In other words the
transformation properties under chiral symmetry of the fermion fields at vanishing
flow-time are propagated to the fermion fields at non-vanishing flow-time.

These variations can be determined considering the functional dependence of the fermion fields
at positive flow-time from the initial conditions of the flow equations, i.e.
evaluating the following expression
\bea
i\frac{\delta \mcO }{\delta \alpha_A^a(x)} =
&-&\int_0^{\infty} dt~\int d^4z~\left\{\left[\psibar(x)\gamma_5 \frac{T^a}{2}\right] \overline{J}(t,z;0,x)
\frac{\delta \mcO}{\delta \chibar_t(z)}  \right. \nonumber \\
&+&\left. \left[\gamma_5 \frac{T^a}{2} \psi(x)\right] J(t,z;0,x)  \frac{\delta \mcO}{\delta \chi_t(z)}\right\} \,.
\label{eq:delta_Ot}
\eea
where the Jacobians are defined as
\be
J(t,z;s,x) = \theta(t-s) \frac{\delta \chi_t(z)}{\delta \chi_s(x)}\,, \qquad
\overline{J}(t,z;s,x) = \theta(t-s) \frac{\delta \chibar_t(z)}{\delta \chibar_s(x)}\,.
\label{eq:jaco}
\ee
Eq.~\eqref{eq:delta_Ot} is a non-local expression due to the presence of the Jacobian matrices
$J$ and $\overline{J}$, thus it would
be rather difficult to study its divergences and renormalization properties.
This problem can be solved if the operator $\mcO$ does not depend on the Lagrange multipliers $\lambda$ and $\lambdabar$.
In fact in this specific case it is possible to evaluate the expression on the r.h.s of eq.~\eqref{eq:delta_Ot}
obtaining
\be
\left\langle i\left[\frac{\delta \mcO}{\delta \alpha_A^a(x)}\right]_R \right\rangle =
- \left\langle \mcO_R \widetilde{P}_R^a(0,x)\right\rangle\,,
\label{eq:delta_Ot2}
\ee
where
\be
\tilde{P}^a(t,x) = \lambdabar(t,x)\frac{T^a}{2}\gamma_5\chi(t,x)
  +\chibar(t,x)\frac{T^a}{2}\gamma_5\lambda(t,x)\,,
\ee
and
\be
\tilde{P}_R^a = Z_{\tilde{P}}\tilde{P}^a\,.
\ee

The renormalization of $Z_{\tilde{P}}$, discussed in ref.~\cite{Luscher:2013cpa},
is multiplicative and it turns out to be equal to one even if the interacting theory
is regularized with Wilson fermions.
If the operator $\mcO$ contains only fermion and gauge fields at positive flow-times
and does not contain Lagrange multipliers it renormalizes multiplicatively~\cite{Luscher:2013cpa}
depending on the total number $n$ of fermion and anti-fermion fields, i.e.
\be
\mcO_R = \left[Z_\chi\right]^{n/2} \mcO\,,
\ee
where $Z_\chi$ is the normalization factor for the fermion fields and Lagrange multipliers at positive flow-time
$\chi_R = Z_\chi^{1/2}\chi$, $\lambda_R = Z_\chi^{-1/2}\lambda$.

Eq.~\eqref{eq:delta_Ot2} is a specific case of a more general result
(see app.~\ref{sec:appC} for the derivation) that we can formally write as
\bea
\frac{1}{\mcZ_{\chi,\lambda}}\int\mcD\left[\chi,\chibar \right] \mcD\left[\lambda,\lambdabar \right]
\mcO \left[\lambdabar(s,x) \Gamma^a(s,x) \right.  &+&  \left.
\overline{\Gamma^a}(s,x)\lambda(s,x) \right] \exp\left\{-S_{F,fl} \right\} = \nonumber \\
&=& \int_s^{\infty} dt~\int d^4z~\left[\overline{\Gamma^a}(s,x) \overline{J}(t,z;s,x) \frac{\delta \mcO}{\delta \chibar(t,z)}
\right. \nonumber \\
&+&\left. \Gamma^a(s,x)J(t,z;s,x)\frac{\delta \mcO}{\delta \chi(t,z)}\right]_0\,.
\label{eq:delta_chiral1}
\eea
where
\be
\mcZ_{\chi,\lambda} = \int\mcD\left[\chi,\chibar \right] \mcD\left[\lambda,\lambdabar \right] \exp\left\{-S_{F,fl} \right\}\,,
\ee
and the index $0$ indicates that the r.h.s of eq.~\eqref{eq:delta_chiral1} has to be evaluated
with fields $\chi$ and $\chibar$ solutions of the GF equations.

Eq.~\eqref{eq:delta_chiral1} is useful because it relates the symmetry transformation of $\mcO$, induced
by the boundary conditions, to a multi-local correlation function whose form is solely dictated by the type of transformation,
here parametrized by the functions $\Gamma$ and $\overline{\Gamma}$.
Additionally the equation is valid if $\mcO$, $\Gamma^a$ and $\overline{\Gamma}^a$ do not depend
on the Lagrange multipliers $\lambda$ and $\lambdabar$.

We are now in the position to derive the chiral WIs using standard techniques obtaining
\be
\left\langle \left[\partial_\mu A^a_{\mu,R}(x) - P^{\{a,M_R\}}_R(x) +
\widetilde{P}_R^a(0,x)\right] \mcO_R\right\rangle = 0\,,
\label{eq:PCACt}
\ee
\be
\left\langle \left[\partial_\mu V^a_{\mu,R}(x) + S^{\left[a,M_R\right]}_R(x)  + \overline{S}_R^a(0,x)\right] \mcO_R\right\rangle = 0\,,
\label{eq:CVCt}
\ee
where
\be
\overline{S}^a_R = Z_{\overline{S}} \overline{S}^a\,,
\ee
and
\be
\overline{S}^a(t,x) = \lambdabar(t,x)\frac{T^a}{2}\chi(t,x)-\chibar(t,x)\frac{T^a}{2}\lambda(t,x)\,.
\ee
The renormalization of $\overline{S}^a$ following the same argumentation used for $\widetilde{P}^a$
is also multiplicative.
For a generic mass matrix the scalar and pseudoscalar densities read
\be
P^{\{a,M_R\}}_R(x) = Z_P \psibar(x) \gamma_5 \left\{\frac{T^a}{2},M_R\right\}\psi(x)\,,
\ee
and
\be
S^{\left[a,M_R\right]}_R(x) = Z_S \psibar(x) \left[\frac{T^a}{2},M_R\right]\psi(x)\,.
\ee

Eq.~\eqref{eq:PCACt} was already derived in ref.~\cite{Luscher:2013cpa}
using the defining equations for the fermion propagators and Wick contractions.
It is interesting to see that the same relation can be obtained using standard
symmetry methods.
This derivation also elucidates the origin of the additional term proportional to $\widetilde{P}^a$,
i.e. it is the variation of the probe operator at positive flow-time
resulting from the chiral transformation of the initial conditions
of the flow equations.

