text:
Merkle RMG Takes Innovation to New Levels with Smartsheet Certification
When Jann Schultz joined Merkle Response Management Group (RMG) as VP of Client Services, she quickly realized her teams weren't working as effectively as they could be. With version-control issues and a high volume of their work stuck in spreadsheets, they needed a cohesive platform to improve processes.
Organization Size
Medium (50 - 999 employees)
To really understand what the commitment was from a learning perspective, I took a step back and completed the five introductory learning modules. I believe if you're going to ask your team to do something, then you need to lead by example, and I quickly discovered how easy Smartsheet would be for my team to learn.
Jann Schultz, VP of Client Services, Merkle Response Management Group (RMG)
Photo Courtesy of Merkle
"I came on board a little over a year ago and spent my early days really assessing needs, considering both the procedures and tools that my teams were using," said Schultz, "If I heard one more person say, 'Oh, we have that on a spreadsheet in the share drive,' I was going to pull my hair out." She began searching for a platform to help streamline processes related to their growing portfolio of over 200 clients at Merkle RMG, one of the nation's largest direct-response processing and fulfillment companies.
Schultz selected Smartsheet to help improve visibility and communications across her two teams — account management and new client implementation — as well as with their clients,
"I was very excited to implement Smartsheet, and rapidly began to provide licenses to my team members. I sat back and thought, 'Wow, such a great tool! This is going to be awesome.' But I quickly realized the 'build it and they will come' mentality wasn't working." says Schultz. Like most busy teams, hers were hesitant to adopt new processes and technologies.
Rethink to Rebuild
Schultz recognized that the team needed a jumpstart to incorporate Smartsheet into their daily activities. "I decided that I needed to do a reset, and rethink how I was going to introduce and approach the use of Smartsheet across my team."
With the Smartsheet Center of Excellence as part of her Enterprise plan, Schultz identified five modules within Center of Excellence to tie to her team's yearly performance goals. "To really understand what the commitment was from a learning perspective, I took a step back and completed the five introductory learning modules. I believe if you're going to ask your team to do something, then you need to lead by example," says Schultz, "and I quickly discovered how easy Smartsheet would be for my team to learn."
After landing on her learning requirements, Schultz began to put her plan in motion. "I held a kickoff meeting that I called 'Sweet Treats and Smartsheet'. I armed myself with doughnuts and candy, and began to set expectations for what I wanted to see from the team going forward."
With the baseline performance goal set to complete the five introductory modules in the Center of Excellence, Schultz invited the team to go beyond that and become a Smartsheet certified user. "The first four certified users would receive a $50 gift certificate, so the race was on!"
The "Council of Champions"
In order to mobilize and encourage her team, Schultz formed an internal Council of Champions comprised of staff members who had participated in the Center of Excellence modules early on.
"I brought the Council of Champions together because they clearly had an interest and were intrigued by Smartsheet," says Schultz, "They set up office hours to make themselves available to their peers, so that team members who were working to complete the modules or go the extra mile to become certified had an internal resource to turn to."
The internal Council of Champions has helped expand the knowledge and awareness of Smartsheet at Merkle RMG. They recently held an information session for operations management to demonstrate how Smartsheet could help to increase efficiencies for their teams, as well.
"We were able to provide the managers with short-term licenses and let them test out Smartsheet's capabilities. We have continued to provide the licenses and the operational staff has started to develop their workflows and processes in Smartsheet," says Schultz, "As we've expanded use across the organization, we encourage them to participate in the Council of Champions office hours, and additional training so that they can develop their skill set and discover additional ways that Smartsheet would be useful to their team members."
A New Level of Sophistication
Schultz now has 15 certified users on her team, and all 25 team members have completed the required five modules in the Center of Excellence. "I have a very high level of knowledge and skill across the team, thanks to these programs," she says. "I think they have really opened up a new understanding of how to use Smartsheet and apply it across the business.
"The new client implementation team has moved from old-school Excel spreadsheet project plans into very nice project plans and dashboards within Smartsheet," says Schultz, "and the account management team has developed really delightful client dashboards that are templates now being used across the team for consistent communication with clients."
As the team has grown, the Center of Excellence continues to be tied to their performance goals. "Certification has become part of our team culture, it's core to our onboarding and really helps new users get up and running," says Schultz.
"Smartsheet Certification has brought a whole new level of sophistication to my team. Instead of asking people to do things, they are coming to me with innovative ideas and new ways they can use Smartsheet to streamline their work."
Learn more about how the Smartsheet Certification program can help you expand your knowledge here.
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text:
New Fat Tire beer tastes like butt so that we take climate change seriously
Allison Robicelli
Photo: AGUSTIN PAULLIER (Getty Images), New Belgium (Getty Images)
Scientists have been warning us about the perils of climate change for decades now, but we just don't listen, do we? We've got companies spewing greenhouse gases into the atmosphere, dumping waste into the oceans, and flinging around plastic products willy-nilly. If humanity can't get its shit together, everything is going to start sucking in whole new ways. We will live in a world where coffee is strictly rationed, our primary source of food is fiber-rich meal replacement shakes, and all our beer tastes like fermented butts. Want proof of that last one? Then check out the new Torched Earth Ale from Fat Tire, a limited-edition beer meant to force your taste buds to face our dark, dystopian future.
The environmentally conscious brewery wants us all to know that unless drastic action is taken to mitigate climate change, hops, grains, and all the other stuff that makes beer so enjoyable will become scarcer and scarcer, forcing the brewers of the future to make do with whatever they can find. Torched Earth Ale is an example of what such future beer might taste like, brewed with smoke-tainted water, dandelions, and drought-resistant grains. Fat Tire says the beer tastes like cold, fizzy garbage, and the brewery hopes to emphasize that bad things are headed our way if we don't all stand together and demand action.
Even if you are doing your personal best to live a cleaner, greener life, it's not enough to address the problem, because the majority of environmental damage is the work of—surprise!—massive global corporations. This is why Fat Tire has launched a "Last Call for Climate" initiative, urging us all to demand better from the companies behind our favorite products. Currently, 70% of Fortune 500 companies don't have a meaningful climate action plan, nor have they committed to reducing greenhouse gas emissions by 2030—the year scientists claim is our planet's point of no return. In addition to this call to action, Fat Tire has launched an online tool that shows you which Fortune 500 companies already have climate plans and supplies the tools for you to reach out directly to those that do not.
In case you're wondering if the company is practicing what it preaches, Fat Tire—the flagship label from Colorado's New Belgium Brewing—became America's first certified carbon neutral beer last year. New Belgium has also pledged to achieve net-zero emissions across the entire company by 2030.
"If you don't have a climate plan, you don't have a business plan," said New Belgium CEO Steve Fechheimer in a press release. "Aggressive action to help solve the climate crisis is not only an urgent environmental and social imperative – it's also a no-brainer for companies seeking to create long-term shareholder value, compete with rivals like China, and create good-paying jobs here at home. As a medium-sized company, New Belgium can only have a medium-sized impact. We need more of the big guys to step up, too."
New Belgium plans to release a detailed blueprint for a green beer industry later this year, which will help brewers measure their carbon footprint and take steps to become carbon neutral. Good on them! If mankind is drinking palatable beer in fifty years, we'll all know who to thank.

redpajama_set_name: RedPajamaCommonCrawl


Tina Aedo/FAASMax, a 2-year-old red wire-haired Chihuahua mix, is the Alameda AnimalShelter's pet of the week.
Warm weather is here, and so are the fleas. Are you ready?
Distinguish between products designed for dogs and cats. Some products labeled for use in dogs can be deadly to cats.
Do your research. Ask your friends and neighbors what flea control treatments have worked for them. Ask pet store employees which products sell best and which get returned frequently for ineffectiveness. Pet store staff often receive training on how different treatments work and may have up-to-date information on the latest products.
Understand that you get what you pay for. Flea control is not the place to pinch pennies. Many of the supermarket brands have the same methods of application and active ingredients as those sold at pet stores, but this does not mean the dosage or content is going to be as effective. Despite their convenience, low cost and availability, flea collars and flea powders do not work and can be irritating to your pet's skin, nose and eyes.
Talk to your vet. Most vets recommend oral flea treatments that not only repel and kill fleas on your pet but interrupt the fleas' life cycle by keeping eggs from hatching. Oral flea treatments create no issues with bathing, no greasy residue on the coat and no risk of exposure to children, other pets or family members.
Give yourself and your pets a break by creating a flea-free season. Your pets will be happier, and so will you.
PET OF THE WEEK: MAX is a 2-year-old red wire-haired Chihuahua mix. This long-legged boy really loves to play fetch with his tennis ball. He would do well in a family that is looking for a snuggly and affectionate dog to be a companion around the house and go for a stroll around the block.
Dog Alley: There are 15 dogs available for adoption. SUKI is a 3-year-old blond-and-white Labrador mix looking for a home without cats. CHARLEE is a 2½-year-old blue pit bull mix. This great girl has been waiting for a family for six months.
Cat lair: There are 10 cats available for adoption. GERTRUDE is a 7-year-old tortie girl that needs to be the only cat in the house. She also prefers the company of adults. ROCKETTE is a 5-year-old white-and-black domestic shorthair. She is a real beauty with her green eyes and pink nose.
RABBIT ROW: We have four rabbits available for adoption.
SMALL PETS: There are two red-eared slider turtles available for adoption. Please call the shelter for more information.
Meggan Stringent is a veterinary technician at Friends of the Alameda Animal Shelter. Visit the shelter at 1590 Fortmann Way at Grand Street, call 510-337-8565 or visit www.alamedaanimalshelter.org to view the animals.


'Democratic Discourse' is a reference to the concept of the contemporary German Philosopher Jürgen Habermas. I cannot claim any deep understanding of his philosophy (despite reading many of his books), but what I take from it is that we will move closer to positions of truth when we engage in discourse on key issues from positions of equality rather than positions of different levels of power.
In the case of education this philosophy goes back to John Dewey (1859-1952) and the concept of education for democracy.
I mention Australian because it provides the point from which the perspective comes. This country's position as a small democratic nation that is a relatively prosperous and peaceful island on the 'underside' of the world helps Australians reflect on some of the big issues confronting others. We claim, and still largely practice, an egalitarianism, but inequality is growing. We also have dark aspects including: our past and continuing treatment of Indigenous Australians; our treatment of refugees; and the insignificant amount our Federal Government spends on aid to the world's underprivileged.
See Australia : An Overview for a background to our Geography, History and Institutions.
Finally I mention the author's name, because I believe each of us must take responsibility for our own opinions and understand that all of us come to discussion with our own preconceptions.
I was born in 1943, at a time when Australia with USA support was fighting the Japanese in the Pacific. I grew up in an industrial suburb some 8 kilometres from the heart of Sydney.
My father was a labourer, first in a paper mill and then a timber yard. In my early years I was very involved in the local Methodist Church and became a Local Preacher. While values remain from this time, I am no longer involved in any particular religion but retain an interest in religion in general.
After school I did a Science degree, majoring in the Biological Sciences at the University of New South Wales. I taught Science and Maths in secondary schools for a few years then did an Arts Degree, with a major in History, at Sydney University. Following that I taught mainly History in Public (Government) schools but also have taught English, Legal Studies and Studies of Religion.
I spent about thirty years teaching, with the last third of this being a Head of Humanities Departments in Secondary Schools in Western Sydney. I have also been a major contributor to many textbooks used in Secondary Schools in New South Wales and Nationally.
I retired from teaching in 2006 but remained involved with Educational and Cultural Associations, which are on the Links page.


