Misconceptions on Effective Field Theories and spontaneous symmetry breaking: Response to Ellis' article

In an earlier paper~\cite{Luu:2019jmb} we discussed emergence from the context of effective field theories, particularly as related to the fields of particle and nuclear physics. We argued on the side of reductionism and weak emergence. George Ellis has critiqued our exposition in~\cite{Ellis:2020vij}, and here we provide our response to his critiques. Many of his critiques are based on incorrect assumptions related to the formalism of effective field theories and we attempt to correct these issues here. We also comment on other statements made in his paper. Important to note is that our response is to his critiques made in archive versions arXiv:2004.13591v1-5 [physics.hist-ph]. That is, versions 1-5 of this archive post. Version 6 has similar content as versions 1-5, but versions 7-9 are seemingly a different paper altogether (even with a different title).

which governs quarks and gluons, from QCD is difficult to do, partly because the degrees of freedom of the emergent phenomena (pions and nucleons) are vastly different from their fundamental constituents. However, we stress that it has been shown that the Greens functions of QCD are indeed exactly reproduced by the ones in chiral perturbation theory [7], which leaves us with the LECs, as different theories can in principle lead to the same operator-structure but are characterized by different LECs. In QCD, the LECs of the EFT are very difficult to extract formally, but can, for example, be determined from nonperturbative numerical methods. Equally valid is the determination of the coefficients from empirical data, which is what is commonly done. It is quite amazing that these days some of the LECs can be determined more precisely from lattice QCD calculations than from phenomenology, which is related to the fact that one the lattice we can vary the quark masses but not in Nature. Either way, the principles are the same: The operators in the EFT share the same symmetry as the fundamental theory, and the separation of scales dictates the rate of convergence of the terms and thus how many terms are required for a desired accuracy.
The EFT is derived from the underlying theory, and in principle can calculate observables to arbitrary precision. Clearly, scale separation is an important ingredient in any EFT. This will be stressed at various points in what follows. Another important remark is that chiral perturbation theory is what is called a non-decoupling EFT, as the relevant degrees of freedom, the Goldstone bosons, are only generated through the spontaneous symmetry breaking as discussed below.
The notion that EFTs are glorified models with multiple fit parameters (LECs) has been a misconception since its first application in nuclear physics by Weinberg [8]. Another historical example is the abandonment of the Fermi theory of the weak interactions because it violates unitarity at about 80 GeV (at that time a dream for any accelerator physicist). Now we know that it is indeed an EFT, with the breakdwon scale given by the mass of the heavy vector bosons, alas 80 GeV. As stated above, the LECs of the theory are not ad hoc (the same cannot be said of many models in other areas of physics!) but are directly connected to the underlying theory. Furthermore, the number of LECs per given order of calculation is predetermined, and because of its strict power counting rules, estimates of the theory's error due to omission of higher order terms can be made. The same cannot be said for any type of model. The survey by Hartmann [4] completely overlooks these facts and thus gives a very poor and inaccurate description of the efficacy of EFTs. For a survey with philosophical connections, the reader should consider [9]. Note also that EFTs are widely used in all field of physics, in atomic, cold atom, condensed matter, astrophysics, you name it. This shows that EFTs are of general interest as they capture the pertinent physics in a certain energy regime. This, however, does not mean that they are all disconnected, often the reduction in energy and thus resolution leads to a tower of EFTs that fulfill matching conditions in the course of the reduction in resolution. A nice example is flavor-diagonal CP violation, where one starts with a beyond the standard model theory, like e.g. supersymmetry, at scales way above the electroweak breaking and runs down through a series of EFTs to the chiral Lagrangian of pions and nucleons including CP-violating operators, see e.g. [10]. Note that while the scale of supersymmetry is about 1 TeV, the one of the chiral Lagrangian is well below 1 GeV, thus one bridges about 3 orders of magnitude by this succession of EFTs.

B. On the applicability of EFTs to other areas of science
Ellis asks if EFTs can be applied ". . . to quantum chemistry, where methods such as the

Born-Oppenheimer approximation and Density functional Theory (DFT) have been used?"
We see no reason why EFTs cannot be applied here. Indeed, both of these methods are mean-field approximations which, from an EFT perspective, are the leading order term of some EFT [11,12]. In this view, an EFT description goes beyond both DFT and Born-Oppenheimer approximations, since it naturally includes dynamics of quasi-particles (emergent phenomena) above the mean-field approximation. To be more specific, the Born-Oppenheimer approximation shares all the features of an EFT, the light (fast) modes are decoupled form the heavy (slow) ones, all symmetries pertinent to the interactions are included and the proper degrees of freedom are identified. What is simply missing is to set up a power counting in which to calculate the corrections. This has recently been achieved in hadron physics, where the Born-Oppenheimer approximation has been in use for quite some time, but has since been surpassed by EFTs. For example, it was heavily utilized in the context of the bag model [13] leading to various model studies like e.g. by one of the authors [14]. More recently, in the context of heavy quark physics, this was even cast in terms of an EFT [12], which clearly shows that even in quantum chemistry the formulation of an EFT embodying the Born-Oppenheimer approximation should be possible. We point out that in quantum chemistry powerful techniques exist to perform extremely precise calcu-lations like the already mentioned DFT approaches or the coupled-cluster scheme, originally invented for nuclear physics, thus the need for setting up an EFT has been less urgent than in strong interaction physics. However, as DFT in chemistry now also enters the stage to accomodate strong electronic correlations ab initio, this will change in the future. We will return to the topic of the Born-Oppenheimer approximation when we discuss phonons in the next section.
Ellis goes on to ask about the applicability of EFTs to signal propagation in neurons, neural networks 2 , Darwinian evolution, and election results. We admit we cannot answer these questions because some of these systems fall well outside our purview of expertise. We say "some" because one of us, however, has embarked in research in modelling neuron dynamics [17], and in fact we are presently setting up a simulation laboratory at Forschungszentrum Jülich that deals with the application of numerical quantum field theory to complex systems in particle and nuclear physics, solid-state physics, and also biological systems like the brain.
Nevertheless, for certain fields, such numerical methods are not yet available or only based on simple modelling, but we do not dismiss the possibility of applicability just because of our ignorance. Ellis, on the other hand, answers with a definite NO! because he claims these are strong emergent phenomena.

