Misconceptions on Effective Field Theories and spontaneous symmetry breaking: Response to Ellis' article
aa r X i v : . [ phy s i c s . h i s t - ph ] J u l Misconceptions on Effective Field Theories and spontaneoussymmetry breaking: Response to Ellis’ article
Thomas Luu
1, 2 and Ulf-G. Meißner
2, 3, 1, 4 Institute for Advanced Simulation (IAS-4),and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, Germany Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Rheinische Friedrich-Williams-Universit¨at Bonn, Germany Center for Science and Thought, RheinischeFriedrich-Williams-Universit¨at Bonn, Germany Tbilisi State University, 0186 Tbilisi, Georgia (Dated: July 22, 2020)
Abstract
In an earlier paper [1] we discussed emergence from the context of effective field theories, particu-larly as related to the fields of particle and nuclear physics. We argued on the side of reductionismand weak emergence. George Ellis has critiqued our exposition in [2], and here we provide ourresponse to his critiques. Many of his critiques are based on incorrect assumptions related to theformalism of effective field theories and we attempt to correct these issues here. We also com-ment on other statements made in his paper. Important to note is that our response is to hiscritiques made in archive versions arXiv:2004.13591v1-5 [physics.hist-ph]. That is, versions 1-5 ofthis archive post. Version 6 has similar content as versions 1-5, but versions 7-9 are seemingly adifferent paper altogether (even with a different title).
PACS numbers: . INTRODUCTION Emergent phenomena are fascinating quantities that when viewed from lower level (morefundamental) constituents are highly complex in nature. The question of whether suchphenomena can be totally derived from their underlying constituents provides an intriguingdebate not only from a philosophical point of view, but also from a scientific point of viewthat spans all areas of science. At issue, from our point of view, is the invocation of strong emergence, which is the notion that certain (or all) properties of some emergent phenomenacannot be deduced from its fundamental constituents. The weaker claim, naturally called weak emergence, states that all properties of an emergent phenomena can, in principle , bededucible from its fundamental constituents . A more detailed definition of these cases isgiven in [3].In our paper [1] we gave an overview of the formalism of Effective Field Theories (EFT)and its relation to emergent phenomena, providing examples prevalent in nuclear and particlephysics. We won’t repeat our arguments here, but only state that in our paper we rejectedthe concept of strong emergence and argued in favor of weak emergence and reductionismin this context. Ellis has since criticised our arguments in [2], and we take this opportunityto respond to his criticisms.Ellis’ criticisms have not changed our views on the matter. Furthermore, many of hiscriticisms are based on incorrect assumptions and an incomplete understanding of the EFTformalism. Thus our response here also serves as a means to clarify any misconceptions thatmight occur from an initial read of Ellis’ critiques by anyone uninitiated in the concepts ofEFTs.Our paper is simply structured. In the next section II we respond to Ellis’ critiques,pointing out incorrect statements he makes related to EFTs that lead to wrong conclusions.We make some further comments in section III that are not directly related to Ellis’ state-ments related to EFTs, but we feel are important nonetheless in view of the debate on strongemergence. We conclude in section IV. Though weak emergent phenomena can in principle be deduced from fundamental constituents, such adeduction may in practice not be possible. I. OUR RESPONSEA. On the connection between an EFT and its lower-level fundamental theory
Ellis states that EFTs “. . . are an approximation and are not strictly derived from amore fundamental theory.”
He goes on to question whether EFTs, because of this limitation,can be used to even defend reductionism. To make his point he pulls quotes from varioussources (Ref. [4] in particular) to demonstrate that EFTs often are no better than some adhoc model of the system in question.
This is categorically wrong.
