Nobel lecture: from weak interactions to gravitation

9 Pages·2000·0.094 MB·English

by  Veltman M.J.G.

Nobel Lecture: From weak interactions to gravitation*
Martinus J. G. Veltman†
DepartmentofPhysics,UniversityofMichigan,AnnArbor,Michigan48109-1120
[S0034-6861(00)00602-4]
INTRODUCTION bosonvertexshowedadisappearanceofmanydivergen-
cies for a properly chosen vector-boson magnetic
This lecture is about my contribution to the proof of
moment. In studying Yang-Mills theories I noted that
renormalizability of gauge theories. There is of course
those theories automatically produced this particular
no perfectly clear separation between my contributions
magnetic moment. I therefore concluded that Yang-
and those of my co-laureate ’t Hooft, but I will limit
Mills theories are probably the best one can have with
myself to some brief comments on those publications
respect to renormalizability. Thus I was led to the study
that carry only his name. An extensive review on the
of renormalizability of these theories.
subjectincludingmoredetailedreferencestocontempo-
(III) Technical progress. Starting then on the study of
rary work can be found elsewhere (Veltman, 1992a).
diagrams in a Yang-Mills theory I established the van-
Asiswellknown,theworkontherenormalizabilityof
ishing of many divergencies, provided the external legs
gauge theories caused a complete change of the land-
of the diagrams were on the mass shell. That by itself is
scapeofparticlephysics.Theworkbroughtcertainmod-
not enough with respect to renormalizability, because
els to the foreground; neutral currents as required by
those models were established and the discovery of the that requires diagrams and Feynman rules of a renor-
J/C was quickly interpreted as the discovery of charm, malizable type. I was thus led to search for a transfor-
part of those models as well. More precisely, we refer mationofthetheorysuchthatnew,renormalizable-type
here to the model of Glashow (1961), the extension to Feynman rules were derived, without changing the S
include quarks by Glashow, Iliopoulos, and Maiani matrix. In this I succeeded up to one loop.
(GIM; 1970) and the model of Weinberg and Salam None of these points is trivial, as can be shown easily
(Weinberg, 1967; Salam, 1968) for leptons including a by considering work in that period. For example, Wein-
Higgs sector. The GIM paper contained discussions on berg (1979) in his 1979 Nobel lecture reports that he
the required neutral hadron currents and also the inclu- interpretedthesuccessoftheAdler-Weisbergerrelation
sionofcharmassuggestedfirstbyHara(1964).Afteran as a property of strong interactions, namely, the validity
analysis by Bardeen, at a seminar in Orsay (see also of chiral SU23SU2. Consequently he continued work-
Bardeen, 1969), the work of Bouchiat, Iliopoulos, and ing on things such as p2p scattering. Feynman is re-
Meyer(1972)establishedthevanishingofanomaliesfor puted to have exclaimed that he never thought of inves-
three color quarks. Without going into details, subse- tigating the renormalizability aspect of Yang-Mills
quentlyquantumchromodynamicscametobeaccepted. theories when he heard of that development. Finally,
InthiswaytheStandardModelwasestablishedinjusta there existed several papers where the non-
few years. renormalizability of Yang-Mills theories was ‘‘proven,’’
for example one by Salam (1962).
In the following I will discuss these three points in
ESSENTIALSTEPS detail as they developed historically.
Let me review here what I consider as my contribu-
tion to the subject. It can be described in three separate
parts. I will try to simplify things as much as possible.
(I) The physics argument. In 1965 Adler (1965b) and THEPHYSICSARGUMENT
Weisberger(1965)establishedwhatisnowknownasthe
Adler-Weisberger relation. This relation, numerically In 1965 Adler and Weisberger derived their famous
agreeing with experimental data, was interpreted by me relation between the axial-vector coupling constant of b
as a consequence of a Ward identity of a non-Abelian decay in terms of a dispersion integral for pion-nucleon
gauge theory (also called Yang-Mills theories), and as scattering. This relation, agreeing well with experiment,
such guided me to take up the study of such theories. was based on Gell-Mann’s current commutator rules
(II) The renormalizability argument. Earlier calcula- (Gell-Mann, 1964). Subsequently an extensive discus-
tions on the radiative corrections to the photon-vector- siondevelopedintheliteratureconcerningtheso-called
Schwinger terms that could invalidate the argument. I
decided to try to derive these same results starting from
*The 1999 Nobel Prize in Physics was shared by Gerard ’t another assumption, and as a starting point I took the
Hooft and Martinus J. G. Veltman. This lecture is the text of well-known conserved-vector-current (CVC) and
ProfessorVeltman’saddressontheoccasionoftheaward. partially-conserved-axial-vector-current (PCAC) equa-
†Electronicaddress:[email protected] tions for the weak currents:
ReviewsofModernPhysics,Vol.72,No.2,April2000 0034-6861/2000/72(2)/341(9)/$16.80 ©TheNobelFoundation1999 341
342 MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation
]mJmV50, tons. The paper of Lee and Yang mainly concentrated
on deriving the Feynman rules for vector bosons. The
]mJmA5iap. troubleatthattimewasthatindoingtheusualcanonical
These equations do not include higher-order electro- derivationoneencounteredcertaincontacttermsforthe
magnetic(e.m.)orweakeffects.AsafirststepItriedto vector-boson propagator. I will not delve any further
include electromagnetic effects by using the well-known into this; later I found a simple way to circumvent these
substitution]m!]m2iqAm, whereq isthechargeofthe problems. In those days, however, these were consid-
object on which ]m operates. Since the currents were ered serious problems.
