Does QCD make sense without a cutoff?

[Demystifier]

This is a kind of continuation of the dispute I had with @A. Neumaier. According to one view, QFT of the Standard Model should fundamentally be viewed as a continuous theory, while the UV cutoff is just an auxiliary tool that eventually should be put to infinity. According to another view, the UV cutoff is a fundamental finite parameter of the theory. The latter point of view seems particularly convincing for QCD, not only because QCD on the lattice makes fine results, but also because there is a $\Lambda_{\rm QCD}$ that seems closely related to the UV cutoff $\Lambda_{\rm UV}$. Tong, in his lectures http://www.damtp.cam.ac.uk/user/tong/gaugetheory/2ym.pdf page 63, says this:
"Quantum field theories are not defined only by their classical action alone, but also by the cut-off $\Lambda_{\rm UV}$ . Although we might like to think of this cut-off as merely a crutch, and not something physical, this is misleading. It is not something we can do without. And it this cut-off which evolves to the physical scale $\Lambda_{\rm QCD}$."

What's your take on this?

 

[PeterDonis]

Moderator's note: Moved to QM interpretations forum.

 

[vanhees71]

As any interacting field theory in 4 dimensions you have to renormalize, and this implies that you need an energy scale, where to renormalize (particularly if you have only massless fields as in pure Yang Mills or QCD in the chiral limit of massless quarks). That's of course not really a cutoff parameter but rather a renormalization scale, but without such a scale the theory is undefined. Whether or not renormalized QCD is well-defined is an open question not yet solved.

 

[A. Neumaier]

According to another view, the UV cutoff is a fundamental finite parameter of the theory. The latter point of view seems particularly convincing for QCD, not only because QCD on the lattice makes fine results, but also because there is a $\Lambda_{\rm QCD}$ that seems closely related to the UV cutoff $\Lambda_{\rm UV}$. Tong, in his lectures http://www.damtp.cam.ac.uk/user/tong/gaugetheory/2ym.pdf page 63, says this:
"Quantum field theories are not defined only by their classical action alone, but also by the cut-off $\Lambda_{\rm UV}$ . Although we might like to think of this cut-off as merely a crutch, and not something physical, this is misleading. It is not something we can do without. And it this cut-off which evolves to the physical scale $\Lambda_{\rm QCD}$."

This latter point of view cannot be correct since there are perturbative formulations of renormalized QCD that do not have any cutoff but are Poincare covariant throughout. That a cutoff appears in Tong's treatment is just because he takes the oldest, Poincare symmetry breaking approach to QCD.

No cutoff appears in the modern treatment via [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL], which is fully rigorous (without any divergences to be swept under the carpet) and covariant at any fixed loop order, and works quite well for QCD. There is also no cutoff in dimensional regularization, the standard renormalization method for nonabelian gauge theories, which is also covariant at any fixed loop order, though not quite rigorous.

Only the choice of a renormalization scale is needed, which controls the energy range where the few loop approximations are best. At the (noncomputable) infinite loop order, which presumably corresponds to an exact but currently intractable QCD, the renormalization scale does not matter.

Note that Tong has this renormalization scale (his $\mu$ on p.61) in addition to the cutoff. The physical scale $\Lambda_{\rm QCD}$ is defined in terms of $\mu$ and $g_0$ (and determines the latter as a function of $\mu$), not in terms of the unphysical cutoff! However, by choosing this cutoff equal to this scale one gets rid of it, while still having the dependence on the renormalization scale $\mu$.

 

[HomogenousCow]

There is also no cutoff in dimensional regularization, the standard renormalization method for nonabelian gauge theories, which is also covariant at any fixed loop order, though not quite rigorous.

Would you mind elaborating on this point, why is Dim-reg not fully rigorous?

 

[vanhees71]

This of course only refers to perturbative QCD. It's well known that it is a perturbatively Dyson-renormalizable QFT and this is independent of any momentum cutoff but only dependent on the renormalization scale. It's independent of how you choose to regularize it before you renormalize and then put the regularization away by taking the appropriate physical limit. In dim. reg., e.g., you take the dimension to 4. You can also renormalize without any intermediate regularization using subtraction schemes (similar to BPHZ though in QCD you need to use a mass-independent renormalization scheme because you have massless gluons in the theory; this necessarily also introduces a renormalization scale).

What I had in mind above was the full theory, and AFAIK it's not yet settled whether it really exists in a mathematically strict sense.

