Bottoms Up: The Standard Model Effective Field Theory from a Model Perspective

During the past decades, BSM searches at the LHC and elsewhere were mostly guided by models that postulate specific new concepts to solve open issues of the SM, e.g. new symmetries, new interactions, or new spatial dimensions at some energy scale $\Lambda$. These concepts are reflected in ${\cal{L}}_{\rm BSM}$ by at least explicitly adding new (non-SM) fields and their interactions. No assumption is made on their UV completeness. Prominent examples of such models are supersymmetry (Wess and Zumino, 1974), a new symmetry between fermions and bosons¹³¹³13Throughout the paper the term ‘supersymmetry’ denotes a field theory with the SM supplemented by the additional supersymmetry., technicolor models (Weinberg, 1976; Susskind, 1979) which assume a new strong interaction, and the Randall Sundrum model (Randall and Sundrum, 1999) with an extra spatial dimension. These new concepts might be realized in a fundamental theory at (much) higher energies than the SM. However, in a top-down approach observable deviations from the SM expectations at LHC energies, like new particles or modifications of cross sections, can be deduced in these models.

Consider, for example, the case of supersymmetry (SUSY). Supersymmetry is the largest possible extension of the Poincaré group. This concept leads to the idea of a symmetry between fermions and bosons. All the SM particles have supersymmetric partners with spins differing by $1/2$. TeV-scale SUSY models have typically about 100 new parameters and could address many outstanding issues of the SM by providing gauge coupling unification, describing the nature of dark matter, and solving the hierarchy problem. However, as of yet, no signal of SUSY has been found for any of the available models.

Due to the complexity of full BSM models, systematic simplifications have been developed in recent years that facilitate searches for BSM physics. The main idea is not to use a whole, concrete BSM model, but focus on one or a few of its new features (see, e.g., Alwall et al., 2009; Alves, 2012). To achieve this formally, one may consider scenarios where almost all the new particles of a BSM model have masses larger than the energies probed at the current collider. Only a few selected new particles with TeV-scale masses are then treated as relevant for experimental and phenomenological considerations.

A prominent example is the decay of a supersymmetric top quark $\tilde{t}$. In a complete SUSY framework, it can decay into a large number of final states. Such a decay can be described by several BSM models. In a simplified model, one restricts, for example, the decays to $\tilde{t}\rightarrow t\chi^{0}$ (see e.g. Aaboud et al. (2017)), while all other particles are assumed to be too heavy to be produced in the decay. Here $\chi^{0}$ is the lightest neutral supersymmetric particle and considered a candidate for dark matter since it interacts at most very weakly with normal matter. Such a simplification has repercussions on the interpretation of a measurement. Restricting SUSY to just one (or two) particles takes these out of the context of a specific model and just provides an experimental signature without any relation to other processes in the model. Here, the simplified model is motivated in a top-down way from the supersymmetric model. While such a signature may be interpreted as a signal for $\tilde{t}$, it can equally well come from other objects in other BSM models with an equivalent decay, e.g. scalar third generation leptoquarks that decay into $t+\bar{\nu}_{t}$ (Aad et al., 2016) or a generic Dark Matter particle with affinity to the top quark. In this spirit, simplified models can also be constructed in a bottom-up way: by building final states comprised of combinations of new particles with different quantum numbers, independent of any theory predicting these particles and their interactions.

Effective field theories (EFTs) are of central importance in particle physics (see, e.g., Georgi, 1993). In this subsection, we will give an overview of their use and development in particle physics and introduce a recent prominent case in the search for deviations from the SM. ¹⁴¹⁴14While the notion of EFT is well-established in contemporary particle physics, some authors, among them Wells (2012) and Rivat and Grinbaum (2020) consider them a continuation of previous research strategies, ranging from effective extensions of classical mechanics to the S-matrix theory and other black-box models in particle physics. Such a broader historical analysis is beyond the scope of the present paper. The Fermi theory discussed in this section fulfils all the characteristics of present EFTs, including the separation of scales, and is often cited as an archetypical EFT by particle physicists.

The physicists’ turn towards using the general SM-EFT formalism is borne out of the conceived dead end of expressing ideas for new physics in terms of concrete models. This attitude can be characterised as such: “In absence of a direct observation of new states [i.e. particles], our ignorance of the EWSB sector can be parametrized in terms of an effective Lagrangian” (Contino et al., 2013, p. 1). It is beyond this paper to discuss in depth the epistemic and pragmatic goals of physicists leading to such a characterisation. Suffice it to note, especially in view of the following discussions, that the bottom-up SM-EFT approach is considered as being ignorant with respect to the nature of a ‘new state’.

What we consider important here is that a new state would have definite properties to be scrutinized experimentally and theoretically, which is not the case for the general SM-EFT approach. If an experiment would find a new state, it would have definite properties that have to be represented in theory, e.g. by a field $\chi$. Vice versa, if theory would propose a new state, its properties would largely be fixed and would provide a clear target for experiments to search for. In this case the field $\chi$ would be a representation of a new, but not yet observed, particle. In contrast, as will be detailed in Sect. 5, the EFT approach does not start from definite properties and thus does not provide a clear target for experimentation. In contrast to SM-EFT, concrete BSM models and simplified models do assume fields or vertices representing entities and processes, and thus offer a well defined search strategy. It is such BSM theories that physicists ultimately strive for (also for other reasons). This does not render EFTs useless, but the “main motivation to use this framework is that the constraints on the EFT parameters can be later re-interpreted as constraints on masses and couplings of new particles in many BSM theories,” (de Florian et al., 2016, p. 305). Thus, while using EFT as a tool to parametrise constraints due to the (non)-observation of deviations from the SM, most physicists want to reach beyond and arrive at a consistent and concrete BSM model. The concrete BSM models and the simplified models provide targets to search for. If no BSM target is defined by a model, physicists characterise their search for BSM models as ‘model independent’. As will be discussed in Section 5, the SM-EFT is considered a tool to support model-independent searches.