isovectors, that could be done easily using isospin nota- Subsequently Lee started a complicated calculation,
tion. For the vector current the equation became namely, the lowest-order radiative corrections to the
~]m1ieAWm3!JWmV50, ]vemc!to]rm-b2oiseoAn–mphinottohnecvoeucptloinr-gb.oTsohne Lusaugarlanrgeipalnaceismneontt
treating the e.m. field as the third component of an is- sufficient to determine the vector-boson magnetic mo-
ovector. Next I used the idea that the photon and the ment; it remains an arbitrary parameter. This is because
chargedvectorbosonsmaybeseenasanisotriplet,thus oftheoccurrenceoftwoderivativessuchas]m]n; when
generating what I called divergency conditions (Velt- making the minimal substitution it matters whether one
man, 1966). For the axial-vector current this gave writes ]m]nor ]n]m, and this causes the arbitrariness in
]mJWmA5iapW1ieAWm3JWmA1igWWmV3JWmA1igWWmA3JWmV. the magnetic moment. Anyway, Lee, concentrating on
theelectricquadrupolemomentofthevectorboson,cal-
As a matter of technical expedience two vector culated the appropriate triangle diagram using a cutoff
bosonswereusedtodenoteavectorbosoncouplingtoa procedure called the j-limiting process.
vector current and a vector boson coupling to an axial I was very interested in this calculation, because like
current. This equation turned out to be adequate to de- many physicists I strongly believed in the existence of
rivetheAdler-Weisbergerrelation.Anaddedbenefitof vector bosons as mediators in weak interactions. This
this derivation was that no difficulties with respect to
belief was based on the success of the V-A theory, sug-
Schwinger terms arose, and the axial vector coupling
gestingavectorstructurefortheweakcurrents.Indeed,
constantwasrelateddirectlytothepion-nucleonscatter-
this led Glashow to his famous 1961 paper. I decided
ing length. The Adler-Weisberger relation evidently
that Lee’s work ought to be extended to other situa-
used an additional relation in which the pion-nucleon
tions, but it was quite obvious that this was no mean
scatteringlengthwasgivenintermsofadispersioninte-
task. Given the j method, and the occurrence of the
gral.
magneticmomentasanarbitraryparameter,thetriangle
In response, John Bell, then at CERN, became very
diagram if calculated fully (Lee limited himself to parts
interested in this derivation. He investigated what kind
relevant to him) gives rise to a monstrous expression
of field theory would generate such divergency condi-
involving of the order of 50000 terms in intermediate
tions, and he found that this would happen in a gauge
stages. There was simply no question of going beyond
theory (Bell, 1967).
the triangle diagram.
Subsequentdevelopmentsweremainlyaboutthecon-
At this point I decided to develop a computer pro-
sequencesofthoserelationsinvolvinge.m.fieldsonly.It
gram that could do this work. More specifically, I con-
is clear that specializing to the third component of the
centratedonthetrianglegraph,butIwrotetheprogram
axialdivergenceconditiontherearenoe.m.corrections.
Following Adler, reading ]mJmA05ap0 in the opposite insuchawaythatotherprocessescouldbeinvestigated.
In other words, I developed a general-purpose symbolic
wayimposedaconditiononthepionfieldincludinge.m.
manipulation program. Working furiously, I completed
effects. This was an extension of earlier work by Adler
thefirstversionofthisprograminaboutthreemonths.I
(1965a), known under the name of consistency condi-
called the program SCHOONSCHIP, among other reasons
tions for processes involving pions. In this case one of
toannoyeverybodynotDutch.Thenamemeans‘‘clean
the conclusions was that p0 decay into two photons was
ship,’’ and it is a Dutch naval expression referring to
forbidden, and without going into a detailed description
clearing up completely a messy situation. In January
this led to the work of Bell and Jackiw (1969) on the
1964,visitingNewYorkinconnectionwithanAmerican
anomaly. Simultaneously Adler (1969) discovered the
Physical Society meeting, I visited Lee and told him
anomaly and in fact used precisely my (unpublished)
about the program. He barely reacted, but I heard later
derivation to connect this to p0 decay. Later I became
that after I left the office he immediately wanted one of
quiteworriedbythisdevelopment,asIsawthisanomaly
the local physicists to develop an analogous program.
as a difficulty with respect to renormalization.
In toying with the calculation I tried to establish what
would be the best value for the vector-boson magnetic
THERENORMALIZABILITYARGUMENT moment with respect to the occurring divergencies.
There was one value where almost all divergencies dis-
Here I must go back to 1962. In that year Lee and appeared,butIdidnotknowwhattodowiththisresult.
Yang(1962)andlaterLee(1962)alonestartedasystem- It remained in my memory though, and it played a role
atic investigation of vector bosons interacting with pho- as explained below.
Rev.Mod.Phys.,Vol.72,No.2,April2000
MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation 343
TECHNICALPROGRESS sum of all diagrams that can be obtained by cutting the
initial diagram in all possible ways.
Toexplainthedevelopmentrequiressomebacktrack- The importance of this work was twofold. Not only
ing. In 1959 I took up the study of the problem of un- did unitarity become a transparent issue, I also learned
stable particles. The problem is of a nonperturbative toconsiderdiagramsdisregardingthewaytheywerede-
character, because a particle is unstable no matter how rived,forexampleusingthecanonicalformalism.Given
small the coupling constant of the interaction that pro- that it is not easy to derive Feynman rules for a Yang-
duces the decay. Thus the (unstable) particle will not Mills theory in the canonical way, that gave me an ad-
appearintheinandoutstatesoftheS matrix.However, vantage in studying that theory. For considering Yang-
for zero value of the coupling constant the particle is Mills theories, the path-integral formalism is quite
stableandmustbepartoftheinandoutstates.Thusthe adequate; there is only one point, and that is that this
limit of zero coupling constant does not reproduce the formalism does not guarantee unitarity. In 1968 the
zero-coupling-constant theory. path-integral formalism had all but disappeared from
It was in principle well known at the time how to the literature, although students of Schwinger did still
handle an unstable particle. Basically one did what is learnfunctionalmethods.Imyselfdidnotknowthefirst
called a Dyson summation of the propagator, and that thing about it.