 

[atyy]

At the informal level, QCD is often argued not to need a UV cutoff because of asymptotic freedom. As @vanhees71 says, this is not yet rigorously proved.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1166630/
"Thus, QCD provides the first example of a complete theory, with no adjustable parameters and with no indication within the theory of a distance scale at which it must break down."

In Tong's notes, he gives 2 equations for $\Lambda_{QCD}$, where it depends on $\Lambda_{UV}$ in Eq (2.59), but can also be stated in the equation after (2.59) with a formula that is valid at any (high) energy scale. Tong says $\Lambda_{QCD}$ is a low energy where $g^{2}(\mu)$ diverges. The Wikipedia article in dimensional transmutation also says $\Lambda_{QCD}$ is an infrared cut-off. So this is more that QCD has a low energy cutoff where perturbation theory fails, not a high energy cutoff like the Landau pole in QED.

 

[A. Neumaier]

Would you mind elaborating on this point, why is Dim-reg not fully rigorous?

Well, it uses nonphysical nonintegral dimensions to remove infinities in integral dimensions, and takes a limit to the physical dimensions at the very end.

 

[Demystifier]

Moderator's note: Moved to QM interpretations forum.

Can you explain why do you think that it belongs to QM interpretations forum? I think it doesn't.

 

[vanhees71]

Of course it doesn't belong to the QM interpretations forum. That's why I answered though it has been moved there ;-)).

 

[PeterDonis]

Can you explain why do you think that it belongs to QM interpretations forum?

Because the conflict between different views that you describe in the OP seems like a conflict of interpretations. Both views you describe, as far as I can tell, make the same experimental predictions, so the conflict you describe cannot be resolved by experiment. That makes it a matter of interpretation.

 

[A. Neumaier]

Because the conflict between different views that you describe in the OP seems like a conflict of interpretations. Both views you describe, as far as I can tell, make the same experimental predictions, so the conflict you describe cannot be resolved by experiment. That makes it a matter of interpretation.

No. Lattice methods make different predictions than covariant methods, and are applicable to different regimes. One cannot predict high energy properties with lattice methods, that would take millions of years to compute.

 

[PeterDonis]

Lattice methods make different predictions than covariant methods, and are applicable to different regimes.

Do any of the predictions of the two methods apply to the same experiments? Or does "different regimes" mean that the "different predictions" made by the two methods are about different experiments altogether, so there's no way to test by experiment which method is more accurate?

 

[vanhees71]

I think that's a very physical question, without any relation to philosophy. It's, of course, an unanswered question.

Lattice QCD (both vacuum and thermal) is first of all a numerical algorithm using a regularization scheme (discrete spactime lattice as a UV and finite volume as an IR cutoff). It's violating of course all kinds of spacetime symmetries. Continuum limits and infinite-volume limits provide approximate results of physical quantities though, where applicable (or course high-energy properties need super-large lattices which are infeasible for practical purposes). Unfortunately of course this continuum and infinite-volume limit is everything than trivial, and AFAIK it's not solidly proven that these limits exist nor whether the spacetime symmetries are restored.

On the other hand, what can be calculated on the lattice (e.g., the hadron mass spectrum) is in good to excellent agreement with the observations, thus at least indicating that (a) QCD is a valid theory of the strong interactions and (b) the approximation works to the extent being successful in describing the physics.

 

[A. Neumaier]

Do any of the predictions of the two methods apply to the same experiments? Or does "different regimes" mean that the "different predictions" made by the two methods are about different experiments altogether, so there's no way to test by experiment which method is more accurate?

The low lying mass spectrum of mesons and baryons can be computed both with noncovariant, discrete lattice methods combined with extrapolation to the continuum limit, and with covariant, continuum Dyson equations. The results differ from each other and from experiment by a few percent. Exprimentally, these masses are known much more accurately.

Which method is more accurate is a matter of guessing from the publications how much hand tuning was done in producing the published results - the straightforward approaches are too inaccurate without making many delicate decisions that depend on who makes the analysis. It is also a matter of time and competition effort, since doing a good analysis of the computed data is highly nontrivial in both cases.

High energy properties such as what lead to the parton picture and from that to the discovery of QCD are accessible only by continuum methods. Had we only lattice QFT then QCD would never have been discovered.

 

[PeterDonis]

The low lying mass spectrum of mesons and baryons can be computed both with noncovariant, discrete lattice methods combined with extrapolation to the continuum limit, and with covariant, continuum Dyson equations. The results differ from each other and from experiment by a few percent.

