Guest blogger Louisa van Eeden-Smit of LexisNexis UK follows her recent article on smart contracts with another future-gazing blog, this time looking at what you need to future proof your career for the law firm of the future.
What will the law firm of the future look like? It's a reasonable question to ask considering how much the legal market has changed already, and the ever-increasing pace of change moving forwards.
PwC's 2018 Law Firm Survey found that 100% of the top 10 firms cited technology as the key challenge to growth over the next two years, but there is an overall optimism about the direction the industry is heading and the ability to stay competitive. The key questions that seem to come up in our experience are: How much will technology change the face of legal service? Will the generalist die out? Will more firms merge, or will the niche outfit emerge triumphant? Will more lawyers become self-employed, providing virtual services to organisations?
Regardless of the outcomes, there are ways that legal professionals – be they part-time, returners, flexi-workers, or full-time in-house – can future-proof their careers.
A degree of agility and flexibility will be necessary regarding how lawyers deliver legal services. This is as true now, with the advent of new regulations and an increasingly tech-savvy and informed client base, as it will be in the future. A willingness to adapt will serve you well, as well as open-mindedness regarding alternatives to the traditional model of working that is fast becoming a relic. From portfolio careers to flexible working, there are more models than ever to suit professionals – and benefit both employer and employee.
Client loyalty isn't a given in a market replete with so many options – it's earned. Today's client is more informed and tech-savvy than ever. They are willing to shop around and they are empowered. When clients demand efficient, tech-led services, for example, legal services providers should listen and adapt. It's also important to be proactive and show how the firm is anticipating future change and preparing to evolve services.
Client loyalty may not reign supreme anymore, so it has become critical that law firms prove themselves to the in-house legal teams they serve. Listening to what they need and learning the business from the inside out will allow them to stand out from the crowd by offering in-house counsel exactly what they want, exactly what they consider to be valuable. "Deep knowledge of the business is what really breeds loyalty," according to Richard Harris, Chief Legal Officer at Robert Walters Group.
In addition to learning about the business in order to provide more nimble and forward-looking advice, a future-proof and commercial legal professional is one that acts proactively. If you can look ahead and anticipate what issues might impact your business and put forward relevant plans of action, you will prove yourself to be indispensable. Such advice is "worth its weight in gold", says Dean Nash, Head of Legal and Compliance at Monzo Bank, and will allow you to retain business in a competitive market.
In-house legal teams consider it to be a win-win. After all, with technology streamlining the service, the process becomes faster, and invoices get lower. Legal tools are just that – tools that can be used to support lawyers, not replace them. Harnessing them to provide efficient service is a no-brainer, especially considering the fact that it can free up lawyers to focus on the parts of their job they actually like.
Overall, legal professionals who are willing and able to jump in and run with the changes – namely, maintaining a lean, agile practice, one that uses technology and offers efficient, business-centric service – will find themselves in a good position – and way ahead of the curve.
It also seems there is a common prediction emerging – part hope, part anticipation based on current trends – that law firms will be more holistic in the future. This applies to both client service and employee care. There is a hope that law firms will take a broader look at the service they provide; addressing the whole commercial picture of the business, as opposed to offering discrete pieces of legal advice, for example.
For consultants or legal professionals moving into this sphere, or laterally moving within it, there are more opportunities than ever to have the kind of career you want. The legal market is constantly evolving and allowing for more diverse working opportunities – the only question remains: what does the legal team or law firm of your future look like?
We are delighted to have Louisa Van Eeden of Lexis Nexis UK join us as a guest blogger on The Attic. Her first post takes a look at how millennials are shaping the future of the legal industry.
Millennials. They're the generation that everybody loves to blame for, well, pretty much anything. They are branded as "snowflakes", and written off as a problem that either needs to be overcome, or ignored until they 'grow up' and become more like the older generations. But who are they really, and what impact are they having on the legal industry?
Much like Generation X back in the 1990s, the term millennial is often used in the media to refer to anything related to a youth culture that other generations generally don't understand that well. Unlike Generation X, the Millennial Generation was typically born between the early 1980s and the late 1990s, and grew up in a dramatically different landscape to their forebears. The world has changed: the internet dominates our lives, job security and home ownership are not a certainty; the future of the planet is at risk through climate change; and it's becoming rapidly apparent that continuing with a business as usual attitude, just because that's the way it's always been, is untenable.
Consequently, millennials simply aren't as compelled by traditional practices as previous generations, both in life and work. They are who they are. Where previous high achievers would chase salary, millennials now typically look for a company that aligns with their values. For the millennial, culture reigns supreme. In fact, a study recently conducted by Fidelity showed that millennials are willing to give up (up to) £7,600 in salary every year for a job that gave them a better environment and culture.
Considering that millennials now form the backbone of staff and client bases, making up to 35% of our current workforce, with that set to increase to 50% of the workforce by 2020, this is not a demographic to underestimate. Millennials are not just a vague notion of youth culture, they are real people progressing into management positions, and are shaping the technological and cultural landscape of every industry, including the law.
This is a key battleground for law firms and one which millennials are well-positioned to approach and understand, especially because, as the Financial Times continues to report, firms are struggling to source and retain talent in today's rapidly changing marketplace. Attracting the best and brightest young talent is more important than ever before, and harder than ever before, with this generation taking a markedly different approach to their careers. Millennials are key to helping law firms communicate their vision of the future, enabling firms to modernise with an eye to the demands of new talent and driving a competitive edge.
Law firms are already changing with the millennial worker in mind. As the Law Journal Newsletter reports: "A number of firms have moved, remodelled or completely overhauled their physical workplaces with millennials in mind, favouring common areas, for example, over large corner offices." But it's not just physical changes, but more fundamental ones as well. As one of the co-founders of the Legal A-Team asserts, the partnership model – one of the traditional prestige markers in law firms – is no longer the aim: "Millennials want what they want and they want it now. The patience factor is not one of their fortes — they're not going to stand around for 12 years." In order to retain and attract millennial talent (and not lose them to agile, tech-forward start-ups), law firms will need to significantly adapt their culture and perhaps even their company structure.
Law firms and lawyers need to think more creatively about their working practices in light of the rise of consumerism in today's legal market. Client power is increasingly dominant, with billing and efficiency becoming ever more important. New and agile practices, from social media to technology, are areas where millennials excel. They also prefer to work collaboratively rather than as a silo, which may well serve firms well moving forwards. Being open to change and engaging in a dialogue with junior members of the team may be beneficial here, in order to ensure that the firm remains stimulated and doesn't fall behind.
While all three areas are important to understand, technology is the one that underpins them all. The agile working practices and lateral knowledge-sharing solutions favoured by millennial legal professionals and legal start-ups are all enabled by technology. Indeed, there are already reports that legal tools are being used with increasing regularity. A recent LexisNexis In-house Insights report, 'Legal Technology – Looking Past the Hype' found that 85% of in-house legal teams surveyed have introduced multiple technology types and almost three quarters (73%) of respondents who have already introduced legal technology tools are making plans to expand their implementation. It is very likely that the next generation of lawyers will practise law in very different ways.
Technology is undoubtedly changing the legal profession, and it's likely that the next generation of lawyers will practise law in very different ways. Indeed, there is already a growing section of the workplace – populated by those who have different life demands and values – who no longer fit the traditional working model. This has led to an increase in portfolio careers as well as flexible working models designed to benefit both employee and employer. This can be seen in the continued success of organisations such as Obelisk Support, which recently joined the FT Future 100 UK list as a diversity leader – the only legal company to do so.
This trend is likely to continue to increase and evolve as more millennials dominate the workforce, bringing with them their approach to work-life balance, their use of technology as a natural enabler, the importance they place on purposeful business (one which looks beyond the profit line), their desire for flexibility, and their alternative definitions of success.
We are entering an exciting era for the legal industry, one in which we wait in watchful anticipation to see who will accept and accelerate this new approach to working culture, and what impact it will have across the legal profession and wider society.
For more insights on how young lawyers are best positioning themselves to weather the upcoming changes, check out our post on "The Legal Profession for Millennials".
How do we deal with upheaval as individuals? We have to adapt, be open to the changes ahead and listen to advice. In the legal profession, it is no different – as the world in which we practice law changes rapidly, lawyers need to be ready to rethink how they work.
That is the premise of a new book by Michele DeStefano, law professor and founder of LawWithoutWalls and MoveLaw. Legal Upheaval: A Guide To Creativity, Collaboration and Innovation in Law introduces readers to 7 essential experiences that lawyers must master to achieve innovation, transform their collaboration with clients, and create solutions at the intersection of law, technology and business.
You are recognised as a 'legal rebel' by the American Bar Association – what does that mean to you?
To me a legal rebel means someone who isn't just talking about what needs to be fixed in legal practice, both in training and practice, but is actively taking risks to do things that are different. Ironically, since the law is slower to change than other industries it's not that hard to be considered a rebel!
There certainly is a lot of talk about innovation at the moment – in the book you define it as 'lasting incremental change that adds value', how much of that are we seeing right now?
There are various ways to define innovation, and it can be a hackneyed word. But there is some consensus in law that innovation is still about small steps – small change is difficult but is easier than asking for 'big bang' innovation, especially in a world of people that like the status quo.
However, lawyers and Heads of Innovation I think still inaccurately focus on the technology side of innovating, and it's starting to frustrate in house teams and clients. Not every innovative solution has to be a technology. Will tech be involved in improvement? Yes probably, but we need to first change the view of the way legal services are provided. The focus needs to shift from what lawyers do, to how we do it; how we are utilising and leverage tech in order to improve our service and provide better legal products. If we look at design thinking, there was a similar trajectory that law is now learning from: there is lots of literature on the design thinker perspective on improving service, and we're starting to see people from a design thinking method background being hired to work with lawyers to help them work through pain points and affect change in their service.
Yes, that is something we typically struggle with. It's a chicken-egg scenario: is the law attracting a certain type of person – those who are more introverted, more risk averse, more sceptical, but are great at complex problem solving – or is it that through the way training and practice churns and burns us that we create them?
That's not to say being risk averse and sceptical are bad things, because in so many ways it's our job to be those things to protect clients. We need those qualities, but it's important to not be that all the time in the way we approach everything we do. Be a human! Use that fantastic lawyer mind but let's work together and build on each other's ideas to create a better service for those we work with.
Tell us about how you try to encourage openness with Law Without Walls?
With Law Without Walls, we have created a learning programme that is multidisciplinary in every way – people of all ages experience levels and type of discipline: academics, public servants, law firms and law schools from across the world come together into teams to co-create a Project of Worth – a practical solution to a real business problem designed to bring value. It's extremely rewarding to watch the teams, especially the lawyers, change and grow in the way they ask questions, think about problems, approach meetings etc. Especially when you hear that their teams back home notice the difference too, so much so they are asking them 'who are you and what have you done with Craig?!' That impact is exactly what we aim to achieve.
How was the process of writing Legal Upheaval? What did you learn from it and was there anything that surprised you in the conversations you had?
It wasn't so hard to write but it was hard to edit! It took two years of interviews and I had enough for three books but had to edit it all down to one. Interviewing is really a tough field – it requires listening beyond listening, there is no 'I' or 'me'! We should be doing more interviewing training in law school.
Obviously I knew going in that the topic of innovation was being pushed, GC and in-house counsels are constantly saying 'innovate or die' but they don't exactly know what it is or what they are asking for. Can you really measure it if you only know it when you see it? There's an analogy to be made there with the diversity movement – calls for diversity initially were very vague, so firms would say 'oh we have a female working with us' – no mention of what level they were at, but okay!
It was only over time that the questions became more focused: what % of minorities are in our organisation, what % on my senior team are diverse etc. Now, clients are asking for your flexi-time policies, because without that you cannot support diversity – diversity doesn't truly exist without creating an inclusive culture and environment.
It's the same with innovation – who is going to lead it? What are you hoping to achieve? If you don't want the same thing to happen as diversity, where you are racing to meet client demand instead of forward thinking and define innovation for yourself, now is the time to be asking serious questions. Part of being a great innovator is self awareness.
Another thing I was surprised by was the amount of in-house counsel complaints on simple matters – particularly over advising. There is a disconnect there, and I don't know why or what is happening. Perhaps it's because we are taught to see the trees not the forest, so many lawyers are missing the bigger picture of how their documents are used in day to day business practices. In-house counsel can read the law, they don't want to receive reams of information that they have to filter and rewrite. We need to spend more time sitting back listening and asking 'why' in what we are doing. Of course, that makes people uncomfortable, especially the more senior we get, as we think we know the answers and we are taught to find answers for ourselves.
From your experience as a professor of law, is there a change in approach to teaching? Are students coming in with different expectations now?
It is a bit like moving the Titanic. I don't know if students are all that different – for hundreds of years they have come in with bright eyes and big hearted missions, that won't change, but the next generation may have different expectations of the culture of law. Unfortunately law schools are much slower to move and the tenure systems that are in place very much encourage status quo and professors keep on teaching the same things they have taught for years. So, though there are some great things being done in law schools across the world in terms of bespoke programmes being created, it's not reaching every student that it should. But it's not just up to the schools, it's going to take the whole village to move and change way we train and retrain lawyers.
What do you hope people will take from the book overall?
My hope is that people will leave with hope. Lots of articles about law as an industry are negative, but we should realise that some of the traditional habits that have made us successful so far are also characteristics that we can utilise to get over hurdles. The mission of the book is to get every lawyer to try a problem solving group project with the mindset of an innovator and try to adopt some of those skill sets and characteristics.
My three rules of engagement are Open Heart, Open Mind and Open Door – it does sound corny, but they are essential. Yes, effective people are good at editing out the nonsense and saying no to things, but innovators are different. They let go of preconceptions and allow themselves to be more open to accept seemingly silly ideas. That's something that those in the legal industry can adopt and build on, and take that brilliant lawyer brain to fine tune and turn them into the really good ideas. We can approach creative problem solving collaboratively through just a small shift in thinking.
Michele's efforts to encourage collaboration struck a real chord with us here at Obelisk. While this change in mindset is a challenge for all lawyers given their traditional education, DeStefano concludes it is overall good news for women lawyers, because on balance they are better at the necessary skills: having an Open Mind, Open Heart and Open Door. It certainly gave us hope that we are on a real cusp of change in the legal industry, and that small actions being taken today are laying the groundwork for a more open and inclusive future.
While cryptocurrency valuations may currently be in a state of decline, confidence around them being a long term prospect seems to be increasing, and more people in the legal industry are taking note. At Obelisk Support, we follow this topic with interest as our clients get on the cryptocurrency bandwagon. Investment fund lawyer John Lore advises clients on cryptocurrencies and investments as part of his firm Capital Fund Law Group, and places great importance on educating the wider industry on the implications of cryptocurrency investment and blockchain technology. He took some time out of his increasingly busy schedule to talk to us about the pressing issues and how lawyers are responding.
What is your legal background and when did you decide to create the Capital Fund Law Group?
I was previously with Akin Gump Strauss Hauer & Feld LLP, where I worked within their Hedge Fund and Private Equity practice. Then in 2010, I launched my own firm to focus exclusively on the investment fund sector and fund manager sector. To begin with, most of our clients were in the United States and it took a few years to build a global clientele.
How did you first come to deal with cryptocurrencies? What were the perceptions and predictions for cryptocurrencies like then, and how have they evolved since?
We started getting some phone calls about cryptocurrencies around 2013. We initially held off were as we were not ready to jump into the asset class, until 2016 when we launched our first cryptocurrency fund. [In those years] we invested effort in getting up to speed on regulations and drafting some of the initial disclaimer language. It was still very new so getting comfortable from a regulatory perspective took some time.
In terms of wider perceptions – honestly, I didn't see much perception at all from legal community early on, pretty much nothing was being discussed back then. The shift in perceptions really happened recently in Spring 2017 where there was an explosion of activity in cryptocurrency due to the initial major surge in price of Bitcoin.
Though perceptions around cryptocurrency are rapidly developing, it's still a very new area. How do you support and advise clients who are interested yet inexperienced?
We emphasise a solid experience in finance. From 2017, we ended up receiving hundreds of phone calls for individuals who wanted to start cryptocurrency funds. We were, and still are, very cautious about representing fund managers who have experience with crypto but don't have a background in the finance sector. We are more on the side of tempering the swift formation of a cryptocurrency fund and teaching emerging managers how to create the proper structure for a fund. There are a lot of people we deal with who are very savvy with technology and are startup focused, so we advise them to partner with career finance people to create a more diversified skillset.
How have lawyers had to adjust their learning and experience to cryptocurrencies and the blockchain?
There is much more interest now than there was, particularly in recognising that certain aspect of blockchain technology will change the future of a number of industries. From the investment side, we've seen a lot more education pop up – there have been some real strong early adopters on the legal side, who lack seasoned experience in finance, that we end up turning away but that gap is now getting filled by others. In general, there is a continued interest in the legal community in understanding the space and serving it. We, of course, only see a small slice of that in an investment context but in terms of the blockchain universe e.g. smart contracts, coin offering token offerings etc. there is a tremendous need for legal counsel in all those fields.
I think we're right in the middle of the shift between curiosity and significant resources being allocated to the area. Again, I can only speak for the investment side, but a lot will depend on emergence of institutional investment and Wall Street involvement. I anticipate along with the rest of the cryptocurrency community that there will be major transitioning to a greater focus from these areas.
What impact are cryptocurrencies and the blockchain having now?
There's a plethora of opportunities that can be matched with technology across many sectors. On the investment side, there are major opportunities for cryptocurrencies as a stored value, and blockchain is already proving to be a very important aspect of the industry going forward.
What is the future looking like? Will certain cryptocurrencies eventually become a way of life for more institutions and individuals?
There's the question of whether it will gain mainstream acceptance and whether it will continue as a convenient form of exchanging value. My predictions are: as an investment vehicle, yes, for next few years it will be limited to hedge funds and to high net worth individuals as a major trading instrument, but that can change as soon as there are infrastructures capability such as custodies and exchanges allowing for exchange traded funds (ETFs) – that would provide a strong avenue for retail level participation. Regarding seeing cryptocurrency as a replacement currency for our existing financial structures, that is more of an academic question at this point.
What developments and changes are still required to ensure longevity?
That's exactly the question I like to be asked! Right now, the question most people have in mind is when and how the problem of custody will be resolved. That is both a technological problem and a legal problem. Getting the technology to a place where we can satisfy the custody requirement to meet financial crime compliance and to satisfy the regulatory requirement of jurisdictions around the world is a big challenge. So, we announced in July at our Cryptocurrency Custody workshop the creation of a working group, an international council of self-regulatory organisations that is going to be co-sponsored by the Stanford Law Blockchain Law and Policy Journal, and the Global Center for Investment Fund Studies, our non-profit research centre. There is a real need for international dialogue. There has been a tremendous interest in London, Dubai, and parts of Asia, and everyone is looking to the U.S. on this – a lot of dialogue is needed on how regulations will be shaped how self-regulatory bodies will play a part in that.
John Lore is a member of the New York State Bar and the Utah State Bar. Mr. Lore represents fund managers and securities issuers throughout the United States. Capital Fund Law Group advises emerging and established hedge fund managers, with a strong focus on cryptocurrency funds on all aspects of fund formation and ongoing operations.


