For a third slogan I’ve chosen:

Nature is fundamentally conformally invariant.

Note the weasel-word “fundamentally”. We know that nature is not conformally invariant, but the kind of thing I have in mind is pure QCD, where the underlying classical theory is conformally invariant, with quantization dynamically breaking conformal invariance in a specific way.

The group of conformal symmetries, its representations, and what this has to do with physics are topics I haven’t written about in the notes I’ve been working on. This is because I suspect we still haven’t gotten to the bottom of these topics, and properly dealing with what is known would require a separate volume. Part of the story is the twistor geometry of four dimensions, which Roger Penrose pioneered the study of, and which recently has found important applications in the calculation of scattering amplitudes.

As a more advanced topic, this slogan would normally have been put off until later, but I wanted to point to a new article by Natalie Wolchover in Quanta magazine which deals with exactly this topic. It describes several different efforts by physicists to rethink the usual story about the hierarchy problem, taking a conformally invariant model as fundamental. For the latest example along these lines, see this arXiv preprint. The whole article is well-worth reading, and it includes a quote from Michael Dine (whose work I’ve been critical of in the past) that I found heart-warming:

“We’re not in a position where we can afford to be particularly arrogant about our understanding of what the laws of nature must look like,” said Michael Dine, a professor of physics at the University of California, Santa Cruz, who has been following the new work on scale symmetry. “Things that I might have been skeptical about before, I’m willing to entertain.”

Perhaps particle theorists are beginning to realize that the landscape is just a dead-end, and what is needed is a re-examination of the conventional wisdom that led to it.

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Peter,

“We know that nature is not conformally invariant, but the kind of thing I have in mind is pure QCD, where the underlying classical theory is conformally invariant, with quantization dynamically breaking conformal invariance in a specific way.”

I am amazed by the number of people who keep repeating this “pure QCD” effect of dynamical breaking of conformal symmetry, with none of them ever actually trying to do the math.

I’ve already made a whole series of comments on one of your previous posts about this — in pure QCD, the symmetry breaking

is nota consequence of quantum corrections. It is a boundary condition that one plugs into the theory by hand. One can equally well choose a different boundary condition, and keep conformal invariance at the quantum level, starting from the very same “pure QCD”. So there is nothing “dynamical” in “generating the scale” from quantum corrections, because the boundary condition is not a consequence of dynamics.I do understand and appreciate the importance and beauty of conformal symmetry. But this symmetry is broken in nature (much like supersymmetry must be). And symmetry breaking is never a consequence of dynamics — be it spontaneous or through quantum corrections — it is always due to a boundary condition, independent of any dynamics.

The fundamental property of nature is the existence of the scale, not lack thereof.

Best, 🙂

Marko

Marko,

I just completely disagree with you here, but there is no point in having the same argument as last time. To stick to undeniable facts, the pure QCD Lagrangian is conformally invariant, the spectrum of the quantized theory (as calculated e.g. in the standard fashion via lattice gauge theory), is not.

By the way, there is a counterexample to your “none of them ever actually trying to do the math”. I started my career doing this kind of QCD calculation…

Whatever you want to call it, the point of the slogan is that the other interactions should get with it and try to behave as nicely as pure QCD.

Peter,

“the pure QCD Lagrangian is conformally invariant, the spectrum of the quantized theory […] is not.”

Of course, I don’t have a problem with the facts. I just have the problem with spinning this as a “dynamically generated” thing. It is a property that you can choose for the theory to have or not have — the spectrum of the quantized pure QCD can also be conformally invariant, if you want it to. This is usually overlooked because it is an uninteresting scenario. But to claim that the conformal symmetry *must* be broken in quantized pure QCD is just wrong.

“Whatever you want to call it, the point of the slogan is that the other interactions should get with it and try to behave as nicely as pure QCD.”

