UTTG-25-2020

On the Development of Effective Field Theory

Steven Weinberg***Electronic address<EMAIL_ADDRESS>
Theory Group, Department of Physics, University of Texas
Austin, TX, 78712

Abstract

This is a lightly edited version of the talk given on September 30, 2020 to
inaugurate the international seminar series All Things EFT. It reviews some of
the early history of effective field theories, and concludes with a discussion
of the implications of effective field theory for future discovery.

What is the world made of? This question is perhaps is the deepest and
earliest in all of science. Greeks were asking this question a hundred years
before the time of Socrates. By the time that I became a graduate student an
answer had apparently been settled. The world is made not of water, earth, air
or fire, but of fields. There is the electromagnetic field that when quantum
mechanics is applied to it is manifested in the form of bundles of energy,
momentum – particles that are called photons. There is an electron field that
similarly when quantized appears as particles called electrons. And there are
other fields that we in the late 1950s knew we did not yet know about. The
weak and the strong interactions were pretty mysterious. It was clear that
there had to be more than just electrons and photons. But we looked forward to
a description of nature as consisting fundamentally of fields as the
constituents of everything.

The quantum field theory of electrons and photons in the late 1940s had scored
a tremendous success. Theorists – Feynman, Schwinger, Tomonaga, Dyson – had
figured out after decades of effort how to do calculations preserving not only
Lorentz invariance but also the appearance of Lorentz invariance at every
stage of the calculation. This allowed them to sort out the infinities in the
theory that had been noticed in the early 1930s by Oppenheimer and Waller, and
that had been the bête noire of theoretical physics throughout the 1930s. They
were able to show in the late 1940s that these infinities could all be
absorbed into a redefinition, called a renormalization, of the electron mass
and charge and the scales of the various fields. And they were able to do
calculations of unprecedented precision, which turned out to be verified by
experiment: calculations of the Lamb shift and the anomalous magnetic moment
of the electron.

More than that, and this particularly appealed to me as a graduate student,
renormalization theory didn’t always work. It wouldn’t work unless the theory
had a certain kind of simplicity. Essentially the only coupling constants
allowed in the theory, in units where Planck’s constant and the speed of light
were 1, had to be dimensionless, like the charge of the electron: $e^{2}/4\pi$
is $1/137$. And this provided not only a means of dealing with the infinities
but a rationale for the simplicity of the theory. Of course we always like
simple theories. But when we have discovered successful simple theories we
always should ask why are they so simple? Renormalizability provided an answer
to that question. Of course it was only an answer if we thought that these
were really the fundamental theories that described nature at all scales.
Otherwise, anything else might intervene to get rid of the infinities.

We hoped that we would see the rest of physics – the mysterious strong and
weak nuclear forces – brought into a similar framework. And that is indeed
what happened in the following decades in the development of the Standard
Model. Once we got past the obscurities produced by spontaneous symmetry
breaking in the weak interactions and color trapping in the strong
interactions, the Standard Model was revealed to us as a theory that was
really not very different from quantum electrodynamics. We had more gauge
fields, not just the electromagnetic field but gluon and $W$ and $Z$ fields.
There were more fermions, not just the electron but a whole host of charged
leptons and neutrinos and quarks. But the Standard Model seemed to be quantum
electrodynamics writ large. One could perhaps have been forgiven for reaching
a stage of satisfaction that, although not everything was answered, although
there were still outstanding questions, that this was going to be part of
nature at the most fundamental level.

Now that has changed. In the decades since the completion of the Standard
Model a new and cooler view has become widespread. The Standard Model, we now
see – we being, let me say, me and a lot of other people – as a low-energy
approximation to a fundamental theory about which we know very little. And low
energy means energies much less than some extremely high energy scale
$10^{15}-10^{18}\;{\rm GeV}\,$. As a low energy approximation we expect
corrections to the Standard Model. These corrections are beginning to show up.
Some of them have already been found.