So it seems like, from an experimental point of view, there is currently no resolution of the dispute being referred to in the OP of this thread. The two methods give different results, and both are different from actual experimental results, so both are clearly approximations, and it's not clear that either approximation is "better".

Can you explain why do you think that it belongs to QM interpretations forum? I think it doesn't.

If the above is a correct summary of the situation (and if you don't think it is, please say so and explain why), I don't see that a useful discussion can be had in the regular QM forum as far as any dispute between the methods is concerned. A useful discussion could possibly be had about what results can be obtained by each method and how those results compare with experiment, but that wouldn't be a "dispute" and wouldn't resolve any dispute about the methods. So if this thread is about such a dispute, I still think it belongs in this forum, since it's basically about different people's opinions regarding the two methods.

 

[atyy]

As stated above, the question of the UV limit is unsolved in the rigorous sense, but lattice QCD (most useful for low energy) and perturbation theory (most useful for high energy in QCD) are generally thought to be related.

https://arxiv.org/abs/hep-lat/0211036
p77: "The range of couplings for which perturbation theory aspires to be a reasonable expansion, that is when $g_{0}$ is small, is closely related to the approach to the continuum limit of lattice QCD. This approach can be described using Callan-Symanzik renormalization group equations which are similar to the continuum ones."

p80: "The two scales $\Lambda_{lat}$ and $\Lambda_{QCD}$ can thus be related. Once the relation between the lattice and continuum coupling constants is known, combining the results of Monte Carlo simulations with the knowledge of $\Lambda_{lat}$ and the quark masses allows in principle to predict all physical quantities."

There are also papers trying to relate perturbation theory ($\overline{MS}$) and lattice calculations, eg.
https://arxiv.org/abs/1004.4613
https://arxiv.org/abs/1004.3997
https://arxiv.org/abs/1408.4169

 

[A. Neumaier]

As stated above, the question of the UV limit is unsolved in the rigorous sense, but lattice QCD (most useful for low energy) and perturbation theory (most useful for high energy in QCD)

... while nonperturbative covariant continuum techniques such as the Dyson equations are useful at both low and high energy.

are generally thought to be related.

https://arxiv.org/abs/hep-lat/0211036
p77: "The range of couplings for which perturbation theory aspires to be a reasonable expansion, that is when $g_{0}$ is small, is closely related to the approach to the continuum limit of lattice QCD. This approach can be described using Callan-Symanzik renormalization group equations which are similar to the continuum ones."

p80: "The two scales $\Lambda_{lat}$ and $\Lambda_{QCD}$ can thus be related. Once the relation between the lattice and continuum coupling constants is known, combining the results of Monte Carlo simulations with the knowledge of $\Lambda_{lat}$ and the quark masses allows in principle to predict all physical quantities."

Yes, with emphasis on ''in principle''. The principle does not apply in practice to the high energy range, not even after enhancements by extrapolation techniques and renormalization group equations. (This has to do with the fact that lattice QCD as practised is a Euclidean quantum field theory, working exclusively in nonphysical imaginary time. There is no Schrödinger equation; instead of frequencies one has decaying correlations. Resolving large energies correspond to resolving high frequencies, hence fast decaying correlations, which require extraordinarily fine grids for their numerical resolution. (Try to estimate the nonrelativistic hydrogen spectrum at zero angular momentum with a Euclidean method and you'll run into the same problem already with one quantum degree of freedom!)

Thus the correct theory which is approximated by all current approaches to QCD is the covariant continuum QCD. Like any approximation method, both lattice QCD and perturbation theory must be considered as approximations, useful precisely in the domain where they approximate well. Use of the renormaliztion group extends the range of usefulness of both approaches, but does not remove the intrinsic limitations of the lattice approach at high energies.

 

[atyy]

Thus the correct theory which is approximated by all current approaches to QCD is the covariant continuum QCD.

Yes, that's the conventional "wisdom" or guess or Bayesian prior with which I agree, but remains to be proved in a rigorous sense.

 

[A. Neumaier]

Moderator's note: Moved to QM interpretations forum.

Can you explain why do you think that it belongs to QM interpretations forum? I think it doesn't.

Of course it doesn't belong to the QM interpretations forum. That's why I answered though it has been moved there ;-)).

I think that's a very physical question, without any relation to philosophy. It's, of course, an unanswered question.

I also think that this has nothing to do with the interpretation of quantum mechanics. It is a matter of discussing approximation aspects of different methods to approach the same topic, which arises in any field of physics where different approaches compete.