Eq.~\eqref{eq:CVCt} is new and is the vector WI with an external probe located
at positive flow-time. In this case the variation of the external operator can be worked out
using the general formula~\eqref{eq:delta_chiral1} obtaining that it is proportional
to $\overline{S}^a$.

\section{Chiral symmetry at positive flow-time}
\label{sec:WI}
A different way to derive chiral WIs at positive flow-time is to extend the chiral
symmetry transformations~\eqref{eq:chiral} to the $(4+1)$-dimensional theory.
This extension is more general and it allows us to find the corresponding
$(4+1)$-dimensional Noether currents and the extension of the WIs for positive
flow-time.

The chiral symmetry transformation of the fermion fields at vanishing flow-time
are the standard ones for a Dirac field. The boundary conditions~\eqref{eq:bc}
extend in a natural way the $SU_L(N_f) \times SU_R(N_f)$ chiral symmetry transformations to the
fermion fields and non-zero flow-time, i.e.
\be
\begin{cases}
\chi(t,x) \rightarrow  {\rm exp}\left\{i\left(\alpha_V^a \frac{T^a}{2} + \alpha_A^a \frac{T^a}{2}\gamma_5\right)\right\} \chi(t,x) \vspace{0.5cm}\\
\chibar(t,x) \rightarrow  \chibar(t,x) {\rm exp}\left\{i\left(-\alpha_V^a \frac{T^a}{2} + \alpha_A^a \frac{T^a}{2}\gamma_5\right)\right\}\,.
\label{eq:chiral_chi}
\end{cases}
\ee

Since the flow equations~\eqref{eq:flow_ferm} together with the initial
conditions~\eqref{eq:bc} are invariant under the chiral transformations~\eqref{eq:chiral},
the time-dependent fields transform in the same way under chiral transformations as
the fields at zero flow-time. For example the scalar and pseudoscalar densities
\be
S^{a}(t,x)=\chibar(t,x)\frac{T^a}{2}\chi(t,x)\,,\qquad P^{b}(t,x)=\chibar(t,x)\gamma_5\frac{T^a}{2}\chi(t,x)\,,
\ee
transform in the same way as the corresponding composite fields at vanishing flow-time~\cite{Luscher:2013vga}.

We want to ask the question how the Lagrange multiplier fields $\lambda$ and
$\lambdabar$ transform under chiral symmetry.
The invariance of the flow equation for fermions can be seen as a consequence
of the invariance under chiral transformation of the bulk action $S_{\rm F,fl}$~\eqref{eq:action_bulk}.
This implies that the chiral transformations are
\be
\begin{cases}
\lambda(t,x) \rightarrow  {\rm exp}\left\{i\left(\alpha_V^a \frac{T^a}{2} - \alpha_A^a \frac{T^a}{2}\gamma_5\right)\right\} \lambda(t,x) \vspace{0.5cm}\\
\lambdabar(t,x) \rightarrow  \lambdabar(t,x) {\rm exp}\left\{i\left(-\alpha_V^a \frac{T^a}{2} - \alpha_A^a \frac{T^a}{2}\gamma_5\right)\right\}\,.
\label{eq:chiral_lambda}
\end{cases}
\ee
We note the opposite sign for the axial transformation coefficient with respect to
the standard transformation.
Another way to understand that these are the correct chiral transformations of the Lagrange multipliers
is to consider the bulk action $S_{\rm F,fl}$ and note that the Lagrange multipliers $\lambda$ and $\lambdabar$
are equivalent to the differential operators
\be
\lambdabar \rightarrow \frac{\delta}{\delta \mcF}\,, \qquad
\lambda \rightarrow \frac{\delta}{\delta \mcFbar}\,,
\ee
where
\be
\mcF(t,x) = \left[\partial_t - \Delta\right]\chi(t,x)\,, \qquad
\mcFbar(t,x) = \chibar(t,x)\left[\overleftarrow{\partial}_t-\overleftarrow{\Delta}\right]\,.
\ee

We use the chiral transformations for the fermions fields~\eqref{eq:chiral_chi} and
for the Lagrange multipliers~\eqref{eq:chiral_lambda} to derive chiral
WIs.
Using standard techniques we consider an infinitesimal local
SU($N_f$) $\times$ SU($N_f$) chiral transformations of the fermionic fields
\be
\begin{cases}
\delta \chi(t,x) = \left[i \alpha_V^a(t,x) \frac{T^a}{2} + i \alpha_A^a(t,x) \frac{T^a}{2} \gamma_5\right]\chi(t,x)\,, \\
\delta \chibar(t,x) = \chibar(t,x)\left[-i \alpha_V^a(t,x) \frac{T^a}{2} + i \alpha_A^a(t,x) \frac{T^a}{2} \gamma_5\right]\,, \\
\delta \lambda(t,x) = \left[i \alpha_V^a(t,x) \frac{T^a}{2} - i \alpha_A^a(t,x) \frac{T^a}{2} \gamma_5\right]\lambda(t,x)\,, \\
\delta \lambdabar(t,x) = \lambdabar(t,x)\left[-i \alpha_V^a(t,x) \frac{T^a}{2} - i \alpha_A^a(t,x) \frac{T^a}{2} \gamma_5\right]\,. \\
\end{cases}
\label{eq:delta_chiral}
\ee
It is interesting to note that the densities $\widetilde{P}^a$ and $\overline{S}^a$
form a multiplet as can be seen performing an infinitesimal axial and vector rotation
\be
\delta_A \widetilde{P}^b = f^{abc} \alpha_A^a\overline{S}^c\,, \quad \delta_A \overline{S}^b = f^{abc} \alpha_A^a \widetilde{P}^c\,,
\ee
\be
\delta_V \widetilde{P}^b = f^{abc} \alpha_A^a\widetilde{P}^c\,, \quad \delta_V \overline{S}^b = f^{abc} \alpha_A^a \overline{S}^c\,.
\ee