text:
\section{Introduction}
\label{Introduction}
Meson masses are important quantities to understand the properties
of Quantum Chromodynamics (QCD) vacuum. For example,
the difference between pion and sigma-meson masses is mainly originated in
the spontaneous breaking of chiral symmetry, so that the restoration
can be determined from temperature ($T$) dependence
of the mass difference.
Similar analysis is possible for the effective restoration of $U(1)_{\rm A}$
symmetry through the difference between pion and $a_0$-meson
(Lorentz scalar and isovector meson) masses.

$U(1)_{\rm A}$ symmetry is explicitly broken by the axial anomaly and
the current quark mass. In the effective model,  the $U(1)_{\rm A}$
anomaly is simulated by the Kobayashi-Maskawa-'t Hooft (KMT) determinant
interaction~\cite{KMK,tHooft}. The coupling constant $K$
of the KMT interaction is proportional
to the instanton density screened by the medium with finite
$T$~\cite{Pisarski-Yaffe-1980}. Hence
$K$ becomes small as $T$ increases: $K=K(T)$.
Pisarski and Yaffe discussed the suppression $S(T) \equiv K(T)/K(0)$
for high $T$, say $T \mathrel{\mathpalette\fun >} 2T_c$ for the pseudocritical temperature $T_c$
of chiral transition,
by calculating the Debye-type screening~\cite{Pisarski-Yaffe-1980}:
\begin{eqnarray}
S(T)&=&
\exp \Big[-\pi^2\rho^2 T^2 \Big( \frac{2}{3}N_c+\frac{1}{3}N_f \Big) \Big]
\nonumber \\
&=&\exp[-T^2/b^2] ,
\label{instanton-suppression-eq}
\end{eqnarray}
where $N_c$ ($N_f$) is the number of colors (flavors) and
the typical instanton radius $\rho$ is about 1/3~fm, and hence
the suppression parameter $b$ is about $0.70 T_c$
for $N_c=N_f=3$ of our interest~\cite{Shuryak:1993ee}; note that
2+1 flavor LQCD simulations show
$T_{\rm c}= 154\pm 9$~MeV~\cite{Borsanyi:2010bp,Bazavov:2011nk,Kanaya}.
This phenomenon is called ``effective restoration of $U(1)_{\rm A}$ symmetry'', since $U(1)_{\rm A}$ symmetry is always broken in the current-operator level
but effectively restored at higher $T$ in the vacuum expectation value.

Figure \ref{Fig-schematic-phase-diagram} shows
the current status of knowledge on 2+1 flavor phase diagram for various
values of light-quark mass $m_l$ and strange-quark mass ${m_s}$.
QCD shows a first-order phase transition associated with the breaking of
chiral ($Z_3$) symmetry at the lower left (upper right) corner
\cite{Pisarski-Wilczek-1984,Vicari:2007ma}. When $m_l$ and ${m_s}$
are finite, these first-order transitions become second order
of 3d Ising  ($Z(2)$) universality class,
as shown by the solid lines \cite{Pisarski-Wilczek-1984,Vicari:2007ma}.

However, the order of chiral transition and its
universality class are unknown on the vertical line of $m_l=0$ and $m_s > 0$,
and it is considered to be related
to the effective $U(1)_{\rm A}$ restoration.
In the two-flavor chiral limit of $(m_l,m_s)=(0,\infty)$ at
the upper left corner, for example, the order may be second order
of $O(4)$ class if
the effective restoration is not completed at $T=T_c$,
because the chiral symmetry becomes
$SU_{\rm L}(2) \times SU_{\rm R}(2)$ isomorphic to $O(4)$ in the situation
and the transition is
then expected to be in the 3d $O(4)$ universality class
\cite{Pisarski-Wilczek-1984,Vicari:2007ma}.
When $U(1)_{\rm A}$ symmetry is restored completely at $T=T_c$,
it was suggested in Ref. \cite{Pisarski-Wilczek-1984} that
the chiral transition becomes the first order.
Recently, however, it was pointed out in Ref. \cite{Vicari:2013}
that the second order is still possible. In this case,
the universality class is not $O(4)$ but $U_{\rm L}(2)\times U_{\rm R}(2)$.
There are many lattice QCD (LQCD) simulations made so far
to clarify the order and its universality class in
the two-flavor chiral limit of $(m_l,m_s)=(0,\infty)$
and the light-quark chiral limit
where $m_l$ vanishes with $m_s$ fixed at the physical value,
but these are still controversial; see Refs.
\cite{Fukugita-PRL1990,Fukugita-PRD1990,Brown-1990,Karsch-1994-1,Karsch-1994-2,Iwasaki-1997,Aoki-1998,Khan-2000,Bernard-2000,D'Elia-2005,Ejiri-2009,Bonati:2014kpa}
and therein.

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.5\textwidth]{columbia_v3.eps}
\end{center}
\caption{A schematic phase diagram of 2+1 flavor QCD as a function of
light-quark mass $m_l$ and strange-quark mass ${m_s}$.
A tricritical point is likely to appear on the $ m_s$ axis; the location
 is shown by $(m_l, m_s)=(0,m_{s}^{\rm tric})$.
The solid lines stand for second-order transitions belonging
to the universality class labeled,
where the labels $Z(2)$ and $O(4)$ mean the 3d Ising and
the 3d $O(4)$ class, respectively.
}
\label{Fig-schematic-phase-diagram}
\end{figure}

Very recently, the effective restoration of $U(1)_{\rm A}$ symmetry was
investigated by pion and $a_0$-meson screening masses calculated with
LQCD with improved p4 staggered
fermions~\cite{Cheng:2010fe} and also by
meson susceptibilities calculated with LQCD with
domain-wall fermions~\cite{Bhattacharya:2014ara,Buchoff:2013nra}.
The effective restoration of $U(1)_{\rm A}$ symmetry thus becomes
an important current issue.