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Everyone seems to have been talking about the weather in the past few days. But what have scientists thought, especially those whose business it is to look at the issue of climate change?
As this ScienCentral News video reports, when looking at the big picture, this week's weather doesn't change anything.
Climate change and weather
If you were one of the people who had to deal with record-breaking snowfall that buried the eastern United States over the President's Day weekend, you might have wondered just how all this might fit in with the issue of global climate change.
The answer, according to climatologists who study the issue, is that very little, if anything about climate change can be read into a single storm, or even a year's worth of weather.
Linda Mearns, senior scientist at the National Center for Atmospheric Research (NCAR), says, "I think the kind of snowstorm we're seeing in the east—there's nothing inconsistent about that kind of storm and a progressive warming of the climate."
Also, when you talk about weather in one place, it might not match the weather elsewhere. For example, the western part of the United States is experiencing the latest in a series of drought years. The President's Day storm dumped more snow on many parts of the east coast than had fallen on Denver during the entire winter up to that time. The day Washington D.C. was digging out of snow, The Denver Post carried stories speculating about upcoming summertime water restrictions.
While 2003 threatens to be especially dry in the west, previous dry years are also contributing to the situation. Is several years worth of weather a sign of climate change?
Many climatologists, while watching the dry weather, know that the drought is important, but are still hesitant to point to even a number of years of unusual weather as climate change.
Robert Harriss, Director of the Environmental and Societal Impact Group at NCAR, says, "The droughts that we're experiencing over the last decade, certainly in my opinion, still fall in the category of climate variability. There is the possibility that this is some emerging impact of climate change, but we haven't been able to come up with a convincing argument for it to be climate change."
But some scientists look at the storm as a good test for the computers and the programs used to forecast climate change. Professor Andrew P. Dobson of Princeton University, who has published extensively on the impact of climate change on disease, looked at the warnings that preceded the storm as validation for the computerized models. "The models that were used to predict that snowfall were exactly the same models that are used to predict climate change," he points out. "So, if you believe there's snow on the east coast, and [you] adjusted your airline travel accordingly, then you should also believe in global warming because it's essentially the same underlying process."
He also notes that "99.9 percent" of scientists worldwide now agree that the planet is undergoing some sort of climate change. Additionally, in 2001 the Intergovernmental Panel on Climate Change linked human activity—most notably, "the use of fossil fuels and chemical fertilizers"—to climate change.
This does not mean that the scientific community is unanimous about the details involving climate change, something Mearns called the "uncertainty factor." But she urged people to not discount the general findings, just because the scientific community is unable at this time to pin down specific temperature changes, exact dates, and detailed impacts.
by Jack Penland
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The Transylvania Times -
Health Trust Launching Fund To Aid Nonprofits, Others Apply For Grants
The Dogwood Health Trust is launching a leverage fund to help Western North Carolina nonprofits, government agencies or collaborations apply for and receive state or national funding from outside the region to support the work of improving health and wellness within the region.
Through the leverage fund, Dogwood Health Trust underwrites the services of professional, experienced grant writers to help individual organizations or collaboratives pursue funding from public and private sources at national and state levels. The leverage fund helps match nonprofits with the right grant writer and supports other costs associated with preparing a grant application, such as travel stipends or meeting costs, according to a press release.
The leverage fund supports efforts to apply for $100,000 or more that will ultimately improve health outcomes for residents, the release said.
"There are billions of dollars available from federal and state funds, as well as from large national foundations," said Antony Chiang, chief executive officer of the Dogwood Health Trust. "However, Western North Carolina has often been overlooked or underrepresented when it comes to attracting those funds to our region. We believe Western North Carolina is home to a host of creative, innovative problem-solvers whose daily demands make it difficult to pay attention to national funding opportunities.
"Further, securing national grants can be a complex and cumbersome process.
"The leverage fund will provide organizations and collaborations in our region with the capacity and technical assistance to overcome those barriers."
To provide more information on the leverage fund, the Dogwood Health Trust will be cosponsoring informational community meetings, including one at the Transylvania County Library at 10 a.m. on Friday, across the region and hosting a webinar on Nov. 20.
For more information email, s.jarvis@pisgahhealthfoundation.org.
The Dogwood Health Trust is a North Carolina nonprofit corporation with the sole purpose of dramatically improving the health and well-being of all people and communities of Western North Carolina, the release said. The Dogwood Health Trust became operational upon the sale of Mission Health's assets to HCA Healthcare and is the recipient of the net proceeds of the sale.
To learn more, visit http://www.dht.org.
'No Dollar In The Holler' Effort Visits Board Again – Transylvania County, NC
New Details Lead To Dropped Charges – Transylvania County, NC
UWTC's Demise Forcing Funding Changes – Transylvania County, NC
Blue Zones Survey Sent Out To Brevard Residents – Brevard, NC
Academy Reviewing Its Curriculum – Brevard, NC
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text:
\section{Introduction}
\label{introduction}
Experimental observations suggest that the Heavy Ion Collisions (HICs) create a novel state of matter consisting of deconfined light quarks and gluons, called the Quark Gluon Plasma (QGP). There are a plethora of theoretical studies investigating various properties
of QGP governed by Quantum Chromo Dynamics(QCD) at high temperature(T). More recent developments indicate production of
a strong magnetic field, with an initial strength few times  the pion mass square, i.e., $eB \sim m_{\pi}^2$ at RHIC  and few tens of pion mass square i.e., $\sim 15 m_{\pi}^2$ at LHC  in a non-central HIC~\cite{ Kharzeev:2007jp,IJMPA24, PRC83, PRC85, PLB710, AHEP2014}. The existence of such a magnetic
field opens up new directions towards the theoretical studies of properties of QGP leading to diverse experimental consequences. The imprint, the magnetic field lays on QGP brings in some of the most important theoretical studies including Chiral Magnetic Effect(CME)
\cite{NPA803, PRD78}, Chiral Magnetic Wave leading to a charge-dependent elliptic flow~\cite{PRD83, JHEP01, PRL107, PRD83Miranssky},
transport properties of QGP in magnetic field~\cite{PRDFeng,PRLfukushima,Kurian:2018dbn}, quarkonia suppression~\cite{EPJC77,PRD97096011}, dilepton and photon production~\cite{Bandyopadhyay:2016fyd,Tuchin:2010gx}, HQ drag and diffusion coefficients~\cite{fukushima,santosh}, jet quenching~\cite{Li:2016bbh} etc. To what extent the magnetic field embosses its influence on the deconfined medium depends on several salient properties of the
HICs at the early stage. Despite high initial strength, the magnetic field eventually decays at a significant rate.

The initial dynamics of the system of deconfined matter plays a pivotal role in deciding the longevity of the magnetic
field at the later stages. It can be hypothesized that the magnetic field can induce an amount of
electrical conductivity which becomes adequate enough for the decaying magnetic field to persist~\cite{Tuchin:2010gx,AHEP2013, PRD92, PRD92Mamo}. This in turn can induce a current which opposes the rate of decrease of the magnetic field as per Lenz's Law~\cite{PRC88Tuchin, PRC93Tuchin, NPA929}. Therefore, it is imperative that the external magnetic field persists long enough so that it can impart crucial effects on various properties of the medium. It is instructive to investigate the effect of this field on the space-time evolution of the medium. Some studies  are trying to see this effect by reconstructing Hydrodynamic evolution in presence of magnetic field, i.e. Magneto hydrodynamics ~\cite{NPA_MHD, EPJC_MHD, PRC_MHD}.

Consequently, it is of utmost importance to explore to what extent the magnetic field inside QGP affects different observables of the deconfined matter. To this end, surveying the in-medium properties of heavy quarks~\cite{PRD88, PRD89, PRD93Hattori} and quarkonia have become
quite relevant in the context of magnetic field~\cite{PRC84044908, PRD88105017, PRL113, PLB751, PRD92054014, PRD91066001, JHEP01052, EPJC77, PRD97096011, EPJC78spin,CS:2018mag,CS:2018jql,Reddy:2017pqp}. However, since the HQs are moving in real time inside the QGP, understanding and estimating the dynamical properties
of HQs are also necessary. In this context, transport coefficients like drag and diffusion of HQ have been estimated in presence of
a strong external magnetic field in some of the recent literatures~\cite{santosh,fukushima}. AdS/CFT has also been employed to have an estimation of the drag
force of HQ\cite{AdSCFT}. Most of the calculations with strong magnetic field have been performed using perturbative QCD (pQCD) techniques in Leading Order (LO) of the strong coupling $\alpha_s$ in the limit $M\gg \sqrt{eB}$ so that the motion of the HQ is not directly affected by the external magnetic field. Nonetheless, the light quarks/anti-quarks are  affected by the magnetic field with the gluons remaining unaffected. The thermally equilibrated light quarks are Landau quantized. The magnetic field also affects the gluon self energy through the quark loop. Further they also affect the HQ light thermal parton scattering cross-sections.

It is well known that a high energy particle created in the initial stages of the heavy ion collision loses its energy in the medium by interacting with the medium partons. This leads to the phenomena of jet quenching, which as anticipated many years ago, is one of the prominent probes of QGP~\cite{Bjorken:Fermi,Xie:2019oxg,Han:2017nfz,Arleo:2017ntr}; for a recent review see Ref.\cite{Cao:2020wlm}. Generally, there are two types of processes that contribute to the energy loss namely; radiative process~\cite{Abir:2012pu} and collisional process. Experimental results for the quenching of heavy flavor suggested that radiative energy loss is not sufficient and one needs to take the collisional energy loss into account~\cite{Bielcik:2005wu}, for a recent review see~\cite{Wicks:2005gt}. In the present investigation, we focus our attention to the collisional energy loss of HQ in the background of a constant magnetic field which may be relevant for the HICs  as predicted in literature~\cite{Wicks:2005gt,Connors:2017ptx,Rohrmoser:2018fkf,Edmonds:2016gys,Coci:2017lhx}.

The present investigation intends to estimate the HQ collisional energy loss ($-dE/dx$) in the low coupling regime and strong magnetic field.  Specifically we will consider the hierarchy in the scales i.e., $\sqrt{\alpha_s eB}\ll T\ll \sqrt{eB}$ and $\sqrt{eB}\ll M$. To do so, we first calculate the resummed gluon propagator in the strong magnetic field  background. Let us note here that only quark loop contributes to the resummed gluon propagator in the limit $eB\gg T^2$. This resummed propagator is used to estimate the collisional energy loss. In this hierarchy of scales, two types of processes contribute to the scatterings of HQ with the light thermal partons  affecting the HQ energy loss. As we shall see, the collisional energy loss increases with the magnetic field and for a given magnetic field, the field dependent contribution to the collisional energy loss could be similar in magnitude to the collisional energy loss in the absence of magnetic field. This can be important for the jet quenching in the heavy ion collisions.

The paper is organized as follows. In Sec.\ref{setup}, we standardize the mathematical notations, followed by a brief description of the real-time formalism. Further, in the real-time formalism of thermal field theory, we also discuss the fermion propagator in Sec.\ref{fprop} and resummed gluon propagator in Sec.\ref{gluonresummed} in LLL approximation . In Sec.\ref{formalism}, we discuss the
formalism to calculate HQ energy loss i.e., $-dE/dx$ with descriptions of both the cases; (a) when HQ is interacting with light quarks (Sec.\ref{HQq}) and (b) HQ scattering with the thermal gluons (Sec.\ref{HQg}). Our findings are presented in Sec.\ref{results} with the relevant plots and the possible
explanation. In Sec.\ref{summary}, we summarize the present findings and discuss the possible outlook. In the appendices, we present the detailed calculations of the gluon self energy in a magnetized thermal medium and scattering amplitudes for the process $Q  g\rightarrow Q g$.