To take into account properly boundary effects we discretize the
flow-time direction with spacing $\epsilon$ and
we treat separately the cases $t=0$ and $t>0$.
Discretizing the flow-time direction the bulk action reads
\be
S_{\rm F,fl} = \epsilon \sum_{t=0}^{T-\epsilon}\int\rmd^4x
  \left[\lambdabar(t,x)
         \left(\partial_t-\Delta\right)\chi(t,x) +\chibar(t,x)
  \left(\overleftarrow{\partial}_t-\overleftarrow{\Delta}\right)
  \lambda(t,x)\right]\,,
\label{eq:action_bulk_lattice}
\ee
where
\be
\partial_t \chi(t,x) = \frac{1}{\epsilon} [ \chi(t+\epsilon,x) - \chi(t,x) ]\,,
\label{eq:partialmu}
\ee
\be
\chibar(t,x) \overleftarrow{\partial}_t = \frac{1}{\epsilon} [ \chibar(t+\epsilon,x) - \chibar(t,x) ]\,.
\ee
The $\Delta$ operator is the lattice Laplacian
\be
\Delta = \nabla_{\mu,t} \nabla_{\mu,t}^*\,, \qquad
\overleftarrow{\Delta} = \overleftarrow{\nabla}_{\mu,t} \overleftarrow{\nabla}_{\mu,t}^*\,,
\ee
where $\nabla_\mu$ and $\nabla_\mu^*$ are the gauge-covariant forward and backward difference operators
in presence of the flow-time dependent gauge field.

Computing the local chiral variation of the action at $t=0$ we obtain the so-called PCAC relation
\be
\partial_\mu A^a_\mu(x) - P^{\{a,M\}}(x) + \widetilde{P}^a(0,x)=0 \,.
\label{eq:AWI0}
\ee
Once again we find an additional term with respect to the standard PCAC.
In the approach we are following in this section
the additional term is the result of the chiral variation of the boundary terms.

We consider now a generic operator
\be
\mcO = \phi_1(t_1,x_1)\cdots\phi_n(t_n,x_n)\,,
\ee
where $\phi_1,\cdots,\phi_n$ can be fermion or gauge fields. The WIs in the interacting theory,
valid for the renormalized fields once the regulator has been removed, is identical to eq.~\eqref{eq:PCACt}.
Following the line of reasoning of the previous section one might wonder why we have not considered
a variation of the operator $\mcO$. The reason exemplifies the differences between the two methods.
In this second approach the fermion fields at positive flow-time are in general {\it not} solutions
of the GF equations but only integration variables. As such there is no complicated dependence
of these fermion fields from other fermion fields located at smaller flow-time.
In the previous section the fermion fields were solutions of the GF equations, i.e.
the integration over the Lagrange multipliers was already performed, thus the non-local dependence on the
initial conditions.
The generalization of the chiral symmetry transformations for the $(4+1)$-dimensional theory
gives us another tool to obtain eq.~\eqref{eq:PCACt}.

Using the transformations in~\eqref{eq:delta_chiral} we can derive bulk chiral WIs.
If we perform a local axial variation at a flow-time $s>0$ and
we assume that $s<t_i$ for all the $t_i$ arguments of $\mcO$ the axial WI after $\epsilon \rightarrow 0$
is
\be
\left\langle \left[ \partial_s \tilde{P}^a(s,x) + \partial_\mu \mcA_\mu^a(s,x) \right]\mcO_R\left(\{t_0\}\right)\right\rangle = 0 \qquad s>0\,, \quad s < \left\{t_0\right\}
\label{eq:AWIt}
\ee
where we have denoted generically with $\{t_0\}=\left\{t_1,\ldots,t_N\right\}$ the flow-time
dependence of the operator $\mcO$. The current $\mcA_\mu^a$ is
\be
\mcA_\mu^a(s,x) = \lambdabar(s,x)\frac{T^a}{2}\gamma_5~\left(\lD_{\mu,t} -D_{\mu,t}\right)\chi(s,x)
  -\chibar(s,x)\frac{T^a}{2}\gamma_5\left(\lD_{\mu,t} - D_{\mu,t}\right)\lambda(s,x)\,.
\ee
Even if we use Wilson fermions as a lattice regulator at the boundary, the chiral symmetry of the bulk action implies that
the WI~\eqref{eq:AWIt} is an exact relation in the bare theory.
Since we expect the same WI to be valid after renormalizing the fields and removing the regulator
we can conclude that the normalization factors of $\tilde{P}^a(s,x)$ and $\mcA_\mu^a(s,x)$ are equal to one
for all positive flow-times and the regulator can be safely removed.
We recall that operators located at positive flow-time are renormalized with their
field content and because $Z_\chi = Z_\lambda^{-1}$
the WI~\eqref{eq:AWIt} for $s>0$ is consistent with this result.

Introducing a $4+1$ axial current $\mathcal{A}_M$ with $M=0,\ldots,4$
where $0$ denotes the flow-time direction
\be
\mathcal{A}_M^a = (\tilde{P}^a,\mcA_\mu^a)\,,
\ee
we can rewrite the WI for $s>0$ in a suggestive way
\be
\left\langle \partial_M \mcA_M^a(s,x) \mcO_R\left(\{t_0\}\right)\right\rangle = 0\,.
\label{eq:AWI_Dp1}
\ee
where
\be
\partial_M = (\partial_t,\partial_\mu)\,.
\ee
We note a dimensional mismatch between the different components of the $(4+1)$-dimensional axial current.
This is a direct consequence of the different dimensionality between the Lagrange multipliers
and the fermion fields that is related to the different dimensionality between the
flow-time and any space-time dimension.

Integration of eq.~\eqref{eq:AWIt} over the 4-dimensional space-time leads to
\be
\int d^4 x\left\langle \partial_s \widetilde{P}^a(s,x) \mcO_R(\{t_0\})\right\rangle = 0\,,
\label{eq:ptilde_ind}
\ee
This result can also be obtained using the GF equations and observing that
\be
\int d^4x~K(t,y;s,x)~K(s,x;0,v) = K(t,y;0,v)\,,
\ee
with $t>s>0$.
Eq.~\eqref{eq:ptilde_ind} is interesting because if we consider the short flow-time expansion
\be
\widetilde{P}^a(s,x) = c_{\tilde{P}}(s) \widetilde{P}^a(0,x) + {\rm O}(s)\,,
\label{eq:sde_ptilde}
\ee
eq.~\eqref{eq:ptilde_ind}, satisfied order by order in $s$,
immediately implies that $c_{\tilde{P}}$ is a constant and because of the renormalization
group equations applied to eq.~\eqref{eq:sde_ptilde} also independent of the coupling.
A tree-level calculation thus shows that $c_{\tilde{P}}(s)=1$.