In moving from syntactic to semantic views, the focus of philosophical analysis shifted from theories to models and this also informed accounts that stressed the autonomy or independence of models from theories, such as the practice-based approaches of Cartwright (1999), and Morgan and Morrison (1999). By and large, accounts of models have become broader and more encompassing. A model no longer needs to be an interpretation of a formal system, a set-theoretic entity, or an isomorphic representation, but can be determined by in part by its function and use in scientific practice. The distinction between model and theory can be one of degree more than kind, where a theory has broader empirical scope and is more highly confirmed. In this Section, we will analyse the distinction in the context of two approaches that have already played a role in discussions of elementary particle physics, the practice-based account of Morgan and Morrison (1999) and the semantic account of Hartmann (1998). While the MaM approach emphasizes the representative and autonomous roles of models in scientific practice, Hartmann’s hierarchy of theories and models attributes a stronger role to framework theories, such as QFT, thus being somewhat closer to Suppes’ version of the semantic view (Suppes, 1962). As representation is a key issue, we also introduce the artefactual view of models, given by Knuuttila (2011, 2017), that explicitly does not require the accurate representation of models.

On the MaM view, “the rough and ready distinction followed by scientists is usually to reserve the word model for an account of a process that is less certain or incomplete in important respects. Then as the model is able to account for more phenomena and has survived extensive testing it evolves into a theory” (Morgan and Morrison, 1999, p. 18). Morgan and Morrison identify four basic elements of models:

1. 1.

   They are partially autonomous from both theory and data.

2. 2.

   They are purpose-built.

3. 3.

   They are representative.

4. 4.

   They enable us to learn about the world.

We take it to be fairly uncontroversial that models allow us to learn and that they are built with a variety of purposes in mind. As such, we will focus our discussion on the remaining two elements. We will first examine autonomy of models by reviewing the relations between models and theories and move to representation in Section 4.2. One can see that models are partially autonomous from data and theory from the way they are constructed and function. Models are not derived entirely from theory or from data, but partly from both and their construction avails itself of scientific practices such as simplification, idealisation, and approximation. A model is autonomous in that it can be an object of study or a tool in investigations of the world independently of its relation to the formal framework of a theory.

Let us now view the model-theory relation from the perspective of Hartmann (1998), who distinguishes two kinds of theories and two kinds of models:

• •

  A Type-A theory is a general background theory, quantum field theory in our case. We understand this theory to include a set of techniques like renormalization and only require that it is consistent in a physical sense.

• •

  A Type-B theory is a fundamental “model-theoretical model of a Type A theory, that is a concrete realization of the general formalism of QFT” (Hartmann, 1998, p. 101). This theory is called fundamental because “at the moment everything that can be said about the respective domain of applicability can be captured” (ibid.) by it. Hartmann’s own examples are QED and QCD.

All other theoretical constructs are models.

• •

  Type-A models are non-fundamental model-theoretical models. They can be simple toy models for a Type-A theory or be as specific as variants or speculative modifications of a Type-B theory. Hartmann’s example is a quartic interaction $\phi^{4}$ theory, but it seems to us that also most BSM models belong in this class as long as they do not fall outside the framework of QFT. The important point is that these models are models from a formal point of view.

• •

  A Type-B model, or phenomenological model, is “a set of assumptions about some object or system. Some of these assumptions may be inspired by a Type-B theory, others may even contradict the relevant theory (in case there is any)” (ibid.). Accordingly, Type-B models are rather a residual category in Hartmann’s fourfold classification. They are not necessarily models in a syntactic or semantic sense, but emerge from scientific practice. Hartmann’s examples are the various models of nuclear and hadronic structure that were developed long before QCD, models which later could be integrated into this formal framework.

Hartmann’s understanding of being ‘fundamental’ involves a relationship between the theoretical constructs and the domain of facts known at the time in such a way that if a fundamental model describes a broader domain of facts, it becomes a theory in line with Morrison’s above quoted terminology. However, Hartmann seems well aware that his understanding of fundamental is quite permissive and does not support reductionism. Discussing non-renormalizable EFTs as Type B models he concludes that, if one interprets the cut-off physically as their domain of applicability, “at a given energy scale E a suitably chosen EFT can be fundamental in the sense that (practically) nothing more can be said about the ongoing physics” (p. 104).

Hartmann’s examples for Type-B models also correspond to phenomenological models in the MaM approach as examples that “make use of a variety of theoretical and empirical assumptions” (Morrison, 1999, p. 45). Their autonomy is based on certain representative features that are not fully derivable from theory or may even go beyond the available background theory, if any. Hartmann’s requirements for Type-B models seem lower than those of MaM because he only requires ‘a set of assumptions’, and not any partially autonomous representative features. Yet his partly hierarchical account emphasizes the role of the background theory and of Type-B theories more strongly, while MaM start from the models’ autonomy and their epistemic function. In their approach, the existence of what Hartmann calls a Type-A theory is not required—as long there is some framework or some tangible way to study the model. Having a well developed framework theory is not excluded in MaM, but the autonomy of models kicks in because the framework theory is insufficient to make clear predictions or to derive the model in a canonical way. This is precisely the situation in particle physics. ¹⁹¹⁹19Note that a hierarchical approach, in the sense of Hartmann and Suppes, also stands behind Karaca’s 2013 distinction between a framework theory, a model theory (the Type-B theories, in Hartmann’s sense) and phemomenological models (Type-A or Type-B models, in Hartmann’s sense). Karaca (2018) also extends, but qualifies, a similar levelling into models of the detector etc.—which are not of concern here.

Putting MaM alongside Hartmann’s version of a semantic approach we find two, only partially overlapping uses of ‘model’ that one can also discern in the broader philosophical literature. One where a model is defined as a restricted or specified part of a scientific theory that is related to a domain of phenomena, and one where a model is built in practice by studying real-world phenomena. Both are equally legitimate and are applied in physical practice. On some accounts, model building begins within an existing theoretical framework while, on others, it proceeds initially without regard for such a theoretical framework, even though models may well become theories. While MaM clearly belongs into the latter category, Hartmann’s account belongs to the former, apart from his introduction of the residual class of phenomenological models.

Let us now turn to the fourth element of MaM, representation. In contrast to the traditional view where a model must be isomorphic to, or somehow mirror, the phenomenon, representation for MaM is instead taken as “a kind of rendering—a partial representation that either abstracts from, or translates into another form, the real nature of the system or theory, or one that is capable of embodying only a portion of a system” (Morgan and Morrison, 1999, p. 27). Models can be representative of either the world (their “target system”) or of theory, they can also represent both, one more than the other. For instance, the hydrodynamic model of the boundary layer discussed by (Morrison, 1999) demonstrates how representation can happen in both directions. Prandtl built a small water tunnel that acted as a representation of the world, which exhibited different behaviours in different regions and allowed him to construct a mathematical model. The mathematical model in turn represented aspects of both the classical theory and the Navier-Stokes equations, neither of which could be applied directly to the real world. Thus, the models represented in different ways, had different targets of representation, and acted as mediators between theories and the world. Morrison contrasts the boundary layer model, where one has to develop a target for mathematically efficient representation, to wit, the boundary layer, with the mathematical pendulum, where the formalism of Newtonian mechanics provides all the formal tools for a de-idealisation, i.e., possible terms for the introduction of friction, finite size of the bob, etc. It is this emphasis on the practical aspects of representation that distinguishes MaM from Hartmann’s more theory-based approach—even though Hartmann’s residual category (Type-B models) is pretty elastic.