Effective field theories, defined as we outlined in [1],are often derived from a fundamental theory (we simplify the discussion for the moment,as cases exist, where this fundamental theory is not (yet) known and even might not beneeded). A prime example is non-relativistic Quantum Electrodynamics (NRQED), whichis an effective field theory of bound states, i.e. emergent phenomena, in Quantum Electro-dynamics (QED) [5]. Here the coefficients of the low-energy effective field theory can bedirectly matched to non-pertubative resummations (i.e. calculations) of the fundamentaltheory, which is QED. NRQED and QED share the same symmetries, but what differ-entiates NRQED from QED is the reshuffling of diagrammatic terms such that operatorsrepresenting the bound state degrees of freedom are explicit in NRQED. Such a reshufflingof terms results in low-energy constants (LECs) with each associated operator in NRQED.To calculate emergent bound states with fundamental QED requires an infinite number ofresummations. Not surprisingly, the number of operators in NRQED is also infinite, butwith a defined separation of scales there is a systematic hierarchy in relevance of these op-erators. One then truncates the number of terms in the EFT to achieve a desired accuracy.The truncation of terms to achieve a desired accuracy can be viewed as an “approximation”of the fundamental theory, but it is in principle possible to perform calculations with theEFT to any desired accuracy. Note also that in the fundamental theory there are practi-cal limitations in performing calculations beyond some accuracy, as best exemplified in thetenth order calculation of the electrons anomalous magnetic moment [6], but this is anotherstory.Coming back to nuclear and particle physics, we admit that an explicit derivation of chiralperturbation theory ( χ EFT), the low-energy EFT of Quantum Chromodynamics (QCD)3hich governs quarks and gluons, from QCD is difficult to do, partly because the degreesof freedom of the emergent phenomena (pions and nucleons) are vastly different from theirfundamental constituents. However, we stress that it has been shown that the Greensfunctions 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 sameoperator-structure but are characterized by different LECs. In QCD, the LECs of theEFT are very difficult to extract formally, but can, for example, be determined from non-perturbative numerical methods. Equally valid is the determination of the coefficients fromempirical data, which is what is commonly done. It is quite amazing that these days someof the LECs can be determined more precisely from lattice QCD calculations than fromphenomenology, which is related to the fact that one the lattice we can vary the quark massesbut not in Nature. Either way, the principles are the same: The operators in the EFT sharethe same symmetry as the fundamental theory, and the separation of scales dictates the rateof 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 observablesto 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 thatchiral perturbation theory is what is called a non-decoupling EFT , as the relevant degreesof freedom, the Goldstone bosons, are only generated through the spontaneous symmetrybreaking as discussed below.The notion that EFTs are glorified models with multiple fit parameters (LECs) hasbeen a misconception since its first application in nuclear physics by Weinberg [8]. Anotherhistorical example is the abandonment of the Fermi theory of the weak interactions because itviolates unitarity at about 80 GeV (at that time a dream for any accelerator physicist). Nowwe know that it is indeed an EFT, with the breakdwon scale given by the mass of the heavyvector bosons, alas 80 GeV. As stated above, the LECs of the theory are not ad hoc (thesame cannot be said of many models in other areas of physics!) but are directly connectedto the underlying theory. Furthermore, the number of LECs per given order of calculation ispredetermined, 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 ofmodel. The survey by Hartmann [4] completely overlooks these facts and thus gives a verypoor and inaccurate description of the efficacy of EFTs. For a survey with philosophical4onnections, the reader should consider [9]. Note also that EFTs are widely used in all fieldof physics, in atomic, cold atom, condensed matter, astrophysics, you name it. This showsthat EFTs are of general interest as they capture the pertinent physics in a certain energyregime. This, however, does not mean that they are all disconnected, often the reduction inenergy and thus resolution leads to a tower of EFTs that fulfill matching conditions in thecourse of the reduction in resolution. A nice example is flavor-diagonal CP violation, whereone starts with a beyond the standard model theory, like e.g. supersymmetry, at scalesway above the electroweak breaking and runs down through a series of EFTs to the chiralLagrangian of pions and nucleons including CP-violating operators, see e.g. [10]. Note thatwhile the scale of supersymmetry is about 1 TeV, the one of the chiral Lagrangian is wellbelow 