Panther Creek Park Trails is a 6.7 mile moderately trafficked loop trail located near Talbott, Tennessee that features a lake and is rated as moderate. The trail offers a number of activity options and is accessible year-round. Dogs are also able to use this trail but must be kept on leash.
I did the knob creek trail; one of the longest in the park but so nice. When you get to the 'loop' part where the mountain meets the water you can jump in right there... the water feels amazing! In total we hiked 6.6 miles. We asked the ladies inside where to go and they didn't know much, but on the map it says to park where the horse trailer parking lot is, then hike on the horse connector trail until you come to the "Knob Creek" Trailhead (which is right past the camping grounds) The horse trailer parking lot is about a quarter mile past the main entrance depending on which direction you come from (right near the boat ramp) But while we were walking on the connector trail we went right by the tennis courts which are in the park so there is probably a way to park in the main lot, then just find your way down to the trail.. it would definitely shorten the walk.
Knob creek itself is about 3 miles. But like I said, definitely worth it at the end to take a jump into Cherokee Lake and cool off!
Great place to spend the day and enjoy nature. Trail is great and views of Cherokee Lake are stunning.
Great trails, it close by and convienant. Wonderful views of Cherokee Lake.
Nice views around the lake (especially if the water is up). Plenty to do around the park.
I like this trail. I've done it a few times. I'd say it's decently difficult since I'm the only one of my friends that made it all the way through.
This is a great trail system something for all skill levels not to crowed and everyone I ran into was very nice and helpful trails were in good shape only seen tree across 1 trail in 2 places I will go back and ride the rest of the trails I didn't have time to get to.
Great park, not many people on the trails.
My brother and I did the point lookout trail, which was pretty steep at some points so a great workout! I do wish that there was a higher platform to stand on or some of the trees were cleared for the view at the summit. It's a pretty narrow window where the trees are thin enough to see through, and even still it's fairly hard to see. The park overall was very well kept and beautiful, and great for a family outing or some camping!
Very peaceful for a weekend hike! I hiked alone so was graced with more wildlife due to the quietness. Trails are great here with all levels of difficulty and rewarding views of Cherokee Lake. Best of all I live right on the edge of the park with a trailhead a block away!!
Amazing park. Great MTB trails. Great camping, and hiking.
Great trail late fall riding you have to pay close attention as you can go off the trail. Beautiful lake views. Wooden bridge, beach, creeks. Rock formations. Day use only. Leave your valuables at home as there was reports of break ins. I have never had trouble though. Lots of pine, all wooded.
My local state park. Fun little area but doesn't compare to the nearby smokies. Trails are also difficult to navigate at times due to terrible marking.
Great place to hike, ride bike, and camp. View of Cherokee Lake is amazing at the end of the park. there is a trail that will take you to the water front where you can enjoy a nice swim.
Great place for hiking, biking, swimming and boating, they even have tennis courts and campsites with playgrounds. Great place for everyone. I went mountain biking there and it was tons of fun.
David Miller recorded Nice Lake Views!


text:
News Highlights
Healthcare Schemes & Subsidies
View all healthcare financing schemes & subsidies available in Singapore such as MediShield Life, CareShield Life, MediSave, CHAS and ElderShield.
Fee Benchmarks and Bill Amount Information
Search for fee benchmarks for private sector professional fees and bill amount information for public and private hospitals.
Healthcare Locator
Search for Healthcare Institutions
Search for Healthcare Professionals
Suspensions/Revocations from Healthcare Financing Schemes
List of suspensions and revocations taken by MOH under CHAS, MediSave and MediShield Life
Enforcement Actions for Breaches of Regulatory Requirements
List of enforcement actions taken by MOH under the PHMCA, MRA and DRA
Upcoming and Completed Healthcare Facilities
View all upcoming and completed healthcare facilities.
Disease Updates
Weekly Infectious Diseases Bulletin
Being Prepared for a Pandemic
Learn more about how Singapore is prepared to prevent & respond to disease outbreaks.
Overview of Diseases
Learn more about the various infectious diseases that Singapore has overcome.
Resources & Statistics
Through E-Consultation, MOH aims to gather feedback and suggestion from members of the public to further improve policies, programmes and services.
Search for various available grants to drive the translation of basic research to advance human healthcare, and to increase the translational and clinical research capabilities of public hospitals, research institutions and medical researchers.
Search for publications, education brochures, statistics on admission rates, waiting times, bed occupancy rates, diseases and more.
Licensing and Regulation
Health Professional Portal
Allied Health Professional
All Heathcare Professionals
Acts regulating drugs, healthcare professionals, medical practices, research and more.
Join Us. Together, we can protect the health of this nation.
Singapore's Healthcare System
MOH website is scheduled for a routine monthly maintenance on 22 Jan (Saturday) from 2200 hrs to 23 Jan (Sunday) 0600 hrs. During that period, you may experience some intermittent downtime while accessing the site. We apologise for any inconvenience caused and look forward to serving you better.
Find speeches, press releases and forum replies.
Click here for E-Consultation.
Summary of local situation
1,464 cases are currently warded in hospital. 310 cases require oxygen supplementation and 62 are in the intensive care unit (ICU).
Over the last 28 days, of the 68,147 infected individuals, 98.6% had no or mild symptoms, 1.1% required oxygen supplementation, 0.1% required ICU care, and 0.2% has died.
As of 15 October 2021, 84% of our population has completed their full regimen/ received two doses of COVID-19 vaccines, and 85% has received at least one dose.
As of 16 October 2021, 12pm, the Ministry of Health has detected a total of 3,348 new cases of COVID-19 infection in Singapore, with 2,688 in the community, 656 in the migrant worker dormitories and 4 imported cases.
Situation update
1. Over the last 28 days, 98.6% of local infected cases were asymptomatic or had mild symptoms, 1.1% required oxygen supplementation, and 0.1% had been in the ICU. Amongst those who required oxygen supplementation and ICU, 49.4% were fully vaccinated and 50.6% were unvaccinated/ partially vaccinated.
Figure 1: Local Cases in the Last 28 Days by Severity of Condition[1]
2. 9 more cases have passed away from complications due to COVID-19 infection [2]. Of these, 4 were male Singaporeans and 5 were female Singaporeans, aged between 60 and 89 years. Amongst them, 5 had been unvaccinated against COVID-19, 2 had been partially vaccinated and 2 had been vaccinated. 8 of them had various underlying medical conditions, while an unvaccinated case had no known medical conditions.
3. Amongst those who have passed away over the last 28 days, 26.3% were fully vaccinated and 73.7% were unvaccinated/ partially vaccinated.
Figure 2: Deaths and Active Cases in ICU, Requiring Oxygen Supplementation or Hospitalised, by Age Groups
4. As of 16 October 2021, 12pm, amongst all COVID-19 community cases who have been onboarded or admitted, 16,580 (76.5%) cases are undergoing Home Recovery. 2,862 (13.2%) cases are in Community Care Facilities, 766 (3.5%) cases are in COVID-19 Treatment Facilities, and 1,464 (6.8%) COVID-19 patients are currently warded in hospital, mostly for observation. 310 require oxygen supplementation, and 62 are in the ICU.
Figure 3: Breakdown of Number of Active Cases in Hospital, COVID-19 Treatment Facilities, Community Care Facilities, and Home Recovery Programme in the Last 7 Days
Vaccination progress
5. As of 15 October 2021, 84% of our population has completed their full regimen/ received two doses of COVID-19 vaccines, and 85% has received at least one dose.
6. We have administered a total of 9,598,504 doses of COVID-19 vaccines under the national vaccination programme (Pfizer-BioNTech Comirnaty and Moderna). In total, 4,616,536 individuals have received at least one dose of vaccine under the national vaccination programme, and 4,555,676 individuals having completed the full vaccination regimen. In addition, 224,870 doses of other vaccines recognised in the World Health Organization's Emergency Use Listing (WHO EUL) have been administered, covering 117,283 individuals.
7. To date, we have invited about 780,000 eligible individuals to receive their booster doses. 554,004 individuals have received their booster shots and another 100,000 have booked their appointments.
Active clusters under close monitoring
8. We are closely monitoring the clusters listed below, which have already been ringfenced through tracing, testing and isolation.
Figure 4: Large Clusters with New Cases
Total[3]
ECON Medicare Centre & Nursing Home (10 Buangkok View Block 5)
Transmission amongst staff and residents. Of the 59 cases, 3 are staff and 56 are residents.
United Medicare Centre (Toa Payoh)
Transmission amongst staff and residents. Of the 123 cases, 17 are staff, 105 are residents and 1 is a household member of a case.
Summary of trends for local cases
9. As of 16 October 2021, 12pm, the Ministry of Health has detected a total of 3,348 new cases of COVID-19 infection in Singapore, comprising 2,688 community cases, 656 dormitory residents cases and 4 imported cases. Amongst the local cases today are 475 seniors who are above 60 years.
Figure 5: Number of Community Cases by Age
Figure 6: Number of Dormitory Cases by Age
10. Please refer to MOH's daily Situation Report (www.moh.gov.sg/covid-19/situation-report) for details.
[1] Includes only new cases reported in the past 28 days. Based on cases' worst-ever condition; deceased cases previously in ICU are counted under 'Deceased' and not under 'Ever in Intensive Care Unit'.
[2] Cases 92570, 94015, 103313, 105159, 128981, 129822, 130150, 133461 and 138258.
[3] Includes new cases added today.
Category: Press Releases Highlights
Schemes & Subsidies
Healthcare Professionals Portal
About MOH
Our Healthcare System
Last Updated DD MM YYYY

redpajama_set_name: RedPajamaCommonCrawl


OTHERS, SNES Giocare The Ren & Stimpy Show - Veediots! Online, Jogar The Ren & Stimpy Show - Veediots! Online, Jugar The Ren & Stimpy Show - Veediots! Online, The Ren & Stimpy Show, The Ren & Stimpy Show - Veediots!, Veediots!. permalink.


The first decision in any water feature is whether to use Salt Water or Fresh Water.
Salt water inhibits algae growth and requires minimum filtration but will not support fish or plant life.
Fresh water supports fish & plants but requires aggressive filtration.
While fish and water plants appeal to many people, choosing to include them in any water feature impacts the issues of design, equipment & maintenance.
Fish will typically require a deeper pond, better filtration and higher water turnover. Fish can also be subject to disease that may require special water treatment. The presence of fish also limits the number of options you have in treating for algae. If you would like some help in selecting fish species or dealing with some disease problems, contact a PSG pond specialist.