I can appreciate this way of looking at things. But in whatever way you formulate interactions, sooner or later you need to break the conformal symmetry. And there is nothing in the theory (classical or quantum) that requires you to do so — you need to

postulatethat conformal symmetry is being broken (for example, via quantum corrections). IMHO, this somehow this goes against the statement of the slogan.Best, 🙂

Marko

Totally off subject, but really not:

https://www.southernnewengland.aaa.com/legacy/TravelDiscounts/discounts-details.php?PROD_ID=295

There’s a new cruise liner in the Caribbean, Quantum of the Seas. No word if it sails through the Bermuda Triangle wormhole, or whatever it is.

Clearly, this quantum business is reaching new audiences.

Pure QCD is badly infrared divergent; in contrast with QED, its infrared divergence survives even in the transition probability. Probably, it is impossible to avoid the appearance of infrared divergence without violating scale invariance. Indeed, in the 2-dimensional free massless scalar field theory, infrared divergence is very serious; scale invariance is spontaneously broken in the consistent quantization.

QCD is classically conformally invariant. By asserting that it is fundamentally conformally invariant, you are claiming that classical physics is more fundamental than quantum physics. I do not agree.

Postulating conformal symmetry as a fundamental one (i.e. at short-distance scales) is the same idea as the fundamental supersymmetry: one wants to stabilize the quantum corrections to the Higgs mass or the cosmological constant by using the symmetry. However, in all these calculations the effect of quantum gravity is completely ignored, or it is assumed that it is not important. In a paper with M. Vojinovic (arxiv:1407.1394) we showed that the cosmological constant problem can be solved when a quantum gravity contributions are taken into account. I believe that the same could be done for the Higgs mass, so that one does not need fundamental supersymmetry or conformal symmetry. However, one needs a fundamental quantum gravity theory, and a good candidate is Regge quantum gravity (a modification of spin-foam models where the basic variables are the edge lengths and a spacetime triangulation).

“Of course, I don’t have a problem with the facts. I just have the problem with spinning this as a “dynamically generated” thing. It is a property that you can choose for the theory to have or not have — the spectrum of the quantized pure QCD can also be conformally invariant, if you want it to. This is usually overlooked because it is an uninteresting scenario. But to claim that the conformal symmetry *must* be broken in quantized pure QCD is just wrong.”

I don’t get that. The spectrum of pure QCD (SU(3) gauge fields only) consists of glueballs and you can compute their mass ratios in lattice QCD. I have done that. So there is no conformal symmetry in the quantum theory. What “property” of the theory can I choose to make these massive glueballs go away?

“The spectrum of pure QCD (SU(3) gauge fields only) consists of glueballs and you can compute their mass ratios in lattice QCD.”

Calculating mass ratios means nothing until you prove that at least one of those glueballs has nonzero mass. Otherwise all masses of all glueballs could be zero, and your mass ratios are satisfied trivially.

Btw, if you have managed to prove that a single glueball must have a nonzero mass, go claim the Millenium prize. 🙂

“What “property” of the theory can I choose to make these massive glueballs go away?”

Naively said, there is usually an implicit (and unproven!) assumption that at least one of the glueballs is massive, because only in that case calculating mass ratios is nontrivial.

More seriously, show me your calculation in full detail, and I’ll probably be able to point you to such an a priori assumption. (Though it might be hard to find, if it is phrased in an obscure way or buried somewhere in the numerical calculations.)

Even more seriously, lattice QCD is an approximate rather than exact calculation method, and can introduce all sorts of artifacts (fermion doubling is the most popular example of this), some of which we may still not be aware of. These artifacts might lead you to a conformally broken result simply as a consequence of the approximation scheme, rather than being the property of the theory itself.

And really, in all these discussions one needs to pay special attention to distinguish the statement “mass gap *can* exist” from the statement “mass gap *must* exist”, in every particular calculation. This can be done only by detailed step-by-step inspection on every paper that deals with conformal breaking in QCD. And it doesn’t make sense to do this in blog comments.

I feel like I am going against the windmills with this. I’ve already stated my point as clearly as I can, and I see no point in further debate.

Best, 🙂

Marko

“Calculating mass ratios means nothing until you prove that at least one of those glueballs has nonzero mass. Otherwise all masses of all glueballs could be zero, and your mass ratios are satisfied trivially.”

