Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
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Studies in History and Philosophy
of Modern Physics
journal homepage: www.elsevier.com/locate/shpsb
The case of the composite Higgs: The model as a ‘‘Rosetta stone’’
in contemporary high-energy physics
Arianna Borrelli
IZWT, Bergische Universität Wuppertal, Gaussstraße 20, D-42097 Wuppertal, Germany
a r t i c l e i n f o
abstract
Article history:
Received 18 July 2011
Received in revised form
15 January 2012
Accepted 30 April 2012
Available online 20 July 2012
This paper analyses the practice of model-building ‘‘beyond the Standard Model’’ in contemporary
high-energy physics and argues that its epistemic function can be grasped by regarding models as
mediating between the phenomenology of the Standard Model and a number of ‘‘theoretical cores’’ of
hybrid character, in which mathematical structures are combined with verbal narratives (‘‘stories’’) and
analogies referring back to empirical results in other fields (‘‘empirical references’’). Borrowing a
metaphor from a physics research paper, model-building is likened to the search for a Rosetta stone,
whose significance does not lie in its immediate content, but rather in the chance it offers to glimpse at
and manipulate the components of hybrid theoretical constructs. I shall argue that the rise of hybrid
theoretical constructs was prompted by the increasing use of nonrigorous mathematical heuristics in
high-energy physics. Support for my theses will be offered in form of a historical–philosophical analysis
of the emergence and development of the theoretical core centring on the notion that the Higgs boson
is a composite particle. I will follow the heterogeneous elements which would eventually come to form
this core from their individual emergence in the 1960s and 1970s, through their collective life as a
theoretical core from 1979 until the present day.
& 2012 Elsevier Ltd. All rights reserved.
Keywords:
High-energy physics
Model-building
Composite Higgs
Nonrigorous mathematics
When citing this paper, please use the full journal title Studies in History and Philosophy of Modern Physics
1. Introduction: Model-building ‘‘beyond
the Standard Model’’
The Standard Model of particle physics is a theoretical framework which emerged in the 1970s and has been eminently
successful at explaining and predicting a wide range of experimental observations in the field of particle physics. Despite its
empirical success, though, the Standard Model is considered by
many physicists to be valid only for phenomena involving
energies up to the order of a few TeV, beyond which a still
unknown ‘‘new physics’’ rules. Even though no indisputable
evidence of failures of the Standard Model exists, theorists have
been developing alternatives to it since many decades, and the
rate of creation of models of physics ‘‘beyond the Standard
Model’’ (BSM-physics) is steadily increasing. Theoretical physicists working on alternatives to the Standard Model are often
collectively referred to as ‘‘model-builders’’, and indeed their
main efforts seem to be directed rather at creating new, often
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http://dx.doi.org/10.1016/j.shpsb.2012.04.003
quite sketchy models, then at focussing on one of them to develop
it further into a full-fledged theory.
The complex, fragmented landscape of models of BSM-physics
and the enthusiasm with which model-builders work in expanding
it appear somehow confusing to the external observer. The present
paper shall offer an assessment of the practice of model-building in
today’s high-energy physics and an analysis of the epistemic
dynamics of the development of BSM-models. Particular attention
shall be devoted to the question, whether and how it may be
possible in this context to draw a distinction between ‘‘models’’
and ‘‘theories’’. I shall argue that such a differentiation is not only
possible, but even necessary and that the development of BSMmodels can only be understood as a multi-centered process in
which a number of long-lived, physically meaningful, but only
scantily outlined ‘‘theoretical cores’’ are tentatively instantiated in
fully expendable and easily manipulable models. This happens
with diverse aims: exploring the phenomenological potential of the
theoretical cores; combining two or more of them with each other;
developing and/or testing new mathematical tools to extract
predictions from the cores; understanding better the formal or
physical implications of theoretical hypotheses. In these and other
cases, succeeding in building a model with certain desired features
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A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
provides a nonrigorous proof (‘‘demonstration’’) of the physical–
mathematical validity of the core, even if the model itself is
patently unrealistic. I will argue that theoretical cores cannot be
understood purely in terms of mathematics because they are
constituted by a mixture of (1) mathematical structures, (2) verbal,
narrative components (‘‘stories’’) and (3) analogies referring back
to accepted theoretical explanations of experimental results
(‘‘empirical references’’)—a mixture whose elements have come
to resonate with each other giving rise to a stable, resilient and
flexible core to guide model-building and be in turn constantly
reshaped by it.
I will also attempt to show how the combination of mathematical and nonmathematical elements into hybrid theoretical
structures is a feature which has characterized theorizing in highenergy physics at least since the 1960s, although it has attained a
dominant role only in the post-Standard-Model era. I will argue
that today, although the ideal of a ‘‘final theory’’ as a coherent,
rigorous mathematical construct still features prominently in the
rhetoric of high-energy physics research, theoretical practitioners
engaged in speculating about BSM-physics are more interested in
exploring the properties and potentials of hybrid theoretical cores
than in trying to construct such a final theory. In this sense, I will
suggest that today’s practices of model-building in high-energy
physics have to be understood as a sign that the notion of
‘‘theory’’ as a coherent mathematical construct expressing natural
order de facto faces erosion. The rise in the epistemic significance
of hybrid theoretical practices can be brought in connection with
the increasing use of nonrigorous mathematical methods in
theoretical physics in general, and in high-energy physics in
particular.
The general theses stated above will be illustrated by an
analysis of the development of the class of models having at their
core the notion of a ‘‘composite Higgs boson’’. In the next section,
I offer a brief overview of the different approaches to BSM-physics
developed in contemporary research papers and introduce the
questions and tentative answers at the centre of this paper. In
Section 3, I will connect these reflections with a first-hand
account of the aims and methods of model-building given by a
practising model-builder, Lisa Randall. In Section 4, I will then go
on to expound more in detail the key notions of the paper
(‘‘theoretical cores,’’ ‘‘stories,’’ ‘‘empirical references’’) using as
reference points ideas introduced by Margaret Morrison, Mary
Morgan, Stephen Hartmann, Isabelle Peschard and Andrew Pickering. Section 5 is devoted to discussing the use and epistemic
significance of nonrigorous mathematical heuristics in today’s
high-energy physics, as these issues play an important role in
understanding how the practice of model-building may in some
cases be regarded as a nonrigorous form of proof. After having
introduced my main theses I will provide some evidence in their
support by reconstructing the development of the theoretical core
centering around the idea of a ‘‘composite Higgs boson’’. This
theoretical core emerged at the end of the 1970s from the
combination of a number of elements which had been present
in the high-energy physics discourse since a decade or longer.
Sections 6–10 will deal with events pre-dating the emergence of
the core, with particular attention to the work of Yoshiro Nambu
and Giovanni Jona-Lasinio on what would later be termed
‘‘spontaneous symmetry breaking’’. This discussion will also
provide evidence of the presence of hybrid theoretical practices
of the 1960s and 1970s, and I shall argue that the notion of
theoretical core can find application also in that context. Sections
11–14 will then sketch the rise and further development of the
composite Higgs approach from the late 1970s until the present
day. Section 15 presents a summary of the results, as well as some
tentative speculations on the factors which might be linked to the
transformations of theoretical practices discussed in the paper.
2. The ‘‘new physics’’ in contemporary research papers
Models of new physics can be tentatively collected into a
number of classes corresponding to different approaches. An
exhaustive discussion of these approaches is beyond the aims of
this paper, but a brief sketch of some of the most popular ones is
necessary to formulate the questions which this study will
address. More extensive reviews with bibliographical references
can be found in (Grojean, 2009; Altarelli, 2010; Bustamante, Cieri,
& Ellis, 2009). Among the most popular avenues to new physics
we find:
1. Extending the Standard Model by additional particles. Without
changing the essential elements of the Standard Model, it is
possible to add a new symmetry or new particles to it. Of
course, to explain the fact that these additional particles have
not yet been detected, they must be assumed to either have
high masses, or be very weakly coupled to known particles, so
that the probability of creating them in a high-energy collision
is very small.
2. Supersymmetry. In quantum field theory, each particle falls into
one of two categories of quantum-statistical behaviour: fermions or bosons. In supersymmetric models the Standard
Model is extended so as to make it symmetrical with respect
to a transformation changing bosons into fermions and vice
versa. Since, however, no such symmetry is actually observed
in nature, it must be assumed that supersymmetry is broken
and that all ‘‘supersymmetric partners’’ of known particles
have such a high mass, that they have not yet been produced
in high-energy experiments.
3. Composite Higgs. In the Standard Model, the Higgs boson is an
elementary particle, but theorists have been producing models
in which it is a composite state. Later on, we shall take a closer
look at this class of models.
4. Extra dimensions. Despite clear observational evidence that our
space–time is 4-dimensional, physicists have long been experimenting with the idea that one or more as-yet-unremarked
dimensions exist. While initially confined to extensions of
general gravitation and to string theory, this notion was taken
up by mainstream high-energy theorists at the end of the
1990s and has since been a valuable model-building technique.
There are very many subcategories to this approach, depending
for example on the number of extra-dimensions or on the way
in which they are made ‘‘invisible’’ under everyday conditions.
These four broad classes by no means exhaust the panorama,
but will suffice for our present purposes. The first thing to note is
that, as already remarked in the introduction, there are no
experimental results contradicting the predictions of the Standard
Model and favouring one or more of the models of new physics. In
fact, a main feature of these models is that, at low energies, they
all mimic the phenomenology of the Standard Model. It is only at
higher, as-yet-unexplored energies, that the ‘‘new physics’’ is
expected to appear. In short, from the empirical point of view
all BSM-models are at present equally speculative. The second
point worth noting is that none of the classes listed above can be
associated to some coherent mathematical construct—a ‘‘theory’’
in a traditional sense—from which the various models are
derived. For example, there is no ‘‘theory of supersymmetry’’,
but only a number of different supersymmetric models (see for
example Drees, Rohini, & Prorbir, 2004). The same applies to the
other approaches, and perhaps the most fruitful way of looking at
the various ‘‘classes’’ of BSM-models is to consider them as
loose clusters whose elements may gravitate around one or
more central ideas such as supersymmetry, extra space–time
dimensions or the notion of a composite Higgs boson. Before
A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
trying to characterize more in detail these ‘‘central ideas’’ it is
important to note that, for the most part, they are not mutually
exclusive. In fact, the last years have seen an increasing tendency
to combine two (or even more) of the overarching approaches in
the same model, building for example supersymmetric models
with extra space–time dimensions. Moreover, if one looks at the
work produced recently by some prominent model-builders, one
notices that most of them have not focussed on developing one
model, but have rather proposed a large numbers of variations on
one or more themes. The start of the Large Hadron Collider at
CERN, which is expected to bring about evidence of new physics,
has apparently increased the pace of model building, as jokingly
remarked in an overview paper:
‘‘The prospect of imminent Higgs discovery concentrates
wonderfully the minds of theorists, and many theorists with
cold feet are generating alternative models, as prolifically as
monkeys on their laptops. These serve the invaluable purpose
of providing benchmarks that can be compared and contrasted
with the SM Higgs’’ (Bustamante et al., 2009, p. 21).
Models are here characterized as ‘‘benchmarks’’, a term often
used to indicate constructs which are not regarded as a serious
candidate for a theory, but allow quantitative predictions which
give at least an approximate idea of how phenomena in that
general direction of new physics might look like.
A further feature that strikes a first-time reader of papers on
BSM-models is how the authors attempt to underscore the
novelty and originality of their models by giving them inventive,
sometimes funny names. In an overview paper on ‘‘New Theories
at the Fermi Scale’’ (2009) the theorist Christoph Grojean comments on the number of attributes which have been invented to
describe hypothetical alternative means of breaking the electroweak symmetry:
‘‘Theorists have always been very good at giving names to
things they do not understand. And clearly the EWSB [electroweak symmetry breaking] sector has been an inspirational
source of creativity to them, as it is evident by collecting the
attributes that have been associated to the Higgs boson over
the last few years [y]: buried, charming, composite, fat,
fermiophobic, gauge, gaugephobic, holographic, intermediate,
invisible, leptophilic, little, littlest, lone, phantom, portal,
private, slim, simplest, strangephilic, twin, un-, unusual,y’’
(Grojean, 2009 p. 3, see there also for the references of the
research papers where the various terms appear).
In general, the attitude of model-builders towards their creations
suggests that, in their eyes, what is being built are ‘‘only’’ models
to be later modified or discarded without much regard. In short,
these models seem to enjoy a lower epistemic status than would
in general parlance be associated to a ‘‘theory’’. For comparison,
we may look at those high-energy theorists who are today usually
regarded as a category distinct from model-builders: string
theorists. The majority of string theorists have a very strong
commitment to strings ontology and their most prominent
representatives do no hesitate to propound them as the ‘‘theory
of everything’’ in the strongest possible terms (Susskind, 2005;
Hawkins & Mlodinow, 2010). Yet, if in string theory there are only
‘‘theories’’, in model building there seem to be only ‘‘models’’ and
therefore one might be justified in regarding them as nothing but
theories under another name—perhaps theories which are still
incomplete or lack empirical backing, but theories nonetheless. In
both cases, the distinction between theories and models would
seem to fade. Is it so? I shall argue that this is not the case, and
that it is possible to draw a very clear distinction between models
and theories of new physics, provided one is ready to give up the
197
idea that theoretical constructs must have a purely mathematical
character. Let us now go back to the question of how ‘‘central
ideas’’ like supersymmetry or the composite Higgs approach may
be characterized. Although they are not coherent mathematical
structures, they do contain mathematical elements, such as the
supersymmetric transformations turning boson into fermions and
vice versa, or the methods used to make extra-dimensions
‘‘disappear’’. However, these elements are only bound into a
(more or less) coherent mathematical whole at the level of
specific, individual models. The central ideas also contain elements which are purely qualitative and expressed only in verbal
terms, such as the generic notion that supersymmetry must be
broken (as opposed to the many specific methods of actually
breaking it) or the concept of extra space–time dimensions (as
opposed to specifying how many of them and with what kind of
metric). Mathematical and nonmathematical elements are only
loosely bound together, so that it is possible to build models
which combine ideas of different origin and therefore belong to
two or more clusters.