To: "Glenn at home" <GlennErvin_at_cableone.net>; "Rockbox"
voice, that may fix it, you never know.
SP2, but I cannot find a fix for it.
> do a thorough scan of the hard drive.
> These steps should be done from time-to-time anyway.
> computer speech engines may work.
> I have some Microsoft voices I could send you if you don't have any.
> Or you could go to Microsoft.com, and download the latest SAPI software.
> error. Does anyone please know how to resolve this?
Next message: tj_at_tjolsen.net: "Re: Shutting down/startup voicing wanted?"
In reply to: Brian Hartgen: "Re: problem with Rockbox speech"
Next in thread: Brian Hartgen: "Re: problem with Rockbox speech"
Reply: Brian Hartgen: "Re: problem with Rockbox speech"


text:
Last month, Dr. Jim Frankel of MusicFirst joined nearly 100 music industry leaders, notable artists and arts education activists to advocate for all school-aged children to have access to quality, comprehensive school music education programs. As part of the National Association of Music Merchants (NAMM) Advocacy Fly-In, held May 22-24, the delegation met with Members of Congress and other policy stakeholders to reinforce the importance of music as part of a well-rounded education.
The issue of music education takes on a special importance this year as the Every Student Succeeds Act (ESSA), signed into law in 2015 by President Obama readies for state-level implementation in 2018, currently faces underfunding. The law specifies music as core part of a well-rounded education for all K-12 students.
"Being on Capitol Hill for the day, representing MusicFirst along with other music industry leaders, advocating for music education for every child in America was one of the most important things I have ever done. Meeting with elected officials and directly pleading the case for fully funding ESSA, especially funding for arts education programs including music education, was an amazing experience."
The Advocacy week began on Monday, May 22 with a Day of Service at Jefferson Middle School Academy. Members from the group provided three hours of music education instruction on drum, ukulele and guitar for the middle school students. NAMM delegates also provided maintenance and repair to many of the school's existing instruments. That evening, he joined The NAMM Foundation in awarding the SupportMusic Champion Award to Senator Robert Casey (PA). The award was presented in recognition of the Senator's long history in music education advocacy and role as a champion of the Every Student Succeeds Act.
On Tuesday, Jim prepared for his efforts on Capitol Hill by participating in advocacy training at the Newseum, during which Jim and fellow music industry leaders were apprised of current issues facing public school music programs and briefed on ESSA and the current political climate, from a variety of policy and arts leaders, including representatives from the United States Department of Education, Americans for the Arts (AFTA), the National Governors Association (NGA), the National Association for Music Educators (NAfME), and the National School Board Association (NSBA), among others. The group also enjoyed a special lunchtime session on current policy trends presented by PBS News Hour contributor, New York Times columnist and Turnaround Arts artist, David Brooks. Dr. Frankel was also trained on developing state-level advocacy efforts for music and arts education to put into practice here in New York and around the country.
On Wednesday, Jim met with Senators Kirsten Gillibrand and Chuck Schumer, Congresswoman Nita Lowey and Congressman Peter King, and other elected officials to advocate for school-level music programs across the nation and to discuss the multitude of benefits music education espouses such as increased brain function, focus and language development, as discussed in a new research study from Northwestern University. The Country Music Association, VH1 Save the Music Foundation, and former New York Yankees centerfielder and three-time World Series winner, music education champion and Turnaround Arts artist Bernie Williams joined the day of advocacy, sharing his own experience in music education and a desire to create a more musical world.
In addition to the advocacy work, Jim was part of a special Congressional reception, held on Wednesday, May 24 in the Mansfield Room at the U.S. Capitol that brought together arts advocates and policy makers.
About NAMM
The National Association of Music Merchants (NAMM) is the not-for-profit association with a mission to strengthen the $17 billion music products industry. NAMM is comprised of approximately 10,300 member companies located in 103 countries. NAMM events and members fund the NAMM Foundation's efforts to promote the pleasures and benefits of music, and advance active participation in music making across the lifespan. For more information about NAMM, please visit www.namm.org, call 800.767.NAMM (6266) or follow the organization on Facebook, Twitter and YouTube.


text:
Q: How does Laravel extract the get parameter and pass to closure I am new to Laravel and coming from PHP wanted to understand some of the aspects of the framework. I have never used a framework before and would like to understand some of the under the hood stuff. I saw this syntax for the route class:
// Second Route method – Root URL with ID will match this method
Route::get('ID/{id}',function($id){
   echo 'ID: '.$id;
});

From my understanding Laravel does pattern matching for the URL and extracts the value from id and assigns it to $id and passes it to the closure.

*

*Is my above understanding correct

*Can someone point to the part in Laravel Code where the value of id is extracted and assigned to $id variable

*Why are curly braces used? It it just to make pattern matching easier?

A: *

*Yes, you are correct

*Right here

*The curly brackets are to let the route parser know that this is a dynamic part that will be able to change in the url. It could have been any other way, but this is how Laravel does things.

redpajama_set_name: RedPajamaStackExchange


text:
Home » Newsline: Ben Cohen Warming to B&J CEO
Dairy Processor News
Newsline: Ben Cohen Warming to B&J CEO
After Ben & Jerry's was purchased by Unilever in 1999, cofounder Ben Cohen distanced himself from his former company. At one point a few years ago he was even openly critical of the company, saying Unilever had done little to foster the unique, altruistic corporate culture that had become as well-known as Chunky Monkey.
But things have changed, and Cohen now says he is pleased with the direction the company is taking under Walt Freese, who became CEO about a year ago.
"Walt Freese happens to agree with the original mission-believes in it personally, and also believes that it's a critical component of the so-called brand of Ben & Jerry's," Cohen recently told the San Francisco Chronicle. "He's actively looking for ways to integrate social concerns into how the company does business."
Cohen backs three issues that Freese has gotten the company involved in: global warming, family farms and federal budget priorities.
Newsline: New Brigham's Ownership Hires B&J Veteran
Newsline: Danone Adds to W-B-D Stake
Newsline: Pepsi to Market Ben & Jerry's Shakes
Newsline: Acquisition Arena Warming in 2006


We don't have to convince anybody who had a chance to try chili pepper that after eating it you get really hot! It turns out that this effect may be used in a functional way to support losing weight! Yes, this is not a mistake. Capsaicin contained in chili is a natura l and strong thermogenic!
Chili peppers are the source of many interesting substances, such as vitamins and minerals. However, capsaicin demonstrates especially interesting effect. It is an organic, chemical compound with strong irritant and thermogenic properties. In old times capsaicin was used to blind enemies for a short time and Indians chewed chili pepper to relieve tooth pain. Today, this alkaloid constitutes the component of pepper sprays.
Capsaicin influences vanilloid receptor (which becomes activated under the influence of high temperature), which is accompanied by the feeling of acute burning or sometimes pain. Body temperature gets physically increased, which is connected to, among others, catecholamine release from adrenal medulla (adrenalin and noradrenalin). By the same token the metabolism is accelerated and lipolysis gets intensified. It is, however, only temporary effect. On the other hand, it turns out that capsaicin may have prolonged thermogenic effect!
If thermogenic effect of capsaicin was limited only to stimulating vanilloid receptors, it would be very short and would pass after a few minutes. In practice, however, thermogenic effect, after taking the substance, remains for longer. The recently published study results indicate that it is connected to the fact that the above mentioned alkaloid changes the activity of protein called SERCA, appearing in muscle tissue. It results in the "dysfunction" of the transport of calcium ions in muscle cells in such a way that a considerable part of energy derived in metabolic changes gets dispersed in the form of heat. It is a bit as if during boing water, we took the lid from the pot – the larger part of the energy gets "wasted".
Taking into consideration the above mentioned capsaicin properties we may risk the statement that spicy meals are more silhouette-friendly than the dishes with delicate taste, without adding hot seasoning. This fact may be used in practice during balancing your own meals in a reducing diet.


Today Gurudeva talks once more about willpower, how some people possess more than others, developed in this life or a past life. He talks about how to get more willpower and store it in the manipura chakra.
Today at Kauai Aadheenam. It is April 5th. We have Sadhaka Thondunatha here to translate what I am going to say into the beautiful Tamil language. So, all of you Tamil-speaking devotees, hang on and tune in. But, first we are going to talk a little bit more about willpower.
Willpower is an actual possession. Many people have a lot of it. Many people have developed it in their last life. Many people have had several past lives and it comes naturally easy to them in this life. If you find you don't have will, you don't have determination, it is hard for you to carry things through to finish that you begin. You are a victim of procrastination. That means you put things off until tomorrow and then never get around to doing it. But, it is always in your mind that you are going to. "I am going to do it." So, if you are a I-am-going-to-do-it person, you need more willpower.
How do you get more willpower? Walk a little further than you think you can walk. Sit a little longer in meditation than you think is possible for you to sit. Be a little nicer than you think you can be. Be more forgiving. Put out extra effort in everything that you do and that extra effort begins to build up within you because that extra effort is a prana, is an energy that accumulates within your manipura chakra. That is just where your belly button is and in that chakra in your astral body, you begin to build a force field of pranic energy, you become strong. Of course, not to create bad karma by using your willpower, as we talked about yesterday, we must have love in the will. We have to do good for people, to do good for ourselves. Think positive thoughts, be positive in words and then, what happens? Our life and future smoothes out.
Now we are going to ask Sadhaka Thondunatha to tell in the beautiful Tamil language everything I had to say and also encourage the youth, especially the youth of Sri Lanka to encourage their friends to tune in to, 'Listen to Gurudeva' today.
Gurudeva: I would like to thank Sadhaka for the beautiful translation. I would also like some of you out there from Sri Lanka to put out a news item in all the Tamil newspapers, radio programs that this service from Kauai Aadheenam is available. Kauai Aadheenam is an extension of our Alaveddy Ashram in Sri Lanka. This service is available to young people. They can ask questions. If they are having problems with drugs, girl friends or boy friends, they can ask questions. We can give either personal or impersonal answers.


text:
Q: RAID Hard drive partition from one motherbord to another? I had ASUS P5K3 Deluxe/WiFi-AP which was working for 4 years. Yesturday it burned out... literaly with smoke and fire... So I plan on  getting  DFI LanParty LT X48-T3RS (Intel X48, DDR3) as a replacement. I had RAID of 5 satas. I do not remember what type of RAID it was but I know for sure that the drives were not mirrored. I have no regrets of loosing it but still I wonder what are my chances of saving the data? Meaning if I take the drives and plug them into a new RAID will I still have access to the data? I do remember the order that the drives were connected to the previous motherboard.

A: There's not really any way to know until you plug it in. If you were replacing your motherboard with the same type of motherboard, then I would say there is a very good chance of recovering your array. However, if your new motherboard supports RAID, there's a chance it will have a slightly different format than your previous motherboard, which would prevent simply mounting the drives. You can search for a raid recovery tool, or simply trial and error of getting the new motherboard to detect the drives (you'll have to figure out the raid level and stripe size, as well as preventing the motherboard from initializing the disks when you try to create the array in the BIOS).
Either way, it's most likely not going to be simple.


text:
Dapo Abiodun Wins Ogun Governorship Election
March 11, 2019 Dimeji Kayode-Adedeji
Dapo Abiodun
The governorship candidate of the ruling All Progressives Congress (APC) in Ogun State, Dapo Abiodun, has emerged the winner of the governorship election held on Saturday.
He led with 241,670 votes against his closest rival, Adekunle Akinlade, of Allied Peoples Movement (APM), who scored 222,153 votes.
The final results released on Monday by the Independent National Electoral Commission (INEC) also showed that Gboyega Isiaka of the African Democratic Congress (ADC) polled 110,422 votes while the Peoples Democratic Party (PDP) Candidate, Buruji Kashamu got 70,290 votes.
The results also showed that Mr Abiodun won in 11 local government, which include; Ijebu Ode, Odogbolu, Odeda, Ijebu North East, Sagamu, Abeokuta South, Ijebu East, Remo North, Ogun Water Side, Ikenne and Obafemi Owode.
Similarly, Mr Akinlade of the APM, won in six local government areas which are; Ewekoro, Abeokuta North, Egbado South, Ipokia, Ifo and Ado-Odo/Ota, while the Governorship Candidate of African Democratic Congress, ADC, Gboyega Isiak won in two local government comprising: Imeko Afon and Egbado North.
Mr Kashamu, a serving senator representing Ogun State at the National Assembly, managed to win a local government, Ijebu North.
The total results of the local twenty local government showed that the APC recorded 241,670, APM recorded 222,153, ADC recorded 110,422 while the PDP recorded 70,290.
The results was announced by the Returning Officer, Idowu Olayinka, of the University of Ibadan, who, thereafter, declared Mr Abiodun of the APC as the winner of the election.
Lagos Assembly confirms three commissioner-nominees, others
Sotitobire: Four charged with killing police sergeant
Passengers trapped as commercial bus catches fire in Ogun
820 buses: Court fixes date for ruling on Ambode's suit against Lagos lawmakers
Fayose: Court to rule on application challenging evidence by EFCC witness February 17
Alleged N11.5bn fraud: Court orders EFCC official, six others to appear in Alao-Akala's trial
Buhari departs for UK on Friday
Again, Nigeria plunged into darkness as national grid collapses for hours
Nyako's Trial: Court dismisses defendant's plea asking EFCC to produce witnesses


Abstract: Sci-Hub's mission is to remove barriers to research and knowledge production, particularly for unaffiliated researchers or researchers in developing countries. Yet, recent research suggests that Sci-Hub is also heavily used by researchers on college campuses, which makes this an important issue for the Davidson community to grapple with.
The panel will address several questions, such as: What is ethical for Davidson students to do in connection with Sci-Hub? Stealing is forbidden by the Honor Code, but does downloading papers from Sci-Hub represent unethical theft? Is what's legal always ethical? How are our value judgments influenced by our own privilege and access to information?
Abstract excerpt: Weeks after you graduate, you will lose access to the important databases, journals, books, and articles that you have utilized through the library for the last four years. Students are shocked to hear this and are often unprepared for finding information outside of the academy. This session will inform you about why this happens and how you, as an information creator, can change the system. The session will also feature free and open tools you can use to find information after leaving Davidson. Librarians will walk you through tips and tricks for retaining your copyright as you start to publish or work in a corporate environment.
Co-created and facilitated a panel (with fellow IL librarian Cara Evanson and Biology faculty member Kevin Smith) on Information Ethics and Privilege. Asked students to consider the ethical complexities of open access and scholarly publishing.


text:
\section{Introduction} \label{intro}

Standard Model(SM) has been very successful so far in describing the physics of
elementary particles. Precision study has played an important role in establishing the
SM through the latest discovery of Higgs boson at the Large Hadron Collider (LHC).
The properties of the Higgs boson is being studied with higher accuracy.
Recent observations at the LHC demonstrate that the systematic precision study is essential
to look for any deviation from the SM in search of new physics beyond the SM (BSM).
While there is no promising sign of new physics signature so far at the LHC,
it is extremely important to know the SM predictions for the standard processes
like Higgs and DY or $Z,W^\pm$ productions to utmost accuracy.
Not only this could help in BSM searches but also help to understand the
perturbative structure of the underlying gauge theory.