\section{Set-up}
\label{setup}
For the present investigation, we assume here that the magnetic field is constant and is along the $\hat{z}$ direction i.e., $\vec{B}=B\hat{z}$. In the subsequent subsection, we shall discuss the quark propagator in the real-time formalism of thermal field theory and in the presence of such a magnetic field. For this purpose we use the following notation. The notations $\parallel$ and $\perp$ represents the components parallel and perpendicular to the magnetic field of the corresponding quantities. For the metric tensor, we use
\begin{equation}
g_{\mu \nu}^{\parallel}=(1,0,0,-1)  \hspace{1cm} g_{\mu \nu}^{\perp}=(0,-1,-1,0).
\end{equation}
The parallel (i.e., $a_{\mu}^{\parallel}=g_{\mu \nu}^{\parallel}a^{\nu}$) and perpendicular (i.e., $a_{\mu}^{\perp}=g_{\mu \nu}^{\perp}a^{\nu}$) components of a four-vector $a_{\mu}$ are represented as
\begin{equation}
a_{\mu}^{\parallel}=(a_{0},0,0,-a_{3}) \hspace{1cm} a_{\mu}^{\perp}=(0,-a_{1},-a_{2},0).
\end{equation}
The four-vector product ($a^{\mu}b_{\mu}=a\cdot b$) can be written as
\begin{equation}
a \cdot b=a_{\parallel}\cdot b_{\parallel}-a_{\perp}\cdot b_{\perp}.
\end{equation}
Similarly, both the components of square of a four-vector is
\begin{equation}
a^2_{\parallel}=a_{0}^2-a_{3}^2  \hspace{1cm} a^2_{\perp}=a_{1}^2+a_{2}^2.
\end{equation}
\section{Real-time formalism}
\label{realtime}
In this section, we first summarize the basic formulation for real time thermal field theory in the presence of magnetic field in a self contained manner that will be used to obtain the results on energy loss. In thermal field theories (TFT), due to the Kubo-Martin- Schwinger conditions, the time argument for the fields varies from 0 to $-i\beta$, $\beta$ being the inverse of the temperature. In the real time formulation of TFT, the time contour can be deformed to go from $t=0$ to $t=\infty$ infinitesimally above the real axis and then back to $t=-i\beta$ below the real axis leading to a $2\times 2$ matrix structure for the propagator. Corresponding to different structure for the propagation along the contour, one can define the following four functions e.g. for fermionic fields $\psi_\alpha(x)$, ($\alpha$=1,2,3,4) as
\begin{equation}
S^{>}(X,Y)_{\alpha \beta}=-i\langle \psi_{\alpha}(X)\bar{\psi}_{\beta}(Y)\rangle
\end{equation}
\begin{equation}
S^{<}(X,Y)_{\alpha \beta}=i\langle\bar{\psi}_{\beta}(Y)\psi_{\alpha}(X) \rangle
\end{equation}
\begin{equation}
S(X,Y)_{\alpha \beta}=-i\langle \hat{T}[\psi(X)_{\alpha} \bar{\psi}_{\beta}(Y)]\rangle
\end{equation}
\begin{equation}
S^{*}(X,Y)_{\alpha \beta}=-i\langle \hat{T}^{*}[\psi_{\alpha}(X)\bar{\psi}_{\beta}(Y)]\rangle
\end{equation}
Similarly, for bosonic fields i.e., $A^{\mu}(x)$ ($\mu=0,1,2,3$), one can define the following propagators
\begin{equation}
D^{>}_{\mu \nu}(X,Y)=-i\langle A_{\mu}(X)A_{\nu}(Y) \rangle
\end{equation}
\begin{equation}
D^{<}_{\mu \nu}(X,Y)=-i\langle A_{\nu}(Y)A_{\mu}(X) \rangle
\end{equation}
\begin{equation}
D_{\mu \nu}(X,Y)=-i \langle \hat{T}[A_{\mu}(X)A_{\nu}(Y)] \rangle
\end{equation}
\begin{equation}
D^{*}_{\mu \nu}(X,Y)=-i \langle \hat{T}^{*}[A_{\mu}(X)A_{\nu}(Y)] \rangle.
\end{equation}
In the above, $\hat{T}$ and $\hat{T^*}$ are the
time ordering and anti-time ordering operators respectively which are defined as
\begin{equation}
\hat{T}[A(X)B(Y)]=\theta(x_{0}-y_{0})A(X)B(Y)\pm \theta(y_{0}-x_{0})B(Y)A(X).
\end{equation}
\begin{equation}
\hat{T}^{*}[A(X)B(Y)]=\theta(y_{0}-x_{0})A(X)B(Y)\pm \theta(x_{0}-y_{0})B(Y)A(X).
\end{equation}
In terms of these four functions, the $2\times 2$ propagator is given by, e.g. for fermion fields
\begin{equation}
S=\begin{pmatrix}
S_{11} & S_{12}\\
S_{21} &S_{22}
\end{pmatrix}\equiv\begin{pmatrix} S &S^<\\ S^> &S^*\end{pmatrix}.
\end{equation}
The $11$ component corresponds to the conventional time ordered propagator while the $22$ component corresponds to anti time ordering as the contour ordering along the contour below the real axis is conversely ordered in time. The off diagonal components correspond to the Wightman propagators. From the definitions of the functions, it is clear that all the four functions are not independent and are related by
\begin{equation}
S_{11}+S_{22}=S_{12}+S_{21}.
\end{equation}
In the Keldysh representation, a linear combination of above causal propagators is used to define the retarded ($S_{R}$), advanced ($S_{A}$) and Feynman ($S_{F}$) propagators as
\begin{equation}
S_R=S_{11}-S_{12},\quad S_A=S_{11}-S_{21},\quad S_F=S_{11}+S_{22}.
\label{allprop}
\end{equation}
Feynman propagator  can also be obtained from the advanced and the retarded propagators as
\begin{equation}
S_{F}(K)=\bigg(\frac{1}{2}-\tilde{f}(k_0)\bigg)[S_{R}(K)-S_{A}(K)]
\end{equation}
where $\tilde{f}(k_0)$ is the distribution function of the fermions. One can invert the relations of Eq.(\ref{allprop}) to re-write the propagators in the Keldysh basis in terms of those in the RA basis as
\begin{equation}
S_{11}(K)=\frac{1}{2}\bigg(S_{F}(K)+S_{A}(K)+S_{R}(K)\bigg),
\label{s11}
\end{equation}
\begin{equation}
S_{12}(K)=\frac{1}{2}\bigg(S_{F}(K)+S_{A}(K)-S_{R}(K)\bigg),
\label{s12}
\end{equation}
\begin{equation}
S_{21}(K)=\frac{1}{2}\bigg(S_{F}(K)-S_{A}(K)+S_{R}(K)\bigg),
\label{s21}
\end{equation}
\begin{equation}
S_{22}(K)=\frac{1}{2}\bigg(S_{F}(K)-S_{A}(K)-S_{R}(K)\bigg).
\label{s22}
\end{equation}
It may be mentioned that these formulations can also be applied to nearly equilibrium systems. For thermal equilibrium systems $\tilde{f}(k_0)$ becomes the Fermi-Dirac distribution function. In the next section we discuss the quark propagator in the background of magnetic field.
\subsection{Fermion propagator in LLL}
\label{fprop}
The retarded and advanced propagators ($S_{R},S_{A}$) of a free quark of electric charge $q_f$ and mass $m$ in the presence of magnetic field $B$  can be given as~\cite{Fukushima:2019ugr}
\begin{equation}
S_{R/A}(K)=\sum_{n=0}^{\infty}\bigg[\frac{i\Xi_{n}(K)}{K^2-m^2}\bigg]_{k_{0}\rightarrow k_0\pm i\epsilon}
\label{propq}
\end{equation}
where the retarded ($R$)/advanced ($A$) corresponds to $+i\epsilon/-i\epsilon$. The sum is over all the Landau levels (LLs) that is represented by $n$. Four momentum squared $K^2=k_{0}^2-k_{z}^2-2n|q_fB|$. All LLs except the lowest ($n=0$) are doubly degenerate. The numerator of Eq.(\ref{propq}) has the Dirac structure as~\cite{Fukushima:2019ugr}
\begin{equation}
\Xi_n(K)=(\slashed{k}_{\parallel}+m)[\mathcal{P}_{+}\Theta_{n}(\zeta)+\mathcal{P}_{-}\Theta_{n-1}(\zeta)]+\slashed{k}_{\perp}\Phi_{n-1}(\zeta)
\end{equation}
where $\zeta=2 \frac{k_{\perp}^2}{|q_fB|}$ and
\begin{equation}
\Theta_{l}(\zeta)=2 e^{-\frac{\zeta}{2}}(-1)^{l}L_{l}(\zeta)
\label{theta}
\end{equation}
\begin{equation}
\Phi_{l}(\zeta)=4 e^{-\frac{\zeta}{2}}(-1)^{l-1}L^{1}_{l-1}(\zeta).
\label{phi}
\end{equation}
$L_{l}(\zeta)$ and $L_{l}^{1}(\zeta)$ are associated Laguerre Polynomials. In Eq.(\ref{propq}), the projection operator ($\mathcal{P}_{\pm}$) that projects the spin in the direction of magnetic field is defined as $\mathcal{P}_{\pm}=(1\pm sgn(q_fB)i\gamma^{1}\gamma^{2})/2$. Note, here that the projection operator depends on the electric charge of the quark as the spin magnetic moment depends on the charge of the quark.  As mentioned earlier,  we take the strong magnetic field limit so that only LLL is relevant. We shall further assume that, at finite temperature $\sqrt{eB}\gg T,m$  so that the dynamics of light quark is governed by the magnetic field. In the LLL approximation the associated Laguerre Polynomials $L_{-1}(\zeta)=0$ and $L_{0}(\zeta)=1$, so that Eq.(\ref{propq}) reduces to
\begin{equation}
S_{R/A}(K)=i\exp\bigg({-\frac{k_{\perp}^2}{|q_fB|}}\bigg)\frac{2 (\slashed{k}_{\parallel}+m)\mathcal{P}_{+}}{k_{\parallel}^2-m^2\pm i\epsilon k_{0}},
\label{propLLL}
\end{equation}
and the Feynman propagator can be obtained from
\begin{eqnarray}
S_F(K)&=&\bigg(\frac{1}{2}-\tilde{f}(k_0)\bigg)\bigg[S_{R}(K)-S_{A}(K)\bigg]\nonumber\\
&\equiv&\bigg(\frac{1}{2}-\tilde{f}(k_0)\bigg)\rho_{F}(K).
\label{propfn}
\end{eqnarray}
where $\tilde{f}(k_0)$ is Fermi-Dirac distribution function and $\rho_{F}(K)$ is quark spectral density. From Eq.(\ref{propLLL}), it is clear that in the limit $k_{\perp}^2\ll q_fB$, the motion of a quark is restricted in the transverse directions and allowed only in the direction parallel to the magnetic field. It can also be observed that for the infrared limit i.e., $m^2,T^2,k_{\perp}^2,k_{0}^2\ll eB$, the dimensional reduction from 3+1-dimension to 1+1-dimension takes place. This dimensional reduction in the LLL approximation suggests that the pairing dynamics of quarks occurs in 1+1 dimension and spontaneous chiral symmetry breaking occurs even at weak interaction between quarks in 3+1 dimension~\cite{Gusynin:1995nb}.
\subsection{Resummed gluon propagator in LLL}
\label{gluonresummed}
Next we consider the resummed retarded/advanced gluon propagator in the presence of magnetic field within the LLL approximation. The resummed propagator is obtained by inserting the self energy corrections in the bare propagator and can be written as
\begin{equation}
D^{R/A}_{\mu \nu}(K)=[(D^{R/A}_{\mu \nu}(K))_{0}^{-1}+\Pi^{R/A}(K)_{\mu \nu}]^{-1}
\label{rprop}
\end{equation}
where the bare gluon propagator $(D^{R/A}_{\mu \nu}(K))_{0}$ in covariant gauge is given as
\begin{equation}
(D^{R/A}_{\mu \nu}(K))_{0}=-\frac{P_{\mu \nu}(K)}{(k_0\pm i\epsilon)^2-k^2}+\xi \frac{K_{\mu}K_{\nu}}{((k_0\pm i\epsilon)^2-k^2)^2}.
\label{bareg}
\end{equation}
In Eq.(\ref{bareg}), the projection operator ($P_{\mu \nu}(K)$ is defined as
\begin{equation}
P_{\mu \nu}(K)=-g_{\mu \nu}+\frac{K_{\mu} K_{\nu}}{(k_0\pm i\epsilon)^2-k^2}
\end{equation}
and $\xi$ is the gauge parameter. The retarded/advanced resummed gluon propagator depends on the retarded/advanced gluon self energy that, in general, can get contribution from both the gluon loop and the quark loop. The magnetic field does not affect the contribution from the gluon loop. However, the magnetic field modifies the quark loop contribution. The leading contribution from the gluon loop to the self energy at finite temperature $T$ is proportional to $g^2T^2$ while we shall see that the leading contribution from the quark loop at finite $T,B$ is proportional to $g^2 |q_fB|$ as given in Eq.(\ref{pir1}). Since we work in the limit $eB\gg T^2,m^2$, we shall drop the gluon loop contribution and keep quark loop contribution in the retarded/advanced gluon self energies. Taking the magnetic field in the the $\hat{z}$ direction, the most general form of gluon self energy at finite $T$ and $B$ can be written in terms of seven independent tensors as
\begin{equation}
\Pi^{R/A}_{\mu \nu}(K)=\sum_{j=\parallel,\perp,T,L}\Pi^{R/A}(K)_{j}P^{j}_{\mu \nu}(K)+\Pi_{P}\frac{K_{\mu}K_{\nu}}{K^2}+\Pi_{n}n_{\mu}n_{\nu}+\Pi_{b}b_{\mu}b_{\nu}
\end{equation}
where $L,T,\parallel,\perp$ respectively are for longitudinal, transverse, parallel and perpendicular components of the gluon self energy. The four-vectors $n^{\mu}=(1,\textbf{0})$ and $b^{\mu}=(0,0,0,-1)$, break Lorentz and rotational symmetry due to thermal medium and the magnetic field respectively. Further, the projection operators are transverse to the momentum i.e., $P_{\mu}P^{\mu \nu}_j=0$. It turns out that only the four projection tensors contribute to the retarded/advanced self energies in the leading order. Explicitly these projection operators are given as
\begin{equation}
P^T_{\mu\nu}(K)=-g_{\mu\nu}+\frac{k_0}{k^2}\bigg[K_{\mu}n_{\nu}+n_{\mu}K_{\nu}\bigg]-\frac{1}{k^2}\bigg[K_{\mu}K_{\nu}+K^2n_{\mu}n_{\nu}\bigg],
\label{PT}
\end{equation}
\begin{equation}
P^L_{\mu\nu}(K)=-\frac{k_0}{k^2}\bigg[K_{\mu}n_{\nu}+n_{\mu}K_{\nu}\bigg]+\frac{1}{k^2}\bigg[\frac{k_0^2}{K^2} K_{\mu}K_{\nu}+K^2n_{\mu}n_{\nu}\bigg],
\label{PL}
\end{equation}
\begin{equation}
P^{\parallel}_{\mu \nu}(k)=-g^{\parallel}_{\mu \nu}+\frac{k_{\parallel}^{\mu} k_{\parallel}^{\nu}}{k_{\parallel}^2},
\end{equation}
\begin{equation}
P^{\perp}_{\mu \nu}(k)=-g^{\perp}_{\mu \nu}+\frac{k_{\perp}^{\mu} k_{\perp}^{\nu}}{k_{\perp}^2}.
\end{equation}
 The parallel and perpendicular components of self energy comes from the quark loop while the longitudinal and transverse components are from the gluon loop. Taking contribution from both quark and gluon loop the resummed retarded gluon propagator is given as~\cite{Hattori:2017xoo}
\begin{eqnarray}
D^{R}_{\mu\nu}(K)&=& -\frac{1}{\Delta(K)}\bigg[(K^2-\Pi^{\parallel}_R(K)-\Pi^{L}_R(K))P^{T}_{\mu\nu}(K)+(K^2-\Pi^{\parallel}_R(K)-\Pi^{L}_R(K))P^{L}_{\mu\nu}(K)\nonumber\\
&+&\Pi^{\parallel}_R(K)P^{\parallel}_{\mu\nu}(K)+D_{\perp}(K)P^{\perp}_{\mu\nu}(K)\bigg]+\xi\frac{K_{\mu} K_{\nu}}{(K^2)^2}
\label{gluonprpagator}
\end{eqnarray}
where
\begin{eqnarray}
\Delta(K)&=&(K^2-\Pi^{T}_R(K))(K^2-\Pi^L_R(K))-\Pi^{\parallel}_R(K)\bigg[K^2-a\Pi^T_R(K)\frac{K^2}{k^2_{\parallel}}\nonumber\\
&-&\Pi^L_R(K)(1-a)\frac{k_0^2}{k^2_{\parallel}}\bigg],
\label{delta}
\end{eqnarray}
and
\begin{eqnarray}
D_{\perp}(K)&=&\frac{1}{K^2-\Pi^T_R(K)-\Pi^L_R(K)}\bigg[\Pi_{\parallel}(K)(\Pi^L_R(K)-\Pi^T_R(K))(1-a) \frac{k_0^2}{k^2_{\parallel}}\nonumber\\
&+& \Pi^{\perp}_R(K)(K^2-\Pi^L_R(K)-\Pi^{\parallel}_R(K))\bigg]
\end{eqnarray}
and $a=k_3^2/k^2$. In the LLL approximation for the propagator as in Eq.(\ref{propLLL}) lead to $\Pi_{\perp}=0$. This is due to the fact that  there is no current in the transverse direction in LLL approximation. All the expressions for the gluon self energies ($\Pi_L, \Pi_T$ and $\Pi_{\parallel}$ ) are explicitly given in the appendix(\ref{selfEn}). As may be noted, the gluon loop contribution to the gluon self energy is of the order of $\alpha_s T^2$ which can also be dropped with respect to the quark loop contribution which is of the order of $\alpha_seB$.  In this approximation, the resummed retarded gluon propagator becomes
\begin{equation}
D^R_{\mu\nu}(K)=-\frac{1}{\Delta}[(K^2-\Pi^{\parallel}_R)(P^{T}_{\mu\nu}+P^{L}_{\mu\nu})+\Pi^{\parallel}_RP^{\parallel}_{\mu\nu}+
D_{\perp}(K)P^{\perp}_{\mu\nu}]
+\xi\frac{K_{\mu}K_{\nu}}{(K^2)^2}
\label{propg}
\end{equation}
where
\begin{equation}
P^L_{\mu\nu}+P^T_{\mu\nu}=-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^2}.
\end{equation}
Since $D_{\perp}(K)\sim \alpha_s^2 eB T^2\ll \Pi_{\parallel}\sim \alpha_s eB$, the third term in Eq.(\ref{propg}) may be dropped. Further as may be  observed from Eq.(\ref{delta}) $\Delta\approx K^2(K^2-\Pi_{\parallel}(K))$ with similar argument. Thus Eq.(\ref{propg}), taking appropriate $i\epsilon$ factors into account, can be approximated as
\begin{equation}
D^{R}_{\mu \nu}(K)=-\frac{P_{\mu \nu}(K)}{(k_0+i\epsilon)^2-k^2}-\frac{\Pi^{\parallel}_{R}(k_0+i\epsilon,\textbf{k})P^{\parallel}_{\mu \nu}(K)}{K^2(K^2-\Pi^{\parallel}_{R}(k_{0}+i\epsilon,\textbf{k}))}+\xi \frac{K_{\mu}K_{\nu}}{((k_0+i\epsilon)^2-k^2)^2}.
\label{rprop1}
\end{equation}
Similarly, the resummed advanced and Feynman gluon propagators can be written as
\begin{equation}
D^{A}_{\mu \nu}(K)=-\frac{P_{\mu \nu}(K)}{(k_0+i\epsilon)^2-k^2}-\frac{\Pi^{\parallel}_{A}(k_0-i\epsilon,\textbf{k})P^{\parallel}_{\mu \nu}(K)}{K^2(K^2-\Pi^{\parallel}_{A}(k_{0}-i\epsilon,\textbf{k}))}+\xi \frac{K_{\mu}K_{\nu}}{((k_0-i\epsilon)^2-k^2)^2},
\label{adprop}
\end{equation}
\begin{equation}
D^{F}_{\mu \nu}(K)=(1+2f(k_0))\bigg[D^R_{\mu \nu}(K)-D^A_{\mu \nu}(K)\bigg].
\label{feynprop}
\end{equation}
We shall use Eqs.(\ref{rprop1}), (\ref{adprop}) and (\ref{feynprop}) to estimate the HQ energy loss.
\section{Formalism}
\label{formalism}
In this section, we shall calculate the energy loss of HQ of mass $M$, moving in a magnetized thermal medium of light quarks/anti-quarks and gluons, with momentum $\textbf{p}$ and energy $E=\sqrt{\textbf{p}^2+M^2}$. The rate of energy loss of a heavy fermion moving with velocity $\textbf{v}$ in a thermal medium  has been considered earlier in Ref.~\cite{Braaten:1991jj,Peigne:2007sd} for QED plasma and for QGP in Ref.~\cite{Peigne:2008nd}. In general the energy loss of the heavy fermion is given by
\begin{equation}
-\frac{dE}{dx}=\frac{1}{v}\int_{M}^{\infty}dE' (E-E')\frac{d\Gamma}{dE'}
\label{eloss}
\end{equation}
where $\Gamma$ is the interaction rate of heavy fermion with the medium particles.  For the collisional energy loss, we confine our attention to $2\rightarrow 2$ processes only as has been considered earlier~\cite{Peigne:2008nd} . For large momentum of HQ, however, the energy loss by radiative processes will also be important. In general, for the case of HQ moving in a QGP medium of light partons there can be two types of scatterings namely the Coulomb scattering i.e., $Qq\rightarrow Qq$ and Compton scattering i.e., $ Qg\rightarrow Qg$. As mentioned, we shall consider the limit $M\gg \sqrt{eB}\gg T$ so that HQ is not directly affected by the magnetic field and the light quarks are populated only in LLL. In the following sections we shall calculate the interaction rate and subsequently estimate the energy loss of the HQ.
\subsection{Energy loss due to scattering with light quark: $ Q q\rightarrow  Q q$}
\label{HQq}
The interaction rate $\Gamma$ of the HQ is related to imaginary part of its retarded self energy in the medium~\cite{Thoma:2000dc}
\begin{equation}
\Gamma(E)=-\frac{1}{2E}(1-\tilde{f}(E))Tr\bigg[(\slashed{P}+M)\Im\Sigma_{R}(P)\bigg]
\label{interaction}
\end{equation}
where $\tilde{f}(E)$ is Fermi-Dirac distribution function and $ \Sigma_R(P)$ is the retarded self energy of HQ as shown in Fig.(\ref{Qq}).