Despite their very vague character, the central ideas implemented in the models command from physicists more respect
and commitment than the models themselves. For example,
supersymmetry became widely accepted as one of the most
promising approaches to BSM-physics in the 1980s and has since
remained one of the prime candidates, despite the fact that longawaited experimental evidence in its favour has until now failed
to turn up (Drees et al., 2004, p. 4). The situation in the case of the
composite Higgs approach is similar, as we shall see later on:
Many models have come and gone, but the central idea survived.
In the end, one cannot help but conclude that although ideas like
supersymmetry, a composite Higgs or extra-dimensions are not
coherent mathematical structures, their elements are seen by
physicists as containing valuable clues to the workings of nature.
It is to characterize these central ideas and their elements that I
shall introduce the notion of a ‘‘theoretical core’’ and its mathematical and non-mathematical components, reserving the term
‘‘theory’’ to indicate coherent mathematical structures which lay
a claim at explaining and predicting a broad, possibly infinite
range of phenomena. Such a clear-cut distinction is valid only in a
first approximation, but is necessary to avoid ambiguities in the
following discussion. When looking at specific examples we shall
however see how the border between ‘‘models,’’ ‘‘theories,’’ and
‘‘theoretical cores’’ is permeable. For example, I will argue that the
Weinberg-Salam model of electroweak interactions emerged as a
model implementing a theoretical core, but eventually ended up
as a theory in its own right. According to the terminology chosen,
though, the Standard Model, the BCS theory of superconductivity
and general relativity are today examples of existing theories,
while the ‘‘theory of everything’’ of which some theoretical
physicists speak is a (still) nonexisting one (‘t Hooft et al. 2005).
BSM-models, on the other hand, although they are usually
coherent, if sketchy, mathematical constructs, are not regarded
by physicists as starting points for a theory to explain and predict
phenomena, but rather as expendable tools to explore the
potentialities of theoretical cores. For example, the minimal
supersymmetric standard model (MSSM) is a quite complex
structure employed by theorists as a convenient means to
formulate predictions and test hypotheses about supersymmetry
(a ‘‘benchmark’’), but is today hardly regarded as a candidate for
the final theory (Drees et al., 2004, pp. 251–252).
3. Model-building as a search for ‘‘nouns and phrases’’
In the previous section, I made some claims about the practice
of model-building in the search for BSM-physics, and these
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A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
reflections will now be compared with an account of this practice
as described by one of its most prominent contemporary practitioners: Lisa Randall, a theoretical physicists from Harvard
University best known for the BSM-model she proposed together
with Raman Sundrum, a model in which space–time is assumed
to be 5-dimensional and to posses a ‘‘warped’’ metric (Randall &
Sundrum, 1999). In her book ‘‘Warped Dimensions’’, an account of
her research addressed at a broad audience, Randall endeavours
to convey to the reader both her enthusiasm for and open
questions about the notion of extra-dimensions (Randall, 2006).
She also offers an open and demystifying description of the
practice of model-building which can provide us with some
valuable insights for the following discussion:
‘‘Particle physics models are guesses at alternative physical
theories that might underlie the Standard Model. If you think of
a unified theory as the summit of a mountain, model builders
are trailblazers who are trying to find the path that connects the
solid ground below, consisting of well-established physical
theories, to the peak—the path that will ultimately tie new
ideas together. [y] Models that go beyond the Standard Model
incorporate its ingredients and mimic its consequences at
energies that have already been explored, but they also contain
new forces, new particles and new interactions that can be seen
only at a shorter distances. [y] Model builders look at the
unresolved aspects of the Standard Model and try to use known
theoretical ingredients to address its inadequacies. [y] Model
builders try to see the big picture so they can find the pieces
that could be relevant to our world.[y] Only some models will
prove correct, but models are the best way to investigate
possibilities and build up a reservoir of compelling ingredients’’
(Randall, 2006, pp. 70–73).
In this passage the search for a ideal ‘‘unified theory’’ features
prominently as the ultimate motivation for model-builders’ work,
yet model-builders are not directly looking for such a theory, but
rather search for a ‘‘path’’ to it and for ‘‘theoretical ingredients’’,
‘‘pieces that could be relevant to our world’’ to be put in a
‘‘reservoir’’. I would like to argue that this intermediate level
between the (many) models and the (one) theory corresponds to
the ‘‘theoretical cores’’ introduced in Section 2.
Randall’s distinction between a ‘‘unified theory’’ on the one
side and the collection of ingredients in a ‘‘reservoire’’ on the
other may be seen as corresponding to the difference between a
mathematically coherent proposal for a final theory and the
central idea(s) shared by the models in a cluster. As I have shown
in the previous section, supersymmetry and other approaches to
BSM-physics are loose collections of elements of different nature
and are indeed similar to ‘‘reservoires’’ of promising ingredients
for a future theory. At the same time, as Randall notes, ‘‘only some
models will prove correct, but models are the best way to explore
possibilities and build up a reservoire of compelling ingredients,’’
so that the relation between models and ‘‘reservoires’’ appears
one of mutual feedback: Models are built from reservoire ingredients, but also serve to enlarge the reservoire. For example,
supersymmetric transformations can be implemented in many
different models to explore new methods of supersymmerybreaking to be collected.
The models’ primary aim is not to become the seed of a
unified theory, and in this point Randall sees the main difference
between model-builders and string-theorists: The latter ones, she
claims, are actually aiming directly at building a unified theory.
To express the difference Randall makes use of a metaphor based
on the age-old notion of the ‘‘language of nature’’:
‘‘You might say that we are all searching for the language of
the universe. But whereas string theorists focus on the inner
logic of the grammar, model-builders focus on the nouns and
phrases that they think are most useful. If particle physicists
were in Florence learning Italian, the model builders would
know how to ask for lodging and acquire the vocabulary that
would be essential to finding their way around, but they might
talk funny and never fully comprehend the ‘Inferno’’’ (Randall,
2006, p. 73).
Randall is here using the language analogy to indicate mathematical structures of different complexity and various degrees of
interconnectedness: While the ‘‘grammar’’ of string theory
imposes strong constraints on their practitioners, model-builders
are free to collect scattered mathematical ‘‘nouns and phrases’’ to
be later variously linked with each other—for example supersymmetric transformations or extra-dimensional metrics. As I
have argued above, though, theoretical cores also contain nonmathematical elements and these, too, should in my opinion be
regarded as ‘‘nouns and phrases’’ in Randall’s sense—although
this was probably not what she had in mind when using this
expression. I shall presently say more about the characteristics of
nonmathematical elements of theoretical cores, and I will argue
that among them verbal narratives constitute an essential component, so that it may not be a pure chance that Randall chose a
linguistic metaphor to express her thoughts. Indeed, as we shall
see, deciphering the book of nature has also been likened by
theorists to the search for a ‘‘Rosetta stone.’’
Randall’s exposition of the results of BSM-research has the same
rhapsodic character as model-building practices: After having
summarized the development of physics in the twentieth century
up to the creation of the Standard Model, she devotes the remaining chapters of the book to various ‘‘names and phrases’’ which
were collected in the last decades: supersymmetry (chapter 13),
strings (chapter 14), ‘‘branes’’ (i.e., multidimensional membranes in
extradimentsional space–time, chapter 15), and finally a number of
alternative ‘‘proposals for extra-dimensional universes’’ like
‘‘sequestering’’ (the localization of some particles on a brane,
chapter 17), ‘‘Kaluza–Klein particles’’ (4-dimensional manifestations of particles existing in extra-dimensions, chapter 18), ‘‘large’’
extra dimensions (i.e., of the order of a millimeter, chapter 19), and
of course the two different models of ‘‘warped’’ extra dimensions
proposed by herself together with Sundrum (chapters 20 and 22).
Beginning each section with a short science-fiction-like narrative
inspired by the notion to be discussed, Randall expounds the pros
and contras of the various approaches and, although she makes
clear her preferences, she does not attempt to present any one of
them as the best candidate for a ‘‘unified theory,’’ leaving the
decision up to experiment and appropriately naming the last
chapter of the book ‘‘(In) Conclusions.’’
Randall clearly sees a contrast between the practices of string
theory and those of model-building and seems to imply that
string theorists have more interest in mathematically coherent
formulations, as opposed to the fragmentary creations of modelbuilders. Indeed, the interest of string theorists for complex
mathematics—and their disinterest for experiment—has often
been noted in the physics community. An analysis of the practices
and values of string theorists has been given by Galison (2004),
who has convincingly argued that in the last decades string
theory has effected a ‘‘shift in values in physical research away
from the constant interplay between theory and experiment’’ and
to a ‘‘realignment of theory towards mathematics’’ (Galison, 2004,
pp. 24–25). However, Galison’s study has also shown that the
string theorists’ approach to mathematics radically differs from
the one of mathematicians, since string theorists—and physicists
in general—make use of nonrigorous methods, following their
‘‘hunches’’ and ‘‘intuitions’’ rather then accepted standards of
mathematical proof (Galison, 2004, p. 35, 53). In short, what
A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
Randall calls the ‘‘grammar’’ of string theory is certainly more
refined than the ‘‘nouns and phrases’’ of model-building, but is
usually no rigorous mathematical structure either. Significantly,
Galison has characterized the practices of string theorists as a
‘‘hybrid’’ of physics and mathematics (Galison, 2004, pp. 25, 53,
60) and, as we shall see in Section 5, the use of nonrigorous
mathematical heuristics on the part of theoretical physicists is a
key element not only in string theory but also in model-building.
In this sense, some of the reflections of the present paper may also
apply to string theory, yet the present investigation is concerned
only with model-building, and I will attempt no assessment of
whether the notion of theoretical core may also be applied in the
context of string theory.
4. Theoretical cores and their mathematical and
nonmathematical components
To offer a more detailed characterization of models, theoretical
cores, and their mutual relationship, I will take as a starting point
the analysis of Margaret Morrison and Mary Morgan, where
‘‘models’’ are seen as ‘‘both a means and a source of knowledge’’,
fulfilling a mediating function between theory and experiment,
and representing an autonomous method of production of scientific knowledge (Morrison & Morgan, 1999, quote on p. 35). I shall
argue that the two levels between which BSM-models mediate
are, on the experiment side, particle phenomena as represented
by the Standard Model and, on the theory side, theoretical cores,
i.e. composite constructs which only display partial mathematical
character. Although the modus operandi of model-builders may
formally still be motivated by the idea of a final theory, so little
actual commitment to the feasibility of that aim is present in the
practice of model-building that one might argue that they
represent an implicit shift in the attitude to what ‘‘scientific
knowledge’’ is taken to be—a shift in which the notion of theory
faces erosion, at least in the traditional form paradigmatically
represented by classical mechanics, electromagnetism, relativity
or quantum mechanics. In other words, I would like to claim
that—de facto if not de jure—in the practice of model-building
the notion of theoretical core has replaced the idea of theory.
However, this transformation does not in any way mean a shift
toward a lesser commitment to some form of realism on the part
of (theoretical) physicists.
As remarked in the previous sections, theoretical cores contain
elements of nonmathematical character and these are of two
kinds: On the one side narratives (‘‘stories’’) and, on the other,
analogical references to empirical results which are not in
themselves relevant for the core, but provide evidence that
similar theoretical approaches have been successful in other fields
(‘‘empirical references’’). While stories are expressed verbally,
empirical references may be formulated using not only words, but
also images, diagrams or mathematical formulas. Let us first turn
to the narrative components: I employ the term ‘‘story’’ to
indicate them, as I believe that they can indeed be described so
according Stephan Hartmann’s terminology (Hartmann, 1999). In
a study on phenomenological models, Hartmann argued that rival
models may enhance their appeal by means of a good story:
‘‘A story is a narrative told around the formalism of the model.
It is neither a deductive consequence of the model nor of the
underlying theory. It is, however, inspired by the underlying
theory (if there is one). This is because the story takes
advantage of the vocabulary of the theory (such as ‘gluon’)
and refers to some of its features (such as its complicated
vacuum structure). Using more general terms, the story fits
the model in a larger framework (a ‘world picture’) in a
199
non-deductive way. A story is, therefore, an integral part of a
model; it complements the formalism. To put it in a slogan: a
model is an (interpreted) formalismþa story’’ (Hartmann, 1999,
p. 344, italics in the original).
Although it was developed for phenomenological models, the
notion of ‘‘story’’ can be very fruitful also when applied at the
‘‘theory’’ level of Morgan and Morrison’s scheme: We may
conceive a theoretical construct as composed both of mathematical elements and of stories complementing them. Out of the
interplay of these two factors, a theoretical construct emerges
which is not a full-fledged mathematical theory, but rather a
theoretical core.
The term ‘‘theoretical core’’ was suggested in 2007 by Margaret Morrison, in a paper asking ‘‘Where have all theories gone?’’
and arguing that a more flexible notion of theory should be
adopted when studying the reality of scientific practice (Morrison,
2007). Morrison stated that, after focussing mostly on the epistemological role of models, ‘‘The time has come to bring theories
back into the picture and attempt a reconstruction of the relation
between models and theories that emphasizes a distinct role for
each’’ (Morrison, 2007, p. 196). Morrison proposed to differentiate
between models and theories using ‘‘the notion of a theoretical
core: a set of fundamental assumptions that constitute the basic
content of the theory, as in the case of Newton’s three laws and
universal gravitation. This core constrains not only the behaviour
but the representation of objects governed by the theory as well
as the construction of models of the theory.’’ (Morrison, 2007,
pp.197–198). According to her view, this characterizations avoids
the problem of having to include in a theory all mathematical
techniques which may be required to expound or apply it, and
‘‘nothing about this way of identifying theories requires that they
be formalized or axiomatized’’ (Morrison, 2007, p. 205). Using the
theory of superconductivity as an example, Morrison stated that
‘‘it refers to a well-defined core that involves the notion of pairing
and a connection with the broader theoretical principles of
spontaneous symmetry breaking, both of which constrain the
way superconducting phenomena can be represented’’ (Morrison,
2007, p. 226).