We integrate now the WI along the flow-time direction from $\epsilon$ and $t$ with $0<\epsilon<t<t_i$
and we obtain
\be
\left\langle \left[\mcA_0^a(\epsilon,x) - \mcA_0^a(t,x)\right] \mcO_R\left(\{t_0\}\right)\right\rangle =
\left\langle \left[\partial_\mu \int_\epsilon^t ds~\mcA_\mu^a(s,x)\right] \mcO_R\left(\{t_0\}\right)\right\rangle\,.
\ee
To perform the limit $\epsilon\rightarrow 0$  we need to make sure that the integral
in the r.h.s has, at most, integrable singularities.
One needs to classify all the operators of dimension $5$ or less that can mix with $\mcA_\mu$.
The first thing we notice is that the operators appearing in the short flow-time
expansion of $\mcA^a_\mu$ should contain at least one Lagrange multiplier.
This because otherwise any two-point function between such operators
and $\mcA^a_\mu$ would vanish.
This implies that there can be operators of dimension
$4$ or $5$, but chiral symmetry, extended to the Lagrange multipliers, charge conjugation and O($4$) symmetry
exclude them. We are left with at most a logarithmic divergence when $s\rightarrow 0$
and this divergence is integrable.

Using eq.~\eqref{eq:PCACt} we can then write
\be
-\left\langle \widetilde{P}^a(t,x) \mcO_R\left(\{t_0\}\right)\right\rangle =
\left\langle \left[\partial_\mu \overline{A}_{\mu,R}^a(t,x) - P^{\{a,M\}}_R(x) \right] \mcO_R\left(\{t_0\}\right)\right\rangle\,,
\label{eq:intPCAC}
\ee
where
\be
\overline{A}_{\mu,R}^a(t,x) = Z_A A_\mu^a(x) + \int_0^t ds~\mcA_\mu(s,x)\,.
\label{eq:A_5D}
\ee
We can interpret eq.~\eqref{eq:intPCAC} as a generalization of the axial WI
at positive flow-time, where the l.h.s. of eq.~\eqref{eq:intPCAC}, following the general result of eq.~\eqref{eq:delta_chiral1},
is the variation of the operator $\mcO$ induced by a chiral transformation
performed on the quark fields at a flow-time $t<t_i$.

We show now with an example the usefulness of this approach
deriving the identity obtained by L\"uscher that
allows to relate the chiral condensate to pseudoscalar correlation functions (cf. with eq. (4.9) of ref.~\cite{Luscher:2013cpa}).
We consider the WI in eq.~\eqref{eq:AWI_Dp1} with an operator $\mcO$
function of fermion fields localized at flow-time
$t$. We do not assume any ordering on the flow-times thus we need to add a contact term to the WI when $t=s$
stemming from the local variation of the operator
\be
\left\langle \left[\partial_s \tilde{P}^a(s,x) + \partial_\mu \mcA_\mu^a(s,x)\right]
\mathcal{O}_R(t,y)\right\rangle = \left\langle i \left[\frac{\delta \mathcal{O}(t,y)}{\delta \alpha_A^a(s,x)}\right]_R\right\rangle \,.
\label{eq:WI2_quant}
\ee
The r.h.s. can be computed in a standard manner if we consider
$\mcO(t,y) = P^b(t,y)$
\be
\langle
\left[\partial_s \tilde{P}^a(s,x) + \partial_\mu \mcA_\mu^a(s,x) \right] P_R^b(t,y)\rangle =
- \langle S_R^{\left\{a,b\right\}}(s,y) \rangle
\delta(t-s) \delta (x-y)\,,
\ee
where
\be
S^{\left\{a,b\right\}}(s,y) = \chibar(s,y)\left\{\frac{T^a}{2},\frac{T^b}{2}\right\}\chi(s,y)\,.
\ee
We recall that $P^a(t,y)$ and $S^a(t,y)$ renormalize multiplicatively with $Z_\chi$~\cite{Luscher:2013cpa}.
Integrating this WI over the whole $4$-dimensional space-time volume we obtain
\be
\int d^4x~\langle
\partial_s \tilde{P}^a(s,x) P_R^b(t,y)\rangle =
- \langle S_R^{\left\{a,b\right\}}(s,y) \rangle \delta(t-s)\,.
\ee
Having included the contact term we can now integrate in the flow-time
in the range $\epsilon < s < T$ with $T>t$ and perform the limit $\epsilon \rightarrow 0$
\be
\lim_{\epsilon \rightarrow 0}\int d^4x~\int_\epsilon^T ds~\langle
\partial_s \tilde{P}^a(s,x) P_R^b(t,y)\rangle =
- \lim_{\epsilon \rightarrow 0} \int_\epsilon^T ds~\langle S_R^{\left\{a,b\right\}}(s,y) \rangle \delta(t-s)\,.
\ee
The boundary term at $T$ does not contribute because propagators $\wick{1}{<*\lambda(T,x) >*{\chibar}(t,x)}$
and $\wick{1}{<*\chi(t,x) >*{\lambdabar}(T,x)}$ vanish for $T>t$~\cite{Luscher:2013cpa}.
We thus obtain
\be
\int d^4x~\langle \tilde{P}^a(0,x) P_R^b(t,y)\rangle = \langle S_R^{\left\{a,b\right\}}(t,y) \rangle\,.
\label{eq:ptilde_cond}
\ee
This is exactly the relation obtained in ref.~\cite{Luscher:2013cpa},
that allows the determination of the chiral condensate (cfr. eq. 4.15 of ref.~\cite{Luscher:2013cpa}).
This example shows that it is possible to use standard techniques based
on chiral symmetry transformations to derive WIs
involving fields at the boundary and solutions of the GF
for positive flow-time.

We remark that in the derivation of eq.~\eqref{eq:ptilde_cond} it is not necessary to send $\epsilon \rightarrow 0$.
If we integrate over the flow-time between $s_0$ and $T$ with $s_0<t<T$ we obtain
\be
\int d^4x~\langle
\tilde{P}^a(s_0,x) P_R^b(t,y)\rangle = \langle S_R^{\left\{a,b\right\}}(t,y) \rangle\,.
\ee
This result is independent of $s_0$ as far as $0<s_0<t$ as can be seen from eq.~\eqref{eq:ptilde_ind}.
It is only to make contact with the physics of pions that it is necessary to place $\tilde{P}^a$
at the boundaries.