indeed removed the pole in the propagator. From the In 1968 I was invited by Pais to spend a month at
Ka¨lle´n-Lehmann representation of the propagator one RockefellerUniversity.Ihappilyacceptedthisinvitation
knows that every pole corresponds to an in or out state, and decided to try to think through the present situa-
so the summation indeed seemed to correspond to re- tion. For two weeks I did nothing but contemplate the
moving the particle from the in and out states. whole of weak interactions as known at the time. I fi-
However, when performing the Dyson summation nally decided to take Bell’s conclusion seriously and
one found that the theory became explicitly nonpertur- therefore assumed that the weak currents were those of
bative, as self-energy diagrams and with them factors g a gauge theory. Thus I started to learn Yang-Mills
(the coupling constant of the destabilizing interaction) theory and tried to find out how that would work in
appeared in the denominator of the propagator of the some simple weak processes. In the process of writing
unstable particle. That propagator looked like this: down the Feynman rules I noted that this theory gave
precisely the ‘‘best’’ vertex (with respect to divergen-
1
. cies) as I had found out doing the work reported above
k21M21g2F~k!
on the photon–vector-boson interaction. This encour-
Obviously, this propagator cannot be expanded as a aged me to concentrate on the renormalizability aspect
function of g in the neighbourhood of k21M250 if the of the theory.
imaginarypartofF(k) isnonzerointhatpoint(thereal AsfarasIrememberIstartedbyconsideringtheone-
part is made zero by mass renormalization). So, instead loop corrections to neutrino-electron scattering. Here
ofapropagatorwithapoletheDysonsummationmade the situation became quickly quite complicated. The
it into a function with a cut in the complex k2 plane. At verticesoftheYang-Millstheoryweremuchmorecom-
this point it is no longer clear that the S matrix is uni- plicated than those that one was used to, and even the
tary, because the usual equation for the S matrix, S simplestdiagramsgaverisetoveryinvolvedexpressions.
5T@exp(iH)#, is no longer valid. In other words, to es- IntheendIdecidedtodropeverythingexceptthebasic
tablish unitarity one had to consider the diagrams by theory of vector bosons interacting with each other ac-
themselves. cordingtoaYang-Millsscheme.Inaddition,ofcourse,I
ThusIattackedthisproblem,essentiallyfinishingitin gavethesevectorbosonsamass,sincethevectorbosons
1961. This was for my thesis, under the supervision of ofweakinteractionswereobviouslymassive.Istartedin
Leon van Hove. The article went its ponderous Dutch blissfulignoranceofwhateverwaspublishedonthesub-
thesiswayandwaspublishedin1963(Veltman,1963),in ject, which was just as well or I might have been con-
a somewhat unusual journal (Physica) for high-energy vinced that Yang-Mills theories are non-renormalizable.
physics.Curiously,aboutthesametimeFeynman(1963) As Feynman said in his Nobel lecture as presented at
considered the same problem, in connection with estab- CERN, ‘‘Since nobody had solved the problem it was
lishing unitarity for the massless Yang-Mills theory, a obviously not worthwhile to investigate whatever they
theory whose diagrams include ghosts. These ghosts had done.’’ I want to mention here that at that time I
make unitarity nonevident. Moreover, the derivation by started to get worried about the anomaly, but I decided
Feynman, done with path integrals, did not guarantee to leave that problem aside for the moment.
unitarity. I am quite sure that he never saw my article, Consider the propagator for a massive vector field:
andIneverdiscusseditwithhimeither.Hetriedtodoit dmn1kmkn/M2
some other way, quite complicated, initially succeeding .
k21M2
onlyuptooneloop.Later,DeWitt(1964,1967a,1967b)
extended Feynman’s proof to any number of loops, but The source of all trouble is of course the kmknterm. So
my proof is much simpler and moreover connects quite anyone starting at this problem tries to eliminate this
directly to physical intuition. In fact, my proof had as a term. In quantum electrodynamics that can indeed be
result that the imaginary part of a diagram equals the done,butforYang-Millsfieldsthisisnotpossible.There
Rev.Mod.Phys.,Vol.72,No.2,April2000
344 MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation
is always some remnant. Now here a simple observation oftherenormalizabletype.Nomatterthatghostsoccur;
canbemade:thisbadtermcomeswithafactor1/M2. In those do not get in the way with respect to the renor-
fact, one can trace the worst divergencies in a diagram malization program.
simplybycountingfactors1/M2. Butgiventhattheywill There is a bonus to this procedure: one can write an
not ever completely cancel, as they indeed do not for a amplitude involving one such scalar field. Because the
Yang-Mills-type theory, then one will never get rid of scalar field is a free field, that amplitude must be zero if
these divergencies unless somehow these factors 1/M2 all other external lines are on the mass shell. That then
cancel out. But where should the necessary factors M2 gives an identity. Using Schwinger’s source technique
come from? There is only one way, and that is through one can extend this to the case in which one or more of
external momenta that are on the mass shell, meaning the other external lines are off the mass shell. I later
that the momentum p of such an external line satisfies called the resulting identities generalized Ward identi-
p252M2. And here is the problem. ties.
Renormalizationmeansthatforadivergentgraphone Thereisanotheraspecttothisprocedure.Becausethe
cannot take the external momentum on the mass shell final diagrams contain a vector-boson propagator that
and then do the necessary subtractions, because the hasnokmknpart,thattheoryisnotevidentlyunitary.At
graph may occur as part of a more complicated graph. that point one would have to use the cutting rules that I
For example, in a box diagram there may be a self- hadobtainedbeforeandshow,usingWardidentitiesfor
energy insertion in one of the internal lines. There mo- thecutscalarlines,thatthetheorywasunitary.Allinall
menta of the lines attached to the self-energy insertion quite a complicated affair, but not that difficult.
are not on the mass shell, thus it is not sufficient to sub- Here the miracle occurred. On the one-loop level al-
tract only those divergencies that remain if those mo- mostalldivergenciesdisappeared.Itwasnotasstraight-
menta are on the mass shell. You would still have to forward as I write it here, because even with the new
showthattheextradivergenciesarisingwhenthoselines rules one needed to do some more work using Ward
are not on the mass shell actually cancel, a gruesome identities to get to the desired result. In any case, I ar-
task. What to do? rived at Feynman rules for one-loop diagrams that were
Well, what I did was to reformulate the theory such by ordinary power-counting rules renormalizable rules.
thatsomehowallcancellationswereimplementedinthe For those who want to understand this in terms of the
rules. In the first instance I took the Stu¨ckelberg tech- modern theory: instead of a Faddeev-Popov ghost (with
nique (Stu¨ckelberg, 1938; see Veltman, 1992a for other a minus sign for every closed loop) and a Higgs ghost
references): I added a scalar field and made couplings (no minus sign, but a symmetry factor) I had only one
involvingderivativessuchthatitappearedtogetherwith ghost, and on the one-loop level that was actually the
the vector-boson propagator difference of the two ghosts as we know them now.
