text:
For debate
Trials and tribulations in the removal of dextropropoxyphene from the Australian Register of Therapeutic Goods
Nick A Buckley and Thomas A Faunce
Med J Aust 2013; 199 (4): 257-260. || doi: 10.5694/mja13.10040
Published online: 2 September 2013
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The Therapeutic Goods Administration determined in November 2011 that dextropropoxyphene should be removed from the Australian Register of Therapeutic Goods. This is consistent with this drug's removal from the market in many other developed countries.
However, dextropropoxyphene is still on the market in Australia owing to a series of appeals made to the Administrative Appeals Tribunal (AAT) by the drug's manufacturer.
There is a difference between the standards by which the AAT judges the safety and efficacy of medicines and the standards used for registering therapeutic goods by regulatory agencies worldwide.
This raises the question as to whether the appeal process against TGA decisions appropriately serves the Australian public interest.
In November 2011, the delegate of the Secretary of the Department of Health and Ageing gave notice that the Therapeutic Goods Administration (TGA) intended to remove dextropropoxyphene (brand names Di-Gesic and Doloxene) from the Australian Register of Therapeutic Goods.1
This followed a review of the safety and efficacy of medicines containing dextropropoxyphene. The TGA "determined that the overall risk of serious adverse reactions outweighs any benefits that may be provided by these medicines".2 The main concern was the risk of sudden death from cardiotoxicity (not shared by other opioid drugs), in the setting of renal impairment, drug interactions and accidental or deliberate overdose.2-6
Evidence against dextropropoxyphene has been accumulating for over 30 years.3 The drug has been removed from the market by medicine regulators in the United Kingdom (2004), European Union (2009), United States (2010), New Zealand (2010) and elsewhere.2 Over a decade ago, the Australian Pharmaceutical Benefits Advisory Committee (PBAC) recommended removal of subsidies for dextropropoxyphene, and the Australian Medicines Handbook and Therapeutic Guidelines advised against using the drug.5,6 However, dextropropoxyphene remains on the market in Australia in 2013, 18 months after the TGA's decision to delist it, owing to a series of legal appeals by the drug's manufacturer before the Administrative Appeals Tribunal (AAT).1
Deregistration of dextropropoxyphene in the AAT
The manufacturer of dextropropoxyphene in Australia, Aspen Pharmacare Australia Pty Ltd (Aspen), asked the Minister for Health and Ageing to reconsider the deregistration decision, but in January 2012 it was reaffirmed. The manufacturer then appealed that decision to the AAT. Both the manufacturer and the TGA accepted that the ultimate issue in the appeal was whether the "quality, safety or efficacy" of each drug "is unacceptable" within the meaning of section 30(2)(a) of the Therapeutic Goods Act 1989 (Cwlth) (the TG Act).
The AAT appeal was heard by Justice Kerr, President of the Tribunal, and Senior Member Dr Teresa Nicoletti. Teresa Nicoletti is also a partner at Piper Alderman, Founding Director of Pharmedica Consulting and a consultant to, and former regulatory and medical affairs manager with, several large pharmaceutical companies. In general terms, the appeal of the manufacturer, Aspen, was based around an argument that the drug was efficacious in some patients for whom other moderate analgesics did not work and that it could be used safely provided patients did not take excessive doses, and if the prescribing indications, contraindications and warnings were all carefully observed.1 The manufacturer also indicated [para 100]1 that it would be prepared to enter into contractual obligations with all pharmacies retailing dextropropoxyphene to supply it only on production of a signed statement from the prescribing doctor and countersigned by the patient about awareness of the risks. To obtain this stay, Aspen undertook to add significant safety warnings to the product information and consumer medicine information for Di-Gesic and Doloxene highlighting, among other things, that products containing dextropropoxyphene had recently been associated with substantial prolongation of the QT interval. Aspen also agreed to write "dear doctor" and "dear pharmacist" letters, in a form approved by the TGA, drawing attention to the additional safety warnings [para 11].1
Evidence was tendered to the AAT that despite regulatory decisions to remove dextropropoxyphene from general availability in the UK and US, it continued to be available under prescription in those jurisdictions [para 94].1 Rejecting arguments from the TGA's counsel that the drug be made available only under the Special Access Scheme and to authorised prescribers (s19 TG Act) (it had only remained available after the UK ban on this sort of scheme), the AAT decided to remit the matter again to the Minister to see if agreement could be reached under s28 of the TG Act for the drug to remain available with "highly visible warnings and in blister packs designed to minimise the risk of misuse" [paras 99 and 103].1 The AAT also indicated that if the manufacturer still felt aggrieved by the TGA decision it could return to the AAT to continue its appeal. The manufacturer ultimately took up this option, and in April 2013 the AAT upheld the appeal in its final decision on this matter.7
Role of the AAT in Australian therapeutic drug regulation
The AAT is usually the final hurdle for the TGA in regard to appeals against drug regulatory decisions in Australia. The role of the AAT is "to provide independent merits review of administrative decisions".8
The Tribunal reviews a wide range of administrative decisions made by Australian Government ministers, departments and agencies . . . The Tribunal aims to provide a review mechanism that is fair, just, economical, informal and quick.8
Most of the AAT's work relates to review of decisions made in regard to social security, worker's compensation and taxation; only 12 of 5682 applications lodged before the AAT in the past financial year related to therapeutic goods, and appeals lodged by pharmaceutical companies often took over a year to resolve.9 Fighting litigation is expensive and time-consuming for the TGA; it takes resources away from its core roles, and the threat of legal appeal may undermine the capacity for regulators to make decisions solely in the best interests of public health.10,11
How well is the AAT equipped to make final judgments concerning the safety and efficacy of medicines? How well does the TG Act support the primary aim of promptly making scientifically sound decisions in the best interests of public health? Under the TG Act, the initial decision for deregistration of a medicine is made by the government on the grounds that safety, quality and efficacy are unacceptable. This can be contested by the manufacturer, who has the right to request the Minister to reconsider the decision; and then, if the manufacturer finds that outcome unsatisfactory, they can apply to the AAT for independent merits review of the decision.
In conducting that review, the AAT is not required to judge safety and efficacy by the usual standards set down for registering therapeutic goods by regulatory agencies worldwide — that is, on evidence of efficacy from clinical trials.12 The interim AAT judgment in this case stated:
There is no case law known to the Tribunal that suggests that the words 'safety or efficacy' should be construed otherwise than as ordinary English words, albeit read in the context of the Act as a whole
and refers to the Macquarie and Oxford dictionaries.1 Efficacy becomes "ability to bring about the intended result". Evidence on efficacy presented to the AAT included patients' and doctors' opinions, 1998 Australian Drug Evaluation Committee minutes, and a 1984 publication from an industry-sponsored symposium involving reanalysis of small numbers of patients from previously published studies.1
Notably, the clinical trial data that would be a requirement for new drugs for which manufacturers are seeking TGA registration were not present in this case. Dextropropoxyphene gained registration through the process of "grandfathering" after the commencement of the TG Act, and the AAT did not compel the production of such data. In contrast, regulators today require that evidence of efficacy comes from clinical trials conducted to established regulatory standards.12
The TGA submitted in the 2012 AAT hearing that after the banning of dextropropoxyphene in the UK "there was a significant reduction in the rate of deaths by suicide and no commensurate increase in the number of deaths associated with other analgesics"1,4 (Box). However, the AAT judgment referred to "the straightforward and inescapable mathematical consequences" of these data being that there had been an increase in suicidal poisoning by other drugs.1 The AAT made the following comments about the expert witnesses who agreed with the TGA's interpretation of these data:
. . . it was plain that both witnesses were highly resistant to conceding any point that might possibly be thought to assist the interests of the Applicant. This was particularly noticeable in both cases in respect of the Hawton et al evidence.1 [para 25]
The figure in the Box shows the data4 referred to in the judgment (the ban was phased in over the 3 years 2005–2007). The reduction in all drug suicides (362 fewer in 2008 than 2004) was greater than that from fatal poisonings attributed solely to dextropropoxyphene (196 fewer), suggesting that there may also have been a large additional reduction in mixed-drug overdoses. The frequency of other analgesic poisonings did not change significantly over 13 years (the rate varied between 209 and 287, but the linear trend was horizontal [− 0.5 deaths/year] and the 2010 data are equal to the long-term median [257]). Scientific thinking understands that minor year-to-year fluctuations are expected and are simply statistical noise rather than signals.
In another example, the manufacturer7 and the AAT [para 44]1 have also made much of the fact that there were more deaths recorded in the past decade in New South Wales coronial inquiries from codeine, paracetamol and tramadol than from dextropropoxyphene; however, no attempt was made to put that into perspective by adjusting for the vastly different rates of use.
The TGA has a statutory obligation under the TG Act to protect Australians against risks posed by unsafe, poor-quality and low-efficacy pharmaceuticals. In legal terms, the TGA administers the TG Act to ensure the quality, safety and efficacy of any therapeutic good supplied or produced in Australia. The TGA is responsible for licensing manufacturers, approving therapeutic products, and regulating such products once they enter the Australian market.
The developments in Australian law, arising from both policy and adjudicative determinations, potentially erode the ability of the TGA to firmly uphold its mandate. For example, the Australian Government's decision to pay compensation rather than fight the civil litigation concerning the TGA's action against Pan Pharmaceuticals has much broader implications for public health, to the extent that the decision could be interpreted as undermining the regulatory authority of the TGA.10
The final AAT decision overruled cancellation of Di-Gesic and Doloxene from the Register of Therapeutic Goods by the TGA but, despite this, summarised the AAT view on the key issues of efficacy and safety in the following statements:
Given that neither Di-Gesic nor Doloxene have been found to have exceptional efficacy and that the Tribunal has found that the risk of accidental overdose is real — such that it was common ground between the parties that the amount of DPP contained in a single blister pack of either Doloxene or Di-Gesic, if combined with alcohol, could in a vulnerable patient be sufficient to create a risk of sudden death — the Tribunal has concluded that the safety and efficacy of Di Gesic and Doloxene would be acceptable if, and only if, additional conditions were imposed which provide a high level of assurance that doctors will prescribe the products (a) only if therapeutically justified; (b) only after they have considered any contraindications and any recent changes to the patient's clinical presentation or medical status; and (c) only after they have discussed with their patient the appropriate use of the products and the risks of overdose.7 (para 44)
. . . the safety and efficacy of Di-Gesic and Doloxene would be acceptable if, and only if, there were conditions put in place directed at minimising the risk that doctors might prescribe the product if a patient's history indicates that he or she is at risk of intentional self-harm".7 (para 45)
The AAT expectation that doctors and patients can comprehensively deal with the consequences of this decision contrasts with clear contemporary evidence that consistent delivery of care according to guidelines does not usually occur in the Australian health system.13
This case highlights the need to reconsider whether this appeal process appropriately serves the Australian public interest. Many developing countries around the world regard Australian registration of a therapeutic good as evidence that the drug has passed thorough scientific review.14 The current market for dextropropoxyphene is now almost entirely in such countries. When registration can be contrary to TGA advice and based around civil court proceedings rather than scientific interpretation of evidence, the international reputation of Australia's drug regulatory system could be at stake.
Regulators need to consider legislation and precedents when undertaking regulatory actions on registered drugs, but ultimately this is a finely nuanced expert scientific judgement about evidence of risk and benefit.15 Evidence of harmful effects is often of lower quality and might take many years to accumulate. An alternative appeal process such as that used by the US Food and Drug Administration, based around independent expert panels and ombudsmen, might be better for reviewing disputed scientific evidence.16 The evidence put forward at appeals should be restricted to that generally accepted by regulatory bodies. The Commonwealth Parliament could also legislate to protect the TGA (and other scientific regulators) from legal challenges against good-faith actions designed to protect public safety; that is, when they are acting in the presence of substantial (but not conclusive) proof of significant harm. Such legislation might have safeguarded the TGA from actions in the Pan litigation; the TGA acted promptly to protect public health after discovery of widespread and serious breaches in manufacturing practice, only for taxpayers to lose over $122 million dollars in subsequent court cases.10
There was a very long lead time during which there were increasingly strong warnings to prescribers that the risk–benefit ratio of dextropropoxyphene was unfavourable2,5,6 before the TGA finally sought delisting (the last of the world's leading drug regulatory agencies to do so).2 We are concerned that the Australian legislative and appeal framework and recent adverse litigation against the TGA contributed to this delay. While acting too early restricts drugs that are potentially still useful in some individuals, late action has often been measured in lives lost (thousands have died worldwide from dextropropoxyphene poisoning,3,4 and several deaths have occurred in Australia even as this appeal has unfolded [National Coronial Information System; unpublished data]). The TGA should be empowered to generate further evidence, or take prompt action on the balance of probabilities.17 There is only one benefit to the Australian public that could come from this current appeal — it should clearly highlight to our government the urgent need to revise the legal appeal processes that in our view inappropriately burden the TGA when it decides to take action to protect the Australian people.
Deaths from Co-proxamol* and other analgesic poisoning, and suicide by drug poisoning, United Kingdom, 1998–2010†
* UK brand name for dextropropoxyphene–paracetamol. † Graph drawn from data from Table 2 in: Hawton K, et al. PLOS Med 2012; 9: e1001213.4 Shading represents the period over which a ban on dextropropoxyphene was phased in.
Provenance: Not commissioned; externally peer reviewed.
View this article on Wiley Online Library
Nick A Buckley1
Thomas A Faunce2
1 Medical Professorial Unit, Prince of Wales Hospital Clinical School, Sydney, NSW.
2 College of Medicine, Biology and Environment and ANU College of Law, Australian National University, Canberra, ACT.
Correspondence: n.buckley@unsw.edu.au
Competing interests:
Nick Buckley was an expert witness at the AAT appeal against the delisting of dextropropoxyphene; he is also a member of the Advisory Committee on the Safety of Medicines and the Chairman of the Editorial Advisory Board of the Australian Medicines Handbook.
1. Aspen Pharmacare Australia Pty Ltd and Minister for Health and Ageing [2012] AATA 362 (15 June 2012). Last updated Aug 2012. http://www.austlii.edu.au/au/cases/cth/aat/2012/362.html (accessed Jan 2013).
2. Therapeutic Goods Administration. TGA to cancel four prescription pain-killers from 1 March 2012. TGA, 2 Dec 2011. http://www.tga.gov.au/newsroom/media-2011-dextropropoxyphene-111122.htm (accessed Jan 2013).
3. Finkle BS. Self-poisoning with dextropropoxyphene and dextropropoxyphene compounds: the USA experience. Hum Toxicol 1984; 3 Suppl: 115S-134S.
4. Hawton K, Bergen H, Simkin S, et al. Six-year follow-up of impact of co-proxamol withdrawal in England and Wales on prescribing and deaths: time-series study. PLOS Med 2012; 9: e1001213.
5. Rossi S, editor. The Australian medicines handbook: AMH. Adelaide: The Australian Medicines Handbook Unit Trust, 2000.
6. Therapeutic Guidelines Expert Writing Group. Therapeutic guidelines: analgesic. Version 4. Melbourne: Therapeutic Guidelines Ltd, 2002.
7. Aspen Pharmacare Australia Pty Ltd and Minister for Health and Ageing [2013] AATA 197 (5 April 2013). Last Updated: 10 Apr 2013. http://www.austlii.edu.au/au/cases/cth/aat/2013/197.html (accessed Apr 2013).
8. Administrative Appeals Tribunal. What we do. http://www.aat.gov.au/AboutTheAAT/IntroductionToTheAAT.htm (accessed Aug 2013).
9. Administrative Appeals Tribunal of Australia. [Database of decisions]. Australasian Legal Information Institute. http://www.austlii.edu.au/au/cases/cth/aat (accessed Jan 2013).
10. Shirlow E, Faunce T. Recent legal developments and the authority of the Australian Therapeutic Goods Administration. J Law Med 2009; 16: 764-769.
11. Hamburg MA. Shattuck lecture. Innovation, regulation, and the FDA. N Engl J Med 2010; 363: 2228-2232.
12. Therapeutic Goods Administration. Note for guidance on good clinical practice (CPMP/ICH/135/95). TGA, 2000. http://www.tga.gov.au/industry/clinical-trials-note-ich13595.htm (accessed Jan 2013).
13. Runciman WB, Hunt TD, Hannaford NA, et al. CareTrack: assessing the appropriateness of health care delivery in Australia. Med J Aust 2012; 197: 100-105. <MJA full text>
14. Ratanawijitrasin S, Wondemagegnehu E. Effective drug regulation: a multicountry study. World Health Organization, 2002.
15. Committee on Ethical and Scientific Issues in Studying the Safety ofApproved Drugs, Board on Population Health and Public Health Practice, Institute of Medicine. Ethical and scientific issues in studying the safety of approved drugs. Washington, DC: National Academies Press, 2012.
16. US Food and Drug Administration. About FDA. CDER Ombudsman. http://www.fda.gov/AboutFDA/CentersOffices/Officeof MedicalProducts andTobacco/CDER/ContactCDER/CDEROmbudsman (accessed Jan 2013).
17. Hamburg MA. The growing role of epidemiology in drug safety regulation. Epidemiology 2011; 22: 622-624.