In LQCD, pole and screening masses are evaluated from
the exponential decay of mesonic correlation functions in the temporal and
spatial directions, respectively, but for finite $T$ the lattice size is
smaller in the temporal direction than in the spatial direction.
This makes LQCD simulations less feasible for pole masses
than for screening masses. The problem is getting serious as $T$ increases.
This is the reason why meson screening masses are calculated
in most of LQCD simulations.
In fact, as mentioned above, state-of-the-art LQCD calculations were done
for meson screening masses with large volumes
($16^3 \times 4$, $24^3 \times 6$, $32^3 \times 8$) in a wide range of
$T$~\cite{Cheng:2010fe}.

The effective model is suitable for qualitative understanding of QCD.
In fact, the QCD phase diagram, the properties of light-meson
pole masses and the $T$ dependence of $U(1)_{\rm A}$ anomaly
have been studied extensively with
the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model
~\cite{Meisinger,Dumitru,Costa:2005,Fukushima1,Ghos,Megias,Ratti1,Ciminale,Ratti2,Rossner,Hansen,Sasaki-C,Schaefer,Kashiwa1,Costa:2009,Ruivo:2012a}.
The PNJL model can treat the deconfinement and chiral transitions
simultaneously, but cannot reproduce their coincidence seen in
2-flavor LQCD data, when the model parameters are properly set
to reproduce $T_c$ calculated with LQCD~\cite{Ratti1}.
This problem is solved by introducing the four-quark vertex depending
on the Polyakov-loop~\cite{Sakai_EPNJL,Sasaki_EPNJL}.
The model with the entanglement vertex is refer to as
the entanglement-PNJL (EPNJL) model.
The EPNJL model is successful in reproducing the phase diagram
in 2-flavor QCD at imaginary chemical potential~\cite{D'Elia3,FP2010} and
real isospin chemical potential~\cite{Kogut2}, and well accounts for
the phase diagram in the $m_l$--$m_s$ plane, i.e., the so-called Columbia plot
in 2+1 flavor QCD~\cite{FP2007}.
So far, $T$ dependence of $U(1)_{\rm A}$ anomaly and low-lying meson pole masses was
studied extensively by both the PNJL and EPNJL
models~\cite{Costa:2005,Costa:2009,Ruivo:2012a,Ruivo:2012b}, but
$T$ dependence of meson screening masses was investigated
by the EPNJL model only in our previous work~\cite{Ishii:2013kaa}.

In NJL-type models, it is well known that the calculation of
meson screening mass $M_{\xi,{\rm scr}}$ is difficult,
where $\xi$ denotes the type of meson.
In fact, only a few trials were made so far~\cite{Kunihiro,Florkowski}.
The first problem is the regularization. The regularization widely
used is the three-dimensional momentum cutoff, but it breaks
Lorentz invariance at $T=0$ and spatial-translation invariance at any $T$.
This generates
unphysical oscillations in the spatial correlation function
$\eta_{\xi\xi}(r)$ ~\cite{Florkowski}.
This refuses us to determine $M_{\xi,{\rm scr}}$ from
the asymptotic form of $\eta_{\xi\xi}(r)$ as
\begin{equation}
M_{\xi,{\rm scr}}
=-\lim_{r\rightarrow \infty}\frac{d \ln{\eta_{\xi\xi}(r)}}{dr}.
\label{scr-mass}
\end{equation}
This problem can be solved~\cite{Florkowski} by introducing
the Lorentz-invariant Pauli-Villars (PV) regularization~\cite{PV}.

Even after the unphysical oscillations are removed,
heavy numerical calculations
are still required to obtain $\eta_{\xi\xi}(r)$
at large $r$~\cite{Florkowski}. This is the second problem.
In the model calculation, the spatial correlation function is obtained
first in the momentum representation $(\tilde{q}=\pm |{\bf q}|)$
as $\chi_{\xi\xi}(0, \tilde{q}^{2})$. Hence we have to make
the Fourier transform
from $\chi_{\xi\xi}(0, \tilde{q}^{2})$ to $\eta_{\xi\xi}(r)$:
\begin{equation}
\eta_{\xi\xi}(r)=\frac{1}{4\pi^{2}ir}\int^{\infty}_{-\infty}d\tilde{q}\hspace{1ex}\tilde{q}\chi_{\xi\xi}(0,\tilde{q}^2)e^{i\tilde{q}r} .
\label{chi_r}
\end{equation}
The $\tilde{q}$ integration is quite hard particularly at large $r$,
since the integrand consists of a slowly damping function
$\tilde{q}\chi_{\xi\xi}(0,\tilde{q}^2)$ and a highly oscillating function
$e^{i\tilde{q}r}$.
If $\chi_{\xi\xi}(0, \tilde{q}^{2})$
has a pole below the cut in the complex $\tilde{q}$ plane,
one can easily determine $M_{\xi,{\rm scr}}$ from the pole location.
In the old formulation of Ref.~\cite{Florkowski},
the condition was not satisfied, since logarithmic cuts appear
in the vicinity of the real $\tilde{q}$ axis in addition to physical cuts.
Very recently we solved the problem
in our previous paper~\cite{Ishii:2013kaa}, showing that
the logarithmic cuts near the real $\tilde{q}$ axis are
unphysical and removable.
In the new formulation based on the PV regularization,
there is no logarithmic cut and
a pole appears below physical cuts, as shown later.

In this paper,
we incorporate the effective restoration of $U(1)_{\rm A}$ symmetry
in the 2+1 flavor EPNJL model
by introducing a $T$-dependent coupling strength $K(T)$
to the KMT interaction.
$T$ dependence of $K(T)$ is well determined
from state-of-the-art 2+1 flavor LQCD results~\cite{Cheng:2010fe}
on pion and $a_0$-meson screening masses.
For the derivation of meson screening mass, we extend the previous
prescription of Ref.~\cite{Ishii:2013kaa} for 2 flavors to 2+1 flavors.
The $K(T)$ determined from the LQCD data
is strongly suppressed near $T_c$. Using the parameter set, we show that
the chiral transition is second order in
the light-quark chiral limit.
This result indicates that there exists a tricritical point
near the ``light-quark chiral-limit'' point in the $m_l$--$m_s$ plane.
We then estimate the location.

We recapitulate the EPNJL model and the method of
calculating meson screening masses in Sec. \ref{Model setting} and
show the results of numerical calculations in Sec. \ref{Numerical Results}.
Section \ref{Summary} is devoted to a summary.

\section{Model setting}
\label{Model setting}

\subsection{EPNJL model}
\label{EPNJL model}

We start with the 2+1 flavor EPNJL model~\cite{Sakai_EPNJL,Sasaki_EPNJL}.
The Lagrangian density is
\begin{align}
 {\cal L}
=& {\bar \psi}(i \gamma_\nu D^\nu - {\hat m_0} )\psi
  + G_{\rm s}(\Phi) \sum_{a=0}^{8}
    [({\bar \psi} \lambda_a \psi )^2 +({\bar \psi }i\gamma_5 \lambda_a \psi )^2]
\nonumber\\
 &- K(T) \Bigl[\det_{f,f'} {\bar \psi}_f (1+\gamma_5) \psi_{f'}
           +\det_{f,f'} {\bar \psi}_f (1-\gamma_5) \psi_{f'} \Bigr]
\nonumber\\
&-{\cal U}(\Phi [A],{\bar \Phi} [A],T)
\label{L}
\end{align}
with quark fields $\psi=(\psi_u,\psi_d,\psi_s)^T$ and
$D^\nu=\partial^\nu + iA^\nu$ with $A^\nu=\delta^{\nu}_{0}g(A^0)_a{t_a / 2}=-\delta^{\nu}_{0}ig(A_{4})_a t_a/2$ for the gauge coupling
$g$, where the $\lambda_a$ ($t_a$) are the Gell-Mann matrices
in flavor (color) space and $\lambda _0 = \sqrt{2/3}~{\bf I}$
for the unit matrix ${\bf I}$ in flavor space. The determinant in
\eqref{L} is taken in flavor space.
For the 2+1 flavor system, the current quark masses
${\hat m_0}={\rm diag}(m_u,m_d,m_s)$ satisfy a relation
$m_s > m_l \equiv m_u=m_d$.
In the EPNJL model, the coupling strength $G_{\rm s}(\Phi)$
of the scalar-type four-quark interaction depends
on the Polyakov loop $\Phi$ and its Hermitian conjugate
${\bar \Phi}$ as
\begin{equation}
G_{\rm s}(\Phi)=G_{\rm s}(0)
\left[
1-\alpha_1\Phi{\bar \Phi}-\alpha_2 (\Phi^3+{\bar \Phi}^3)
\right].
\label{EPNJL}
\end{equation}
This entanglement coupling is charge-conjugation and $Z_{3}$ symmetric.
When $\alpha_1=\alpha_2=0$, the EPNJL model is
reduced to the PNJL model.
We set $\alpha_2=0$ for simplicity, since
the $\alpha_2$ term yields the same effect as the $\alpha_1$ term
in the present analysis.
As shown later in Sec. \ref{Numerical Results},
the value of $\a_1$ is determined
from LQCD data on pion and $a_0$-meson screening masses;
the resulting value is $\a_1 = 1.0$.

For $T$ dependence of $K(T)$, we assume the following form phenomenologically:
 \begin{eqnarray}
 K(T)=\left\{ \begin{array}{ll}
 K(0) & (T < T_1) \\
 K(0) e^{-(T-T_1)^2/b^2}& (T \ge T_1) \\
 \end{array} \right. .
 \label{T-dependent-K}
 \end{eqnarray}
For high $T$ satisfying $T \gg T_1$, the form  \eqref{T-dependent-K}
is reduced to \eqref{instanton-suppression-eq}.
As shown later in Sec. \ref{Numerical Results}, the values of
$T_1$ and $b$ are well determined
from LQCD data on pion and $a_0$-meson screening masses;
the resulting values are
$T_1=0.79T_{c}=121$~MeV and $b = 0.23T_c = 36$~MeV.

After the Pisarski-Yaffe discussion on $S(T)$,
$T$ dependence of the instanton density was estimated theoretically by the
the instanton-liquid model \cite{Shuryak:1993ee},
but the estimation is applicable only for $T \mathrel{\mathpalette\fun >} 2T_c$. For this reason,
in Ref.~\cite{Ruivo:2012b}, a Woods-Saxon form $(1+e^{(T-T_1')/b'})^{-1}$
with two parameters $T_1'$ and $b'$ was used phenomenologically
for $K(T)/K(0)$. The present form \eqref{T-dependent-K} has $T$ dependence
similar to the Woods-Saxon form.