That sounds a bit like demanding a proof that the International Prototype Kilogram has a non-zero mass.

vmarko,

There is no fermion-doubling problem in pure QCD on the lattice. There is very strong numerical evidence for the conventional conjectures about the existence of a continuum limit of the lattice with a non-zero mass gap. Yes, there is no proof, and it’s possible the usual conjectures are just wrong, but I know of no evidence pointing to that.

Confused,

“That sounds a bit like demanding a proof that the International Prototype Kilogram has a non-zero mass.”

Sigh… Are you trying to compare a glueball to a proton? One lives in pure QCD, while the other (arguably) lives in the Standard Model — where the conformal symmetry is explicitly broken by the Higgs mass, already at the classical level. Apples and oranges.

The proton mass can (in principle, ultimately) be expressed as the Higgs mass times a dimensionless coefficient. This coefficient is a complicated function of electroweak-, QCD- and Yukawa-couplings, some large and some small, combining into the renormalized grand-total of 1/125. While one can argue that the dominant contribution comes from QCD, it still ends up multiplying the Higgs mass (this is on purely dimensional grounds). On the other hand, in pure QCD there is no Higgs, and unless you *assume* the existence of some other nonzero mass parameter, the coefficient of 1/125 is multiplied by a zero, keeping all glueballs massless.

Peter,

“There is no fermion-doubling problem in pure QCD on the lattice.”

Sure, but there might be other (unknown) artifacts of the lattice. Fermion doubling is just an example and a warning that such things may appear where you don’t expect them a priori.

“There is very strong numerical evidence for the conventional conjectures about the existence of a continuum limit of the lattice with a non-zero mass gap. Yes, there is no proof, and it’s possible the usual conjectures are just wrong, but I know of no evidence pointing to that.”

That’s because nobody is looking for such evidence — everyone is trying to calculate a nonzero mass gap. It’s a social effect. Whenever I speak to QCD people, they always eventually say something on the lines of “Sure, you can keep conformal invariance, but why would you want that? It doesn’t correspond to experiment.”. And then people like Alessandro Strumia go on to claim that conformal invariance *must* be broken at the quantum level, because this effect was “proven” in QCD. But there is no proof. And there is no evidence to the contrary simply because QCD folks are just not interested in preserving conformal invariance, and they always implicitly discard that scenario as trivial.

That’s why I have a problem with things like agravity and fundamental conformal invariance — they are blinded by a social effect, and have no solid basis.

Best, 🙂

Marko

Marko,

are you saying that in the end all mass is generated by the Higgs mechanism ?

But what if the Higgs mass is actually due to quantum corrections (Coleman-Weinberg mechanism) breaking the fundamental conformal symmetry of nature ?

http://arxiv.org/abs/1401.4185

“are you saying that in the end all mass is generated by the Higgs mechanism ?”

Possibly. If not, you have to postulate other sources of mass. Mass is never automatically enforced by quantum corrections (or dynamics in general).

“But what if the Higgs mass is actually due to quantum corrections (Coleman-Weinberg mechanism) breaking the fundamental conformal symmetry of nature ?”

That always involves a postulate (a suitable boundary condition for the corresponding RGE). The CW mechanism does not *necessarily* break conformal symmetry — it does only if you choose appropriate boundary conditions. Alternatively, you can choose to keep the minimum of the Higgs potential at zero, with no spontaneous symmetry breaking at all.

Best, 🙂

Marko

Marko,

1. It’s not true that the mass of the proton mostly comes from the Higgs. It is the other way around – most of the mass comes from QCD. Most of the proton’s mass is not from the quark masses (the latter does come from the Higgs, of course). Turning off the Higgs has little effect on the proton mass.

We used to call a fraction of the proton mass the “constituent” quark mass – but it is not really the quark mass – it comes from QCD.

2. “That’s because nobody is looking for such evidence — everyone is trying to calculate a nonzero mass gap. It’s a social effect.” Not really. The numerical evidence has been done very carefully. As a matter of fact, it’s been done carefully for decades.

There were serious analytic and numerical efforts by Patrasciou and Seiler to prove there was no confinement and gap, challenging the conventional wisdom. In the end, their program did not succeed, but they managed to do some interesting things in the attempt.