\subsection{Vector Ward identities}

Using the procedure outlined in the previous section we can derive also the so-called PCVC relation at the boundary
$t=0$ and the corresponding relation in the bulk
\be
\begin{cases}
\partial_\mu V^a_\mu(x) + \psibar(x) \left[\frac{T^a}{2},M\right]\psi(x) + \overline{S}^a(0,x)=0  & t=0 \\
\partial_t \overline{S}^a(t,x) + \partial_\mu \mcV_\mu^a(t,x)=0  & t > 0
\end{cases}
\label{eq:WI1}
\ee
where
\be
\overline{S}^a(t,x) = \lambdabar(t,x)\frac{T^a}{2}\chi(t,x)-\chibar(t,x)\frac{T^a}{2}\lambda(t,x)\,,
\ee
and
\be
\mcV^a_\mu(t,x) = \lambdabar(t,x)\frac{T^a}{2}\left(\lD_{\mu,t} - D_{\mu,t}\right)\chi(t,x) +
\chibar(t,x)\left(\lD_{\mu,t} - D_{\mu,t}\right)\frac{T^a}{2}\lambda(t,x)\,.
\ee

As for the axial case we can rewrite the second relation in~\eqref{eq:WI1} for $t>0$
introducing a $4+1$ vector current $\mathcal{V}_M$ with $M=0,\ldots,4$
where the $0$ index refers to the flow-time direction
\be
\mathcal{V}_M^a = (\overline{S}^a(t,x),\mcV^a_\mu(t,x))\,.
\ee
For $t>0$ the vector current $\mathcal{V}_M^a$ is conserved
\be
\partial_M \mathcal{V}_M^a=0\,.
\ee
Using as a probe $\mcO\left(\{t_0\}\right)$, the VWI with $t_0>s$ reads
\be
\left\langle \left[\partial_t \mcV_0^a(s,x) + \partial_\mu \mcV_\mu^a(s,x)\right]\mcO_R(\{t_0\})\right\rangle=0\,,
\ee
and the generalization of the standard VWI is
\be
-\left\langle \overline{S}^a(t,x) \mcO(t_0,y)\right\rangle =
\left\langle \partial_\mu \overline{V}_\mu^a(t,x) \mcO(\{t_0\})\right\rangle\,,
\label{eq:intCVC}
\ee
where
\be
\overline{V}_\mu^a(t,x) = Z_V V_\mu^a(x) + \int_0^t ds~\mcV_\mu(s,x)\,.
\ee

\section{Twisted mass QCD}
\label{sec:tmQCD}
In this section we consider the GF with twisted mass fermions~\cite{Frezzotti:2000nk}.
For simplicity we consider a flavour doublet of mass degenerate twisted
mass fermions. The extension to non-degenerate twisted mass fermions~\cite{Frezzotti:2003xj}
does not pose any additional complication.
For reviews about twisted mass fermions see refs.~\cite{Sharpe:2006pu,Sint:2007ug,Shindler:2007vp}.
In the twisted basis the continuum twisted mass action
\be
S_{\rm F,tmQCD} = \int d^4x~\psibar(x)\left[\gamma_\mu D_\mu + m + i \mu_{\rm q}\gamma_5 \tau^3\right]\psi(x)\,,
\ee
is obtained by a chiral rotation of a twist angle $\omega={\rm atan}\left(\mu_q/m\right)$
\be
\begin{cases}
\psi(x) \rightarrow  {\rm exp}\left\{i\left(\omega \frac{\tau^3}{2}\gamma_5\right)\right\} \psi(x) \vspace{0.5cm}\\
\psibar(x) \rightarrow  \psi(x) {\rm exp}\left\{i\left(\omega \frac{\tau^3}{2}\gamma_5\right)\right\}\,.
\end{cases}
\ee
of the continuum QCD action.
To obtain the fermionic part of the action
for the $(4+1)$-dimensional theory we perform the same twist rotation
on the fermion fields at positive flow-time
\be
\begin{cases}
\chi(t,x) \rightarrow  {\rm exp}\left\{i\left(\omega \frac{\tau^3}{2}\gamma_5\right)\right\} \chi(t,x) \vspace{0.5cm}\\
\chibar(t,x) \rightarrow  \chibar(t,x) {\rm exp}\left\{i\left(\omega \frac{\tau^3}{2}\gamma_5\right)\right\}\,.
\label{eq:twist}
\end{cases}
\ee
The Lagrange multipliers are also a flavor doublet and the twist rotation
is obtained using the chiral symmetry transformations defined in eq.~(\ref{eq:chiral},\ref{eq:chiral_lambda})
\be
\begin{cases}
\lambda(t,x) \rightarrow  {\rm exp}\left\{i\left(- \omega \frac{\tau^3}{2}\gamma_5\right)\right\} \lambda(t,x) \vspace{0.5cm}\\
\lambdabar(t,x) \rightarrow  \lambdabar(t,x) {\rm exp}\left\{i\left(- \omega \frac{\tau^3}{2}\gamma_5\right)\right\}\,.
\label{eq:twist_lambda}
\end{cases}
\ee
The exact chiral symmetry of the bulk term in the $4+1$ theory implies that this term of the action
stays invariant under a twist rotation, i.e. the total twisted mass action is
\be
S_{\rm F} = S_{\rm F,tmQCD} + S_{\rm F,fl}\,,
\label{eq:twisted_action}
\ee
where $S_{\rm F,fl}$ is given by eq.~\eqref{eq:action_bulk}.
This is not surprising because the GF equations~\eqref{eq:flow_ferm} and the initial
conditions~\eqref{eq:bc} are invariant under a twist rotation.

The twist rotations in eqs.~(\ref{eq:twist},\ref{eq:twist_lambda}) define the
operators at positive flow-time in the twisted basis.
Fermions bilinears formed only by fermion fields or only
Lagrange multipliers transform in a standard manner.
Bilinears made of fermion and Lagrange multiplier fields
twist in a different way and, as an example, we show how the twist rotation applies to $\tilde{P}^a$ and
$\overline{S}^a$
\be
\begin{cases}
\tilde{P}^a \longrightarrow \cos \omega \tilde{P}^a + \sin \omega \overline{S}^a\,, \qquad a=1,2 \\
\tilde{P}^a \longrightarrow \tilde{P}^a\,, \qquad a=0,3\,,
\end{cases}
\label{eq:ptilde_twist}
\ee
\be
\begin{cases}
\overline{S}^a \longrightarrow \cos \omega \overline{S}^a  + \sin \omega \tilde{P}^a\,, \qquad a=1,2 \\
\overline{S}^a \longrightarrow \overline{S}^a\,, \qquad a=0,3\,.
\end{cases}
\label{eq:sbar_twist}
\ee
Using the action in eq.~\eqref{eq:twisted_action} and the chiral
transformations~(\ref{eq:chiral_chi},\ref{eq:chiral_lambda})
we can derive the PCAC and PCVC relations
\be
\begin{cases}
\partial_\mu A_\mu^a(x) - 2mP^a(x) -i \mu_q \delta^{a3} S^0(x) + \tilde{P}^a(0,x)=0 & t=0  \\
\partial_t \tilde{P}^a(t,x) + \partial_\mu \mcA_\mu^a(t,x)=0 & t>0 \,,
\end{cases}
\label{eq:WI_tm}
\ee
and
\be
\begin{cases}
\partial_\mu V_\mu^a(x) + 2 \mu_q \epsilon^{3ab}P^b(x) + \overline{S}^a(0,x) = 0 & t=0  \\
\partial_t \overline{S}^a(t,x) + \partial_\mu \mcV_\mu^a(t,x) = 0 & t>0 \,.
\end{cases}
\label{eq:VWI_tm}
\ee