While Morrison only suggested as an aside that stories may
provide a relevant element of theoretical cores, Isabelle Peschard
focussed precisely on this aspect and introduced ‘‘the notion of a
theoretical story as a resource and source of constraint for the
construction of models of phenomena,’’ aiming to ‘‘show the
relevance of this notion for a better understanding of the role
and nature of values in scientific activity’’ (Morrison, 2007,
pp. 220, 224; Peschard, 2007, p. 151). Peschard argued that
conceiving theories (also) as stories allows to characterize the
theoretical coherence of the models connected to them without
depending on the existence of mathematical principles, and also
to account for ‘‘the way in which a theoretical framework guides
and constrains the construction of models’’ (Peschard, 2007,
p. 156). Her conclusion is that theoretical stories represent
cognitive values:
‘‘The theoretical story guides and constrains the construction
of models by prescribing what should be accounted for, what
should be taken into account, what are the things that are
relevant to the understanding of a domain of phenomena,
what features they have, what kind of behaviour these things
can have or kind of relations there can be between the features
of the different things that are involved’’ (Peschard, 2007,
p. 166).
The cases studied by Peschard are much different from those
which are the subject of the present work, and my study is no
rigorous application of her approach. However, I find her ideas
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highly inspiring and in the following pages I shall underscore how
the stories which came to be associated to the composite Higgs
approach played a key role in defining what features the models
of a class should possess and how stories contribute to unify the
many different models despite the lack of a rigorously defined
common mathematical element.
Beside the mathematical and narrative components there is a
third element which is essential for the coming-to-be of a
theoretical core in today’s high-energy physics: The reference to
observed phenomena whose generally accepted explanation can
be interpreted in analogy to the ideas characterizing the core in
question. I will refer to this kind of element as an ‘‘empirical
reference.’’
To introduce this notion, let us again consider the case of
supersymmetry: Its theoretical core contains the idea that
particle interactions are approximately invariant with respect to
supersymmetric transformations and that therefore the ‘‘superpartners’’ of all known particles await discovery at higher energies. Since the mechanism of supersymmetry-breaking is
unknown, the masses of these hypothetical particles cannot be
exactly predicted, yet their interaction properties are for the most
part determined by those of Standard Model particles. This
picture combines the mathematics of supersymmetry with the
story of its breakdown and of the existence of a supersymmetric
world awaiting discovery. Its plausibility is supported by the
knowledge that, since the 1960s at the latest, identifying groups
of particles as quasi-multiplets of a broken symmetry has led to
successful predictions of ‘‘missing’’ members of such multiplets
and eventually to a deeper understanding of elementary interactions. The phenomenological successes of the ‘‘broken symmetry’’
idea do not imply that this approach will work also in the case of
supersymmetry, but they help such a scenario appear more
plausible. In this sense they constitute an important ‘‘empirical
reference’’ for the theoretical core of supersymmetry.
The importance of analogical references to previous empirical
successes had already been discussed by Pickering (1981), who
used Thomas Kuhn’s remarks on ‘‘exemplars’’ as a starting point:
‘‘Kuhn used the term ‘exemplar’ as a shorthand for ‘shared
example’. He emphasized that such a shared example derives
from the concrete demonstration in some practical situation of
the utility of a cultural product—a new experimental technique, a new theoretical model, or whatever—and that it is
precisely through such demonstrations that new concepts are
related to the natural world and acquire meaning [y] For the
purposes of the present work, an exemplar can be regarded as
an analogy drawn between some new aspect of the subject
matter of the field and some other field of discourse’’
(Pickering, 1981, p. 108, italics in the original).
Pickering expounded his idea using an example from high-energy
physics of the 1970s: The choice between ‘‘colour’’ and ‘‘charm’’
as an explanation for an observed phenomenon, i.e., the resonance J/psi. It would bring us too far to summarize here Pickering’s historical case study, but it is worth mentioning his
conclusions: Around 1975, neither the ‘‘colour’’ nor the ‘‘charm’’
hypothesis was clearly favoured by experimental evidence, but
the latter idea was supported in the eyes of most physicist by the
fact that the J/psi could be represented as a bound state of a
charmed quark with its antiquark by using a simple mathematical
model (‘‘charmonium’’) analogous to the one employed in nonrelativistic quantum mechanics:
‘‘In particular, the charmonium model depended upon an
image of a charmed quark non-relativistically orbiting its
antiparticle under the influence of a central potential, in exact
analogy with the atomic system of an electron orbiting a
positron (the antiparticle of the electron). The latter system
is known as positronium, and is one of the textbook applications of quantum mechanics to atomic physics, being almost
identical to the hydrogen atom in its formal treatment. The
name charmonium was coined to make explicit the parallel
between the envisaged structure of the new particles and the
simplest and best understood atomic structure [y] The
charmonium model illustrates perfectly the role of an exemplar as the concrete embodiment of an analogy relating a new
field of research back to an established body of practice. The
model was intuitively transparent to any trained physicist, and
this had two important consequences. Firstly, whenever the
model encountered a mismatch with reality the resources
were available to essentially anyone to attempt to fix it up
[y] Another source was less direct but more fundamental:
namely, the charmonium model made quarks themselves
‘real’’’ (Pickering, 1981, pp. 124–125).
Although I do not wish to fully identify ‘‘empirical references’’
with Pickering’s ‘‘exemplars,’’ which have a broader scope, I
believe that in both cases analogies ‘‘referring back’’ to established knowledge play a key role in promoting new theoretical
notions in the physics community, as the analogies suggest ‘‘old’’
ways of dealing with ‘‘new’’ concepts and let the latter appear
more intelligible. Moreover, in the case of charmonium we also
find another element which, as we shall see, has an important
function in supporting theoretical cores: the suggestive choice of
the names of their elements. Finally, like charmonium, empirical
references often involve both mathematical structures and stories, but their function within the core is primarily that of offering
a physically certified, analogical template for the physical significance of speculative ideas of the core. Therefore, an empirical
reference has to be regarded as a whole and as such plays a role in
supporting the cognitive values of stories. As we shall see in detail
later on, the explanation of the pion as a bound state of a quark
and an antiquark provided a very important template for conceiving the Higgs boson as a composite state of two hypothetical
elementary fermions.
To sum up, I would like to suggest that the various approaches
to new physics should be regarded as theoretical cores characterized by a combination of mathematics, empirical references and
stories. The notion of a theoretical core shall allow us not only to
draw a distinction between the level of theories and that of
models, but also to understand how the practice of modelbuilding contributes to the development of theoretical structures
in high-energy physics. Models function as mediating instances
between theoretical cores and the Standard Model, which in turn
represents well-established experimental results. Models are
built as explorative tools working at different levels: to extract
from a core quantitative estimates about observable parameters;
to understand whether a mathematical method leads to physically relevant results when employed under the guidelines of a
given story; or to test whether and how elements from a core can
be implemented or combined with those of another one. A word
of caution has to be added at this point: The distinction between
‘‘mathematical’’, ‘‘verbal’’ and ‘‘empirical’’ elements of a theoretical core is a powerful heuristic tool, but is not to be understood
in the sense that a core can be taken apart into three groups of
components. Quite the contrary: The theoretical core only
emerges from the interplay of all three elements. Despite their
hybrid character theoretical cores are coherent constructs commanding respect and commitment on the part of physicists. This
fact hardly comes as a surprise to those familiar with today’s
theoretical practices in high-energy physics and cosmology,
where the use of nonrigorous mathematical methods is the rule
rather than the exception. The next section shall be devoted to
A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
discussing some testimonies by theoretical physicists and mathematicians on this point, as well as on the various non-mathematical elements which are constitutive ingredients of physical
theorizing. Understanding the kind of practice in which modelbuilding is embedded allows to grasp how a hybrid theoretical
core may play a role in many ways equivalent to that of a
mathematically well-defined theory. Moreover, it is only by
taking into account nonrigorous practices that the epistemic
function of the models being built can be understood: Their value
lies primarily in the fact that they can deliver nonrigorous, but
convincing ‘‘proofs’’ of the validity of hypotheses concerning
theoretical cores. This point will be the subject of the next section.
5. Nonrigorous mathematical methods and the probative
value of models
According to what we have discussed in the previous sections,
models of BSM-physics could in a way be thought of as toymodels, to be set aside soon after they have served their
immediate aim. However, it would be wrong to believe that
model-builders consider their creations as ‘‘toys’’ in the sense of
constructions helpful only for an approximate assessment of a
situation, or to be used for didactic purposes: As far as their
specific explorative task is concerned, models represent the most
serious instance of proof available to researchers. To understand
better how models may come to be regarded as having probative
value with respect to hybrid theoretical cores, we have to take a
closer look at the employment of nonrigorous mathematical
methods in theoretical physics. In the construction of physical
knowledge there is a long tradition of using mathematical
methods which are considered nonrigorous by the standards of
the time, both in defining expressions and in proving their
validity (Grosholz, 2007). Prominent examples can be found in
celestial mechanics (George William Hill’s variational methods),
electromagnetism (Oliver Heaviside’s operational calculus) and
quantum theory (infinite matrices, d-funtion) (Bernkopf, 1968,
pp. 318–320; Borrelli, 2010; Daley, 2003; Lützen 1979; Peters,
2011). In many—yet not in all—cases physically significant
results which were obtained non-rigorously could eventually be
reformulated in mathematically acceptable terms, and these
successes have bolstered the physicists’ rather lax attitude to
mathematical proof:
‘‘Mathematicians are far more interested in finding rigorous
proofs, whereas physicists, who use mathematics as a tool, are
usually happy with a convincing argument for the truth of a
mathematical statement, even if that argument is not actually a
proof. The result is that physicists, operating under less stringent constraints, often discover fascinating mathematical phenomena long before mathematicians do’’ (Gowers, 2008, p. 7).
In fact, one may add that theorists often have little choice but
accept mathematical risks if they want to pursue their physics
interests.
While it is not my intention to discuss here (or elsewhere)
normative issues regarding the use of mathematics in physics, I
wish to note how physicists, although they do not in general
disregard mathematical rigor and apply it whenever possible, are
also at times ready to employ ill-defined expressions or base their
work on ‘‘folk theorems,’’ which in the words of Steven Weinberg
are ‘‘things that have never been proven, but are known to be
true’’ (Weinberg, 1985, p. 125; Schweber, 1993, p. 162). Weinberg
attributes this expression to Arthur Wightman, the mathematical
physicists who devoted himself to the development of a rigorous
version of quantum field theory, yet mathematicians use this
term rather to indicate theorems whose proof has not yet been
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published, and it was in this sense that Wightman for example
spoke of a ‘‘folk lemma’’ (Wightman, 1973, p. 473). Weinberg was
instead largely responsible for the diffusion of the term among
theoretical physicists to indicate statements that the community
regards a true, even though no proofs are known for them: ‘‘These
theorems can probably be formulated in precise terms, and
proved, though I haven’t done it’’, wrote Weinberg of two folk
theorems (Weinberg, 1985, p. 125; see also Weinberg, 1997). This
attitude is ambiguous, since the existence of a proof for folk
theorems is not doubted, and is indeed regarded as a necessary
requirement for the validity of the physical theories built upon
them, while at the same time the grounds on which the conviction of such existence rests remain obscure.
The use of folk theorems and of nonrigorous heuristics has
become increasingly common among high-energy physicists, and
is generally regarded by them as an unproblematic issue to be
either glossed over or underpinned with unconditional statements of belief in the truth-content of science such as the one
quoted above. However, by the 1990s at the latest, such attitudes
to mathematical rigor had started spreading also among mathematicians: As discussed by Galison (2004), one of the main factors
in this development was the rise of string theory, whose practitioners combined the nonrigorous methods of theoretical physics
(e.g., path integrals, Feynman diagrams) to physical arguments in
the search for new mathematical structures which they believed
would provide a key to understanding nature. As we saw in
Section 3, Galison has underscored how the development of string
theory created a new ‘‘border region’’ between physics and
mathematics characterized by hybrid research objects:
‘‘With the new sense of theoretical physicist and geometer
came also a new object of inquiry: in its present form not quite
mathematical and not quite physical, either. One day pieces of
such entities may be folded back into physics or into geometry,
but at the century’s end they were conceptual objects, hugely
productive and yet seen with discomfort by purists in both
camps.’’ (Galison, 2004, p. 60, italics in the original).
Galison explained at length how the new hybrid conceptual
objects were perceived by ‘‘purists’’ as differing both from
mathematical structures and from empirically-based physical
notions, but he did not discuss in detail whether they were
expressed only through nonrigorous mathematical formalisms,
or also in fully nonmathematical (e.g., verbal or visual) terms. I
would like to suggest that these string-theory hybrids are closely
related to—although not identical with—theoretical cores of
BSM-physics. Both kinds of theoretical constructs are born out
of the humus of nonrigorous mathematical practices which has
come to play an increasingly central role in the whole of highenergy physics. Under these premises, to shed some light on the
practice of model-building it is important to learn more about the
epistemic status of nonrigorous heuristics among theorists and
mathematicians. This issue is rarely if ever explicitly addressed
within the context of model-building, but we can learn something
about it thanks to a dispute between mathematicians and
theoretical physicists whose starting point was a paper written
in 1993 by Artur Jaffe, a former PhD. student of Wightman who,
like his advisor, had devoted much effort to the development of a
rigorous version of quantum field theory, and Frank Quinn, a
topologist (Jaffe & Quinn, 1993). The paper was published in the
Bulletin of the American Mathematical Society with the title:
‘‘Theoretical mathematics: towards a cultural synthesis of mathematics and theoretical physics’’. As the place of publication shows,
the article was directed at mathematicians and mathematical
physicists, and was prompted by the spread in the mathematical
community of the non-rigorous practices of ‘‘proof’’ common
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A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
among theoretical physicists and computer scientists. The authors
stated their views clearly in the abstract:
Morris Hirsch, a topologist who had been Thurston’s PhD
advisor, also made some remarks which are very relevant for
our subject, stating:
‘‘Is speculative mathematics dangerous? Recent interactions
between physics and mathematics pose the question with
some force: Traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice
there can be benefits, but there can also be unpleasant and
destructive consequences. Serious caution is required, and the
issue should be considered before, rather than after, obvious
damage occurs. With the hazards carefully in mind, we
propose a framework that should allow a healthy and positive
role for speculation’’ (Jaffe & Quinn, 1993, p. 1).