Drell-Yan production has been a standard candle at the hadron colliders and is extremely important
for luminosity monitoring. This is one of the hadronic processes which is well understood theoretically.
For example, next to next to leading order (NNLO) quantum chromodynamics (QCD)
correction \cite{Hamberg:1990np,Matsuura:1990ba,Harlander:2002wh} to this
process was computed three decades ago.
DY is also an important process experimentally for several BSM searches.
Experimentally, one has a very clean environment for precise measurements in terms of
the kinematics of the final state lepton pairs.
Higher order perturbative QCD corrections to DY provides ample opportunity
to explore the structure of the perturbation series. Thus DY serves as an important process
in collider experiments.  At the LHC, the strong interaction dynamics dominates over the others and hence
There have been attempts to go beyond NNLO accuracy in order to improve the precision from the theoretical
side.

The calculation of complete N$^3$LO cross-section is extremely
difficult due to increasing number of subprocesses involved, however there have been significant
progress to obtain third order contribution to this process in QCD. Very recently the first result at complete N$^3$LO from only virtual photon mediator has been calculated in \cite{Duhr:2020seh}.  From the theory side,
DY is seen to be extremely stable with respect to factorization and renormalisation scales
already at NNLO. The scale uncertainty has been seen to be reduced to $2\%$ for a canonical
variation of factorization and renormalisation scales compared to NLO where uncertainty is about $9.2\%$, whereas the K-factor seem to improve marginally from $1.25$ at NLO to $1.28$ at NNLO. However keeping in mind the importance of this process, it is worth studying the results from next orders and devise methods
to incorporate more and more higher order corrections.
Since a complete calculation beyond NNLO level is difficult,  the soft-virtual (SV)
contributions is often computed as first step.
In addition, the later constitutes a significant part of the cross-section
in the region where the partonic scaling variable $z\to1$, called
the threshold region.
The SV cross-sections are known for many SM processes {\it e.g.}
Higgs production \cite{Anastasiou:2014vaa,Moch:2005ky,Laenen:2005uz,Ravindran:2005vv,
Ravindran:2006cg,Idilbi:2005ni,Li:2014afw}, associated production \cite{Kumar:2014uwa},
bottom quark annihilation  \cite{Ahmed:2014cha}, pseudo-scalar Higgs production  \cite{Ahmed:2015qda}.For DY production, using the three loop quark form factor \cite{Gehrmann:2010ue}, exploring the
universal structure of the soft part \cite{Ravindran:2006bu} of SV cross
section to Higgs production \cite{Anastasiou:2014vaa},
the dominant soft-virtual (SV) corrections for DY at third order was obtained
\cite{Ahmed:2014cla} and later it was confirmed in \cite{Li:2014bfa}.

The SV contributions dominate at every order in the perturbation theory
through large logarithms spoiling the reliability of
the fixed order predictions.   The resolution to this is to resum these large logarithms to all orders.
Resummation of these large logarithms is thus very important to correctly describe the cross
section in the threshold region.
In \cite{Sterman:1986aj,Catani:1989ne,Catani:1990rp}, a systematic
approach was proposed to resum these logarithms to all orders.
The large logarithms arise in the hard partonic cross section when the total available center-of-mass energy ($\hat{s}$) becomes close to the invariant mass ($Q$) of the  final state, in other words the partonic scaling
variable $z=q^2/\hat{s} \to 1$.
This results from the soft gluon emissions, as a consequence of which the cross-section is enhanced
by the large logarithms that appear as distributions namely Dirac delta $\delta(1-z)$ and $+$ distributions:
\begin{eqnarray}
\label{eq:calD}
{\cal D}_j(z) = \left( { \log^j(1-z) \over 1-z }\right)_+
\end{eqnarray}
In Mellin space these singular
terms are transformed into powers of logarithms of
the Mellin variable $N$. In Mellin $N$ space, these contributions can be systematically resummed
to all orders and they display a nontrivial pattern of exponentiation. In the threshold region, the fixed order predictions often fail to describe the cross-section well and hence the resummation of these large logarithms becomes very important to correctly describe the region. Moreover, it has been very well established that the resummed contributions give a sizeable contributions to the cross-section. In fact many SM fixed order calculations have been improved with the corresponding resummed results, for example, the inclusive scalar Higgs production in gluon fusion \cite{Catani:2003zt,Moch:2005ky,Catani:2014uta,Bonvini:2014joa,Bonvini:2016frm} (see  also \cite{Ahmed:2015sna} for renormalisation group improved prediction) as well as in bottom quark annihilation \cite{H:2019dcl}, deep inelastic scattering \cite{Moch:2005ba,Das:2019btv}, DY production \cite{Moch:2005ky,Idilbi:2006dg,Catani:2014uta}, pseudoscalar Higgs production \cite{Ahmed:2016otz,Schmidt:2015cea,deFlorian:2007sr}, spin-2 production \cite{Das:2019bxi,das:rs} etc.
Threshold resummation not only improves the inclusive fixed order results but also differential observables like rapidity \cite{Catani:1989ne,Westmark:2017uig,Banerjee:2017cfc,Banerjee:2018vvb,Lustermans:2019cau} and in the context of LHC precision measurements, it is important to include these corrections and they are
shown to improve the fixed order results.

In the resummed predictions for the cross-sections, there is an intrinsic ambiguity on what is
exponentiated and what is not. In the standard approach, one exponentiates only large-$N$ pieces coming
from the soft function which are enhanced in the threshold region. However one can also define large logarithms in terms of a new variable $\overline{N} = N \exp(\gamma_E)$, $\gamma_E$ being the {\it Euler-Mascheroni} constant. Theoretically this is allowed, since $\gamma_E$ arises as a mathematical artifact due to dimensional regularization in $d$-space time dimensions. Moreover, this does not spoil the fact that the large-$N$ pieces are exponentiated in the threshold region. In this terminology, one exponentiates $\overline{N}$ instead of $N$. Numerically, however this makes a difference already at the leading logarithmic accuracy. It has been already seen in \cite{Das:2019btv} that the perturbative convergence is improved if one exponentiates the large-$\overline{N}$ terms. Apart from the standard threshold exponentiation, one can in fact exponentiate the complete soft function \textit{i.e.,} all the large-$\overline{N}$ terms as well as the $\delta(1-z)$ terms  arising from the soft function. We call this `Soft exponentiation' which renders some part of the $N$-independent constant ($\overline g_0$) for the standard $\overline{N}$-threshold.
In addition to these procedures, one can also exponentiate the complete form factor along with the soft function. This was studied in the context of the SM Higgs production \cite{Bonvini:2014joa,Bonvini:2016frm} and was shown to improve the scale uncertainty better than the standard threshold approach.
The form factor is process dependent and therefore is non-universal unlike the soft function. However, the form factor as well as the soft function both satisfy the similar Sudakov K+G type equation \cite{Sudakov:1954sw, Mueller:1979ih, Collins:1980ih, Sen:1981sd, Ravindran:2005vv,Ravindran:2006cg}.  Hence the solution to K+G equation for the form factor is
an exponential $N$ independent constant justifying the exponentiation.
The numerical impact of this has already been studied in the past for the DY production in \cite{Eynck:2003fn} where the authors show that both in DIS scheme and in $\overline{\rm MS}$ scheme the complete form factor exponentiates to the orders currently known.

The goal of the present article is to study the effect of threshold logarithms at N$^3$LL accuracy and match it to the known NNLO results. We perform this study for the neutral DY production as well as for onshell $Z$ and $W^\pm$ productions. The paper is organized as follows. In  Sec.\ \ref{sec:theory}, we collect the useful formulae required for the invariant mass distribution for DY and total cross-section for  $Z, W^\pm$ productions at the LHC.  Next we discuss the theoretical set up in the context of resummation. Here we describe in detail the factorization of soft-virtual coefficient in Sec.\ \ref{sec:softvir}.  Next we set up in Sec.\ \ref{sec:resummation} different resummation prescriptions as well as derive some useful formulas needed.  Sec.\ \ref{sec:numerical} we study in detail the effect of threshold logarithms for different prescriptions and present our results along with the estimation of uncertainties. We finally conclude in Sec.\ \ref{sec:conclusion}.
\section{Theoretical Framework}

\subsection{Drell-Yan and $Z,W^\pm$ production}\label{sec:theory}
The hadronic cross-section for DY or onshell $Z,W^\pm$ production at the LHC can be written as

\begin{align}\label{eq:hadronic-xsect}
\sigma
&=\sigma^{(0)}\sum_{ab={q,\overline q,g}} \int_0^1 dx_1
\int_0^1 dx_2~ f_a(x_1,\mu_f^2) ~
f_b(x_2,\mu_f^2)
\int_0^1 dz \,\,
\Delta_{ab}(z,q^2,\mu_f^2)
\delta(\tau-z x_1 x_2)\,,
\end{align}
where $\sigma =  {d \sigma \over d Q}\left(\tau,q^2\right)$ for DY production,
with $Q$ being the invariant mass of the di-lepton pair.
Here $f_a(x_1,\mu_f^2)$ and $f_b(x_2,\mu_f^2)$
are the non-perturbative parton distribution functions (PDFs) of the partons $a,b$
carrying momentum fractions $x_1,x_2$ of the incoming protons at the factorization scale
$\mu_f$. These PDFs are appropriately
convoluted with perturbatively calculable partonic coefficients $\Delta_{ab}(z,q^2,\mu_f^2)$.
For the on-shell $Z,W^\pm$ production, $\sigma = \sigma_V, V=Z/W^\pm$ and $Q=M_V$,
the mass of the vector boson.
The partonic coefficients are obtain from the partonic cross section using perturbation theory.
For the DY production $V$ we include contributions from $\gamma$ and $Z$ as well as their interference.