No one will know the elation I felt when obtaining
dmn1kmkn/M21kkmkn/M2. this result. I could not yet get things straight for two or
k21M2 k21M2 more loops, but I was sure that that would work out all
right. The result was for me a straight and simple proof
The second term would be due to the exchange of the
scalar particle. The parameter kwas introduced to keep that my ideas were correct. A paper presenting these
track of the counter term. Eventually kwas taken to be results was published (Veltman, 1968; see also Veltman,
21. Now this new field is physically undesirable, be- 1992b).
The methods in that paper were clumsy and far from
cause to have k521 is actually impossible. The scalar
transparentorelegant.Theideas,however,wereclear.I
field would have to have indefinite metric, or some such
cannot resist quoting the response of Glashow and Il-
horrible thing. In an Abelian theory it is easy to show
iopoulos (1971). After my paper appeared they decided
that the field is noninteracting, but not in the non-
to work on that problem as well, and indeed, they
Abelian case. Then I had an idea: introduce further in-
showed that many divergencies cancelled, although not
teractions of this new scalar field in such a way that it
anywhere to the renormalizable level. For example, the
becomes a free field. The result, hopefully, would be a one-loopboxdiagramisdivergentlikeL8 intheunitary
new theory, involving a well-behaved vector-boson gauge; their paper quoted a result of L4. I of course
propagator and furthermore an interacting scalar field obtained the degree of divergence of a renormalizable
that would then be a ghost. Indeed, being a free field it theory, i.e., log(L). Here then is a part of the footnote
could appear in the final state only if it was there in the theydevotetothispoint:‘‘ThedivergenciesfoundbyM.
initialstate.AtthispointonewouldhavenewFeynman Veltman go beyond the theorem proven in this paper,
rules, presumably much less divergent because of the but they only apply to on-mass-shell amplitudes... .’’
improvedvector-bosonpropagator.Itwasallamatterof Indeed!
what Feynman rules would result for this scalar field. If In present-day language one could say that I made a
they were those for a renormalizable field, then we transformation from the unitary gauge to a ‘‘renormal-
would be in business! izable’’ gauge. As I had no Higgs the result was not
So here is the important point: the theory must be perfect.Buttheideaisthere:theremaybedifferentsets
formulatedintermsofdiagramswhichwouldhavetobe of Feynman rules giving the same S matrix.
Rev.Mod.Phys.,Vol.72,No.2,April2000
MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation 345
INTERIM workwithVanDamwasactuallymyfirstexerciseinthe
quantum theory of gravitation.
Intheyears1969–1971Iexpendedconsiderableeffort So, by the end of 1970 I was running out of options. I
trying to go beyond the one-loop result. There were started to think of studying the difference between the
many open problems, and they had to be considered. In masslessandmassivecases,moreexplicitly,totrytosort
thebeginningof1970Istreamlinedthederivationtothe ofsubtractthemasslesstheoryfromthemassivetheory,
point that it became transparent. This was done by de- diagramwise. That would have produced a hint to the
riving Ward identities using Schwinger’s source tech- Higgs system. Indeed, the theory with a Higgs particle
niques(Veltman,1970).Thisisreallymuchliketheway allows a continuous approach to the massless theory
one derives Ward identities today, now called Slavnov- with, however, four extra particles. Furthermore, I was
Taylor identities. The Becchi-Rouet-Stora transforma- toyingwiththeideatoseeiftheremaininginfinitieshad
tion is a sophisticated form of the free-field technique a sign that would allow subtraction through some fur-
(using anticommuting fields). I remember being upset ther interactions. Conceivably, all this could have re-
when I first heard a lecture by Stora on what he called sultedintheintroductionofanextraparticle,theHiggs,
the Slavnov-Taylor identities. I told him that they were withinteractionstunedtocancelunwanteddivergencies,
anothervariantofmygeneralizedWardidentities.How- or to readjust the one-loop counter terms to be gauge
ever, Stora is not a diagram man, and I am sure that he invariant (in my paper the four-point counter terms
never understood my paper. werenot).Theresultwouldhavebeenintheworstkind
Another issue was the limit of zero mass of the mas- of ‘‘Veltmanese’’ (a term used by Coleman to describe
siveYang-Millstheory.InJanuary1969therewasacon- the style of ’t Hooft’s first article). However, that devel-
ference at CERN and I announced that two-loop dia- opment never happened, and hindsight is always easy.
That kind of work needed something else: a regulariza-
gramsforthemassiveYang-Millstheorydidnotgoover
tion procedure. Not only was the lack of a suitable
into those of the massless theory, in other words, the
method impeding further investigation or application of
massless theory is not the limit of zero mass of the mas-
the results obtained so far to practical cases, but there
sive theory (Veltman, 1969). This argument was spelled
wasalsothequestionofanomalies.Itisatthispointthat
outinanarticlewithJ.Reiff(ReiffandVeltman,1969).
’t Hooft entered into my program.
Themainpartofthatarticlewastotieupanotherloose
’t Hooft became my student somewhere in the begin-
end: the Feynman rules for vector bosons in the unitary
ningof1969.Hisfirsttaskwaswritingwhatwascalleda
gauge. The argument is quite elegant and superseded
‘‘scriptie,’’asortofpredoctoralthesis,orthe`setroisie`me
the article by Lee and Yang mentioned before. This cycle(inFrance).Thesubjectwastheanomalyandthes
made it clear to me that spurious contact terms related
model.Thatbeingfinishedinthecourseof1969hethen
to that part of the theory were not responsible for the
started on his Ph.D. work. At the same time he took
two-loop problems. part in my path-integral enterprise, so let me describe
Somewhere in the first half of 1970 I heard via that.