text:
Next Match:
Leeds United in a Pre-Season Friendly.
Wednesday 17th July 2019, KO 12:00 BST.
JOSE- Man Utd material or not.
Forum List Football U-N-I-T-E-D
marcus leong
Re: JOSE- Man Utd material or not.
Relly wrote:
marcus leong wrote: ↑
Relly wrote: Just let him go, or we will be dealing with this shit until the day he retires. And frankly I am sick of it.
World class players will always attract buyers.
It's just a matter of time with him, it makes so much sense. Keepers don't make or break a team. We can find a decent replacement and not deal with this distraction every season.
Why? It's not like he keep saying he wanted to join them like Ronaldo. In fact he had came out to support Atletico.
I don't see how Real's interest is his fault.
Rick: "Weddings are basically funerals with cake."
Relly
I didn't say it was. Although he hasn't come out and categorically denied it.
"Make America Great Again!"
Relly wrote: It's just a matter of time with him, it makes so much sense. Keepers don't make or break a team. We can find a decent replacement and not deal with this distraction every season.
Probably fishing for a new contract.
redmaria
First Team Player
He's two years into a 4 year contract
He is the best in the world
He should not need to fish for a new contract
He should get an annual 10% increase and put his mind to rest tp concentrate on football
redmaria wrote: ↑
Whether they follow the advice of their agent or their own decision, players been doing that forever. Both past captain, Keane and current captain Rooney did it too.
I am more and more convinced this manager is losing the plot
Did anybody see his press conference
A couple of questions caught my attention
From the Irish journalist trying to build the atmosphere for the Irish fans,even though they rarely attend any more
"So Jose are you really excited,is the excitement really building up,are you looking forward to Wednsday/"
"No i am not excited.I am not thinking about the game.I am only thinking about Sunday
I want to do the same as i did against Arsenal,Tottenham and Southampton"(FFS we lost 2 and drew one and didn't score a goal.Indeed barely a shot on target) and he wants to do the same again
"After the game on Sunday i will not think about the game on Wednesday.We will train on Tuesday and i will then get on a plane"
Irish journalist not giving up on behalf of the Irish fans that are left
"So Jose.After all the fixture congestion and all the games you have had to play are you ready and looking forward to the game"
"Not really.We have played so many games we have been living in hotels for 160 days this season.This is not ideal"
(FFS he lives in a hotel 365 days as he won't commit and buy a house.He talks more contradictory shit that Donald Trump)
Karen-"One last question and not from you"(the Irish journalist)
"I have heard that every player has to go to this game"
"Yes we are all going.Some will be on crutches but we are one big family"
(Jose are you sure?.Ask Smalling and Jones and Shaw and Martial and DDG)
Jounalist
"Even Zlatan?"
"Yes Zlatan will be with us.He should get a medal"
Must be a few badged up shirts left in the megastore!!!
He is losing the plot
The above is not the United way i am afraid
Especially no party after the game and no street parade to celebrate if we win as United can't afford either.What is this club coming to
SUBXERO
International Player
redmaria wrote: I am more and more convinced this manager is losing the plot.
This is the thought I feared most from day one of his tenure. Sadly, I'm slowly coming to this conclusion myself. The last month or so his behaviour was way off the (United) mark.
UNITED > ENGLAND
Jose is becoming more irrational by the day
I have now found out that immediately after doing his press interview along with "one last question Karen" he was scheduled to do a personal interview with MUTV as part of the UEFA final build up
They had a whole crew of cameramen,sound engineers and interviewers who had spent a long time setting up for a prearranged interview
He had just spent half an hour talking about harmony,togetherness and companionship in the club
After a fairly soft press interview when nobody asked him anything hard except for one Irish interviewer who tried in vain to get him to show excitement for next Wednesday he stormed past the MUTV crew and without giving a reason refused to do an interview with them now or before the big game
They are not sure if he is rude,obnoxious,a pain in the arse or a total wanker
Either way i am sure one of you will find a way to convince themselves he is somehow playing a mind game to upset Ajax
The general feeling is that he might be wishing he wasn't at this club and it was also pointed out that in his contract there is a break clause that United can get rid if he is not in the Champions league
Either way next season is going to be critical for both results and style
But oh yes.
Next season is going to be a great season isn't it
Not like the last 4
joze ruis
interesting article :
Jose Mourinhohas challenged the "fragile" members of Manchester United's squad to develop a winner's mentality after admitting they risk losing his trust amid his persistent struggles to understand players who do not share his own fierce will to win.
The United manager said his biggest failing revolved around his difficulties working with players who have "a different mentality to what I have" and that he has to lean on his assistants to try to coax the best out of such players.
Mourinho cited Marcus Rashford as having the sort of mentality he relishes working with, but in pointed reference to the likes of Anthony Martial, Luke Shaw and Chris Smalling, whom he battled with throughout the campaign, the Portuguese conceded that certain players can lose their appeal to him.
Speaking to Rio Ferdinand in a revealing BT Sport documentary, Old Trafford Uncovered – with Jose Mourinho, that was filmed before United's Europa League final win over Ajax, Mourinho said: "For me what's more difficult [as a manager] is the fragile mentality.
"I think it's probably my weakness as a manager, that it's difficult for me to understand people with a different mentality to what I have. It's difficult to understand so it takes me time to understand and sometimes I'm not able to feel attracted again [to the player].
"Sometimes I ask my assistants to help me on that because maybe they have a different profile to me. Because for me I want to be in love with the player with this character, with this personality and that kid Marcus is the best example of it, especially in this club."
Mourinho's remarks are likely to increase the pressure on Shaw and Smalling, both of whom face uncertain futures at Old Trafford, as well as Martial and even Henrikh Mkhitaryan ahead of next season.
Mourinho has made several references this season to the need for Martial and Mkhitaryan to toughen up mentally and deliver more consistently on their talents and used England striker Rashford, still only 19, as an example of the attitude and mentality he is looking for.
United's Europa League success guaranteed qualification for next season's Champions League and earned the club a third trophy after the League Cup and Community Shield but Mourinho believes players must raise their game next term.
"As players, you have to have that responsibility every day and it's something my boys have to learn," Mourinho said. "Some of them they can win a big match. What I think they can't do is win it and win it and win it and cope with that intensity, especially that mental intensity, the focus, the concentration, the responsibility, the determination. This is something you only learn when you are at the highest level.
"I try to do it [teach and transfer it] in relation to the personality of the players. For example, what I love about this kid Marcus is he copes with the pressure. I can press him, I can be very demanding with him, he's the kind of kid that when the training session finishes it doesn't finish for him. He wants more. He lives for football.
"He was, of course, a tornado, when he arrived last season without pressure. No responsibility. And this season when the pressure was on him, when he was not scoring goals for one, two, three, four months, he coped with that pressure. This is the kind of character I like. I like to squeeze the player and the player to cope with it and react to it."
Mourinho said he wanted his players to grasp the scale of United's history but not feel overburdened by it. "I think it's very, very important but at the same time they shouldn't be over pressurised by that," he said.
"That's why I asked at the training ground, which is full of [pictures of] the previous history, to do a wall with the last FA Cup and last League Cup on it. And people can say this is not the Champions League, this is not Premier League. but this is their cup. They are the two cups they won.
"So of course it's an amazing history and they have to feel proud with the same shirt of these people. The No. 11 has to know that shirt was Ryan Giggs' shirt and the No. 7 must know that was Best and Beckham and Cristiano's shirt, they have to know this and be proud of it but they cannot be over pressurised by it. There is not another Giggs, there is not another Scholes, there is not another Rio, there is not another Best. They have to be what they are and they have to cope with it. It's part of my job that I have to be responsible for."
Mourinho also admitted it can be harder motivating young players now who earn riches almost instantly.
"In the past you could think about being at the club for the rest of your life," Mourinho said. "You became rich at the end of your career. Now they become rich at the beginning of their careers. Yes [that makes the job harder]. You need to be lucky with the players, you need to be lucky with their parents, with the people around them. Maybe it's harder to motivate them and harder to make them focused on what they have to do. If they're not the right mentality they can lose a bit of appetite ,if they have the right mentality and are rich even better but if you don't have the mentality it makes it hard."
joze ruis wrote: ↑
Colin Francis
And if he continues this non-Utd type of football then he should leave! First season and perhaps not his squad but hopefully second season will see our style back again.
...and the Truth shall make you free.....
Colin Francis wrote: ↑
Academy Player
All we can do is trust him and our squad!
JonSnow wrote: ↑
I trust him with our squad.Well sort of.
It is the squad,that is chosen by Jose but negotiated by Ed Woodward where the worry is for me and a high percentage of fans
The silence continues to be deafening whilst others are starting to move
I know City are moving quickly because they have too
Strong rumours are that they will get a sanctions of signing on players in the very near future which is why they are dealing now
However if Liverpool start signing quality players and/or Chelsea then it will probably be as in previous years
Penny upping a transfer fee on some shirt seller
Never mind i am a pessimist with this shower of thieves
I know some of you are getting excited about incoming transfers and Champions league football
Next season will be a great season
Harley Ben
Dude who are you comparing Mourinho to? Alex Ferguson era is long gone. He won't come out to win PL titles for the fans anymore. We have to trust our manager. David Moyes was not a worthy successor of Ferguson. LVG had tiring and stale training methods. But under LVG, we were performing well in bigger matches. Mourinho is a tactical genius and has been the only Manager who gave tough time to Ferguson when he was at Chelsea. We should have faith in him. It is because of him, we got Ibra, Pogba and Mikhi. We will continously improve. He needs time and he needs players at his will.
Playing attractive is important but winning has to be our top priority. I like the question above about playing attractive and not winning?
We shall introduce fast pace game in the coming season. You see, Ibra is offloaded, we are probably getting Morata who is a bit pacy.
dbecks07
Great read : https://www.theguardian.com/sport/blog/ ... erpunching
esobeso
dbecks07 wrote: ↑
+1 Thanks
ArizonaRed
I have always felt that Mourinho was the most prepared manager in football. He not only knows what his players are capable of, but what the opposition players can do and what his side needs to do to neutralize the opposition. Very rarely does he get caught in the trap of asking his players to do something they are not capable of doing, he adjusts his game-plan to take full advantage of the skill sets of his squad and his game changing substitution are second to no-one.
When he joined United I had some trepidation about his off the field antics but since joining he has been a model manager as far as I am concerned.
I can't believe it. I can't believe it. Football. Bloody hell.
ArizonaRed wrote: ↑
This. He seems to calm a lot, which I appreciate.
I wonder if it's because this is the job he's always wanted.
All I can say is I told you guys so and frankly when it comes to picking a manager 'style' needs to be criteria.
Jose negative bus parking style had me pulling my hair out of my head.
Thank God the nightmare is over and the flair is back.
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Two pieces of toast were hanging out looking for a marshmallow to eat. Just when they got tired and gave up, they saw a muffin and decided to take a bite. The muffin, however, was the king's best friend, and he died when they ate his head. The king of all the marshmallows took an oath to kill the pieces of toast and everything he loves. Can you help the toast survive?