The partonic cross-section can be decomposed as
\begin{align}\label{eq:sv-decompose}
\Delta_{ab}(z,q^2,\mu_f^2) &=
\Delta^{{\rm(sv)}}_{ab}(z,q^2,\mu_f^2)
+ \Delta^{{\rm(reg)}}_{ab}(z,q^2,\mu_f^2) \,.
\end{align}
The first term $\Delta^{{\rm(sv)}}$ is called the SV partonic coefficient and it contains
distributions such as $\delta(1-z)$ and ${\cal D}_+$, whereas the second term $\Delta^{{\rm(reg)}}$
contains those terms that are regular in the scaling variable $z$.
The prefactors for DY and $Z,W^\pm$ production  are given as below:
\begin{align}
\sigma_{DY}^{(0)} &= \frac{2\pi }{n_c } \bigg[ \frac{Q}{S} {\cal F}^{(0)} \bigg]\,, \nonumber\\
\sigma_Z^{(0)} &=  \frac{2\pi }{n_c } \bigg[ \frac{\pi\alpha}{8s_w^2 c_w^2 S}  \bigg]\,, \nonumber\\
\sigma_{W^\pm}^{(0)} &= \frac{2\pi }{n_c } \bigg[ \frac{\pi\alpha}{4 s_w^2 S}  \bigg]\,,
\end{align}
where $S$ is the hadronic centre-of-mass energy and $n_c=3$ in QCD. For DY production, the factor ${\cal F}^{(0)}$ is found to be,
\begin{align}
\begin{autobreak}
{\cal F}^{(0)} =
{4 \alpha^2 \over 3 q^2} \Bigg[Q_q^2
- {2 q^2 (q^2-M_Z^2) \over  \left((q^2-M_Z^2)^2
+ M_Z^2 \Gamma_Z^2\right) c_w^2 s_w^2} Q_q g_e^V g_q^V
+ {Q^4 \over  \left((q^2-M_Z^2)^2+M_Z^2 \Gamma_Z^2\right) c_w^4 s_w^4}\Big((g_e^V)^2
+ (g_e^A)^2\Big)\Big((g_q^V)^2+(g_q^A)^2\Big) \Bigg]\,.
\end{autobreak}
\end{align}
Here $\alpha$ is the fine structure constant, $c_w,s_w$ are sine and cosine of Weinberg angle respectively. $M_Z$ and $\Gamma_Z$ are the mass and the decay width of the $Z$-boson.
\begin{align}
 g_a^A = -\frac{1}{2} T_a^3 \,, \qquad g_a^V = \frac{1}{2} T_a^3  - s_w^2 Q_a \,,
\end{align}
$Q_a$ being electric charge and $T_a^3$ is the weak isospin of the electron or quarks.

In the threshold region, the SV terms which consist of distributions
contribute significantly at the hadron level.  After mass factorization, the partonic coefficient in the threshold region experiences further factorization in terms of the form factor and soft-collinear function. In the next section we will discuss in detail on the structure of distributions in the SV coefficient which will form the basis for the resummation.

\subsection{Soft-virtual cross-section}\label{sec:softvir}
In the following, we briefly describe the theoretical
set up that is required to study the impact of
threshold corrections within the framework of resummation a la
Sterman, Catani and Trentedue \cite{Sterman:1986aj,Catani:1989ne}.
We do this in order to understand the role of various pieces that
contribute to the resummed result.
Exploiting the factorization of infrared sensitive contributions
and gauge and renormalisation group invariances,
inclusive cross section for the DY and on-shell $Z/W^\pm$ productions in the threshold limit can be
expressed in terms of
form factor of the neutral/charged current, soft distribution function and Altarell-Parisi kernels
(see \cite{Ravindran:2005vv,Ravindran:2006cg}).
The resulting expression expressed in $z$ space is free of both ultraviolet and infrared divergences and
captures the
distributions ${\cal D}_j$ with given logarithmic accuracy to all orders in perturbation theory.
In the Mellin $N$ space, we can achieve the same and in addition, one has the advantage to
reorganize the series in such a way that order one contributions of the form $a_s \beta_0 \log(N)$ can
be resummed systematically to all orders in the large $N$ limit.
Here, $a_s$ is defined by
$a_s = g_s^2(\mu_r^2)/16 \pi^2$ with $g_s$ begin the strong coupling constant and $\mu_r$ the
renormalisation scale and $\beta_0$ is the first coefficient of the coupling constant
beta function.  Note that in Mellin N space, the convolutions in $z$ space become
simple products.  The $z$ space result can be used to compute the soft-virtual contributions
in power series expansion of strong coupling constant $a_s$.

In $d=4+\epsilon$ space time dimensions, the threshold enhanced partonic
soft-virtual cross-section to all orders in perturbation theory
in $z$ space can be written \cite{Ravindran:2005vv,Ravindran:2006cg} as
\begin{align}
  \label{eq:sigma}
  \Delta^{{\rm{(sv)}}} (z, q^2, \mu_r^{2}, \mu_f^2) =
  {\cal C} \exp \Big( \Psi \left(z, q^2, \mu_r^2, \mu_f^2,
  \epsilon \right)  \Big)  \Big|_{\epsilon = 0} \,.
 \end{align}
Here $\Psi$ is a distribution function which is finite in the limit $\epsilon \to 0$.
The symbol $ {\cal C}$ denotes the Mellin convolution (denoted below as $ \otimes$) which in the above expression should be treated as
\begin{align}
  \label{eq:conv}
  {\cal C} \exp \Big( {f(z)} \Big) = \delta(1-z) + \frac{1}{1!} f(z) + \frac{1}{2!} f(z) \otimes f(z) + \cdots \,,
\end{align}
with $f(z)$ being a function containing only $\delta(1-z)$ and plus distributions. The finite exponent in the above is refactorized in the threshold limit and  gets contribution from the form factor \big(${\hat{\cal F}} (\hat{a}_s, Q^2, \mu^2, \epsilon)$\big) with $q^2=-Q^2$, soft-collinear function \big(${\cal \varPhi} (\hat{a}_s, z, q^2, \mu^2, \epsilon)$\big) (later called as soft function) as well as mass factorization kernels  \big($\Gamma (\hat{a}_s, z, \mu_f^2, \mu^2, \epsilon)$\big) and takes the following form in dimensional regularization:
\begin{align}\label{sv:psi}
  \Psi \Big(z, q^2, \mu_r^2, \mu_f^2, \epsilon \Big)
  = &\bigg( \ln \Big[ {\cal Z} (\hat{a}_s, \mu_r^2, \mu^2, \epsilon) \Big]^2
      + \ln \Big|  {\hat{\cal F}} (\hat{a}_s, q^2, \mu^2, \epsilon)   \Big|^2 \bigg) \delta(1-z)
      \nonumber\\
    & + 2 {\cal \varPhi} (\hat{a}_s, z, q^2, \mu^2, \epsilon)
      - 2 {\cal C} \ln \Gamma (\hat{a}_s, z, \mu_f^2, \mu^2, \epsilon) \, .
\end{align}
 $\mu$ keeps the strong coupling ($\hat{a}_s$) dimensionless in the $d=4+\epsilon$ dimensions. ${\cal Z} (\hat{a}_s, \mu_r^2, \mu^2, \epsilon)$ denotes the overall UV renormalization constant which for the processes considered here is unity due to conserved current.

The bare quark form factor satisfies the Sudakov K+G equation \cite{Sudakov:1954sw, Mueller:1979ih, Collins:1980ih, Sen:1981sd, Ravindran:2005vv,Ravindran:2006cg} which follows as a consequence of the gauge invariance as well as renormalisation group invariance,
\begin{align}\label{eq:FFKG}
\frac{d \ln \hat{\cal F}}{d\ln q^2} =   \frac{1}{2} \bigg[ {{\cal K}}(\hat{a}_s, \frac{\mu_r^2}{\mu^2},\epsilon) + {{\cal G}}(\hat{a}_s, \frac{q^2}{\mu_r^2},\frac{\mu_r^2}{\mu^2},\epsilon)\bigg] \,.
\end{align}
The function ${\cal K}$ contains all the infrared poles in $\epsilon$ whereas the function ${\cal G}$ is finite in the limit $\epsilon \to 0$. The renormalisation group invariance leads to the following solutions of these functions in terms of cusp anomalous dimensions ($A$):
\begin{align}
\frac{d{\cal K}}{d\ln\mu_r^2} = - \frac{d{\cal G}}{d\ln\mu_r^2}  = A(a_s(\mu_r)) = \sum_{i=1}^{\infty} a_s^i(\mu_r) A_{i} \,.
\end{align}
The cusp anomalous dimensions are known to fourth order \cite{Moch:2004pa,Lee:2016ixa,Moch:2017uml,Grozin:2018vdn,Henn:2019rmi,Davies:2016jie,Lee:2017mip,Gracey:1994nn,Beneke:1995pq,Moch:2017uml,Moch:2018wjh,Henn:2019rmi,Bruser:2019auj,Lee:2019zop} and are collected in Appendix \ref{app:anodim}. The $\mu_r$ independent piece of the ${\cal G}$ can be written in perturbative series as
\begin{align}
{\cal G}(a_s(q),\epsilon) = \sum_{j=1}^{\infty} a_s^j(q) {\cal G}_{j}(\epsilon)\,,
\end{align}
where the coefficients ${\cal G}^{(j)}(\epsilon)$  can be decomposed as
\begin{align} \label{eq:G}
 {\cal G}_i(\epsilon) = 2 B_{i} + f_{i} + C_{i}  + \sum_{k=1}^{\infty} \epsilon^k G_{ik}    \,,
\end{align}
where
\begin{align}
C_1  &=   0 \,, \nonumber\\
C_2  &= - 2 \beta_0 G_{11} \,, \nonumber\\
C_3  &= - 2 \beta_1 G_{11} - 2 \beta_0 \Big( G_{21} + 2 \beta_0  G_{12} \Big) \,
\end{align}
The coefficients $G_{ik}$ are the finite coefficients found in terms of QCD color factors and can be extracted from explicit calculation of quark form factor. Note that up to the third order one also needs coefficients $G_{22},G_{31}$ and thereby needs the three-loop calculation of the form factor \cite{Gehrmann:2010ue}. We have collected them in the Appendix \ref{app:anodim}.
Similar to the cusp anomalous dimension, the coefficients $f_{i}$ have been found to be maximally non-abelian to third order in strong coupling {\it i.e.} they satisfy
\begin{align}
f^g_{i} = \frac{C_F}{C_A} f^{q}_{i} \,.
\end{align}

The initial state collinear singularities are removed using the Altarelli-Parisi (AP) splitting kernels $\Gamma (\hat{a}_s, \mu_f^2, \mu^2, z, \epsilon)$. They satisfy the well-known DGLAP evolution given as,
 %
 \begin{align}
\label{eq:APRGE}
 \frac{d\Gamma (  z, \mu_f^2, \epsilon)}{d\ln \mu_f^2} = \frac{1}{2} P(z,\mu_f^2) \otimes \Gamma (  z, \mu_f^2, \epsilon)\,,
 \end{align}
 where $P(z,\mu_f^2) $ is the AP splitting functions. The perturbative expansion for these splitting functions has the following form:
 %
 \begin{align}
 P(z,\mu_f^2) = \sum_{i=0}^{\infty} a_s^{i+1}(\mu_f) P^{(i)}(z) \,.
 \end{align}
 As already discussed, only the $q\bar{q}$ channel contributes to the SV cross-section and thus we find that, only the diagonal terms of the splitting functions contribute to the SV cross-section. The diagonal part of the splitting functions is known to contain the $\delta(1-z)$ and distributions and can be written as,
 %
\begin{align}
P_{II}^{(i)} = 2 \Big[B_{i+1} \delta(1-z) + A_{i+1} {\cal D}_0 \Big] + P_{II}^{(reg,i)}(z) \,.
\end{align}
The splitting functions are known exactly to four loops \cite{Vogt:2004mw,Moch:2004pa,Moch:2014sna,Henn:2019swt}.

