Disadvantages of dimensional regularization as a regularization method

Question

Is there any disadvantage or symmetry violation caused by choosing such regularization method? Like, Hard cut-off regularization that violates gauge symmetry in QED. Is there such a practical instance, apart from all the objections based on the problem of rigor(Like the number of Gamma matrices or their representation etc.)?

If not, can one deduce that DR is a "Physically" (and not mathematically!) unbeatable regularization method?

@Triatticus Sure but the problem about %Gamma_5% can be solved case by case using functionals. https://arxiv.org/abs/1403.4212 Also this still is a mathematical issue I guess. Look, to me, a physical objection is like losing Lorentz invariance or gauge invariance as one faces with hard cut-off method. — Bastam Tajik, Jun 21 '20 at 17:35
A problem with dim reg is that it does not mean anything nonperturbatively. — Abdelmalek Abdesselam, Jun 22 '20 at 13:54
One additional problem with dim reg that it is insensitive to odd divergences — Artem Alexandrov, Jul 04 '20 at 21:05

score 1 · Answer 1 · answered Apr 21 '21 at 22:42

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If not, can one deduce that DR is "physically" (and not mathematically!) unbeatable regularisation method?

Why do you say that dimensional regularisation is 'not mathematically [an] unbeatable regularisation method?

The Connes-Kreimer renormalisation theory makes mathematical sense out of dimensional regularisation. The theory is geometric and is related to the cosmic Galois group. In fact, the renormalisation group acting on structures in renormalisation in QFT is a 1-parameter subgroup of the cosmic Galois group.

answered Apr 21 '21 at 22:42

Mozibur Ullah

12,994

I did not say it's the "only" mathematically correct one. – Bastam Tajik Apr 21 '21 at 22:44
@Bastam Tajik: I didn't think that you had said that and nor did I imply that in my answer. What I was simply pointing out was that dimensional regularisation had a rigorous mathematical definition, (contra your assertion). – Mozibur Ullah Apr 21 '21 at 22:48
I see your point. But at the time I wrote the question I could only see mathematical naive objections to the method that was all around the correct definition of ${\gamma}_5$ matrix. Though I've attached an article which tries to solve the mathematical inconvenience! This means personally could not see any mathematical problem with it. But even in that case, my interest was merely physical objections, like symmetry violations, etc. But thanks for your answer. – Bastam Tajik Apr 21 '21 at 22:56

Qmechanic · Answer 2 · 2020-06-21T18:10:48.390

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Well, dimensional regularization has issues with the representation theory for spinors, Clifford algebras, gamma matrices, gamma-five & the Levi-Civita symbol, cf. e.g. this & this Phys.SE posts and links therein.

edited Jun 21 '20 at 18:10

answered Jun 21 '20 at 17:13

Qmechanic

201,751

The inconsistency arises as the answer of the link you attached, from dimensions less than 3. I was thinking if such inconsistencies arise in the case of d=4? And does it lead to any physical problem? Like, violation of Lorentz invariance or gauge invariance etc? – Bastam Tajik Jun 21 '20 at 17:41
You find that description of our world with non-integer dimensions can be “physically” unbeatable, rather than a mathematical trick? – Oбжорoв Jun 25 '20 at 18:14
Is there any observable quantity that ensures us that the dimension of spacetime is an absolute integer? If not it cannot be objected in this regard, unless one finds some observables that whose dependence on epsilon is not "renormalizable" in other words epsilon can't be deleted to find an observable theoretically. – Bastam Tajik Jun 26 '20 at 14:59
In addition to this answer, I would like to add my own experience: dim-reg scheme does not work for odd divergences, see this – Artem Alexandrov Apr 21 '21 at 22:25

score 0 · Answer 3 · answered Apr 21 '21 at 22:10

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The most unphysical feature of DR is its infrared behavior for the running of the mass parameter, while the electron mass is infrared fixed in reality!

answered Apr 21 '21 at 22:10

Bastam Tajik

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Disadvantages of dimensional regularization as a regularization method

3 Answers3