This whole point of view goes by the name of effective field theory. It’s had
applications outside elementary particle physics in areas like
superconductivity. I am not going to in this talk try to bring the subject up-
to-date, including all the applications of effective field theory to hadronic
physics, and to areas of physics outside particle physics, like
superconductivity. That’s going to be done by subsequent lecturers in this
series by physicists who played a leading role in the development of effective
field theory beyond anything that I knew about it in the early days. They are
true experts in the field. I won’t dare to try to anticipate what they will
say. I’ll talk about a subject on which I am undoubtedly the world’s expert
and that is my own history of how I came to think about these things. I’m a
little bit unhappy that I am putting myself too much forward. Other people
came to effective field theories through different routes. I’m not going to
survey anyone else’s intellectual history except my own.

From my point of view, it started with current algebra. The late 1950s and
early 1960s were a time of despair about the future – about the practical
application of quantum field theory to the strong interactions. Although we
could believe that quantum field theory was at the root of things we didn’t
know how to apply quantum field theory to the strong interactions.
Perturbation theory didn’t work. Instead, a method was developed called
current algebra in which one concentrated on the currents of the weak
interactions, the vector and axial vector currents, using their commutation
relations, their conservation properties and in particular a suggestion made
by Nambu that the divergence of the axial vector current was dominated by one
pion states. This current algebra was used in a very clunky way, requiring
very detailed calculations that just called out for a simpler approach to
derive useful results, including the Goldberger-Treiman formula for the pion
decay amplitude and the Adler-Weisberger sum rule for the axial vector
coupling constant.

After a while some of us began to think that although these results were
important and valuable, perhaps we were giving too much attention to the
currents themselves, which of course play a central role in the weak
interactions. We ought to concentrate on the symmetry properties of the strong
interactions which made all this possible. In particular, the existence of a
symmetry which gradually emerged in our thinking, chiral $SU(2)\times SU(2)$,
which is just isotopic spin symmetry applied separately to the left handed and
right handed parts of what we would now say are quark fields.

This symmetry is a property of the strong interactions which would be
important even if there weren’t any weak interactions. And it was employed to
derive purely strong interaction results, like for example, the scattering
lengths of pions on nucleons and pions on pions and more complicated things
like the emission of any number of soft pions in high energy collisions of
nucleons or other particles. When this was done using these, as I said, clunky
methods of current algebra, looking at the results, they seemed to look like
the results of a field theory. You could write down Feynman diagrams just out
of the blue which would reproduce the results of current algebra.

And so the question naturally arose, is there a way of avoiding the machinery
of current algebra by just writing down a field theory that would
automatically produce the same results with much greater ease and perhaps
physical clarity? Because after all in using current algebra one had to always
wave one’s hands and make assumptions about the smoothness of matrix elements,
whereas if you could get these results from Feynman diagrams you could see
what the singularity structure of the matrix elements was and make only those
smoothness assumptions that were consistent with that.

At the beginning this was done using a standard theory with the chiral
symmetry that we thought was at the bottom of all these results, the linear
sigma model, and then re-defining the fields in such a way that the results
would look like current algebra. The effect of the redefinition was the
introduction of a non-linearly-realized chiral symmetry. Eventually the linear
sigma model was scrapped; instead the procedure was simply to ask, what kind
of symmetry transformation for the pion field alone, some transformation into
a nonlinear function of the pion field, would have the algebraic properties of
chiral symmetry, based on the Lie algebra of $SU(2)\times SU(2)$.

That theory had the property that in lowest order in the coupling constant
$1/F_{\pi}$ the results reproduced the results of current algebra. Why did it?
Well, it had to, because a theory having chiral and Lorentz invariance and
unitarity and smoothness satisfied the assumptions of current algegra and
therefore had to reproduce the same results. For example current algebra
calculations gave a pion-pion scattering matrix element of order
$1/F_{\pi}^{2}$, where $F_{\pi}$ is the pion decay amplitude, so if you use
this field theory and just threw away everything except the leading term which
is of order $1/F_{\pi}^{2}$, this matrix element had to agree with the results
of current algebra.

In this way phenomenological Lagrangians were developed that could be thought
of as merely labor-saving devices, which were guaranteed to give the same
results as current algebras because they satisfied the same underlying
conditions of symmetry and so on, and that could be used in lowest order
because current algebra said the result was of lowest order in the $1/F_{\pi}$
coupling – that is, $1/F_{\pi}^{2}$ for $\pi\pi$ scattering and also for pion-
nucleon scattering. If you had more pions you would have more powers of
$1/F_{\pi}$. But the results would always agree with current algebra.