The finiteness of the soft-virtual cross-section demands that the soft-collinear function ${\cal \varPhi}$ will also satisfy similar Sudakov type equation like the form factor {\it i.e.} one can write
\begin{align}\label{eq:PHIKG}
\frac{d{\cal \varPhi}}{d \ln q^2} =   \frac{1}{2} \bigg[ \overline{{\cal K}}(\hat{a}_s, z, \frac{\mu_r^2}{\mu^2},\epsilon) + \overline{{\cal G}}(\hat{a}_s, z, \frac{q^2}{\mu_r^2},\frac{\mu_r^2}{\mu^2},\epsilon)\bigg] \,,
\end{align}
where $\overline{{\cal K}}(\hat{a}_s, z, \frac{\mu_r^2}{\mu^2},\epsilon)$ contains all the poles and $\overline{{\cal G}}(\hat{a}_s, z, \frac{q^2}{\mu_r^2},\frac{\mu_r^2}{\mu^2},\epsilon)$ is finite in the dimensional regularization such that $\Psi$ becomes finite as $\epsilon \to 0$. The solution to the above equation has been found \cite{Ravindran:2005vv,Ravindran:2006cg} to be
\begin{align}\label{eq:phi-sol}
{\cal \varPhi} =  \sum_{j=1}^{\infty} \hat{a}_s^j       \frac{j\epsilon}{1-z}  \bigg(  \frac{q^2(1-z)^2}{\mu^2}  \bigg)^{j\epsilon/2}   {\cal S}_{\epsilon}^j ~\hat{{\cal \varPhi}}^{(j)}(\epsilon) \,.
\end{align}
$\hat{{\cal \varPhi}}^{(j)}$ can be found from the solution of the form factor by the replacement  as $A \to -A, {\cal G}(\epsilon) \to \overline{{\cal G}}(\epsilon) $. Notice that $ \overline{{\cal G}}(\epsilon) $ are now new finite $z$-independent coefficients coming from the soft function whereas the $z$ dependence has been taken out in Eq.\ (\ref{eq:phi-sol}). This can be found by comparing the poles and non-pole terms in $\hat{{\cal \varPhi}}^{(j)}$ with those coming from the form factors, overall renormalisation constants, splitting kernel and the lower order SV terms.
The coefficient $\overline{{\cal G}}$ has same structure as the form factor in Eq.\ (\ref{eq:G}) after setting $f_{i} \to - f_{i}, B_{i} \to 0, \gamma_{i} \to 0 $,
\begin{align}\label{eq:Gt}
\overline{{\cal G}}_{i} = - f_{i}  +    \tilde{C}_{i}    + \sum_{k=1}^{\infty} \epsilon^k \tilde{G}_{ik} \,,
\end{align}
where
\begin{align}
\tilde{C}_{1} &= 0, \nonumber\\
\tilde{C}_{2} &= -2 \beta_0 \tilde{G}_{11}, \nonumber\\
\tilde{C}_{3} &= -2 \beta_1 \tilde{G}_{11} - 2 \beta_0 \Big( \tilde{G}_{21} + 2 \beta_0 \tilde{G}_{12}    \Big) \,.
\end{align}
The coefficients $f_{i}$ are same as those appear in the quark form factor in Eq.\ (\ref{eq:G}).
The coefficients $\tilde{G}_{ij}$ required up to three loops have been extracted in \cite{Ahmed:2015qia} and also collected in the Appendix \ref{app:anodim}.
Note that one has to perform the following expansion in Eq.\ (\ref{eq:phi-sol}) in order to  get all the distributions and delta function coming from the soft function,
\begin{align}
\frac{1}{(1-z)} \Big[ (1-z)^2 \Big]^{j\epsilon/2} = \frac{1}{j\epsilon} \delta(1-z) + \sum_{k=0}^{\infty} \frac{(j\epsilon)^k}{k!} {\cal D}_k   \,.
\end{align}
It is worth noting that $\overline{{\cal G}}$ as well as the complete soft function ${\cal \varPhi}_I$ satisfy the maximally non-abelian property up to three loops. Moreover ${\cal \varPhi}_I$ is also universal in the sense that it only depends on the initial legs and is completely unaware of the color neutral final state.
Expanding $\Delta^{(\rm{sv})}$ in powers of $a_s$ as
\begin{align}
\Delta^{{\rm(sv)}}_{ab}  = \delta_{a \overline b} \sum_{i=0}^{\infty} a_s^i \Delta^{(i)} \,,
\end{align}
with the born contribution being $\Delta^{(0)} = \delta(1-z) $.
The SV correction at the three loops are known \cite{Ahmed:2014cla} which we collect here for completeness in the Appendix \ref{app:svn}.

In the following, we will study the numerical impact of resummed result resulting from
$\Delta^{{\rm(sv)}}_{ab}$ after performing the Mellin transformation in the large $N$ limit.
We start with $\Psi$ which is finite while the individual contributions to it contain
UV and IR singularities.  Decomposing the later ones as sum of singular and finite parts as
\begin{eqnarray}
\ln \big|  {\hat{\cal F}} \big|^2(q^2) &=& {\cal L}_{\cal F}^{\rm{sing}}(q^2,\mu_r^2) + {\cal L}_{\cal F}^{\rm{fin}}(q^2,\mu_r^2)  \nonumber\\
\Phi(z,q^2) &=& \Phi^{\rm{sing}}(z,q^2,\mu_f^2,\mu_r^2)
+ \Phi_{\cal D}^{\rm{fin}}(z,q^2,\mu_f^2,\mu_r^2)
+ \Phi_{\delta}^{\rm{fin}}(q^2,\mu_f^2,\mu_r^2)  \delta(1-z)
\nonumber\\
{\cal C} \ln \Gamma (z,\mu_f^2)& =& {\cal L}_\Gamma^{\rm{sing}}(z,\mu_f^2,\mu_r^2)
+ {\cal L}_{\Gamma {\cal D}_0}^{\rm{fin}} (\mu_f^2,\mu_r^2){\cal D}_0
+ {\cal L}_{\Gamma\delta}^{\rm{fin}} (\mu_f^2,\mu_r^2) \delta(1-z)
\end{eqnarray}
Substituting the above equations in Eq.(\ref{sv:psi}), we can easily show
that all the singular terms in the limit $\epsilon\rightarrow 0$ cancel among
themselves.  In addition, ${\cal D}_0$ terms in finite part of ${\cal C} \ln\Gamma$
go away when added to $\Phi_{\cal D}^{\rm{fin}}$ resulting in a finite distribution.
Substituting the $\Psi$ in Eq.(\ref{eq:sigma}), we obtain
\begin{eqnarray}
\label{eq:resumz}
\Delta^{\rm{sv}}(z,q^2) = C_0(q^2) \otimes {\cal C} e^{
G_+\left(z,q^2\right)}
\end{eqnarray}
where (supressing dependence on $\mu_f$ and $\mu_r$), the $N$ independent constant $C_0$ is given by
\begin{eqnarray}
C_0(z,q^2)&=& \exp\left({\cal L}_{\cal F}^{\rm{fin}}
+ 2 \Phi^{\rm{fin}}_\delta-2 {\cal L}_{\Gamma \delta}^{\rm{fin}}\right) \delta(1-z)
\\
G_+(z,q^2) &=&\left({1 \over 1-z} \left[\int_{\mu_f^2}^{q^2(1-z)^2} \frac{d\mu^2}{\mu^2}2~ A(a_s(\mu^2)) + D(a_s(q^2(1-z)^2))\right] \right)_+
\end{eqnarray}
and $D$ in $G_+$ is related to $\overline {\cal G}$
by $D = 2 \overline {\cal G}$
and $\overline {\cal G} (a_s(q^2 (1-z)^2))$ is $\overline {\cal G}$ in Eq.(\ref{eq:PHIKG})
evaluated at $\mu_r^2=\mu_f^2=q^2$.

So far, we showed how various collinear soft gluon emissions as well as
the wide angle soft emissions can be systematically summed to all orders to obtain Eq.(\ref{eq:resumz})
in the $z$ space when partonic variable $z\rightarrow 1$.
Note that $C_0$ is obtained by first collecting those terms that are proportional to
$\delta(1-z)$ terms of $\Psi$ and then expanding the exponential of them in powers of $a_s$.
The remaining function $G_+$ contains only distributions ${\cal D}_j$.
Hence, one can predict the following structure for $G_+$:
\begin{eqnarray}
\label{eq:expz}
G_+(z,q^2) = G_1 (q^2) \otimes {\cal D}_0 + G_2(z,q^2) + a_s G_3(z,q^2) + \cdots \,.
\end{eqnarray}
where each $G_1$ sums certain terms of the $a_s^i {\cal D}_{i-1}$ to all orders,
and $G_2$ sums $a_s^i {\cal D}_{i-2}$ terms to all orders, etc etc.
The result $\Delta^{\rm{sv}}$ expressed in terms of $C_0$ and the exponential of
$G_+$ using Eq.(\ref{eq:expz}) systematically sums the distributions
${\cal D}_j$ to all orders and hence can predict
these distributions to all orders provided $A$ and $D$ are known to desired accuracy in
$a_s$.  For example, knowing $A_{1}$, we can predict
all the terms $a_s^i {\cal D}_{i}$ with $i=1,2,...,\infty$ in $\Phi$, similarly given $A_{1}$ and
$D_{1}$, we can predict $a_s^i {\cal D}_{i-1}$ with $i=1,2,...,\infty$ etc.
Hence, expression given
in Eq.(\ref{eq:resumz}) has the predictive power for $\Delta^{\rm{sv}}$ to all orders in
$a_s$ given the logarithmic accuracy in $z$ space, quantified by terms of the form $a_s^i {\cal D}_{j}$.
Note that when the exponential of $\Phi$ is expanded
using convolution rules given in Eq.(\ref{conv}), we will get not only ${\cal D}_j$ but also $\delta(1-z)$.
In other words, $\delta(1-z)$ terms in $\Delta^{\rm{sv}}$ can come from both $exp(G_+)$ as well as $C_0$.

Often in certain kinematic regions, these contributions can be enhanced when convoluted with the
parton distribution functions spoiling the reliability of the perturbation theory.  Hence we need to
include these potentially large terms to all orders in perturbation theory for any sensible predictions.
Such an exercise in the $z$ space is technically challenging due to the complexity involved in
computing the convolutions of ${\cal D}_j$.  However, in the Mellin $N$ space, the convolutions
become simple products allowing us to study the impact of these large logarithms to all orders
in a systematic fashion.  In the following, we will describe how this can be done in Mellin $N$ space.

\subsection{Threshold Resummation}\label{sec:resummation}
In the last sub-section, we showed that threshold effects
for partonic coefficients can be obtained
near threshold as a product of well-defined functions, each organizing a class of infrared and
collinear enhancements as can be seen from Eq.\ (\ref{sv:psi}).
This refactorization is valid up to corrections which are nonsingular at threshold
when partonic $z \to 1$.  While the $z$ space result captures the entire underlying
infrared dynamics in the threshold limit, it can be better described in the Mellin-$N$ space
where the threshold limit $z\to 1$ translates into $N\to \infty$.
We found that the form in Eq.\ (\ref{sv:psi}) was already suitable for all order study,
however complications arise in performing the convolution.  On the other hand any
such convolution becomes simple product in Mellin space and all the distributions coming
from the soft function are thus translated into large logarithms in Mellin $N$.