text:
Datasheet - Dhaka Division
Suicide Attacks
Arms Recovery
Number of Terrorism Related Incidents
Major Incidents (Text Data)
Data View Graph View Chart View Map view
Total Number of Incidents
Initiated By Security Forces
Initiated By Militant
Initiated By NS
Initiated By Civilians
2001 6 45 1 0 10 56 175 0 0 100 275 0 0 4 2
2002 1 18 0 0 0 18 299 0 0 0 299 0 0 1 0
2003 1 8 0 0 0 8 15 0 0 0 15 0 0 1 0
2016 4 20 2 25 0 47 0 0 1 53 54 0 3 1 0
2017 1 0 0 7 0 7 0 0 0 0 0 0 1 0 0
Total** 20 142 7 41 10 200 885 0 9 153 1047 0 5 11 4
*Data since March 6, 2000, ** Data till , January 21, 2020
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text:
\section{Introduction}

Supersymmetry (SUSY) is not only a candidate for physics beyond the standard model
but also a possible framework (or a part of it) with definite ultraviolet behaviors in quantum field theories.
If we can construct supersymmetric theories nonperturbatively,
we shall obtain deeper understandings of vacuum property and
 other  nonperturbative behaviors\cite{Feo2003,Kaplan2004,Kaplan-Catterall}.
Although the SUSY algebra is generally incompatible with a discrete lattice space,
there is no obstruction to formulate a free SUSY theory on lattice,
if we replace an infinitesimal translation with a lattice translation.

Our real difficulty  to overcome
is Leibniz rule (LR) on lattice  in  interacting SUSY theories.
Unfortunately, the no-go theorem for the rule on lattice has been proved in our previous papers\cite{KSS,KSS2}.
The essential point of the proof is the incompatibility of the translational invariance,
 the locality and the LR on lattice. Under a certain circumstances, SLAC type difference operator  satisfies the LR on lattice\cite{KSS,SLAC} but it is non-local.

As another approach, we could utilize a Nicolai map for constructing
a lattice SUSY theory\cite{Sakai-Sakamoto}.
For supersymmetric quantum mechanics, a lattice model with
a Nicolai map was proposed in Ref.\cite{LSUSY1},
\begin{equation}
S_{N} = \sum_{n}\frac{1}{2}
        \Bigl( (\Delta\phi)_{n} - W_{n}(\phi) \Bigr)^{2}
        + \sum_{m,n} i\bar{\psi}_{m}
          \left( \Delta_{mn} + \frac{\partial W_{n}(\phi)}{\partial \phi_{m}}
          \right) \psi_{n}\,,
\label{LSUSY_action1}
\end{equation}
where $\Delta_{mn}$ is a difference operator and $W_{n}(\phi)$
is an arbitrary function of $\phi_{m}$.
The lattice action (\ref{LSUSY_action1}) possesses
an exact supersymmetry\cite{LSUSY2,LSUSY3} but
contains the unconventional term
\begin{equation}
- \sum_{n}(\Delta\phi)_{n} W_{n}(\phi)\,.
\label{surface_term}
\end{equation}
It should be noticed that the above term is absent from the
action of supersymmetric quantum mechanics and
would vanish in the continuum as a surface term.
It cannot be, however, written into the form of total divergences
due to the lack of the Leibniz rule on lattice.
The author\cite{LSUSY4} has discussed,
in a topological point of view, an off-shell action
\begin{equation}
S_{B} = \sum_{n}\frac{1}{2}(B_{n})^{2}
       + \sum_{n}iB_{n}\Bigl( (\Delta\phi)_{n} - W_{n}(\phi) \Bigr)
        + \sum_{m,n} i\bar{\psi}_{m}
          \left( \Delta_{mn} + \frac{\partial W_{n}(\phi)}{\partial \phi_{m}}
          \right) \psi_{n}\,,
\label{LSUSY_action2}
\end{equation}
which reduces to (\ref{LSUSY_action1})
after eliminating the auxiliary field $B_{n}$.
The above lattice action was rederived in a superfield
formulation\cite{LSUSY5}.
Although the lattice action has an exact fermionic symmetry,
it may be regarded as a lattice version of the stochastic
action\cite{stochastic} rather than the
standard supersymmetric quantum mechanics,
\begin{equation}
S_{\rm{SQM}}
  = \int dt
     \left\{ \frac{1}{2}\left(\frac{d\phi}{dt}\right)^{2}
             + \frac{1}{2} F^{2} + iF W(\phi)
             + i\bar{\psi} \left( \frac{d}{dt}
             + \frac{\partial W(\phi)}{\partial \phi}\right) \psi \right\}
\label{SQM_action}
\end{equation}
with a Euclidean time $t$.
Hence, it will be worth looking for a lattice formulation
which leads to a lattice version of
$S_{\rm{SQM}}$ and also does not rely on the existence of
local Nicolai maps without unwanted terms like (\ref{surface_term}).

In this paper, we propose a novel criterion for the lattice SUSY realization which we call cyclic Leibniz rule (CLR), that is not a genuine LR on lattice but still
a natural lattice analog of the Leibniz rule in the continuum theories in the sense that it is reduced to the corresponding relation in the continuum limit. We construct sets of the lattice difference operator and the associated symmetric field product which satisfy the CLR. With these set of difference operator and field product, we formulate the concrete example of lattice SUSY quantum mechanics. Although only a part of SUSY could
be realized exactly, yet they have several advantages over the existing approaches.
One of the advantages is that our lattice model is free from
the \lq\lq surface\rq\rq term (\ref{surface_term}) and is just
a lattice analog of the standard supersymmetric action (\ref{SQM_action}).
This property with CLR allows us to construct a superfield formulation,
whose realization is different from that given in Ref.\cite{LSUSY5}.
In the action of our model the kinetic term and the interaction terms are separately invariant under SUSY transformation. Because of this property we can apply a localization technique to obtain some exact results in our model.
Since our lattice formulation mimics the continuum supersymmetric quantum
mechanics and does not rely on the existence of local Nicolai maps,
it may shed a new light on the construction of higher-dimensional
supersymmetric lattice models.

In section 2, we propose the cyclic Leibniz rule (CLR) as a criterion for the SUSY by using a concrete example, for definiteness, of $D=1$ ${\cal N}=2$ Wess-Zumino model on lattice.
The lattice difference opertors and associated symmetric product is found as a solution to the CLR.
Based on these, in section 3, we construct exactly (half-)supersymmetric quantum mechanics on lattice with arbitrary interactions.
The Witten index is exactly calculated by the Nicolai map and the localization technique.
Multicomponent cases are discussed in section 4 and we find a local solution  which is a natural generalization of one-component case.
In section 5, we give an attempt to a higher-dimensional case letting spinor index aside, while we so far could not find any suitable solution of the CLR for euclidean field theories.
Section 6 is devoted to conclusion  and discussions.
In appendies, we find the general solution of CLR with locality, the properties of the associated product,
and  explicit multi-field product examples based on CLR.

\vspace{5mm}
\section{A criterion for lattice supersymmetry}

In this section, we begin by recapitulating the difficulty in constructing interacting supersymmetric theories on lattice where the important key concept is the Leibniz rule.
Our proposal to avoid the difficulty will appear subsequently in this section.

For concreteness, we consider $D=1$ model, {\it i.e.}~supersymmetric quantum mechanics, which consists of a multiplet
\begin{equation}
\Big(\phi_n,\psi_n, F_n\Big)
\label{multiplet-0}
\end{equation}
\noindent
where $\phi$ is a real scalar, $F$ is a real auxiliary field and $\psi$ is a complex fermion.  Index $n$ stands for a lattice site. $na$ is a coordinate
but hereafter we set lattice constant $a=1$ for simplicity.
The ${\cal N}=2$ SUSY transformation is described as
\begin{eqnarray}
\delta \phi_n &=& \epsilon\bar{\psi}_n-\bar{\epsilon}\psi_n,  \nonumber\\
\delta \psi_n &= &\epsilon (i (\Delta\phi)_n + F_n),  \nonumber\\
\delta \bar{\psi}_n &=& \bar{\epsilon} (-i (\Delta\phi)_n + F_n),  \nonumber \\
\delta F_n &=& -\epsilon i (\Delta\bar{\psi})_n-\bar{\epsilon}i(\Delta\psi)_n,
\label{SUSY-1}
\end{eqnarray}
\noindent
where  a difference operator $\Delta$ is defined as
\begin{equation}
(\Delta A)_n \equiv \sum_m  \Delta_{nm} A_m, ~~\sum_m \Delta_{nm}=0,
\label{diff}
\end{equation}
\noindent
for arbitray lattice field $A_n$. The second relation states that the difference of a constant field vanishes.
The algebra  follows from the above definition:
\begin{equation}
[\delta_1,\delta_2]\Big(\phi_n, \psi_n, F_n \Big)  = 2i(\bar{\epsilon}_1\epsilon_2-\bar{\epsilon}_2\epsilon_1)
\Big( (\Delta\phi)_n,  (\Delta\psi)_n, (\Delta F)_n\Big).
\label{algebra-1}
\end{equation}
\noindent

The kinetic terms $S_0$ and the mass term $S_m$ of the free action
\begin{equation}
S_0= \sum_n \big(   \frac{1}{2}(\Delta\phi)_n^2  + i\bar{\psi}_n(\Delta\psi)_n+\frac{1}{2}F_n^2 \big),
\label{free-1}
\end{equation}
\noindent
\begin{equation}
S_m=im\sum_n \big( F_n\phi_n+\bar{\psi}_n\psi_n \big)
\label{mass-1}
\end{equation}
\noindent
are  invariant under (\ref{SUSY-1}) provided
\begin{equation}
\Delta_{nm}=-\Delta_{mn}
\label{symmetric diff}
\end{equation}
\noindent
which  is a generalized symmetric difference operator.