In our use of representation, we follow MaM and make no strong demands on isomorphism or accurate representation and gladly accept the important role of modelling practices, such as idealisation, approximation, and abstraction. Exactly how the elements of a system are mathematically represented in a given model can only be justified by some story made in a case-by-case basis (Hartmann, 1999, cf. ). We are thus willing to endorse a rather liberal notion of representation where a representation in a scientific model is a mathematical description of a target or target system. A target (or target system) can be whatever the model is aiming at describing, which may be actual, as in the case of beta decay, or merely potential, as in models of BSM physics. Targets may be objects, such as fields, or descriptions of how objects interact, such as interaction vertices, and target systems may include both. Targets may be as accurate as possible, or they may be idealised, abstracted, and approximate. Again, we make no strong demands on the ‘mapping’ relation. However, a bare minimum condition for being a model is that it must be a model of something, and as such it must at least have some target of representation. Furthermore, its target, whatever it is, needs to be distinct and well defined, such that we can meaningfully say that it is being modelled. Having a specifiable target is then a bare necessary condition for modelhood, though it is of course not a sufficient condition. In our specific context, these targets are elements are of physics beyond the SM, such as new fields and vertices. For a genuine model of BSM physics, its target is some purported phenomenon beyond the SM and what it represents needs to be distinct from the SM. If a BSM model has a distinct and well-defined representational target it will allow physicists to make specific predictions about new physics. Making numerical predictions alone, however, is not sufficient for having a well-defined target. Lagrangians like the one in Eq. 4 contain a large number of parameters that can be tweaked to absorb any deviation. The essential point of Eqs. 1 and 4 is thus the schematic separation into a well-defined background model—the SM whose sizeable number of parameters are considered fixed by experiment—and deviations from it that are parametrized in terms of an EFT.

While the MaM approach requires that a model has some target or even representative features, others have argued that representation is not a characteristic feature of a scientific model. Some practice-based approaches consider representation an accomplishment of model-users rather than a feature of the model itself. Knuuttila (2011, 2017) and Teller (2001) have argued that there is no general account of how and in virtue of what scientific models can be said to be adequate representations. In a sense, anything can stand for anything and what represents something is what is chosen to represent it. This representation can be successful or not for a given end, but whether it represents or not is not decided by features of the model or the target. On accounts such as Knuuttila’s, the key features of models are epistemic: they are tools that allow us to learn, to make good inferences, to convey useful information about the world, and the role of representation is no longer central. Knuuttila argues that representationalist views of modelling come up against problems because they treat models as distinct objects that need to be mapped to the world in order to justify their success. By contrast, she develops an artefactual account that attempts to dissolve these problems by conceiving of models as epistemic tools rather than abstract objects.

This “amounts to regarding [models] as concrete artefacts that are built by specific representational means and are constrained by their design in such a way that they facilitate the study of certain scientific questions, and learning from them by means of construction and manipulation” (Knuuttila, 2011, p. 262). She lists five characteristics of models as epistemic tools: “(i) the constrained design of models, (ii) non-transparency of the representational means by which they are constructed, (iii) their results-orientedness, (iv) their concrete manipulability and (v) the way their justification is distributed so as to cover both the construction and the use of models” (Knuuttila, 2011, p. 267). The epistemic characteristic of models does not include accurate representation, but neither is it precluded. Representation is always an accomplishment. But a model’s success cannot be assessed without “the culturally established external representational tools that enable, embody and extend scientific imagination and reasoning” (Knuuttila, 2017, p. 1). They “form an ineliminable part of the model itself,” (ibid., p. 3) not just of its description. “What distinguishes models from many other kinds of representations is often their systemic, autonomous nature,” (ibid., p. 17) which they share with fictions. Knuuttila illustrates her artefactual account with the example of highly-idealised mathematical models that are applicable in unrelated empirical contexts.

There already exists considerable philosophical literature on effective field theories. Philosophical attention turned to EFTs in the late 1980s and early 1990s with the seminal papers by Teller (1989), Cao and Schweber (1993), and Huggett and Weingard (1995). Cao and Schweber (1993) used EFTs to articulate anti-foundationalist metaphysics where there is no final theory, but only an infinite tower of EFTs. From these discussions various metaphysical and ontological arguments were picked up in Castellani (2002) and Fraser (2009), and more recently in (Bain, 2013; Crowther, 2016; Rivat and Grinbaum, 2020; Ruetsche, 2018; Williams, 2019). Rivat and Grinbaum (2020) provide an excellent overview of the debates about EFTs in philosophy and note that they are primarily about issues of ontology, emergence and inter-theory relations, and fundamentality. A substantial part of these debates focuses on whether they are robust or fundamental enough to be worthy of philosophers’ attention as compared to axiomatized approaches—which, for Wallace (2006) and Williams (2019) are still in the focus of philosophical orthodoxy. Williams’ ‘effective realism’ about EFT is certainly one way to justify the value of EFTs through the robustness of entities, properties, or relationships that are ”accessible (detectable, measurable, derivable, definable, producible, or the like) in a variety of independent ways,” (Williams, 2019, p. 219). Fraser (2020) defends a more refined selective account of realism based on the renormalization group. A detailed criticism of the various ways to identify the real parts of EFTs has been given by (Rivat, 2020). Another, more instrumentalist, view was developed, initially by Huggett and Weingard (1995), in response to the controversial views of Cao and Schweber (1993), whereby EFTs are merely tools for extending the scope of physics. Rivat and Grinbaum (2020) describe Huggett and Weingard as saying that “one should not draw conclusions about the hierarchical structure of the world, the existence of a fundamental level, or the prospects of a completely unified theoretical description” based on EFTs. This pragmatic view is picked up on by Hartmann (2001), for whom EFTs do not suggest anti-foundationalist metaphysics, nor will they eclipse models or theories, but rather that they have a role play in the search for new physics alongside models and theories. According to Butterfield (2010), the use of EFTs come from “an opportunistic or instrumentalist attitude to being unconfident” about the applicability of QFT at high energies. Grinbaum (2008) also endorses this kind of view and likens the current interest in EFTs to the widespread use of S-matrix theory in the 1950s, where a clear consensus on the way forward was missing. With considerable foresight, Grinbaum suggests that “perhaps something like this is happening today with EFT and the model-independent analysis of new physics” (ibid., p. 45). Indeed, today, physicists are unsure what the next accepted theory will be like and once again we find them turning to pragmatic benefits in the form of EFTs.