Also concerning 2.: there are a few people who do fool themselves into thinking they have demonstrated the existence of mass gap. They are generally not taken seriously (especially after the fiftieth paper). Most of us think that it will take a lot of creativity, hard work and (importantly) luck to understand the gap.

Correction – almost all the constituent mass comes from QCD (flavor dependence comes from the Higgs).

Peter Orland,

“Turning off the Higgs has little effect on the proton mass.”

How can you know this? You can’t turn it off in experiment, and you can’t evaluate the proton mass analytically and take the limit, because we cannot solve SM QCD equations in the IR. Moreover, the argument that the two masses are proportional goes on purely dimensional grounds — besides the Higgs mass, there is no other fundamental dimensionful parameter in the SM that can be proportional to the proton mass. People often claim that effective QCD scale can serve for this purpose, but you should keep in mind that this scale is a phenomenological parameter, measured in the experiment, where the Higgs mass is always “turned on”. So the QCD scale can also be considered proportional to the Higgs mass, and it can also go to zero if Higgs mass goes to zero.

As for numerical evidence, everything I ever saw were the calculations of mass ratios between hadrons. I’ve never seen an ab initio calculation from the fundamental SM parameters (if you know a reference for such an attempt, I’d like to see it). So numerical evidence basically says nothing about the nonzero-ness of hadron masses.

I am really frustrated by this being such an uphill struggle — everyone is just repeating the conventional wisdom, while nobody is able to provide serious proofs. And most often people assume that I don’t know what I am talking about. I am really being tempted to write a serious paper with all the gory details of integrating beta-functions in various models, just to point out that a differential equation always comes with boundary conditions, and debunk the conventional wisdom about this.

Best, 🙂

Marko

Peter Orland, could you say a little more (or point to a reference) about the failure of Patrascioiu & Seiler’s program? I only know a little of the story and that the question still appears to open on a (mathematically) rigorous level.

Peter,

“Nature is fundamentally conformally invariant.”

Wouldn’t that imply that “fundamentally” gravity is Weyl gravity (instead of Einstein gravity) as the Weyl tensor squared (more or less) uniquely leads to a conformally invariant gravitational action ? (Which wouldn’t be all that far fetched, see http://www.ejtp.com/articles/ejtpv11i30p1.pdf)

Hi Marko,

“How can you know this? You can’t turn it off in experiment, and you can’t evaluate the proton mass analytically and take the limit, because we cannot solve SM QCD equations in the IR. ”

The experimental evidence is that the actual quark masses can be found through current algebra (pseudo-Goldstone Boson masses, for example). These are the masses given to the quarks by the Higgs. They are much smaller than the effective quark mass used to find other masses. You don’t have to solve QCD to know this. This is fairly well-established physics. An experimentalist might laugh at our theories, but she/he takes this sort of thing very seriously.

Dimensional arguments don’t work here. Any energy is proportional to any other energy. It doesn’t mean both energies come from the same physics.

“I am really frustrated by this being such an uphill struggle — everyone is just repeating the conventional wisdom, while nobody is able to provide serious proofs. ”

In the case, the conventional wisdom is clearly right. We really can say something about nature. The uphill struggle is due to QCD being a hard problem (hence no proofs), but it is wrong to say we don’t know anything about it.

Hish,

I suggest you find P. and S.’s many papers at INSPIRE. There is far too much to summarize here.

Regards,

Peter O.

Here is the link:

http://inspirehep.net/search?ln=en&ln=en&p=a+patrascioiu+and+seiler&of=hb&action_search=Search&sf=&so=d&rm=&rg=25&sc=0

I can have conformal invariance in QCD even with the scale anomaly?

Marko,

“That always involves a postulate (a suitable boundary condition for the corresponding RGE). The CW mechanism does not *necessarily* break conformal symmetry — it does only if you choose appropriate boundary conditions. ”

Aren’t these boundary conditions coming from experiments, namely that one measures and fixes the relevant parameters at some scale and then “runs the RGE equations”. Now, as experiments tell us that scale symmetry is (obviously) broken, we can exclude the case of radiative corrections maintaining it. Isn’t this a peculiar/”fine tuned” case anyway ?

