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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?

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    $\begingroup$ How about the nature of $\gamma^5$? $\endgroup$
    – Triatticus
    Commented Jun 21, 2020 at 17:07
  • $\begingroup$ @Triatticus Sure but the problem about %Gamma_5% can be solved case by case using functionals. 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. $\endgroup$
    – Bastam Tajik
    Commented Jun 21, 2020 at 17:35
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    $\begingroup$ A problem with dim reg is that it does not mean anything nonperturbatively. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Jun 22, 2020 at 13:54
  • $\begingroup$ One additional problem with dim reg that it is insensitive to odd divergences $\endgroup$
    – Artem Alexandrov
    Commented Jul 4, 2020 at 21:05

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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.

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  • $\begingroup$ I did not say it's the "only" mathematically correct one. $\endgroup$
    – Bastam Tajik
    Commented Apr 21, 2021 at 22:44
  • $\begingroup$ @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). $\endgroup$
    – Mozibur Ullah
    Commented Apr 21, 2021 at 22:48
  • $\begingroup$ 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. $\endgroup$
    – Bastam Tajik
    Commented Apr 21, 2021 at 22:56
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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.

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  • $\begingroup$ 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? $\endgroup$
    – Bastam Tajik
    Commented Jun 21, 2020 at 17:41
  • $\begingroup$ You find that description of our world with non-integer dimensions can be “physically” unbeatable, rather than a mathematical trick? $\endgroup$
    – Oбжорoв
    Commented Jun 25, 2020 at 18:14
  • $\begingroup$ 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. $\endgroup$
    – Bastam Tajik
    Commented Jun 26, 2020 at 14:59
  • $\begingroup$ In addition to this answer, I would like to add my own experience: dim-reg scheme does not work for odd divergences, see this $\endgroup$
    – Artem Alexandrov
    Commented Apr 21, 2021 at 22:25
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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!

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