The obstruction for SUSY realization on lattice appears
in the interaction parts. For an illustration we consider here cubic interaction such as
\begin{eqnarray}
S_{{\rm int}}= i\frac{g}{2}\sum_{\ell m n} \big(  M_{\ell mn}F_{\ell}\phi_m\phi_n + 2N_{\ell mn}\phi_{\ell}\bar{\psi}_m\psi_n \big) ,
\label{interaction-1}
\end{eqnarray}
\noindent
where $g$ is a coupling constant, and $M_{\ell mn}$ and $N_{\ell mn}$  are  coefficients for  fields products. We can impose $M_{\ell mn} = M_{\ell nm}$ without the loss of generality.
Under (\ref{SUSY-1}), the interaction terms are transformed  as follows,
\begin{eqnarray}
\delta S_{{\rm int}} &=& i\frac{g}{2}\sum_{\ell m n} \Big(
-2\bar{\epsilon}N_{\ell mn}\psi_{\ell} \bar{\psi}_m \psi_n
 +2 \epsilon N_{\ell mn}\bar{\psi}_{\ell}\bar{\psi}_m\psi_n
\nonumber \\
&&
-2\bar{\epsilon}( M_{\ell mn}-N_{m \ell n})F_{\ell} \psi_m \phi_n
 +2 \epsilon(M_{\ell mn}-N_{n m \ell})F_{\ell}\bar{\psi}_m\phi_n
\nonumber \\
&&
-i\bar{\epsilon}(M_{\ell mn} (\Delta\psi)_{\ell} \phi_m  \phi_n
+  2N_{\ell mn} \phi_{\ell}  (\Delta\phi)_m \psi_n)  \nonumber \\
                     & & -i \epsilon (M_{\ell mn}(\Delta\bar{\psi})_{\ell} \phi_m \phi_n
                     +2N_{\ell mn} \phi_{\ell} \bar{\psi}_m  (\Delta\phi)_n)
                     \Big )  .
\label{tranform of int-1}
\end{eqnarray}
\noindent
In order to accomplish the invariance under  (\ref{SUSY-1}), we must require
\begin{eqnarray}
N_{\ell mn} &=& N_{m \ell n}, \nonumber  \\
M_{\ell mn}&=&N_{nm \ell}, \nonumber \\
\sum_{k}(\Delta_{k\ell}M_{kmn} &+& N_{n\ell k}\Delta_{km} + N_{m\ell k}\Delta_{kn})=0 .
\label{condition-1}
\end{eqnarray}
for $\epsilon$ invariance and
\begin{eqnarray}
N_{\ell mn}&=&N_{n m \ell},   \nonumber \\
M_{\ell mn} &=& N_{m \ell  n},  \nonumber \\
\sum_{k}(\Delta_{k \ell}M_{kmn} &+& N_{nk\ell}\Delta_{km} + N_{m k\ell}\Delta_{kn})=0, \label{condition-1bar}
\end{eqnarray}
for $\bar\epsilon$ invariance.
These conditions can be combined into a simpler form
\begin{equation}
M_{\ell mn} = M_{\ell nm}=M_{n m\ell}=N_{\ell mn}= N_{\ell nm}=N_{n m\ell},
\label{condition-2-0}
\end{equation}
\begin{equation}
\sum_{k}(\Delta_{ k \ell}M_{kmn} +M_{\ell k n}\Delta_{km} + M_{\ell mk}\Delta_{kn})=0.
\label{condition-2}
\end{equation}
\noindent
With  (\ref{symmetric diff}), the relation (\ref{condition-2}) implies
a Leibniz rule on lattice
\begin{equation}
\sum_{k}\Delta_{\ell k}\sum_{m,n}M_{kmn}\phi_m\psi_n=
\sum_{k,n}M_{\ell kn}(\Delta\phi)_k\psi_n+
\sum_{m,k}M_{\ell mk}\phi_m(\Delta\psi)_k,
\label{LR}
\end{equation}
\noindent
where $\phi,\psi$ are any lattice fields.

Before  going into the further analysis, we describe  the translational invariant and local  difference operator  and  product-coefficients as  holomorphic functions\cite{KSS}.
From translational invariance, these objects depend only on the difference of site indices, so we denote conveniently
\begin{equation}
\Delta(k) \equiv \Delta_{m\: m-k}, ~~  M(k,\ell)\equiv M_{m\: m-k\: m-\ell}.
\end{equation}
\noindent
By transforming them to the momentum representation ($w-$expression), the complex functions
\begin{equation}
\hat{\Delta}(w) \equiv \sum_{n=-\infty}^{\infty} w^n \Delta(n),~~
\hat{M}(w,z) \equiv \sum_{m,n=-\infty}^{\infty} w^mz^n M(m,n) = \hat{M}(z,w)
\label{w-expression}
\end{equation}
are defined.  Substituting $w=\exp(ipa),~z=\exp(iqa)$,
the expression (\ref{w-expression}) exactly corresponds to its Fourier expansion
 and $p,q$ become momentum variables when $p,q$ are real.
 The locality of the difference operator  is
exactly equivalent to the holomorphism of the $w$-expression,
and the holomorphic domain ${\cal D}$ is an annulus in the complex $w$-plane which includes  a unit circle
around the origin.   For  the product $\hat{M}(w,z)$, the domain  is
${\cal D} \otimes {\cal D}$ as  a two-variables function.
Namely, we proved that  the locality, {\it i.e.} long distance behaviors
\begin{equation}
|\Delta_{m\,n}| \leq C\exp{(-K_0|m-n|)},~~|M_{\ell m n}| \le C_1\exp{(-K_1|\ell - m|-K_2|\ell -n|)}
\end{equation}
\noindent
with $C,~C_1,~K_0,~K_1,~K_2 >0$ are equivalent to the holomorphism of $\hat{\Delta}(w)$ and $\hat{M}(w,z)$\cite{KSS}.
From the second relation of (\ref{diff}), the correspondence $\hat{\Delta}$ has the property,
\begin{equation}
\hat{\Delta}(1)=0 .
\end{equation}
\noindent
Turning the relation (\ref{condition-2}) into the $w$-expression,  we can find
\begin{equation}
\hat{M}(w,z)\hat{\Delta}(\frac{1}{wz})+\hat{M}(w,z)\hat{\Delta}(w)+
\hat{M}(w,z)\hat{\Delta}(z)=0.
\label{LR3}
\end{equation}
\noindent
The only holomorphic solution around $w=z=1$ of (\ref{LR3}) is
$\hat{\Delta}(w)\sim \log w$  {\it i.e.} a SLAC-type difference operator besides
a trivial solution with $\hat{M}(w,z)=0$.
The SLAC-type operator is, however, not  holomorphic in the annulus ${\cal D}$.
Thus there is no set of $\Delta$ and $M(\ne 0)$ with
translational invariance  and locality  that satisfies (\ref{condition-2}). This is an essence of
the no-go theorem for the Leibniz rule on lattice proved in our previous work\cite{KSS}.
So the lack of lattice Leibniz rule is an obstruction
for lattice supersymmetry with  interaction.

To overcome the above problem, we demand only a subset of supersymmetry.
For example, if we impose the invariance only for
 $\epsilon$ in (\ref{SUSY-1}),  then
 we have  the half of the condition, {\it i.e.} (\ref{condition-1}) without (\ref{condition-1bar}). Thus, instead of (\ref{condition-2-0}) and (\ref{condition-2}), we obtain
\begin{equation}
M_{\ell mn}=M_{\ell nm}=N_{nm\ell}=N_{mn\ell},
\label{symmetric Leibniz 0}
\end{equation}
\begin{equation}
\sum_k (\Delta_{k\ell}M_{kmn}+M_{kn\ell}\Delta_{km}+M_{k\ell m}\Delta_{kn})=0 .
\label{symmetric Leibniz}
\end{equation}
We call  (\ref{symmetric Leibniz}) as {\it cyclic  Leibniz rule} (CLR).
This relation cannot be factorized, like (\ref{LR3}), into a separate condition for a difference operator, but must be solved by the combination of a difference  operator and  a product.
It is  worth noting that   the symmetry in  the product  $M_{k\ell m}$ of ({\ref{condition-2-0}) is different from
 that of  (\ref{symmetric Leibniz 0}); the former is totally-symmetric for the three indices
 and the latter is only symmetric for the second and the third indices.
Although  the conventional  LR (\ref{condition-2}) seems equivalent to the CLR (\ref{symmetric Leibniz})  in the naive continuum limit because  the  limit implies $M_{k\ell m} \rightarrow \delta_{k\ell} \delta_{km} $,
they clearly have different property for the finite lattice constant.
In contrast to (\ref{LR}), the CLR can be expressed as
\begin{eqnarray}
\sum_{k}\Delta_{\ell k}\sum_{m,n}M_{kmn}\phi_m\psi_n
= \sum_{n,k}   M_{kn\ell}(\Delta \phi)_k\psi_{n} +   \sum_{k,m}  M_{k \ell m}   \phi_{m}(\Delta \psi)_k ,
\end{eqnarray}
\noindent
whose indices are indeed summed in a different way from (\ref{LR}).
Although the CLR is found in constructing a specific model with cubic interaction,
it will turn out to be very useful for the analysis of more general
supersymmetric theories in later sections.
Instead of the usual Leibniz rule (\ref{condition-2}), we can adopt the CLR
as the key  rule for the exact lattice supersymmetry.
As seen later, we can construct a superfield formalism and find a Nicolai map
with vanishing surface terms in interacting cases.
 This framework with the exact symmetry  are  a real advantage of our approach .

In $w$-represenation,
the cyclic  Leibniz rule (\ref{symmetric Leibniz}) can be expressed as
\begin{equation}
\hat{M}(w,z)\hat{\Delta}(\frac{1}{wz})+\hat{M}(z,\frac{1}{wz})\hat{\Delta}(w)+\hat{M}(\frac{1}{wz},w)\hat{\Delta}(z)=0 .
\label{hat-CLR}
\end{equation}
\noindent
We give here an example solution for
(\ref{hat-CLR})
keeping translational invariance  and  locality
\begin{equation}
\hat{M}(w,z) =\frac{1}{6}\Big(2wz+wz^{-1}+zw^{-1}+2(wz)^{-1}\Big), ~~\hat{\Delta}(w)=\frac{w-w^{-1}}{2} .
\end{equation}
\noindent
This corresponds to a real space expression
\begin{eqnarray}
M_{\ell mn} &=&\frac{1}{6}(2\delta_{\ell, m-1} \delta_{\ell, n-1}
+ \delta_{\ell, n-1} \delta_{\ell, m+1}+ \delta_{\ell, n+1}\delta_{\ell, m-1}
+ 2 \delta_{\ell, m+1}\delta_{\ell, n+1}), \nonumber \\
\Delta_{mn} &=& \frac{1}{2}(\delta_{m, n-1} - \delta_{m, n+1}).
\label{eg-1}
\end{eqnarray}
\nonumber
This simple example has doublers which can be resolved by a supersymmetric Wilson term or a
supersymmetric overlap-type mass.
We shall see these terms explicitly in the subsection \ref{construction}.
More generic other solutions to (\ref{hat-CLR}) are given in Appendix \ref{appendix_solution}.

\vspace{5mm}
\section{Analysis of supersymmetric lattice quantum mechanics}
In the previous section,  we have found the important relation, the CLR.  Its solution consists of a local difference operator and a symmetric field product   smeared over the lattice sites keeping locality.
In this section, we explicitly construct
general supersymmetric quantum mechanics using the CLR as a guiding principle.
And we will analyze the model by localization technique.

\subsection{Construction of supersymmetric action}\label{construction}
We saw in the previous section that a cubic action can be half-supersymmetric if the product satisfies the CLR. This can be immediately extended to other order terms including quadratic, quartic or higher.
An important point is that the CLR for each order guarantees the invariance of each order term independently. In other words, the coupling constant for each order term can be taken independently. This naturally leads us to the superfield formalism with half-supersymmetry to construct generic action.
Also we are able to utilize this property to make an exact calculation with localization technique which will be discussed in the next subsection.

Let us first define a 2-body product of lattice fields using the coefficient $M_{\ell mn}$
\begin{equation}
 \{ \phi, \psi  \}_{\ell} \equiv  \sum_{m n} M_{\ell m n}\phi_{m}\psi_n =
  (-1)^{\epsilon_{\phi} \epsilon_{\psi}}\{ \psi, \phi  \}_{\ell},
\label{smeared s-product}
\end{equation}
where ${\epsilon_{\phi}}$ and ${\epsilon_{\psi}}$ are the Grassmann-parity for $\phi$ and $\psi$.
We  refer  this product as {\it a smeared symmetric product}, because this is a symmetric product smeared over lattice sites. The locality of the product is guaranteed by the holomorphic property of its $w$-expression $\hat M(z,w)$.
We also define an inner product of two lattice fields
\begin{equation}
(\phi, \psi) \equiv  \sum_n \phi_n  \psi_n.
\label{inner product}
\end{equation}
Then a simple 3-body summed product of lattice fields  is expressed as
\begin{equation}
(\phi, \{ \psi, \chi  \}) \equiv  \sum_{\ell m n} M_{\ell m n}\phi_{\ell}\psi_m \chi_n.
\label{3-body inner product}
\end{equation}
With these notations, as well as (\ref{symmetric diff}), (\ref{symmetric Leibniz}), the CLR can be rewritten into a compact form
\begin{equation}
(\Delta\phi,\{\psi,\chi\} )+ (\Delta\psi,\{\chi,\phi\} )+(\Delta\chi,\{\phi,\psi\} )=0.
\label{2-product-0}
\end{equation}
\noindent
In the case of three same  fields,  (\ref{2-product-0}) is reduced to
\begin{equation}
(\Delta\phi,\{\phi,\phi\} )=0.
\label{2-product-1}
\end{equation}
\noindent
The expression (\ref{2-product-1}) is equivalent to (\ref{2-product-0})
if we take the symmetry of the product $M$ into account.  These simple expressions are useful in
further SUSY analysis.

Now we construct various terms in the action using superfield formalism.
As stated before, the kinetic term is fully supersymmetric with $\epsilon$ and $\bar\epsilon$, while the interaction term is half-supersymmetric with, say $\epsilon$. Therefore we can use ${\cal N}=2$ superspace for the former, but
must use ${\cal N}=1$ for the latter.