‘‘The nonrigorous use of mathematics by scientists, engineers,
applied mathematicians and others, out of which rigorous
mathematics sometimes develops, is in fact more complex than
simple speculation. While sloppy proofs are all too common,
deliberate presentation of unproven results as correct is fortunately rare. Much more frequent is the use of mathematics for
narrative purposes [italics in the original]. An author with a story
to tell feels it can be expressed most clearly in mathematical
language. In order to tell it coherently without the possibly
infinite delay rigor might require, the author introduces certain
assumptions, speculations and leaps of faiths [y] In such cases
it is often irrelevant whether the mathematics can be rigorized,
because the author’s goal is to persuade the reader of the
plausibility or relevance of a certain view about how some real
world system behaves.’’ (Atiyah et al., 1994, p. 186).
Jaffe and Quinn advocated a clear-cut, explicit separation between
‘‘rigorous’’ and ‘‘theoretical’’ mathematics, where the latter was to
be understood as encompassing both the conjectures and folk
theorems of theoretical physicists and the numerical demonstrations of hypotheses practised by computer scientists. They shared
Weinberg’s (and indeed most physicists’) view that the ultimate
aim of theoretical physics research is the construction of theories
of rigorous mathematical character. However, other than Weinberg, they were not ready to belittle or even ignore the gap
between rigorous ideals and nonrigorous research practices.
The editors of the Bulletin invited a large number of mathematicians and physicists to respond to the paper, and their
answers were published in a subsequent issue of the journal,
together with a reply by Jaffe and Quinn. Those texts addressed all
possible facets of the matter: mathematical, physical, historical,
epistemological, sociological and institutional (Atiyah et al., 1994;
Jaffe & Quinn, 1994; Thurston, 1994). Galison has discussed some
of these texts with particular attention to the interplay of
different sets of ‘‘values,’’ here I will instead focus on some
passages shedding light on the hybrid character of the constructs
with which theorists and some mathematicians work, and on the
ambiguous attitude of physicists with respect to rigorous proofs
and their nonrigorous surrogates. William Thurston—a topologist
and computer scientist whose conjectural ‘‘geometrization theorem’’ Jaffe and Quinn had used as a ‘‘cautionary tale’’ against the
use of nonrigorous methods—wrote not just a letter, but a whole
paper ‘‘On proof and progress in mathematics,’’ where he discussed in depth and detail the issue of communication strategies
within mathematics (e.g., verbal, visual, kinaesthetic, logical and
more) and their decisive role in shaping the notion of validity of
mathematical statements. He related his own experience of
difficulties in communicating results to different mathematical
communities and recounted how he had come to recognize that
‘‘mathematical knowledge and understanding were embedded in
the minds and in the social fabric of the community of people
thinking about a particular topic. This knowledge was supported
by written documents, but the written documents were not really
primary.’’ (Thurston, 1994, p. 169). He also claimed that ‘‘Most
mathematicians adhere to foundational principles that are known
to be polite fictions,’’ quoting as an example the well-ordering of
real numbers and problematized the notion of ‘‘proof,’’ opposing
‘‘formal proofs’’ to human understanding of mathematical validity: ‘‘For the present, formal proofs are out of reach and mostly
irrelevant: we have good human processes for checking mathematical validity’’ (Thurston, 1994, pp. 170–171). Thurston characterized formal proofs as ‘‘mostly irrelevant,’’ but did not in
principle doubt their existence (they were just ‘‘out of reach’’):
This is the same ambiguous attitude we already encountered in
Weinberg’s statements because, as underscored by Jaffe and
Quinn, it can never be sure that a formal proof exists until it
has been found.
This passage can be read as a describing those aspects of the
theoretical practices which are the subject of the present study:
Tn the theoretical cores of BSM-physics mathematical fragments
such as supersymmetric transformations or extra-dimensional
metrics are embedded into ‘‘narratives’’ containing also elements
of nonmathematical character, such as stories and empirical
references. The aim is to persuade the audience of the physical
relevance of a given theoretical core and, as we shall see, the
method is indeed fruitful. Thus we see how a connection exists
between the employement of nonrigorous mathematical heuristics and the increased importance of theoretical constructs involving nonmathematical elements.
There is a further detail in Hirsch’s statement which is worth
noting: Unlike Thurston and Weinberg, he states explicitly that it
can be ‘‘irrelevant whether the mathematics can be rigorized.’’
From this point of view, nonrigorous demonstrations are not only
qualitatively different, but also epistemically independent from the
rigorous proofs which may or not exist. In my opinion Hirsch’s
words offer an accurate characterization of the actual practices of
theoretical physicists, where nonrigorous proofs are perceived as
valid independently of the existence of formal ones, despite the
fact that most physicists, when asked, stick to the idea that their
constructs can be transformed into rigorous ones. To underscore
this distinction, in the following pages I shall make use of the terms
demonstration/demonstrate—as opposed to proof/prove—to indicate arguments which have probative value in the context of
model-building, although they do not constitute a proof in the
rigorous mathematical sense. The perceived epistemic independence of demonstrations from proofs is of central importance for
our subject, because it is within this framework that the practice of
model-building becomes endowed with its own specific kind of
validating power. The act of building a model which implements
elements from a core and possesses some desired features (e.g., a
Higgs boson of a given mass) ‘‘demonstrates’’ that a theory with
those features is possible, even though the model itself may have
other characteristics that are clearly unrealistic. In the following
sections I shall discuss a case study spanning the period from the
1960s until today, and we shall see how already in the early years
model building was a powerful means to demonstrate the validity
of theoretical hypotheses and of theoretical cores.
6. The ‘‘composite Higgs’’ approach to BSM-physics as a
theoretical core: A case study
The reflections and notions introduced in the previous sections
will now be illustrated and supported by a case study focussing
A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
on the theoretical core linked to the notion of a composite Higgs
boson. The best way to grasp how the unity of a theoretical core
emerges from the interplay of various elements is to reconstruct
the historical path of its development. In this way, it will become
clear how—under specific conditions—words, formulas and
experimental results may resonate with each other in the perception of the physics community, leading scientists to regard the
network of mutual references between them as a theoretical
unity—a core—endowed with cognitive values and worthy of
being used as a guideline for explorative model-building.
Although experiments play an important role, the emergence of
a theoretical core of BSM-physics cannot be attributed to positive
experimental indications in its favour. At the same time, the
success of a core is not purely due to abstract considerations of
mathematical beauty or to choices of career-opportunitism:
Although these factors certainly have their weight, the process
through which a theoretical core establishes itself is an epistemic
phenomenon involving all facets of the scientific enterprise and
characterized by a peculiar kind of interplay between abstract and
empirical elements. In my case study, I will focus on reconstructing these aspects, leaving aside for the moment the broader social
and scientific context in which they are embedded. This restriction is necessary in the context of the present, exploratory study.
Before discussing the composite Higgs approach, a few words
about the main features and the history of the Standard Model are
in order. The Standard Model is made out of two parts, one
dealing with electroweak interactions (the ‘‘Weinberg–Salam
model’’), the other with strong forces (‘‘quantum chromodynamics’’, short: QCD) ( Hoddeson, Brown, Riordan, & Dresden,
1997; Zee, 2010). In both cases the theory is formally defined by
assigning a function (‘‘Lagrangian’’) which—at least in principle—uniquely determines the elementary particles contained in
it, as well as their properties (masses, mutual coupling constants,
transformation properties with respect to various symmetries).
Despite the definiteness of the Lagrangian, physical predictions
can be extracted from it only in a limited number of cases and,
even then, usually only with the help of approximations and of
additional assumptions—and without a rigorous mathematical
proof that the procedures involved are correct. For example, most
Standard Model predictions rely on perturbative expansions
whose convergence is not ensured (Cao & Schweber, 1993,
pp. 33–34). Moreover, in QCD perturbation theory has a very
limited application and theorists are forced onto even less safe
ground when attempting to compute the consequences of nonperturbative effects, such as the properties of observed hadrons as
bound states of elementary quarks. These difficulties have to be
kept in mind, since nonperturbative effects play a central role in
the composite Higgs approach.
In the Standard Model there is one particle to which special
significance is attached: an elementary scalar (i.e., spinless) particle
usually referred to as ‘‘the Higgs boson’’. We need not discuss here
the technical details making the Higgs boson so important, since for
our purposes it is sufficient to quote the verbal statements explaining its role: The Higgs boson ‘‘breaks spontaneously’’ the local gauge
symmetry of electroweak interactions through the ‘‘Higgs mechanism’’ and, in doing so, allows itself and all other particles of the
Standard Model to have a non-zero mass without spoiling the
‘‘renormalizability’’ of the whole theory. The notion of ‘‘spontaneous
symmetry breaking’’ will be discussed later on. As to ‘‘renormalizability’’, it is a property of a theory which ensures that the divergent
terms appearing in perturbative computations of observable quantities can be formally subtracted by imposing a finite number of
empirically determined conditions, such as requiring that the
observable electron charge be equal to its measured value. Renormalization procedures are a prominent example of nonrigorous
mathematical methods in quantum field theory. In 1971, Gerhard
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‘t Hooft proved the renormalizability of the class of models to which
both the Weinberg-Salam model and QCD belong and, by the end of
the 1970s, the Standard Model had established itself as a phenomenologically successful theory. However, the Higgs boson has not yet
been observed and, indeed, the ‘‘Higgs mechanism’’ is the feature of
the Standard Model which has most often been called into question.
The theoretical core which has the notion of a ‘‘composite Higgs’’
at its centre shares with the Standard Model the idea that particle
masses are linked to the spontaneous breakdown of electroweak
symmetry. However, the mathematical structures and the story
associated to this general notion diverge from those of the Standard
Model. From the mathematical point of view, the main distinction is
the fact that the Lagrangian of composite Higgs models does not
contain any elementary scalar particles like the Higgs boson.
The symmetry breakdown is attributed to nonperturbative effects
in the interaction of as-yet-unobserved fermions, which eventually
form a bound state having the same properties (mass, charge etc.) as
the Higgs boson. Although there are no mathematical tools which
allow to accurately describe nonperturbative dynamics, it is nonetheless possible to convincingly argue that they can lead to the
spontaneous breakdown of electroweak symmetry.
7. Yoichiro Nambu’s story of symmetry lost and recovered
Historically, spontaneous symmetry breaking both through an
elementary Higgs boson and by way of nonperturbative effects
can be traced back to a paper by Yoichiro Nambu devoted to the
phenomenon of superconductivity and its relation to symmetry
(Nambu, 1960).1 In Nambu’s paper the formal methods developed
in quantum electrodynamics were used to open up new physical
perspectives in the study of superconductivity. Starting point of
his reflections was the Bardeen–Cooper–Schrieffer (BCS) theory of
superconductivity, in which nonperturbative effects of the electromagnetic interactions were shown to give rise in certain
materials to a ground state which was separated from excited
states by a finite energy gap. It was thanks to the presence of this
energy gap that the material could exists in a stable superconducting state. The BCS theory had been very successful from
the phenomenological point of view, but had initially raised eyebrowses in the solid-state community because of its apparent lack
of invariance with respect to the local gauge symmetry of
electromagnetism—a fact which seemed to imply a violation of
electric charge conservation. Indeed, in the formalism developed
by Bogliubov, Tolmachev, & Shirkov (1959). This violation of one
of the most sacred principles of physics had however been shown
to be only apparent because, again thanks to nonperturbative
effects, quasi-particles combined to form ‘‘collective excitations’’,
which in turn insured that all observable quantities would be
gauge-invariant and charge-conserving. These results had been
known already in the late 1950s, but in 1960 Nambu used the
techniques developed in quantum electrodynamics to pursue a
more detailed investigation of the relationship between quasiparticles, collective excitations and gauge invariance: ‘‘If such
collective states are essential to the gauge-invariant character of
the theory—he wrote—then one might argue that the former is a
necessary consequence of the latter. But this point has not been
clear so far’’ (Nambu, 1960, p. 649).
1
I will summarize only those historical–philosophical aspects of spontaneous
symmetry breaking which are relevant to the present subject. Those who wish to
learn more about Nambu’s work are referred to the detailed discussion in (Brown
& Cao, 1991), where the authors however attribute to the notion of ‘‘spontaneous
symmetry breaking’’ a universal character which is not to be found in historical
sources. Exhaustive historical–philosophical discussion of spontaneous symmetry
breaking in the Standard Model with references can be found in: (Brown, Brout,
Cao, Higgs, & Nambu 1997; Karaca, forthcoming; Borrelli, forthcoming).
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Thus, Nambu’s central aim was not to derive new observable
consequences of the BCS theory, but rather to use new mathematical tools to investigate the physical implications of the
already existing formalism. The new mathematical presentation
of an already known subject lent itself to a particularly appealing
narrative interpretation (a story) in terms of a symmetry apparently lost and recovered: Since the underlying theory was symmetric and the superconducting state was not—so Nambu
argued—collective excitations had to appear to compensate. In
other words, the combined existence of the underlying symmetry
and of the energy gap was verbally interpreted as the cause of the
appearance of collective excitations. Despite Nambu’s refined and
highly innovative methods, no compelling mathematical argument corresponded to the generality of this narrative, but his
results were enough to provide a nonrigorous demonstration of
the story of apparent loss and recovery of symmetry through
nonperturbative effects. Here we see an early example of how
fragmentary, nonrigorous mathematical arguments can be combined with a story to build a hybrid theoretical construct, whose
validity cannot be rigorously proven, but only demonstrated in
special cases. It is here worth noting that, although the work of
Nambu would later be described as a case of ‘‘spontaneous
symmetry breaking’’, the author never spoke of a ‘‘breaking’’ of
symmetry, but rather of its preservation.