In the EPNJL model, the time component of $A_\mu$ is treated as a homogeneous
and static background field, which is governed by the Polyakov-loop potential
$\mathcal{U}$. In the Polyakov gauge, $\Phi$ and ${\bar \Phi}$
are obtained  by
\begin{align}
\Phi &= {1\over{3}}{\rm tr}_{\rm c}(L),
~~~~~{\bar \Phi} ={1\over{3}}{\rm tr}_{\rm c}({L^*})
\label{Polyakov}
\end{align}
with $L= \exp[i A_4/T]=\exp[i {\rm diag}(A_4^{11},A_4^{22},A_4^{33})/T]$
for real variables $A_4^{jj}$
satisfying $A_4^{11}+A_4^{22}+A_4^{33}=0$.
For zero quark chemical potential where $\Phi={\bar \Phi}$,
one can set $A^{33}_4=0$ and determine the others as
$A^{22}_4=-A^{11}_4={\rm cos}^{-1}[(3\Phi -1)/2]$.

We use the logarithm-type Polyakov-loop potential of Ref.~\cite{Rossner}
as $\mathcal{U}$. The parameter set in $\mathcal{U}$ has already been
determined from  LQCD data at finite $T$ in the pure gauge limit.
The potential has a parameter $T_0$ and yields a first-order deconfinement
phase transition at $T=T_0$. The parameter used to be set to $T_0=270$~MeV,
since LQCD data show the phase transition at $T=270$~MeV in
the pure gauge limit. In full QCD with dynamical quarks, however,
the PNJL model with this value of $T_0$ is found not to explain LQCD
results. Nowadays, $T_0$ is then rescaled to reproduce the LQCD results.
In the present case, we take $T_0=180$~MeV so that the EPNJL model
can reproduce LQCD results for the pseudocritical temperature
$T_c^{\rm deconf}$ of deconfinement transition;
actually, $T_c^{\rm deconf}=165$ MeV in the EPNJL model and
$170 \pm 7$ MeV in LQCD \cite{Aoki:2009sc}.

Making the mean field approximation (MFA) to \eqref{L} leads to
the linearized Lagrangian density
\begin{eqnarray}
  {\cal L}^{\rm MFA}
= {\bar \psi}S^{-1}\psi  - U_{\rm M} - {\cal U}(\Phi [A],{\bar \Phi} [A],T)
\label{linear-L}
\end{eqnarray}
with the quark propagator
\begin{eqnarray}
S=(i \gamma_\nu \partial^\nu - \gamma_0A^0 - \hat{M})^{-1},
\end{eqnarray}
where $\hat{M}={\rm diag}(M_u,M_d,M_s)$ with
\begin{eqnarray*}
M_u &=& m_u -4G_{\rm s}(\Phi)\sigma_u +2K(T)\sigma_d \sigma_s ,\\
M_d &=& m_d -4G_{\rm s}(\Phi)\sigma_d +2K(T)\sigma_s \sigma_u ,\\
M_s &=& m_s -4G_{\rm s}(\Phi)\sigma_s +2K(T)\sigma_u \sigma_d ,
\end{eqnarray*}
and $\sigma_f $ means the chiral condensate $\langle \bar
\psi_f \psi_f\rangle$ for flavor $f$. The mesonic potential $U_{\rm M}$ is
\begin{eqnarray*}
U_{\rm M}
= 2G_{\rm s}(\Phi)(\sigma^2_u+\sigma^2_d+\sigma^2_s)
-4K(T) \sigma_u \sigma_d \sigma_s.
\end{eqnarray*}
Making the path integral over quark fields, one can get
the thermodynamic potential (per unit volume) as
\begin{align}
&\Omega= U_{\rm M}+{\cal U}-2 \sum_{f=u,d,s} \int \frac{d^3 {\bf p}}{(2\pi)^3}
   \Bigl[ 3 E_{p,f} \notag \\
&+ \frac{1}{\beta}
           \ln~ [1 + 3(\Phi+{\bar \Phi} e^{-\beta E_{p,f})}
           e^{-\beta E_{p,f}}+ e^{-3\beta E_{p,f}}] \notag\\
&+ \frac{1}{\beta}
           \ln~ [1 + 3({\bar \Phi}+{\Phi e^{-\beta E_{p,f}}})
              e^{-\beta E_{p,f}}+ e^{-3\beta E_{p,f}}]
	      \Bigl]
\label{PNJL-Omega}
\end{align}
with
$E_{p,f}=\sqrt{{\bf p}^2+M^2_f}$ and $\beta = 1/T$. We determine the mean-field variables
($X=\sigma_l,\sigma_s,\Phi,\bar{\Phi}$)
from the stationary conditions:
\begin{equation}
\frac{\partial \Omega}{\partial X}=0,
\end{equation}
where isospin symmetry is assumed for the light-quark sector,
i.e., $\sigma_l \equiv \sigma_u=\sigma_d$.

On the right-hand side of (\ref{PNJL-Omega}), the first term (vacuum term)
in the momentum integral diverges. We then use
the PV regularization~\cite{Florkowski,PV}.
In the scheme, the integral $I(M_f,q)$ is regularized as
\begin{eqnarray}
I^{\rm reg}(M_f,q)=\sum_{\alpha=0}^2 C_\alpha I(M_{f;\alpha},q) ,
\label{PV}
\end{eqnarray}
where $M_{f;0}=M_f$ and the $M_{f;\alpha}~(\alpha\ge 1)$ mean masses
of auxiliary particles. The parameters $M_{f;\alpha}$ and $C_\alpha$
should satisfy the condition
$\sum_{\alpha=0}^2C_\alpha=\sum_{\alpha=0}^2 C_\alpha M_{f;\alpha}^2=0$.
We then assume $(C_0,C_1,C_2)=(1,1,-2)$ and
$(M_{f;1}^2,M_{f;2}^2)=(M^2_{f}+2\Lambda^2,M^2_{f}+\Lambda^2)$.
We keep the parameter $\Lambda$ finite
even after the subtraction \eqref{PV}, since the present model is
non-renormalizable.
The parameters are taken from Ref. \cite{Bernard} and they are
$m_l=6.2$ MeV, $m_s=175.0$ MeV, $G_{\rm s}(0)\Lambda^2=2.35$ and
$K(0)\Lambda^5=27.8$ for $\Lambda =795$ MeV.
This parameter set reproduces mesonic observables at
vacuum, i.e., the pion and kaon decay constants
($f_{\pi}=92$ MeV and $f_K=105$ MeV) and their masses
($M_{\pi}=141$ MeV and $M_K=512$ MeV) and the $\eta'$-meson mass
($M_{\eta'}=920$ MeV).
In the present work, we analyze LQCD results
of Ref. \cite{Cheng:2010fe} for pion and $a_0$-meson screening masses.
In the LQCD simulation, the pion mass $M_{\pi}(0)$ at vacuum ($T=0$)
is 175~MeV and a bit heavier than the experimental value 138~MeV.
We then change $m_l$ to $9.9$~MeV in the EPNJL model in order to reproduce
$M_{\pi}(0)=175~{\rm MeV}$. This parameter set yields
$M_{a_0}(0)=711$ MeV, $M_{\eta}(0)=481$ MeV and $M_{\sigma}(0)=537$ MeV
as $a_0,\eta$ and $\sigma$ meson pole masses at vacuum.

\subsection{Meson pole mass}
We derive the equations for pion and $a_0$-meson pole masses,
following Ref~\cite{Hansen,KHKB}.
The current corresponding to a meson of type $\xi$ is
\begin{equation}
  J_{\xi}(x) = \bar \psi(x)\Gamma_\xi \psi(x)
             - \langle\bar \psi(x) \Gamma_\xi \psi(x)\rangle ,
\label{source}
\end{equation}
where
$\Gamma_\pi=i\gamma_5\lambda_3$ for $\pi$ meson and
$\Gamma_{a_0}=\lambda_3$ for $a_0$-meson.
We denote the Fourier transform of the mesonic correlation function
$\eta_{\xi\xi} (x) \equiv \langle 0 | {\rm T} \left( J_\xi(x) J^{\dagger}_{\xi}(0) \right) | 0 \rangle$ by $\chi_{\xi\xi} (q^2)$ as
\begin{align}
\chi_{\xi\xi} (q^2) = \chi_{\xi\xi} (q^2_0, {\tilde q}^2)
=  i \int d^4x e^{i q\cdot x} \eta_{\xi\xi} (x) ,
\end{align}
where $\tilde{q}=\pm |{\bf q}|$ for $q=(q_0,{\bf q})$ and
${\rm T}$ stands for the time-ordered product.
Using the random-phase (ring) approximation,
one can obtain the Schwinger-Dyson equation
\begin{eqnarray}
  \chi_{\xi\xi}
= \Pi_{\xi\xi} +2 \sum_{\xi'\xi''} \Pi_{\xi\xi'}G_{\xi'\xi''}\chi_{\xi''\xi}
\label{SD}
\end{eqnarray}
for $\chi_{\xi\xi}$, where
$G_{\xi'\xi''}$ is an effective four-quark interaction and
$\Pi_{\xi\xi'}$ is the one-loop polarization function defined by
\begin{eqnarray}
  \Pi_{\xi\xi'}(q^2)  \equiv  (-i) \int \frac{d^4 p}{(2\pi)^4}
  {\rm Tr} \left( \Gamma_\xi iS(p'+q) \Gamma_{\xi'} iS(p') \right)
\end{eqnarray}
with $p'=(p_{0}+iA_4,{\bf p})$, where the trace ${\rm Tr}$ is taken
in flavor, Dirac and
color spaces. Here the quark propagator $S(p)$ in momentum space
is diagonal in flavor space: $S(p)={\rm diag}(S_u,S_d,S_s)$.
For $\xi=\pi$ and $a_0$, furthermore, $G_{\xi\xi'}$
and $\Pi_{\xi\xi'}$ are diagonal
($G_{\xi\xi'}=G_{\xi}\delta_{\xi\xi'}$, $\Pi_{\xi\xi'}=\Pi_{\xi}\delta_{\xi\xi'}$), because we impose isospin symmetry for the light-quark sector and
employ the random-phase approximation. One can then easily
solve the Schwinger-Dyson equation for $\xi=\pi$ and $a_0$:
\begin{eqnarray}
\chi_{\xi\xi} &=&
\frac{\Pi_{\xi}}{1 - 2G_{\xi}\Pi_{\xi}}
\label{chi_pia0}
\end{eqnarray}
with the effective couplings $G_{\pi}$ and $G_{a_0}$ defined by
\begin{eqnarray}
G_{a_0} &=& G_{\rm s}(\Phi) + \frac{1}{2}K(T) \sigma_s,
\label{g_a0}
\\
G_{\pi} &=& G_{\rm s}(\Phi) - \frac{1}{2}K(T) \sigma_s.
\label{g_pi}
\end{eqnarray}

As for $T=0$,  $\Pi_{\pi}$ and $\Pi_{a_0}$ have the following explicit forms:
\begin{eqnarray}
\Pi_{a_0}&=&i\sum_{f,f'}(\lambda_3)_{f'f}(\lambda_3)_{ff'}\nonumber\\
&\times&
\int {d^4p\over{(2\pi )^4}}{\rm tr}_{\rm c,d}\Bigl[{\{\gamma_\mu (p'+q)^\mu +M_f\}(\gamma_\nu p'^\nu +M_{f'})\over{\{(p'+q)^2-M_f^2\}(p'^2-M_{f'}^2)}}\Bigr]
\nonumber\\
&=&4i[I_{1}+I_{2}-(q^2-4M^2)I_{3}],
\label{Pi_SS}
\\
\Pi_{\pi}&=&i\sum_{f,f'}(\lambda_3)_{f'f}(\lambda_3)_{ff'}\nonumber\\
&\times&\int {d^4p\over{(2\pi )^4}}{\rm tr}_{\rm c,d}\Bigl[(i\gamma_5){\{\gamma_\mu (p'+q)^\mu +M_f\}\over{\{(p'+q)^2-M_f^2\}}}
\nonumber\\
&&\times (i\gamma_5)
{(\gamma_\nu p'^\nu +M_{f'})\over{(p'^2-M_{f'})^2}}\Bigr]
\nonumber\\
&=&4i[I_{1}+I_{2}-q^2I_{3}],
\label{Pi_PP}
\end{eqnarray}
and
\begin{eqnarray}
I_{1}&=&\int {d^4p\over{(2\pi )^4}}{\rm tr_c}\Bigl[{1\over{p'^2-M}}\Bigr],
\label{I1}
\\
I_{2}&=&\int {d^4p\over{(2\pi )^4}}{\rm tr_c}\Bigl[{1\over{(p'+q)^2-M^2}}\Bigr],
\label{I2}
\\
I_{3}&=&\int {d^4p\over{(2\pi )^4}}{\rm tr_c}\Bigl[{1\over{\{(p'+q)^2-M^2\}(p'^2-M^2)}}\Bigr] ,
\nonumber\\
\label{I3}
\end{eqnarray}
where ${\rm tr}_{\rm c,d}$ (${\rm tr}_{\rm c}$) means
the trace in color and Dirac spaces (color space) and $M=M_u=M_d$.
For finite $T$, the corresponding equations are
obtained by the replacement
\begin{align}
&p_0 \to i \omega_n = i(2n+1) \pi T,
\nonumber\\
&\int \frac{d^4p}{(2 \pi)^4}
\to iT\sum_{n=-\infty}^{\infty} \int \frac{d^3{\bf p}}{(2 \pi)^3}.
\label{finte_T_mu}
\end{align}

The meson pole mass $M_{\xi,{\rm pole}}$ is a pole of
$\chi_{\xi\xi}(q_0^2,{\tilde q}^2)$ in the complex $q_0$ plane.
Taking the rest frame $q=(q_0,{\bf 0})$ for convenience,
one can get the equation for $M_{\xi,{\rm pole}}$ as
\begin{align}
\big[1 - 2G_{\xi}\Pi_{\xi}(q_0^2,0)\big]\big|_{q_0=M_{\xi,{\rm pole}}-i\frac{\Gamma}{2}}=0 ,
\label{mmf}
\end{align}
where $\Gamma$ is the decay width to $q\bar{q}$ continuum. The method of
calculating meson pole masses is well established in the PNJL
model~\cite{Hansen, KHKB}.

\subsection{Meson screening mass}
We derive the equations for pion and $a_0$-meson screening masses, following
Ref.~\cite{Ishii:2013kaa}.
This is an extension of the method of Ref.~\cite{Ishii:2013kaa}
for 2 flavors to 2+1 flavors.

As mentioned in Sec. \ref{Introduction}, it is not easy to
make the Fourier transform from $\chi_{\xi\xi} (0,\tilde{q}^{2})$
to $\eta_{\xi\xi}(r)$ particularly at large $r$.
When the direct integration on the real $\tilde{q}$ axis is difficult,
one can consider a contour integral in the complex $\tilde{q}$ plane
by using the Cauchy's integral theorem.
However, $\chi_{\xi\xi} (0,\tilde{q}^{2})$ has logarithmic cuts
in the vicinity of the real $\tilde{q}$ axis \cite{Florkowski},
and it is reported in Ref. \cite{Florkowski} that
heavy numerical calculations are necessary for evaluating the cut effects.
In our previous work~\cite{Ishii:2013kaa}, we showed
that these logarithmic cuts are unphysical and removable. Actually, we have no logarithmic cut, when analytic continuation is made
for the $I_3({\bf q})$ after ${\bf p}$ integration.
Namely, the Matsubara summation over $n$ should be taken
after the ${\bf p}$ integration in \eqref{finte_T_mu}.
We then express
$I_3^{\rm reg}$ as an infinite series of analytic functions:
\begin{eqnarray}
&&I_{3}^{\rm reg}(0,\tilde{q}^{2})=iT\sum_{j=1}^{N_c}\sum_{n=-\infty}^\infty\sum_{\alpha=0}^2C_{\alpha}
\nonumber\\
&\times & \int {d^3{\bf p}\over{(2\pi )^3}}
\Bigl[{1\over{{\bf p}^2+M_{j,n,\alpha}^2}}{1\over{({\bf p}+{\bf q})^2+M_{j,n,\alpha}^2}}\Bigr]\nonumber\\
&=&{iT\over{2\pi^2}}\sum_{j,n,\alpha}C_{\alpha}\int_0^1dx\int_0^\infty dk{k^2\over{[k^2+(x-x^2)\tilde{q}^2+M_{j,n,\alpha}^2]^2}}
\nonumber\\
&=&{iT\over{4\pi \tilde{q}}}\sum_{j,n,\alpha}C_{\alpha}  \sin^{-1}{\Bigl({{\tilde{q}\over{2}}\over{\sqrt{{\tilde{q}^2\over{4}}+M_{j,n,\alpha}^2}}}\Bigr)}
\label{I_3_final}
\end{eqnarray}
with
\begin{equation}
M_{j,n,\alpha}(T)=\sqrt{M_{\alpha}^2 + \{(2n+1)\pi T+A_{4}^{jj}\}^2 },
\label{KK_mode}
\end{equation}
where $M_\alpha=M_{u;\alpha}=M_{d;\alpha}$. We have numerically confirmed
that the convergence of $n$-summation is quite fast in (\ref{I_3_final}).
Each term of $I_3^{\rm reg}(0,\tilde{q}^2)$
has  two physical cuts on the imaginary axis, one is an upward vertical line
starting from $\tilde{q} = 2iM_{j,n,\alpha}$ and the other is
a downward vertical line from $\tilde{q} = - 2iM_{j,n,\alpha}$.
The lowest branch point is $\tilde{q}=2iM_{j=1,n=0,\alpha=0}$.
The value is the meson screening mass in which there is no interaction
between a quark and an antiquark, i.e., $G_{\xi}=0$.
Hence we may call $2M_{j=1,n=0,\alpha=0}$ ``the threshold mass''.

We can obtain the meson screening mass $M_{\xi,{\rm scr}}$ as a pole of
$\chi_{\xi\xi}(0,\tilde{q}^{2})$,
\begin{align}
\big[1 - 2G_{\xi} \Pi_{\xi}(0,\tilde{q}^2)\big]\big|_{\tilde{q}=iM_{\xi, {\rm scr}}}=0  .
\label{meson_screening}
\end{align}
If the pole at $\tilde{q}=iM_{\xi, {\rm scr}}$ is well isolated from the cut,
i.e., $M_{\xi,{\rm scr}}<2M_{j=1,n=0,\alpha=0}$, one can determine
the screening mass from the pole location
without making the $\tilde{q}$ integral.
In the high-$T$ limit, the condition tends to $M_{\xi,{\rm scr}}<2 \pi T$.

\subsection{Meson susceptibility}
We consider meson susceptibilities
$\chi_{\xi}^{\rm sus}$ for $\xi=\pi,a_0,\eta$ and $\sigma$.
In LQCD simulations of Refs. \cite{Bhattacharya:2014ara,Buchoff:2013nra},
the $\chi_{\xi}^{\rm sus}$ are defined
in Euclidean spacetime $x_{\rm E}=(\tau,{\bf x})$ as
\begin{eqnarray}
\chi_{\xi}^{\rm sus}
= \frac{1}{2}\int d^{4}x_{\rm E}~
\langle J_{\xi}(\tau,{\bf x})J_{\xi}^\dagger(0,{\bf 0})\rangle .
\end{eqnarray}
In the simulations, $J_\sigma$ and $J_\eta$ are assumed to have no
s-quark component for simplicity: namely,
$J_\sigma=\sum_{f=u,d}\bar{\psi}_f\psi_f - \langle\sum_{f=u,d}\bar{\psi}_f\psi_f\rangle$ and $J_\eta=\sum_{f=u,d}\bar{\psi}_fi\gamma_5\psi_f -\langle\sum_{f=u,d}\bar{\psi}_fi\gamma_5\psi_f\rangle$.
For consistency, we take the same assumption also in the present
analysis, and denote the mesons with no s-quark (ns) component by
$\sigma_{\rm ns}$ and $\eta_{\rm ns}$.
The factor $1/2$ is introduced to define the $\chi_{\xi}^{\rm sus}$
as single-flavor quantities.

The $\chi_{\xi}^{\rm sus}$ is related to the
the Matsubara Green's function $\chi_{\xi\xi}^{\rm E}(q_4^2,{\bf q}^2)$
in the momentum representation as
\begin{eqnarray}
\chi_{\xi}^{\rm sus} = \frac{1}{2}\chi_{\xi\xi}^{\rm E}(q_4^2,{\bf q}^2)
\big|_{q_4=0, {\bf q}=0} ,
\end{eqnarray}
and $\chi_{\xi\xi}^{\rm E}$ is obtainable from \eqref{chi_pia0}
for $\pi$ and $a_0$ mesons. For $\eta_{\rm ns}$ meson, we have to consider
a mixing between $\eta_{\rm ns}$ and
$\eta_{\rm s}=\bar{\psi}_si\gamma_5\psi_s$. As a result, one can
obtain $\chi_{\eta_{\rm ns}\eta_{\rm ns}}$ as \cite{KHKB}
\begin{equation}
\chi_{\eta_{\rm ns}\eta_{\rm ns}}=\frac{(1-2G_{\eta_{\rm s}\eta_{\rm
 s}}\Pi_{\eta_{\rm s}\eta_{\rm s}})\Pi_{\eta_{\rm ns}\eta_{\rm
 ns}}}
{\det{\left[{\boldsymbol I} - 2{\boldsymbol G}{\boldsymbol
       \Pi}\right]}},
\label{chi_eta}
\end{equation}
where ${\bf I}$ is the unit matrix and
\begin{eqnarray}
{\boldsymbol G}
=
\left(
\begin{array}{cc}
G_{\eta_{\rm s}\eta_{\rm s}}& G_{\eta_{\rm s}\eta_{\rm ns}}\\
G_{\eta_{\rm ns}\eta_{\rm s}}& G_{\eta_{\rm ns}\eta_{\rm ns}}
\end{array}
\right),~~
{\boldsymbol \Pi}
=
\left(
\begin{array}{cc}
\Pi_{\eta_{\rm s}\eta_{\rm s}}& 0\\
0& \Pi_{\eta_{\rm ns}\eta_{\rm ns}}
\end{array}
\right)~~
\label{gpi_eta}
\end{eqnarray}
for the elements
\begin{eqnarray}
G_{\eta_{\rm s}\eta_{\rm s}}   &=& G_{\rm s}(\Phi) , \\
\label{gelement_eta1}
G_{\eta_{\rm ns}\eta_{\rm ns}} &=& G_{\rm s}(\Phi) + \frac{1}{2}K(T)\sigma_{s}, \\
\label{gelement_eta2}
G_{\eta_{\rm s}\eta_{\rm ns}}  &=& G_{\eta_{\rm ns}\eta_{\rm s}} = \frac{\sqrt{2}}{2}K(T)\sigma_l.
\label{gelement_eta3}
\end{eqnarray}
In the isospin symmetric case we consider, the polarization functions
$\Pi_{\eta_{\rm s}\eta_{\rm s}}$ and $\Pi_{\eta_{\rm ns}\eta_{\rm ns}}$
have the same function form as $\Pi_{\pi}$:
\begin{eqnarray}
\Pi_{\eta_{\rm s}\eta_{\rm s}}  &=& \Pi_{\pi}(M_{s}), \\
\label{pielement_eta1}
\Pi_{\eta_{\rm ns}\eta_{\rm ns}}&=& \Pi_{\pi}(M),
\label{pielement_eta2}
\end{eqnarray}
where note that $\Pi_{\pi}(M_{s})$ is a function of not $M$ but $M_s$.
Similarly, $\chi_{\sigma_{\rm ns}\sigma_{\rm ns}}$ is obtainable from
(\ref{chi_eta}) with $K(T)$ replaced by $-K(T)$ and $\Pi_{\pi}$ by
$\Pi_{a_0}$.

\section{Numerical Results}
\label{Numerical Results}

\subsection{Meson screening masses}
\label{Meson screening masses}

The EPNJL model has three adjustable parameters,
$\a_1$ in the entanglement coupling $G_{\rm s}(\Phi)$ and
$b$ and $T_1$ in the KMT interaction $K(T)$.
These parameters can be clearly determined from
LQCD data~\cite{Cheng:2010fe} for
pion and $a_0$-meson screening masses,
$M_{\pi,{\rm scr}}$ and $M_{a_0,{\rm scr}}$, as shown below.

Figure~\ref{Fig-screening} shows $T$ dependence of
$M_{\pi,{\rm scr}}$ and $M_{a_0,{\rm scr}}$.
Best fitting is obtained, when
$\a_1=1.0$, $T_1=0.79T_{c}=121$~MeV and $b = 0.23T_c = 36$~MeV.
Actually, the EPNJL results (solid and dot-dash lines)
with this parameter set well account for LQCD data~\cite{Cheng:2010fe}
for both $M_{\pi,{\rm scr}}$ and $M_{a_0,{\rm scr}}$.
The parameters thus obtained indicate the strong suppression of $K(T)$ in the vicinity of $T_c$.
The mass difference
$\Delta M_{\rm scr}(T)=M_{a_0,{\rm scr}}(T)-M_{\pi,{\rm scr}}(T)$
is sensitive to $K(T)$ because of \eqref{g_a0} and
\eqref{g_pi}, and hence
the values of $b$ and $T_1$ are well determined from $\Delta M_{\rm scr}(T)$.

When $\a_1=0$, the EPNJL model is reduced to the PNJL model.
The results of the PNJL model are
shown in Fig.~\ref{Fig-screening-K(T)-PNJL} for comparison.
The PNJL results cannot reproduce LQCD data particularly in the region
$T \mathrel{\mathpalette\fun >} 180$~MeV. The slope of the solid and dot-dash lines in the region
is thus sensitive to the value of $\a_1$. Namely,
the value of $\a_1$ is well determined from the slope.

In Fig.~\ref{Fig-screening},
the solid and dot-dash lines are lower than the threshold mass
$2M_{j=1,n=0,\alpha=0}$ (dotted line). This guarantees
that the $M_{\pi, {\rm scr}}$ and $M_{a_0, {\rm scr}}$ determined
from the pole location in the complex-$\tilde{q}$ plane
agree with those from the exponential decay
of $\eta_{\xi\xi}(r)$ at large $r$.

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.4\textwidth]{screening.eps}
\end{center}
\caption{$T$ dependence of pion and $a_0$-meson screening masses,
$M_{\pi,{\rm scr}}$ and $M_{a_0,{\rm scr}}$.
The solid (dot-dash) line denotes
$M_{\pi,{\rm scr}}$ ($M_{a_0,{\rm scr}}$) calculated by
the EPNJL model, whereas
the dotted line corresponds to the threshold mass.
LQCD data are taken from Ref.~\cite{Cheng:2010fe};
closed squares (open circles) correspond to
the 2+1 flavor data
for $M_{\pi,{\rm scr}}$ ($M_{a_0,{\rm scr}}$).
In Ref. \cite{Cheng:2010fe},
$T_{c}$ was considered $196~{\rm MeV}$,
but it was refined to $154\pm 9$ MeV
\cite{Borsanyi:2010bp,Bazavov:2011nk}.
The latest value is taken in this figure.
}
\label{Fig-screening}
\end{figure}

\begin{figure}[t
\begin{center}
 \includegraphics[width=0.4\textwidth]{screening-K-Tdep-PNJL_v2.eps}
\end{center}
\caption{Effects of $T$-dependent KMT interaction on
pion and $a_0$-meson screening masses.
The solid (dot-dash) line denotes
$M_{\pi,{\rm scr}}$ ($M_{a_0,{\rm scr}}$) calculated by
the PNJL model with $T$-dependent coupling $K(T)$.
See Fig. \ref{Fig-screening} for LQCD data.
}
\label{Fig-screening-K(T)-PNJL}
\end{figure}

In the EPNJL model with the present parameter, the chiral susceptibility
$\chi_{ll}$ for light quarks has a peak at $T=163$ MeV,
as shown later in Fig. \ref{Fig-condensate-chiral-limit}(a).
This indicates $T_c=163$ MeV.
The model result is consistent with LQCD data
$T_{\rm c}= 154\pm 9$~MeV of Refs. \cite{Borsanyi:2010bp,Bazavov:2011nk}.
For the deconfinement transition, meanwhile,
the parameter $T_0$ is adjusted to reproduce LQCD data
on $T_c^{\rm deconf}$, as already mentioned in Sec. \ref{Model setting}.
In fact, the Polyakov-loop susceptibility $\bar{\chi}_{\Phi {\bar \Phi}}$
has a peak at $T=165$ MeV in the EPNJL model, as shown in Fig.
\ref{Fig-condensate-chiral-limit}(b).
The model result $T_c^{\rm deconf}=165$ MeV is consistent
with LQCD data $T_c^{\rm deconf}=170 \pm 7$ MeV of Ref. \cite{Aoki:2009sc}.

Figure~\ref{Fig-condensate} shows $T$ dependence of
the renormalized chiral condensate $\Delta_{l,s}$ defined by
\begin{equation}
\Delta_{l,s}\equiv\frac{\sigma_l(T)-\frac{m_l}{m_s}\sigma_s(T)}
{\sigma_l(0)-\frac{m_l}{m_s}\sigma_s(0)},
\end{equation}
and the Polyakov loop $\Phi$.
The present EPNJL model well reproduces LQCD data~\cite{Borsanyi:2010bp}
for the magnitude of $\Delta_{l,s}$ in addition to the value of $T_c$.
The present model overestimates LQCD data for the magnitude of
$\Phi$, although it yields a result consistent with LQCD
for $T_c^{\rm deconf}$.
The overestimation in the magnitude of $\Phi$ is a famous problem
in the PNJL model.
Actually, many PNJL calculations have this overestimation.
This is considered to come from the fact that
the definition of the Polyakov loop is different between LQCD
and the PNJL model~\cite{Braun:2007bx,Marhauser:2008fz}.
In LQCD the definition is
$\Phi_{\rm LQCD}
=\langle {\rm tr}_{{\rm c}}~\mathcal{P}\exp[i\int_{0}^{1/T} d\tau A_4(\tau,{\bf x})] \rangle/3$,
while in the PNJL model based on the Polyakov gauge and
the mean field approximation the definition is $\Phi_{\rm PNJL}
= {\rm tr_c} \exp[i \langle A_4 \rangle/T] /3$,
although both are order parameters of $Z_3$
symmetry~\cite{Braun:2007bx,Marhauser:2008fz}; see for example Ref.~\cite{Megias,Arriola:2014} as a trial to solve this
problem.

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.4\textwidth]{condensate_lat_v2.eps}
\end{center}
\caption{$T$ dependence of $\Delta_{l,s}$ and $\Phi$.