\begin{figure}[h]
\centering
\includegraphics[width=.41\textwidth,origin=c,angle=0]{hq.pdf}
\caption{\label{Qq} Feynman diagram for heavy quark self energy. The bold solid line refers to HQ while the curly line corresponds to gluon. The black blob here shows the gluon resummed propagator in the presence of a magnetic field background. In the LLL, the contribution to the resummed gluon propagator comes from the quark loop only.}
\end{figure}

In the RTF, similar to the retarded propagator Eq.(\ref{allprop}), the retarded self energy can be written in terms of $11$ and $12$ components of self energy as
\begin{equation}
\Sigma_{R}(P)=\Sigma_{11}(P)-\Sigma_{12}(P).
\end{equation}
Using the propagators for quark and gluon in the Keldysh basis the retarded self energy of HQ can be written as
\begin{equation}
\Sigma_{R}(P)=ig^2 (t_a t_b)\int \frac{d^4K}{(2\pi)^4}\bigg[D_{11}^{\mu\nu}(K)\gamma_{\nu}S_{11}(P')\gamma_{\mu}-D_{12}^{\mu\nu}(K)\gamma_{\nu}S_{12}(P')\gamma_{\mu}\bigg].
\label{selfR}
\end{equation}
The HQ propagators are $S_{11}(K)=(\slashed{K}+M)\Delta_{11}(K)$ and $S_{12}(K)=(\slashed{K}+M)\Delta_{12}(K)$ where $\Delta_{11}(K)$ and $\Delta_{12}(K)$ are given as
\begin{equation}
\Delta_{11}(K)=\frac{1}{K^2-M^2+i\epsilon}+2\pi i\tilde{f}(k_0) \delta(K^2-M^2)
\label{delta11}
\end{equation}
and
\begin{equation}
\Delta_{12}(K)=2\pi i\tilde{f}({k_0})\delta(K^2-M^2).
\label{delta12}
\end{equation}
Here, $\tilde{f}({k_0})$ is the fermion distribution function of the HQ. With these simplifications, Eq.(\ref{selfR}) becomes
\begin{eqnarray}
\Sigma_{R}(P)&=&ig^2 (t_a t_b)\int \frac{d^4K}{(2\pi)^4}\bigg[D_{11}^{\mu\nu}(K)\gamma_{\nu}(\slashed{P'}+M)\gamma_{\mu}\Delta_{11}(P')\nonumber\\
&-&D_{12}^{\mu\nu}(K)\gamma_{\nu}(\slashed{P'}+M)\gamma_{\mu}\Delta_{12}(P')\gamma_{\mu}\bigg].
\label{selfR1}
\end{eqnarray}
It is easier for the evaluation of the retarded self energy to convert the Keldysh propagators to the RA basis using the relations as given in Eqs.(\ref{s11}) and (\ref{s12}) for the propagators $D^{\mu \nu}_{ab}$ and $\Delta_{ab}$. Let us note that in Eq.(\ref{selfR1}), the propagators $D_{11}^{\mu \nu}$ and $D_{12}^{\mu \nu}$ are resummed gluon propagators and $S_{11},S_{12}$ for the HQ and hence the corresponding $\Delta_{11},\Delta_{12}$ are bare propagators. Eq.(\ref{selfR1}) for the retarded self energy then can be rewritten as
\begin{eqnarray}
\Sigma_R(P)&=&\frac{ig^2 (t_a t_b)}{8}\int \frac{d^4K}{(2\pi)^4}\bigg[D^{\mu\nu}_{F}(K)\lambda_{\mu\nu}\Delta_{R}(P')+D^{\mu\nu}_{A}(K)\lambda_{\mu\nu}\Delta_{R}(P')\nonumber\\
&+&D^{\mu\nu}_{R}(K)\lambda_{\mu\nu}\Delta_{F}(P')+D^{\mu\nu}_{R}(K)\lambda_{\mu\nu}\Delta_{A}(P')\bigg],
\label{sigmar}
\end{eqnarray}
where $\lambda_{\mu\nu}=\gamma_{\mu}(\slashed{P'}+M)\gamma_{\nu}$. The leading contribution to the self energy comes from the gluon propagator arising in the soft momentum transfer  limit i.e., $|\textbf{k}|\sim g\sqrt{eB}$. In the limit $E\gg\sqrt{eB}\gg T\gg |\textbf{k}|$, out of the four terms in Eq.(\ref{sigmar}) only the first term becomes dominant. This is because $D_F^{\mu \nu}$ involves gluonic distribution function. Thus Eq.(\ref{sigmar}) reduces to
\begin{equation}
\Sigma_{R}(P)=\frac{ig^2 (t_a t_b)}{8}\int \frac{d^4K}{(2\pi)^4}\bigg[D_F^{00}\gamma_{0}(\slashed{P'}+M)\gamma_{0}+2 D_F^{0i}\gamma_{0}(\slashed{P'}+M)\gamma_{i}+D_F^{ij}\gamma_{i}(\slashed{P'}+M)\gamma_{j}\bigg]\Delta_{R}(P').
\label{selfE}
\end{equation}
Using Eqs.(\ref{rprop1}),(\ref{adprop}) and (\ref{feynprop}) in the Eqs.(\ref{interaction}) and (\ref{sigmar}), the interaction rate of HQ can be given as
\begin{eqnarray}
\Gamma(E)&=&\frac{g^2}{8 E}(1-\tilde{f}(E))\int\frac{d^4K}{(2\pi)^4}(1+2f(k_0))Tr\bigg[(\slashed{P}+M)\gamma_{0}(\slashed{P'}+M)\gamma_{0}P_{\parallel}^{00}\nonumber\\
&+&2(\slashed{P}+M)\gamma_{0}(\slashed{P'}+M)\gamma_{i}P_{\parallel}^{0j}+(\slashed{P}+M)\gamma_{i}(\slashed{P'}+M)\gamma_{j}P_{\parallel}^{ij} \bigg]\rho_L\Im(\Delta_R(P'))
\end{eqnarray}
where
\begin{equation}
\rho_L=\frac{2 \Im \Pi_{R}^{\parallel}(K)}{(K^2-\Re\Pi_{R}^{\parallel}(K))^2+(\Im \Pi_{R}^{\parallel}(K))^2}.
\label{rho}
\end{equation}
In the above equation, $\Im \Pi^{R}_{\parallel}(K)$ and $\Re\Pi^{R}_{\parallel}(K)$ respectively are the imaginary and the real parts of retarded gluon self energy as defined in Eqs.(\ref{pir1}) and (\ref{a13}). Further, let us note that the imaginary part in the retarded self energy of HQ comes from the propagator $\Delta_{R}(P')$ which can be written as
\begin{equation}
\Im(\Delta(P'))=-\frac{\pi}{2E'}\bigg(\delta(E-k_0-E')+\delta(E-k_0+E')\bigg).
\label{imgdelta}
\end{equation}
The second delta function in Eq.(\ref{imgdelta}) does not contribution due to kinematic reasons ($k_0\ll M$). Therefore, we drop this term and continue with the first delta function. Thus, the interaction rate becomes
\begin{eqnarray}
\Gamma(E)&=&-\frac{g^2}{8 E}(1-\tilde{f}(E))\int \frac{d^4K}{(2\pi)^4}(1+2f(k_0))\bigg[(4EE'+4p^2-4\textbf{p}\cdot \textbf{k}+4M^2)k_z^2+8(E(\textbf{p}'\cdot \textbf{k})\nonumber \\
&+&E'(\textbf{p}\cdot \textbf{k}'))k_0+4(\textbf{p}\cdot(\textbf{p}-\textbf{k})+E E'-M^2)k_0^2\bigg]\frac{\rho_L\pi}{2E'k_{\parallel}^2}\delta(E-k_0-E').
\label{intrate}
\end{eqnarray}
Since we are working in the limit where $|\textbf{k}|\sim g\sqrt{eB}\ll T$ so in this limit $E\gg |\textbf{k}|$ and  one can write $E'=\sqrt{(\textbf{p}'^2+m^2)}\equiv E-\textbf{v}\cdot \textbf{k}$. The delta function in the above equation can be simplified as $\delta(E-k_0+E')\sim \delta(k_0-\textbf{v}\cdot \textbf{k})$. Further, we assume that HQ moves parallel to the magnetic field. The integration over azimuthal angle in Eq.(\ref{intrate}) can be done trivially and the integration over the polar angle can be done by using the energy delta function. The energy loss is thus obtained as
\begin{eqnarray}
-\frac{dE}{dx}\bigg|_{qQ\rightarrow qQ}&=&-\frac{g^2}{2 E}(1-\tilde{f}(E))\int_{0}^{\infty}\frac{k dk}{(2\pi)^2}\int_{-kv}^{kv} \frac{k_0 dk_0}{2\pi}(1+2f(k_0))\bigg[4(2 E^2-2E k_0)k_0^2 k_0\nonumber\\
&-&4v^2(2 E^2-E k^2)k_0+4v^2(2E^2-2 M^2-2 Ek_0)k_0^2\bigg]\frac{\rho_L\pi}{2E k_0^2(v^2-1)}.
\label{qeloss}
\end{eqnarray}
In the above equation we have used $1/E\approx 1/E'$ as the momentum of the gluon is soft. The magnetic field dependence in the energy loss for $Qq\rightarrow Qq$ scattering comes through $\rho_L$ as defined in Eq.(\ref{rho}) which depends on the real and the imaginary parts of retarded self energy of gluon.
\subsection{HQ gluon Scattering: $ Q g\rightarrow  Q g$}
\label{HQg}
The other contribution to the HQ energy loss comes from the scattering of the gluons off the the HQ i.e., $Q(P)+g(K)\rightarrow Q(P')+g(K')$ as shown in Fig.(\ref{Qg}). We shall discuss this process in some detail.
\begin{figure}[h]
\centering
\includegraphics[width=.28\textwidth,origin=c,angle=0]{gQ.pdf}
\caption{\label{Qg} Feynman diagram for heavy quark and gluon scattering. In the strong magnetic field limit the contribution to the resummed gluon propagator comes from the quark loop only. }
\end{figure}
The interaction rate is given as
\begin{eqnarray}
\Gamma(E)&=&\frac{1}{2E}\int\frac{d\textbf{p}'}{(2\pi)^32E'}\int\frac{d\textbf{k}}{(2\pi)^32k}f(k)
\int\frac{d\textbf{k}'}{(2\pi)^32k'}(1+f(k'))\nonumber\\
&\times&(2\pi)^4\delta^4(P+K-P'-K')|\mathcal{\bar{M}}|^2
\label{rateHQg}
\end{eqnarray}
where $|\mathcal{\bar{M}}|^2$ is the matrix element squared averaged over initial spin and color degrees of freedom and summed over final spin and color for the $Qg\rightarrow Qg$ scattering and $f(k)$ is the gluonic thermal distribution function. Generally, there are three types of processes i.e., through $s, t$ and $u$ channels that can contribute to the total scattering amplitude for this process. However, we are considering the strong magnetic field limit (LLL approximation) i.e., $M\gg \sqrt{eB}\gg T$ and assume that the effect of the  magnetic field on HQ is suppressed due to its large mass so that the $s$ and the $u$ channels, where HQ propagator arises, does not give any additional contribution arising from the magnetic field.
Therefore, the only contribution to Eq.(\ref{rateHQg}) comes from the $t$-channel scattering as shown in Fig.(\ref{Qg}). Let us note that as mentioned earlier the resummed gluon propagator has thermal contribution ($\sim \alpha_sT^2$) from gluon loop  and the magnetic field contribution ($\sim \alpha_s{eB}$) from quark loop in the gluon self energy. In LLL approximation, the thermal contributions are negligible as compared with that of the magnetic field. Consequently, we shall use the resummed gluon propagator as given in Eq.(\ref{rprop1}). The $t-$channel scattering amplitude can be written as
\begin{eqnarray}
\mathcal{M}&=&-ig^2f_{acb}t^a_{ji}[\bar{u}_j(P')\gamma^{\alpha}u_i(P)]D^{R}_{\delta\alpha}(Q)C^{\delta\mu\nu}(Q,K,-K')\epsilon_{\mu}(K)\epsilon^{*}_{\nu}(K')
\label{M}
\end{eqnarray}