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 theBorn-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 aremean-field approximations which, from an EFT perspective, are the leading order termof some EFT [11, 12]. In this view, an EFT description goes beyond both DFT andBorn-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 aredecoupled form the heavy (slow) ones, all symmetries pertinent to the interactions are in-cluded and the proper degrees of freedom are identified. What is simply missing is to setup a power counting in which to calculate the corrections. This has recently been achievedin hadron physics, where the Born-Oppenheimer approximation has been in use for quitesome time, but has since been surpassed by EFTs. For example, it was heavily utilized inthe context of the bag model [13] leading to various model studies like e.g. by one of theauthors [14]. More recently, in the context of heavy quark physics, this was even cast interms of an EFT [12], which clearly shows that even in quantum chemistry the formulationof an EFT embodying the Born-Oppenheimer approximation should be possible. We pointout that in quantum chemistry powerful techniques exist to perform extremely precise calcu-5ations like the already mentioned DFT approaches or the coupled-cluster scheme, originallyinvented for nuclear physics, thus the need for setting up an EFT has been less urgent thanin strong interaction physics. However, as DFT in chemistry now also enters the stage toaccomodate strong electronic correlations ab initio , this will change in the future. We willreturn to the topic of the Born-Oppenheimer approximation when we discuss phonons inthe next section.Ellis goes on to ask about the applicability of EFTs to signal propagation in neurons, neu-ral networks , Darwinian evolution, and election results. We admit we cannot answer thesequestions 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 dynam-ics [17], and in fact we are presently setting up a simulation laboratory at ForschungszentrumJ¨ulich that deals with the application of numerical quantum field theory to complex systemsin 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 basedon simple modelling, but we do not dismiss the possibility of applicability just because ofour ignorance. Ellis, on the other hand, answers with a definite NO! because he claims theseare 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 state-ment that EFTs cannot capture spontaneous symmetry breaking (SSB), or explicit brokensymmetries in general, and therefore cannot describe the emergent phenomena (which heclaims are strong emergent) that ensue from these reduced symmetries. He uses the solid-state example of the reduced point symmetry of a lattice that ultimately leads to the creationof phonons. Indeed, he claims most of condensed matter and solid-state physics is off-limitsto EFTs, because much of their emergent phenomena is due to SSB or explicit symmetrybreaking.We point out, however, that spontaneous symmetry breaking, and its ensuing conse-quences, is not solely relegated to the fields of condensed matter and solid-state physics, Already statistical field theoretical techniques are being applied to neural networks (see for example [15,16]). Such theories are amenable to EFTs. N f ) L × SU( N f ) R , in the QCD vacuum this symmetry is “hidden”, that is, the global symmetry of SU( N f ) L × SU( N f ) R is spontaneously broken to SU( N f ) V . As Ellis cor-rectly points out, this symmetry breaking leads to emergent phenomena, which are gaplesslong-distance (pseudo-)scalar modes in the theory with reduced symmetry. The number ofgapless, or massless, states is given by the much celebrated Goldstone theorem [22], whichstates that for each generator of the hidden symmetry that is broken there corresponds along-distance massless (pseudo-)scalar particle. These states are more commonly known asNambu-Goldstone bosons. In QCD these bosons are the three (nearly) massless pions (for N f = 2) observed in Nature. The basis of chiral perturbation theory is exactly this theorem- it provides the EFT of light-quark QCD with its proper degrees of freedom (for the heavyquark sector, a different type of EFT comes into play, which we will not discuss here). Theemergent phenomena of pions due to SSB (and coupled eventually to nucleons, which act asmatter fields) are the explicit degrees of freedom in this theory. Thus SSB and its subsequentreduced symmetry are not only captured but exactly reproduced (as stated above) in thisEFT, as well as any other unaffected symmetries.The same happens when, during the formation of a crystal, whether natural or synthetic,the underlying global Lorentz and Galilean symmetries are spontaneously broken to some re-duced point symmetry of the crystal lattice. There are some differences between relativisticand non-relativistic formulations of the Goldstone theorem , but the central point remainsthe same: The broken generators of the underlying continuous symmetries result in theformation of (pseudo-)scalar Nambu-Goldstone bosons, in this case the so-called phonons.Furthermore, Goldstone’s theorem, and