It is best to move the thread back and perhaps to delete the corresponding meta-discussion.

 

[Demystifier]

Because the conflict between different views that you describe in the OP seems like a conflict of interpretations. Both views you describe, as far as I can tell, make the same experimental predictions, so the conflict you describe cannot be resolved by experiment. That makes it a matter of interpretation.

Well, it could be resolved by experiment in principle. For instance, if the cutoff is at the Planck scale, then an experiment which accelerates particles to Planck energies could resolve it. Of course, we don't have such an accelerator, but we could have it in principle. Otherwise, almost all discussions on quantum gravity would belong to quantum interpretations forum.

 

[vanhees71]

I think all this are well-posed physical problems and have nothing to do with interpretations of a theory which you can choose at your taste without changing anything of the physics content of this theory (as are the discussions about the right interpretation of quantum theory or more generally the "physical meaning" of probabilities).

I would say QCD is pretty well established as a very good description of the phenomena related to the strong interaction. In the low-energy sector the properties of hadrons and in the high-energy sector the phenomena related to, e.g., deep-inelastic scattering and the related parton model.

Much of this is of course still subject of current research. The full understanding of confinement is, in my opinion, still among the most interesting questions on the boundary of high-energy particle and high-energy nuclear physics, which includes also the understanding of the properties of "bulk matter" from the underlying fundamental microscopic laws of the Standard Model (and maybe a future more comprehensive theory beyond the Standard Model). That's in some analogy with condensed-matter physics, which is also from a fundamental point of view the aim to understand the properties of bulk matter from the underlying microscopic theory, which here is mostly QED. The only difference is that it is easier to do experiments than in strong-interaction-matter physics. What I envy most is that it's easy to prepare everyday matter and also quite exotic materials in thermal equilibrium and being able to compare to all kinds of effective models in thermal equilibrium to it...

 

[PeterDonis]

I also think that this has nothing to do with the interpretation of quantum mechanics.

What doesn't? What is this thread about? The OP says it's about a continuation of a dispute between him and you, which, as I've said, looks to me like a matter of opinion. Perhaps it's not a matter of opinion about "interpretation of quantum mechanics", but it's still about a matter of opinion. If discussion about matters of opinion regarding QM belongs anywhere, it belongs in this forum. It certainly doesn't belong in the regular QM forum.

It is a matter of discussing approximation aspects of different methods to approach the same topic, which arises in any field of physics where different approaches compete.

But, as I said, that's not a dispute, and would not resolve any dispute. So again, what is this thread about? Is it about a dispute? Or just about discussing different approximation methods?

it could be resolved by experiment in principle

But not in practice for the foreseeable future. So there would not seem to be any point in trying to have a discussion about a "dispute" in the meantime, since there can't be any resolution until we can do the experiment.

Otherwise, almost all discussions on quantum gravity would belong to quantum interpretations forum.

Discussions about "disputes" over which quantum gravity method is better would indeed belong here in the interpretations forum, if they belong anywhere. But discussions about what the different quantum gravity methods say, with no "dispute" involved, can be had just fine in the regular QM forum.

So, again, what is this thread about?

 

[vanhees71]

I think it's a thread about the current understanding of the strong interaction from the point of view of a mathematician vs. the point of view of a physicist. If there is any dispute it's a dispute about "no-nonsense physics" and mathematics and not philosophy. But so what? Whether we discuss this in the QM or the QM foundations forums doesn't really matter a lot, does it?

 

[PeterDonis]

Whether we discuss this in the QM or the QM foundations forums doesn't really matter a lot, does it?

Yes, it does. If it's about matters of opinion, it doesn't belong in the regular QM forum.

 

[vanhees71]

It's not about matters of opinion but about the status of our understanding of the strong interaction within the Standard Model of elementary-particle physics. It's of course subject of current research, because for sure we do not understand the strong interaction completely. So there are open questions about it, but it doesn't mean that it belongs in a section in the forums about philosophy of quantum mechanics, because it's physics not philosophy!

 

[A. Neumaier]

So, again, what is this thread about?

The subject title asks the question ''Does QCD make sense without a cutoff?'', which I take to be the basic question of the thread.

The dispute menioned in the first post is peripheral to this. It refers to various other discussions on related topics that we had. For example: Is QED nonrelativistic?, where the focus was on QED, and which currently is not in the interpretations section.

Some of the points relevant for QED no longer apply to QCD, so it is worthwhile to collect the evidence specifically for QCD, where (unlike in QED) lattice methods have some real impact.