The equivalence between QCD and tmQCD in the interacting theory has been proven in~\cite{Frezzotti:2000nk} using as a regulator
overlap fermions. We note that the exact chiral symmetry of the bulk action does not add any complication
and the result of~\cite{Frezzotti:2000nk} is valid for correlation functions involving
local operators also at positive flow-time.

\section{Wilson twisted mass fermions}
\label{sec:auto}
The GF equations can be applied to a fermion field independently on the
fermionic lattice action of choice. Here we choose to study the interplay
between the GF and Wilson twisted mass (Wtm) fermions. In particular we want to analyze
the O($a$) cutoff effects of this formulation for operators defined
at positive flow-time and we want to see if it is possible to extend the
property of automatic O($a$) improvement.

Since we will only be concerned with O($a$) effects of the $(4+1)$-dimensional theory
we do not need to specify the exact form of the gauge action.
The form of the unimproved fermionic lattice action at the boundary $t=0$ is
\be
  S_{\rm Wtm}[\psi,\psibar,U] =a^4\sum_x\psibar(x)\Big[D_{\rm W} + m_0 + i\mu_0\gamma_5\tau^3\Big]\psi(x)\,,
\label{eq:WtmQCD2}
\ee
where $D_{\rm W}$ is the usual Wilson massless operator
\be
D_{\rm W} = \frac{1}{2}\left[\gamma_\mu\left(\nabla_\mu + \nabla_\mu^*\right) - a\nabla^*_\mu\nabla_\mu\right]\,.
\ee
The bulk action is the one defined in~\eqref{eq:action_bulk_lattice} and $m_0$, $\mu_0$ are the bare
untwisted and twisted quark mass.

The long distance properties of Wtm close to the continuum limit
may be described in terms of a local effective theory with action
\be
S_{\rm eff} = S_0+aS_1+a^2S_2+\ldots
\label{eq:eff_action}
\ee
where the leading term, $S_0$, is the action of the target continuum theory~\eqref{eq:twisted_action}
plus the gauge action with properly renormalized parameters.
The higher order terms are linear combination of higher-dimensional operators. For example
\be
S_1 = \int_0^{\infty} dt \int d^4x \sum_i c_i(g_0^2) \mcO_{R,i}(t,x)\,,
\ee
where $\mcO_i(t,x)$ respect the symmetries of the lattice action.
We omit for simplicity the dependence on the renormalization scale.

A correlation function of products of a multiplicatively renormalizable lattice fields, here denoted by $\phi_R=Z_\phi \phi$,
at separated points $(t_i,x_i)$
\be
G(t_1,x_1;\ldots;t_n,x_n) = \langle \phi_R(t_1,x_1) \cdots \phi_R(t_n,x_n) \rangle \equiv \langle \Phi_R \rangle
\label{eq:phi}
\ee
takes the form
\be
\left\langle \Phi_R \right\rangle =  \langle \Phi_0 \rangle_0 - a \langle \Phi_0
     S_1 \rangle_0 + a \langle \Phi_1 \rangle_0 + {\rm O}(a^2)\,,
\label{eq:sym_exp1}
\ee
where
\be
\langle \Phi_0 \rangle_0 \equiv \langle\phi_0(t_1,x_1) \cdots \cdots \phi_0(t_n,x_n) \rangle_0\,,
\ee
\be
\langle \Phi_1 \rangle_0 \equiv \sum_{k=1}^{n}\langle\phi_0(t_1,x_1) \cdots \phi_1(t_k,x_k) \cdots \phi_0(t_n,x_n) \rangle_0\,,
\ee
and $\phi_0,\phi_1$ are renormalized continuum fields. $\phi_1$ is a linear combination of
local operators of dimension $d_\phi + 1$ that depends on the specific operator $\phi$ and
are classified according to the lattice symmetries transformation properties of $\phi$.
The expectation values on the right hand side of eq.~\eqref{eq:sym_exp1} are to be taken in the continuum theory
with action $S_0$

In the massless limit Wilson and Wtm fermions have the same lattice action
thus the higher-dimensional operators in the Symanzik effective action are the same.
Using the equations of motion for the quark fields in the chiral limit a possible list of O($a$) terms
is~\cite{Luscher:1996sc,Luscher:2013cpa}
\be
 \qquad {\mathcal O}_1 =
  i \psibar(x)\sigma_{\mu\nu}F_{\mu\nu}\psi(x)\,, \qquad
{\mathcal O}_2 =
  \lambdabar(0,x) \lambda(0,x)\,.
\label{eq:sym_op}
\ee

It is important to notice that there are no $t$-dependent terms in the effective action.
The additional operator $\mcO_2$ is a boundary term.
From the point of view of the chiral symmetry transformations~(\ref{eq:chiral_chi},\ref{eq:chiral_lambda})
the absence of O($a$) terms for $t>0$ is a consequence of the chiral invariance of the bulk action.

Additional terms in the effective theory at non-zero quark mass for Wtm fermions
have been classified in~\cite{Luscher:1996sc,Frezzotti:2001ea}. The gradient flow
adds the operators $\mcO_7$ and $\mcO_8$ to the list
\bea
\mcO_3 &=& m \tr \{ F_{\mu\nu}(x) F_{\mu\nu}(x) \}\,,\quad
\mcO_4 = m\psibar(x) \psi(x)\,, \quad
\mcO_5 = m\mu_{\rm q} i \psibar(x)\gamma_5\tau^3\psi(x)\,, \nonumber \\
\mcO_6 &=& \mu_{\rm q}^2\psibar(x)\psi(x)\,, \quad
\mcO_7 = m\left(\lambdabar(0,x)\psi(x) + \psibar(x)\lambda(0,x)\right)\,, \nonumber \\
\mcO_8 &=& i \mu_{\rm q} \left(\lambdabar(0,x)\gamma_5 \tau^3\psi(x) + \psibar(x)\gamma_5 \tau^3\lambda(0,x)\right)\,.
\label{eq:op_mass}
\eea
The operator $\mcO_7$ is already given in ref.~\cite{Luscher:2013cpa}.
The operator $\mcO_8$ is new and is the one needed to remove O($a \mu_{\rm q}$) effects to the fermion fields and Lagrange multipliers.