An ${\cal N=}2$ supermultiplet for a superspace $(n,\theta,\bar{\theta})$
\begin{equation}
{\Xi}_n(\theta,\bar{\theta})= \Phi_n(\theta)- \bar{\theta}\Psi_n(\theta)
\label{multiplet-1}
\end{equation}
\noindent
is decomposed into two ${\cal N=}1$ multiplets,
\begin{equation}
\Phi_n(\theta)= \phi_n + \theta \bar{\psi}_n
\label{multiplet-2}
\end{equation}
\noindent
which is Grassmann even and
\begin{equation}
 \Psi_n(\theta)= \psi_n + \theta F_n
 \label{multiplet-3}
\end{equation}
\noindent
which is Grassmann odd.

A kinetic term of the action can be written as
\begin{equation}
S_0 = \frac{1}{2}\int d\bar{\theta} d\theta (\bar{D}{\Xi}(\theta,\bar{\theta}),D{\Xi}
(\theta,\bar{\theta}))  ,
\label{free-action}
\end{equation}
\noindent
where  supercovariant difference operators are defined as
\begin{equation}
\bar{D}_{mn} \equiv  -i \frac{\partial}{\partial \theta} \delta_{mn} - \bar{\theta} \Delta_{mn} , \  \
 D_{mn} \equiv  -i \frac{\partial}{\partial \bar{\theta}} \delta_{mn} - \theta \Delta_{mn} .
\label{supercovariant}
\end{equation}

To formulate mass term and Wilson term supersymmetrically, we consider a generic bilinear form
\begin{equation}
\label{2-body}
(\phi, G \psi )  \equiv \sum_{k\ell}\phi_k G_{k\ell} \psi_\ell ,
\end{equation}
where $G_{mn}$ is a certain operator on a lattice.
The translational invariance implies that $G$ is a function of the difference of its indices $G_{mn}=G(m-n)$, and satisfies $\Delta G = G \Delta$ with translationally-invariant difference operator $\Delta$. Thus we have
\begin{equation}
(\Delta\phi, G\psi )=-(\phi, G\Delta \psi) ,
\label{DeltaG}
\end{equation}
with
$\Delta^T=-\Delta$.
Then
\begin{equation}
S_2= i \int d\theta \, (\Psi(\theta), G \Phi(\theta)  )
 \label{bilinear}
\end{equation}
gives a supersymmetric quadratic action invariant under $\epsilon$-transformation.
For example, a simple mass term with Wilson term is given by choosing
\begin{equation}
\hat{G}(w)\equiv \sum_n w^nG(n) = im+ ir\frac{2-w-w^{-1}}{2}
\end{equation}
where $m$ is a bare mass and $r$ is a standard Wilson term parameter.
For more generic doubler-suppressing term, we can write
\begin{equation}
\hat{G}(w) = im+i\hat H(w,M) ,
\end{equation}
\noindent
where $\hat H(w,M)$ is a holomorphic function in ${\cal D}$ with a parameter $M>0$ and corresponds to $w$-expression of a translationally invariant operator $H_{mn}(M)$, {\it i.e.}~$\hat H(w,M)=\sum_mw^mH_{\ell\,\ell+m}(M)$.
We further impose that $H$ is symmetric, {\it i.e.}~$H^T=H$ or $\hat H(w)=\hat H(1/w)$, throughout this paper.
If we need an overlap-type action, we choose $\hat H$ to have the poles at $w=0,\infty$ and the branch points at $w=-1/M, -M$ with $M>1$, as well as the property $\hat H(w=1,M)=0$.
The explicit form for the overlap-type term\cite{overlap-1, overlap-2}  is given as
\begin{equation}
\frac{\hat H(w,M)}{M+1}= 1- \frac{(M+\frac{w+w^{-1}}{2})}{\sqrt{-(\frac{w-w^{-1}}{2})^2+(M+\frac{w+w^{-1}}{2})^2}}
= 1-\frac{(M+\frac{w+w^{-1}}{2})}{\sqrt{(M+w)(M+w^{-1})}} ,
\end{equation}
\noindent

Note that, together with an explicit solution for the CLR
\begin{equation}
\hat\Delta(w)=\frac{(M+1)(w-w^{-1})}{2\sqrt{(M+w)(M+w^{-1})}} ,
\end{equation}
\begin{equation}
\hat{M}(w,z)=\frac{1}{6(M+1)}\sqrt{(M+w)(M+w^{-1})}\Big(2wz+wz^{-1}+zw^{-1}+2(wz)^{-1}\Big),
\end{equation}
the above $H$ satisfies an analog of the Ginsparg-Wilson relation\cite{Ginsparg-Wilson,Neuberger},
\begin{equation}
\hat\Delta(w)\hat\Delta(w^{-1}) + \hat H(w,M)\hat H(w^{-1},M)=  (M+1)\Big(\hat H(w,M)+\hat H(w^{-1},M)\Big),
\end{equation}
\noindent
where  the Dirac operator  is given by
$\hat{D}_{Dirac}(w) = i\hat\Delta(w)+i\hat H(w,M)$.

The quadratic terms including mass and doubler-suppressing term and the cubic interaction term of the supersymmetric action are now obtained as
\begin{equation}
S_2+S_{{\rm int}} = i \int d\theta \Big(  (\Psi(\theta), G \Phi(\theta)  )
+ g_2(\Psi(\theta),\{ \Phi(\theta), \Phi(\theta)  \} )
 \Big)
 \label{int-action}
\end{equation}
\noindent
with a coupling constant $g_2$ (here subscript 2 stands for 2-body product).
The action (\ref{int-action}) is clearly
invariant under SUSY transformation
\begin{eqnarray}
 \delta \phi_n&=& \epsilon \bar{\psi}_n,\nonumber \\
 \delta \psi_n&=& \epsilon (i(\Delta\phi)_n+F_n), \nonumber \\
 \delta \bar{\psi}_n&=& 0,\nonumber \\
 \delta F_n&=&  -i\epsilon  (\Delta\bar{\psi})_n  .
\label{1/2SUSY}
\end{eqnarray}
\noindent
This transformation can be realized on the superfield as follows,
\begin{equation}
{\delta} \Xi_n(\theta,\bar{\theta}) = \sum_m
\epsilon \Big(\frac{\partial}{\partial \theta}\delta_{nm}
+ i\bar{\theta} \Delta_{nm}\Big)
\Xi_m(\theta,\bar{\theta}) =\epsilon \Big(\frac{\partial}{\partial \theta} \Phi_n(\theta)
+\bar{\theta}\frac{\partial}{\partial \theta}\Psi_n(\theta) +i \bar{\theta} (\Delta\Phi)_n(\theta)\Big)  ,
\label{N=1-1}
\end{equation}
\noindent
or equivalently on the ${\cal N}=1 $ superfields
\begin{eqnarray}
\delta\Phi_n(\theta)&=&\epsilon \frac{\partial}{\partial \theta}\Phi_n(\theta),  \nonumber \\
\delta \Psi_n(\theta) &=& \epsilon\frac{\partial}{\partial \theta}\Psi_n(\theta)+i\epsilon (\Delta \Phi)_n(\theta) .
\label{N=1-2}
\end{eqnarray}

Thus the total action
\begin{eqnarray}
S_{{\rm t}}&=& S_0+S_2+S_{{\rm int}}\nonumber\\[5pt]
&=& \frac{1}{2}(\Delta \phi,  \Delta \phi)+ i (\bar{\psi}, \Delta \psi)+
\frac{1}{2}(F,F)\nonumber\\[3pt]
&+&(F,(im+iH)\phi) +
(\bar{\psi},(im+iH)\psi)+ig_2(F,\{\phi,\phi\})-2ig_2(\psi,\{\phi,\bar{\psi}\})
\label{total action}
\end{eqnarray}
is also invariant under (\ref{N=1-1}) or (\ref{N=1-2}),
\begin{equation}
 \delta S_{\rm t} =0,
\label{invariance}
\end{equation}
due to the relation (\ref{DeltaG}) and the CLR (\ref{2-product-1}).
It is worth emphasizing here that our formulation keeps locality; the holomorphic property of the $w$-expression of various operators and product guarantees the locality.

To extend to multi-body supersymmetric interactions, we define  the smeared   $N$-body products for a bosonic superfield
\begin{equation}
\{\Phi,\ldots, \Phi  \} _n  \equiv  \sum_{m_1,\ldots,m_{N}} M_{n;m_1,\ldots,m_{N}} \Phi_{m_1}  \cdots  \Phi_{m_{N}} .
\label{N-Product-1}
\end{equation}
\noindent
where $m_1,..,m_N$  are totally symmetric indices
(we have been omitting semicolon for $N=2$, {\it i.e.} $M_{n;m_1,m_2} = M_{nm_1m_2}$.)

If this product satisfies the $N$-body CLR
\begin{equation}
(\Delta \Phi, \{\Phi,\ldots, \Phi \} ) =0 ,
\label{N-Product-2}
\end{equation}
then the action
\begin{equation}
\int d\theta\,i g_{N} \left( \Psi, \{\Phi,\ldots,\Phi \} \right)
\label{N-Product-interaction}
\end{equation}
is invariant under the supersymmetry transformations (\ref{N=1-2}) in exactly same way as the 2-body case.
It should be noticed that the $R$-symmetry
\begin{eqnarray}
\theta &\to& e^{i\alpha} \theta\,,\nonumber\\
\Phi &\to& \Phi\,,\nonumber\\
\Psi &\to& e^{i\alpha}\Psi\,,
\label{Rsymmetry}
\end{eqnarray}
allows only the type of interaction terms (\ref{N-Product-interaction}),
which consists of $\Phi$ and $\Psi$
and is necessary  for constructing supersymmetric
$2N$-body interactions for scalar fields with a coupling constant $g_N$.
Also note that $N$-body product satisfying the CLR can be constructed in terms of 2-body products which will be discussed in Appendix~B.

Before closing this subsection, it may be helpful to understand the exact SUSY invariance in an alternative way.
We write the total action $S_{\rm t}$  as a sum of kinetic term $S_0$, bare mass term $S_m$, doubler-suppressing term $S_{\rm ds}$ and interaction term $S_{\rm int}$
\begin{equation}
S_{\it t}=S_0+S_m+S_{\rm{ds}}+S_{{\rm int}}.
\label{total action'}
\end{equation}
Then each term can be expressed as a supersymmetric transform of something
\begin{eqnarray*}
 S_0&=& \frac{1}{2}\delta'(\psi, (-i\Delta \phi +F)), \\
 S_m&=& im\delta'(\psi, \phi ),   \\
 S_{\rm{ds}}& = & i\delta'(\psi, H \phi ),
 \end{eqnarray*}
 \begin{equation}
 S_{{\rm int}}=S_{3}+S_{4}+ \cdots =ig_2\delta'(\psi, \{\phi, \phi\} )
 + ig_3 \delta'(\psi, \{\phi, \phi, \phi\} ) +\cdots ,
\end{equation}
\noindent
with coupling constants $g_2,g_3,\cdots$. Here we denote the transformation without parameter by $\delta'$, {\it i.e.}  $\delta = \epsilon \delta'$,
or explicitly
\begin{eqnarray}
\delta'\phi_n&=&\bar{\psi}_n,  \nonumber \\
\delta' \psi_n &=& i(\Delta \phi)_n+F_n,  \nonumber \\
\delta' \bar{\psi}_n &=&0, \nonumber \\
\delta'F_n &=& -i(\Delta  \bar{\psi})_n .
\label{N=1'-2}
\end{eqnarray}
Due to the nilpotencey of the transformation $\delta'^2=0$,
the invariance of each term is manifest,
\begin{equation}
\delta' S_{0}=\delta' S_m=\delta'S_{\rm{ds}}=\delta' S_{{\rm int}}=0.
\end{equation}

\subsection{Localization technique and calculation of Witten index}

With the action (\ref{total action'}), we can find a Nicolai map for our system, and apply the localization technique\cite{local-1,local-2} to calculate the Witten index (the partition function with periodic boundary condition for all variables).

Before going into the detail, it would be useful to compare with the previous attempts in the existing literature.
In the references \cite{Kaplan-Catterall,LSUSY5} the partition function has been calculated in the ultra local
limit, where the $N$-site partition
function $Z_{N}$
reduces to $N$ copies of the one-site one, {\it i.e.} $Z_{N} = (Z_{1})^{N}$.
This result indicates that every degrees of freedom may contribute
to the Witten index that is quite different from the usual understanding of its topological nature. Also it depends on whether the number of lattice sites is even  or odd for $Z_1=-1$ case of which we could not understand the validity.
On the other hand, our lattice formulation is suitable to use
the localization technique as will be seen below.
It will turn out that the Witten index is determined by only zero modes
and all fluctuation modes are irrelevant in it.