where $Q=P-P'=K'-K$ is four-momentum vector for the exchanged gluon and we have kept explicitly the color indices as in Fig.(\ref{Qg}). The resulting  matrix element squared can be written after performing the color and spin sum over the final state and averaging over the initial state as
\begin{eqnarray}
|\mathcal{\bar{M}}|^2&=&\frac{1}{4}g^4C_{F}Tr[(\slashed{P'}+M)\gamma^{\alpha}(\slashed{P}+M)\gamma^{\alpha'}]D^R_{\delta\alpha}(Q)D^R_{\delta'\alpha'}(Q)
C^{\delta\mu\nu}(Q,K,K')\nonumber\\
&\times&C^{\delta'\mu'\nu'}(Q,K,K')
\sum\epsilon_{\mu}(K)\epsilon_{\mu'}(K)\sum\epsilon_{\nu}(K')\epsilon_{\nu'}(K')\nonumber\\
&=&\frac{1}{4}g^4C_{F}Tr[(\slashed{P'}+M)\gamma^{\alpha}(\slashed{P}+M)\gamma^{\alpha'}]D^R_{\delta\alpha}(Q)D^R_{\delta'\alpha'}(Q)\nonumber\\
&\times&C^{\delta\mu\nu}(Q,K,-K')C^{\delta'}_{\mu\nu}(Q,K,-K')
\label{Msquared}
\end{eqnarray}
where $C_F=1/2$ is the color factor and $C^{\mu \nu \alpha}(P,Q,R)=(P-Q)^{\alpha}g^{\mu \nu}+(Q-R)^{\mu}g^{\nu \alpha}+(R-P)^{\nu}g^{\mu \alpha}$. To further simplify Eq.(\ref{Msquared}), we will use the transversality condition $\epsilon(K)\cdot K=\epsilon(K')\cdot K'=0$ for the gluons to obtain
\begin{eqnarray}
C^{\delta\mu\nu}(Q,K,-K')&=&-2g^{\delta\mu}K^{\nu}+g^{\mu\nu}(K+K')^{\delta}-2g^{\nu\delta}K'^{\mu}\nonumber\\
C^{\delta'}_{\mu\nu}(Q,K,-K')&=&-2g^{\delta'}_{\mu}K_{\nu}+g_{\mu\nu}(K+K')^{\delta'}-2g^{\delta'}_{\nu}K'_{\mu}
\end{eqnarray}
so that,
\begin{equation}
C^{\delta\mu\nu}(Q,K,-K')C^{\delta'}_{\mu\nu}(Q,K,-K')=4(K^{\delta}K'^{\delta'}+K'^{\delta}K^{\delta'}).
\label{gluon vertex}
\end{equation}
The product of propagators that appear in Eq.(\ref{Msquared}) can be simplified to
\begin{eqnarray}
D^R_{\delta\alpha}(Q)D^R_{\delta'\alpha'}(Q)&=&\frac{g_{\delta\alpha}g_{\delta'\alpha'}}{Q^4}-\frac{\Pi^{\parallel}_R(Q)g_{\delta\alpha}P^{\parallel}_{\delta'\alpha'}+\Pi^{\parallel}_R(Q)g_{\delta'\alpha'}P^{\parallel}_{\delta\alpha}}{Q^4(Q^2-\Pi^{\parallel}_R(Q))}\nonumber\\
&+&\frac{(\Pi^{\parallel}_R(Q))^2P^{\parallel}_{\delta\alpha}P^{\parallel}_{\delta'\alpha'}}{Q^4(Q^2-\Pi^{\parallel}_R(Q))^2}.
\label{gluon propagator}
\end{eqnarray}
The first term in Eq.(\ref{gluon propagator}) corresponds to the vacuum contribution. Since we are interested in the medium contribution (i.e., $T$ and $B$), we will not consider this term. The medium dependent term that appears in Eq.(\ref{Msquared}) can be written in compact manner as
\begin{equation}
(D^R_{\delta\alpha}(q)D^R_{\delta'\alpha'}(q))(C^{\delta\mu\nu}(K,K')C^{\delta'}_{\mu\nu}(K,K'))=\mathcal{A}_{\alpha\alpha'}+\mathcal{B}_{\alpha\alpha'}+\mathcal{C}_{\alpha\alpha'}
\label{gluonvertexpropagator}
\end{equation}
where
\begin{eqnarray}
\mathcal{A}_{\alpha\alpha'}&=&-\frac{4\Pi^{\parallel}_{R}(q)}{q^4(q^2-\Pi^{\parallel}_{R}(q))}(K^{\delta}K'^{\delta'}+K'^{\delta}K^{\delta'})
g_{\delta\alpha}P^{\parallel}_{\delta'\alpha'}
\label{A1}
\end{eqnarray}
\begin{eqnarray}
\mathcal{B}_{\alpha\alpha'}&=&-\frac{4\Pi^{\parallel}_{R}(q)}{q^4(q^2-\Pi^{\parallel}_{R}(q))}(K^{\delta}K'^{\delta'}+K'^{\delta}K^{\delta'})
g_{\delta'\alpha'}P^{\parallel}_{\delta\alpha}
\label{A2}
\end{eqnarray}
\begin{eqnarray}
\mathcal{C}_{\alpha\alpha'}&=&\frac{4(\Pi^{\parallel}_{R}(q))^2}{q^4(q^2-\Pi^{\parallel}_{R}(q))^2}(K^{\delta}K'^{\delta'}+K'^{\delta}K^{\delta'})P^{\parallel}_{\delta\alpha}
P^{\parallel}_{\delta'\alpha'}
\label{A3}
\end{eqnarray}
So Eq.(\ref{Msquared}), can be written as
\begin{equation}
|\mathcal{\bar{M}}|^2=\frac{1}{4}\times\frac{1}{2}g^4(\mathcal{T}_1^{\alpha\alpha'}+\mathcal{T}_2^{\alpha\alpha'})(\mathcal{A}_{\alpha\alpha'}+\mathcal{B}_{\alpha\alpha'}+\mathcal{C}_{\alpha\alpha'})
\label{Msquared2}
\end{equation}
where $\mathcal{T}_1^{\alpha\alpha'}=Tr[\slashed{P'}\gamma^{\alpha}\slashed{P}\gamma^{\alpha'}]$ and $\mathcal{T}_2^{\alpha\alpha'}=M^2Tr[\gamma^{\alpha}\gamma^{\alpha'}]$ are traces over Dirac space. Six tensor contracted terms of Eq.(\ref{Msquared2}) are simplified in appendix(\ref{contraction}). Further simplification leads to the final form of scattering amplitude as
\begin{eqnarray}
|\mathcal{\bar{M}}|^2&=&4g^4\frac{(\Pi^{\parallel}_R(Q))^2}{Q^4(Q^2-\Pi^{\parallel}_R(Q))^2}[(P.P_{\parallel}.K)(P'.P_{\parallel}.K')+(P.P_{\parallel}.K')(K.P_{\parallel}.P')\nonumber\\
&-&(P.P')(K.P_{\parallel}.K')]-\frac{4g^4\Pi^{\parallel}_R(Q)}{Q^4(Q^2-\Pi^{\parallel}_R(Q))}[(P.K)(P'.P_{\parallel}.K')+(P.K')(K.P_{\parallel}.P')\nonumber\\
&+&(K.P')(P.P_{\parallel}.K')+(P'.K')(P.P_{\parallel}.K)-(P.P')(K.P_{\parallel}.K')]\nonumber\\
&-&4g^4M^2\bigg[\frac{2\Pi^{\parallel}_R(Q)}{Q^4(Q^2-\Pi^{\parallel}_R(Q))}(K.P_{\parallel}.K')+\frac{(\Pi^{\parallel}_R(Q))^2}{Q^4(Q^2-\Pi^{\parallel}_R(Q))^2}(K.P_{\parallel}.K')\bigg]
\label{Msquaredfinal}
\end{eqnarray}
where the tensor product is
\begin{eqnarray}
P.P_{\parallel}.K&=&P_{\mu}P^{\mu\nu}_{\parallel}K_{\nu}=\frac{(P.q_{\parallel})(K.q_{\parallel})}{q^2_{\parallel}}-(P.k_{\parallel})
\end{eqnarray}
The expression for the retarded self energy $\Pi^{\parallel}_{R}$ is given explicitly in appendix (\ref{selfEn}). This completely defines the matrix element squared.
\subsection*{Energy loss of HQ due to thermal gluons:}
The contribution of $t-$channel Compton scattering i.e, $Qg\rightarrow Qg$, to the HQ energy loss can be obtained by using Eq.(\ref{eloss}) and the interaction rate ($\Gamma$) as given in Eq.(\ref{rateHQg}). Hence, one can write~\cite{Braaten:1991jj}
\begin{eqnarray}
-\frac{dE}{dx}\bigg|_{Qg\rightarrow Qg}&=&\frac{1}{2vE}\int\frac{d\textbf{p}'}{(2\pi)^32E'}\int\frac{d\textbf{k}}{(2\pi)^32k}f({k}_0) \int\frac{d\textbf{k}'}{(2\pi)^32k'}(1+f({k'}_0))\nonumber\\
&\times&(2\pi)^4\delta^4(P+K-P'-K')(E-E')|\mathcal{\bar{M}}|^2
\label{dEdxHQgluonic}
\end{eqnarray}
The energy and momentum transfer in each scattering are, $\omega=E-E'=|\textbf{k}'|-|\textbf{k}|$ and $\textbf{q}=\textbf{p}-\textbf{p}'=\textbf{k}'-\textbf{k}$. A comment regarding the use of resummed gluon propagator may be relevant here. The hard contributions to the energy loss ($|\textbf{q}|\sim T$) can be obtained by using the bare gluon propagator for the $Qg\rightarrow Qg$ scatterings since the self energy corrections are negligible at leading order (LO). We confine our attention here for the soft momentum transfer in the range $g\sqrt{eB}\leq |\textbf{q}|\ll T\ll \sqrt{eB}$~\cite{fukushima}. This requires the resummation of the gluon propagator  in the LLL approximation as we have used here. The $|\textbf{p}'|$ integration in Eq.(\ref{dEdxHQgluonic}) can be performed with the help of momentum delta function. In the soft momentum transfer limit the energy  $E'\approx E-\textbf{v}\cdot\textbf{q}$ so that the energy delta function reduces to  $\delta(\omega-\textbf{v}.\textbf{q})$. With these simplifications, Eq.(\ref{dEdxHQgluonic}) becomes
\begin{eqnarray}
-\frac{dE}{dx}\bigg|_{Qg\rightarrow Qg}&=&\frac{(2\pi)}{16vE^2}\int\frac{d\textbf{k}}{(2\pi)^3k}f(k)\int\frac{d\textbf{k}'}{(2\pi)^3k'}
(1+f(k'))\delta(\omega-\textbf{v}\cdot\textbf{q})\omega|\mathcal{\bar{M}}|^2.
\label{elossg}
\end{eqnarray}
Now, the simplification of the above 6-dimensional integration requires a proper choice of the co-ordinate system which should also be compatible with the terms appearing in the matrix element squared. For this purpose, we choose the direction of the momentum of the incoming HQ  along the $\hat{z}$-axis
which is also the direction of the magnetic field. We denote the angle made by $\textbf{k}$ and $\textbf{k}'$ with the $\hat{z}$-axis as $\theta_k$ and $\theta_{k'}$
respectively. Further, the azimuthal angles made by the two momenta of the incoming and outgoing gluon are $\phi_k$ and
$\phi{k'}$ respectively. Therefore, the energy loss can be written as
\begin{eqnarray}
-\frac{dE}{dx}\bigg|_{Qg\rightarrow Qg}&=&\frac{1}{(2\pi)^516vE^2}\int kdkf(k)\int k'dk'(1+f(k'))\int^{1}_{-1}dx\int^{1}_{-1}dy\int^{2\pi}_{0}d\phi_k\nonumber\\
&\times& \int^{2\pi}_{0}d\phi_{k'}|\mathcal{\bar{M}}|^2\omega\delta(\omega-vk'y+vkx)
\label{elosscoord}
\end{eqnarray}
where, $x=\cos\theta_k$ and $y=\cos\theta_{k'}$. Now, we introduce a new integration variable $\omega$ which will take care of one of the angular integrations $x$ or $y$.
\begin{equation}
\int^{\infty}_{0}d\omega \delta[\omega-k(1-vx)]=1
\end{equation}
Using these two $\delta$-functions, we perform $x$ and $y$ angular integrations by writing the $\delta$-functions as
$\delta[\omega-k(1-vx)]=\frac{1}{vk}\delta(x-\frac{k-\omega}{vk})$ and $\delta(\omega-vk'y+vkx)=\frac{1}{vk'}\delta(y-\frac{\omega+vkx}{vk'})$.
So, the final expression of the energy loss becomes:
\begin{eqnarray}
-\frac{dE}{dx}\bigg|_{Qg\rightarrow Qg}&=&\frac{1}{(2\pi)^516vE^2}\int^{\infty}_{0}\omega d\omega\int^{\frac{\omega}{1-v}}_{\frac{\omega}{1+v}}f(k)dk
\int(1+f(k'))dk'\int^{2\pi}_{0}d\phi_k\nonumber\\
&\times& \int^{2\pi}_{0}d\phi_{k'}|\mathcal{\bar{M}}|^2
\label{geloss}
\end{eqnarray}
The explicit form of the four-vector product and the tensor contractions in the matrix element squared are elaborately given in the appendix(\ref{ac}).