Zumino that Faddeev (and Slavnov, as I learned later; Ispenttheacademicyear1968–1969attheUniversity
SlavnovandFaddeev,1970)hadestablishedthatalready at Orsay, near Paris. During the summer of 1968 I was
at the one-loop level the massless theory was not the already there for the most part and met Mandelstam,
limitofthemassivetheory.Thedifferencewashidingin who had been working on Yang-Mills theories as well.
a symmetry factor for the one-loop ghost graphs: they He had his own formalism (Mandelstam, 1968), and we
have a factor of 1 as compared to the Faddeev-Popov compared his results with mine. We did not note the
2
ghost loops of the massless case. In the summer of 1970 notoriousfactorof2mentionedabove:Mandelstamhad
H. Van Dam and I reproduced and understood the ar- studiedthemasslesscase,whilemyresultsappliedtothe
gument and went further to consider gravitation (Van massive theory. Boulware was also there. As a student
Dam and Veltman, 1970). Here we found one of the ofSchwingerheknewaboutfunctionalintegrals,andhe
more astonishing facts in this domain: for gravity the laterappliedhisskillstothesubject(Boulware,1970).It
limit from massive gravitons to zero mass is not the became clear to me that there was no escape: I had to
sameasthemasslesstheory(ofEinstein).Thusatheory learn path integrals. At the end of my stay at Rock-
of gravitation with a massive spin-2 particle of exceed- efeller University somebody had already told me that
inglysmallmass(forexample,oftheorderofaninverse therewasworkbyFeynman(1963)andalsoFaddeevon
galactic radius) would give a result for the bending of the massless theory. The article by Faddeev and Popov
lightbythesunthatwasdiscreetlydifferent(byafactor (1967) was, as far as I was concerned, written in Vol-
of 3)fromthatofthemasslesstheory.Thusbyobserving apuk. It also contained path integrals, and although I
4
thebendingoflightinoursolarsystemwecandecideon had accepted this article in my function as editor of
the range of the gravitational field on a galactic scale PhysicsLetters,Ihadnoinklingwhatitwasaboutatthe
and beyond. Many physicists (I may mention Kabir time (summer 1967). I accepted it then because of my
here) found this result hard to swallow. The discontinu- respect for Faddeev’s work. Just as well!
ity of the zero-mass limit as compared to the massless My method of learning about path integrals was lec-
case has always been something contrary to physical in- turing on it, in Orsay. Ben Lee happened to be there as
tuition. Indeed, for photons there is no such effect. The well, and he was also interested. With some difficulty I
Rev.Mod.Phys.,Vol.72,No.2,April2000
346 MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation
obtainedthebookofFeynmanandHibbs.(Thiswasnot are perhaps useful in doing quantum electrodynamics,
easy; the students were busy making revolution and had but completely impractical for a Yang-Mills theory. I
notimeforsuchfrivolousthingsaspathintegrals.Ithus needed a good tool.
sentaroundanoteaskingthemtoreturnthebookprior Inanappendixtohispaper’tHooftpresented,within
to making revolution, which indeed produced a copy. the path-integral scheme, a gauge-choosing method. I
This gave me the reputation of an arch reactionary, did not recognize this at all, but later, backtracking, I
which I considered a distinction, coming as it did from discovered that this was an evolved version of my origi-
Maoists.) Somewhere during these lectures a Polish
nal attempt at a change of gauge, including the ‘‘free-
physicist (Richard Kerner, now in Paris) produced an-
field’’ technique. Russian physicists (Faddeev, Slavnov,
otherarticlebyFaddeev,inRussian,andIaskedhimto
Fradkin, and Tyutin) took it over into the path-integral
translateit.Ihaveneverreadthatarticle;BenLeetook
formalism, mangled, cleaned, and extended the method,
it with him. I was simply not up to it and I still felt that
mainly applying it to the massless case as well as (mass-
I did not understand path integrals. So, returning to the
less) gravitation. The actual scheme proposed by ’t
Netherlands I decided to do it once more, and in col-
Hooft in the quoted appendix is the method that is
laboration with Nico Van Kampen we did set up a
mostly used today.
courseinpathintegrals(autumn1969).Mythenstudent
I am not going to describe the (substantial) Russian
’t Hooft was asked to produce lecture notes, which he
contributions here. This despite the fact that, as almost
did. I would say that then I started to understand path
everywhereelse,doingfieldtheorywasnotverypopular
integrals, although I have never felt comfortable with
in the Soviet Union in that period. I believe that a fair
them. I distrust them. ’t Hooft had no such emotional
account has been given in Veltman (1992a).
ballast, and he became an expert in the subject. So, by
I am also skipping a description of the second paper
the end of 1969 ’t Hooft had been educated in the s
of ’t Hooft (1971b), introducing spontaneous symmetry
model, anomalies, and path integrals.
breakdownandthusarrivingattherenormalizabletheo-
ries with massive vector bosons as known today. There
are only two points that I would like to mention: I in-
sisted that as much as possible the results should not
’tHOOFT depend on the path-integral formalism, i.e., that unitar-
ity should be investigated separately, and secondly, that
At this point ’t Hooft showed unhappiness with the theissueoftherebeingsomethinginthevacuumnotbe
provisional subject that I had suggested, namely, the made into a cornerstone. Indeed, once the Lagrangian
double-resonance peak of Maglic. He wanted to enter including spontaneous symmetry breaking has been
into the Yang-Mills arena. I then suggested that he in- written down you do not have to know where it came
vestigatethemasslesstheory,withemphasisonfindinga from. That is how I wanted the paper to be formulated.
regulator method. This was so decided during a dinner, Isuspectedthattheremightbetroublewiththisvacuum
also attended by Van Kampen. field, and I still think so, but that does not affect in any
In studying the massless case ’t Hooft used combina- way ’t Hooft’s second paper. He sometimes formulates
torialmethods(diagrammanipulation)toestablishvari- this as me opposing the cosmological constant, but at
ous identities (’t Hooft, 1971a). He could have used the thattimeIdidnotknoworrealizethatthishadanything
Ward identities of my earlier paper, but I think he todowiththecosmologicalconstant.ThatIrealizedfor
wantedtoshowthathecoulddobetter.Thusitcameto the first time during a seminar on gravitation at Orsay,
pass that he never wrote the Slavnov-Taylor identities, at the beginning of 1974 (see Veltman, 1974, 1975).