But we have to note that most of these authors have focused on EFTs in general and not specifically on the bottom-up approach of the SM-EFT. Rivat and Grinbaum (2020, p. 3) even allude to SM-EFT but consider it as one among many black-box models. Crowther (2016, p. 72) considers bottom-up EFTs as the main tool for constructing a tower of EFTs in the style of Cao and Schweber, which she does not consider “as a threat to the possibility of there being an ultimate fundamental theory of physics” (p. 74). One exception is Wells (2011) who examines bottom-up EFTs as a part of a general method, or mind-frame, for scientific research that could have been used, he argues, to point the way towards general relativity by predictions about the perihelion precession of Mercury already in the late 19th century. The in-principle feasibility of such a bottom-up approach, to his mind, suggests that physicists should focus on improving theories through EFTs and not be exclusively concerned with principles and justifications. Wells does not discuss whether this makes EFTs or effective theories in general similar to models.

We will largely avoid issues of realism or reductionism because physicists availing themselves of SM-EFT are openly granting the reality of the SM in order to look for deviations of any kind. Their eventual goal is to find constraints on possible models of such new physics. We are interested in the use of EFTs in practice and the importance of the bottom-up approach and find the modelling perspective to be more illuminating. Thus we take our discussion to pick up on questions introduced by Hartmann (2001), who takes a detailed look at the history and development of EFTs in the context of models and theories. The metaphysical and ontological theorizing based on EFTs, which Hartmann was reluctant to do, has been strongly taken up in the last two decades, but his discussion of the roles of EFTs, models, and theories has been more quietly received. In looking at case studies in hadron physics, he argues that EFTs produce scientific understanding of the processes they are modelling; they are simple, intuitive, and satisfy a variety of pragmatic and cognitive goals. There is rather an interplay between theories, models, and EFTs in physics research and distinguishing them is not always easy. “EFTs share many of the functions of theories and models. Like models, they provide a local, intuitive account of a given phenomenon in terms of the degrees of freedom which are relevant at the energy scale under consideration. They are relatively easy to solve and to apply, and they are heuristically useful…Like theories, EFTs are part of a bigger picture or framework, from which they can be derived in a controlled way. They help to make predictions and to test the theory they relate to” (2001, p. 294). Among Hartmann’s examples are the above-discussed Fermi theory and the V–A theory of the weak interaction. Our analysis similarly finds that the SM-EFT has complex relations to theories and models. While accordingly this double relationship remains, there are significant differences and we will argue that its nature may even depend on the experimental success of the SM-EFT.

Differences between these stages highlight issues that will become relevant for our discussion of the model character of SM-EFT. Let us first remark that these stages are meant as analytical categories and do not have to be marched through sequentially in actual scientific practice. For example, in the early 1980s, without reference to a systematic development of the SM-EFT, Eichten et al. (1983) proposed a model for fermion substructure (‘preons’) in analogy to the Fermi theory.

The major difference between Stages 1 and partly Stage 2 versus Stage 3 is the (absence) of an underlying specific physics problem: the SM-EFT of Stage 1 is motivated only by the general and unspecific idea that the SM should be embedded in a larger theoretical framework and the restrictions of Stage 2 are largely motivated by considerations of scientific practice, Stage 3 is built either to understand a measurement beyond the SM, as will be detailed for a current example in Sect. 6, or by a concrete idea of how BSM should look like, be it the preon-hypothesis of Eichten et al. (1983) or the violation of baryon number in the framework of Grand Unified Theories (GUTs), as, for example, in Weinberg (1979). As already discussed in Sect. 3.4, SM-EFT at Stage 1 works as an accounting scheme to quantify deviations from SM distributions and possibly correlate different operators. As such, the SM-EFT is a convenient tool to globally assess the status of the SM. In contrast, selecting one or a few operators in Stage 3 is a step towards the goal of physicists to arrive at a concrete BSM model involving new entities. The roles of target, prediction, testability, and representation are therefore significantly different in these stages, which can be gleaned, for example, from the paper of Weinberg (1979).

In his paper, Weinberg addresses the specific problem of B–L (non-)conservation (B denoting the baryon number, L the lepton number) with GUTs in mind. Thus, he explicitly neglects the other operators of the SM-EFT because they are of no relevance for his specific target, but he is careful to select dimension-6 operators that do not conserve baryon number. From those he derives general constraints for B–L violation, including relations among observables like partial decay widths of baryons. Weinberg thus starts from Stage 2; from a well-defined theoretical target, which implies an experimental target, viz. proton decay. Both of these were considered in concrete models before his paper.

On the other hand, Weinberg also makes clear that limitations exist in his EFT approach. He immediately emphasizes that B–L non-conservation should be understood better by specific gauge models of baryon decay (Weinberg, 1979, p. 1569). A derivation of such a model is hardly possible if it should simultaneously address all 2499 operators of the SM-EFT. Furthermore, Weinberg’s paper is sometimes used as an argument for the ability of SM-EFT to predict: in focussing on two specific dimension-5 operators, Weinberg states that these would produce a neutrino mass between roughly 10^-5 and 10^-1 eV. Here, Weinberg argues from what we call Stage 3, which we agree, has a target and representational content. This, however does not mean that an EFT per se allows one to predict: to actually arrive at the values for neutrino masses, Weinberg assumes certain Wilson coefficients, a cut-off scale of 10¹⁴ GeV, suggested by the merging of the electromagnetic, weak and strong couplings, and a coupling of $\cal{O}$(1). Thus, the EFT simpliciter points out at most possibilities. To provide a quantitative prediction, it has to be supplemented by an additional assumption that comes closer to a concrete model. The same is true for the preon assumption of Eichten et al. (1983): it motivates searches in certain processes, which were anyhow motivated with parametrisations other than EFT, it provides a framework to set constraints on fermion substructure, but, in the absence of a quantitative estimate of observable effects, it does not allow one to definitely rule out such substructure.