Following \cite{Catani:1989ne}, the resummed partonic SV coefficient function can be organized as follows:
\begin{align}\label{eq:res}
\hat{\sigma}_N (q^2)&= \int_0^1 dz z^{N-1} \Delta^{\rm{sv}}(z,q^2)
\nonumber\\
&= \overline g_0(q^2) \exp \Big( G_{\oline{N}}(q^2) \Big) \,,
\end{align}
where $G_{\oline{N}}$ is obtained by computing the large $N$ limit of Mellin moment of $G_+$ and then
by decomposing as
\begin{eqnarray}
\label{eq:resGNb}
\lim_{N \to \infty}
\int_0^1 dz z^{N-1} G_+(z,q^2)
= \overline G_0(q^2) + G_{\oline{N}}(q^2)
\,, \quad {\rm {with}} \quad G_{\oline{N}}(q^2)\big|_{\oline{N}=1} = 0
\end{eqnarray}
where $\oline{N} = N\exp(\gamma_{E})$ and $\gamma_E$ is E-M constant.
The $N$ independent constant $\overline g_0$ is given by
\begin{eqnarray}
\overline g_0(q^2) =\exp\left({\cal L}_{\cal F}^{\rm{fin}}
+ 2 \Phi^{\rm{fin}}_\delta-2 {\cal L}_{\Gamma \delta}^{\rm{fin}}+\overline G_0(q^2) \right)
\end{eqnarray}
$G_{\oline{N}}$ is function of the universal coefficients $A$ which are known to fourth order and
$D$ known to third order in $a_s$.  $G_{\oline{N}}$ collects and resums all the large-$N$
logarithms to all orders and it can be expressed as a
resummed perturbative series which takes the following form:
\begin{align}\label{eq:gnb}
G_{\oline{N}}(q^2) = \ln \overline{N} ~\overline g_1(\oline{N},q^2) + \overline g_2(\oline{N},q^2) + a_s ~\overline g_3(\oline{N},q^2) + a_s^2~ \overline g_4(\oline{N},q^2) + \cdots \,.
\end{align}
Following \cite{Catani:2014uta,Catani:2003zt}, we computed $\overline g_i$ up to $i=4$
(for $\overline g_i$ up to $i=3$, see \cite{Idilbi:2006dg})
and they are given in Appendix \ref{app:NBres}. Note that $\overline g_i$ coefficients are universal in the sense that it depends only on whether the born process is $q\bar{q}$ channel or $gg$ channel.
In the Mellin $N$ space, the $\delta(1-z)$ in $z$-space directly translates into $N$ independent piece
whereas the plus-distributions give rise to the $\ln(\oline{N})$ as well as $N$ independent constants
in the large $N$ limit.
Part of these constant pieces, namely $\overline G_0$, is absorbed into the
coefficients $\overline g_0$ in the standard resummation approach.
Hence, $\overline g_0$ contains only $N$ independent pieces
which come from the form factor, soft distribution function, AP kernels and $N$ independent part
of the Mellin moment of $G_+(z,q^2)$.
The condition $G_{\oline{N}}=0$ for ${\oline{N}=1}$ allows the constants $\overline g_i$ to contain $N$ independent terms.
Note that the expressions for $\overline g_0$ and $\overline g_i$ obtained this way depend on the condition
$G_{\oline{N}}=0$ for ${\oline{N}=1}$.  In other words, there is an ambiguity in
treating the $N$ independent terms in the resummed results.  Exploiting this, in \cite{Catani:1989ne},
the $N$ independent constants were defined by demanding $G_{\oline{N}}=1$ when $\oline{N}=1$.
With this, $\overline g_0$ has the following perturbative expansion:
\begin{align}\label{eq:gg0b}
\overline g_0(q^2) = 1+\sum_{n=1}^{\infty} a_s^i \overline g_{0i}(q^2) \,.
\end{align}
The successive terms in the resummed exponent Eq.\ (\ref{eq:gnb}) along with the corresponding terms in Eq.\ (\ref{eq:gg0b}) define the accuracies leading logarithmic (LL), next to LL (NLL), NNLL and N$^3$LL etc.
Terms independent of $N$ can be treated, in principle, by the same methods that resum terms enhanced by logarithms of $N$.

In summary, the resummed result will differ depending on how we treat the $N$-independent constants.
We define various
schemes that differentiate how these constants are treated in our
numerical implementation for the phenomenological
studies.
This allows us to investigate numerical impact of the various resummed results in detail.
\begin{itemize}
\item {\bf Standard $\oline{N}$ exponentiation:} This is the case we have discussed so far where we define
large logarithms are functions of $\oline{N} = N\exp(\gamma_{E}^{})$, where $\gamma_E$ is E-M constant.
The $N$ dependent functions
$G_{\oline{N}}$ in this case can be computed by simply performing the Mellin moment of $G_+(z,q^2)$ in
the large $\oline{N}$ limit and keeping only those terms that vanish when $\oline{N}=1$.

\item {\bf Standard $N$ exponentiation:} This approach differs from the previous one in the
definition of large-$N$ variable. In this case the large logarithm is simply $\ln N$ and these terms are exponentiated to all orders through the resummed exponent. It is evident that this only accounts for reshuffling of $\gamma_E^{}$ between $\overline g_0$ and $G_{\oline{N}}$ in Eq.\ (\ref{eq:res}) which now takes the following form:
\begin{align}\label{eq:resN}
\hat{\sigma}_N(q^2) = g_0(q^2) \exp \Big( G_{N}(q^2) \Big) \,.
\end{align}
 The resummed exponent $G_N$ also takes a different form  compared to the standard $\oline{N}$ exponent,
\begin{align}
G_N(q^2) = \ln N ~g_1(N,q^2) + g_2(N,q^2) + a_s ~g_3(N,q^2) + a_s^2~ g_4(N,q^2) +\cdots \,.
\end{align}
The resummed coefficients $g_i$ in  the above equation which defines the resummed accuracy,
differs from $\overline g_i$ in Eq. (\ref{eq:gnb}).  The present scheme is defined by demanding $G_N=0$ when $N=1$.
\begin{eqnarray}
\label{eq:resGN}
\lim_{N \to \infty}
\int_0^1 dz z^{N-1} G_+(z,q^2)
= G_0(q^2) + G_{N}(q^2)
\,, \quad {\rm {with}} \quad G_{N}(q^2)\big|_{N=1} = 0
\end{eqnarray}
With this definition, the rest of the $N$ independent terms from the Mellin moment of $G_+$
is combined with finite parts of form factor, soft distribution function and the AP kernels
as
The $N$ independent constant $g_0$ is given by
\begin{eqnarray}
g_0(q^2) =\exp\left({\cal L}_{\cal F}^{\rm{fin}}
+ 2 \Phi^{\rm{fin}}_\delta-2 {\cal L}_{\Gamma \delta}^{\rm{fin}}+G_0(q^2) \right)
\end{eqnarray}
and the above result is expanded in powers of $a_s$:
\begin{align}\label{eq:gg0b}
g_0(q^2) = 1+\sum_{n=1}^{\infty} a_s^i g_{0i}(q^2) \,.
\end{align}
Numerically this can make a difference and it was seen in the context of DIS previously.
In case of DY also we find such differences which will be discussed in the next section.
Up to N$^3$LL accuracy, the resummed exponents $g_i, i=1,..,4$ for both quark as well as
for gluon initiated process in $N$ exponentiation scheme can be found in
\cite{Catani:2003zt,Moch:2005ba} and we computed the results for
the $g_{0i}$ coefficients up to $i=3$ which are listed in Appendix \ref{app:Nres} along with $g_i$.

\item {\bf Soft exponentiation:} In the standard $\oline{N}$ ($N$) exponentiation, one exponentiates
$\ln \oline{N}$ ($\ln N$) and certain $\oline{N}$ ($N$) independent terms which arise from $G_+$, subjected
to the condition $G_{\oline{N}}=0$ $(G_N=0)$ when $\oline{N}=1$ ($N=1$).  The remaining $\oline{N}$ ($N$) independent
terms in the Mellin moment of $G_+$ along with $C_0$ give
the coefficient $\overline g_0$ ($g_{0})$.  In principle, we can define a scheme
wherein entire $\oline{N}$ ($N$) independent terms of $G_+$ can be
kept in the exponent.  More specifically, we define the scheme (relaxing $G_{\oline{N}}=1$ ($G_N=0$) for
$\oline{N}=1$ ($N=1$)) wherein we exponentiate all the terms coming
from the finite part of soft distribution function and those from the AP kernels.  That is,
the exponential contains
\begin{eqnarray}
\label{GNsoft}
G_{\oline{N}}^{\rm{Soft}} = G_{\oline{N}} + 2 \Psi_\delta^{\rm{fin}} - 2 {\cal L}_{\Gamma \delta}^{\rm {fin}}
\end{eqnarray}
that is,
\begin{align}\label{eq:resN}
\hat{\sigma}_N(q^2) = g^{\rm {Soft}}_0(q^2) \exp \Big( G^{\rm {Soft}}_{\oline{N}}(q^2) \Big) \,.
\end{align}
with
\begin{align}\label{eq:gnb}
G_{\oline{N}}^{\rm {Soft}}(q^2) = \ln \overline{N} ~g^{\rm Soft}_1(\oline{N},q^2) + g^{\rm Soft}_2(\oline{N},q^2) + a_s ~
g^{\rm Soft}_3(\oline{N},q^2) + a_s^2~ g^{\rm Soft}_4(\oline{N},q^2) + \cdots \,.
\end{align}
The remaining $N$ independent terms define $g^{\rm{Soft}}_0$ that is obtained by
expanding $\exp({\cal L}_{\cal F \delta}^{\rm{fin}})$ in power series expansion
in $a_s$:
\begin{align}\label{eq:gs0b}
g^{\rm{Soft}}_0(q^2) = 1+\sum_{n=1}^{\infty} a_s^i g^{\rm{Soft}}_{0i}(q^2) \,.
\end{align}
$G_{\oline{N}}^{\rm Soft}$ and $ \overline g_0^{\rm Soft}$ have similar expansion as Eq.\ (\ref{eq:resGNb})
and Eq.\ (\ref{eq:gnb}) respectively and the corresponding coefficients are calculated and
presented in Appendix \ref{app:softres}.

\item {\bf All exponentiation:}  The soft function and the form factor
satisfy K+G type Sudakov integro-differential equations given in
Eqs.\ (\ref{eq:FFKG}),  and (\ref{eq:PHIKG}) and the AP kernels satisfy renormalisation
group equation Eq. (\ref{eq:APRGE}) governed by AP splitting functions.
Hence, their solutions given the boundary conditions demonstrate exponential.  The $z$ space solutions
that we obtained carry all order information on the distribution ${\cal D}_j$ in terms of
universal cusp $A$, soft $f$ and collinear $B$ anomalous dimensions and certain process dependent
constants resulting from the form factor.
Hence it is natural to study the numerical impact of the entire contribution
in the Mellin space without imposing any condition on the $N$ dependent terms.
This can be easily achieved and the result for $\hat \sigma_N$ takes the following form
\begin{align}\label{eq:resall}
\hat{\sigma}_N = \exp \Big( G_{\oline{N}}^{\rm All} \Big) \,,
\end{align}
where
\begin{eqnarray}
\label{eq:resGAb}
G_{\oline{N}}^{\rm All}(q^2)= {\cal L}_{\cal F}^{\rm{fin}}(q^2)
+ 2 \Phi^{\rm{fin}}_\delta(q^2)-2 {\cal L}_{\Gamma \delta}^{\rm{fin}}
+\overline G_0(q^2) + G_{\oline{N}}(q^2)
\end{eqnarray}
where $G_{\oline{N}}^{\rm All}$ is expanded as
\begin{align}
G_{\oline{N}}^{\rm All}(q^2) = \ln \oline{N} ~g_1^{\rm All}(q^2) + g_2^{\rm All}(q^2) + a_s ~g_3^{\rm All}(q^2) + a_s^2~ g_4^{\rm All}(q^2) \,.
\end{align}
The present scheme was already explored in \cite{Bonvini:2014joa,Bonvini:2016frm}
for studying inclusive cross section for the production of Higgs boson at the LHC.
For similar study for the DY in DIS and $\overline{\rm MS}$ schemes, see \cite{Eynck:2003fn}.
Here we will extend it to the N$^3$LL accuracy. The relevant resummed exponent has been
provided in Appendix \ref{app:allres}.
\end{itemize}