 

[PeterDonis]

It's not about matters of opinion

The thread as it stands looks to me like it is. The OP talks about continuing a dispute, and asks for people's opinions about which of two viewpoints on QCD is right. But, as seems apparent from the ensuing discussion, both viewpoints are approximations and both viewpoints make predictions that are different from the actual experimental results (and from each other). So neither one is "right" as a matter of experimental confirmation. That means any claim about which one is right can only be a matter of opinion.

I agree that there are plenty of things that could be discussed about either of these two approaches to QCD without involving any opinions. But this thread does not seem to me to be framed as that kind of discussion. It seems to me to be framed as asking for opinions.

 

[PeterDonis]

For example: Is QED nonrelativistic?, where the focus was on QED, and which currently is not in the interpretations section.

That's because the interpretations forum didn't even exist when that thread took place. Now that you've brought it to my attention, I'll take a look to see whether it should be moved.

The subjdect title asks the question ''Does QCD make sense without a cutoff?''

And as far as I can tell, the answer is that it's a matter of opinion. Some people say yes, some people say no. Experiment can't resolve the question because the energy scales we would have to get to to do so are not accessible now or in the foreseeable future.

If the question were "how is QCD done without a cutoff?" or "what role does the cutoff play in QCD treatments that use it?" or "how do the predictions of QCD models with a cutoff/without a cutoff compare with experiment?", those questions would not be matters of opinion. But that's not what this thread is about.

 

[vanhees71]

It's not a matter of dispute, it's just an unanswered question. As I said, I don't care, where we discuss this. Just let's discuss it and not in which section of PF this should be done!

 

[PeterDonis]

It's not a matter of dispute, it's just an unanswered question. As I said, I don't care, where we discuss this.

What is there to discuss if we can't do the experiment?

 

[vanhees71]

Well, we are doing experiments all the time to learn about all aspects of strongly interacting matter in heavy-ion collisions (LHC with the highest available energies, RHIC, GSI) and construct new big experiments (FAIR, NICA, the EIC at BNL,...) or investigate them by astronomical observations of neutron stars and neutron-star mergers in all ranges of the em. spectrum and for some years with gravitational waves.

We learn more and more about all aspects of the strong interaction and strongly interacting matter (and thus also QCD).

 

[A. Neumaier]

And as far as I can tell, the answer is that it's a matter of opinion.

The subforum is called 'Quantum Interpretations and Foundations' and not 'Opinions on quantum physics'. The main forum expresses lots of opinions about quantum physics (perhaps more than 50%). Statements by experts are still opinions, and often challenged in the main forum.

Based on its current contents, excepting the present thread, the subforum is about the interpretation of the foundational aspects of quantum mechanics, not about comparing different approaches to get numbers from theory (which is fully shut-up-and-calculate).

 

[vanhees71]

It's also not an issue of opinions but an open question about solid research in physics and mathematics in contradistinction to philosophical issues of belief in the one or the other interpretation of QM!

 

[PeterDonis]

Based on its current contents, excepting the present thread, the subforum is about the interpretation of the foundational aspects of quantum mechanics

Yes, that's what it was originally split off for.

However, that split also clarified what the regular QM forum is for: it's for discussions that fall within the 7 basic rules of QM, the Insights article for which you authored.

The present thread, at least based on its title and stated topic in the OP, does not fall within those guidelines. See below.

comparing different approaches to get numbers from theory (which is fully shut-up-and-calculate).

The title and OP of the present thread did not ask about comparing different approaches to get numbers from theory. It asked about whether one approach (the one without a cutoff) "makes sense". That question is not a "shut up and calculate" question that falls within the 7 basic rules referred to above.

It seems like your position is that the answer to the title question of this thread is a simple "yes". If @Demystifier, who started this thread, agrees with that answer, we can simply close this thread with that answer, and then if someone wants to start a new thread in the regular QM forum that does fall within the regular QM forum guidelines, particular posts in this thread could be moved to that one upon request. I am perfectly willing to help facilitate that.

If, OTOH, @Demystifier does not agree with that "yes" answer to the title question, any argument about it is not going to fall within the guidelines of the regular QM forum. It doesn't seem like this thead has contained much of an argument about it, which leads me to believe that the simple "yes" answer is in fact correct. But that just puts us back to the situation I described in the last paragraph: we can simply close this thead and someone can start a new one in the regular QM forum whose stated topic falls within the guidelines for that forum.