\subsection{Automatic O($a$) improvement}

Automatic O($a$) improvement~\cite{Frezzotti:2003ni} is the property of Wtm that physical correlation functions
with a finite continuum limit made of multiplicatively renormalizable fields are free from O($a$) effects,
when the lattice parameters are tuned to obtain in the continuum
limit that the renormalized untwisted quark mass vanishes,$m_{\rm R} = 0$.

From the lattice perspective this corresponds to set the bare untwisted mass $m_0$ to its critical
value $\mcr$. The exact way this is achieved is not relevant for what follows, but for
a discussion and further references on this topic see~\cite{Shindler:2007vp}.

To parametrizes the O($a$) uncertainties
stemming from the determination of the critical mass,
i.e. $m_{\rm q} = m_0 - \mcr = $ O($a$) in the Symanzik effective theory
one adds the operator
\be
\mcO_0 =  \Lambda^2\psibar(x) \psi(x)\,,
\ee
where $\Lambda^2$ is some energy scale squared which depends on the way the critical mass is determined,
e.g. it could be of the order of the QCD scale $\Lambda_{\rm QCD}^2$, or it could
be something proportional to $\Lambda_{\rm QCD}\mu_{\rm q}$.
This allows to check if potential O($a$) uncertainties in the critical mass contradict
the property of automatic O($a$) improvement.

The proof of automatic O($a$) improvement for correlation functions containing fields at positive flow-
time follows the proof for the $t=0$ case~\cite{Frezzotti:2005gi,Shindler:2005vj,Aoki:2006nv}.
We extend the relevant symmetries for the Lagrange multipliers
\be
\mcR^{1,2}_{5,t} \colon
\begin{cases}
\chi(t,x) \rightarrow i \gamma_5 \tau^{1,2} \chi(t,x) \\
\chibar(t,x) \rightarrow  \chibar(t,x) i \gamma_5 \tau^{1,2} \\
\lambda(t,x) \rightarrow -i \gamma_5 \tau^{1,2} \lambda(t,x) \\
\lambdabar(t,x) \rightarrow  - i \lambdabar(t,x) \gamma_5 \tau^{1,2} \,,
\end{cases}
\label{eq:R512_t}
\ee
and
\be
\mcD_t \colon
\begin{cases}
V(t,x;\mu) \rightarrow V^{\dagger}(t,-x-a\hat{\mu};\mu), \\
\chi(t,x) \rightarrow {\rm e}^{3 i \pi/2} \chi(t,-x) \\
\chibar(t,x) \rightarrow  \chibar(t,-x){\rm e}^{3 i \pi/2}\\
\lambda(t,x) \rightarrow {\rm e}^{5 i \pi/2} \lambda(t,-x) \\
\lambdabar(t,x) \rightarrow  \lambdabar(t,-x){\rm e}^{5 i \pi/2}\,.
\end{cases}
\ee
The equivalent transformations for continuum fields,
that with abuse of notation we indicate in the same way, are the same for the fermion
fields, while for the gauge fields the $\mcD_t$ transformation is
$B_\mu(t,x) \rightarrow -B_\mu(t,-x)$.
In the $\mcD_t$ transformation we do not reverse the sign in the $t$ direction
because the flow-time has a space-time dimension of $2$. Thus any derivative
with respect to the flow-time contained in a generic operator will not change the
parity related to its dimensionality.
To include the twisted mass in the counting of the dimensions of the
operators appearing in the lattice and continuum Lagrangian one introduces the spurionic
symmetry
\be
\widetilde{\mathcal{D}_t} =  \mathcal{D}_t\times [\mu_{\rm q} \rightarrow -\mu_{\rm q}]\,.
\ee

The lattice action~\eqref{eq:WtmQCD2} at the boundary, and~\eqref{eq:action_bulk_lattice}
in the bulk is invariant under the $\mcR^{1,2}_{5,t} \times \widetilde{\mathcal{D}_t}$ transformation.
If the target continuum theory has a vanishing renormalized untwisted mass, $m_R=0$, it
is invariant separately under the $\mcR^{1,2}_{5,t}$ and the $\widetilde{\mathcal{D}_t}$
transformations.
This immediately implies that all the higher-dimensional operators in the Symanzik expansion
contributing to $S_1$ are odd under $\mcR^{1,2}_{5,t}$. This is not true for operators proportional
to the untwisted quark mass, but the maximal twist condition $m_R=0$ forces them to contribute
to higher orders in the Symanzik expansion.
For example the new terms $\mcO_2$ and $\mcO_8$,
containing at least one Lagrange multiplier are odd under $\mcR^{1,2}_{5,t}$, thus they
vanish once inserted in $\mcR^{1,2}_{5,t}$ even correlation functions.
The same argument applies for the higher-dimensional operators
appearing in the effective theory representations of local operators such as axial currents
or pseudoscalar densities at positive flow-time.

Let us see with specific examples how this works.
The list of operators contributing at O($a$) to the axial current $A_\mu^a$ in the Symanzik effective theory
is given by
\be
\left(\delta A^a_\mu\right)_1 = \partial_\mu P^a(0,x)\,,\quad  \left(\delta A^a_\mu\right)_2=\tilde{A}_\mu^a(0,x)\,,\nonumber
\ee
\be
\left(\delta A^a_\mu\right)_3 =  m A_\mu^a(x)\,,\quad
\left(\delta A^a_\mu\right)_4 =  \mu_{\rm q} \epsilon^{3ab}V_\mu^b(x)\,,
\ee
where
\be
\tilde{A}_\mu^a(0,x) = \lambdabar(0,x) \gamma_\mu \gamma_5 \frac{T^a}{2} \psi(x) + \psibar(x) \gamma_\mu\gamma_5 \frac{T^a}{2} \lambda(0,x)\,.
\ee
It is easy to check that all these operators have opposite $\mcR_{5,t}^{1,2}$ parity with respect to $A_\mu^a$.
The only exception being the term proportional to the untwisted mass $m$ that vanishes at maximal twist.
Consistently also the $\widetilde{\mcD}_t$ parity is different between $A_\mu^a$ and the higher-dimensional
operators. This means that their contribution to the effective correlator
vanishes if the continuum limit of the lattice correlator containing $A_\mu^a$ is $\mcR_{5,t}^{1,2}$ even.