Note that a detailed comparison between the $N$-exponentiation and
$\overline{N}$-exponentiation has been done in \cite{Das:2019btv} for the charge and neutral DIS processes.
There, one finds that the $\overline{N}$-exponentiation shows a faster convergence compared to
the $N$-exponentiation. In fact, the convergence has already been achieved at NLO+NLL
order in the threshold region in the case of $\overline{N}$-exponentiation,
whereas in $N$-exponentiation, this occurs after the NLO+NLL order.
Notice that the leading logarithmic term also differs between these two approaches.
In the case of $\overline{N}$-exponentiation, all the $\gamma_{E}^{}$ terms
are exponentiated through the variable $\overline{N} = N \exp(\gamma_E^{})$;
but in the $N$-exponentiation these $\gamma_E^{}$ terms are distributed among the exponent
and the $N$ independent term $g_0$. As a result the deviation starts already at the LL accuracy.
In the next section, we will discuss how various schemes discussed so far can affect the predictions.
Note that they all give same result at the LL accuracy, however from NLL they differ.
At NNLO level, we have the contributions from all the channels and at N${}^3$LO only SV contribution
is known so far.  Hence, our numerical predictions will be based on fixed order N$^3$LO${}_{\rm {sv}}$
results for the parton coefficients and on parton distribution functions known to NNLO accuracy.
Note that the resummed result has to be matched to the fixed order result
in order to avoid any double counting of threshold logarithms.
Hence, the matched result which is usually denoted by N$^n$LL is
computed by by taking the difference between the resummed result and the same truncated
up to order $a_s^n$. Hence, it contains contributions from the threshold
logarithms to all orders in perturbation theory starting from $a_s^{n+1}$:
\begin{align}\label{eq:matched}
\sigma_V^{N^nLO+N^nLL} &=
\sigma_V^{N^nLO} +
\sigma_V^{(0)}
\sum_{ab\in\{q,\bar{q}\}}
  \int_{c-i\infty}^{c+i\infty} \frac{dN}{2\pi i} (\tau)^{-N} \delta_{a\overline b}f_{a,N}(\mu_f^2) f_{b,N}(\mu_f^2) \nonumber\\
&\times \bigg( \hat{\sigma}_N \bigg|_{N^nLL} - \hat{\sigma}_N\bigg|_{trN^nLO}     \bigg)  \,.
\end{align}
The Mellin space PDF ($f_{i,N}$) can be evolved using QCD-PEGASUS \cite{Vogt:2004ns}.
Alternatively they can be related to the derivative of $z$-space PDF as prescribed
in \cite{Catani:1989ne,Catani:2003zt}. The contour $c$ in the Mellin
inverse integration can be chosen according to {\it Minimal prescription} \cite{Catani:1996yz} procedure.
Notice that the second term in Eq.\ (\ref{eq:matched}) represents the resummed result truncated
to N$^n$LO order, i.e. the same order to which singular SV results are available.
In the next section we present the numerical results for the DY production as well as
on-shell $Z,W^\pm$ production for LHC where we match the existing N$^3$LO fixed order SV results with the
N$^3$LL resummation derived in this article.

\section{Numerical Results}\label{sec:numerical}
In this section, we present the numerical impact of resummed threshold corrections
for neutral DY production as well as on-shell $Z/W^\pm$ production at the LHC.
For neutral DY production we consider all the partonic channels at the FO up to
NNLO with off-shell $\gamma^{*}, Z$ intermediate states.
Detailed analysis is done for 13 TeV LHC,
however it can be extended to other energies as well as to other colliders.

\subsection{Soft-virtual correction for neutral DY invariant mass}
We start our discussion by examining the SV corrections at N$^3$LO. For our numerical study,
we use the following electro-weak parameters for the vector boson masses and widths, Weinberg
angle ($\theta_w$) and the fine structure constant ($\alpha$):
\begin{align}\label{eq:parameters}
m_Z &= 91.1876 ~ \text{GeV},  ~~ \Gamma_Z=2.4952~ \text{GeV} \,,\nonumber\\
m_W &= 80.379  ~ \text{GeV}, ~~ \text{sin}^2\theta_w=0.2311 ~~ \alpha=1/128 \,.
\end{align}
We present our results for the default choice of hadronic center of mass energy $13$ TeV at the LHC. The parton distribution functions (PDFs) are directly taken from the $\tt{lhapdf}$ \cite{Buckley:2014ana} routine. Except for studying the PDF uncertainties, we use $\tt{MMHT2014}$ \cite{Harland-Lang:2014zoa} parton densities throughout.
The $(n+1)$-loop strong coupling constant is used for computing N$^n$LO order cross sections with $\alpha_s=0.120 (0.117)$ at NLO(NNLO) respectively.

\begin{figure}[ht]
\centerline{
\includegraphics[width=7.5cm, height=7.0cm]{plots/sm_inv_fo.pdf}
\includegraphics[width=7.5cm, height=7.0cm]{plots/sm_k_fo.pdf}
}
\vspace{-2mm}
\caption{\small{Invariant mass distribution (left panel) of di-lepton pair for $13$ TeV LHC
and the corresponding $K$-factors (right panel) to N$^3$LO$_{sv}$ level. The renormalization and factorization scales are set to be same as the di-lepton invariant mass.}}
\label{fig:foinv_kfac}
\end{figure}
In \fig{fig:foinv_kfac}, we present  the invariant mass distribution (left panel) of the
di-lepton production for the neutral case to N$^3$LO$_{sv}$ in QCD for $13$ TeV LHC as well as the corresponding  K-factors  (right panel). It is worth noting here that at ${\cal O}(\alpha_s^3)$ level  the $\delta(1-z)$ contribution is comparable but opposite in sign to the sum of  logarithmic contributions as is mentioned in \cite{Ahmed:2014cla}.  The $3$-loop SV corrections are found to  be positive up to around $Q=400$ GeV and remain negative for $400\text{ GeV} < Q < 2200 \text{ GeV}$ and become positive thereafter as threshold logarithms dominate in the high Q region. At around 3500 GeV, the $3$-loop SV corrections contribute by about 2\%. The observed values of $Q$ where this change in the sign happens are not fixed but can change with the center of mass energy of the hadrons.

\begin{figure}[ht]
\centerline{
\includegraphics[width=12.5cm, height=7.0cm]{plots/7-point-scale-sm-fo.pdf}
}
\vspace{-2mm}
\caption{\small{Sevel-point scale variation is plotted against hadronic $\tau$ variable up to N$^3$LO$_{sv}$ order. All the figures are normalised to LO contribution evaluated at the central scale $\mu_r=\mu_f=Q$.}}
\label{fig:scalevarFO}
\end{figure}
While the perturbation series is asymptotic and the higher orders terms are very small, the reliability of the theory predictions depends somewhat on the uncertainties due to the unphysical factorization ($\mu_F$) and renormalization ($\mu_R$) scales as well as those due to choice of PDFs. To this end, we estimate the $7$-point scale uncertainties in the invariant mass distribution at various orders in the perturbation theory by varying the scales $\mu = \{ \mu_f, \mu_r \}$ in the range $\frac{1}{2} \leq \frac{\mu}{Q} \leq 2$. The scale uncertainties are conveniently presented in terms of the invariant mass distribution at higher orders normalized with respect to LO ones. In \fig{fig:scalevarFO} we present these normalized distributions up to N$^3$LO$_{sv}$ as a function of $\tau = Q^2/S$. At LO, there is no dependence on $\mu_r$, hence the observation that these scale uncertainties are minimum around $\tau=0.001$ (corresponding to about $Q=400$ GeV) can be directly related to the behavior of the corresponding quark fluxes. At higher orders, the dependence on $\mu_r$ and $\mu_f$ is known and the scale uncertainties are found to increase with Q in the region $Q > 400$ GeV.  For $Q=1500$ GeV, they are found to be $12.55\%$, $6.23\%$, $1.50\%$ and $1.91\%$ respectively at LO, NLO, NNLO and N$^3$LO$_{sv}$. For the $3$-loop SV case, the scale uncertainties are expected to get further reduced only after including the regular terms that are yet to be computed in the fixed order perturbation theory.  However, as we increase Q value, even N$^3$LO$_{sv}$ show reasonable reduction in scale uncertainty as threshold logarithms  dominate over the regular terms for larger Q values. For completeness, we note that the scale uncertainties for $Q=3000$ GeV are found to be 21.39\%, 10.95\%, 3.04\% and 2.16\% at LO, NLO, NNLO and N$^3$LO$_{sv}$ respectively.

\subsection{Resummed prediction for neutral DY invariant mass}

\begin{figure}[ht]
\centerline{
\includegraphics[width=12.5cm, height=6.5cm]{plots/Fig_NNbone.pdf}
}
\vspace{-2mm}
\caption{\small{
The comparison between Standard $N$ and $\bar{N}$ approaches are presented up to N$^3$LL setting $\mu_r=\mu_f=Q$.}}
\label{fig:NNbcompare1}
\end{figure}
We have studied the impact of different resummation schemes as described in the previous section. First we compare the resummed results between two approaches: the Standard $\oline{N}$ and Standard $N$ prescriptions. We find that the perturbative convergence is better in the case of $\oline{N}$ exponentiation for the scale choice $\mu_r=\mu_f=Q$. This can be clearly seen from \fig{fig:NNbcompare1} where the convergence is already achieved at NLO+NLL whereas in $N$ exponentiation it happens only after NLO+NLL order. At $Q=2500$ GeV, we see the corrections received in Standard $N$ exponentiation is $21.6\%$ at NLO+NLL, $2.2\%$ at NNLO+NNLL whereas in the Standard $\oline{N}$ exponentiation these are $6.7\%$ and $2.3\%$ respectively. This observation is also true for different scale choices. This is expected since naively one can expect that as we exponentiate more and more terms the convergence becomes better.  In the rest of the discussion we will mention `Standard' only in the context of $\oline{N}$ exponentiation unless otherwise stated.

\begin{figure}[ht]
\centerline{
\includegraphics[width=14.5cm, height=7.0cm]{plots/ratio.pdf}
}
\vspace{-2mm}
\caption{\small{Comparison between the Soft (left panel) and All exponentiation (right panel) with Standard $\oline{N}$ approach. The renormalisation and factorization scales are set to $Q$. }}
\label{fig:SAcompare}
\end{figure}
We now investigate the differences resulting from
two approaches {\it viz.} the Soft exponentiation and All exponentiation to study their perturbative behavior.   To illustrate this, we show \fig{fig:SAcompare} where we took the ratio with respect to the Standard $\oline{N}$ results at each order. Notice that LO+LL results are same for all these three approaches by construction. To this end one sees that at lower orders the resummed cross-sections are improved over $\oline{N}$ exponentiations. At NNLL the Soft exponentiation gets additional $0.12\%$ corrections compared to the Standard $\oline{N}$ approach at $Q=100$ GeV. However at N$^3$LL level the Soft exponentiation does not improve over the Standard $\oline{N}$ results and both approaches provide almost same results. On the other hand,  All exponentiation still gets some contribution from higher orders through the exponentiation of complete $\overline g_0$ even at N$^3$LL order.  The increment is however very small giving only $0.12\%$ corrections over the Standard $\oline{N}$ scenario.

\begin{figure}[ht]
\centerline{
\includegraphics[width=14.5cm, height=12.0cm]{plots/kfac_resum.pdf}
}
\vspace{-2mm}
\caption{\small{The K-factors are shown for resummed results up to N$^3$LL level for all different resummed approaches consider. (see text) }}
\label{fig:resumKfactor}
\end{figure}
We have quantified the impact of resummed results through $K$-factor.
In \fig{fig:resumKfactor} we present the resummed $K$-factors (K$_{NLO+NLL}$, K$_{NNLO+NNLL}$, K$_{NNLO+N3LL}$) up to order N$^3$LL. We define the $K$-factor as $\frac{d\sigma^{resum}}{dQ}/\frac{d\sigma^{LO}}{dQ}$, where $resum$ represents all the resummed corrections up to NNLO+N$^3$LL. One observes that the perturbative convergence is improved in the case of All exponentiation compared to others although marginally. The $K$ factor defined this way will be useful to directly compare against the experimental results. For All exponentiation case, we find that the $K$-factor is $1.294$ at $Q=100$ at NNLL which changes to $1.286$ at N$^3$LL. The $K$-factor increases with $Q$. At higher $Q=2500$ GeV the $K$-factors  become $1.362$ at NNLL and $1.350$ at N$^3$LL.