Moreover, doesn’t conformal symmetry imply that one is in a fixed point ?

(P.S. If these questions are naive, I apologize; I only have a quite superficial understanding of this topic).

“Alternatively, you can choose to keep the minimum of the Higgs potential at zero, with no spontaneous symmetry breaking at all.”

Never heard of this kind of scenario. Is there a good reference where I can read about it ?

AC,

I am also also confused by those remarks.

Generally (but not always) conformal-invariant actions lead to breaking of conformal invariance. The scale appears, not because of some special boundary condition, but because of the LACK of a special boundary condition. Generic boundary conditions have broken conformal invariance. There is a conformal limit, obtained by taking the scale to zero (which is the special bc).

Anyway it’s the conformal-invariant limit which is non-generic. It’s best not to learn such things from remarks on comment threads (including my own).

vmarko: “As for numerical evidence, everything I ever saw were the calculations of mass ratios between hadrons. ”

What I recall, and it has been almost 30 years since I worked on this, is that in pure SU(3) YM on the lattice you compute some glueball masses as well as another “mass” which is related to a “string tension”. Everything depends singularly on the lattice spacing but only mass-ratios are meaningful quantities (in ANY theory) so we report the glueball masses in units of the string tension “mass”. The latter is now the new “dimensional” free parameter, instead of the original dimensionless coupling constant g (“dimensional transmutation”). We can measure the string tension related “mass” Ms, which means really we can measure the ratio of this “mass” and that standard block of mass in France which we call 1 kg, so this allows you to come up with actual predictions for the glueball masses.

All you seem to be saying is that you can set Ms=0, which is true but as trivial as saying you can set g=0 in the SU(3) theory and get 8 massless particles instead of the usual QCD.

Obviously no theory without mass parameters can predict a particle mass in kg, if only because the laws of nature don’t know anything about that object in France which we call kg.

vmarko,

“Even more seriously, lattice QCD is an approximate rather than exact calculation method, and can introduce all sorts of artifacts (fermion doubling is the most popular example of this), some of which we may still not be aware of. These artifacts might lead you to a conformally broken result simply as a consequence of the approximation scheme, rather than being the property of the theory itself.”

I do not intend to pick upon you, rather the above statement. This is a rather naive viewpoint, especially today. Aside from the lack of proof of lattice QCD calculations belonging to the same universality class (a proof which is not special to QCD, but to any theory requiring a non-perturbative regulator), all approximations made in numerical lattice QCD calculations are well understood, and systematically improvable, meaning these approximations can be removed with refined calculations to, in principle, any desired level of precision. Fermion doubling is not one of these approximations. Fermion doubling is a well understood consequence of the “naive” implementation of fermions. This problem has been resolved for decades, with the original solution by Wilson himself, with a technique now known as “Wilson fermions”. There are several other solutions that are all employed in today’s calculations.

The three big systematics are 1) the continuum limit, 2) the infinite volume limit and 3) the limit of physical quark masses. All three of these systematics can be cast into the language of Effective Field Theory and hence, order by order corrections due to being away from these limits can be systematically described and parameterized in a model independent way, and then these corrections can be extrapolated away. The order in perturbation theory needed to control this extrapolation depends upon the desired level of precision of the final result. State of the art lattice calculations are now performed with the light (up,down) and strange quark masses basically at their physical values. Three or more lattice spacings are used and several volumes are used.

These calculations have demonstrated, for example, that ~95% of the mass of the proton comes from glue – meaning the calculations were performed at several values of the current quark masses (and lattice spacings and volumes), and then the proton mass was extrapolated to the point where these quark masses are zero, with still ~900 MeV for the mass of the proton. This is not a pen-and-paper proof that the mass of the proton comes from QCD (and not the Higgs vev), but this is a numerical proof. Taken alone, you may have your doubts, but lattice QCD calculations have now been used to make several predictions (not post-dictions) that have all been experimentally confirmed. Most of these lie in the realm of heavy quark physics. There is an extreme preponderance of numerical evidence now that QCD is the correct theory describing the IR realm of strong interactions (hadronic and nuclear physics).