The solid (dot-dash) line corresponds to results of the EPNJL model
for $\Delta_{l,s}$ ($\Phi$).
LQCD data for 2+1 flavors are taken from
Ref.~\cite{Borsanyi:2010bp}.
}
\label{Fig-condensate}
\end{figure}

Now we investigate effects of $T$-dependent KMT interaction $K(T)$ on
$M_{\pi,{\rm scr}}$ and $M_{a_0,{\rm scr}}$.
In Fig. \ref{Fig-screening-K(0)}, $T$-dependence of $K(T)$ is switched off;
namely, results of the EPNJL model with $K(T)=K(0)$ are shown.
One can see that
$T$-dependence of $K(T)$ reduces the mass difference
between $M_{\pi,{\rm scr}}$ and $M_{a_0,{\rm scr}}$
significantly in a range $150 \mathrel{\mathpalette\fun <} T \mathrel{\mathpalette\fun <} 180$~MeV,
comparing Fig. \ref{Fig-screening-K(0)} with Fig. \ref{Fig-screening}.
At $T = 176$ MeV where
first-order chiral and deconfinement transitions take place,
$M_{\pi,{\rm scr}}$ has a jump while $M_{a_0,{\rm scr}}$ has a cusp.
Meson screening mass is thus a good indicator
for a first-order transition.

\begin{figure}[t
\begin{center}
 \includegraphics[width=0.4\textwidth]{screening-K-const_v2.eps}
\end{center}
\caption{Effects of $T$-dependent KMT interaction on
pion and $a_0$-meson screening masses.
The solid (dot-dash) line denotes
$M_{\pi,{\rm scr}}$ ($M_{a_0,{\rm scr}}$) calculated by
the EPNJL model with $K(T)=K(0)$.
See Fig. \ref{Fig-screening} for LQCD data.
}
\label{Fig-screening-K(0)}
\end{figure}

In Fig. \ref{Fig-screening-K(0)-PNJL},
both the $T$ dependence of $K(T)$ and the entanglement of
$G_{\rm s}(\Phi)$ are switched off. Namely, the results of
the standard PNJL model with a constant $K$ are shown.
The model cannot reproduce LQCD data, as expected.

\begin{figure}[t
\begin{center}
 \includegraphics[width=0.4\textwidth]{screening-K-const-PNJL_v2.eps}
\end{center}
\caption{Effects of the entanglement coupling
$G_{\rm s}(\Phi)$ on pion and $a_0$-meson screening masses.
The solid (dot-dash) line denotes
$M_{\pi,{\rm scr}}$ ($M_{a_0,{\rm scr}}$) calculated by the standard PNJL model with constant $K$, i.e., the EPNJL model with $K(T)=K(0)$ and $\a_1=0$.
See Fig. \ref{Fig-screening} for LQCD data.
}
\label{Fig-screening-K(0)-PNJL}
\end{figure}

Figure \ref{Fig-screening-difference} shows three types of
EPNJL calculations for the mass difference $\Delta M_{\rm scr}(T)$.
The mass difference plays a role of the order parameter of
the effective restoration of $U(1)_{\rm A}$.
The full-fledged EPNJL calculation (solid line)
with both $T$-dependent $K$ and
the entanglement coupling $G_{\rm s}(\Phi)$ well
reproduces LQCD data, while the standard PNJL model
(dotted line) with constant $K$ largely overestimates the data.

The present model has $T$ dependence implicitly in $G_{\rm s}(\Phi)$
through $\Phi$ and explicitly in $K(T)$.
As a model opposite to the present one, one may consider
the case that $K(T)=K(0)$ and $G_{\rm s}$ has $T$ dependence explicitly,
i. e., $G_{\rm s}=G_{\rm s}(T)$. We can determine $G_{\rm s}(T)$ so as to reproduce LQCD data for $\Delta_{l,s}$, however, this model overestimates LQCD data for $\Delta M_{\rm scr}$.
Thus the present model is well designed.

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.4\textwidth]{screening-mass-difference_v2.eps}
\end{center}
\caption{Mass difference $\Delta M_{\rm scr}(T)$
between pion and $a_0$-meson screening masses.
The solid, dot-dash and dotted lines denote results of
the EPNJL model, the EPNJL model with $K(T)=K(0)$ and
the standard PNJL model with $K(T)=K(0)$, respectively.
See Fig. \ref{Fig-screening} for LQCD data.
}
\label{Fig-screening-difference}
\end{figure}

\subsection{Meson susceptibilities}
\label{Meson susceptibilities}

The validity of $K(T)$ is investigated by comparing
LQCD data with the  model results for
meson susceptibilities $\chi_{\xi}^{\rm sus}$ ($\xi=\pi,a_0,\eta_{\rm ns},
\sigma_{\rm ns}$).
LQCD data based on domain-wall fermions~\cite{Bhattacharya:2014ara}
are available for two cases of pion mass $M_\pi(0) $ at vacuum
being the physical value $135$ MeV and a slightly heavier value $200$ MeV.
In order to reproduce these values with the EPNJL model,
we take $m_l=5.68$ MeV for the first case and $12.8$ MeV for the second one.

We consider the difference
$\Delta_{\pi,a_0}=\chi_{\pi}^{\rm sus}-\chi_{a_0}^{\rm sus}$
as an order parameter of the effective $U(1)_{\rm A}$-symmetry restoration.
Figure \ref{Fig-meson-susceptibility-U1A}
shows $T$ dependence of $\Delta_{\pi,a_0}/T^2$ for two cases of
$M_\pi(0)=135$ and $200$ MeV.
Since the $\chi_{\xi}^{\rm sus}$ have ultraviolet divergence,
they are renormalized with the $\overline{\rm MS}$ scheme in LQCD.
For this reason, one cannot compare the LQCD data with the results
of the EPNJL model directly. We then multiply the model results by a constant
so as to reproduce LQCD data at $T=139$ MeV
for the case of $M_\pi(0)=135$ MeV.
The model results thus renormalized well reproduce
LQCD data for any $T$ in both cases of $M_\pi(0)=135$ and $200$ MeV.

A similar analysis is made for $T$ dependence of
$\Delta_{\pi,\sigma}=\chi_{\pi}^{\rm sus}-\chi_{\sigma_{\rm ns}}^{\rm sus}$
and
$\Delta_{\eta,a_0}=\chi_{\eta_{\rm ns}}^{\rm sus}-\chi_{a_0}^{\rm sus}$ that
are related to $SU_{\rm L}(2)\times SU_{\rm R}(2)$ symmetry.
Figure \ref{Fig-meson-susceptibility-chiral-1}
(\ref{Fig-meson-susceptibility-chiral-2}) shows $T$ dependence of
$\Delta_{\pi,\sigma}/T^2$ and $\Delta_{\eta,a_0}/T^2$ for
$M_\pi(T)=135$ ($200$) MeV. In both the figures, the EPNJL model well
reproduces $T$ dependence of LQCD results. The present model
with the $K(T)$ of \eqref{T-dependent-K}  is thus reasonable.

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.4\textwidth]{meson_sus_U1A_v3.eps}
\end{center}
\caption{$T$ dependence of the difference $\Delta_{\pi,a_0}$
between $\pi$ and $a_0$ meson susceptibilities
for two cases of $M_\pi(0)=135$ and $200$ MeV.
}
\label{Fig-meson-susceptibility-U1A}
\end{figure}

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.4\textwidth]{meson_sus_chiral_mpi135_v2.eps}
\end{center}
\caption{$T$ dependence of $\Delta_{\pi,\sigma}$ and
$\Delta_{\eta,a_0}$ for $M_\pi(0)=135$ MeV.
}
\label{Fig-meson-susceptibility-chiral-1}
\end{figure}

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.4\textwidth]{meson_sus_chiral_mpi200_v2.eps}
\end{center}
\caption{$T$ dependence of $\Delta_{\pi,\sigma}$ and $\Delta_{\eta,a_0}$ for $M_\pi(0)=200$ MeV.
}
\label{Fig-meson-susceptibility-chiral-2}
\end{figure}

\subsection{The order of chiral transition near the physical point}
\label{The order of chiral transition}

Finally we consider the order of chiral transition near the physical point
{\bf
$(m_l^{\rm phys},m_s^{\rm phys})=(6.2{[\rm MeV]},175{[\rm MeV]})$} in the $m_l$--$m_s$ plane, First we vary $m_l$ from 9.9 to 0 MeV with $m_s$ fixed
at 175~MeV.

Figure \ref{Fig-condensate-chiral-limit}
presents $T$ dependence of the chiral susceptibility $\chi_{ll}$
for light quarks and the Polyakov-loop susceptibility
$\bar{\chi}_{\Phi {\bar \Phi}}$ in three points,
``simulation point (S-point)'' of $(m_l,m_s)=(9.9{[\rm MeV]},175{[\rm MeV]})$,
``physical point (P-point)'' of $(m_l,m_s)=(6.2{[\rm MeV]},175{[\rm MeV]})$
and ``light-quark chiral-limit point (C$_l$ point)'' of
$(m_l,m_s)=(0{[\rm MeV]},175{[\rm MeV]})$.
In general, $T_c$ and $T_c^{\rm deconf}$
determined from peak positions of $\chi_{ll}$ and $\bar{\chi}_{\Phi {\bar \Phi}}$
depend on $m_l$ and $m_s$. However, as shown in panel (a),
the $T_c$ thus determined is $163$~MeV at
S-point and $160$~MeV at P-point,
and hence the value little varies between the two points.
At C$_l$ point, $\chi_{ll}$ diverges at $T=T_c=153$~MeV.
The chiral transition is thus second order at C$_l$-point at least
in the mean-field level. This result suggests that
the effective $U(1)_{\rm A}$ restoration is not completed at $T=T_c$.
This suggestion is supported by LQCD data at S-point in Fig.
\ref{Fig-screening-difference} where
$\Delta M_{\rm scr}(T_c)$ is about a half of $\Delta M_{\rm scr}(0)$.

As shown in panel (b), $m_l$ dependence of $T_c^{\rm deconf}$ is even smaller;
namely, $T_c^{\rm deconf}=165$~MeV for S-point and C$_l$-point and 163~MeV
for P-point. At C$_l$-point, $\bar{\chi}_{\Phi {\bar \Phi}}$ has a sharp peak
at $T=153$~MeV. It is just a result of the propagation of divergence
from $\chi_{ll}$ to $\bar{\chi}_{\Phi {\bar \Phi}}$ \cite{Kashiwa:2009zz},
and never means
that a second-order deconfinement takes place there.

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.4\textwidth]{sus_chiral_v3.eps}
   \includegraphics[width=0.4\textwidth]{sus_polyakov_v3.eps}
\end{center}
\caption{
$T$ dependence of (a) chiral susceptibility $\chi_{ll}$ and
(b) Polyakov-loop susceptibility $\bar{\chi}_{\Phi {\bar \Phi}}$
at S-point, P-point and C$_l$-point.
Here $\chi_{ll}$ and $\bar{\chi}_{\Phi {\bar \Phi}}$ are dimensionless and
their definition is the same as in the LQCD formulation.
Calculations are done by
the EPNJL model with the present parameter set.
The dotted, dot-dash and solid lines stand for the results
at S-point, P-point and C$_l$-point, respectively.
At C$_l$-point, $\chi_{ll}$ is divided by 10 and diverges
at $T=T_c=153$~MeV
}
\label{Fig-condensate-chiral-limit}
\end{figure}

Next, both $m_l$ and $m_s$ are varied near P-point.
Figure \ref{Fig-Columbia-plot} shows the value of
$\log[\chi_{ll}(T_c)]$ near P-point in the $m_l$--$m_s$ plane.
The value is denoted by a change in hue.
Three second-order chiral transitions (solid lines)
meet at $(m_l^{\rm tric},m_s^{\rm tric})
\approx (0,0.726m_s^{\rm phys}) = (0{[\rm MeV]},127{[\rm MeV]})$.
This is a tricritical point (TCP) of chiral phase transition.

\begin{figure}[t
\begin{center}
   \includegraphics[width=0.45\textwidth]{CP_v2.eps}
\end{center}
\caption{
Order of chiral transition near physical point in the
$m_l$--$m_s$ plane.
The value of $\log[\chi_{ll}(T_c)]$ is shown by a change in hue.
Simulation point, physical point, light-quark chiral-limit point and
tricritical point are denoted by S, P, C$_l$ and TCP.
The solid lines stand for second-order chiral transitions.
}
\label{Fig-Columbia-plot}
\end{figure}

\section{Summary}
\label{Summary}

In summary,
we incorporated the effective restoration of $U(1)_{\rm A}$ symmetry
in the 2+1 flavor EPNJL model by introducing a $T$-dependent coupling
strength $K(T)$ to the KMT interaction.
The $T$ dependence was well determined from
state-of-the-art 2+1 flavor LQCD data
on pion and $a_0$-meson screening masses.
To derive the meson screening masses in the EPNJL model,
we extended our previous prescription of Ref.~\cite{Ishii:2013kaa}
for 2 flavors to 2+1 flavors. The strength $K(T)$ thus obtained is
suppressed in the vicinity of the pseudocritical temperature of chiral
transition.