One year later, in a paper written together with Giovanni JonaLasinio, Nambu applied the construct developed in solid state
physics to quantum field theory (Nambu & Jona-Lasinio, 1961a).
The authors established an analogy between the phenomenon of
superconductivity and the masses of strongly interacting particles. Their idea was that the mass of observed hadrons (protons,
neutrons, pions etc.) resulted from nonperturbative effects in the
interactions of some as yet undetected, massless elementary
fields. The unknown fundamental dynamics was assumed to be
invariant with respect to a symmetry which, as in the case of
superconductivity, was (apparently) lost and (equally apparently)
recovered due to nonperturbative effects. More precisely,
nucleons (i.e., protons and neutrons) were regarded as analogous
to quasi-particles, with their finite masses playing the role of the
superconductivity energy gap. In the same way in which quasiparticles violated electromagnetic gauge invariance (and with it
electric charge conservation), nucleons broke a fundamental
symmetry (‘‘chiral symmetry’’) which would have required them
to be massless. Overall invariance was however recovered thanks
to the contribution of ‘‘collective excitations,’’ which in strong
interactions took the form of pions (and eventually also other
hadrons). Once again, the whole construct was not a rigorous
mathematical theory, but rather a composite of rigorous and
nonrigorous mathematics, as well as verbal narratives. At the
beginning of the paper, the authors explained that their analogy
was primarily a qualitative one, which they ‘‘would like to pursue
mathematically.’’ (Nambu & Jona-Lasinio, 1961a, p. 346) They
were able to deliver some mathematical arguments in favour of
their thesis, but the chosen form for the underlying interaction
remained purely speculative, and they could only offer a demonstration of their thesis through model-building: ‘‘Our model
Hamiltonian, though very simple, has been found to produce
results which strongly simulate the general characteristics of real
nucleons and mesons’’ (Nambu & Jona-Lasinio, 1961a, p. 357).
Thus, the hybrid theoretical construct proposed by Nambu to
explain superconductivity in terms of a hidden symmetry was
now being extended to the field of strong interactions thanks to a
mathematical analogy based on the formalism of quantum field
theory. Model-building was used to ‘‘demonstrate’’ the validity of
the extension. The fact that Nambu’s construct arguably
explained (nonsymmetric) superconductivity in terms of (symmetric) electromagnetism provided an ‘‘empirical reference’’
further supporting the idea that nonperturbative effects linked
to a hidden symmetry could explain observed variety (differences
in hadron masses) in terms of hidden simplicity (a small number
of massless particles).
As we see, Nambu and Jona-Lasinio’s explanation of some
features of strongly interacting particles can be interpreted as a
theoretical core like those which dominate today’s BSM-modelbuilding. In proposing their theory Nambu and Jona-Lasinio were
partly building upon previous work by Julian Schwinger and
Werner Heisenberg, yet Schwinger and Heisenberg had developed
their arguments only at the formal, quantum-field-theoretical
level (Schwinger, 1957; Heisenberg, 1957; Dürr, Heisenberg,
Mitter, Schlieder, & Yamazaki, 1959). Nambu and Jona-Lasinio
instead brought in a new, decisive element into play: The
interdisciplinary analogy linking speculative hypotheses in particle physics to well-established solid state theories and phenomena. This empirical reference was essential in supporting the
validity of the core. However, Nambu and Jona-Lasinio soon had
more to offer than analogies: In the same year, they published a
follow-up paper in which they tentatively developed some
phenomenological consequences of their ideas (Nambu & JonaLasinio, 1961b). In particular, they showed how one could derive
from their model the so-called ‘‘Goldberger–Treiman relation’’
linking the pion mass to the pion decay constant. The formula was
known to be phenomenologically successful and had later found a
justification on the basis of a mathematical assumption on the
form of weak interactions (the ‘‘partially conserved axial current’’
(PCAC) hypothesis) whose physical significance remained however unclear (Pickering, 1984, pp. 112–113). The derivation
of that formula by Nambu and Jona-Lasinio was based on a
number of additional assumptions, but nonetheless delivered a
phenomenological underpinning for the conjectured analogy
between superconductivity and hadron masses. The derivation of
the Goldberger–Treiman relation was for Nambu and Jona-Lasinio’s theoretical core much more than an analogical empirical
reference, as it provided a directly relevant evidence of the
predictive power of the construct. In this sense, their theoretical
core was much less speculative than those of today’s BSM-physics.
8. Enter ‘‘spontaneous symmetry breaking’’
Responses to the work of Nambu and Jona-Lasinio were prompt
and positive. In 1961, Jeffrey Goldstone took up the story of the
existence of ‘‘superconductor solutions’’ in particle physics and
implemented it anew in a simple model in which the collective
excitations were represented by an elementary scalar field put in
by hand (Goldstone, 1961). Goldstone explicitly acknowledged the
explorative character of the model he was building and the nonbinding, yet interesting nature of his results:
‘‘The present work merely considers models and has no direct
physical applications, but the nature of these theories seems
worthwhile exploring. The models considered here all have a
boson field in them from the beginning. It would be more
desirable to construct bosons out of fermions and this type of
theory does contain that possibility. The theories of this paper
have the dubious advantage of being renormalizable, which at
least allows one to find simple conditions in finite terms for
the existence of ‘superconducting solutions’’’ (Goldstone, 1961,
pp. 154–155).
For Goldstone, the scalar boson effectively represented the nonperturbative effects of the unknown underlying interactions and
allowed to ‘‘black-box’’ them, investigating the consequences of the
assumed existence of ‘‘superconducting solutions’’. The new class
of models implementing the theoretical core proposed by Nambu
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and Jona-Lasinio were explicitly presented as exploratory tools
with ‘‘no direct physical application,’’ but this approach to modelbuilding proved heuristically fruitful and was subsequently
employed by a number of authors to study the spontaneous
breakdown of a special kind of symmetry (local gauge symmetry),
leading to the formulation of what is today referred to as the
‘‘Higgs mechanism’’: a spontaneous breakdown of local gauge
symmetry linked to an elementary scalar boson (the Higgs boson).
While Goldstone’s introduction of an elementary scalar was
very important as a mathematical tool for model-building in
connection with the theoretical core proposed by Nambu and
Jona-Lasinio, another relevant step was taken in 1962 by Marshall
Baker and Sheldon Glashow, who introduced the term ‘‘spontaneous symmetry breaking’’ to characterize Nambu’s story of
symmetry loss and recovery (Baker & Glashow, 1962; Brown
et al., 1997, p. 512). The question of naming might at first appear
as a secondary issue, but, as far as stories and hybrid constructs
are concerned, it may weight heavily in favour or disfavour of their
success in the community, and physicists are well aware of this
fact. In Section 2 we have seen the creative efforts to ‘‘name’’
alternative Higgs models, and in the next sections I will underscore how physicists often strove to find suggestive, catchy
nicknames for their creations or, if possible, choose names
suggesting an empirical reference, as in the case of ‘‘charmonium’’.
In 1967, spontaneous symmetry breaking through the Higgs
mechanism was used by Weinberg and Salam in (independently)
formulating their model of electroweak unification (Weinberg,
1967; Salam, 1968). Both authors expressed the hope that the
spontaneous nature of the breakdown of electroweak symmetry
would lead to the renormalizability of the theory, but had no
proof to offer for their conjecture. Once again, a story had been
introduced into particle physics, and it would later prove to be a
very successful one. Weinberg and Salam’s model could be seen
as the implementation of a theoretical core combining mathematical elements (Higgs mechanism, local gauge invariance, the
methods of renormalization) with stories (the ‘‘spontaneity’’ of
the electroweak symmetry breaking and the idea that it would
guarantee renormalizability) and empirical references (superconductivity, Nambu and Jona-Lasinio’s explanation of the Goldberger—Treiman relation). In 1967, there was no indication that
the connection proposed by Weinberg and Salam between spontaneous symmetry breaking and renormalizability might be real,
as they could not offer any mathematical arguments in support of
this hypothesis. Their papers were practically ignored by the
scientific community showing how, in this period, a theoretical
core without any direct phenomenological or rigorous mathematical support did not necessarily appear attractive to theorists.
In 1971, though, ‘t Hooft delivered a solid mathematical underpinning for the story connecting the ‘‘Higgs mechanism’’ to
renormalizability (‘t Hooft, 1971a, 1971b; Koester, Sullivan, &
White, 1982; Veltmann, 1997), and this development suddenly
made the Weinberg–Salam model appealing. Within a few
years, experimental evidence in its favour had accumulated. Yet,
while experimental physicists were busy finding evidence in
favour of the Weinberg-Salam model, many theorists had already
started building models which went beyond it (Pickering, 1984,
pp. 180–187). I shall briefly discuss this trend in the next session.
9. Model-building beyond the Weinberg–Salam model
in the early 1970s
In a review article on effective field theory (1993), Howard
Georgi wrote:
‘‘In the early 70’s, after Gerhard ‘t Hooft’s explanation of the
renormalizability of spontaneously broken gauge theories, but
205
before the ascendancy of the standard model, many physicists
engaged in model building, exploring the huge new space of
renormalizable models that ‘t Hooft opened up to us. This now
seems a little naive, but a tremendous amount of effort was
expended understanding the range of possibilities for spontaneous symmetry breaking with elementary scalar field. Much
of this effort was devoted to two goals: 1. to determine the
precise form of the gauge structure of the partially unified
theory of electroweak interactions; 2. to further unify the
electroweak interactions by incorporating its gauge structure
into something more comprehensive and simpler. The first
goal, as it turned out, was bootless. Somehow, Glashow,
Weinberg and Salam had written down the right electroweak
gauge group the first time’’ (Georgi, 1993, pp. 217–218).
Looking at the number of proposals made in those years for
alternatives to the Weinberg–Salam model, it becomes clear how
that specific model was not regarded with the respect due to a
promising candidate for describing fundamental interactions, but
was rather seen as one of the countless expendable models which
had come and gone in the previous years. Only the general
principles it embodied (renormalizability, gauge invariance, spontaneous symmetry breaking) were expected to eventually survive. In
other words, the hybrid theoretical core proposed by Weinberg and
Salam was regarded as being phenomenologically successful, while
the specific model they had formulated was expected to be only a
short-lived preliminary proposal. As Georgi noted, things turned out
differently, but of course model-builders of that period could not
know that. Even Weinberg conceived his own model as one ‘‘of a
general class’’ and stated: ‘‘There are many possible presumably
renormalizable models, based on different underlying symmetries
and different patterns of symmetry breaking’’ (Weinberg, 1972,
pp. 1962–1963). He, too, regarded the renormalizability of his own
model as a positive test not of that specific mathematical form, but
rather of the theoretical core embodied in it. By 1974, experimental
evidence in favour of the Weinberg–Salam model had lead to the
demise of its early rivals, as extensively documented in (Koester
et al., 1982; Sullivan, Koerster, White, & Kern, 1980). However, these
developments by no means meant that model-builders had given up
their trade. Quite the contrary: The second half of the 1970s saw the
rise both of the Standard Model and of some among the most
significant approaches to physics beyond it: grand unified theories,
supersymmetry and, last but not least, the idea of a composite
Higgs boson.
A feeling for the attitude of model-builders already in this early
period may be gleamed from a paper published in 1974 by Abraham
Pais and Howard Georgi, in which they tried to develop some
general guidelines for model-building (Georgi & Pais, 1974). I shall
not discuss here the contents of the paper, as they are not directly
relevant for the present discussion, but I would like to offer from it
two quotes which well represent the spirit of model-building, then
as now. The first one is from the beginning of the work:
‘‘Many gauge models of weak and electromagnetic interactions
have been devised in the last few years. The basic strategy for
their construction consists in a reconciliation of field-theoretical and phenomenological requirements. From the side of
field theory one insists on the renormalizability of the scheme
as the principal predictive theoretical tool. From the side of
phenomenology one attempts to incorporate all the known
regularities of the weak interactions, What is known here
almost entirely concerns the rather low-energy and lowmomentum-transfer domain. Indeed, it is our ignorance of
high-energy weak phenomena which allows, at this stage, for
so much play in model building.’’ (Georgi & Pais, 1974,
pp. 539–540).
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The second was in the conclusions:
‘‘We could not foresee the labyrinth of technical difficulties
into which this problem has led us. [y] We feel there must be
general strategies for the construction of natural models and
further that such general results would be a very important
advance in the study of the structure of gauge theories’’
(Georgi & Pais, 1974, p. 557).
The ‘‘general strategies’’ of which Georgi and Pais spoke might be
seen as similar to Randall’s ‘‘nouns and phrases’’: not proper
theories, but possible elements of a theoretical core to put in a
‘‘reservoire.’’ As in Randall’s account, the primary function of
model-building was not to construct a theory, but rather to find
‘‘general strategies’’ for further model-building.
10. Dynamical symmetry breaking from the late
1960s to 1978
As we saw in the previous section, by the early 1970s
spontaneous symmetry breaking had become a hugely successful
hybrid notion in the study of particle interactions, but it had come
to be associated primarily with an elementary scalar boson. Yet
the original idea of Nambu had not been forgotten and, since the
late 1960s, it had been receiving new impulses from the emergence of another new idea: the quark model. In the 1960s,
theorists had found a way of at least formally making sense of
the plethora of observed strongly interacting particles by regarding them as composites of elementary fermion, the ‘‘quarks’’
(Pickering, 1984, pp. 85–124). Although quarks were unobserved
and their dynamics remained unspecified, they provided a new
means for implementing Nambu and Jona-Lasinio’s theoretical
core. In 1968, Gell-Mann, Robert J. Oakes and R. Benner had
shown how the reasoning which had led Nambu and Jona-Lasinio
to derive the Goldberger–Treiman relation from a hidden chiral
symmetry could be instantiated also using the quark model as
the fundamental interaction, and a similar suggestion was also
made by Glashow and Weinberg, although they did not mentions
quarks explicitly (Gell-Mann, Oakes, & Renner, 1968; Glashow &
Weinberg, 1968). However, at this time quarks were being
regarded by the majority of physicists as purely formal devices,
and so the idea that some unknown underlying symmetry of an
equally unknown quark dynamics might be the origin of phenomenological relations was also regarded rather as a useful
mathematical constructions than as a physical reality. For example,
Roger Dashen and Marvin Weinstein wrote:
‘‘The reader who finds it difficult to believe that this symmetry
[i.e. the symmetry of quark dynamics] really exists in a
physically meaningful sense can take comfort in the fact that
PCAC and current algebra lead to exactly the same formulas for
soft-mesons theorems. The language of approximate symmetry, nevertheless, is quite useful and allows us to give a precise
meaning to PCAC in a natural way’’ (Dashen & Weinstein,
1969, p. 1261).
