# Triviality Pursuit: Can elementary scalar particles exist?

Originality
+ 1 - 1
Accuracy
+ 2 - 1
Score
1.00
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Referee this paper: Phys.Rept. 167 (1988) 241 RU-87-B1-20 by David J.E. Callaway

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

Great effort is presently being expended in the search for elementary scalar Higgs'' particles. These particles have yet to be observed. The primary justification for this search is the theoretically elegant Higgs-Kibble mechanism, in which the interactions of elemetary scalars are used to generate gauge boson masses in a quantum field theory. However, strong evidence suggests that at least a pure phi 4 scalar field theory is trivial or noninteracting. Should this triviality persist in more complicated systems such as the standard model of the weak interaction, the motivation for looking for Higgs particles would be seriously undermined. Alternatively, the presence of gauge and fermion fields can rescue a pure scalar theory from triviality. Phenomenological constraints (such as a bounded or even predictable Higgs mass) may then be implied. In this report the evidence for triviality in various field theories is reviewed, and the implications for high energy physics are discussed.

summarized
paper authored Mar 6, 1988
retagged Jul 29, 2015

This is a review, so originality is not a concern.

It seems someone has upvoted for originality here as well. Downvoted to counter-vote. edit: removed my comments regarding accuracy, see below.

@Dimension10 I think older papers that correctly made use of the state of the art knowledge valid at the time they have been written and have therefore been correct at the time of publishing, should not be downvoted because our knowledge has now changed/deepend/improved, etc ...

Otherwise, all phenomenological papers writing about BSM models that are excluded by now, would get downvoted too for example ...

This would be not a good idea, one can not expect authors to know and include the future state of the art into their work or to know in advance what will be known in the future etc....

@Dilaton OK, you're right here (although the phenomenology analogy is not meaningful) - a review paper is supposed to reflect the existing literature, if it doesn't it gets an originality score, right. Reverted my accuracy downvote.

@dimension10, even by today's knowledge $\phi^4$ is quite possibly trivial, it's more about the mathematical structure than phenomenology.

I haven't read the paper, but it's a bit strange for the abstract to say

Should this triviality persist in more complicated systems such as the standard model of the weak interaction, the motivation for looking for Higgs particles would be seriously undermined.

Wouldn't the same comment apply to QED? By 1988 people should already know QED is quite possibly trivial, nonetheless describes reality very well, @RonMaimon could you comment on this?

@JiaYiyang You're right - it's the sentence that followed (which you quoted above) which puzzled me a little. Besides, the paper is from 1988, which isn't too far back.

The title "Can elementary scalar particles exist?" contains the word "elementary". It implies some sort of triviality, doesn't it? Now, here "elementary scalar particle" is also implied self-interacting as $\varphi^4$. What this self-interaction should describe? Decay of a free elementary scalar particle into several other free elementary scalar particles, which in turn will decay on and on?

This self-action needs counter-terms, so the "true interaction" is different from $\varphi^4$, as a matter of fact. How behaves the corresponding (renormalized) scalar quantum? Decays too or moves freely? Maybe some interaction remains and the "renormalized quantum" is not so "elementary".

Then what can it be physically? It can be a system of some interacting constituents with a "free" center of mass (energy) motion and "free" elementary normal modes of the relative motion. The free equations for the center of mass and internal normal modes are decoupled from each other, look non-interacting and in this sense they are trivial, but they may describe a non-trivial physical entities.

As to Landau pole, I am not sure that only summing up the "leading terms" does not spoil the properties of the exact function (containing sub-leading terms too), so the conclusion on QED triviality may be based on a wrong sum.

@ VladimirKalitvianski: Not every interaction means decay. It can just be scattering, as for a repulsive potential in the nonrelativistic case.

Right, Arnold, that is why I put a question mark while asking about evolution of the initially "free" scalar quantum ("particle").

I have finally skimmed through the review. This is a pretty comprehensive(possibly all-encompassing) review of renormalization and triviality/Landau pole problem, from abstract theories to their phenomenologies. I don't think I understand all the content, but the parts I do understand(which is a non-negligible portion) are fairly accurate, +1 to accuracy.

@JiaYiyang Do not hurry to vote for accuracy if there is still a part you do not understand. If you are excited, then write "So far I am excited with this review".

@JiaYiyang: Wouldn't the same comment apply to QED? By 1988 people should already know QED is quite possibly trivial, nonetheless describes reality very well

Unlike the standard model, QED has no scalar particle. Therefore the triviality issues, though nontrivial for QED, too, are different than those for $\Phi^4_4$ or the standard model.

By the way, I believe that neither $\Phi^4$ theory nor QED are trivial; the techniques that suggest it are based on particular constructive approaches that are likely to fail. But there are alternative constructive principles where the verdict is still out. E.g., Klauder's affine approach has not been investigated from the rigorous point of view and might well lead to a construction.

@ArnoldNeumaier, Yuh I'm aware of your take on the triviality issue, but I don't understand what gets you there. The majority of the theoretical evidences, though arguably all premature, seem to suggest triviality in both pure QED and pure $\phi^4$. If we don't have any relevant evidence, then it's rational to say there's a 50% chance for the existence of triviality, but given the evidences we have, wouldn't it be rational to think there's at least a 51% chance for the existence of triviality?

@JiaYiyang:  If there is no evidence for a probablility, one has epistemic uncertainty, and the rational approach is to assign no probability at all, since it can be anything between 0% and 100%. Probabilities make rational sense only for aleatoric uncertainty (where the result depends randomly on the context). [I did significant research on uncertainty quantification - it is a terrible practical mistake to assume 50% chances for either possibility if there is no evidence for either side. Would you assume 50% chances to any binary question you don't know the answer? The chances to win with a single ticket some prize, a big prize, the biggest prize in a lottery are different, you can assign 50% at most to one of the situations, and you'd probably be quite mistaken in all cases.]

But in fact for QFT we have evidence either way: Some negative theoretical evidence based on the exploration of very approximate approaches (lattice constructions and low order renormalization enhanced perturbation theory), some positive theoretical evidence based on other approximations in which the Landau poles is absent, and overwhelming positive experimental evidence that QED is nontrivial. Together with my belief in a rational creator of the universe this make me 99.9% certain.

@ArnoldNeumaier,

Would you assume 50% chances to any binary question you don't know the answer?

Well if it's truly a binary question, and if I have absolutely no clue of which answer is right, then yes I'd bet on 50-50. Or I'll just refrain from betting at all. In the lottery case we already have enough information to know it's not a 50-50 game when formulated as a win/lose binary question.

 But in fact for QFT we have evidence either way: Some negative theoretical evidence based on the exploration of very approximate approaches (lattice constructions and low order renormalization enhanced perturbation theory), some positive theoretical evidence based on other approximations in which the Landau poles is absent I've seen much more theoretical evidence pointing to one direction than the other, but of course I might be biased by my limited reading, and I can only change my mind after seeing enough opposite evidence of at least equal strength. overwhelming positive experimental evidence that QED is nontrivial.  Of course QED is nontrivial when you look at it from infrared, so is $\phi^4$. But in this picture current experiments simply don't say much about the existence of Landau pole, since we are far away from that energy scale.

@JiaYiyang:  In the absence of information, only refraining form betting is rational.

IF QED exists at low energies the Wightman axioms imply that it exists at all energies. If you are interested in other positive theoretical evidence for the nontriviality of QED, ask a corresponding question!

@ArnoldNeumaier, done, it's here.

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