Since this is already a bit long – look at any of the proceedings of the annual international lattice conference in the last few years. To come up with an alternate theory that shows all the properties observed in nature, can reproduce all the results of lattice QCD, yet does not poses “dimensionful transmutation” would really require a “conspiracy theory”.

Regards,

André

confused:

“Obviously no theory without mass parameters can predict a particle mass in kg, if only because the laws of nature don’t know anything about that object in France which we call kg.”

In fact, this is precisely the point. The QCD Lagrangian with all quark masses=zero (pure QCD? or is pure QCD just Yang Mills – I am unfamiliar with this jargon), is conformally invariant. But the dynamics of the theory generate a mass scale, Lambda_QCD. This length scale is approximately the confinement scale of the theory, and all states of this theory will have a mass proportional to Lambda_QCD – hence the generation of a mass scale. In physics, this was coined “dimensional transmutation”, I believe by Sidney Coleman. This mechanism, as noted above, is how QCD solves its IR problem. Without confining the gluons, QCD would be IR divergent, and thus not a sensible theory.

Now, relating this mass to something in kg requires more subtle questions involving atomic physics and units. But the point is that a theory with a conformally invariant Lagrangian can have a mass scale and be used to predict the mass of states of the theory. These predictions may require the aid of a computer, but they can still be made.

Regards,

André

Oh, wow, it appears that we are finally getting somewhere!

AC,

“Aren’t these boundary conditions coming from experiments, namely that one measures and fixes the relevant parameters at some scale and then “runs the RGE equations”. Now, as experiments tell us that scale symmetry is (obviously) broken, we can exclude the case of radiative corrections maintaining it. Isn’t this a peculiar/”fine tuned” case anyway ?”

YES! You are RIGHT! It is experimental data that provides us with the correct boundary conditions. My only point is that the theory itself *doesn’t*! The theory *allows* a maintained conformal symmetry, but this is not interesting (or, as you say, a peculiar case) because in the real world conformal symmetry is broken! The theory doesn’t say it must be broken, the experiment does! And this is in direct contradiction with agravity approach (and the like) which claims that breaking of conformal symmetry is required by the theory. It isn’t — you have to postulate the breaking (in the theory), because experiment says it is broken. And this is in direct contradiction with Peter’s slogan. Thanks for noticing!

Peter Orland,

“Generic boundary conditions have broken conformal invariance. There is a conformal limit, obtained by taking the scale to zero (which is the special bc).”

Thank you too! We agree that the conformal limit can exist, and that it is not excluded by the theory itself (but rather by experimental data). As I explained above, this is in contradiction with the agravity approach and Peter’s slogan.

Confused,

“All you seem to be saying is that you can set Ms=0, which is true but as trivial as saying you can set g=0 in the SU(3) theory and get 8 massless particles instead of the usual QCD.”

And thanks to you as well! That is indeed all I am saying! The theory allows for the case Ms=0. Trivial or not, the theory itself does not exclude this possibility. The fact that Ms is not zero in nature is therefore not a consequence of the theory, but comes from experiment. Therefore, to make theory match experiment, one needs a theoretical postulate that Ms is not zero. This contradicts agravity approach and Peter’s slogan.

Walkloud,

“But the dynamics of the theory generate a mass scale, Lambda_QCD. This length scale is approximately the confinement scale of the theory, and all states of this theory will have a mass proportional to Lambda_QCD – hence the generation of a mass scale.”

My statement is that the theory (the theory alone) allows for the case where Lambda_QCD iz zero. This would correspond to to strong coupling asymptotically flowing to one as we run it further and further into the IR regime. As unusual as it might seem, the only thing I want to say is that there is nothing in the theory that forbids this case. It is rather an *experimental* fact that the value of Lambda_QCD is nonzero, and consequently we have a mass scale via dimensional transmutation. And as I explained above several times already, this is in contradiction with the point of view of Strumia, Woit and others, who claim that nature is in fact conformally invariant, but that this symmetry must be broken because the theory dictates it. My whole point throughout this long discussion is that the theory *does not* dictate it, rather the experiment does. Hence the contradiction with Peter’s slogan.