Now let us begin by defining the Nicolai map
\begin{equation}
\xi_n=-(\Delta \phi)_n +  m \phi_n + (H\phi)_n +g_2\{ \phi,  \phi \}_n + g_3 \{ \phi,  \phi, \phi \}_n + \cdots.
\label{Nicolai_map}
\end{equation}
\noindent
With this, the Witten index can be  calculated  as
\begin{eqnarray}
Z=\int D\phi  D\bar{\psi}D\psi DF \,e^{-S_{\rm t}}&=&\int D\xi D\bar{\psi}D\psi
|\det \eta|^{-1} \,e^{-\sum_n \frac{1}{2}\xi_n \xi_n-i\sum_{kl}\bar{\psi}_{l}\eta_{lk}\psi_{k}}   \nonumber \\
 & =  &    \int D\xi
|\det \eta|^{-1}  \det \eta~
e^{-\sum_n \frac{1}{2}\xi_n \xi_n} ,
\label{path-integral-1}
\end{eqnarray}
\noindent
where all variables follow a periodic boundary condition and
\begin{equation}
\eta_{\ell k} \equiv \frac{\partial \xi_k}{\partial \phi_{\ell}}=\Delta_{\ell k} + m\delta_{\ell k}+{(H)}_{\ell k}
+g_2\frac{\partial }{\partial \phi_{\ell}}\{\phi, \phi  \}_{k} +
g_3\frac{\partial }{\partial \phi_{\ell}}\{ \phi,  \phi,  \phi \}_{k} + \cdots .
\end{equation}
\noindent
The prefactor of (\ref{path-integral-1})
\begin{equation}
|\det \eta|^{-1}\det \eta  ,
\end{equation}
\noindent
is not generally a constant but $\det\eta$ can vanish for certain cases.
Also the path of the integral  (\ref{path-integral-1}) is not definite.
Therefore we must carry out more detailed analysis.

In order to control the infrared behavior, we restrict a size of the system to finite $N$. Also we assume the highest power of the potential in (\ref{Nicolai_map}) is $p$, for definiteness.
We rescale our kinetic and doubler-suppressing term with parameters $t$ and $t_{\rm ds}$ respectively,
\begin{equation}
S_0 \rightarrow tS_0 ,~~S_{\rm{ds}} \rightarrow t_{\rm{ds}}S_{\rm{ds}}.
\end{equation}
\noindent
Note that $t_{\rm{ds}}$ looks like a Wilson term parameter for Wilson fermion case. Then we integrate out $F_n$ and obtain
 a $t$-scaled partition function,
\begin{eqnarray}
 Z (t,t_{\rm{ds}})&\equiv& \int D\phi D\bar{\psi}  D\psi DF\,
 e^{-S(t,t_{\rm{ds}})} \nonumber \\
  &\equiv&  \int D\phi D\bar{\psi}  D\psi DF\,
  e^{-tS_0-t_{\rm{ds}}S_{\rm{ds}} -S_m -S_{{\rm int}}} \nonumber
  \\
 & = &t^{-N/2}\int D\phi D\bar{\psi}  D\psi~ \nonumber \\
 &\times& e^{-\big(tS^b_0
  + t^2_{\rm{ds}}t^{-1} S^b_{\rm{ds}} + t^{-1} S^b_{m+{\rm int}}
  + t_{\rm{ds}}t^{-1}S^b_{\rm cross}\big)
- \big(tS^f_0+  t_{\rm{ds}}S^f_{\rm{ds}}+S^f_{m + {\rm int}}\big)} .
\label{t-scaled Z}
\end{eqnarray}
Here we denote  $S^b_0, S^b_m, S^b_{\rm{ds}}$ and $S^b_{{\rm int}}$ as bosonic (without fermion) parts of kinetic,  mass, doubler-suppressing   and interaction terms, respectively.
$S^f_0, S^f_m, S^f_{\rm{ds}}$ and $S^f_{{\rm int}}$  are  fermionic (with fermion) counterparts.
\begin{table}[hb]
\caption{Bosonic and fermionic parts of the action after integrating of $F_n$}
\begin{center}
$$
\begin{array}{|c|c|c|}
\hline
S_0^b & S^b_{\rm ds} & S^b_{m+{\rm int}}\rule[-6pt]{0pt}{18pt}\\ \hline \hline
\frac{1}{2}(\Delta \phi, \Delta \phi)  & \frac{1}{2}(\phi, H^2 \phi) &
\frac{1}{2}(m\phi+g_2\{\phi,\phi\}+\cdots, m\phi+g_2\{\phi,\phi\}+\cdots) \rule[-6pt]{0pt}{18pt}\\ \hline
\end{array}
$$
$$
\begin{array}{|c|c|c|} \hline
S_0^f & S^f_{\rm ds} & S^f_{m+{\rm int}}\rule[-6pt]{0pt}{18pt}\\ \hline \hline
i(\bar{\psi}, \Delta \psi) &  i(\bar{\psi},H \psi) & im(\bar{\psi},  \psi)-i(\psi, 2g_2\{ \phi , \bar{\psi}\}+\cdots) \rule[-6pt]{0pt}{18pt}\\ \hline
\end{array}
$$
\end{center}
\label{table2}
\end{table}
We summarize each part of our action in Table~\ref{table2}.
In addition, the cross terms $S^b_{\rm cross}$ for doubler-suppressing term and mass plus interaction terms are defined as
\begin{equation}
S^b_{\rm cross} \equiv (H\phi\,,\,m\phi+g_2\{\phi,\phi\}+\cdots).
\end{equation}

 It is clear that this $Z(t,t_{\rm{ds}})$
 is $t$-independent,
because before $F_n$-integration one can show
\begin{eqnarray}
\frac{\partial}{\partial t}Z(t,t_{\rm{ds}})&=&-\int D\phi D\bar{\psi}  D\psi DF ~S_0~e^{-tS_0 - S_m - t_{\rm{ds}}S_{\rm{ds}} -S_{{\rm int}}} \nonumber \\
 &=& -\int D\phi D\bar{\psi}  D\psi  DF\,\delta' (Xe^{-tS_0 -S_m -t_{\rm{ds}}S_{\rm{ds}} -S_{{\rm int}}}) =0 ,
\end{eqnarray}
\begin{eqnarray}
\frac{\partial}{\partial t_{\rm{ds}}}Z(t,t_{\rm{ds}})&=&-\int D\phi D\bar{\psi}  D\psi DF ~S_{\rm{ds}}~e^{-tS_0 - S_m - t_{\rm{ds}}S_{\rm{ds}} -S_{{\rm int}}} \nonumber \\
 &=& -\int D\phi D\bar{\psi}  D\psi  DF\,\delta' (Ye^{-tS_0 -S_m -t_{\rm{ds}}S_{\rm{ds}} -S_{{\rm int}}}) =0 ,
\end{eqnarray}
\noindent
where $X\equiv\frac{1}{2}\Big(\psi, -i\Delta \phi + F\Big),~Y\equiv \Big(\psi,H\phi  \Big)$.
Here we used the invariance of the path-integral measure under $\delta'$ and $\delta' (S(t,t_{\rm{ds}}))=0$.

In the following calculations of the Witten index, we take
two typical limiting cases.  One is analogous to the continuum theory in which  important modes are only zero-modes. The second case  is   peculiar to
lattice theories, which has doubling modes in addition to zero-modes.\footnote{Hereafter we assume the number of sites $N$ is even for simplicity.
There is no doubler mode for odd $N$, so that it gives the same result
 as Case~(i) below. }
Although  the two cases are physically and mathematically different,
after using localization technique, we can obtain
the exactly same results.
We first decompose each field into zero mode (script $0$), doubler mode (script $d$) and the other modes:
\begin{eqnarray}
\phi_n&=&\frac{\phi_0}{\sqrt{N}}+ \frac{(-1)^n}{\sqrt{N}}\phi_d +a_n ,  \nonumber \\
\psi_n&=&\frac{\psi_0}{\sqrt{N}} + \frac{(-1)^n}{\sqrt{N}}\psi_d +\chi_n, \nonumber \\
\bar{\psi}_n&=&\frac{\bar{\psi}_0}{\sqrt{N}} +
 \frac{(-1)^n}{\sqrt{N}}\bar{\psi}_d +\bar{\chi}_n
\end{eqnarray}
Then we rescale each part with $t$ or $t_{\rm ds}$ in such a way that the only important mode survives in the infinitely scaling limit.

\vspace{0.5cm}

\underline{Case (i)}~~~We make the scaling of each field as
\begin{eqnarray}
\phi_n&=&\frac{\phi_0}{\sqrt{N}}+ \frac{(-1)^n}{\sqrt{N}\sqrt{t_{\rm{ds}}}}\phi_d +\frac{1}{\sqrt{t}}a_n ,  \nonumber \\
\psi_n&=&\frac{\psi_0}{\sqrt{N}} + \frac{(-1)^n}{\sqrt{N}\sqrt{t_{\rm{ds}}}}\psi_d +\frac{1}{\sqrt{t}}\chi_n, \nonumber \\
\bar{\psi}_n&=&\frac{\bar{\psi}_0}{\sqrt{N}} +
 \frac{(-1)^n}{\sqrt{N}\sqrt{t_{\rm{ds}}}}\bar{\psi}_d +\frac{1}{\sqrt{t}}\bar{\chi}_n ,
\end{eqnarray}
then we take $t,t_{\rm{ds}} \rightarrow \infty$  with a fixed $t/t_{\rm{ds}}$.
This case is similar to  the continuum theory, because
the real zero mode (not doubler) is only $t,t_{\rm{ds}}$-unscaled.
A scalar zero mode  $\phi_0$ contributes to
$t^{-1}S^b_{m+{\rm int}}+S^f_{m+{\rm int}}$
and a scalar doubler mode $\phi_d$ contributes to
\begin{equation}
\frac{t^2_{\rm{ds}}}{t}S^b_{\rm{ds}}=\frac{t_{\rm{ds}}}{tN}\sum_{m}\Big(\sum_n H_{mn} (-1)^n \phi_d\Big)^2
=\frac{t_{\rm{ds}}}{t}\big(\hat{H}(-1)\big)^2\phi_d^2,
\end{equation}
\noindent
where $\hat{H}(w)\equiv \sum_n w^n H(n)$ for translational invariant $H_{mn}=H(m-n)$.
On the other hand, the fermion zero modes contribute to
$S^f_{m+{\rm int}}$,
and   a fermion doubler mode contributes to
\begin{equation}
t_{\rm{ds}}S^f_{\rm{ds}}=i \frac{1}{N}\sum_{mn}(-1)^m\bar{\psi}_d H_{mn} (-1)^n \psi_d =
 i \hat{H}(-1)\bar{\psi}_d \psi_d   .
\end{equation}
\noindent
Any non-zero modes  can be integrated out in the  limit  $t,t_{\rm{ds}} \rightarrow  \infty$  with fixed $t/t_{\rm{ds}} $, and
the $t$ factor of the partition function (\ref{t-scaled Z}) is exactly canceled.
$(t_{\rm{ds}}/t) S^b_{\rm{cross}}$ terms in the action
vanish  in the limit.

Therefore
\begin{eqnarray}
Z(t,t_{\rm{ds}}) &=&\frac{\sqrt{t_{\rm{ds}}}}{t}\int d\phi_0 d\phi_d
d\bar{\psi}_0d\psi_0 d\bar{\psi}_d d\psi_d  ~e^{-(t_{\rm{ds}}^2t^{-1}S^b_{\rm{ds}}+t^{-1}S^b_{m+{\rm int}})-(t_{\rm{ds}}S^f_{\rm{ds}}+
S^f_{m+{\rm int}})}     \nonumber \\
 &=&\frac{\sqrt{t_{\rm{ds}}}}{t}  \int d\phi_d \bar{\psi}_d \psi_d ~e^{-t_{\rm{ds}}^2t^{-1}S^b_{\rm{ds}}-t_{\rm{ds}}S^f_{\rm{ds}}}
 \int d\phi_0 d\bar{\psi}_0d\psi_0 ~e^{-t^{-1}S^b_{m+{\rm int}}-S^f_{m+{\rm int}}}
\nonumber \\
 & = & \frac{\sqrt{t_{\rm{ds}}}}{t} \sqrt{\frac{t}{t_{\rm{ds}}
 \big(\hat{H}(-1)\big)^{2}}} \,\hat{H}(-1) \sqrt{t}
 \int d\xi |\eta_0|^{-1}\eta_0 ~e^{-\frac{1}{2}\xi^2} ,
 \label{Z-i}
 \end{eqnarray}
\noindent
where
\begin{eqnarray}
\xi(\phi_0)&\equiv&\frac{1}{\sqrt{t}}\left(m\phi_0+ \frac{g_2}{\sqrt{N}} \hat{M}(1,1)\phi_0^2  +\frac{g_3}{N} \hat{M}(1,1,1)\phi_0^3 + \cdots
+\frac{g_{p}}{\sqrt{N}^{p-1}} \hat{M}(1,\cdots,1)\phi_0^{p}\right) , \nonumber \\
\eta_0(\phi_0) &\equiv& m+ \frac{2g_2}{\sqrt{N}} \hat{M}(1,1)\phi_0+ \frac{3g_3}{N} \hat{M}(1,1,1)\phi_0^2 + \cdots +
\frac{pg_{p}}{\sqrt{N}^{p-1}} \hat{M}(1,\cdots,1)\phi_0^{p-1} \qquad.
\end{eqnarray}
with $w$-expression of translaionally invariant multi-body product
\begin{equation}
 \hat M(w_1,\cdots,w_{n-1})=\sum_{m_1,\cdots,m_{n-1}}
 w_1^{m_1}\cdots w_{n-1}^{m_{n-1}}
 M_{\ell;\ell+m_1,\cdots,\ell+m_{n-1}} .
 \end{equation}
The final expression (\ref{Z-i}) results in $Z(t,t_{\rm{ds}})=0$ for $p=$even and  $Z(t,t_{\rm{ds}})=\pm 1$ for $p=$odd (its sign depends on the asymptotic behavior of the potential).