As a future work, it is quite interesting to clarify
how the present suppression is explained by instantons.

 In order to check the validity of $K(T)$, we analyze
$\pi,a_0, \eta_{\rm ns},\sigma_{\rm ns}$-meson susceptibilities
obtained by state-of-the-art LQCD simulations with domain-wall
fermions~\cite{Bhattacharya:2014ara}. The EPNJL model with
the $K(T)$ of \eqref{T-dependent-K}
well reproduces $T$ dependence of LQCD data.
The present model building is thus reasonable.

Using the EPNJL model with the present parameter set, we showed that,
at least in the mean field level,
the order of chiral transition is second order
at the light-quark chiral-limit point
of $m_l=0$ and $m_s=175$ MeV (the physical value).
This result indicates that there exists a tricritical point
near the light-quark chiral-limit point in the $m_l$--$m_s$ plane.
We then estimated the location of the tricritical point as
$(m_l,m_s) \approx (0{[\rm MeV]},127{[\rm MeV]})$.

In conclusion, we present a simple method for calculating
meson screening masses in PNJL-like models. This allows us to compare model
results with LQCD data on meson screening masses.
Meson screening masses are quite useful to determine model parameters.
In particular, the mass difference between
pion and $a_0$-meson is effective to determine $T$ dependence of the KMT
interaction.
The EPNJL model with the present parameter set is useful
for estimating the order of chiral transition at
the light-quark chiral-limit point and the location of the tricritical
point, since it is hard to reach the chiral regime directly with LQCD.

The present model consists of $\Phi$-dependent four-quark interactions
and $T$-dependent six-quark interactions. Meanwhile, the importance of
eight-quark interactions was pointed out in
Ref.~\cite{Osipov:2006}, since it makes the thermodynamic potential bounded from
below. Furthermore, it is reported in Ref.~\cite{Osipov:2013} that current-quark-mass
dependence of quark-quark interactions
is effective to reproduce meson pole masses with
good accuracy. Therefore, further
inclusion of these interactions
is interesting as a future work.

\noindent
\begin{acknowledgments}
M. I, J. T., H. K., and M. Y. are supported by Grant-in-Aid for Scientific Research (
No. 27-3944, No. 25-3944, No. 26400279 and No. 26400278) from the Japan Society for the Promotion of Science (JSPS).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
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All Your Friday Links!
* The itself.blog Star Trek: Discovery event is underway! Star Trek: Discovery Is Optimism, But Not for Us. "Can you bury your heart"? Having feelings about Discovery. Star Trek: Discovery as the End of Next Generation Triumphalism.
* CFP: Activist Speculation and Visionary Fiction (MLA 19).
* Jaimee Hills is officially a dangerous woman.
* The Male Glance.
* The university in ruins: UW Stevens Point. The administration clearly doesn't even understand what it's proposing:
When releasing the plan, university officials said that English majors for teacher certification would continue. But Williams said that under the state Department of Public Instruction's certification criteria, a person looking to become an English teacher has to have been an English major.
"They just both have to exist, or both have to be eliminated," Williams said. "One depends directly on the other."
The most salient fact of academia today is that low-cost humanities classes subsidize every other aspect of university operations. You will never hear a single administrator acknowledge this basic fact and indeed they insist that the money flows the other way.
* Professors earn about 15 percent less than others with advanced degrees, finds a study circulated Tuesday by the National Bureau of Economic Research. Though perhaps some of us make it up in job satisfaction (really).
* It's gotten so bad in the humanities even the Yale grads don't get jobs handed to them as soon as they graduate.
* Why Creative-Writing Programs Have Been Havens for Harassment.
* It has taken me two and a half decades to recognize that my experience of having a senior male nominal adviser and a female (usually more junior) actual adviser is common throughout academe. On Ghost Advising.
* Abusers and enablers in faculty culture.
* 5% raises in West Virginia. Onward to Oklahoma.
Today's front page is one worth saving pic.twitter.com/ig9SNqbpuZ
— Jake Zuckerman (@jake_zuckerman) March 7, 2018
* Marquette vs. Trauma.
* Snowflake students claim Frankenstein's monster was 'misunderstood' — and is in fact a VICTIM.
* On the Blackness of the Panther.
* Loved this from Barbara Ehrenreich: Body Work: The curiously self-punishing rites of fitness culture.
If anything, the culture of fitness has grown more combative than when I first got involved. It is no longer enough to "have a good workout," as the receptionist at the gym advises every day; you should "crush your workout." Health and strength are tedious goals compared to my gym's new theme of "explosive strength," achieved, as far I can see, through repeated whole-body swinging of a kettleball. If your gym isn't sufficiently challenging, you might want to try an "ultra-extreme warrior workout" or buy a "home fitness system" from P90X, which in 2016 tweeted a poster of an ultra-cut male upper body, head bowed as if in prayer, with the caption "A moment of silence please for my body has no idea of what I'm about to put it through." Or you could join CrossFit, the fastest-growing type of gym in the world, and also allegedly the most physically punishing. The program "prepares trainees for any physical contingency," the company boasts, "not only for the unknown, but for the unknowable, too. Our specialty is not specializing," and the latter category includes the zombie apocalypse. The mind's stuggle for mastery over the body has become a kind of mortal combat.
* Too big to tax.
* In this economy you're either burned out, or you're boxed out.
* The Secret NYPD Files: Officers Can Lie And Brutally Beat People — And Still Keep Their Jobs.
* A prosecutor who obtained a wrongful conviction that sent a Houston man to death row for nearly 10 years didn't just withhold evidence but also denied under oath that he had information that supported Alfred Dewayne Brown's alibi, court records show.
* Twilight of the cult film.
* Could Trump get a White House job if he weren't president? Didn't we already know Trump couldn't get a loan before he was president?
* Beware Liberal-Baiting "White House in Chaos" Stories. Trump's Position Is As Solid As It's Ever Been.
For that reason win or lose I think the fights on the liberal-left will be a lot nastier after 2020 than they've been before. A huge part of the Democratic base is still living in denial about what this country has become.
Well, I'd like to see ol Donny Trump wriggle his way out of THIS jam!
*Trump wriggles his way out of the jam easily*
Ah! Well. Nevertheless,
— jesse farrar (@BronzeHammer) October 2, 2016
* Gratuitous cruelty by Homeland Security. Lying to the immigrant soldiers you promised citizenship.
* A Dozen Democrats Want To Help Banks Hide Racial Discrimination In Mortgages.
* Guess Who's Not Coming To America? International Students.
* Kobe Steel's Chief to Step Down as It Discloses Wider Quality Problems Fifty-Year Conspiracy to Defraud Customers.
* Toddler feelings helpline.
— If you still feel pretty messed up about how they were just going to burn the Velveteen Rabbit, please mash all of the keys but mostly 2.
* White flight remains a reality.
* Liberate our schools.
[future history class]
Teacher: How did World War 3 start? Anyone? Yes, Khaleesi.
Girl: It started bec-
Teacher: No, I meant Khaleesi M. She had her hand up first.
Girl 2: It started because president Trump was hangry.
Teacher: Correct. [holsters gun]
— OhNoSheTwitnt (@OhNoSheTwitnt) March 3, 2018
* '50 or 60. If I get lucky maybe 150.'
* The grim reality of job hunting in the age of AI.
* The Marvel Cinematic Universe only really makes sense when you remember Captain America is a trans man.
* Wait, what exactly was Luke Skywalker's plan in Return Of The Jedi?
* The opioid crisis has become an "epidemic of epidemics." Meanwhile, a new study suggests opioids are no better than Tylenol for treating some kinds of pain.
* Kentucky's 'child bride' bill stalls as groups fight to let 13-year-olds wed.
* U.N. Report On Magical Realism Warns Of Increased Incidences Of Women's Tears Flooding The Entire World.
* False news stories travel faster and farther on Twitter than the truth.
* There's no idea so terrible there isn't someone in favor of it.
* York University philosophy professor and team submit brief supporting chimpanzee personhood.
* Ok, but you're on a very short leash.
* Her name was Kanga and she was trouble.
I thought of how it all used to be, back when C.R. was young, when we still all believed things would get better. When we still had hope. In a way, hope is as powerful a drug as honey.
— Lavie Tidhar (@lavietidhar) March 7, 2018
* I Am the Very Important Longread Everyone Is Talking About.
* The United States of Middle-earth.
* Understanding Fight Club.
* And the arc of history is long, but.
Tagged with academia, activism, actually existing media bias, advising, Amazon, America, animal personhood, animal rights, animals, Barbara Ehrenreich, Black Panther, capitalism, Captain America, chimps, class struggle, consent, creative writing, cult film, dangerous women, deportation, Disney, Donald Trump, English departments, fake news, Fight Club, film, fitness, Frankenstein, fraud, games, graduate student life, guns, harassment, homeland security, How the University Works, humanzees, hyperexploitation, ice, immigration, international students, Jaimee Hills, job satisfaction, Kentucky, kids, liberals, Lili Loofbourow, longreads, Lord of the Rings, magic realism, Marquette, Marvel, Marvel Cinematic Universe, Mary Shelley, McSweeney's, Middle-Earth, Milwaukee, MLA, mortgages, Nintendo, Nintendo Switch, NYPD, Oklahoma, opioids, over-educated literary theory PhDs, pain, parenting, perjury, poetry, police brutality, police corruption, politics, race, racism, rape culture, Return of the Jedi, school shootings, science fiction, segregation, sex, sexual harassment, Sopranos, Star Trek, Star Trek: Discovery, Star Wars, steel, strikes, Super Smash Brothers, taxes, teachers, Teju Cole, tenure, the gym, the humanities, the male glance, the university in ruins, the Wisconsin Idea, there is nothing intelligent to say about a massacre, Tolkien, trauma, true crime, Twitter, unions, University of Wisconsin, University of Wisconsin Stevens Point, West Virginia, white flight, Winnie the Pooh, Yale