No one took these theories seriously as true quantum field theories at the
time. (I am talking about the late 1960s.) No one would have dreamed at this
point of using the phenomenological Lagrangian in calculating loop diagrams.
What would be the point? We knew that it was the tree approximation that
reproduced current algebra. As I said, these phenomenological Lagrangians were
simply labor-saving devices.

The late 1960s and 1970s saw many of us engaged in the development of the
Standard Model. During this time I wasn’t thinking much about current algebra
or phenomenological Lagrangians. The soft pion theorems had been successful,
not only in agreeing with experiment, but also in killing off a competitor of
quantum field theory known as S-matrix theory. S-matrix theory had been the
slogan of a school of theoretical physicists headed by Geoff Chew at Berkeley.
I had been there at Berkeley in its heyday but had never bought on to it.

Their idea was that field theory is hopeless. It deals with things we will
never observe like quantum fields. What we should do is just to study things
that are observable, like S-matrix elements: apply principles of Lorentz
invariance, analyticity and so on, and get results like dispersion relations
that we can compare with observation. It was even hoped that stable or
unstable composite particles like the $\rho$ meson provide the force that
produces these composites, so that using this bootstrap mechanism one could
actually do calculations. This never really worked as a calculational scheme,
but was extremely attractive philosophically because it made do with very
little except the most fundamental assumptions, without introducing things
like strongly interacting fields that we really didn’t know about. But the
chiral symmetry results, the soft-pion results, showed that some of the
approximations assumed in using S-matrix theory, like strong pion-pion
interactions at low energy, just weren’t right. Chiral symmetry provided
actual calculations of processes like $\pi\pi$ and $\pi$-nucleon scattering,
which made the ideas of S-matrix theory seem unnecessary, attractive as the
philosophy was.

S-matrix theory had been largely killed off and chiral symmetry had been put
in the books as a success, but we were all involved in applying the ideas of
quantum field theory in the weak and the electromagnetic interactions and then
the strong interactions, building up the Standard Model, which was beginning
to be very successful experimentally. It was a very happy time for the
interaction between theory and experiment. During this period of course I was
teaching. It was in the course of teaching that my point of view changed,
because in teaching quantum field theory I had to keep confronting the
question of the motivation for this theory. Why should these students take
seriously the assumptions we were making, in particular the formalism of
writing fields in terms of creation and annihilation operators, with their
commutation relations. Where did this come from? The standard approach was to
take a field theory like Maxwell’s theory and quantize it, using the rules of
canonical quantization. Lo and behold, you turn the crank, and out come the
commutation relations for the operator coefficients of the wave functions in
the quantum field.

I found that hard to sell, especially to myself. Why should you apply the
canonical formalism to these fields? The answer that the canonical formalism
had proved useful in celestial mechanics in the 19th century wasn’t really
very satisfying. In particular, suppose there was a theory that in other ways
was successful but couldn’t be formulated in terms of canonical quantization,
would that bother us? In fact, we have quantum field theories like that.
They’re not realistic theories. They’re theories in six dimensions, or
theories we derive by compactifying six dimensional theories. We have quantum
field theories that apparently can’t be given a Lagrangian formalism – that
can’t be derived using the canonical formalism. So I looked for some other way
of teaching the subject.

I fastened on a point of view that is really not that different from S-matrix
theory. One starts of course by assuming the rules of quantum mechanics as
laid down in the 1920s, together with special relativity and then one makes an
additional assumption, the cluster decomposition principle, whose importance
was emphasized to me by a colleague at Berkeley, Eyvind Wichmann, while I was
there in the 1960s. The cluster decomposition principle says essentially that
unless you make special efforts to produce an entangled situation the results
of distant experiments are uncorrelated. The results of an experiment at CERN
are not affected by the results being obtained by an experiment being done at
the same time at Fermilab. The natural way of implementing the cluster
decomposition principle is by writing the Hamiltonian as a sum of products of
creation and annihilation operators with non-singular coefficients. Indeed,
this had been done for many years by condensed matter physicists not because
they were interested in quantum field theory as a fundamental principle, but
in order to sort out the volume dependence of various thermodynamic
quantities. They were managing this by introducing creation and annihilation
operators long before I began to teach courses in quantum field theory.