The inability of an EFT at any stage to predict without further model assumptions implies that it cannot be meaningfully tested. While one can certainly look for any kind of deviation the SM-EFT allows, something that is anyhow in the spirit of expecting BSM somewhere, any non-observation of a deviation would not falsify SM-EFT since it does not allow for a quantitative prediction. In consequence, the SM-EFT cannot be tested in a meaningful sense; a single operator, supplemented by a physics model, can.

We thus diagnose a major distinction between Stages 1 and 2 and Stage 3 that will be relevant for determining whether and at what stage the EFT is a model. At first glance it may be surprising that the SM-EFT, if interpreted as a collection of operators, does not produce a target or have representational character, whereas each individual operator has. However, we understand this as a breaking of an (virtually) infinite amount of possibilities into just a single one. As we know from many examples, such a breaking significantly changes the character of the related objects.

To explain the apparent LFV with a concrete model, new particles with different couplings to leptons are assumed. Furthermore, since LFV has only been observed for the bottom quark, it is suggestive to assume such a particle to have an affinity to bottom quarks, or in general to quarks of the third generation. Attempts to explain the B anomaly include SUSY (Altmannshofer et al., 2017), strongly coupled models, such as composite Higgs (Greljo et al., 2015), or additional (heavy) Z’s or W’s (Boucenna et al., 2016).

While these models may be easily adjustable to work for LFV in bottom hadron decays, they also have implications for other processes that cannot be accommodated easily and require a high level of fine tuning. Furthermore, some of these hypotheses, although starting from a different underlying concept, arrive at a similar phenomenology for LFV effects, and thus cannot be experimentally distinguished from each other. Therefore, physicists currently turn to EFTs and simplified models, which provide general constraints on a viable model, despite the fact that concrete models would have been more satisfying. This is why several authors prefer a bottom-up approach using the SM-EFT (see e.g. Buttazzo et al. (2017)).

The physics of bottom hadrons involves different scales, the QCD scale of some 100 MeV, the scale of the bottom quark of some 4 GeV and the electroweak scale of some 100 GeV. It is thus adequate to use an EFT to describe bottom quark decays. The corresponding Lagrangian, containing only transitions of the bottom quark into other SM particles, is a subset of the SM-EFT and has been used in analyses of bottom decays and transitions for some 20 years.

While the indication of LFV can be analysed with a full EFT for bottom quarks, the observed properties indicate only certain operators to be relevant as pointed out in Buttazzo et al. (2017). Conveniently the (potential) BSM effects can be represented as an additional contribution to the SM Lagrangian. For example in Buttazzo et al. (2017) the effective Lagrangian for $b\rightarrow sl^{+}l^{-}$

is postulated for the production of $e^{+},~e^{-}$ (or $\mu^{+}\mu^{-}$ by replacing $e_{L}$ by $\mu_{L}$).²⁰²⁰20Instead of general quark transitions, here only the one relevant for the indication of LFV, namely the $b\rightarrow s$ transition is used It contains two four-fermion operators which describe new interactions between left-chiral quark and lepton fields $(s,b)_{L}$ and $(e,\mu)_{L}$, respectively, already basically known from Fermi theory. The coefficients $C_{T}$ and $C_{S}$ are flavour blind and encode the new physics scale $C_{T,S}\propto\ v^{2}/\Lambda^{2}$, where $v\approx 246$ GeV is the vacuum expectation value of the SM Higgs field. The indices $T$ and $S$ refer to the colour-triplet and colour-singlet structure of the corresponding operators, respectively. The flavour structure of the new interactions is determined by a $U(2)_{q}\times U(2)_{l}$ flavour symmetry and encoded in the Hermitian matrices $\lambda^{q}_{ij}$ and $\lambda^{l}_{\alpha\beta}$. The $C_{i}$ and $\lambda$ coefficients are free parameters.

Describing the BSM contribution in the EFT approach allows one to combine the apparent flavour violation in $b\to sl^{+}l^{-}$ with other measurements both in the bottom quark sector but also in LHC production of pairs of leptons of different flavour. These can be used to further constrain the free parameters. A global analysis of existing flavour data (see, e.g., Albrecht, 2018) indicates non-zero coefficients $C_{T}$ and $C_{S}$, as shown as the green, yellow and grey regions (1, 2, and 3$\sigma$ contours, respectively) in Fig. 4 (Buttazzo et al., 2017). This is an example of how the SM-EFT acquires representative character: Committing to the existence of non-zero values of $C_{T}$ and $C_{S}$ corresponds, in the chosen basis of the EFT, to the existence of two new vertices of SM fields, namely two different types of a direct coupling between four different fermions which is not present in the SM: two different quark flavours and two leptons per vertex. Assuming the existence of these two new vertices is analogous to the assumption of the existence of the Fermi interaction before the SM was conceived. Also, these vertices can then be specifically studied for their effects on precision measurements in other experiments, or e.g. for their predictions of distortions of kinematic spectra away from the SM in LHC measurements involving the same quarks and leptons as the B-physics measurements.

The allowed regions in $C_{T}$ and $C_{S}$ from the EFT results are not transparent in terms of an underlying physics idea. However, they can both be interpreted as new interactions in a Stage 3 SM-EFT and can be related to elements of the concrete BSM models of Sect. 6.1 in a simplified model approach. As discussed in Sect. 3.2, the idea of simplified models is to focus on just part of a BSM model, say a single type of particle and evaluate its constraints and potential consequences. This approach starts from a Lagrangian consisting of the SM and a term representing a hypothetical new particle with a free coupling strength. The outcome of this Lagrangian is then matched to Eq. 5. Assuming $B^{\prime}(1,1,0)$ or $W^{\prime}(1,3,0)$; (colour-triplet scalar particles) $S_{1}(\bar{3},1,1/3)$ or $S_{3}(\bar{3},3,1/3)$; (colour-triplet vector particles), $U_{1}^{\mu}(3,1,2/3)$ or $U_{3}^{\mu}(3,3,2/3)$²¹²¹21The numbers in brackets indicate the colour, weak, and hypercharge quantum numbers., one obtains a certain correlation between $C_{T}$ and $C_{S}$, shown as straight lines in Fig. 4. While in these simplified forms the particles are only defined by their quantum numbers, they could be interpreted as new electroweak bosons, scalar or vector leptoquarks, which already were discussed in the context of concrete BSM models. Comparing the analysis performed within the EFT, and the predictions of the different simplified models, the model with a coloured vector mediator, $U_{1}^{\mu}(3,1,2/3)$, is clearly preferred.