\section{Results and discussion}
\label{results}
The two scatterings that contribute to the HQ energy loss are $Qq\rightarrow Qq$ evaluated in Eq.{\ref{qeloss}} and $t$ channel $Qg\rightarrow Qg$ evaluated in Eq.{\ref{geloss}}. as mentioned earlier, we have confined our attention to the case where the typical momentum transfer from light partons to HQ is soft i.e., $g\sqrt{eB}\leq |\textbf{k}|\ll T \ll M$. Therefore, we have used the resummed gluon propagator in the evaluation of the matrix element squared for these two processes. To simplify our calculations for energy loss we take HQ momentum $\vec{p}=(0,0,p_z)$ and magnetic field $\vec{B}=B\hat{z}$. For the numerical purpose we take HQ as the charm quark with mass $M=1.2$ GeV, temperature $T=0.25$ GeV and magnetic field $eB=0.1$ GeV$^2$ ($\sim 5m_{\pi}^2$) so that the condition $eB\gg T^2$ is satisfied. Since $eB\gg T^2$, only LLL will be populated by the light quarks. We also take finite mass ($m_f$) of light quarks in the gluon self energy diagram. The energy loss contributions using Eqs.(\ref{geloss}) and (\ref{qeloss}) has been plotted in Fig.(\ref{eloss1}). In the left side of Fig.(\ref{eloss1}), we have shown the variation of the energy loss taking contributions from both the scattering processes as a function of velocity of HQ. We have taken here $m_f=10$ MeV for the light quark mass and number of flavors $n_f=2$. At low velocity the collisional energy loss is very small $\sim 10^{-6}$ GeV$^2$, however with increase is HQ velocity the energy loss increases.
While both the scattering processes contribute to the energy loss, it is observed numerically that the $Qg\rightarrow Qg$ gives the dominant contribution to the energy loss for a given value of velocity of the HQ essentially due to larger color factor compared to the Coulomb scattering.