It gradually appeared to me in teaching the subject that although individual
quantum field theories like quantum electrodynamcis certainly have content
quantum field theory in itself has no content except the principles on which
it’s based, namely quantum mechanics, Lorentz invariance, and the cluster
decomposition principle, together with whatever other symmetry principles you
may want to invoke, like chiral symmetry and gauge invariance.

This means that if we think we know what the degrees of freedom are – the
particles we have to study – like for example, low energy pions, that if we
write down the most general possible theory involving fields for these
particles, including all possible interactions consistent with the symmetries,
which in this case are Lorentz invariance and chiral $SU(2)\times SU(2)$, if
we write down all possible invariant terms in the Lagrangian, and we work to
all orders in perturbation theory, the result can’t be wrong, because it’s
just a way of implementing these principles. It is just giving you the most
general possible matrix element consistent with Lorentz invariance, chiral
symmetry, quantum mechanics, cluster decomposition, and unitarity.

Now, this may not sound as if it’s a very useful realization. If you tell
someone to calculate using the most general possible Lagrangian with an
unlimited number of free parameters and calculate to all orders in
perturbation theory they’re likely to seek some advice elsewhere on how to
spend their time. But it’s not that bad, because even if the theory has no
small dimensionless couplings, you can use this approach to generate a power
series in powers of the energy that you’re interested in. For instance, if
you’re interested in low-energy pions and you don’t want to consider energies
high enough so that $\pi\pi$ collisions can produce nucleon-antinucleon pairs,
you will be dealing with typical energies $E$ well below the nucleon mass.
This very general Lagrangian gives you a power series in powers of $E$.
Specifically, aside from a term that depends only on the nature of the process
being considered, the number of powers of $E$ arising from a given Feynman
diagram is the total number of derivatives acting at all the vertices, plus
half the number of nucleon lines connected to all the vertices, plus twice the
number of loops.

Now, chiral symmetry dictates that the number of derivatives plus half the
number of nucleon fields in each interaction is always equal to or greater
than two. The diagrams that give the lowest total number of powers of $E$ are
those constructed only from interactions where that number equals two, with no
loops. For pion-pion scattering the leading term comes from a single vertex
with just two derivatives. For pion-nucleon scattering there is a diagram with
a single vertex with one deerivative to which are connected two nucleon lines.
Thsee diagrams are the ones that we had been using since the mid 1960s to
reproduce the results of current algebra.

But now the new thing was that you could consider contributions of higher
order in energy. If you look for terms that have two additional powers of
energy they could come from diagrams where you have one interaction with the
number of derivatives plus half the number of nucleon fields equaling not two
but four, plus any number of interactions with this number equal to two, and
no loops. Or you could have only interactions with the numbers of derivatives
plus half the number of nucleon fields equal to two, that is, just the basic
interactions that reproduce current algebra, plus one loop. The infinity in
the one loop diagram could be canceled by that one additional vertex that has,
say, not two derivatives but four derivatives. In every order of perturbation
theory as you encounter more and more loops, because you are allowing more and
more powers of energy, you get more and more infinities, but there are always
counterterms available to cancel the infinities. A non-renormalizable theory,
like the soft pion theory, is just as renormalizable as a renormalizable
theory. You have an infinite number of terms in the Lagrangian, but only a
finite number are needed to calculate S-matrix elements to any given order in
$E$.

Similar remarks apply to gravitation, which I think has led to a new
perspective on general relativity. Why in the world should anyone take
seriously Einstein’s original theory, with just the Einstein-Hilbert action in
which only two derivatives act on metric fields? Surely that’s just the lowest
order term in an infinite series of terms with more and more derivatives. In
such a theory, loops are made finite by counterterms provided by the higher
order terms in the Lagrangian. This point of view has been actively pursued by
Donoghue and his collaborators.