As noted in Section 4 above, it is rather uncontroversial that models are constructed to fulfil specific purposes and that they help us learn about a theory or a target system, respectively. These basic elements of the MaM approach not only apply to BSM models and simplified models, but also to SM-EFT, whose purpose is efficiently accounting for possible deviations in order to possibly place constraints on models of BSM physics. The partial autonomy from data and theory, another element of MaM, makes for a more interesting analysis.²²²²22We will discuss the fourth element, representation, in Sec. 7.3. Let us start with the autonomy of explicit BSM models. In (Maettig and Stoeltzner, 2020) it was argued that BSM models stand in a twofold mediating relationship because they are placed within the framework of quantum field theory or a more general successor theory and they must simultaneously recover the SM as a low-energy limit. Or put otherwise, they must recover the SM as a top-down EFT. From this, we see that SM-EFT is at least partly autonomous from data. While it features all the SM fields whose properties must accord with data, the many additional terms are only determined by general theoretical considerations, such as the SM symmetries.

McCoy and Massimi (2018) have argued that simplified models are autonomous from data and ‘theory’ (here meaning full BSM models, like supersymmetry) because a simplified model is partly independent from any particular BSM model. Due to their autonomy, they argue, simplified models can play a role mediating between data and BSM models analogously to the mediating role played by models in the original MaM account. Yet, this autonomy must be qualified because the particle content of a simplified model is borrowed from a potentially large class of BSM models. For example, a simplified model may feature a new scalar top-like particle, but such a particle has originally been proposed within a class of full BSM models. The main point is that simplified models do not specify all properties of the additional fields, admitting a large space of physical possibilities. McCoy & Massimi call such modelling perspectival because it does not require representation in the sense of a mapping onto a target or “onto an actual worldly-state-of-affairs (or suitable parts thereof) but representation has instead a modal aspect: it is about exploring and ruling out the space of possibilities in domains that are still very much open-ended for scientific discovery” (McCoy and Massimi, 2018, p. 338). The initial space of possibilities in the SM-EFT is much larger than the one of simplified models because, at least initially, it is no longer motivated by any properties of classes of models, outside of the SM. At Stage 1, there is no clear and distinct perspective to take. Thus, the perspectival modelling cannot be further extended to consider SM-EFT a model.

SM-EFT is not autonomous from theory in that it uses a theoretically well-established procedure in expanding the Lagrangian into operators of higher dimensions. Only fields of the SM theory are used and the basic symmetries of the SM are retained. Hence the SM-EFT is, at Stage 1, determined by general theoretical principles and a well-confirmed theory, the SM, but not by any particular BSM models; all formally possible terms are taken into consideration. Here, one operates in a generic space of possibilities that parametrizes our ignorance and can be used for exploratory experimental searches. According to Wells (2012), this generic approach is even a requirement of mathematical consistency, to avoid divergences and maintain the independence of arbitrary scale cut-offs. Leaving out terms destroys the renormalizability of the theory, and if one picks out certain terms the corrections might be associated with terms that had been left out. Thus at Stage 1, SM-EFT has little autonomy from the SM and the framework theory. In Stage 2, any such autonomy is only apparent because it arises as the product of pragmatic decisions for the sake of tractability or simplicity. But even if the mathematical consistency is lost, a Stage 2 EFT for special sectors is largely determined by the SM in its field contents and symmetries.

A Stage-3 EFT is, however, sufficiently autonomous from theory that it could be considered a model. This is not only the case in terms of MaM’s notion of autonomy. Also Crowther (2016), more specifically, argues that “the idea of autonomy (or, rather, quasi-autonomy) in EFT comes from the fact that the low-energy theory is largely independent of the details of the high-energy theory. There is extra information contained in the high-energy theory, far over-and-above that required in order to describe the low-energy behaviour of significance.” This autonomy makes the EFT robust against changes in the microphysics and together with the fact that is a novel description, implies for her that EFTs are emergent. A Stage-3 EFT can simply be written down to model some new interaction, in which case, its autonomy from the full theory (as well as a clear target of representation, and hence modelhood) is introduced by hand. However, one may also arrive at a Stage-3 EFT based on the systematic expansion in SM-EFT, where the relevant EFT operators are selected by deviations from the SM observed in measurements. In this sense, SM-EFT is an example of a potentially successful bottom-up strategy that accomplishes autonomy from the theoretical description in terms of products of SM-operators. In such a case, the SM-EFT approach would become the proverbial Wittgensteinian ladder; a formal theory to be thrown away and replaced by a concrete BSM model introducing a new field, a simplified model, or a vertex model of some new phenomenon in the same style as the Fermi theory had done for beta-decay.

Let us now return to the problem handed to us by Hartmann (2001), namely the difficulty in disentangling EFTs from models and theories and address it in the context of Hartmann’s (1995) hierarchical approach discussed in Section 4.1. Theory here has to be understood in the double sense of a Type-A theory (such as QFT) and a Type-B theory (such as the SM). All explicit BSM models can be subsumed under Hartmann’s classification, although it is sometimes a matter of how ‘fundamental’ their aspirations are and whether there are presently any real or purported phenomena they describe.²³²³23Recall that Hartmann’s notion of ‘fundamental’ is defined with respect to the present experimental knowledge and does not involve any reductionist aspirations. Supersymmetry (SUSY) e.g. may be considered a Type-B theory that captures all that can be said in its intended domain of applicability; it is true, key features of SUSY have not been empirically validated. However, it can also be interpreted as a Type-A model if one interprets SUSY just as one way to provide inputs for a more fundamental theory. This can be motivated by the different symmetry breaking mechanisms invoked in special realizations of SUSY. Other concrete BSM models, like technicolor and Randall-Sundrum models, do not have the rather fully worked out fundamental structure, such that they would be considered as capturing their total domain of applicability. In this sense, they can be classified as Type-A models in virtue of their serving as ‘input for more fundamental constructs’. However, there may also be an emphasis on the phenomenological aspects of these models in the sense of Type-B models introducing a ‘structure that cannot be deduced from that theory’. Such a characterization is certainly appropriate if models (partly) work outside the framework of QFT; variants of extra-dimensional modes can be counted as such. Given that simplified models are developed from BSM models, they should be classified as Type-A or Type-B models depending on how their space of possibilities is configured. Recall that, as mentioned in Section 4.1, Hartmann’s requirements for a Type-B model are lower than in the MaM approach; thus it seems to us plausible that the reasoning of McCoy and Massimi concerning representation can be copied to obtain a perspectivalist generalization of Hartmann’s approach.