This passage illustrates the interplay between formal and empirical elements in attributing physical significance to mathematical
methods: The spontaneously broken symmetry was regarded as a
mathematical trick with no physical significance, yet at the same
time it was recognized that it did provide a useful ‘‘language’’ (a
story) which might allow to bridge the gap between mathematics
and nature (‘‘give a precise meaning to PCAC in a natural way’’).
Later on, when quarks came to be regarded as physical entities on
a par with electrons and protons, the ‘‘symmetry story’’ would
become the main carrier of physical meaning, and the derivation
of pion properties would come to provide an empirical reference
supporting the notion of a composite Higgs. Yet this development
still lay in the future.
Let us now go back to model-building in the early 1970s: In
Section 9 we have seen how model-builders like Glashow and
Georgi were searching for alternative implementations of the
theoretical core proposed by Weinberg and Salam. In this context
some of them attempted to build models where the spontaneous
breakdown of electroweak symmetry was implemented using the
nonperturbative approach of Nambu and Jona-Lasinio. An early
suggestion in this sense was made in 1972 by Heinrich Saller, who
was working in the context of Heisenberg’s (non-mainstream)
approach to quantum field theory which had been among the
inspirations for the work of Nambu and Jona-Lasinio (Saller,
1972). Yet Saller’s work was practically ignored and the idea of
a dynamical breakdown of electroweak symmetry was effectively
launched in 1973 independently by John Cornwall and Richard
Norton (‘‘Spontaneous Symmetry Breaking Without Scalar
Mesons’’) and Kenneth Johnson and Roman Jackiw (‘‘Dynamical
Model of Spontaneously Broken Symmetry’’). Cornwall and Norton stated:
‘‘In this paper we present a simple model field theory in which
the spontaneous symmetry breaking and the consequent
massiveness of a vector meson occur in a manner similar to
the violation of electric current conservation and the consequent Meissner effect in the theory of superconductivity.
Indeed, our efforts to construct a theory of this kind were
inspired by the work of Nambu concerning the gauge invariance of the BCS theory of superconductivity’’ (Cornwall &
Norton, 1973, p. 3338).
Like Nambu and Jona-Lasinio, they did not offer a realistic theory,
but used model-building to demonstrate that such a theory was
possible:
‘‘It will be evident that this model is not intended as a realistic
theory of weak or electromagnetic interactions. Rather, it is
only an example of what we feel is probably a large class of
theories in which the spontaneous symmetry breaking derives
from general features of an apparently symmetric interaction’’
(Cornwall & Norton, 1973, p. 3338).
As stated already in the title, the authors did not suggest the
existence of a composite Higgs boson, but rather claimed that one
might do without any scalars at all. Jackiw and Johnson, too,
acknowledged that they were building upon the work of Nambu,
Jona-Lasinio and Goldstone and used model-building to demonstrate their claim that ‘‘a theory consisting of massless Fermi and
vector-meson fields can lead to an excitation spectrum of solely
massive particles’’ (Jackiw & Johnson, 1973, p. 2386). However,
they concluded: ‘‘The physical relevance of this mechanism is not
apparent at the present time’’ (Jackiw & Johnson, 1973, p. 2395).
Their model did not explicitly include a massive, composite
scalar, but they commented that the presence of such a bound
state was a possibility and that such a theory would be ‘‘an
example of the conventional Higgs mechanism’’ (Jackiw&Johnson,
1973, p. 2395).
The earliest explicit suggestion of a ‘‘composite Higgs’’ in the
Weinberg–Salam model seems to be in a paper published in 1974
by Terrance Goldman and Patrizio Vinciarelli (‘‘Composite Higgs
Fields and Finite Symmetry Breaking in Gauge Theories’’), to
which the web of knowledge attributes only two citations
(Goldman & Vinciarelli, 1974). In the following years, a number
of authors engaged in investigating dynamical symmetry breaking, but they focussed on the general mathematical aspects
of the problem and did not speculate on their physical relevance
and on the nature of the nonperturbative interactions underlying
A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
the symmetry-breaking dynamics. In short, no theoretical core
focussed on the idea of a composite Higgs emerged, although
many of its future elements were already there (the mathematical
structures used by Nambu and Jona-Lasinio, the stories of a
dynamical spontaneous breakdown of symmetry and of a composite Higgs boson, the empirical reference to superconductivity).
Let us pause to summarize the situation: In the middle of the
1970s, the Weinberg–Salam model of electroweak interactions
with an elementary Higgs field had established itself as a
coherent, phenomenologically successful theory, but had at the
same time become the starting point of a increasing activity of
model-building. The idea of a dynamically generated, composite
Higgs had been explicitly formulated, but had failed to attract
much attentions and, in particular, no one had addressed the
question of the underlying nonperturbative interactions responsible for the generation of the bound state. As already anticipated,
the decisive contribution which would give rise to a new, selfsustaining theoretical core was precisely a template to conceive
such nonperturbative interactions, and this element came in the
form of Quantum Chromodynamics. In the years between 1976
and 1978 QCD established itself as a viable model for the strong
interactions from which all hadrons emerged as composite states
(Pickering, 1984, pp. 207–230, 309–346). Once the quarks had
come to be regarded as more that formal devices, the idea of
considering the pion as a composite state associated to the
dynamical breaking of a symmetry of quark interactions provided
the empirical reference needed to support the story of dynamical
symmetry breaking of electroweak interactions. In 1976, Steven
Weinberg explored the ‘‘Implications of dynamical symmetry
breaking’’ and explicitly focussed on its possible physical significance (Weinberg, 1976). He did not introduce composite scalar
fields, but he did explicitly ask what physical interactions might
be at the origin of a dynamic symmetry breaking of electroweak
symmetry. He showed that, due to a difference in energy scales, it
was not possible to assume that the breakdown was due to the
strong interactions of QCD, but he showed that QCD provided a
viable template for the new, unknown forces:
‘‘One way to approach these problems is to suppose that in
addition to the color SU(3) associated with the observed strong
interactions, there is another gauge group [%] associated with
a new class of ‘‘extra strong’’ interactions, which act on leptons
as well as other fermions’’ (Weinberg, 1976, p. 992).
However, Weinberg did not find the phenomenology of such
models very attractive (Weinberg, 1976, p. 992) In the years
immediately following its publication Weinberg’s work did not
attract much attention: According to the web of knowledge it was
quoted three times in 1976, seven times in 1977, and no times at
all in 1978. From 1979 onward, however, the paper achieved
popularity because it came to be associated with a work by
Leonard Susskind to be discussed in the next section.
11. Leonard Susskind and the emergence of the composite
Higgs theoretical core
In 1979, Leonard Susskind published a paper whose impact on
model-building in high-energy physics can hardly be overestimated (Susskind, 1979). The paper bore the title ‘‘Dynamics of
Spontaneous Symmetry Breaking in the Weinberg-Salam Theory’’,
yet its focus was not dynamical symmetry breaking in general,
but rather the idea that the Higgs boson should be regarded as a
bound state of more fundamental fermions whose interactions
caused a dynamical breakdown of electroweak symmetry. In this
paper, Susskind combined a number of pre-existing elements into
a new theoretical core. He began by introducing the idea of a
207
composite Higgs by claiming that elementary scalars fields were
not ‘‘natural.’’ I shall not discuss here his argument, except for two
remarks. The first one is that, as all theorists are ready to
recognize, the argument had no compelling character, but was
rather a hybrid construct combining some mathematical elements with a story according to which the fine-tuning of parameters is an aesthetic flaw of the Higgs mechanism. The second
remark is that, despite its non-compelling nature, the ‘‘naturalness problem’’ soon became a main point of criticism against the
Standard Model (Galison, 2004, p. 31; Drees et al., 2004,
pp. 8–10). While the ‘‘naturalness’’ argument gave generic support for the thesis that the Higgs boson was a composite particle,
it provided no indication as to how a composite Higgs model
would look like. The author therefore went on to tentatively fill
up his story with mathematical meaning and it is worth following
his argument step-by-step, as it provides a clear example of
how the different elements of a theoretical core were combined
in an act of model-building demonstrating the core’s validity.
Susskind’s first step was to implement his composite Higgs story
in a simple model (‘‘a warm up example’’) referring back to the
interpretation of the pion as a composite state of quarks which
acquired mass thanks to the spontaneous breakdown of a symmetry of the underlying interactions (Susskind, 1979, pp. 2620–
2622). The stated aim of the warm-up example was this: ‘‘We
would like to know if the strong interactions can somehow
replace the Higgs scalars and provide masses for the intermediate
vector bosons’’ (Susskind, 1979, p. 2621). Susskind considered the
pion as a quark-composite resulting from the dynamical breakdown of the chiral symmetry of quark interactions, and asked
whether it might ‘‘replace the Higgs scalars’’ in giving masses to
electroweak vector bosons. The answer was yes, although the
model was not a realistic one. As we saw in the previous chapters,
this result was nothing new: Following Nambu and Jona-Lasinio—
whom Susskind mentioned, but did not quote (Susskind, 1979,
p. 2621)—the pion had been regarded since the late 1960s as a
quark composite linked to the dynamical breakdown of chiral
symmetry. The idea that QCD might be behind electroweak
symmetry breaking, on the other hand, had already been explored
by Weinberg (1976). Yet Susskind had shrewdly combined the
two elements in his warm-up example, whose aim was not to
explore a realistic possibility, but to help build up a hybrid
narrative in which not QCD itself, but rather a new interaction
would be behind the dynamical breakdown of electroweak
symmetry. What was fundamental for his rhetorical strategy
was to establish a close connection between the (purely fictional)
composite Higgs and the pion, which by that time was unanimously regarded as a quark-composite whose mass could indeed
be linked to a story of spontaneous symmetry breaking. This
connection would be of the greatest importance for the emerging
theoretical core.
Having thus warmed up, Susskind proceeded to formulate ‘‘a
more realistic example’’, in which he postulated the existence of a
‘‘new undiscovered strongly interacting sector, similar to ordinary
strong interactions’’ (Susskind, 1979, p. 2622). The new QCD-like
interaction caused a new type of elementary fermions to bind into
a symmetry-breaking scalar particle and eventually reproduce the
particle content of the Weinberg–Salam model. Susskind used a
language maximally underscoring the analogy with QCD: Since
the strong charge of quarks was ‘‘color’’, he called his new charge
‘‘heavy-color’’ and spoke of ‘‘heavy-color quarks’’ and ‘‘heavycolor pions’’ (Susskind, 1979, p. 2623). After having shown how
the composite scalar would be able to give mass to the electroweak vector bosons like the elementary Higgs did, the author was
however forced to admit that his construction was not fully
realistic: ‘‘The model described here is certainly incomplete.
As it stands it cannot account for the masses of leptons and
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quarks’’ (Susskind, 1979, p. 2623). In the last section of the paper
he addressed in more detail the main problem plaguing his
construction: If the composite Higgs boson gave mass also to
fermions like leptons and quarks, then the relevant Lagrangian
had to contain terms (four-fermion interactions) making it nonrenormalizable. (Susskind, 1979, p. 2624). Still, this result was in
no way seen as a failure, but rather as a first insight into the
potential and problems of a new theoretical approach to BSMphysics: a theoretical core having at its centre the notion of a
composite Higgs. Susskind’s paper was an immediate success,
both for his naturalness argument and for the idea of the
composite Higgs: The Web of Science attributes to it 27 citations
for 1980, 84 for 1981 and 85 for 1982. Keeping into account the
delay between a preprint and its publication, this citation record
indicates an almost immediate reception, much different than
what we saw for Weinberg’s (1976) paper. However, in an
addendum to his own earlier paper Weinberg referred to Susskind’s work—quoted as a preprint—and pointed out how he
himself had already anticipated its ideas, thus ensuring himself a
role as co-creator of the composite Higgs approach (Weinberg,
1979, p. 1279).
Susskind’s paper can be identified as the origin of the core
associated to the idea of a composite Higgs boson and comprising
a mixture of (1) mathematical statements on the renormalizability of the Weinberg-Salam model and the nonperturbative behaviour of quantum-field-theoretical expressions, (2) stories
connecting these elements to qualitative notions such as the
compositeness of a particle or the spontaneity of a symmetry
breakdown and its relationship to renormalizability, and
(3) empirical references to how similar arguments had proved
phenomenologically successful in other cases (superconductivity,
Weinberg–Salam model, Goldberger–Treiman relation, compositeness of the pion and of hadrons in general). None of the
individual elements was in itself a novelty, but Susskind
combined them for the first time into a physically suggestive
and rhetorically unitary theoretical core, using model-building to
demonstrate both the viability and potential usefulness of
the new construct. He admitted openly that none of the models
he wrote down could be the starting point for a future theory,
but this fact did not detract from the success of his approach in
the physics community. It is also important to note that the
theoretical core did not allow any new explanation or predictions,
but only solved a noncompelling problem which Susskind
himself had formulated for the first time: the lack of ‘‘naturalness’’ of the Weinberg–Salam model—an issue which in the
following decades would become a main motivation for BSMmodel-building.
12. Technicolor, supercolor, ultracolor: The many avatars of
the composite Higgs
Susskind’s paper was the starting point of model-building
activities in which he himself played a prominent role: Together
with Savas Dimopoulos he introduced an extension of heavycolor called ‘‘technicolor’’: ‘‘We attempt to show that fundamental scalar fields can be eliminated from the theory of weak and
electromagnetic interactions. We do this by constructing an
explicit example in which the scalar field sectors are replaced
by strongly interacting gauge systems’’ (Dimopoulos & Susskind,
1979, p. 237). In this paper the authors addressed the problem of
fermion masses, but also explicitly underscored the limited scope
of the models they built and their purely exploratory function:
‘‘We will present some examples which we argue are capable
of generating a set of mass scales with realistic orders of
magnitude for both fermions and bosons. The examples in
their present form are too simple to take completely seriously.
