Michael Creutz has a remarkable new preprint out this evening, entitled The Saga of Rooted Staggered Quarks. It explains what has been going on in a rather bitter controversy within the lattice gauge theory community over the last few years.

While lattice gauge theory provides a quite beautiful way of discretizing gauge fields, preserving their geometric significance, fermions have always been much more problematic. Here the geometry is spin geometry, which doesn’t appear to have a natural formulation on a lattice. What does have a natural formulation is not a spinor field S, but End(S), the linear maps from S to itself, which can be identified with the exterior algebra, and naturally put on the lattice by assigning degrees of freedom to points, 1-simplices, 2-simplices, etc. The problem is that if you do this, you don’t get a theory of a single fermionic field, but instead multiple copies. This geometrical argument is just one aspect of the problem, which appears in other more convincing ways, but this all adds up to making chiral symmetry especially problematic on the lattice.

There are many possible ways of dealing with this, but one popular one has been “rooting” some of the fermionic degrees of freedom that have been staggered on neighboring vertices of the lattice. One ends up with four copies of what one wants, so the argument has been that the thing to do is to take the fourth root of this to get a calculation that tells one about a single fermion. The problem is that this is a quite non-analytic thing to do, and it is not clear that it gives one something sensible. A debate between Creutz and people using this method has raged for the last few years, with Creutz claiming that the rooting procedure gives the wrong answer, while proponents of rooting argue that the problems involved will go away in the continuum limit.

Creutz’s preprint describes the conclusion he has been led to about this, and his problems getting some of them published:

This led me to question whether there was some physical measurement one could make to determine if a quark mass was indeed zero. I could think of none, and proposed that a single vanishing quark mass might not be a physical concept. This paper was submitted to Physical Review D.

This is where the shit started hitting the fan. There was a common lore that if the up quark mass were to vanish, then the problem of why theta appeared to be phenomenologically very small would be solved. I was saying that this lore might be wrong. This drove the referees nuts, with statements like “I am somewhat concerned that the errors are so obvious.” After numerous similar scathing remarks the paper went to a divisional editor for PRD, who upheld their opinion. On rejection I took the paper and split it into two parts, one on the phase diagram and the second on the vanishing mass issue. These both appeared in Physical Review Letters, Phys.Rev.Lett.92:201601,2004 (hep-lat/0312018) and Phys.Rev.Lett.92:162003,2004 (hep-ph/0312225). I do derive some visceral pleasure from having turned a rejected PRD paper into two PRL’s…

Eventually the claims of the staggered advocates became so outrageous that I felt I had to be more aggressive. I was pushed further by statements that if someone had issues with staggered quarks, they needed to write them up. At the time I was too naive to appreciate how the stubborn nature of some personalities involved would mean that these arguments would be dismissed without serious discussion. As with the up quark mass issue, this is one of those situations where a person without tenure would be ill advised to challenge conventional lore.

So I submitted a paper (hep-lat/0603020) pointing out the inconsistencies between rooting and the expected chiral behavior. This was quickly rejected by PRL which has a policy of not publishing interesting and controversial papers. After transferring it to PRD, things got stuck, with numerous referees simply refusing to respond. After about a year and eight referee reports, some positive and some negative, PRD decided that they don’t publish interesting and controversial papers either. I did not take this delay kindly and rewrote the paper with the provocative title “The evil that is rooting.” This was fairly quickly accepted by Physics Letters (Phys.Lett.B649:230-234,2007; hep-lat/0701018), although the title was mollified at the editor’s suggestion…

The staggered community has continued to ignore these problems. I feel their stranglehold on the US lattice effort approaches scientific dishonesty. As an example of the prevailing vindictiveness, a recent paper of mine on a completely different topic was rejected from a prominent US journal on the basis of a single negative referee report stating that “It is puzzling that the author ignores all these highly relevant lessons that have been learned long ago in the context of the staggered fermion formalism.” It was overlooked because I wanted to avoid the ongoing controversy, of which the referee was certainly aware. After I did add remarks on the comparison with staggered, the paper was rejected without further review by a divisional associate editor representing the staggered community. He raised some symmetry issues based on comments by the Maryland group, to which I was never given a chance to respond. This paper was then submitted to a European journal where I hoped for a more equitable treatment. There it was quickly published.

Beyond the international ridicule this this controversy brings on the USQCD community, other aspects are particularly upsetting from a scientific point of view. First, enormous amounts of computer time continue to be wasted on generating lattice configurations from which any non-perturbative information will be questionable. About 38 percent of the current computer time allocated by the USQCD collaboration is going to continue these efforts. Second, young people associated with this project are taught to repeat, without question, the party line that all will be okay in the continuum limit. Third, the practitioners are such a powerful force that most outsiders are unwilling to look into the problems despite the fact that the underlying physics is so fascinating. And finally, I find it extremely unsettling that some physicists widely regarded as experts in chiral symmetry and lattice gauge theory can so casually and thoroughly delude themselves with bad science.