Teaching came to my aid again. At the blackboard one day in 1990 it suddenly
occurred to me that one of the kinds of interaction that has the number of
derivatives plus half the number of nucleon fields equal to two is the
interaction with no derivatives at all and four nucleon fields. It had taken
me a decade to realize that four divided by two is two. This sort of
interaction is just the kind of hard-core nucleon-nucleon interaction that
nuclear physicists had always known would be needed to understand nuclear
forces. But now we had a rationale for it. In a calculation of nuclear forces
as a power series in energy, the leading terms are just the ones that the
nuclear physicists had always been using, pion exchange plus a hard core. This
point of view has been explored by Ordoñez and van Kolck and others.

In the theories that I have been discussing, the chiral symmetry theory of
soft pions and general relativity, the symmetries don’t allow a purely
renormalizable interaction. In a theory of gauge fields and quarks and leptons
and scalars you can have a renormalizable Lagrangian. Throwing away everything
but renormalizable interactions you have just the Standard Model, a
renormalizable theory of quarks and leptons and gauge fields and a scalar
doublet. But now we can look at the Standard Model as one term in a much more
general theory, with all kinds of non-renormalizable interactions, which yield
corrections if higher order in energy.

In these corrections the typical energy $E$ must be divided by some
characteristic mass scale $M$. For the chiral theory of soft pions this mass
scale is $M\approx 2\pi F_{\pi}$, about $1200\,{\rm MeV}$. For the theory of
weak, strong and electromagnetic interactions $M$ is probably a much higher
scale, perhaps something like the scale of order $10^{15}{\rm GeV}\,$ where
Georgi, Quinn and I found the effective gauge couplings of the weak, strong,
and electromagnetic interactions all come together. Or perhaps it’s the
characteristic scale of gravitation, the Planck scale $10^{18}\;{\rm GeV}\,$.
Or perhaps somewhere in that general neighborhood. It is the large scale of
$M$ that made it a good strategy in constucting the Standard Model to look for
a renaormalizable theory .

We now expect that there are corrections to the renormalizable Standard Model
of the order of powers of $E/M$. How in the world are we ever going to find
corrections that are suppressed by such incredibly tiny fractions? The one
hope is that those corrections can violate symmetries that we once thought
were inviolable, but that we now understand are simply accidents, arising from
the constraint of renormalizability that we imposed on the Standard Model.

Indeed, quite apart from the development of effective field theory, one of the
great things about the Standard Model was that it explained various symmetries
that could not be fundamental, becasue we already knew they were only partial
or approximate symmetries. This included flavor conservation, such as
strangeness conservation, a symmetry of the strong and electromagnetic
interactions that was manifestly violated in the weak interactions. Another
example was charge conjugation invariance – likewise a good symmetry of strong
and electromagnetic but violated by weak interactions. The same was true of
parity, although in this case you have to make special accommodations for non-
perturbative effects. All thse were accidental symmetries. imposed by the
simplicity of the Standard Model necessary for renormalizability plus other
symmetries like gauge symmetries and Lorentz invariance that seem truly exact
and fundamental. Chiral symmetry itself is such an accidental symmetry, though
only approximate. It becomes an accidental exact symmetry of the strong
interactions in the limit in which the up and down quark masses are zero, as
does isotopic spin symmetry. Since these masses are not zero but relatively
small chiral symmetry is an approximate accidental symmetry.

Now, coming back to effective field theory, there are other symmetries within
the Standard Model that are accidental symmetries of the whole renormalizable
theory of weak, strong, and electromagnetic interactions: in particular baryon
conservation and lepton conservation are respected aside from very small non-
perturbative effects (well, very small at least in laboratories, though maybe
not so small cosmologically). If baryon and lepton conservation are only
accidental properties of the Standard Model maybe they are not symmetries of
nature. In this case there is no reason why baryon and lepton conservation
should be respected by non-renormalizable corrections to the Lagrangian, and
so you would expect terms of ${\cal O}(E/M)$ or ${\cal O}((E/M)^{2})$ or
higher order as corrections to the Standard Model that violate these
symmetries.