the subsequent phonon EFTs based off this theo-rem, go on to describe the interactions between phonons [24, 25]. One can imagine addingelectron degrees of freedom as matter fields in such an EFT, all the while ensuring that theelectrons respect the relevant point symmetry of the lattice, analogous to what is done withnucleons in χ EFT. The LECs corresponding to electron-phonon interactions in the EFT The interested reader might consult e.g. [19–21] for a detailed description of this process and the meaningof these symmetries. See e.g. the lucid discussion in Ref. [23] in principle , calculated directly from QED with appropri-ate boundary conditions. This EFT captures, at its lowest orders, the Born-Oppenheimerapproximation for electrons. Once the LECs are determined, the phonons themselves canbe integrated out of the theory, which results in an effective electron-electron attractive interaction [26] .With regards to Ref. [26] written by Polchinski in 1992, Ellis quotes various passages fromthis reference that seemingly lend credence to his claim that EFTs cannot be applied to high- T c superconductors. We point out that this article was written as a series of introductorylectures on EFTs and the renormalization group. The underlying premise of these lectureswas that the electrons were approximated as a fermi liquid. Non-fermi liquid behavior, whichmost certainly underlies unconventional superconductors like high- T c superconductors, isthus not captured by these EFTs. From an EFT point of view, the separation of scalebetween the Debye length and inverse fermi momentum becomes less clear as electronsbecome more strongly correlated in non-fermi liquids. This signifies that the degrees offreedom of the EFTs in Polchinski’s lectures are, at the very least, incomplete. Indeed,Polchinski points this out, stating the possibility that other degrees of freedom like anyons,or spin fluctuations, are required in a successful EFT of high- T c superconductors. ThusEllis’ chosen quotes, when taken out of context, belie this point. Though there still doesnot exist a successful EFT that captures high- T c superconductors, there has been progressin formulating EFTs for finite-density systems [28] with quasi-particle excitations [29, 30],like magnons [31], as additional degrees of freedom.Ellis also states that topological effects, prevalent in low-dimensional condensed matterand solid-state systems such as topological insulators and superconductors, cannot be cap-tured by EFTs because they cannot be derived from a “bottom-up” framework. Again thisis incorrect. One need only consider the decay process π → γγ and understand how thisprocess is described in an EFT. On classical grounds, the decay of the neutral pion to twophotons is not allowed, but quantum fluctuations allow such a decay through what is calledthe axial anomaly [32] . This process is non-local and is captured in an EFT by the Wess- The analog of this process in chiral perturbation theory χ EFT, that is integrating out pions, results inpionless EFT, see e.g. [27]. A satisfactory description of this anomaly, and its ramifications goes beyond the point of this article, buta nice overview is given in [33] for the interested reader. bottom-up fashion [36]. This action istopological in nature. Its geometrical interpretation relies on the fact that the physics it de-scribes is confined to the surface (where our world “resides”) of a five-sphere S . The WZWformalism for anomaly physics is universal and thus not constrained to the realm of particleand nuclear physics. Indeed, its lower-dimensional analogs are utilized in the classificationof topological insulators and superconductors [37] which Ellis claims is beyond the purviewof EFTs. Its application naturally leads to the Chern-Simons effective Lagrangian [38] thataccounts for the integer quantum hall effect [39].Another area of interplay of spontaneous symmetry breaking, topology (here the oneof the QCD vacuum) and EFT is axion physics. The QCD vacuum exhibits a non-trivialtopology, with different sectors given in terms of an integer winding number. Instantoneffects allow for transitions between these sectors, which are, however, exponentially sup-pressed [40]. Another ramification is the appearance of the so-called θ term in QCD, whichleads to CP violation. However, the value of the θ parameter accompanying this term issmaller than 10 − as deduced e.g. from the experimental upper limit on the neutron elec-tric dipole moment [41]. This constitutes the so-called “strong CP problem”. An elegantsolution was proposed by Peccei and Quinn [42], who elevated this parameter to a dynam-ical variable related to a new U(1) symmetry, nowadays called U(1) PQ . This symmetry isspontaneously broken with the appearance of a Goldstone-like particle, the axion, whoseinteractions with light and matter can be cast into an EFT. For a recent high-precisioncalculation of the axion-nucleon coupling, see [43] which also contains references to earlierwork in this important area of beyond-the-standard-model (BSM) physics, that has becomeone of the major playgrounds for EFTs.Thus, EFTs can indeed capture both SSB and topological effects, and therefore theirsubsequent emergent phenomena. 9 II. ADDITIONAL COMMENTSA. Spontaneous Symmetry Breaking at the micro versus the macro scale