anoversightthatthesetwogentlemenquicklycorrected Soletmegoontotheautumnof1971.’tHooftdived
(Taylor, 1971; Slavnov, 1972). ’t Hooft derived mass- into massless Yang-Mills theory studying the issue of
shell identities, presumably enough for renormalization asymptotic freedom; I think that Symanzik put him on
purposes. that track. I devoted much attention to the dimensional
Perhaps the main point that we argued about was the regularization scheme (’t Hooft and Veltman, 1972a).
necessityofagauge-invariantregularizationscheme.He Again I refer the interested reader to Veltman (1992a)
took the point of view that no matter what scheme one for details, including the independent work of Bollini
usesonesimplyadjuststhesubtractionconstantssothat and Giambiagi.
the Ward identities are satisfied, and that is all that is After dimensional regularization was developed to an
neededtorenormalizethetheory.Well,thatistruepro- easily workable scheme I decided that it would be a
vided there are no anomalies, and after some time he good idea to write two papers: (i) a paper clearly show-
accepted the point. He developed a gauge-invariant ing how everything worked in an example, and (ii) a
method that worked up to one loop. A fifth dimension reasonably rigorous paper in which renormalizable
wasused.Later,tryingtogobeyondoneloopwedevel- gauge theories were given a sound basis, using diagram
opedthedimensionalregularizationscheme;inthesum- combinatorial techniques only. The result was two pa-
mer of 1971 we had a rough understanding of that pers, one entitled ‘‘Example of a gauge-field theory’’ (’t
method. I should say that at all times I had an ulterior Hooft and Veltman, 1972b), the other ‘‘Combinatorics
motive: I very much wanted an actually usable scheme. of gauge-field theories’’ (’t Hooft and Veltman, 1972c).
The existing methods (such as the Pauli-Villars scheme) The first was presented at the Marseille conference,
Rev.Mod.Phys.,Vol.72,No.2,April2000
MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation 347
summer1972,whereapreliminaryversionofthesecond I will not enter into the argument on the number of
paper was also presented. I have no idea how many neutrinos from astrophysics. Such arguments are less
physicists read the ‘‘Example’’ paper; I think it is a pity than airtight, because they build on the whole body of
that we published it only in the conference proceedings our understanding of the big bang and evolution of the
and not in the regular literature. In that paper all one- Universe. Concerning the third generation, an interest-
loop infinities of the simple SU2 model with a two- ing argument developed: what is the mass of the top
parameter gauge choice were computed, and the in- quark? It would fill an amusing article to list all articles
formed reader may without any trouble use the counter
that made claims one way or the other, but I leave that
Lagrangian given in that paper to deduce the bparam-
to someone else. I realized that without a top quark the
eter for that theory (including a Higgs). That is the pa-
theory would be non-renormalizable, and therefore
rameter relevant for asymptotic freedom. The calcula-
thereoughttobeobservableeffectsbecominginfiniteas
tions for this paper were fully automated and done by
the top-quark mass goes to infinity. To my delight there
SCHOONSCHIP.When’tHooftaskedmeaboutthediver- was such a correction to the rparameter and further-
gencies of the massless theory as a check on his own
more it blows up proportional to the top-quark mass
calculations there was no problem doing that. I did not
squared (Veltman, 1977a). This is the first instance in
knowaboutasymptoticfreedomanddidnotunderstand
particle physics in which a radiative correction becomes
the relevance of this particular calculation at that mo-
largerasthemassofthevirtualparticlesincreases.That
ment.HereportedhisresultattheMarseilleconference.
is our first window to the very-high-energy region. This
radiative correction became experimentally better
RADIATIVECORRECTIONS knownandeventuallyproducedapredictionforthetop-
quark mass of 175 GeV, which agrees with the result
Much of my effort after 1972 was directed towards foundwhenthetopwasdiscovered.Thisagreementalso
applying the theory, i.e., towards radiative corrections. seems to indicate that there are no more generations,
In1975therewasstillconsiderableargumentaboutneu- because there is little or no room for any quark mass
tral currents. Most people thought that the precise con- differencesin(hypothetical)newgenerations.Giventhe
figuration contained in the Weinberg model was a must, patternofmassesthatweobservenow,thatappearsun-
not knowing that by choosing another Higgs sector one likely, although it is strictly speaking not impossible.
couldadjusttheZ masstoanyvalue.Thiswasclearlya FromthebeginningIwasveryinterestedintheHiggs
0
critical point, and Ross and I set out to investigate this sector of spontaneously broken theories. I started to
issue (Ross and Veltman, 1975). This led to the intro- look for a way to establish a limit on the Higgs mass;
ductionofanewparameter,nowcalledtherparameter, after all, if the Higgs is an essential ingredient of renor-
thattakesonthevalueof1forthesimplestHiggssector malization there must be terms in perturbation theory
as chosen by Weinberg. The rparameter is essentially that cannot be renormalized away and that would be
thesquareoftheratioofthechargedvector-bosonmass sensitive to the Higgs mass. It can easily be argued that
to the neutral vector-boson mass, with a correction re- the place to look for that is in the radiative corrections
lated to weak mixing. This parameter has become an to the vector-boson masses, and the relevant parameter
important part of today’s physics, because it is the most there is the rparameter, introduced in the paper with
sensitivelocationforradiativeeffectsofheavyparticles, Rossmentionedbefore.Asithappens,whiletherecould
quarksorHiggs.AttheParisconferenceonneutralcur- have been an effect proportional to the square of the
rents of 1974 or 1975 I presented a very short contribu- Higgsmass,itturnedoutthatthatpiececancelsoutand
tion,consistingof,Ibelieve,onlytwotransparencies.All only a logarithmic dependence remains (Veltman,
I said was this: the neutral vector-boson mass can be 1977b).ThismakesitverydifficulttoestimatetheHiggs
anything. Here is a convenient way to parametrize that. mass on the basis of radiative corrections, and I have
To this day I am flabbergasted that nobody, but nobody introduced the name ‘‘screening theorem’’ in this con-
at that conference seemed to have gotten the message. nection.Natureseemstohavebeencarefulinhidingthe
They kept on thinking that finding the precise quantita- Higgsfromactualobservation.Thisandotherfactshave
tive amount of neutral-current effects as predicted by led me to believe that something else is going on than
the Weinberg model (extended to quarks according to the Higgs sector as normally part of the Standard
Glashow, Iliopoulos, and Maiani) was crucial to the ap- Model.