Since for Hartmann the Fermi theory is an EFT and a phenomenological model, one may wonder whether SM-EFT overall can be considered as a model in a non-trivial sense, that is, beyond the fact that any SM-EFT is formulated in terms of a Lagrangian and thus a formal model of QFT. As argued in Section 7.1, SM-EFT is, at Stage 1, primarily influenced by theoretical considerations, such as symmetries and renormalization. It is a formal technique allowed within the context of a Type-A framework theory and strongly dependent on the existence of a well-established and well-tested model, or a Type B theory, such that the formal operators of higher order can be experimentally analysed as deviations from the SM.

But in contrast to concrete BSM models or simplified models, SM-EFT at Stage 1 is neither a toy model (see Reutlinger et al., 2018) that could provide understanding of a Type-A or Type-B theory, nor a variant of a Type-A or Type-B theory. By integrating all formal possibilities, SM-EFT is also too unspecific to provide meaningful input for fundamental constructs. It merely provides a theoretical space for experimental exploration. Similar arguments apply for Stage 2, at least in general and unless going to Stage 2 was motivated by certain assumptions about candidate models. Merely truncating the Lagrangian expansion in Eq. 4 or focusing on the top quark or the Higgs sector does not provide a sufficiently well-defined target for BSM searches. It is only at Stage 3, once significant deviations from the SM have been found, or are assumed to be present, that the SM-EFT can be seen as a phenomenological model of Type-B, i.e., as about some object or system. One may then understand SM-EFT in the same way as the early Fermi theory, i.e., as a description of some physical process that is based on vertices instead of force-mediating fields, and that could be described within some Type-A theory. We thus find that SM-EFT, given its close relation to Type-A and B theories, in principle falls within Hartmann’s hierarchical conception, but that it is not possible to interpret the first two stages as models even though they are formally expressed as theories. The whole point of SM-EFT is to use the formal machinery of a well-confirmed theory to search for deviations, that is, to explore its limits in order to search for new physics. To move forward from this mixed outcome, it is helpful to recall Hartmann’s distinction between the global perspective of a theory and the local understanding provided by models (Hartmann, 2001). Some theories, such as SM-EFT, may fail to provide local understanding, even though typically theories unify the local understanding of the models that fall under them.

We outlined in Sect. 3.4 the importance of representation for physicists searching for BSM effects. While we thus think that representation plays a significant role in the practice of particle physics, we have seen in Section  4.2 that scholars disagree on how crucial it is in characterising modelhood. While representation figures among the four basic elements of the MaM approach, and in McCoy & Massimi’s application of MaM to simplified models, other practice-based approaches, among them Knuuttila’s, treat representation as an asset in successful modelling practice but not as an essential precondition of being a model. We will discuss these two approaches in the next two sections in order to investigate whether SM-EFT represents at all stages, within the context of these approaches, and to what extent the outcome of this analysis matches the research practice in present-day elementary particle physics.²⁴²⁴24Hartmann’s papers discuss the representation within the context of idealisation. Models represent some object system in an idealised fashion, and there is a variety of idealisation strategies on all four levels. Without going into further detail, we believe that his diagnosis about the representation of SM-EFT would largely agree with our conclusion concerning the MaM approach. There is nothing about which SM-EFT could be an idealisation at Stage 1. The key problem of our analysis will not lie in the specifics of the concept of representation applied, but in whether SM-EFT at all stages has a target that is specific enough to allow us to speak meaningfully about representation at all. Having a target appears to us a necessary condition for representation.

Representation in bottom-up EFTs is complex and depends on the stage. Let us start with the Fermi theory that may be considered as a Stage-3 EFT and discuss it in the context of MaM’s notion of representation. Even without having an idea of the larger SM framework, the original theory represented the phenomenon of beta-decay in terms of the input (down quark) and the output (electron, neutrino, and up quark) by respective wave-functions. ²⁵²⁵25In agreement with Eq. 2 we use modern notation. Similarly, one can today write down BSM approaches by using a few EFT operators, as exemplified in Sect. 6, in which each term is a representation of a field and its properties. Such representational terms can be motivated by factual indications, by having a BSM physics idea like preons, leading to almost an identical equation, or just by a parametrization of a possible deviation from the SM without a definite physics idea like $e^{+}e^{-}\rightarrow W^{+}W^{-}$. Thus, at Stage 3, a clear target for experimental and theoretical evaluations is identified. This is also the case if such a target appears without any previous phenomenon or physics idea as a result of experimental investigations of deviations from the SM conducted within the context of the SM-EFT strategy that result in one or two terms of the initial expansion having non-zero coefficients. This suffices for representation according to MaM, because models can be representative of theory or the world, or both. A model at Stage 3 either arises as representing a certain physics idea in terms of a mathematical expression in the EFT or as representing an experimentally discovered deviation in terms of a model whose description in terms of SM-EFT is still in a very preliminary form.

At Stage 2, only some broader area of interest is specified, e.g. the Higgs sector, which, out of pragmatic and epistemic reasons, physicists deem the most likely sector to reveal deviations from the SM. However, there is no specific target defined and there is no representation of a BSM effect. The space of required observations is significant and the Higgs-EFT is just a parametrization of all possible deviations one may think of, taking into account the theoretical constraints discussed in Section. 3.

This lack of a clear target and its representation is compounded in Stage  1. Any target of representation, if there were to be one, would be too vague to be meaningful. If we treated the SM-EFT or some part of it as a formal model—after all it is still a Lagrangian—it would not be clear of what it would be a model. It is an explicit aim of the approach to remain silent on the nature of any new physics because its objective is to be sensitive for experimental deviations from the SM of whatever nature. Thus, it is inherent to the bottom-up approach that there is no specific target of representation in Stages 1 and 2.

To us, this indicates that SM-EFT does not exhibit all four elements identified in the MaM approach. The characteristics are, however, exhibited in a Stage-3 EFT, when a target of representation is specifiable, even if there is no new field posited. But it is clear that if there were evidence of a deviation this is precisely the next step that the physicists would take, either by devising an explicit model or a simplified model (see Section 3.4). For this reason, SM-EFT as an epistemic strategy is somewhat orthogonal to the objectives of the MaM approach that starts from a given phenomenon that cannot be effectively explained by theory, if there is any, without the mediation of autonomous models. In the case of SM-EFT, the theory provides only the formal means for a rather open-ended experimental investigation. This might be seen in analogy to Morrison’s example of the mathematical pendulum discussed in Section 4.2 that provides the theoretical means to introduce additional effects, but in contrast to this well-understood de-idealisation, particle physicists have few empirical indications of what the new physics could be like.