In short, the lattice has been very good to me. It is extremely painful to see it abused so blatantly

One would like to think that this issue will get sorted out over time as more work makes it clear whether or not rooting is as serious a problem as Creutz thinks it is. But the progress of science is not always smooth…

One way to settle the controversy would be to carefully formulate order-by-order renormalization (of physical operators) with rooted fermions. Has anyone studied this?

P.S. It clearly isn’t easy, since there doesn’t seem to be a local (within a small number of lattice spacings) Lagrangian. So I guess I am asking if rooted Fermions can be formulated (in some complicated way) so that there is manifest locality.

Whether Creutz is right or wrong and whether there is bad scientific practice, one good point about hep-ph and hep-lat community is that they still debate, whereas in String theory nobody argues.

“Whether Creutz is right or wrong and whether there is bad scientific practice, one good point about hep-ph and hep-lat community is that they still debate, whereas in String theory nobody argues.”

I don’t think that “It is puzzling that the author ignores all these highly relevant lessons that have been learned long ago in the context of the staggered fermion formalism.” can be considered as a healthy debate situation, and yes, it is a journal context, but I don’t think the arxiv one goes much better. All this only reminds me of an african animation movie in which a young child asks his mom “Mom, why is the Sorceress so evil?”, and the mom replies “She’s not more evil than others: she just has more power.”

I reformulated my arguments in terms of the ‘t Hooft vertex, something crucial to the understanding of how the theta dependence of QCD works. While previously I had not claimed to have a proof, here I showed specific non-perturbative effects that must come out wrong, even in the continuum limit. This proof appears in the proceedings. A more extensive discussion of the issues appears in Annals.How difficult is it to assign a couple of graduate students to check this proof?

Arun,

The problem with physics “proofs” is that they typically involve lots of assumptions, often not made explicit. I don’t know anything about the argument Creutz is referring to, but I’d guess that it evaluating possible objections to it would require quite a lot of expertise, not the sort of thing a typical graduate student has the background for.

In Europe, our community no longer debates Creutz’s ‘proof’ because it is known to be incorrect.

For the most recent paper debunking the criticism, see Kronfeld [2007] http://arxiv.org/abs/0711.0699 that also agrees with previous points by Bernard and his collaborators [2006] http://arxiv.org/abs/hep-lat/0603027

These papers do not quite prove that rooting works perfectly, as indicated by formal manipulations and by the numerical data. But they show that Creutz’s arguments are incorrect. For instance, the Kronfeld paper has a section dedicated to each aspect of Creutz’s criticism (cutoffs, ranks of groups, and so on). I am slightly worried that the new Creutz’s preprint, submitted as HTML, won’t be enough to convince me or my German and other European colleagues.

There are actually two distinct but seldom disentangled issues related to rooting. The same two issues arise with many respect to many of the techniques used in lattice calculations. The issue is whether what you’re calculating approaches continuum QCD in the right limit. The second is whether the technique gives useful/accurate results at the physical lattice spacing you are using.

It’s not clear whether the first issue is a problem with rooting. Neither side has produced a definitive argument. If I had to guess, I’d bet that the continuum theory is not QCD, but that’s just a guess. What gets obscured in this debate over rooting is that there things done in lattice calculations that we know for sure do not lead to the right limiting theory. For example, lattice calculations are now done using an irreversible Markov chain. (Lattice people would refer to this as “not actually using the Metropolis alogrithm.”) In this case, the continuum limit of any calculation cannot even be defined, because the calculation does not, in principle, converge.

But that’s only in principle. We know that practically, the improved, non-Metropolis algorithms do converge (faster even!), so the problem is not worth worrying about. This brings us to the second issue. The lack of Markov reversibility does not introduce any meaningful errors at the level of accuracy of current calculations. Indeed, it improves the accuracy. Coming back to rooting, while whether it is formally valid is an open question, whether it is practically useful is not. Rooting does an excellent job of dealing with low-mass dynamical quarks at the present time. Maybe it will need to be re-evaluated when calculations get a lot more precise, but all indications are that the lattice community will keep an eye on this matter.

Brett, this is way too technical for me – but I wonder if the current argument about rooting is analogous to the old debates about validity of renormalization (“sweeping the problem under carpet” etc), a suspect calculation trick that worked, for reasons that were understood only afterwards

Milkshake,

There were no debates about renormalization. It took people a while to realize that it was deeper than just “sweeping…”, but few competent people doubted it was valid, even in the the late 40’s. The reason was that it gave stunningly good results for measurable quantities in atomic physics.

The argument about rooting is different. I was asking above if a rooted lattice theory might have an (albeit complicated) local formulation. If so, it probably can be renormalized, and therefore should be standard QCD. If not, it isn’t so clear.

but Brett claims that “rooting does excellent job” – and if this is so, does this make the observable things come out better? And how much hand-adjusting is done in these calculations to make things come out right? Also, the most authoritative people at any given time in science history are not always the most competent ones in hindsight. So if Creutz claims he has been unfairly censored, is it because he is crackpotty – or did he become unpopular for repeating a valid complaint that no-one wants to hear?

milkshake,

I don’t see Creutz complaining of censorship. He’s a leading figure in the field, and his problems with referees haven’t kept his papers from being published. The scientific issues he is concerned about are getting a full airing, while, as is often the case in science, the way the debate is taking place may be less than the fully rational process one would hope for.