plicability of gauge theories. In reality, had they found After that I started to set up a systematic scheme for
deviantresults,theonlyconsequencewouldhavebeena the calculation of radiative corrections, together with
different Higgs sector. Passarino. As he, together with Bardin, has written a
In 1976 it became reasonably clear that the Standard bookthathasjustcomeoutIrefertheinterestedreader
Model including the simplest possible Higgs sector was to that book (Bardin and Passarino, 1999).
the right model. Meaningful calculations on radiative There was another motive in doing the radiative cor-
correctionswerenowpossible,andIsetouttodothem. rections. I wanted ultimately to compute the radiative
It appeared that there were at least three families. The corrections to W-pair production at the Large Electron-
following issues were of immediate importance to me: Positron collider (LEP), because it was clear to me that
(i) How many generations are there? those corrections would be sensitive to the Higgs mass.
(ii) Is there an upper limit on the Higgs mass? This then would suggest a value for the LEP energy: it
Rev.Mod.Phys.,Vol.72,No.2,April2000
348 MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation
should be high enough that radiative corrections to theory. The precise meaning of that will become clear
W-pair production were sufficiently large and could be shortly. However, there is trouble at the two-loop level,
studiedexperimentally.OnewouldeitherfindtheHiggs soatthetimeIthoughtthattherehadtobesomecutoff
or see important radiative corrections. That calculation mechanism that would control the (observable) diver-
was done together with Lemoine (Lemoine and Velt- gencies occurring beyond the one-loop level. Actually,
man, 1980) and finished in 1980. I did not succeed in the Higgs system can be seen as such a cutoff mecha-
making the case sufficiently strong: no one understood nism. The Higgs mass becomes the cutoff parameter,
theimportanceofsuchconsiderationsatthattime.Thus and indeed this cutoff parameter is observable (which is
the LEP energy came out at 200 GeV, too low for that the definition of non-renormalizability of the theory
purpose.Asthevectorbosonswerestilltobefound,few withoutaHiggssystem).Thisparameterenterslogarith-
were prepared to think beyond that. Also, I have no micallyincertainradiativecorrections(theZ mass,for
0
ideaifa250-to300-GeVLEPwouldhavebeenpossible example), and from the measurement of these correc-
from an engineering point of view, let alone financially. tions follows some rough estimate of the value of this
parameter. However, part of the input is that the Higgs
sector is the simplest possible; without that assumption
PRESENTSTATUS
thereisnosensitivityattheone-looplevel(becausethen
the Z mass is not known and the radiative correction
The Higgs sector of the Standard Model is essentially 0
becomes a renormalization of that mass). That is the
untested. Customarily one uses the simplest possible
meaning of one-loop renormalizable. Let us, however,
Higgs system, one that gives rise to only one physical
assumethatfromasymmetrypointofviewthingsareas
Higgs particle. With that choice the Z mass is fixed to
0
be equal to the charged W mass divided by cos(u), if the Higgs system were the simplest possible. Then
where uis the weak mixing angle. Let us first establish from the radiative corrections the cutoff parameter (the
Higgs mass) can be estimated.
here a simple fact: by choosing the appropriate Higgs
Fromthispointofviewthequestionistowhatextent
sectoronecanensurethattheZ massisunconstrained.
0
we can be sure today that the cutoff system used by
Furthermore the photon mass need not be zero and can
Nature is the Higgs system as advertised. Evidence
be given any value.
would be if there is indeed a particle with a mass equal
In the early days a considerable amount of verbosity
totheestimatefoundfromtheradiativecorrections.But
was used to bridge the gap between the introduction of
if there is none, that would simply mean that Nature
the models in 1967 and the later-demonstrated renor-
uses some other scheme, to be investigated experimen-
malizability. Two of the terms frequently used to this
tally.
day are ‘‘electroweak unification’’ and ‘‘spontaneous
In all of this discussion the notion of spontaneous
symmetrybreakdown.’’AsIconsiderthesetermshighly
symmetry breakdown does not really enter. In the be-
misleading, I would like to discuss them in some detail.
ginningthiswasaquestionthatIkeptonposingmyself.
Towhatextentareweakandelectromagneticinterac-
Spontaneous symmetry breakdown usually implies a
tions unified? The symmetry used to describe both is
SU23U1, and that already shows that there is really no constant field in the vacuum. So I asked myself: is there
any way one could observe the presence of such a field
unification at all. True unification, as in Maxwell’s
in the vacuum? This line of thought led me to the ques-
theory, leads to a reduction of parameters; for example,
tionofthecosmologicalconstant(Veltman,1974,1975).
in Maxwell’s theory the propagation velocities of mag-
Indeed, if nothing else, surely the gravitational interac-
netic and electric fields are the same, equal also to the
tionscanseethepresenceofafieldinthevacuum.And
speedoflight.Intheelectroweaktheorythereisnosuch
here we have the problem of the cosmological constant,
reduction of parameters: the mixing angle can be what-
as big a mystery today as 25 years ago. This hopefully
ever, and that makes the electric coupling constant e
5gsin(u) a free parameter. If the Higgs sector is not alsomakesitclearthat,withtheintroductionofsponta-
neous symmetry breakdown, the problem of the cosmo-
specified,thentheZ massandthephotonmassarealso
0
logicalconstantentersanewphase.Ihavearguedthata
free parameters. There is really no unification (apart
solution to this problem may be found in a reconsidera-
from the fact that the isovector part of the photon is in
tionofthefundamentalrealityofspace-timeversusmo-
the same multiplet as the vector bosons).
mentum space (Veltman, 1994), but this is clearly not
However, if one specifies the simplest possible Higgs
the place to discuss that. Also, the argument has so far
system then the number of free parameters diminishes.
not led to any tangible consequences.