BSM models, which virtually all hypothesize new fields and interactions, provide a target with fairly well defined properties. These new fields are considered a representation of a possible real phenomenon. Similarly, simplified models select a few entities to represent an observable phenomenon, even though it may have different physical interpretations if eventually expressed within the context of an explicit BSM model. Simplified Models put representation centre-stage, as was emphasized by McCoy and Massimi, because one has to properly define what representation means for entities that are only partially specified. Massimi (2018) has argued more generally that such perspectival modelling avoids contradictions insofar as the same target may well be represented by incompatible models. She identifies four aspects of perspectival modelling: i) there is a plurality of such models; ii) each model is only a partial description; iii) the models are complementary; and iv) as we noted above, they are modal models in that they represent merely possible rather than actual states of affairs. A similar view cannot be applied to SM-EFT which, at Stage 1, does not represent even possible states of affairs, but merely parametrises physicists’ ignorance. At this stage, there is no group of partial, complementary models along which one can take various incomplete perspectives on a space of possibilities.

There are also approaches that most likely would consider SM-EFT as a model at all stages. This is rather obvious for a syntactic account because SM-EFT is a formal model of quantum field theory. Considering SM-EFT as a theoretical tool for experimental searches for BSM physics renders it as an epistemic tool in the sense of Knuuttila’s (2011; 2017) artefactual account discussed in Section 4.2. There can be little doubt that SM-EFT allows us to learn from data in specific ways and to obtain constraints on possible physics. It allows one to make inferences at all stages and has a specific design that allows it to be manipulated as an artefact rather than as an abstract object that accurately represents a target.²⁶²⁶26See the characteristics mentioned in Section 4.2 above. Its justification is distributed over theoretical principles and experimental strategies—while traditional particle physics models are justified by their resolving a certain problem of the SM or by introducing some idea of new physics. The widespread use of SM-EFT in the community documented in Section 2 and the existence of a longer tradition of effective theories (Wells, 2012) indicate that the method is indeed ‘culturally established’. The bottom-up EFTs are based on the same experimental and theoretical techniques as studies of the SM and simplified models and explicit models.

At each stage, the bottom-up SM-EFT approach is described by a Lagrangian obeying the principles also used for the SM and the constraints on the EFT parameters are obtained in the same experiments that test BSM models using the same experimental techniques. However, there are differences of strategy: theoretical and experimental studies of bottom-up EFT approaches emphasise deviations, whereas studies of BSM models mostly look for the footprint of a new particle against smooth background. Thus, for epistemic accounts, there seems to be no principal difference between explicit models and the bottom-up approach that eventually aims to arrive at them. Since in epistemic accounts of models, representation is a welcome add-on and its absence does not automatically disqualify something from being a model, the bottom-up EFTs, including Stages 1 and 2, would be classified as models.

Fair enough, but the problem we see in Knuuttila’s epistemic account is that it does not reflect a key element of the scientific practice of most particle physicists. While Knuuttila considers representations and its tools as all of a piece, the SM-EFT strategy puts an extra value on the goal of representation. As outlined in Sect. 3.4, physicists expect a model to have a well defined target of representation. While, as discussed in Sect. 7.3 such a target exists at Stage 3, it is absent at Stages 1 and 2, as exemplified in Sect. 6. For particle physicists, the Stages 1 and 2 are just (if at all) stepping stones towards a representational model of BSM physics. If induced by experimental findings, the transition between Stages 2 and 3 would signify a major breakthrough.

Our analysis of how three philosophical accounts of models address bottom-up EFT strategies, in particular SM-EFT, puts Hartmann’s main question, the relationship between model and theory in assessing EFT, back into the focus and gives it a somewhat surprising twist. SM-EFT at Stage 1 can fail to be a model, at least on the MaM account, and yet still qualify as a theory, while it may have acquired, at Stage 3, the characteristics of a model. SM-EFT is a theoretical tool that helps experimental physicists to parametrize their constraints or observations of deviations from the SM. Such searches may or may not be guided by any explicit models or aspects of models.

A bottom-up EFT is certainly not guided by a model or a theory in the usual sense. But our case study clearly shows that there exists a role for theory in scientific practice without the mediation of models. This is somewhat at odds both with the general idea of models as mediators between theory and data, and the hierarchical approach of Hartmann and advocates of the semantic approach. To be sure, theory in this case does not predict, unify, or explain. It is primarily a tool for the parametrization of experimental results that involve research motives that are not connected to specific models or theories.

It seems to us that this differentiation also bears important lessons for the present debates about EFT. As mentioned in Sect. 4.3, most philosophers understand bottom-up EFTs either as ways to discover a new fundamental level in the tower of theories or as a way to study the limits of an already specified EFT and support its realist interpretation through a robustness argument. If there are any effects of unknown physics at higher energy, they can be absorbed into the parameters of the EFT on the basis of experiments conducted on the scale of its validity. In doing so, some interpreters consider the limit of validity, the cut-off, as an element of physical reality. When physicists are applying SM-EFT, realism works the opposite way. The SM—or in Stage 2: certain sectors of it—are assumed to be real and its parameters are locked in, which allows physicists to look for deviations from those values of whatever kind and without any commitments about their nature. They are looking for constraints on BSM physics or some target of further analysis—for some thing to model. It seems to us that philosophically this goal to obtain a target and eventually a model that represents new physics is more modest in kind than the realist discourse that present discussions about EFT are embedded in and physicists applying SM-EFT may remain agnostic about realism vs. instrumentalism or reduction vs. emergence. This extends to their attitude about the cut-off. It has to allow potential new physics to reveal itself, but not enough to dominate the physics at the low energy level. Emphasizing this more modest goal, we do not deny that there are relations between the issues of representation and realism as regards EFT. ²⁷²⁷27We thank an anonymous referee for pointing this out to us. To give an example, Williams’ 2019 interpretation of effective realism in terms robustness (of the fields and parameters of the standard model) can well be understood as the goal of the entire SM-EFT strategy in the sense that while the parameters of the SM are considered robust and fixed from the start, physicists try to find a robust description of BSM physics in terms of one or a few coefficients in the Stage 3 EFT. After all, robustness is also a concept describing experimental data.