We offer them as an ‘‘existence proof’’ for a family of theories
which can produce reasonable scales without unnatural
adjustments. We also feel that the examples offer valuable
clues to the various mechanisms which may be needed’’
(Dimopoulos & Susskind, 1979, p. 237).
These words again illustrate the modus operandi of model
builders and fit with Randall’s analogy of looking for ‘‘nouns’’
and ‘‘phrases’’, i.e., for some ‘‘mechanism’’ which might be needed
in constructing theories. The models are not to be ‘‘taken
seriously,’’ but provide an ‘‘existence proof’’ for a ‘‘family of
theories’’ of a given kind which may eventually arise out of the
elements of the core.
Model-building around the composite Higgs core continued in
this vein, for example with the development by Susskind, Dimopoulos and Stuart Raby of models for ‘‘tumbling gauge theories’’,
in which dynamical symmetry breakdown did not occur all at
once, but rather in series of steps (Raby, Dimopoulos, & Susskind,
1980). In this paper, too, the authors made a conscious effort to
associate their quite complex and rather sketchy mathematical
constructions to a good verbal formulation and to a catchy name:
‘‘In this paper we will describe a class of gauge theories in
which a wide variety of length scales may naturally occur.
These scales are associated with the sequential spontaneous
symmetry breakdown of the gauge symmetry in a series of
steps. The breakdowns occur without the aid of fundamental
scalars. [y] We refer to such behaviour as ‘tumbling’.’’ (Raby
et al., 1980, p. 373).
Another versions of the composite Higgs model was ‘‘supercolor’’,
where the idea of technicolor was combined with the element of
another emerging theoretical core: supersymmetry (Dimopoulos
& Raby, 1981). Later on, also models named ‘‘hypercolor’’, ‘‘ultracolor’’ and ‘‘metacolor’’ were proposed, but, eventually, the term
‘‘technicolor’’ came to be used to describe all variations on the
theme (Kaplan, Georgi, & Dimopoulos, 1984; Kaplan & Georgi,
1984). We have now arrived in the early 1980s, a period of
particular importance for the construction of today’s landscape of
‘‘new physics,’’ because it was in the years between 1979 and
1985 that the some of the most important concepts and
approaches in the discourse on ‘‘new physics’’ established themselves: not only technicolor and naturalness, but also the ‘‘hierarchy problem,’’ supersymmetry, grand unified theories, strings,
as well as issues connecting particle physics and cosmology, such
as dark matter or inflation. Like dynamical symmetry breaking, all
of these notion had been present in the scientific discourse
already in the 1970s, but it was only in the early 1980s that they
attained centre stage in the community. In all cases—and not only
in string theory—nonrigorous mathematical heuristics and hybrid
theoretical constructs played a key role, although in string theory
this development was particularly evident. This constellation
would deserve a detailed historical, philosophical and sociological
analysis which cannot be carried out in the context of the present
explorative investigation, but the broad picture has to be kept in
mind as the framework within which the events discussed in the
following pages were embedded.
In the early 1980s supersymmetry overcame the composite
Higgs as a theoretical core worth exploring in model-building,
and remains today the most prominent candidate for BSMphysics. However, the composite Higgs did not die out: In the
Proceedings of the 18th Solvay Conference on Physics Susskind
admitted that technicolor ‘‘runs into grave difficulties’’ when
dealing with quark and lepton masses, but listed tentative wayouts, among them the possibility of considering also quarks and
A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
leptons as composite states (Susskind, 1984, pp. 185–187). In the
same conference Haim Harari offered a review of such ‘‘Composite models for quarks and leptons’’ and concluded:
‘‘There is no satisfactory theoretical model of composite
quarks and leptons. However, the models proposed so far
contain many interesting new ideas. Each of these ideas should
be investigated on its own merit, regardless of the detailed
model which may have led to it. Several correct ingredients of
the correct theory may already be with us now’’ (Harari, 1984,
p. 175).
Although no viable models for a composite Higgs approach which
could treat quarks and leptons had been found, the theoretical
core was still regarded as worthy of attention, as it might contain
‘‘correct ingredients of the correct theories,’’ i.e., ‘‘nouns and
phrases’’ useful for further model-building. Beyond the specific
issue of the composite Higgs, Susskind’s and Hariri’s texts are
interesting because, given the prominent occasion they were
prepared for, they may be seen as expressing contemporary
attitudes in the physics community: High-energy theorists were
ready to commit themselves to investigate ‘‘ideas [y] each in its
own merit regardless of the model that may have led to it.’’ This is
the same kind of commitment to fragmentary theoretical constructs found in Randall’s description of the practices of modelbuilding: The belief that collecting ‘‘nouns and phrases’’ without
bothering with ‘‘grammar’’ was the immediate aim in the search
for new physics.
Among the most active contributors to technicolor modelbuilding in the 1980s was Howard Georgi, who published ‘‘A Tool
Kit for Builders of Composite Models’’ (Georgi, 1986) and,
together with Andrew Cohen, played an important role in adding
a new ingredient to the tool kit: ‘‘walking technicolor’’ (Cohen &
Georgi, 1989). The idea behind this new class of models had been
developed by a number of authors in the years 1985–1987, but in
their paper Cohen and Georgi combined the previous approaches
into a powerful story: To solve the by now well-known problem
of nonrenormalizability of four-fermion interactions present in
composite Higgs models the relevant coupling constant had to
vary very slowly with energy—a story supported by an empirical
reference to the explanation of lepton-hadron deep inelastic
scattering given by QCD in the early 1970s (Pickering, 1984,
pp. 125–158, 207–215). The authors demonstrated the validity of
their proposal by building a simple but significant model, and by
supplying a catchy name: walking technicolor—a play on words
based on the fact that the variation of coupling constants with
energy is usually referred to as ‘‘running’’ (Cohen & Georgi, 1989,
p. 7). As had been the case in the works discussed before, the
paper did not offer a model which could be used as a starting
point to construct a full-fledged theory: Model-building was
aimed at supporting the validity of the general idea by showing
that it worked in a special case. Cohen and Georgi constructed a
‘‘gap equation’’ whose solutions they studied by making use of a
series of approximations heavily relying on perturbative methods.
Their conclusions offer additional evidence of the hybrid nature
of the theoretical constructs with which they were working:
‘‘The really important question about all of this is how much of
the story is an artefact of the brutal approximations that we
have made to the gap equation. Should we believe it at all in a
strongly interacting theory in which, in general, perturbation
theory is not a reliable guide to the physics? Obviously, such
details as the value of the critical coupling constant will not
survive. However, we think that the most important part of the
walking technicolor idea is, in fact, very likely to survive
beyond perturbation theory’’ (Cohen & Georgi, 1989, p. 23).
209
Indeed, the idea of ‘‘walking technicolor’’ survived to become a
new element for the theoretical core of the composite Higgs.
13. Technicolor walks beyond LEP results
With the start of the LEP experiment in 1989 precision data on
electroweak interactions became available, and they were not
particularly favourable to technicolor, as they ruled out many of
the simple models, while signatures of more complex variations
seemed hard to detect (Peskin & Takeuchi, 1990). Yet the
technicolor approach was not abandoned: Already in 1991
Kenneth Lane and M. V. Ramana published a paper on ‘‘Walking
technicolor signatures at hadron colliders’’, in which they looked
forward to data forthcoming from the Tevatron, a proton–proton
collider at Fermilab which had been active since 1983, from the
Superconducting Supercollider, which at the time was planned,
but was eventually cancelled in 1994, and from the LHC, whose
planning was already under way (Lane & Ramana, 1991). Lane
and Ramana argued that walking technicolor offered better
chances than older models and, indeed, this direction of modelbuilding would later become prominent within the composite
Higgs approach.
Although new experimental results were partly problematic
for the composite Higgs core, they also had a positive impact on it,
and this was linked to developments in the search for a particle
which had long been assumed to exist, but had not yet been
detected: the top quark. For theoretical reasons, no doubts on the
existence of the top quark were held since 1977, when evidence
for its partner, the bottom quark, had been found (Liss & Tipton,
1997; Staley, 2004). The mass of the top quark was initially
assumed to be of the same order of magnitude as that of the
bottom quark (ca. 4.2 GeV), but high-energy experiments failed to
find it and, by the end of the 1980s, the lower limit for the top
quark mass had reached the value of 77 GeV. This result led some
theorists, among them Nambu, to speculate along a new path:
until then, the relatively low values of quark masses had prevented regarding them as the particles whose dynamics broke
electroweak symmetry and gave rise to the composite Higgs.
Now the situation had changed and it seemed possible that
the breakdown of electroweak symmetry might be due to a
top-quark condensate, and not to some ‘‘new’’ elementary fermions (Miransky, Tanabashi, & Yamawaki 1989; Nambu, 1989;
Bardeen, Hill, & Lindner, 1990). As usual, model-building was the
tool for testing the hybrid combination of new experimental
results on the top quark and the story of a top-quark condensate
which spontaneously broke electroweak symmetry: V. A. Miransky, Masaharu Tanabashi and Koichi Yamawaki proposed a
‘‘new class of models of dynamical electroweak symmetry breaking in which the t quark is responsible for the formation of the
condensate generating masses of the W and Z boson’’ (Miransky
et al., 1989, quote from p. 180). Once again, not a single model,
but a whole class was being built at the same time to demonstrate
the validity of the new element added to the core. Along similar
lines William Bardeen, Christopher Hill and Manfred Lindner
spoke of a ‘‘minimal dynamical symmetry breaking’’ of electroweak symmetry through a top quark condensate (Bardeen et al.,
1990). They noted how ‘‘several authors, most notable Nambu,
have recently experimented with this idea,’’ proceeded to build
models to ‘‘make precise the definition of the minimal dynamicalsymmery-breaking scheme’’ and then went on to argue that their
simple model ‘‘may be generalized in several directions.’’
(Bardeen et al., 1990, quotes from p. 1647, 1651). Finally, they
derived some predictions for the values of the Higgs and top mass.
At the end of the paper, though, they pointed out how their
models suffered from naturalness problems similar to
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A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
those attributed by Susskind to the Standard Model with an
elementary Higgs.
The idea that a top-quark condensate might be responsible for
electroweak symmetry breaking became known as ‘‘topcolor’’ and
Christopher Hill eventually combined it with standard technicolor
to obtain ‘‘topcolor assisted technicolor’’ (Hill, 1995). After
expounding his new idea Hill noted that the combination of
topcolor and technicolor opened up a number of new avenues for
model-building:
‘‘We note that a number of new models is suggested by this
approach. In model building we have several options:
(I) technicolor breaks both the electroweak interactions and
the topcolor interactions; (II) technicolor breaks electroweak,
and something else breaks topcolor; (III) technicolor breaks
only topcolor and something else drives electroweak symmetry breaking [y] We believe these models offer new insights
into the dynamical origin of fermion masses and electroweak
symmetry breaking, and merit further study’’ (Hill, 1995,
pp. 487–488).
This example illustrates well both the flexibility of a theoretical
core whose different elements could be variously combined, and
its capability to embed new elements of experimental origin. It
also shows how the commitment of physicists to a theoretical
core (the composite Higgs) prompted them to use model-building
to tentatively modify it in different, at times mutually incompatible directions, adding new elements to it.
When the top quark was finally observed (1995), no clear
evidence for or against topcolor emerged. Although the top quark
had failed to provide direct experimental evidence in favour of
the composite Higgs core, its existence could still be used as an
empirical reference supporting that approach, since it showed
how heavy, strongly interacting fermions similar to the hypothetical techni-quarks were indeed possible. By the end of the 1990s,
although the composite Higgs core had failed to produce a
concrete candidate for a viable theory, it had not been abandoned: models were disposable tools, theoretical cores were not.
In his review of ‘‘Avenues for Dynamical Symmetry Breaking’’
(1999) Sekhar Chivukula hoped that forthcoming experimental
results would give a boost to the enterprise:
‘‘Technicolor, topcolor, and related models provide an avenue
for constructing theories in which electroweak symmetry
breaking is natural and has dynamical origin. Unfortunately,
no complete and consistent model of this type exist. [y] If
electroweak symmetry breaking is due to strong dynamics at
energy scales of order a TeV, experimental directions will be
crucial to construct the correct theory. With luck, the necessary clues will begin to appear at the Tevatron in Run II!’’
(Chivukula, 1999, p. 8).
Thus, twenty years after its emergence the composite Higgs was
still thriving, despite the fact that no experimental evidence in its
favour had been found. The same applied to supersymmetry and,
somehow later, to the theoretical cores linked to various extradimensional approaches. This development stands in contrast to
the cases of Nambu and Jona-Lasinio’s ideas on hidden symmetries and hadron masses and of Weinberg and Salam’s electroweak unification: In the first case, the authors had been able to
provide at least a tentative phenomenological support for their
ideas in the derivation of the Goldberger–Treiman relation; in the
second case, the physics community had only taken notice of the
theoretical core proposed by Weinberg and Salam after ‘t Hooft
had provided a rigorous proof underpinning the story connecting
renormalizability and spontaneous symmetry breaking. Instead,
Susskind’s proposal of a composite Higgs boson had been
immediately taken up and expanded upon by other authors even
without direct experimental support or rigorous mathematical
proof. In short, it is remarkable how long-lived this and other
BSM-physics approaches from the late 1970s have turned out to
be. As I shall discuss more in detail in Section 15, part of the
explanation for this longevity lies certainly in the lack of new,
decisive experimental results in particle physics, which has
prompted—if not outright forced—theorists to rely increasingly
on arguments of ‘‘mathematical aesthetics’’ such as Susskind’s
naturalness problem. However, I would like to suggest the the
shift in theoretical practices may also be linked to a long-term
transformation of epistemic values implicit in the trend toward
nonrigorous heuristics discussed in Section 5. Theoretical highenergy physicists are used to assume in principle that rigorous
mathematical proofs exist, but substitute them in practice with
folk theorems and usually deny the epistemic gap between the
two. In a similar way, theories as coherent mathematical constructs remain the ideal aim of research, but its practical focus are
hybrid constructs, and the gap between the two is left unspoken
also in this case.