\begin{figure}[ht]
\centerline{
\includegraphics[width=16.5cm, height=12.0cm]{plots/mu_var_resum.pdf}
}
\vspace{-2mm}
\caption{\small{Renormalisation and factorization scale uncertainties have been estimated through 7-point scale variation around the central scale choice $(\mu_r,\mu_f) = (1,1)Q$}.}
\label{fig:scaleresum}
\end{figure}
Next we study the uncertainties resulting from unphysical scale in these approaches. We follow the canonical variation of $\mu_f$ and $\mu_r$ around the final state invariant mass $Q$ within $[1/2,2]Q$ imposing additional constraint $1/2 \leq \mu_r/\mu_f \leq 2$ as was done in the third order SV prediction in the previous section.  We notice that different approaches for resummation provide a systematic scale reduction at lower invariant mass of the di-lepton pair. For example, in the Standard $\oline{N}$ case, the scale uncertainty reduces from $13.37\%$ at NLO+NLL to $1.99\%$ at NNLO+NNLL and $0.56\%$ at N$^3$LO$_{sv}$+ N$^3$LL.  Similar pattern is seen for the Soft and All exponentiation as well as seen in \fig{fig:scaleresum}. However, when we compare among themselves, we see that in the case of All exponentiation the scale uncertainty is reduced to $1.65\%$ at NNLO+NNLL compared to $1.99\%$ for $\oline{N}$ exponentiation and $2.09\%$ for Soft exponentiation at the same order. At the N$^3$LO$_{sv}$+ N$^3$LL, however All exponentiation gives relatively larger scale uncertainty compared to the other two approaches.
At some high invariant mass (say $Q=2500$ GeV), we see a better scale estimate at order NNLO+NNLL where we observe that the scale uncertainty systematically improved from $\oline{N}$ exponentiation from $0.53\%$ to $0.51\%$ for Soft exponentiation and $0.43\%$ for All exponentiation. However at N$^3$LO$_{sv}$+ N$^3$LL order we observe an over-estimation of scale uncertainty which gets larger for different approaches and can reach the size of NLO scale uncertainty. This shows that the sub-leading regular pieces are also important to capture the scale dependence properly.  This behavior is unlike the Higgs case where one sees a certain scale improvement for exponentiation of complete $\overline g_0$.   We will again come back on this discussion at the end of this section.

\begin{figure}[ht]
\centerline{
\includegraphics[width=8.0cm, height=7.0cm]{plots/pdf_resum.pdf}
}
\vspace{-2mm}
\caption{\small{PDF uncertainty has been estimated at NNLO+NNLL level taking $\mu_r=\mu_f=Q$. }.}
\label{fig:pdfresum}
\end{figure}
We have also estimated in our resummed predictions the uncertainties from the non-perturbative PDFs.  We convolute the resummed coefficient at N$^3$LL level with $n$ different  sets of a given PDF group and estimate the uncertainty from the {\tt lhapdf} routines.  We use the PDFs provided by {\tt ABMP16} (n= 30) \cite{Alekhin:2016uxn} , {\tt CT14} (n=57) \cite{Dulat:2015mca}, {\tt MMHT2014} (n=51) \cite{Harland-Lang:2014zoa}, {\tt NNPDF31} (n=101) \cite{Ball:2017nwa} and {\tt PDF4LHC15} (n=31) \cite{Butterworth:2015oua} groups. These results are shown in \fig{fig:pdfresum} in terms of $\delta\sigma/\sigma$ where $\delta\sigma$ is the difference between the extrema obtained from $n$ different sets and $\sigma$ is the one obtained from central set $n=0$. These PDF uncertainties in general are found to increase with the invariant mass of the di-lepton pair and, for the range of $Q$ considered here,we find that they are smallest in the low $Q$-region for {\tt AMP16} and are largest for {\tt CT14} case. These uncertainties for $Q=1500$ GeV are found to be $6.14\%$ ({\tt AMBP16}), $16.99\%$ ({\tt CT14}), $6.17\%$ ({\tt MMHT2014}), $4.21\%$ ({\tt NNPDF31}) and $7.43\%$ ({\tt PDF4LHC15}).

\begin{figure}[ht]
\centerline{
\includegraphics[width=7.5cm, height=7.5cm]{plots/central_old_high_tau.pdf}
\includegraphics[width=7.5cm, height=7.5cm]{plots/central_new_high_tau.pdf}
}
\vspace{-2mm}
\caption{\small{
Comparison between two different way of matching with the fixed order. All scales are set same as $Q$. Left: the matching is done with threshold logarithms kept in distribution space. Right: matching is done with threshold logarithms in Mellin-$N$ space.  }}
\label{fig:matchcompare}
\end{figure}
Finally, we discuss the matching relation presented in \eq{eq:matched}. We notice that the matching relation \eq{eq:matched} can be interpreted in two ways. One can match the N$^3$LO$_{sv}$ fixed order results (with $n=3$) with the resummed results subtracted up to ${\cal O}(a_s^3)$ (with $n=3$) in order to avoid any double counting from the fixed order. So far, we have followed this approach.  Instead we can match the complete NNLO fixed order result  with the resummed result subtracted up to ${\cal O}(a_s^2)$, which also avoids double counting and retains the threshold terms at ${\cal O}(a_s^3)$ in $N$-space in the threshold limit $N\to \infty$. The difference in these two approaches is sub-leading and has to be related with the fact that $N$-space threshold results when transformed back into distribution space produces sub-leading logarithms in addition to the plus distributions. In \fig{fig:matchcompare} we compare these two approaches setting all the scales same as $Q$ in Standard $\oline{N}$ approach. We see that the threshold terms defined in Mellin-N space provide much better perturbative convergence compared to the $z$-space definition. This is a well-known observation which shows that the sub-leading pieces are also important at this order. As far as scale uncertainty is concerned, this approach gives better estimate of scale uncertainty at N$^3$LL level reducing in some cases by a factor of two, however the general behavior does not change much.

\subsection{Resummed prediction for $Z/W^\pm$ productions}
In this section we present the resummed results for on-shell $Z$ and $W^\pm$ productions to N$^3$LO$_{sv}$+N$^3$LL accuracy. We use 13 TeV as centre of mass energy at the LHC. We set all the parameters same as the previous section. For pdf, we chose the central value from {\tt MMHT2014} set at the corresponding order. At the LHC, the underlying parton fluxes for $W^+$ production are larger than for  $W^-$ case, consequently the production cross sections for the former case are larger than the latter one. This is true also for higher centre of mass energies. In tab.\ (\ref{table3}), (\ref{table1}), (\ref{table2}), we present the central predictions for on-shell $Z$, $W^+$ and $W^-$ respectively with the corresponding percentage scale uncertainties. Note that the scale uncertainties are calculated again using the same procedure i.e. the seven-point scale variation around the central scale which is now the vector boson mass i.e. the central scale has been chosen as $(\mu_r,\mu_f) = (1,1) M_V$, with $V=Z$ for $Z$ production and $V=W^\pm$ for $W$-boson production.  In all the cases we observed that the fixed order scale uncertainties are systematically reduced while going to higher orders, however at N$^3$LO$_{sv}$, it again increases which is due to the fact that, at this order there is still missing pieces which are essential to the scale uncertainty. Similar observation is also seen to the matched resummed prediction.
We see that compared to the fixed order, the resummed results provide better perturbative convergence. The scale uncertainty is also seen to improve starting from NNLO level compared to fixed order.
The resummed $K$-factors as defined before, however increases from NNLO+NNLL to N$^3$LO$_{sv}$+N$^3$LL for all the cases. The absolute size of the perturbative corrections however decreases at N$^3$LO$_{sv}$+N$^3$LL compared to the previous orders confirming the reliability of perturbation theory.
\input{table3.tex}
\input{table1.tex}
\input{table2.tex}

\section{Conclusions}\label{sec:conclusion}
We have studied the Drell-Yan production as well as on-shell $Z,W^\pm$ productions in the context of threshold resummation. We have used all the necessary ingredients available to perform this task, in particular the threshold enhanced large-$N$ as well as the $N$-independent constants. The standard threshold resummation heavily reuses the results of the SV cross-section at a particular order. In particular we showed how the the large $N$-independent constants can be found at N$^3$LL level  using the existing SV results. We also explore other possibilities of doing resummation where we exponentiate the complete soft pieces coming from the soft distribution function and also exponentiate the complete $g_0$ coefficients including the form factor. All these different approaches show a systematic behavior of the resummed perturbative series which gets better when more and more terms are being exponentiated in terms of perturbative convergence. We have matched our resummed N$^3$LL results with the existing NNLO(N$^3$LO$_{sv}$) cross-section and presented results for 13 TeV LHC. We observe a systematic decrease of the size of the corrections at the third order.  At this accuracy however the missing regular pieces are also important to tame the scale uncertainty. The results for inclusive DY and $Z,W^\pm$ production demonstrate the ambiguity on exponentiation of
$N$-independent terms in the resummed results.
\section*{Acknowledgements}
G.D. would like to thank G. Bell, C. Duhr, Y. Ji, L. Magnea, S. Moch and A. Vogt for useful discussions. The algebraic computations have been done with the latest version of the symbolic manipulation system {\sc Form}~\cite{Vermaseren:2000nd,Ruijl:2017dtg}. The research of G.D. is supported partially by DESY postdoctoral fellowship and the DFG within the Collaborative Research Center TRR 257 (``{\it Particle Physics Phenomenology after the Higgs Discovery}'').

redpajama_set_name: RedPajamaArXiv


A new benchmark in wire EDM performance.
GF Machining Solutions' latest AgieCharmilles Cut P Series of wire EDM machines provides users with precise positioning and contouring capabilities that ensures high part quality and unrivalled productivity.
The Cut P 350/550/800/1250 machines are equipped with efficient power generators and feature robust design and build, intuitive human-machine-interfaces (HMIs) as well as several automation options for lights-out/unattended operations.
With Intelligent Power Generator (IPG) digital technology, the EDM machines deliver ultra-fine surface finishes (Ra 0.10 µm) and can improve cutting speeds by up to 20 per cent.
Several Expert systems further optimise the IPG's cutting performance, especially when undertaking demanding and challenging applications.
POWER-Expert monitors and changes the power levels required to machine parts with variable heights. The system protects the parts from unstable conditions while the machine maintains high cutting speeds. Simultaneously, this system controls the spark parameters during finishing operations to provide high-quality surface finishes on variable-height parts.
This maximises part finishes and significantly reduces the need for secondary benchwork.
Regardless of workpiece height, WIRE-Expert dynamically controls wire wear to achieve continuous geometrical accuracy from all directions.
For part profile accuracy, PROFIL-Expert automatically adapts the machining parameters and the cutting path for rough and skim passes to provide excellent control of fine details and to ensure positioning and contour accuracies of ±2 µm, while TAPER-Expert technology cuts angles from 0 to 30 degrees (45 degrees as an option) to maximum Z heights.
TAPER-Expert corrects shifts of the Z reference position in real time, even when the taper angle is changing. The taper angle is also held to within 10 seconds on average, providing taper accuracy and geometry reference accuracy throughout the taper cut.
Fixed tables on the machines provide rigidity and stability that ensure accuracy when machining heavy workpieces. Integrated temperature regulation systems consistently maintain temperatures within ±0.2 degrees Centigrade, which guarantees repeatability over long periods of time. However, in shops where the ambient temperature may vary throughout the day, a thermo-stability option provides an extra layer of temperature control protection from thermal effects on the machines' axes.
The CUT P machines also feature GF Machining Solutions' collision protection technology in the X, Y, U, V and Z axes that prevents potential machine damage and eliminates costly down time and maintenance, even at maximum axis travel speeds of 3 m/min.
The Windows 7-based AC Cut HMI 2 user interface on the machines simplifies file management and maximises operator efficiency with its 19 in. touch screen and latest i7 CPU technology.
This system also supports intuitive programming directly on the control as well as imported CAD/CAM files with built-in AC CAM Easy CAD/CAM software. The EDM series is also ready for the Internet of Things/Industry 4.0 with eConnectivity features that connect the machine to larger shop management systems for effective monitoring and increased machine utilisation.
New automation solutions, including Automatic Slug Management (ASM), Automatic Slug Welding (ASW) and Renishaw probing systems, reduce processing time by up to 20 per cent in lights-out production environments.
Additionally, the Cut P Series 350 and 550 can integrate with robots such as the System 3R WorkPartner 1+ that automatically loads and unloads parts for a single machine or within a two-machine automation cell that doubles machine production capacities.
With over 600 dedicated, pre-programmed cutting processes onboard, the machines optimise production for electronics, automotive, medical and mould and die manufacturers.
The machines can handle parts ranging from less than 1mm up to 800mm in height, weighing up to 6 000 kg in weight and made from a diverse range of materials including steel, carbide and copper to aluminium, titanium and graphite.