The Z mass is fixed if the weak mixing angle and the
0
So,whiletheoreticallytheuseofspontaneoussymme-
charged vector-boson mass are fixed, and the photon
try breakdown leads to renormalizable Lagrangians, the
massisnecessarilyzero.Soherethereseemstobesome
question of whether this is really what happens in Na-
unification going on. It seems to me, however, utterly
ture is entirely open.
ridiculous to speak of ‘‘electroweak unification’’ when
choosing the simplest possible Higgs system.
The question of spontaneous symmetry breakdown is CONCLUSION
more complicated. From my own perspective the situa-
tion is as follows. In 1968 I showed what I termed the The mind-wrenching transition of field theory in the
one-loop renormalizability of the massive Yang-Mills sixtiestopresent-daygauge-fieldtheoryisnotreallyvis-
Rev.Mod.Phys.,Vol.72,No.2,April2000
MartinusJ.G.Veltman: NobelLecture:Fromweakinteractionstogravitation 349
ible anymore, and is surely hard to understand for the Salam, A., 1968, in Elementary Particle Theory: Relativistic
present generation of field theorists. They might ask: Groups and Analyticity, edited by N. Svartholm (Coronet,
why did it take so long? Perhaps the above provides Philadelphia).
some answer to that question. Slavnov, A.A., 1972, Teor. Mat. Fiz. 10, 153 [Theor. Math.
Phys.10,99(1972)].
Slavnov,A.A.,andL.D.Faddeev,1970,Teor.Mat.Fiz.3,18.
REFERENCES Stu¨ckelberg,E.C.G.,1938,Helv.Phys.Acta11,299.
Taylor,J.C.,1971,Nucl.Phys.B33,436.
Adler,S.L.,1965a,Phys.Rev.137,B1022. ’tHooft,G.,1971a,Nucl.Phys.B33,173.
Adler,S.L.,1965b,Phys.Rev.Lett.14,1051. ’tHooft,G.,1971b,Nucl.Phys.B35,167.
Adler,S.L.,1969,Phys.Rev.177,2426. ’tHooft,G.,andM.Veltman,1972a,Nucl.Phys.B44,189.
Bardeen,W.,1969,Phys.Rev.184,1848. ’t Hooft, G., and M. Veltman, 1972b, in Renormalization of
Bardin,D.,andG.Passarino,1999,TheStandardModelinthe Yang-Mills fields and applications to particle physics,
Making(Clarendon,Oxford). Marseille Conference June 19–23, edited by C.P. Korthals-
Bell,J.S.,1967,NuovoCimentoA50,129. Altes(Universite´ d’Aix-Marseille)p.37.
Bell,J.S.,andR.Jackiw,1969,NuovoCimentoA60,47. ’tHooft,G.,andM.Veltman,1972c,Nucl.Phys.B50,318.
Bouchiat, C., J. Iliopoulos, and Ph. Meyer, 1972, Phys. Lett. VanDam,H.,andM.Veltman,1970,Nucl.Phys.B22,397.
38B,519. Veltman,M.,1963,Physica(Utrecht)29,186.
Boulware,D.,1970,Ann.Phys.(N.Y.)56,140. Veltman,M.,1966,Phys.Rev.Lett.17,553.
DeWitt,B.S.,1964,Phys.Rev.Lett.12,742. Veltman,M.,1968,Nucl.Phys.B7,637.
DeWitt,B.S.,1967a,Phys.Rev.160,1113. Veltman, M., 1969, in Proceedings of the Topical Conference
DeWitt,B.S.,1967b,Phys.Rev.162,1195,1239. on Weak Interactions, CERN, Geneva, January 14–17,
Faddeev,L.D.,andV.N.Popov,1967,Phys.Lett.25B,29. CERNyellowreport69-7,p.391.
Feynman,R.P.,1963,ActaPhys.Pol.24,697. Veltman,M.,1970,Nucl.Phys.B21,288.
Gell-Mann,M.,1964,Physics(N.Y.)1,63. Veltman, M., 1974, ‘‘Cosmology and the Higgs mechanism,’’
Glashow,S.L.,1961,Nucl.Phys.22,579. RockefellerUniversitypreprint,May1974.
Glashow,S.L.,andJ.Iliopoulos,1971,Phys.Rev.D3,1043. Veltman,M.,1975,Phys.Rev.Lett.34,777.
Glashow,S.L.,J.Iliopoulos,andL.Maiani,1970,Phys.Rev.D Veltman,M.,1977a,Nucl.Phys.B123,89.
2,1285. Veltman,M.,1977b,ActaPhys.Pol.B8,475.
Hara,Y.,1964,Phys.Rev.134,B701. Veltman, M., 1992a, ‘‘The path to renormalizability,’’ invited
Hoddeson, L., L. Brown, M. Riordan, and M. Dresden, Eds., talkattheThirdInternationalSymposiumontheHistoryof
1997,TheRiseoftheStandardModel(CambridgeUniversity, Particle Physics, June 24–27, 1992; Printed in Hoddeson
Cambridge,England). etal.,1997.
Lee,T.D.,1962,Phys.Rev.128,899. Veltman,M.,1992b,inGaugeTheory—PastandFuture,edited
Lee,T.D.,andC.N.Yang,1962,Phys.Rev.128,885. by R. Akhoury, B. de Wit, P. Van Nieuwenhuizen, and H.
Lemoine,M.,andM.Veltman,1980,Nucl.Phys.B164,445. Veltman(WorldScientific,Singapore).
Mandelstam,S.,1968,Phys.Rev.175,1580. Veltman,M.,1994,ActaPhys.Pol.B25,1399.
Reiff,J.,andM.Veltman,1969,Nucl.Phys.B13,545. Weinberg,S.,1967,Phys.Rev.Lett.19,1264.
Ross,D.A.,andM.Veltman,1975,Nucl.Phys.B95,135. Weinberg,S.,1979,Nobellecture.
Salam,A.,1962,Phys.Rev.127,331. Weisberger,W.I.,1965,Phys.Rev.Lett.14,1047.
Rev.Mod.Phys.,Vol.72,No.2,April2000