14. The composite Higgs in the new millennium: The model as
a ‘‘Rosetta stone’’
The new millennium has seen many more composite Higgs
models being built, while the elements of this theoretical core
have increasingly often been combined with those from other
ones (e.g., Andersen et al., 2011; Dietrich, Sannino, & Tuominen,
2005; Piai, 2010). In October 2010, in a paper entitled ‘‘Technicolor and Beyond: Unification in Theory Space’’ Francesco Sannino
proposed to ‘‘marry’’ technicolor to supersymmetry to ‘‘provide a
unification of different extensions of the Standard Model. For
example, this means that one can recover, according to the
parameters and spectrum of the theory, distinct extensions of
the standard model, from supersymmetry to technicolor and
unparticle physics’’ (Sannino, 2010, p. 1)2. Today, model-builders
await new experimental results not simply to test the products of
their work against observation, but also to obtain new material to
enrich their speculations. To quote one example, in April 2011,
Estia Eichten, Kenneth Lane and Adam Martin published a preprint on ‘‘Technicolor at the Tevatron’’ in which they proposed to
regard a deviation from Standard Model predictions observed at
Tevatron by the CDF experiment as an indication in favor of
technicolor, pointing the way to further theoretical developments
in the field (Eichten, Lane, & Martin, 2011). The authors stated
that the new experimental data might give indications in favour
of a theory of the walking technicolor variety, but devoted their
paper to the construction of a different kind of model (‘‘low-scale
technicolor’’) which was supposed to represent the low-energy
phenomenology of the ‘‘real’’ theory. The authors stressed both
the transitional nature of the new model and its key role in
connecting experiment and the higher level of theoretical cores:
‘‘If experiments at the Tevatron and LHC reveal a spectrum
resembling these predictions [i.e. those of low-scale technicolor], it could well be that low-scale technicolor is the ‘‘Rosetta
stone’’ of electroweak symmetry breaking. For it will then be
possible to know its dynamical origin and discern the character of its basic constituents, the technifermions. The masses
and quantum numbers of their bound states will provide
stringent experimental benchmarks for the theoretical studies
2
‘‘Unparticle physics’’ is yet another approach to BSM-physics and was
introduced in by Georgi (2007).
A. Borrelli / Studies in History and Philosophy of Modern Physics 43 (2012) 195–214
of the strong dynamics of walking technicolor just now getting
started’’ (Eichten et al., 2011, p. 4).
Here we see a scenario very similar to that found in Randall’s
text: A ‘‘real’’ theory is identified in principle as the aim of
research, but in practice the efforts of model-builders concentrate
on formulating expendable models which will provide ‘‘benchmarks’’ for further theoretical and experimental studies and allow
to collect new ‘‘ingredients’’ for theoretical cores. To characterize
their model, the authors use like Randall a striking linguistic
analogy, suggesting that it might turn out to be a ‘‘Rosetta stone.’’
This image expresses well the role of models as tools to explore
and expand hybrid theoretical cores, tentatively linking them
with phenomena, and individuating those ‘‘nouns and phrases’’
out of which in a (far) future a ‘‘real’’ theory might be constructed.
15. Conclusions and outlook: Hybrid theorising and the
epistemic significance of Rosetta stone models
In the last decades no model of BSM-physics has managed to
rise to the role of a clear-cut, phenomenologically viable alternative to the Standard Model—yet no model-builder had ever
expected that. Models of BSM-physics are built to serve as
expendable implementations of one or more non-expendable
theoretical notions of mathematical and nonmathematical
character, which have to be tested and explored as potential
ingredients of future theories. I have argued that in today’s
BSM-model-building - and to some extent in all of high-energy
physics - theoretical constructs have come to display increasingly
often a hybrid character: They are not ‘‘theories’’ in the sense of
coherent mathematical structures, but rather ‘‘theoretical cores’’
comprising a closely knit network of mathematical notions, verbal
narratives (‘‘stories’’), and analogies referring back to established
explanations of phenomena from other fields (‘‘empirical references’’). The models which are continuously built and discarded
are neither failed attempts at a final theory nor toy models
derived from a more complex overarching theoretical structure:
They are the constantly refreshed links in a network connecting,
on the one side, the elements of one or more theoretical cores
and, on the other, experimental data. Following the approach of
Morgan and Morrison, we can regard model-building as mediating between the theoretical and empirical level, yet the theoretical level cannot be anymore conceived purely in mathematical
terms. Other than theoretical cores, model are usually mathematically coherent constructs, yet they are not theories as they are
not regarded by physicists as in themselves describing or predicting phenomena, but only as giving an approximate estimate of
how the predictions of a hypothetical real theory would look like.
Using the suggestive metaphor employed by Eichten, Lane and
Martin, models can be seen as ‘‘Rosetta stones’’, which are not
important because of their immediate content, but because they
provide a chance of adding new mathematical and nonmathematical ‘‘nouns and phrases’’ to theoretical cores. These consideration apply to the approaches to BSM-physics listed in Section 2,
as well as to others.
The case of the composite Higgs has been used as an example
to clarify the notion of theoretical core and the function of
models. In reconstructing the emergence and development of this
theoretical core particular attention has been devoted to showing
how theorists already in the 1960s were fully aware of the
importance of fragmentary theoretical constructs in their work.
When they built a mathematical model, they did not try to pass it
off as a candidate for a theory, but were instead ready to point out
that models were not to be ‘‘taken seriously’’, as they were rather
exploratory tools to individuate ‘‘strategies’’ and ‘‘ingredients’’ for
211
a future theory. Although such epistemic practices had their roots
in pre-Standard-Model high-energy physics, I have argued that
they gained a prominent role only in the last two or three
decades. Since the 1980s, theoretical cores such as supersymmetry, extra-dimension or the composite Higgs have kept on being
developed despite the fact that no experimental evidence of their
validity has been found. Yet it would be incorrect to state that
empirical elements play no role in model-building: As we have
seen in many cases, analogical empirical references have great
importance in supporting the validity of theoretical cores. For
example, the original idea of a dynamical symmetry breaking
involving no elementary scalar particles scarcely played a role in
electroweak physics until QCD became available as an empirical
reference supporting the story of a composite Higgs boson.
Leonard Susskind was the first one to make extensive use of this
argument and it is because of his work that the theoretical core
centring on the notion of a composite Higgs emerged, to evolve in
the following three decades thanks to an uninterrupted activity of
creative model-building.
If one accepts this case study as representative for the
dynamics of model-building in high-energy physics, one may
wonder whether this situation should be seen as a consequence of
the increasingly different time-scales of theoretical and experimental practice in this field. Since the early 1990s. theorists have
been starved of experimental input—except for what came from
astrophysics experiments—and have tried to make the best out of
the few data available. As Pais and Georgi wrote in 1974: ‘‘It is
indeed our ignorance of high-energy weak phenomena which
allows, at this stage, for so much play in model building’’ (Georgi
& Pais, 1974, p. 540). Indeed, in the last decades hardly any new
experimental results came about to guide theorists in developing
theoretical cores. However, as we have seen, the main features of
today’s practice of model-building were already in place in the
early 1970s, when the experimental results on which the phenomenological success of the Standard Model would be built still
lay in the future. Those successes notably failed to establish the
Standard Model as ‘‘the’’ final theory of particle physics. Indeed,
theorists since the 1970s seem to have been more preoccupied
with doubting the Standard Model than with committing to it.
To quote the CERN theorist Guido Altarelli: ‘‘The Standard Model
is a low energy effective theory (nobody can believe it is the
ultimate theory)’’ (Altarelli, 2010, p. 1). It is therefore legitimate
to speculate that the removal into the far distance (for example
on the top of a mountain) of a ‘‘theory’’ conceived as a coherent,
complete mathematical framework explaining a large range of
phenomena may be due to other factors than the paucity of
experimental input. To answer this question conclusively, a more
detailed study of the sociology of knowledge in high-energy
physics is needed, yet in concluding this essay I would like to
tentatively address the issue.
In 1993, Silvan Schweber noted how, thanks to the method of
the ‘‘renormalization group,’’ the renormalizability of a theory has
been linked to the idea that phenomena at different energy scales
are described by different ‘‘effective theories’’, all having the same
right to claim to represent the laws of nature as any ‘‘final theory’’
(Schweber, 1993). Moreover, the techniques of the renormalization group allow under certain circumstances to extract from a
given quantum field theory some informations on how the
effective field theories at higher or lower energies should look
like. On this basis, Schweber concluded:
‘‘The successes of the standard model of electroweak and
strong interactions has been very impressive. These theories,
when interpreted as effective field theories, have lent support
to a positivistic and antireductionist viewpoint. The physical
world could be considered as layered into quasiautonomous
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domains, each layer having its ontology and associated laws.
A considerable number of high-energy physicists have adopted
such a stance’’ (Schweber, 1993, p. 155).
Schweber’s conclusion might be seen as providing at least a
partial explanation for the practices of today’s model-building
as reconstructed in the present paper: If no theory is the final one,
one may be content with learning only a few ingredients from it,
using model-building to try and advance from one theoretical
level to the next one. Like Schweber, other authors, too, have
pointed out the possible epistemological relevance of the renormalization group (Cao, 1993; Fraser, 2011; Hartmann, 2001;
Huggett & Weingard, 1995; Wallace, 2011). However, Elena
Castellani has recently noted that the use of renormalization
group methods does not in itself compel to give up the idea of a
final theory, since they can be interpreted in different ways
(Castellani, 2002). In particular, under an ‘‘extreme interpretation’’ of the renormalization group, one may understand the
layers of which Schweber spoke as ‘‘related in a precise way’’
(Castellani, 2002, p. 262). From this point of view, the ‘‘final
theory’’ can be regarded as existing on top of the layered tower of
effective theories. Indeed, as we have seen, Randall was very
positive about the existence of a ‘‘unified theory’’ as ultimate aim
of physics research, an attitude widely shared in the physics
community by undermining the idea that a final theory exists. In
conclusion, the methods of the renormalization group cannot be
in themselves regarded as the decisive factor promoting the
employment of hybrid constructs in theoretical practice, although
they certainly play a role in letting such practices appear as
epistemically equivalent to the search for a theory of everything. I
would like instead to suggest that the rise of hybrid theoretical
cores is primarily linked to the increasing use of nonrigorous
heuristics in high-energy physics.
As already discussed in chapter 5, the use by theoretical
physicists of methods which contemporary mathematicians
regard as nonrigorous is nothing new, and has often been
considered by physicists, philosophers and historians as a transient phenomenon to be straightened out in the course of theory
development. However, quantum field theory has a special status
in this respect, as it has at its core singularly long-lived examples
of phenomenologically successful methods whose mathematical
foundations have until now escaped a rigorous reformulation:
Renormalization techniques, which involve mathematically questionable manipulations of ‘‘infinities,’’ but have allowed astonishingly precise predictions of observed phenomena (Brown, 1993)3.
The necessity of relying on such nonrigorous methods may be
regarded as an important factor promoting among theorists the
belief that demonstrations are in some sense as good as proofs. In
turn, as discussed in Section 5, demonstrations may involve not
only nonrigorous mathematics, but also thoughroughly nonmathematical elements which, are not only acceptable, but even
necessary, when theorists want to extend a mathematical ‘‘narrative’’ with other means. Indeed, renormalization techniques are
also associated with those tools (path integrals, Feynman integrals) which provided the basis for the hybrids of mathematics
and physics used by string theorists and analysed by Galison
(Galison, 2004, p. 54–55).
In conclusion, the modus operandi of model-builders which
has been the subject of the present study could be regarded as a
symptom not so much of the abandonment of the ideal of a ‘‘final
theory,’’ but rather of a crisis of the notion that rigorous
mathematical structures stand alone as the ultimate goal of
3
(Scharf, 2001) has been able to offer an infinity-free formalization of the
renormalization procedures, but his results are still based on a perturbative
expansion whose convergence is not ensured.
theoretical research in physics. In principle, the foundational
myth of theories as pure and rigorous mathematical constructs
continues thriving in today’s high-energy physics community, but
in practice mathematical theories are usually replaced by hybrid
constructs such as theoretical cores. Using Yehuda Elkana’s
terminology, one might suggest regarding this phenomenon as a
case of ‘‘two-tier thinking’’ linked to the gradual rise of an ‘‘image
of knowledge’’ (i.e., a socially determined view on knowledge) in
which mathematical proof does not anymore represent alone the
highest level in the hierarchy of methods of knowledge legitimization (Elkana, 1981). It is in this context that the building and
reshaping of Rosetta-stone models becomes a epistemic practice
of central importance, as it appears to be the appropriate tool to
grasp and combine elements of the hybrid theoretical level and
confront them with empirical claims demonstrating their validity
(or lack thereof). Physicists often admit to the importance of
verbal components, non-compelling empirical analogies and
numerical methods in their everyday practice, but at the same
time they play down the epistemic role of these factors, and are
ready to assume that all such elements will eventually be filled up
with rigorous mathematical meaning. For philosophers and historians it is however imperative to go beyond the study of
mathematical theories and take into account the interplay of
hybrid strategies of communication and representation in contemporary science.
Acknowledgements
The roots of this paper lie in my collaboration with Michael
Stöltzner, to whom I am deeply indebted. My reflections on the
epistemic implications of non-rigorous mathematical heuristics
were greatly enriched by comments from Erhard Scholz, whom I
warmly thank also for making me aware of the dispute around
the Jaffe&Quinn paper. Finally, I would like to warmly thank the
anonymous reviewer, whose acute comments much helped me
clarify the arguments in this paper. The present study has been
developed in context of the DFG-Project ‘‘Epistemic dynamics of
model-development at the LHC: an empirical investigation’’ at the
University of Wuppertal (Germany).
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