Brett,

In my experience (limited to solving PDEs for derivatives pricing), if you do not have a handle on errors/convergence properties you might as well not bother … better to use the same computer power for playing “Doom” instead – at least you know what is going on there. So I guess, to use Susskind’s terminology, I would never have “made it” as a Lattice Gauge Theorist.

Peter O,

I agree with you about renormalization. After 1950 only retards like Landau, Feynman and Dirac ever expressed doubts.

hi peter orland,

yes, locality is the crucial issue.it has been proven, that a) at m=0 there is no local formulation b) that at m>0 there is no local formulation, but the nonlocality is a cutoff effect, so it vanisches in the continuum limit. so at m=0 rooted staggered fermions are in the wrong universality class (which is easily seen by the fact that the symmetry content is wrong). but outside m=0 the topic is quite subtle if you reflect a bit about what it means that you have a nonlocality that is a cutoff effect. there is the issue of the order in which you take the three limits (m->0, cutoff->00, V->00).

in fact, order by order in PT it has been proven that things are fine, but this is not enough. also there is strong numerical evidence that supports the view that everything is ok if you stay in the correct regime, but a full analytical proof is somethig that the whole community tried to hunt for at least half a decade and it is not in sight.

chris –

Thanks for the information. It’s clearly worth looking into!

Chris O.

>I agree with you about renormalization. After 1950

>only retards like Landau, Feynman and Dirac ever

>expressed doubts.

They doubted the interpretation of renormalization. They weren’t sure whether the idea was really right at a deeper level (in the meantime, nonperturbative techniques have shown that it is). I heard Feynman express these misgivings (he was teaching QED and QCD in 1977-78). But he and others never doubted its validity for leading orders of QED.

Mike Creutz is not expressing only misgivings, but openly claiming rooting is valueless.

In the continuum limit, staggered fermions yield four species, called tastes. To reduce the number of tastes to one (per flavor)…Holy crud! Terry Pratchett was right all along!

So, I have what I hope is not a stupid question:

Here the geometry is spin geometry, which doesn’t appear to have a natural formulation on a lattice. What does have a natural formulation is not a spinor field S, but End(S), the linear maps from S to itselfWhat I am immediately curious here about is, is there a reason the spin networks incidentally used in LQG cannot be made use of here? They aren’t lattices, but they are a method of discretizing space, and at least in their original pre-lqg formulation they by design affixed a spin to each spacetime “point”. Looking up spin geometry online however (which if I’m understanding this right is basically just the study of spinor bundles?), I don’t see anything at all indicating people have even tried to link that subject with spin networks. Is there some obvious reason I’m missing why it would not be natural to try to do spin geometry with spinnets?

Or, put another way: Is it known whether some of the advantages with lattice gauge theory (like computability) can be retained if one moves to some other discrete structure which doesn’t have the problem with accommodating fermions?

Only adding to my confusion, the wikipedia article for “Lattice Gauge Theory” tosses off this cryptic factoid:

> wikipedia … tosses off this cryptic factoid:

Perhaps it refers to

this paper.

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Has anyone asked the Aussie LQCD community for their opinion on rooting.

“Has anyone asked the Aussie LQCD community for their opinion on rooting.”

Do they disapprove of rooting?

Do they refrain from engaging in rooting?

hi all,

some answers to Peter Osland’s technical points — plus a humble opinion.

——————————————————————————————————————

# Peter Orland Says:

One way to settle the controversy would be to carefully formulate order-by-order renormalization (of physical operators) with rooted fermions. Has anyone studied this?

P.S. It clearly isn’t easy, since there doesn’t seem to be a local (within a small number of lattice spacings) Lagrangian. So I guess I am asking if rooted Fermions can be formulated (in some complicated way) so that there is manifest locality.

——————————————————————————————————————

there has been some work along this direction, interested readers may look for papers by Joel Giedt and David H. Adams in SPIRES. the problem is, in perturbation theory the rooting procedure is essentially ok, precisely because PT fails to tackle long-distance couplings at any given order, and non-locality would have a non-perturbative origin. indeeed, as observed by Creutz, even the theorists supporting the rooted staggered approach do currently think that there is no local formulation of the regularised theory; needless to say, this makes it very hard to understand how to take a continuum limit that defines QCD consistently. (cf. work by Bernard, Golterman, Shamir, Sharpe.)

on the other hand, as an Anonymous lattice points out, Creutz’s argument is widely regarded as insufficient (“incorrect” is maybe too precise a word in this context) to show that rooted staggered quarks do not lead to QCD in the continuum limit. yet, the overwhelming majority of the community still thinks that the rooted staggered approach is at best controversial, and at worst just an elaborated model that cannot be plausibly claimed to produce first-principles results. the rift between the two fields is deep indeed…