* Marquette English Spring 2015 courses! I'm teaching a section of 3000 (our new intro to major — mine is themed around magic) and the second round of my NEH "Cultural Preservation" course. I'm also doing a honors seminar on "video game culture" that I'm really excited about, GamerGate notwithstanding.
* A rare spot of optimism: Lockheed announces breakthrough on nuclear fusion energy.
* But don't hang on to it: It's time to push the panic button on the global economy. Markets are panicking again. What's going on?
* Sea Level Rising Faster Than Anytime In 6,000 Years, Study Finds.
* WHO: 10,000 new Ebola cases per week could be seen. The CDC is apparently taking the over. One thing is certain: it's time to panic.
* Another Obama triumph for the left: let a thousand wage thefts bloom.
* The Assassination of Detroit.
* Charter School Power Broker Turns Public Education Into Private Profits. Neoliberalism, Higher Education, and the Rise of Contingent Faculty Labor.
* Identifying The Worst Colleges In America.
* Could Oculus Rift be the next great higher education boondoggle?
* In Taste of Autonomy, Sports Programs Now Battle for Athletes' Bellies.
* The most alarming thing I've heard from friends who've had miscarriages is their surprise (only upon miscarrying) at hearing about how many of their friends, aunts, cousins, sisters, mothers and grandmothers have had them, too. If miscarriages are so common, why do we hide them behind a wall of shame and silence?
* What It's Really Like to Have an Abortion.
* The radical teamsters of Minneapolis showed what democratic unionism looks like.
* "Most schools' internal judicial systems are the worst of both worlds," Berkowitz said. "They don't give the accused the protections of the criminal justice system, and they mistreat the victims, too."
* For example, even into the 1980s, some doctors didn't believe that babies felt pain and so routinely did surgery on them using just muscle relaxants to keep them still. Pain and medicine.
* Guy Debord's The Muppets. More links below Gonzo.
* "You had one job" screwup of the week.
* South Carolina governor levels outrageous accusation against the nation's CEOs, says they're all white supremacists. Huge if true.
* Study claims that whales and dolphins can speak to one another.
* DC has a bit hit on its hands with The Flash, so of course the smart move here is to recast for the film.
* Father, there's a gateway to Narnia in the closet!
* The Absolute Weirdest Thing Ever To Happen At A Political Debate.
* How A California Man Was Forced To Spend 100 Days In Prison For Being An Atheist.
* Next week: Civilization: Beyond Earth.
* Behold! The Counter-Intuitivist!
* And we are all Bartleby now.
Tagged with AA, abortion, academia, addiction, adjunctification, Amazon, animals, atheism, austerity, babies, Barack Obama, Bartleby, boondoggles, California, CDC, CEOs, Charlie Crist, charter schools, Civilization: Beyond Earth, class struggle, climate change, college sports, contrarianism, cultural preservation, DC Comics, Detroit, digital humanities, doctors, dolphins, Ebola, ecology, English, epidemics, Florida, fusion, Gamergate, games, Great Recession, Guy Debord, Herman Melville, How the University Works, I would prefer not to, magic, Marquette, medicine, Minneapolis, miscarriage, Muppets, Narnia, NCAA, NEH, neoliberalism, nuclear energy, Oculus Rift, pain, politics, pregnancy, prison, race, racism, rape, rape culture, real estate, reverse college rankings, Rick Scott, rising sea levels, Saturday Morning Breakfast Cereal, scams, South Carolina, superheroes, teaching, the Confederacy, the economy, the Flash, theory of mind, This Modern World, time to panic, Title IX, unions, wage theft, war on education, whales, what it is I think I'm doing, World Health Organization, worst financial crisis since the last one, you had one job
Or We Could Just Abolish Capitalism
According to KnoxNews, Tennessee legislators are attempting to pass legislation to cut the welfare benefits of parents with children who don't meet attendance and performance requirements. The bill, SB 132, is sponsored by Sen. Stacey Campfield, R-Knoxville, and Rep. Vance Dennis, R-Savannah, and has passed committees in both the House and Senate, and now heads to another House committee, and to the Senate floor for vote.
Tennessee's Terrifying New Plan to Punish the Families of Underperforming Kids.
Tagged with austerity, capitalism, crazy people run the country, neoliberalism, pain, Tennessee, war on education, welfare, Won't somebody think of the children?
* The Center for 21st Century Studies has announced its postdoc theme for 2013-2014: "Changing Climates." Applications due March 1.
* What's coming out with this UNC rape case is astounding. UNC's Former Dean of Students Says She Was Forced to Underreport Sexual Assault Cases. And then this, from the assistant vice underprovost of sickening analogies:
"When I went to report my assault in 2007, I asked an administrator what the process would look like," Clark said. "Instead, that person told me, 'Rape is like a football game, Annie. If you look back on the game, and you're the quarterback and you're in charge, is there anything that you would have done differently in that situation?'"
* Being Married Helps Professors Get Ahead, but Only if They're Male: A new study of history professors shows that married men get promoted faster than their single colleagues, while the opposite is true for women.
* Man Has Alarming Level Of Pride In Institution That Left Him $50,000 In Debt, Inadequately Prepared For Job Market.
* The union at Kalamazoo Valley Community College launches a food drive for its own adjuncts.
* UC aims to bleed its grad students.
* "Fear and loathing in academia" and "Some historical notes on the decline of the universities," from anthropologies issue 16: The Neoliberalized, Debt-plagued, Low Wage, Corporatized University. Also: Passing with Pills: Redefining Performance in the Pharmaceuticalized University.
* The CEO of Whole Foods is laughing at you.
* Naked Capitalism on Hayek's Delusion: The Origins of Neoliberalism: 1, 2, 3. Via MeFi.
* As I say, I have no dog in this race, except a belief that no one, in this sea of riches, should have to be poor. But staring dumbfounded at the lessons unlearned in Britain, Europe and the United States, it strikes me that the entire structure of neoliberal thought is a fraud. The demands of the ultra-rich have been dressed up as sophisticated economic theory and applied regardless of the outcome. The complete failure of this world-scale experiment is no impediment to its repetition. This has nothing to do with economics. It has everything to do with power.
* Theater of Pain: Tom Junod on injury in the NFL.
The perspective of pain is what this story is about. For fans, injuries are like commercials, the price of watching the game as well as harrowing advertisements for the humanity of the armored giants who play it. For gamblers and fantasy-football enthusiasts, they are data, a reason to vet the arcane shorthand (knee, doubtful) of the injury report the NFL issues every week; for sportswriters they are kernels of reliable narrative. For players, though, injuries are a day-to-day reality, indeed both the central reality of their lives and an alternate reality that turns life into a theater of pain. Experienced in public and endured almost entirely in private, injuries are what players think about and try to put out of their minds; what they talk about to one another and what they make a point to suffer without complaint; what they're proud of and what they're ashamed by; what they are never able to count and always able to remember
* An oral history of Fringe: 1, 2, 3, 4.
* Scandal in Lance-ville! Scandal in Gleetown!
* Well, it's been a month since @dronestream started and we're up to January 2011. Two years left.
* Claire Danes performs The Handmaid's Tale.
* The kids are all right: Barbara Walters interviews a twelve-year-old transgender teen she first interviewed in 2007, when Jazz was six.
* A Lawyer's Amazingly Detailed Analysis of Bilbo's Contract in The Hobbit.
* Rules for kids: The book, discovered by a 20-year-old Walmart employee, Raymond Flores, became an Internet sensation after Flores contacted the media to try to find its owner and its touching rules – including the rules "Don't bite the dentist" and "If you're going to wet your bed, wear a pull-up" – went viral.
* Two years before his death, legendary science and science fiction writer Isaac Asimov kicked off a TV pilot dedicated to exploring the faint and ever-shifting boundary separating science from science fiction.
* And Saturday Morning Breakfast Cereal, on robots.
Tagged with academia, adjuncts, Big Pharma, capitalism, Center for 21st Century Studies, class struggle, climate change, college, concussions, drones, ecology, football, Fringe, Glee, Handmaid's Tale, headbrick, How the University Works, Isaac Asimov, Jonathan Coulton, kids today, Lance Armstrong, Margaret Atwood, married people, misogyny, neoliberalism, NFL, Occupy Cal, only if you accept the ethical implications, over-educated literary theory PhDs, pain, politics, postdocs, psychopharmacology, rape, rape culture, robots, rules, rules are rules, science fiction, sports, television, The Hobbit, the kids are all right, the law, theory, transgender issues, UNC, unions, violence, what it is I think I'm doing
(Un)Breakable
Ashlyn Blocker, the girl who feels no pain.
Tagged with disability, medicine, pain
Tuesday Links Quickly
* A refreshingly honest business model: For-Profit College Recruiters Taught To Use 'Pain,' 'Fear,' Internal Documents Show.
* The new Journal of American Studies issue on Oil Culture is free to read. Looks great.
* UWM's C21 has announced its fellows for next year.
* Cracked suggests all contemporary science fiction descends from a bad unauthorized sequel to War of the Worlds, Edison's Conquest of Mars.
* Another KSR interview.
* Snoop the Dogg is no more. I am Snoop the Lion, and I come to you now at the turn of the tide.
* Rest in peace, Dominar.
* And the Democrats figure out how to get the magic of malicious bullshitting working for them for a change.
Tagged with 2312, academia, advertising, Center for 21st Century Studies, Democrats, Edison's Conquest of Mars, fan fiction, Farscape, fear, for-profit schools, Gandalf, general election 2012, Harry Reid, Kim Stanley Robinson, malicious bullshitting, Milwaukee, Mitt Romney, obituary, oil, pain, politics, science fiction, sequels, Snoop Dogg, taxes, War of the Worlds

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