Ellis differentiates SSB that occurs at the micro scale and SSB that occurs at the macro scale. He labels them SSB(m) and SSB(M), respectively. As examples, he mentions theHiggs mechanism as a source for SSB(m) while the SSB that leads to crystallization is anexample of SSB(M). Presumably the fact that the crystal exhibits long-range order at themacro scale is the reason why it falls under SSB(M).Ellis uses this distinction to strengthen his arguments for strong emergence. We questionthe correctness of it, however, and thus the basis for such a classification. The mechanismfor any SSB, whether in QCD due to the scalar condensate or in cyrstallization due tospontaneous nucleation, originates at the micro, or local, scale, but the symmetries thatare broken are global . This means that the ramifications of the broken symmetries extendinto the macro scale. How do we know this? In the case of the crystal we can definitelyobserve 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 alreadymentioned 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 oneSSB. 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 scientificprediction based off strong emergence that could be tested. Our goal here was to applyPopper’s
Fasifiability criterion [44]. Ellis accepted our challenge and answered: “Neither LM (Luu & Meißner) nor any of their nuclear physics or particle physics col-leagues will be able to derive the experimentally tested properties of superconductivity orsuperfluidity in a strictly bottom up way. In particular, they will be unable to thus derive asuccessful 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 since10he 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 inabilityto 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 progressin deriving bottom-up theories in areas that Ellis claims belongs to strong emergence. Forexample, 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 C, which is notonly making life on Earth possible, but has also been a barrier for nuclear theory calculationsuntil 2011 [46]. Such problems were considered intractable but are now soluble due toadvances in high-performance computing and the ingenuity of our colleagues. Theoreticalphysics requires optimism rather than a “can’t do” attitude.
IV. CONCLUSIONS
Conclusions based off misconceptions of EFT have historically lead to erroneous physicalclaims and added confusion about its ability to explain emergent phenomena as well asits connection to underlying theories. Over time many misconceptions have diminished,but unfortunately some still persist and one must remain vigilant to correct them and theconclusions based off them. In this article we corrected the misconceptions of EFT that Ellisuses in his arguments for the case of strong emergence.Admittedly, because our imaginations are still too limited (and may be bounded!), werely heavily on Nature to tell us how to define our physical boundaries that lead to exoticemergent phenomena, especially in cases where the environment is synthetic, like supercon-ductivity in Yttrium-Barium-Copper Oxide. A proponent of strong emergence would saywe “cheated”, we “peaked” at Nature to tell us what to do. But we make no excuses forthis since physics is an experimental science after all!A common argument that Ellis makes related to strong emergent phenomena is that suchphenomena would never be realized from a bottom-up procedure because the environmentwhich 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 adamantlyclaims are strong emergent. If that is the case, then how do we classify neutron superflu-11dity in the crust of neutron stars , or quark-color superconductivity predicted to occurdeep within dense compact stars [47, 48] ? The physical mechanisms that underly thesephenomena are analogous to those in solid-state systems, but clearly here the environmentis not synthetic. So are they strong emergent because they share the same physical mech-anism, or are they weak emergent because we predicted the phenomena ourselves? If weinsist 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, thatwe humans who created the materials are special? Such hubris is best avoided, even by us. Acknowledgments
TL thanks J.-L. Wynen, C. Hanhart, T. L¨ahde, and M. Hru˘ska for insightful commentsand discussions. This work was supported in part by the Deutsche Forschungsgemein-schaft (DFG) through funds provided to the Sino-German CRC 110 “Symmetries and theEmergence of Structure in QCD” (Grant No. TRR110), by the Chinese Academy of Sci-ences (CAS) through a President’s International Fellowship Initiative (PIFI) (Grant NO.2018DM0034) and by the VolkswagenStiftung (Grant NO. 93562) [1] T. Luu and U.-G. Meißner, [arXiv:1910.13770 [physics.hist-ph]].[2] G. F. R. Ellis, [arXiv:2004.13591 [physics.hist-ph]].[3] David J. Chalmers. Strong and weak emergence. In P. Davies and P. Clayton, editors,
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