Wilczek and Zee and I independently did a catalog of the leading terms of this
type. Some of them – those involving baryon number non-conservation – give you
corrections of ${\cal O}((E/M)^{2})$. They have not been yet been discovered
experimentally. But there are other terms that produce corrections of ${\cal
O}(E/M)$ that violate lepton conservation, and they apparently have been
discovered, in the form of neutrino masses. I wish I could say that the
effective field theory point of view had predicted the neutrino masses.
Unfortunately it must be admmitted that neutrino masses were already proposed
by Pontecorvo as a solution to the apparent deficit of neutrinos coming from
the Sun. So though I can’t say that the effective field theory approach had
predicted neutrino masses, I do think that it gets a strong boost from the
fact that we now have evidence of a correction to the renormalizable part of
the Standard Model. But of course there already was a known correction.
Gravitation was always there, warning us that renormalizable quantum field
theory can’t be the whole story.

I expect that sooner or later we will be seeing another departure from the
renormalizable Standard Model in the discovery of proton decay, or some other
example of baryon nonconservation. In a sense baryon nonconservation has
already been discovered, because we know from the present ratio of baryon
number to photon number that in the early universe before the temperature
dropped to a few GeV there was about 1 excess quark over every $10^{9}$ quark-
antiquark pairs. This has to be explained by the non-renormalizable
corrections to the Standard Model, and indeed it has been explained,
unfortunately not just by one such model but by many different models. We
don’t know the actual mechanism for producing baryon number in the early
universe, but I have no doubt that it will be found.

There are still unnatural elements in the Standard Model. I said that you
would expect the leading terms that describe physical reactions to be given by
the renormalizable theory – here the Standard Model – with the effects of non-
renormalizable terms suppressed by powers of $E/M$. Those corrections come
from interactions that have coupling constants whose dimensions have negative
powers of mass, $1/M$, $1/M^{2}$, and so on. But what about interactions that
have positive powers of mass? Why aren’t they there at ${\cal O}(M^{n})$? Well
unfortunately we don’t have a good explanation.

The cosmological constant is such a term. It has the dimensions of energy per
volume, in other words $M^{4}$. We don’t know why it’s as small as it is.
There is also the bare mass of the Higgs boson. That’s the one term in the
Standard Model Lagrangian whose coefficient has the dimensionality of a
positive power of mass. We don’t know why it is not ${\cal O}(10^{15}{\rm
GeV}\,)$. These are great mysteries that confront us: Why are the terms in our
present theory that have the dimensions of positive powers of mass so small
compared to the scale that we think is the fundamental scale, somewhere in the
neighborhood of $10^{15}-10^{18}\;{\rm GeV}\,$? We don’t know.

With the new approach to the Standard Model I think we have to say that this
theory in its original form is not what we thought it was. It’s not a
fundamental theory. But at the same time I want to stress that the Standard
Model will survive in any future textbooks of physics in the same way that
Newtonian mechanics has survived as a theory we use all the time applied to
the solar system. All of our successful theories survive as approximations to
a future theory.

There’s a school of philosophy of science associated in particular with the
name of Thomas Kuhn, that sees the development of science – particularly of
physics – as a series of paradigm shifts in which our point of view changes so
radically that we can barely understand the theories of earlier times. I don’t
believe it for a minute. I think our successful theories always survive, as
Newtonian mechanics has, and as I’m sure the Standard Model will survive, as
approximations.

Now we have to face the question, approximations to what? We think the
Standard Model is a low energy approximations to a theory whose constituents
involve mass scales way beyond what we can approach in the laboratory, scales
on the order of $10^{15}$, $10^{18}{\rm GeV}\,$. It may be a field theory. It
may be an asymptomatically safe field theory, which avoids coupling constants
running off to infinity as the energy increases. Or it seems to me more likely
that it’s not a field theory at all, that it’s something like a string theory.
In this case we will understand the very successful field theories with which
we work as effective field theories, embodying the principles of quantum
mechanics and symmetry, applied as an approximation valid at low energy where
any theory will look like a quantum field theory.

ACKNOWLEDGMENTS

This work is supported by the National Science Foundation under grant number
Phy-1914679 and also with support from the Robert A. Welch Foundation, Grant
No. F-0014.