\vspace{0.5 cm}

\underline{Case (ii)}~~~An alternative way of rescaling of fields is given by
\begin{eqnarray}
\phi_n&=&\frac{1}{\sqrt{N}}\phi_0 + \frac{(-1)^n}{\sqrt{N}}\phi_d +\frac{1}{\sqrt{t}}a_n , \nonumber \\
\psi_n&=&\frac{1}{\sqrt{N}}\psi_0 + \frac{(-1)^n}{\sqrt{N}}\psi_d +\frac{1}{\sqrt{t}}\chi_n, \nonumber \\
\bar{\psi}_n&=&\frac{1}{\sqrt{N}}\bar{\psi}_0+ \frac{(-1)^n}{\sqrt{N}}\bar{\psi}_d +\frac{1}{\sqrt{t}}\bar{\chi}_n.
\end{eqnarray}
Then we take the limit $t \rightarrow \infty$ with a fixed $t_{\rm{ds}}$.
In this case not only zero mode but also doubler mode are $t$-unscaled.
A $\phi_0$  contributes to
\begin{equation}
t^{-1}S^b_{m+{\rm int}} ,  ~~S^f_{m+{\rm int}},
\end{equation}
\noindent
and a $\phi_d$  contributes to
\begin{equation}
t_{\rm{ds}}^2 t^{-1}S^b_{\rm{ds}},~~        t^{-1}S^b_{m+{\rm {\rm int}}} ,  ~~t_{\rm{ds}}t^{-1}S^b_{\rm{cross}}, ~~S^f_{m+{\rm int}}.
\end{equation}
\noindent
Fermion zero modes $\psi_0, \bar\psi_0$ contribute to
\begin{equation}
S^f_{m+{\rm {\rm int}}},
\end{equation}
\noindent
and fermion doubler modes $\psi_d, \bar\psi_d$ contribute to
\begin{equation}
t_{\rm{ds}}S^f_{\rm{ds}},~~S^f_{m+{\rm {\rm int}}}.
\end{equation}
\noindent
Any non-zero modes can be integrated out similarly to the case (i)  in the limit $t \rightarrow  \infty$.

Therefore $Z$ is evaluated as
\begin{eqnarray}
Z(t,t_{\rm{ds}}) &=&\frac{1}{t}\int d\phi_0 d\phi_d d\bar{\psi}_0 d\psi_0d\bar{\psi}_d
 d\psi_d  \nonumber \\
&&~~~\times ~e^{-(t^{-1}t_{\rm{ds}}^2S^b_{\rm{ds}}+t^{-1}t_{\rm{ds}}S^b_{\rm cross} +t^{-1}S^b_{m+{\rm {\rm int}}})-(t_{\rm{ds}}S^f_{\rm{ds}}+
S^f_{m+{\rm {\rm int}}})}    \nonumber \\[7pt]
 & = &   \int d\xi_0 d\xi_d |\det \eta(\phi_0(\xi),\phi_d(\xi))|^{-1} \det \eta(\phi_0(\xi),\phi_d(\xi))
 e^{-\frac{1}{2}\xi_0^2-\frac{1}{2}\xi_d^2}   ,
 \label{Z-ii-2}
\end{eqnarray}
\noindent
where
\begin{eqnarray}
\xi_\ell(\phi_0,\phi_d) &\equiv&  \frac{1}{\sqrt{t}}\left(\frac{t_{\rm{ds}}}{\sqrt{N}} \sum_n H_{\ell n} (-1)^n \phi_d
 + \frac{1}{\sqrt{N}}\sum_n m\delta_{\ell n} (\phi_0 + (-1)^n \phi_d) \right.
 \nonumber \\
&&+ \cdots + \frac{g_p}{\sqrt{N}^{p}}\sum_{n_1,\cdots,n_p} M_{\ell ;n_1,n_2,\cdots,n_{p}}\left.\rule{0pt}{18pt}
(\phi_0 + (-1)^{n_1} \phi_d) \cdots(\phi_0 + (-1)^{n_{p}} \phi_d)\right) \nonumber \\
 &\equiv& \frac{1}{\sqrt{N}}\left(\xi_0(\phi_0,\phi_d)+(-1)^\ell \xi_d(\phi_0,\phi_d)\right) ,
\end{eqnarray}
and
\begin{equation}
\eta(\phi_0,\phi_d) =\left(
\begin{array}{cc}
\displaystyle \frac{\partial\xi_0(\phi_0,\phi_d)}{\partial\phi_0} &
\displaystyle \frac{\partial\xi_0(\phi_0,\phi_d)}{\partial\phi_d} \\[11pt]
\displaystyle \frac{\partial\xi_d(\phi_0,\phi_d)}{\partial\phi_0} &
\displaystyle \frac{\partial\xi_d(\phi_0,\phi_d)}{\partial\phi_d}
\end{array}
\right).
\end{equation}
The final expression (\ref{Z-ii-2}) results in $Z(t,t_{\rm{ds}})=0$ for $p=$even and  $Z(t,t_{\rm{ds}})=\pm 1$ for $p=$odd.

\vspace{0.5cm}

 In summary, these two limiting procedures give exactly the same result,
although the doubler mode survives in the expression (\ref{Z-ii-2}) of the limit (ii). The final results reproduce known results for the supersymmetric quantum mechanics in the continuum. Also there is no subtle dependence on the number of lattice sites.

\section{Multicomponent case}

In order to consider a higher-dimensional extension with spinorial index or a multi-flavor extension, we need a framework for multi-component fields.
A Wilson term or an overlap-type term can be made more realistic in the model with multi-component indices.
The CLR for the multi-component case, however, is slightly modified and should be treated with care.
In this section we will explain this in a simple situation where each field has a single index which we do not specify with either spinor or flavor at the beginning.

For the fields $(\phi_n^a, \psi_n^a, \bar\psi_n^a, F_n^a)$ with single index $a$, we consider the following supersymmetry transformation
\begin{equation}
\begin{array}{rll}
\delta \phi^a_n &= \epsilon \bar{\psi}^a_n & = \epsilon\delta' \phi_n^a,   \nonumber \\[3pt]
\delta \psi_n^a &=  \epsilon (i(\Delta \phi \Gamma)^a_n+F^a_n)&=\epsilon \delta'\psi_n^a,   \nonumber \\[3pt]
\delta \bar{\psi}^a_n &= 0 &=\epsilon \delta' \bar{\psi}_n^a, \nonumber \\[3pt]
\delta F_n^a&= -i\epsilon (\Delta \bar{\psi}\Gamma)_n^a&=\epsilon \delta' F_n^a ,
\label{SUSY-spinor}
\end{array}
\end{equation}
where $\Gamma$ is a certain constant matrix on the index with property $\Gamma =\Gamma^T,~\Gamma^2=1$.
We can easily obtain  a free action including mass and doubler-suppressing terms invariant under the above transformation,
\begin{eqnarray}
S_0 &=& \frac{1}{2}(\Delta\phi^a,\Delta\phi^a)
+ i (\bar{\psi}^a,(\Gamma)_{ab}\Delta \psi^b)
+\frac{1}{2} (F^a, F^a) \nonumber \\
&=& -\frac{i}{2}\delta' (\psi^a, (\Delta\phi \Gamma)^a)+\frac{1}{2}\delta' (\psi^a, F^a) ,\\
S_m &= &im(\phi^a,F^a)+ im (\bar{\psi}^a,\psi^a)= im \delta' (\phi^a, \psi^a), \\
S_{\rm{ds}}&=&i(\phi^a, H_{ab}F^b)+i(\bar{\psi}^a,H_{ab}\psi^b)=i\delta' (\phi^a, H_{ab}\psi^b),
\end{eqnarray}
\noindent
where $H_{ab}$ is a multi-component extension of $H$ in the previous sections and satisfies $\sum_m (H_{ab})_{mn}=\sum_m H_{ab}(m-n)=\hat{H}_{ab}(1)=0 $
and $((H\Gamma\Delta)_{ab})_{mn}=-((H\Gamma\Delta)_{ba})_{nm}$.
The interaction term
\begin{equation}
S_{{\rm int}} = ig_2 \Big((F^a,\{\phi,\phi\}^a) - 2(\psi^a,\{\phi,\bar{\psi}\}^a)\Big)
= ig_2  \,\delta' (\psi^a,\{\phi,\phi\}^a) ,
\end{equation}
\noindent
is also invariant under (\ref{SUSY-spinor})
where we used $\displaystyle\{ \phi, \psi \}^a_k \equiv \sum_{b,c,m,n}M^{abc}_{kmn}\phi^b_m \psi^c_n$ and
\begin{equation}
\sum_a ((\Delta\phi\Gamma)^a,\{\phi,\phi\}^a)=0.
\end{equation}

The translational invariance implies that the $\Delta$ and $M$ only depend on the difference of site indices, as before,
\begin{equation}
\Delta_{mn}=\Delta(m-n), ~M^{abc}_{kmn}=M^{abc}(k-m,k-n).
\end{equation}
\noindent
Their $w-$expressions are
\begin{equation}
\hat{\Delta}(w)\equiv \sum_mw^m\Delta(m), ~\hat{M}^{abc}(w,z)\equiv
\sum_{mn}w^mz^nM^{abc}(m,n) .
\end{equation}
\noindent
Using  the combined quantity
\begin{equation}
P^{abc}(w,z)\equiv \hat{\Delta}(1/(wz))\Gamma^{ad}\hat{M}^{dbc}(w,z),
\end{equation}
\noindent
the multicomponent CLR is expressed as
\begin{equation}
\begin{array}{c}
P^{abc}(w,z)+P^{bca}(z,1/(wz))+P^{cab}(1/(wz),w)=0,\\[5pt]
P^{abc}(w,1/w)=0,\qquad P^{abc}(w,z)=P^{acb}(z,w) .
\end{array}
\label{multi-CLR}
\end{equation}
\noindent
Although solutions for the relation (\ref{multi-CLR}) are complicated in general, a simple and special solution is given by
\begin{equation}
P^{abc}(w,z)=f^{abc}P(w,z),
\end{equation}
\noindent
where $f^{abc}$ is a totally symmetric constant and $P(w,z)=\hat{\Delta}(1/(wz))\hat{M}(w,z)$
 is a solution  of
single component CLR  (\ref{hat-CLR}). This means
\begin{equation}
\hat{M}^{abc}(w,z)=\Gamma^{ad}f^{dbc}\hat{M}(w,z).
\end{equation}

We comment here that when the above  indices $a,b,c$ are somehow interpreted  as spinorial indices (like dimensional reduction to 1-dim.),
the Wilson and overlap-type\cite{overlap-1,overlap-2} fermions appear in analogous way with higher dimension.
The Wilson fermion case is realized by a doubler-suppressing term $\bar{\psi} H \psi$ where
\begin{equation}
H_{mn}^{ab}=\delta^{ab}ir\frac{2-\delta_{m,n-1}-\delta_{m,n+1}}{2},
\qquad\hat{H}(w)^{ab}=r\frac{2-w-w^{-1}}{2}\delta^{ab} .
\end{equation}
\noindent
The Wilson-Dirac operator is written as
\begin{equation}
i\hat{D}_{{\rm WD}}(w) = \frac{w-w^{-1}}{2}\Gamma + r\frac{2-w-w^{-1}}{2} ,
\end{equation}
\noindent
and  the bosonic inverse propagator with the doubler-suppressing term is
\begin{equation}
\hat{D}_B(w) =  \Big(\frac{w-w^{-1}}{2}\Big)^2
-r^2\Big(\frac{2-w-w^{-1}}{2}\Big)^2
=-\gamma_5\hat{D}_{{\rm WD}}(w)\gamma_5\hat{D}_{{\rm WD}}(w),
\end{equation}
where $\gamma_5$ is a matrix satisfying $\{ \gamma_5, \Gamma \} =0$ and $\gamma_5^2=1$.