 This behavior is similar to the case of vanishing magnetic field.  It ought to be mentioned here that in the presence of magnetic field the energy loss is of similar order as compared to the case of vanishing magnetic field as estimated in Ref.{\cite{Peigne:2008nd}}. Indeed, the asymptotic value for the energy loss ($v\rightarrow 1$) of Ref.{\cite{Peigne:2008nd}} is given by
\begin{equation}
\frac{dE}{dx}=\frac{4 \pi}{33-2n_f}m_D^2
\end{equation}
with $m_D^2=4\pi\bigg(1+\frac{n_f}{6}\bigg)\alpha_s T^2$ which with the parameters $\alpha_s=0.3, n_f=2$ and $T=0.25$ GeV turns out to be $0.15$ GeV$^2$.
\begin{figure}[h]
\centering
\includegraphics[width=.44\textwidth,origin=c,angle=0]{elossv.pdf}
\includegraphics[width=.46\textwidth,origin=c,angle=0]{elossB.pdf}
\caption{\label{eloss1} \textbf{Left panel:} Magnetic field dependent contribution to the collisional energy loss as function of HQ velocity. Here we have taken the light quark mass $m_f=10$ MeV, flavor $n_f=2$, magnetic field $eB=0.1$ GeV$^2$($\sim 5m_{\pi}^2$) and temperature $T=0.25$ GeV. The blue, red and black curves corresponds to the coupling constant $\alpha_s=0.2$, $\alpha_s=0.3$ and $\alpha_s=0.4$ respectively. \textbf{Right panel:} Variation of the collisional energy loss scaled with $T^2$ as a function of scaled variable $eB/m_{\pi}^2$ for light quark mass $m_f=10$ MeV,  temperature $T=0.25$ GeV and coupling constant $\alpha_s=0.2$.  }
\end{figure}
This may be compared with the magnetic field contribution given by the red curve in the left panel of Fig.(\ref{eloss1}) which reaches to the value $0.1$ GeV$^2$ in the same limit. Further, it may be relevant to  note that in the limit of vanishing  light quark mass i.e., $m_f=0$, the magnetic field contribution to the gluon self energy vanishes as $\Pi_{\parallel}^{R}$ is proportional to $m_f^2$. This will lead to vanishing of the magnetic field dependent contribution to the energy loss. This is similar to the vanishing of magnetic field contribution for the dilepton production~\cite{Bandyopadhyay:2016fyd} in the same limit.

In the right panel of Fig.(\ref{eloss1}), we display the energy loss scaled with $T^2$ as a function  of magnetic field scaled by pion mass square i.e., $eB/m_{\pi}^2$, where $m_{\pi}$ is pion mass. We have taken here $T=0.25$ GeV and the magnitude of the heavy quark velocity $v=0.6$. As may be  observed in the figure the energy loss increases with the increase in the magnetic field. Numerically, it  is seen that $Qg\rightarrow Qg$ contribution is not affected too much with the magnetic field. For this process, the magnetic field dependence arises from the resummed gluon propagator with the field dependent contribution of the quark loop. This quark loop contribution increases with the magnetic field leading to a mild decrease of the energy loss due to this process as quark loop contribution lies in the retarded propagator.  On the other hand, for the Coulomb scattering process i.e., $Qq\rightarrow Qq$ the contribution increase with increase in the magnetic field. This can be understood as follows; The energy loss is proportional to  $\rho_L$ that depends on real and imaginary parts of retarded self energies and related to the spectral function. The spectral function increases with increase in the magnetic field. This increase of $\rho_L$ with the magnetic field was also observed in Ref.\cite{Bandyopadhyay:2016fyd}. It turns out that this increase is significant and the contribution becomes similar order as the $Qg\rightarrow Qg$ for larger magnetic field leading to increase of the total energy loss with magnetic field as observed in Fig.(\ref{eloss1}).

\section{Summary and Conclusion}
\label{summary}
In the present investigation, we have studied the effect of the magnetic field on the HQ collisional energy loss in a thermalized QGP medium. We have done the analysis for the case of strong field limit i.e., $\sqrt{eB}\gg T$ so that the light quarks are populated only in the LLL. On the other hand, the heavy quark mass $M$ is much larger than the strength of the magnetic field (i.e., $M\gg \sqrt{eB}$), so that heavy quark is not Landau quantized. The effect of the magnetic field manifests through resummed retarded/advanced gluon propagator through quarks loop. Since the gluon loop contribution to the gluon self energy is proportional to $\alpha_s T^2$ and the quark loop contribution is proportional to $\alpha_s eB$, in the limit $\sqrt{eB}\gg T$ the gluon loop contribution in the gluon resummed propagator is not taken into account. For the scattering of HQ with the thermalized light partons we have considered the soft momentum transfer limit i.e., $g\sqrt{eB}\leq |\textbf{k}|\ll T\ll \sqrt{eB}$. With $M\gg \sqrt{eB}$ and the in the LLL approximation the relevant scattering processes are Coulomb scattering i.e., $Qq\rightarrow Qq$ and $t$-channel Compton scattering $Qg\rightarrow Qg$. The $u$ and the $s$ channels of the Compton scatterings are not affected by the magnetic field. For a given magnitude of the heavy quark velocity, the Compton scattering process is dominant over the Coulomb scattering process for the range of the magnetic field considered here.

It is observed that of the two processes, the Coulomb scattering process  is more sensitive to the magnetic field as its contribution to the energy loss is proportional to the spectral function which increases with increase in the magnetic field. This leads to a net increase in the energy loss with increase in the magnetic field. It turns out that in this strong field limit, the magnetic field dependent collisional energy loss is comparable to the same in the vanishing field limit and therefore could be important for the jet quenching phenomena in HICs.

We would like to mention here that the present investigation is a  first attempt to include the effect of magnetic field in the collisional energy loss. However, for the large momentum of heavy quark, the radiative contribution to the energy loss may also be relevant and could be affected by the magnetic field. This apart, for a moderate value of magnetic field $eB\sim T^2$ the contributions  arising from the higher Landau levels could also be important for the energy loss and hence jet quenching. In this case, one must take the contribution of gluon loops in the gluon self energy for the resummed gluon propagator as well as the contribution from the $s$ and $u$ channels of Compton scattering. Some of these problems are relegated to future investigations. We also emphasize on the fact that our results are reliable only in the strong magnetic field limit where the LLL approximation is reasonable.

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Keag, 13 Dec 2010Does anyone know how to upgrade to OS6? I'm being told that you can't??Do you guys that there is an official patch release for Ver 5.0 for Curve 3G.
Does anyone know how to upgrade to OS6? I'm being told that you can't??
I mean should I have processor to install different application for bb like google talk, msn msnger etc,???
there is big diff in price is it worh to pay more fr the bold 9700 ??
These questions are for you.
1. What is BlackBerry Server for?
2. I want to set up corporate email, where can I can the code prompted when trying to set up?
3. Can I download Skype/Fring/Ebuddy?
4. Will BB Messenger support YM/Skype/Microsoft Communicator?
how do i upgrade my BB OS to the newest version?
You can NOT make video calls with any of BB handsets. None of them has a secondary camera(not even Torch) & they do NOT support video calling.
I got mine 2 weeks ago and I love it so much! I upgraded my BB to OS 6..it's pretty neat but nothing too special..but I love it just the same! I just don't know how am I be able to download themes that would be compatible with OS 6!
hai want t0 kn0w did bb 9300 have video call??
can we open .pdf .doc .xls .ppt files on this fone ???
Can You Advise about The Processor ( is it 624 MHZ )?
Where can i download itunes for my bb????
Pak, 18 Nov 2010ok, thanks alot, One more question, everything seems to be set up fine, except is there a wa... morego to your sound profiles on the top left hand corner and scroll down to edit profiles.
Hey, I have been using this phone since last 1 month, its an awesome phone that can cater to your different needs. It has a good processor, works very fast, ok ok camera(i guess, digicam are to be used for capturing those special moments, let digicam be digicam and mobile be a mobile). The battery life is above average, i am all time connected to internet, talk for 2-3hrs and yes do chat, smsing etcc. that battery can run for a day in a single full charge. internet surfing is too good, i have vodafone connection.
I am waiting for BB OS6.0... Anyone know about its upgrade date to be made available for us?
hwz dis fone ?? nd wich one is beter dis one or pearl 3G does pearl 3G luks girlish ??


text:
Q: Just a basic clicker but keep getting error Not sure what I am doing wrong here.
Couple things to keep in mind.

*

*Cells B7 and B8 are merged.

*The code is started by a VB button

*This is a protected sheet however the cells in question are not protected. When I run the script I get this error:
Run-time error '1004':
Application-defined or object-defined error
        Sub OneClickLetterSend()
    With Worksheets("Menu").Range("B7")
        Select Case .Value
            Case 802
                .Value = .Value + 1
                Range("B7").Interior.Color = RGB(0, 0, 255)
            Case 803
                .Value = .Value + 1
                Range("B7").Interior.Color = RGB(0, 255, 0)
            Case 804
                .Value = .Value - 2
                Range("B7").Interior.Color = RGB(180, 0, 255)
            Case Else
        End Select
    End With
    End Sub

A: Protect Sheet feat. Format Cells

*

*If you haven't already, unlock cell B7 and under
Allow all users of this worksheet to:
tick Format cells.

*If for any reason that is not possible, play with the out-commented Unprotect and Protect lines in the code.

The Code
Option Explicit

Sub OneClickLetterSend()
    With Worksheets("Menu").Range("B7")
        '.Worksheet.Unprotect
        Select Case .Value
            Case 802
                .Value = .Value + 1
                .Interior.Color = RGB(0, 0, 255) ' Blue
            Case 803
                .Value = .Value + 1
                .Interior.Color = RGB(0, 255, 0) ' Green
            Case 804
                .Value = .Value - 2
                .Interior.Color = RGB(180, 0, 255)
            Case Else
        End Select
        '.Worksheet.Protect
    End With
End Sub

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text:
GTA IV PC details and screenshots emerge
By Chris | 20th Aug 2008 at 22:50 GMT in GTA IV | 86 Comments
The PC version of Grand Theft Auto IV was shown off at the Leipzig Games Convention today by PR man Steve Hahnel. New features are detailed below:
Much improved visuals, they'll be "polished well beyond what they were on the 360 and PS3".
The number of multiplayer slots may be expanded for the PC version.
The maximum resolution is 2560x1600.
The replay feature has been improved. Players can record 30 second clips and they can now add filters and adjust the camera angle using the replay editor. They can also make a montage of clips and upload them to the R* Social Club for others to view.
Players can finally filter multiplayer games more specifically (e.g. a deathmatches using a particular weapon).
The Xbox 360 controller is compatible with the PC version.
It looks like fingerless gloves have finally been added! Much to the delight of pretty much every player of the game! See this shot
10 new screenshots have been released, click the thumbnails below to see the large versions. These have also been added to our new GTA IV PC screenshot gallery.
Related articles: Eurogamer and IGN previews
Interview with Sam Houser
By Gerard | 16th Aug 2008 at 02:29 GMT in GTA IV | 6 Comments
Developmag have got themselves a very rare interview with Rockstar Games President, Sam Houser. They cover the impact of GTA IV, the direction of the industry and Rockstar's internal ethics and beliefs. Here's a few interesting quotes:
In order to be successful, we are going to have to continue to do what we have always done � make games that we would want to play, and hope there is audience for them.
Most of the people who hate us are people it is truly an honor to be hated by � reactionary creeps with strange agendas � and the Daily Mail. Most people who know about modern pop culture know about GTA and like or dislike it on its own merits.
We don�t want to get too much bigger in terms of teams, because the games need to have the personal touch and production values that are in them, and we can�t do that at a much bigger scale, but we will always look to work closely with excellent people when we come across them.
The problem with academia and prizes and awards and all the things the movie industry has wrapped itself in is that it constrains things too much. 99 per cent of movies are three acts told over two hours with very similar plot structures, camera work and so on. We can rip up the rule book every time we make a game as every aspect of every process gets reconsidered.
Read the full interview at Developmag: Part 1, Part 2
Release date for Japanese version of GTA IV announced
Good news for readers in Japan still waiting to get their hands on Grand Theft Auto IV. A press release from Take2 today announced that the localized Japanese version of the game would be released on 30th October, 2008 for the PlayStation 3 and Xbox 360 consoles.
New York, NY � August 15, 2008 � Rockstar Games, a publishing label of Take-Two Interactive Software, Inc. (NASDAQ: TTWO), announced today an October 30, 2008 release date for the localized Japanese version of Grand Theft Auto IV for the Xbox 360� video game and entertainment system from Microsoft and the PLAYSTATION�3 computer entertainment system.
Developed by series creator Rockstar North, Grand Theft Auto IV was released in North America and Europe on April 29, 2008, selling 3.6 million units on the first day of release, and 6 million units in the first week. Critics from across the globe have showered the game in unanimous praise for its immersive storytelling; culturally eclectic soundtrack; an unparalleled level of detail in Liberty City; and sixteen player multiplayer with more than a dozen online modes for endless replayability.
Interview with Little Jacob's voice actor
By Chris | 13th Aug 2008 at 13:56 GMT in GTA IV | 8 Comments
GTA4Mods.com recently managed to score an interview with Coolie Ranx, best known among GTA fans for playing the part of Little Jacob in Grand Theft Auto IV. There's a number of good questions that were asked, and some interesting answers from Ranx.
On how he got the part of Little Jacob:
I'm still wondering about that feat. My audition I thought was horrible. For me it was about the art because I really wanted to do acting. I didn't have a clue what I was auditioning for, all I knew was that it was an acting job. I wanted to do it so bad that I crumbled at the audition and asked to come back again the next day and try it again. They said yes but to be honest the next day wasn't that much better lol. So I'm still wondering what they saw in me.
Read the rest of the Q&A over at GTA4Mods.com
Thanks to Alan for the heads up.
GTA IV PC release confirmed!
By rockstarrem | 6th Aug 2008 at 15:22 GMT in GTA IV | 140 Comments
Take 2 announced through a press release today that GTA IV will indeed be released on the PC, arriving on November 18th and 21st in North America and Europe, respectively.
"We are very excited to be releasing the PC version of Grand Theft Auto IV," said Sam Houser, Founder of Rockstar Games. "The whole team is dedicated to bringing an amazing gaming experience to the PC. The game looks and plays beautifully on PC and we can't wait for people to play it."
Developed by series creator Rockstar North and set in Liberty City, the latest installment in the enormously successful Grand Theft Auto series features a painstakingly detailed and life-like city for players to explore; a rich, immersive narrative experience; an original soundtrack highlighting the cultural eclecticism of Liberty City; and newly expanded multiplayer just for the PC.
Stay tuned to The GTA Place for the latest GTA IV news!
Link: T2 Press Release
Related note: Wondering whether or not your PC will be able to run GTA IV? We've now set up a dedicated topic for asking those sort of questions. Post there to get advice from our PC experts and find out if you need to upgrade.