I can see now a small glimmer of hope that I am actually getting my point through to people here… 🙂

Best, 🙂

Marko

Marko, I think everyone is well aware of what you’re saying but considers it too trivial to mention every time. By analogy, when someone states that QED explains the Coulomb force people usually leave out the obvious caveat that the coupling alpha=1/137 is not predicted by the theory and there is no a-priori reason why it could not be zero with no Coulomb force.

But maybe I’m not understanding something subtle. For example I am reading in the statement of the millennium problem (http://www.claymath.org/sites/default/files/yangmills.pdf):

“Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R^4 and has a mass gap ∆ > 0”

I don’t understand the question. In what units is \Delta supposed to be expressed?

The only computable quantities are mass (or energy) ratios.

Hello Marko and confused,

“My statement is that the theory (the theory alone) allows for the case where Lambda_QCD iz zero. This would correspond to to strong coupling asymptotically flowing to one as we run it further and further into the IR regime. As unusual as it might seem, the only thing I want to say is that there is nothing in the theory that forbids this case. It is rather an *experimental* fact that the value of Lambda_QCD is nonzero, and consequently we have a mass scale via dimensional transmutation.”

This is not correct. You DO NOT need experiment to be able to know if the theory develops a mass gap (is Lambda_QCD > 0). You need to be able to solve the theory in the IR, which is what no one knows how to do with pen-and-paper. All SU(N>1) gauge theories are known in perturbation theory to exhibit properties indicative of dimensional transmutation – ie their beta functions are all negative, indicating they do develop a mass gap (Lambda_QCD > 0). Of course, this is perturbation theory, so not a proof in the IR regime.

This is where (numerical) lattice field theory enters. The lattice calculations are of course performed in terms of dimensionless variables (to confused’s question – you can only compute ratios of energy scales without further input). What lattice calculations unambiguously show is that the theory does develop a mass gap, with a highly non-trivial spectrum. Add quarks to the calculation, and the spectrum becomes significantly richer. The calculations show the spectrum does not become trivial as the systematics are removed. This does not require experimental input.

Experimental input is needed to fix the absolute scale, so that the resulting dimensionless spectrum determined with the computer can be converted into units we are familiar with. In case one is concerned that somewhere between the scale at which the calculations are performed, and the UV, there is a non-trivial fixed point where the theory becomes conformal, the calculations can be, and are performed at scales where perturbation theory is reliable, so that analytic perturbative results can be matched to the non-perturbative numerical calculations, thus providing a smooth transition between the UV and IR. Yang-Mills SU(N) gauge theories, including those with small-enough number of dynamical fermions (such as QCD), do not have non-trivial fixed points where they become conformal. There is, however, an active area of research where people are adding more fermions to the theory, to see if they become nearly conformal.

The theory (Yang-Mills + small enough number of mass-less fermions), defined as the Lagrangian, does not tell you if Lambda_QCD is zero or non-zero. However, the solution to the theory does tell you Lambda_QCD is non-zero.

Regards,

André

Andre and others,

Thanks for trying to get this straight, but I don’t think anyone is going to convince vmarko on this question of our understanding of QCD. So, no more about QCD’s breaking of conformal symmetry, that topic has been beaten to death.

I lately read part of Penrose’s new book “Cycles of Time” which is also concerned with conformal invariance.

There is one thing that troubles me, which he addresses in a in a footnote, namely the role of the conformal anomaly. He states:

“There can, however, be an issue with regard to what is referred to as a conformal anomaly, according to which a symmetry of the classical fields (here the strict conformal invariance) may not hold exactly true in the quantum context. This will not be of relevance at the extremely high energies that we are concerned with here, though it could perhaps be playing a role in the way that conformal invariance ‘dies off’ as rest-mass begins to be introduced.”

Unfortunately I do not understand his argument which seems crucial for thee whole scenario to work. Can anybody help me ?

Astonished that Dilbert of Sunday Aug 24 has not received a mention yet.