For the overlap-type fermion,
we take
\begin{equation}
\hat{\Delta}(w)=\frac{w-w^{-1}}{2}K_{\rm OD}(w),
\end{equation}
\noindent
\begin{equation}
\hat{M}^{abc}(w,z) =\frac{f^{abc}}{6 K_{\rm OD}(w)}\Big(2wz+wz^{-1}+zw^{-1}+2(wz)^{-1}\Big),
\end{equation}
\noindent
where
\begin{equation}
K_{\rm OD}(w)=\frac{M+1}{\sqrt{(M+w)(M+w^{-1})}}.
\end{equation}
\noindent
The overlap-Dirac operator is written as
\begin{equation}
 i\hat{D}_{\rm OD}   = \Gamma \hat\Delta(w) + \hat{H}(w).
\end{equation}
\noindent
The Ginsparg-Wilson relation\cite{Ginsparg-Wilson,Neuberger} is written as
\begin{equation}
\hat{D}_{\rm OD}\gamma_5+ \gamma_5 \hat{D}_{\rm OD}=      \hat{D}_{\rm OD}
  \gamma_5 \hat{D}_{\rm OD} .
\label{GWR}
\end{equation}
\noindent
This relation leads us to
\begin{equation}
 \hat\Delta(w) \hat\Delta(w^{-1}) + \hat{H}^2(w) = 2i  \hat{H}(w)  .
\end{equation}
\noindent
Then,  the bosonic inverse propagator with the doubler-suppressing term
is expressed as
\begin{equation}
\hat{D}_B(w) = -\Big(  \frac{(1-w^2)K_{\rm OD}(w)}{2w}\Big) + \hat{H}^2(w) =2i \hat{H}(w) .
\end{equation}
\noindent
It is a remarkable fact  that the inverse propagator of a  bosonic variable is  determind by
the fermion doubler's mass  term  due to the supersymmetry.

\section{Higher-dimensional case}
In the previous section we consider a multi-component extension which will give a starting point for higher dimensional case. In this section, we consider a contrary case, namely two dimensional extension of the CLR without spinor index. Although it is not a real higher dimensional extension, we can readily see the difficulty which may exist in the real case. Real higher dimensional extension will be discussed elsewhere.

Let us denote a two dimensional lattice sites by $\bm{m} \equiv (m_1,m_2)$ and complex coordinates for two-variables $w$-expression by $\bm{w}\equiv(w_1,w_2)$.
Then we use the following simplified notations:
\begin{equation}
\bm{w} \bm{z}\equiv(w_1z_1, w_2z_2)\quad{\rm for}\quad
\bm{w}\equiv(w_1,w_2)\quad{\rm and}\quad\bm{z}=(z_1,z_2),
\end{equation}
also we write
\begin{equation}
{\bf 1}\equiv(1,1)\quad{\rm and}\quad\bm{w}^{-1}\equiv(w_1^{-1},w_2^{-1}).
\end{equation}
\noindent

Now we consider the following CLR for any bosonic fields
\begin{equation}
(\bm{\Delta}\Phi,\{ \Psi,   \Xi  \}) + (\bm{\Delta}\Psi,\{ \Xi,   \Phi  \})
+ (\bm{\Delta}\Xi,\{ \Phi,   \Psi  \}) =0,
\label{2-dim-CLR-1}
\end{equation}
\noindent
where the two-dimensional difference operator and the symmetric product are defined by
\begin{equation}
\bm{\Delta} \equiv (\Delta_1,\Delta_2) ,\qquad
\{ \Psi,   \Xi   \}_{\bm{k}} \equiv \sum_{\bm{m}\bm{n}}M_{\bm{k}\bm{m}\bm{n}}\Psi_{\bm{m}}
 \Xi_{\bm{n}} .
\end{equation}
\noindent
Here $\Delta_1$ and $\Delta_2$ operate the first and second site index $(m_1,m_2)$ of the field respectively.
By imposing  the  translational invariance,
\begin{equation}
\bm{\Delta}_{\bm{m}\bm{n}} = \bm{\Delta}(\bm{m}-\bm{n}),\qquad
M_{\bm{k}\bm{m}\bm{n}}=M(\bm{k}-\bm{m},\bm{k}-\bm{n}) ,
\end{equation}
\noindent
two  holomorphic functions can be defined as two-variables $w$-expression
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}) \equiv \sum_{\bm{n}}\bm{w}^{\bm{n}} \bm{\Delta}(\bm{n}),
\quad\hat{\bm{\Delta}}(\bm{1})=0,
\end{equation}
\begin{equation}
\hat{M} (\bm{w},\bm{z}) \equiv \sum_{\bm{m},\bm{n}} \bm{w}^{\bm{m}}\bm{z}^{\bm{n}}
M(\bm{m},\bm{n}) = \hat{M} (\bm{z},\bm{w}) ,
\end{equation}
\noindent
where simplified notation $\bm{w}^{\bm{m}} \equiv w_1^{m_1}w_2^{m_2}$ is used.
Then, the $w-$expression of the CLR  can be  obtained  as
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}^{-1}\bm{z}^{-1})\hat{M}(\bm{w},\bm{z})+
\hat{\bm{\Delta}}(\bm{w})\hat{M}(\bm{z},\bm{w}^{-1}\bm{z}^{-1})+
\hat{\bm{\Delta}}(\bm{z})\hat{M}(\bm{w}^{-1}\bm{z}^{-1},\bm{w})=0  .
\label{2-dim-CLR-2}
\end{equation}
\noindent
Setting $\bm{z}=\bm{1}$,  we obtain
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}^{-1}) =
 - \frac{\hat{M}(\bm{1},\bm{w}^{-1})}{\hat{M}(\bm{1},\bm{w})}
 \hat{\bm{\Delta}}(\bm{w}) ,
\end{equation}
\noindent
and
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}\bm{z})= \frac{\hat{M}(\bm{1},\bm{wz})}{\hat{M}(\bm{w},\bm{z})
\hat{M}(\bm{1},(\bm{wz})^{-1})}
\Big( \hat{M}(\bm{z},(\bm{wz})^{-1})\hat{\bm{\Delta}}(\bm{w})
+ \hat{M}(\bm{w},(\bm{wz})^{-1})\hat{\bm{\Delta}}(\bm{z}) \Big) .
\label{2-difference-1}
\end{equation}
\noindent
From eq.(\ref{2-difference-1}), we have
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}^2)= 2\frac{\hat{M}(\bm{1},\bm{w}^2)}{\hat{M}(\bm{w},\bm{w})
\hat{M}(\bm{1},(\bm{w})^{-2})}
\hat{M}(\bm{w},(\bm{w})^{-2})\hat{\bm{\Delta}}(\bm{w})  .
\label{2-difference-2}
\end{equation}
\noindent
By the use of the Weierstrass preparation theorem\cite{hormander} on multi-variable analytic functions,
the above difference operator can be parametrized as
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}) = \bm{K}(\bm{w})\Big( w_1-1 + (w_2-1)u(w_2) \Big)  ,
\label{2-diff-param}
\end{equation}
\noindent
where $\bm{K}$ and $u$ are nonzero and holomorphic near $\bm{w} \sim \bm{1}$.

In the following, we shall prove the CLR (\ref{2-dim-CLR-2}) has only a solution
$\hat{\bm{\Delta}}(\bm{w}) \propto w_1w_2 -1$ with
the local difference operator $\bm{\Delta}$ and the local product coefficient $M$.
Since also in this two-dimensional case, the holomorphism is equivalent to the locality, we discuss the relation (\ref{2-dim-CLR-2}) based on
a theory of  multi-variable holomorphic functions.

Using eqs. (\ref{2-difference-1}),(\ref{2-difference-2}) repeatedly,
we obtain the following recursion formula,
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}^n)=n  F(\bm{w},n)\hat{\bm{\Delta}}(\bm{w}) ,
\label{recursion}
\end{equation}
\noindent
where $F$ is a holomorphic function of $\bm{w}$ and
becomes unity in the naive continuum limit, $\bm{w} \rightarrow \bm{1}$.
From the parametrization $(\ref{2-diff-param})$, the recursion formula is rewritten by
\begin{eqnarray}
 w_1^n-1 + (w_2^n-1)u(w_2^n)&=&
 \frac{nF(\bm{w},n) K(\bm{w})}{ K(\bm{w}^n)}\Big( w_1-1 + (w_2-1)u(w_2) \Big) \nonumber \\
 &=& \prod_{k=1}^n \Big( w_1-w^{(k)}(w_2^n)  \Big)  ,
\label{factor-1}
\end{eqnarray}
\noindent
where
\begin{equation}
w^{(k)}(w_2^n) \equiv \omega_n^k\Big(   1-  (w_2^n -1)u(w_2^n)
 \Big)^{\frac{1}{n}},~~~~\omega_n=e^{\frac{2i\pi}{n}}  .
\end{equation}

\noindent
From the uniquness of the factorization near the naive continuum limit $\bm{1}$,
\begin{equation}
w_1-w^{(n)}(w_2^n) =  w_1-1 + (w_2-1)u(w_2)
\label{factor-2}
\end{equation}
\noindent
because any considered function is holomorphic around the limit.
The equation (\ref{factor-2}) can be exactly solved as
\begin{equation}
u(w_2)= \frac{1-w_2^{-u(1)}}{w_2-1} .
\end{equation}
\noindent
Imposing the single-valued condition on $u(w_2)$, $u(1)$ must be an integer, $\ell$.
The resulting expression is
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}) = \bm{K}(\bm{w})(w_1-\frac{1}{w_2^{\ell}})
= \frac{\bm{K}(\bm{w})}{w_2^{\ell}}(w_1w_2^{\ell}-1    ) .
\end{equation}
\noindent
Instead of $w_1$, we can argue the same thing about $w_2$ of $\hat{\bm{\Delta}}(\bm{w})$.
Since  both cases hold, $\ell$ must be unity and we can obtain a general form of
two-dimensional difference operator,
\begin{equation}
\hat{\bm{\Delta}}(\bm{w}) = \bm{K}(\bm{w})(w_1-\frac{1}{w_2})
= \frac{\bm{K}(\bm{w})}{w_2}(w_1w_2-1    ) .
\label{2-diff-op}
\end{equation}
\noindent
This general form (\ref{2-diff-op}) is a trivial extension of
one-dimensional difference operators and the general solution
 of CLR can be found easily
by substitution: $w\rightarrow w_1w_2, ~z\rightarrow z_1z_2$ in one-dimensional case.
This solution is, however, unsuitable for euclidean field theory because, with this,
the kinetic term  is  always written as a form of $w_1w_2$ or, in the momentum representation, $p_1+p_2$.

\vspace{1 cm}

\section{Conclusion and discussion}

When we naively tried to realize the full supersymmetry on lattice, we were required to realize a Leibniz rule (LR) on lattice. The no-go theorem, however, prevented us from proceeding further. In this sense, the theorem is indeed an origin of the difficulty in lattice supersymmetry~\cite{KSS,bergner-1,bergner-2}. On the contrary, realizing exact half-supersymmetry leads us a new criterion, {\it i.e.}~the cyclic Leibniz rule (CLR). The important thing is that the CLR actually has solutions which are realized by the set of local difference operator and local symmetric field products. Unlike the LR, the CLR do not contradict with the locality and the translational invariance on lattice. Yet in the naive continuum limit, CLR cannot be distinguished with LR. Because of these features we consider the CLR may be an important key ingredient for further constructing supersymmetric lattice models beyond the quantum mechanics model discussed in this paper.

Using a difference operator and field products which satisfy the CLR, we can easily construct exact half-supersymmetric action with superfield formalism. We can also find the Nicolai map without ``surface term''. Since each term of our superspace action is independently invariant under supersymmetry, we can introduce deformation parameters with no difficulty. This enable us to utilize a localization technique to compute, for  instance, the Witten index of our supersymmetric quantum mechanics models, which reproduces the known results in the continuum.

In order to construct supersymmetric field theoretical model, we have to adapt spinor index for higher dimensional CLR. In this paper, as a first step, we consider two-dimensional CLR without spinor index, in which we find a difficulty that the difference operator has a particular form unsuitable for the kinetic term. We hope to report a further analysis which properly includes spinor index elsewhere.

\vspace{5mm}
\vspace{10mm}
\acknowledgments
This work is supported in part by the Grant-in-Aid for Scientific
Research (No.20340048 (M.K.), No.22540281(M.S.) and No.20540274 (H.S.))
by the Japanese Ministry of Education, Science, Sports and Culture.

\vspace{10mm}

redpajama_set_name: RedPajamaArXiv


text:
Home Countries Australia 'It's a new Test, a new challenge'
'It's a new Test, a new challenge'
South Africa opener Dean Elgar insists there will be no complacency from the Proteas for the second Test against Australia in Hobart, as the series is not won.
The Proteas lead one nil after winning by 177 runs in Perth, and Elgar, who scored a ton in that match, feels his lads will be properly fired up.
He was quoted on Cricinfo as saying: "We know that the Australian side is still a very dangerous side.
"Them being beaten in the first Test is working in their favour because they can bounce back, and we're aware of their bouncebackability.
"It's just human nature to withdraw into yourself because of what you're going through. But they can bounce back hard.
"Complacency is a big thing for us. That's something we've addressed as a team. It's a new Test, a new challenge and a new venue for us."
The opener went on to discuss the Proteas' pace attack and their ability to reverse swing the ball at the WACA, which Kagiso Rabada was especially good at.
He added: "It is an asset for us if the ball can reverse but also, for a bowler to be able to bowl with the reversing ball is a massive skill.
"We're fortunate if that most if not all of our bowlers can bowl with the reversing ball, which works in our favour quite nicely.
"In Perth the ball reversed in both innings for us when we were bowling and it proved to be a massive asset for us."
The second Test starts on 12 November.