Why not use E for Extreme??? This is a new requirement and any 8. Test of a 26' 7. Does anyone else have an 8. I am nervous about running the oil 2. Re: Oil capacity problem: 8. Test of a 28' 8. Please observe that the information available varies depending on the product and year of production.
The one with the most torque at the point that the boat has to struggle to get up and out, will have the better hole shot. Nevertheless, by studying the data you will discover patterns and trends that should help giude you in your engine selection. Personally I would prop it to 4600 or as close as I could get. We invite you to view all of the results. Thank you in advance, Jer Re: Oil capacity problem: 8. I also have heard that suction pumps can't get all of the oil out to begin with.
Write this down… First off, they all start with G, like in Gas. Then in every manual since they have made the same mistake. The odd thing is that I only removed 6 quarts. There are several things that are noteworthy when looking at the overall picture of this engine, including but not limited to. From basic maintenance and troubleshooting to complete overhauls, our Volvo Penta Stern Drive manuals provide the information you need. Anyone know why the Volvo 8. Sorry I can't give you a difinitive answer but I'm sure someone will follow up with one.
Acceleration was crisp with no hesitation upon rapid advancement of the throttle. What I think is odd though is that if you go to Volvo's site now it advises 9 quarts as well. Volvo's same engines in Europe were utilizing private label 15w-50 which was a simple re-brand of Mobil 1 Extended Performance 15w-50 packaged by the big red O into a Volvo container at their Lube Plant in Graveson, France. Our test boat was powered by a pair of 375-hp 8. Depending on how long she cooled off, some of the oil was still in the passages. Hatchman and Richard W, Just like Mercruiser, Volvo Penta has changed their oil spec at least in the owners manual for a 2012 8. Once Volvo makes a typo, it lasts forever.
So to sum it all up, an 8. Didn't know if there were any additives, etc. This assumes all other things are equal, boat, weight, drive, props etc. Cruise modes, tow modes, trim assist, and single lever modes… all are available at the push of a button. In high power turns there was no ventilation of the props. Re: Oil capacity problem: 8. Best cruise was at 3000 rpm where she went 29.
Testing conditions also bear on the results. Yesterday I changed the oil for the first time since ive owned it. Hello all, I have a Penta 8. A truly robust design and an advanced monitoring system help to ensure reliability. Most prominent among those are weight, beam and deadrise. Both oils were suitable, and were not specialized blends, just packaged in Volvo plastic. Use Volvo oil when the new version finially appears if that makes you comfortable, but for me I much prefer Mobil 1 15w-50 as it is available at any Walmart, exceeds our requirements by a bunch, and cost less.
One factor of these Volvo Penta gas engines that many find confusing, including us, is the nomenclature of each engine type. Not looking for a debate here. That's my take on it. This is for lower end 4. The 15w indicates that it acts as a 15 weight oil would at low temperature, and as a 50 weight would at elevated temperatures.
The three models of the 8. In summary, we found the test boat handled extremely smoothly to power inputs. At the low end of this range I am a little concerned about longevity. I was trying to be specific as far as the question of longevity is concerned. I had good oil pressure and the engine sounded and ran fine. There are several Mobil 1 viscosities that do. Where did you read that it takes 6 quarts? So I sucked everything out that would have been read by the dip stick.
I could not find any specific limitations in the owners manual about a P spec or even mention of the catalytic converter. In this case availability dictated the spec, not the other way around as we would assume. Merc may have had a bigger budget for the higher rating, so they spent more time on it. It was equipped with a Duoprop F5 lower unit with a 1. As a comparison, the same 8.


Pastoral appointments will be made according to one of the following tracks. The Conference Superintendent or his appointee shall meet with the Local Church Administrative Council to determine the best method of selection. Pastoral candidates may be considered from any Conference in the International Pentecostal Holiness Church. In all situations, Senior Pastors and churches are subject to the appointing powers of the Conference. Unless an alternative structure has been approved by the Conference Executive Council, the office of Senior/Lead Pastor shall be held by only one person, who must be a credentialed IPHC clergy member.
Track 1: The Conference a. Executive Council may appoint Senior Pastors to local churches in consultation with the Local Church Administrative Council.
Track 2: The Conference Executive Council may appoint Senior Pastors to local churches in consultation with the Local Church Administrative Council and election by the local congregation.
Track 3: The Local Church Administrative Council may appoint a pastoral search committee to offer candidates to the local congregation. This committee shall initiate its work with the Conference Superintendent and work through the Conference Superintendent in making contacts with its candidates.
The process as described above is defined in its entirety in the IPHC Manual. If you would like more information about one or more of the following churches, please contact the Cornerstone Conference Ministry Center at 336-656-7936.
Available Churches: The following is a list of churches actively seeking a Pastor.
Church Planting: We ask that you join with us to pray for new congregations / churches to be birthed and established in Cornerstone Conference in 2019! Also, we would like to see new congregations in our West Elkin, NC and Mt. Holly, NC properties. If you are interested in starting a new congregation, please contact us today!
Christian View PHC is seeking a Godly and energetic youth leader to minister to a small group. This group is looking to grow in Christ and in number. Duties would include weekly bible study and quarterly outings under the guidance of the Lead Pastor. Great opportunity for someone looking to intern or grow their skills for a career in ministry. Part-time paid position. If interested, please contact Pastor Cas Horton at 336-552-7599.


text:
"""Module pyhtslib with common types and routines"""

class GenomeInterval:
    """Zero-based genome interval."""

    def __init__(self, seq, begin_pos, end_pos):
        self.seq = seq
        self.begin_pos = begin_pos
        self.end_pos = end_pos

    def __str__(self):
        return '{}:{:,}-{:,}'.format(self.seq, self.begin_pos + 1,
                                     self.end_pos)

redpajama_set_name: RedPajamaGithub


text:
The 1979 NSWRFL season was the 72nd season of Sydney's professional rugby league football competition, Australia's first. Twelve New South Wales Rugby Football League clubs competed for the J.J. Giltinan Shield and WD & HO Wills Cup during the season, which culminated in a grand final between the St. George and Canterbury-Bankstown clubs. NSWRFL teams also competed in the 1979 Amco Cup.

Season summary
Twenty-two regular season rounds were played from March till August, resulting in a top five of St. George, Parramatta, Cronulla-Sutherland, Western Suburbs and Canterbury-Bankstown who battled it out in the finals.

Parramatta's Ray Price was the 1979 season's Rothmans Medallist, and also won Rugby League Week's player of the year award.

The 1979 season also saw the retirement from the League of future Australian Rugby League Hall of Fame inductee, Bob Fulton.

Teams
Twelve clubs, including six of 1908's foundation teams and another six from around Sydney competed for the NSWRFL premiership.

Ladder

Finals

Chart

Grand final
St. George finished 22 rounds as minor premiers with 17 wins and were straight into the grand final after defeating Parramatta in the major semi. Their Grand final opponents Canterbury, qualified after defeating Wests, Cronulla and Parramatta.

Television Coverage Dispute
Under agreements in place at the time, the broadcast rights to the season saw one to two games on ABC TV on Saturday and/or Sunday afternoons, and a Sunday Night primetime replay and highlights of other matches on ATN-7 (part of the Seven's Big League programming). Under this agreement, the networks had rights to each broadcast and/or simulcast all six of the finals series games, under which they could either accept or decline the matches.

The scene was set by an industrial relations dispute between camera and production crews and commercial broadcasting networks towards the end of 1979. This forced ATN-7 and the ABC into a shared feed arrangement using ABC production crew, but separate network commentary teams calling the game.

ATN-7, wore the financial costs as the designated broadcaster of the preliminary final, and provided the ABC with a clean feed of the match as agreed. However, the ABC refused to offer a clean feed to ATN-7 for the grand final, which caused massive controversy. ATN-7 then exercised their right as the primary broadcaster of the competition to broadcast the match between St George and Canterbury exclusively, deciding to source their own freelance crew and cover the match. The ABC then sued ATN-7, and the Australian Broadcasting Tribunal became involved, but ultimately the match remained exclusive to Seven's Big League.

Match
The pre-game talk had been about the match-up of opposing half-backs Steve Morris and Steve Mortimer and the game didn't disappoint, beginning with a 40-metre flying break on a last tackle by Morris from deep in Dragons territory before he was cut down in a classic covering tackle by Mortimer.

The first half belonged to St. George with new fullback Brian Johnson scoring an easy opening try after taking a pass 10 metres out from the Canterbury line. Soon after, Rod Reddy, who was subduing his opponents with his experience and ruthlessness, put Morris into a gap. "Slippery" kicked ahead and the race was on. Mortimer had a head start and got there first but Morris tackled him into touch.

Soon afterwards Canterbury put a bomb up and toward the St. George in-goal. Brian Johnson was on hand to field the ball and returned it with a 60-metre run ended by a copybook tackle from his opposing fullback Stan Cutler. However the defence was opening up and it wasn't long before the Dragons scored their second try, with winger Mitch Brennan venturing infield and finding a gap courtesy of centre Robert Finch who slipped a short ball to Brennan 30 metres out. The strapping winger made a powerful run to score.

By now, Saints were carving Canterbury up with Morris, on his 20-metre line, putting Graeme Wynn into a gap. The lanky back rower ran 40 metres, leaving defenders in his wake. Next it was "Rocket" Reddy's turn when Morris, Wynn and Reddy combined down the right flank and exposed the Bulldogs defence. 'Rocket' handled twice in scoring a try under the posts. At half time the score was St George 17 – Canterbury 2.

Canterbury fought back gamely after the break. Steve Gearin and then Peter Mortimer crossed, with two further tries being disallowed. The St. George defence rallied and held from that point until Stan Cutler scored a third try out wide late in the match. St. George's goal-kicking second rower George Grant was the difference on the day, kicking four to ensure a 17–13 victory. Grant had kicked brilliantly all season, scoring 211 points in 1979.

The Dave Brown Medal for the best player on field, went to Steve Morris.

St. George 17 (Tries: Johnson, Brennan, Reddy. Goals: Grant 4.)

Canterbury-Bankstown 13 (Tries: Gearin, Mortimer, Cutler. Goals: Gearin 2.)

Player statistics
The following statistics are as of the conclusion of Round 22.

Top 5 point scorers

Top 5 try scorers

Top 5 goal scorers

See also
 Mortimer family

References

External links
Rugby League Tables – Season 1979 The World of Rugby League
Results:1971-80 at rabbitohs.com.au
1979 J J Giltinan Shield and WD & HO Wills Cup at rleague.com
NSWRFL season 1979 at rugbyleagueproject.org
1979 Grand final at Dragons History site

New South Wales Rugby League premiership
NSWRFL season

redpajama_set_name: RedPajamaWikipedia