Why is imposing a symmetry on a theory considered more "natural" than fine-tuning its couplings?

Question

Theories whose behavior would qualitatively change if their couplings were not fine-tuned to particular values are often dismissed as "unnatural" (in high-energy physics) or "unrealistic" (in condensed-matter physics), while theories whose couplings are constrained by symmetry requirements are happily accepted. Why is this? A symmetry constraint can be thought of as just a collection of fine-tunings that has some unifying pattern.

For example, a complex scalar field Lagrangian $$\mathcal{L}_1 = \partial_\mu \varphi^\dagger \partial^\mu \varphi - m^2 \varphi^\dagger \varphi - \lambda \left( \varphi^\dagger \right)^2 \varphi^2 \tag{1}$$ with a global $U(1)$-symmetry $$\varphi \to e^{i \theta} \varphi$$ can be thought of as a general Lagrangian $$\mathcal{L}_2 = \partial_\mu \varphi^\dagger \partial^\mu \varphi - m^2 \varphi^\dagger \varphi - m'^2 \left( \left( \varphi^\dagger \right)^2 + \varphi^2 \right) - \lambda \left( \varphi^\dagger \right)^2 \varphi^2 - \lambda' \left( \left( \varphi^\dagger \right)^4 + \varphi^4 \right) - \dots\tag{2}$$ in which the primed couplings have all been fine-tuned to zero. (Things are admittedly more complicated in the case of gauge quantum field theories, because in that case the gauge symmetry requires you to modify your quantization procedure as well). This kind of "fine-tuning" is a little less arbitrary than the usual kind, but arguably it's not much less arbitrary.

Answer 1

1. If one has a theory $S[\alpha]$ that depends on some parameters $\alpha$, one can always introduce new artificial parameters $\beta$, and claim for free that the theory $S[\alpha,\beta]:=S[\alpha]$ has a symmetry $\beta\to \beta+ b$. This is of course not a very interesting symmetry.

2. A symmetry, say $\alpha \to\alpha + a$, is only interesting if different values of $\alpha$ are physically meaningful/realizable/accessible. In contrast, if there is a physical principle/superselection rule that completely fix $\alpha$ to a certain value, then we are essentially back to pt. 1.

3. How do we quantify naturalness? It seems relevant to mention 't Hooft's definition of technical naturalness, cf. Refs. 1 & 2. At energy scale $\mu$, two small parameters of the form $$m^{\prime 2}~\sim~ \varepsilon^{\prime} \mu^2 , \qquad \lambda^{\prime}~\sim~ \varepsilon^{\prime}, \qquad|\varepsilon^{\prime}|~\ll~ 1, $$
   in the Lagrangian density $${\cal L}_2~=~{\cal L}_1 - m^{\prime 2} \left( \left( \varphi^{\dagger} \right)^2 + \varphi^2 \right) - \lambda^{\prime} \left( \left( \varphi^{\dagger} \right)^4 + \varphi^4 \right)$$ is technical natural, since the replacement $\varepsilon^{\prime}=0$ would increase the symmetry of the system, namely it would restore the $U(1)$-symmetry $\varphi \to e^{i \theta} \varphi$, cf. the Lagrangian density ${\cal L}_1$.

4. Similarly, at energy scale $\mu$, two small parameters of the form $$m^2~\sim~ \varepsilon \mu^2 , \qquad \lambda ~\sim~ \varepsilon , \qquad|\varepsilon |~\ll~ 1, $$
   in the Lagrangian density $${\cal L}_1~=~ \partial_{\nu}\varphi^{\dagger} \partial^{\nu} \varphi - m^2 \varphi^{\dagger} \varphi - \lambda \left( \varphi^{\dagger} \varphi \right)^2$$ is technical natural, since the replacement $\varepsilon=0$ would increase the symmetry of the system, namely it would restore the translation symmetry $\varphi \to \varphi+a$.

References:

1. G. 't Hooft, Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking, NATO ASI Series B59 (1980) 135. (PDF)

2. P. Horava, Surprises with Nonrelativistic Naturalness, arXiv:1608.06287; p.2.

Answer 2

I would argue exactly the opposite : picking a symmetry requires much less fine-tuning.

Indeed, when picking a symmetry, your are fine-tuning one "parameter", the fact that the model has a given symmetry, but effectively killing an infinite number of coupling in one sweep.

On the other end, standard fine-tuning forces you to put an infinite number of of coupling constants to zero. Since one is much smaller than infinity, I think that choosing a symmetry amounts to much less fine-tuning.

Answer 3

Why is imposing a symmetry on a theory considered more “natural” than fine-tuning its couplings?

Take a one carat diamond (200 milligrams ofcarbon). It can be described by a simple symmetry

Unit cell of the diamond cubic crystal structure

Which is "simpler" : imagining the build up of the crystal by using the symmetry or listing all the (x,y,z,t) coordinates of the atoms, even the few ones in this unit cell?

Symmetries "simplify" concepts. Even your argument about "fine tuning of the primed couplings" falls into the category of simplification. One does not have to impose it to each of them, the symmetry does it.

I suppose if we were computers it would make no difference, but man is a pattern recognition animal, and finding patterns and repetitions of patterns is inherent in our tools of accumulation of knowledge, but this is outside physics.