C. Spontaneous Symmetry Breaking and Topological Effects in EFTs
Arguably the most egregious error that Ellis makes, from our point of view, is the statement that EFTs cannot capture spontaneous symmetry breaking (SSB), or explicit broken symmetries in general, and therefore cannot describe the emergent phenomena (which he claims are strong emergent) that ensue from these reduced symmetries. He uses the solidstate example of the reduced point symmetry of a lattice that ultimately leads to the creation of phonons. Indeed, he claims most of condensed matter and solid-state physics is off-limits to EFTs, because much of their emergent phenomena is due to SSB or explicit symmetry breaking.
We point out, however, that spontaneous symmetry breaking, and its ensuing consequences, is not solely relegated to the fields of condensed matter and solid-state physics, Ellis uses this distinction to strengthen his arguments for strong emergence. We question the correctness of it, however, and thus the basis for such a classification. The mechanism for any SSB, whether in QCD due to the scalar condensate or in cyrstallization due to spontaneous nucleation, originates at the micro, or local, scale, but the symmetries that are broken are global. This means that the ramifications of the broken symmetries extend into the macro scale. How do we know this? In the case of the crystal we can definitely observe the long-range order of the crystal's point symmetry, as Ellis correctly points out.
In QCD, on the other hand, we observe pions everywhere. Another example is the already mentioned Higgs field, that is generated in SSB at the electroweak scale and penetrates the whole universe.
Therefore, there is no distinction between SSB(m) and SSB(M). There is only just one SSB. Ellis' ensuing arguments based off this distinction are thus non sequiturs.

B. We are making progress
In our original paper we challenged proponents of strong emergence to make a scientific prediction based off strong emergence that could be tested. Our goal here was to apply Popper's Fasifiability criterion [44]. Ellis accepted our challenge and answered: "Neither LM (Luu & Meißner) nor any of their nuclear physics or particle physics colleagues will be able to derive the experimentally tested properties of superconductivity or superfluidity in a strictly bottom up way. In particular, they will be unable to thus derive a successful theory of high-temperature superconductivity." He argues that we will never be able to do this since these phenomena are strong emergent.
Let us be the first to admit that we (Luu & Meißner) have not derived such a theory since the publication of his challenge, and even with the amount of hubris that we already have, we would never claim that we ourselves will ever do so. We stress, however, that our inability to derive such a theory does not provide confirmation of strong emergence.
But we will point out that some of our colleagues of similar "ilk" have made progress in deriving bottom-up theories in areas that Ellis claims belongs to strong emergence. For example, in [45] a relativistic formulation of the fractional Quantum Hall effect was derived.
Another example is the ab initio calculation of the so-called Hoyle state in 12 C, which is not only making life on Earth possible, but has also been a barrier for nuclear theory calculations until 2011 [46]. Such problems were considered intractable but are now soluble due to advances in high-performance computing and the ingenuity of our colleagues. Theoretical physics requires optimism rather than a "can't do" attitude.

IV. CONCLUSIONS
Conclusions based off misconceptions of EFT have historically lead to erroneous physical claims and added confusion about its ability to explain emergent phenomena as well as its connection to underlying theories. Over time many misconceptions have diminished, but unfortunately some still persist and one must remain vigilant to correct them and the conclusions based off them. In this article we corrected the misconceptions of EFT that Ellis uses in his arguments for the case of strong emergence.
Admittedly, because our imaginations are still too limited (and may be bounded!), we rely heavily on Nature to tell us how to define our physical boundaries that lead to exotic emergent phenomena, especially in cases where the environment is synthetic, like superconductivity in Yttrium-Barium-Copper Oxide. A proponent of strong emergence would say we "cheated", we "peaked" at Nature to tell us what to do. But we make no excuses for this since physics is an experimental science after all! A common argument that Ellis makes related to strong emergent phenomena is that such phenomena would never be realized from a bottom-up procedure because the environment which enables the phenomena does not occur naturally in Nature, rather it is synthetic.
He refers to superfluidity and superconductivity in solid-state physics, which he adamantly claims are strong emergent. If that is the case, then how do we classify neutron superflu-idity in the crust of neutron stars 7 , or quark-color superconductivity predicted to occur deep within dense compact stars [47,48] 8 ? The physical mechanisms that underly these phenomena are analogous to those in solid-state systems, but clearly here the environment is not synthetic. So are they strong emergent because they share the same physical mechanism, or are they weak emergent because we predicted the phenomena ourselves? If we insist that the solid-state examples are strong emergent, while the others are weak emergent, than doesn't that imply that the synthetic materials are special? And by extension, that we humans who created the materials are special? Such hubris is best avoided, even by us.