Another example is given by the local operator $\tilde{P}^a$. The list of operators contributing at
O($a$) in the Symanzik effective theory is given by
\be
\left(\delta \tilde{P}^a\right)_1 = \partial_\mu \tilde{A}_\mu^a(0,x)\,, \quad \left(\delta \tilde{P}^a\right)_2 = \hat{P}^a(0,x)\,, \nonumber
\ee
\be
\left(\delta \tilde{P}^a\right)_3 = m \tilde{P}^a(0,x)\,,\quad \left(\delta \tilde{P}^a\right)_4 = \mu_{\rm q} \epsilon^{3ab}\overline{S}^b(0,x)\,,
\ee
where
\be
\hat{P}^a(0,x) = \lambdabar(0,x) \gamma_5 \frac{T^a}{2} \lambda(0,x)\,.
\ee
As before all the higher-dimensional operators have opposite $\mcR_{5,t}^{1,2}$ parity with respect to $\tilde{P}^a$,
excluding the term proportional to $m$ that anyhow vanishes at maximal twist.

\subsection{Chiral condensate}

As an application of the WIs we have derived, following the strategy suggested by L\"uscher in~\cite{Luscher:2013cpa},
we show how to compute the chiral condensate with Wtm fermions at maximal twist, retaining
the property of multiplicative renormalization and absence of mixing with lower-dimensional operators.
Additionally with Wtm at maximal twist one does not need to compute any improvement coefficient but having leading O($a^2$) effects
discretization errors.

In the basis we have chosen for the fermion fields, the so-called twisted basis,
the quark condensate at positive flow-time is related to the expectation value of $P^{3}$
\be
\Sigma_t = \frac{\Sigma_t^{uu} + \Sigma_t^{dd}}{2} = - i \langle P^3(t,x) \rangle\,,
\ee
that renormalizes multiplicatively with $Z_\chi$.
To relate the $\Sigma_t$ to the physics of pions we use the VWIs at vanishing and positive flow-time~\eqref{eq:VWI_tm}
and as external probe operator at positive flow-time we choose a pseudoscalar density, i.e
\be
\begin{cases}
\left\langle \left[\partial_\mu V_{\mu,R}^1(0,x) + 2 \mu_R P_R^2(0,x) + \overline{S}_R^1(0,x)\right]P_R^2(t,y)\right\rangle = 0 \\
\left\langle \left[\partial_s \overline{S}^1(s,x) + \partial_\mu \mcV_\mu^1(s,x)\right] P_R^2(t,y) \right\rangle =
i \langle P^3(t,y) \rangle \delta(t-s) \delta(x-y) \,.
\end{cases}
\ee
Following the same strategy we have used in sec.~\ref{sec:WI} we integrate the first VWI over space-time and
the second one over space-time and over the flow-time direction from $0$ to $T>t$.
We then obtain
\be
2 \mu_R \int d^Dx \langle P_R^2(0,x) P_R^2(t,y)\rangle = i \langle P_R^3(t,y) \rangle\,.
\label{eq:int_VWI_tm}
\ee
This equation relates the charged pion correlation function with the chiral condensate
at positive flow-time. The existence of a conserved vector current guarantees that we can choose the
normalization of twisted mass such that $\mu_R P_R^2$ is renormalization group invariant. This insures that
eq.~\eqref{eq:int_VWI_tm} is valid with bare correlators up to discretization errors.
Automatic O($a$) improvement guarantees that the leading discretization errors are of O($a^2$).
Using the standard VWI with an external probe at vanishing flow-time
\be
\left\langle \left[\partial_\mu V_{\mu,R}^1(0,x) + 2 \mu_R P_R^2(0,x) \right]P_R^2(0,y)\right\rangle = 0\,, \quad x \neq y \\
\ee
and considering the spectral decomposition of the charged pseudoscalar correlators we obtain
for the renormalized chiral condensate
\be
\Sigma_{\rm R} = \lim_{\mu_R \rightarrow 0}Z_P \frac{\Sigma_t G_\pi}{G_{\pi,t}}\,,
\ee
where $G_\pi$ and $G_{\pi,t}$ are the vacuum-to-pion matrix elements of the
charged pseudoscalar density respectively at vanishing and positive flow-time.
Given that the renormalization between $\Sigma_t$ and $G_{\pi,t}$ simplifies
the only renormalization constant needed to compute the chiral condensate is $Z_P$.
It is not surprising that we reproduce the same equation obtained by L\"uscher in~\cite{Luscher:2013cpa}.
The additional advantage of using Wtm at maximal twist is automatic O($a$) improvement
and the fact that there is no need to compute any improvement coefficient.
In fact the expectation value $i \langle P^3(t,x) \rangle$
is even under $\mcR_{5,t}^{1,2}$ as the charged pseudoscalar correlators thus
the chiral condensate not only renormalizes multiplicatively
but it is also not affected by any O($a$) cutoff effects.

With standard or clover Wilson fermions to remove completely O($a$) cutoff effects
it is necessary to add an additional counterterm to the lattice action proportional to
$\lambdabar \lambda$. The tunable coefficient of this term has been labeled in~\cite{Luscher:2013cpa}
$c_{\rm fl}$. We can write the quark condensate lattice correlator with the insertion of this term
\bea
\langle \ubar(t,x) \gamma_5 u(t,x) &-& \dbar(t,x) \gamma_5 d(t,x)  \rangle = \nonumber \\
&-& a^8 \sum_{y,z} \langle \Tr \left\{ K(t,x;0,y) \gamma_5\left[S_u(y,z) -a c_{\rm fl} \delta_{y,z}\right]K(t,x;0,z)^\dagger \right\}  \nonumber \\
&-& \Tr \left\{ K(t,x;0,y) \gamma_5\left[S_d(y,z) -a c_{\rm fl} \delta_{y,z}\right]K(t,x;0,z)^\dagger \right\}\rangle_{\rm G}\,,
\eea
where $\langle \rangle_{\rm G}$ indicates the gauge average including the fermionic determinant.
The terms proportional to $c_{\rm fl}$ are independent of the flavor and they cancel out.
This is just a reflection of the different $\mcR_{5,t}^{1,2}$ parities between the observable and the O($a$) term.

redpajama_set_name: RedPajamaArXiv