Research Profile

*by Roberto Lalli***Martinus J. G. Veltman ****Nobel Prize in Physics 1999 together with Gerardus 't Hooft**

**"for elucidating the quantum structure of electroweak interactions in physics".**

The Dutch theoretical physicist Martinus J. G. Veltman has contributed to the construction of the Standard Model of particle physics by supplying the technical and conceptual basis underlying Gerardus ‘t Hooft’s demonstration that the Glashow-Weinberg-Salam model of the electroweak unification was renormalizable. Veltman worked on the programme of renormalizing the massive Yang-Mills gauge field theories in a period when the majority of his colleagues believed such theories to be non-renormalizable. Working in relative isolation, Veltman developed innovative theoretical techniques and pursued a project that was eventually completed by his PhD student G. ‘t Hooft in 1971. Developing Veltman’s idiosyncratic methods, ‘t Hooft independently derived the Higgs mechanism - discovered by various physicists around 1964 - and the gauge symmetry structure of the electroweak theory of Weinberg and Salam. ‘t Hooft’s proof that the gauge field theory of the electromagnetic and weak interactions was renormalizable was a fundamental factor in the change of attitude of theoretical physicists towards relativistic quantum field theories. As soon as ‘t Hooft’s proof became public, theoretical physicists began calculating empirical predictions, and teams of experimental physicists planned experimental tests of those predictions. Less than ten years after the publication of ‘t Hooft’s work, the agreements between theoretical predictions and empirical data led to the acceptance of the unified quantum field theory of electromagnetic and weak interactions. At the same time, the change of climate surrounding relativistic quantum field theory also resulted in the development of the theory of strong interactions called quantum chromodynamics. By 1979, thanks to this development the Glashow-Weinberg-Salam electroweak theory plus quantum chromodynamics had been broadly accepted as the Standard Model of particle physics.

**Becoming a CERN Theorist **

Martinus Veltman was born on June 27, 1931, in Waalwijk - a small town of 20 000 inhabitants in the southern region of the Netherlands. In his youth, Veltman was very interested in electronics and had some experience as the local radio repairman. After he passed his final high school exam, Veltman’s physics high school teacher had a meeting with his parents and suggested that they send Veltman to university - a very unusual step for a teenager living in Waalwijk because universities were still very exclusive. In order to make his university life economically affordable, he enrolled at the State University of Utrecht, which he could reach by train with a daily 90-minute trip. After a period in which Veltman focused on experimental physics, he decided to switch to theoretical physics. His early involvement in electronics and experimental physics, however, proved to be essential in shaping Veltman’s approach to theoretical practices.

In the Netherlands, the community of theoretical physicists was principally interested in statistical mechanics – a subject that did not catch Veltman’s interest. An important change in his student life occurred in 1955, when the Belgian theoretical physicist Leon van Hove joined the Utrecht University faculty. Although van Hove was involved in statistical mechanics like his colleagues at Utrecht, he encouraged Veltman’s interest in particle physics and accepted Veltman as a PhD student. In the late 1950s, theoretical particle physics was still a relatively novel field, and European universities did not yet offer formal courses in this subject. In order to keep the students updated with the current advancements it was common to organise summer schools inviting researchers who were contributing to the theoretical understanding of particle behaviour. Veltman attended a couple of summer schools dedicated to theoretical particle physics and met some of the protagonists who were building what was to become the Standard Model of particle physics in the next decades, including S. Glashow, N. Cabibbo, and B. Zumino. Apart from these experiences, Veltman worked in relative isolation during his PhD studies.

In 1960, van Hove became the director of the Theory Division at CERN, and Veltman followed him in 1961. Veltman’s work at CERN was fundamental for many reasons. There, he became aware of the central problems of particle physics that were under investigation in High Energy Accelerators. Secondly, he had the opportunity to collaborate with other theorists involved in particle physics investigations, including J. Bell and S. Berman. When he arrived at CERN, Veltman was working on the field theoretical problems concerning unstable particles. Between 1937 and 1940, J. A. Wheeler and, independently, W. Heisenberg had introduced the S-matrix (scattering matrix) that related the initial and final states of scattering experiments in quantum mechanical formalism. Whatever the underlying theoretical approach, in the 1950s the S-matrix was understood as the central link between theory and experiment: the square of the absolute values of the S-matrix constituted the transition probabilities in particle interactions. For such a reason, the S-matrix had to be unitary. Veltman’s dissertation aimed at finding proofs of the unitarity of the S-matrix in the case of renormalizable quantum field theory with unstable particles. During his work, Veltman began employing the Feynman diagrams without any reference to the Hamiltonian. This diagrammatic approach to the resolution of theoretical problems was rather unusual and shaped his future work on the renormalization of gauge field theories providing the conceptual basis of his commitment to the Feynman diagrams. His stay at CERN made Veltman extend the topics of his dissertation in order to include the Coulomb corrections for vector bosons production by neutrinos. Between 1961 and 1963, groups of experimenters at CERN were planning the observation of high-energy neutrino properties, and the CERN theoretical division was involved in doing the related calculations. Veltman worked mainly in the phenomenological tradition until 1966, when he decided to address more fundamental questions.

**Renormalizability of Quantum Field Theories Between the 1940s and the 1960s**

In the late 1940s, J. Schwinger, R. Feynman, S.-I. Tomonaga – who would receive the 1965 Nobel Prize in Physics - and R. Dyson had put forward the procedures for the renormalization of the relativistic quantum field theory of electrodynamics (QED). Their work demonstrated that the divergences affecting higher-order corrections of QED cancelled out provided that a finite number of terms – including mass and coupling constant – of the original Lagrangian were substituted with the observed renormalized values. After the successes of renormalized QED, other physicists attempted to extend the range of quantum field theory to include weak and strong interactions. However, this extension was far from being straightforward. The dimension of the Fermi coupling constant made the quantum field theory of weak interactions unrenormalizable. In addition, the strength of the coupling constant made the perturbation theory useless for strong interactions. In the 1950s, the quantum field theory approach had lost consensus and many theorists were exploring different ways to look at issues of particle interactions favouring a phenomenological outlook. After the experimental discovery of parity violation in weak interactions in 1957, it was proposed that weak interactions might be described by a theory that combined vector and axial currents. The V-A theory – as the universal theory of weak interactions based on a vector-axial vector structure of the Lagrangian has since been called – strongly suggested the existence of vector bosons as mediators of weak interactions. Because of some features of the weak interactions, the intermediate vector bosons were believed to be electrically charged (both positively and negatively) and massive. This last requirement was particularly troubling because it led to divergences in the perturbation series that were not renormalizable.

At the same time the V-A theory was put forward, Schwinger developed a unifying quantum field theory of weak and electromagnetic interactions based on the existence of gauge bosons mediating the interactions. In the following years, Schwinger’s PhD student S. Glashow pursued this idea, and in 1961 proposed that the electroweak unification should be based on the gauge symmetry group SU(2)xU(1) – the structure of the theory that has been accepted as the Standard Model electroweak theory since the 1970s. Many attempts to build a quantum field theory of weak interactions or, alternatively, of unified electroweak interactions were based on the gauge theory formulated by Yang and Mills in 1954.

Yang and Mills wished to extend the gauge principle – on which QED was based - to include strong isospin interactions employing the gauge symmetry SU(2), and formulated a general non-Abelian gauge theory. Such a theory soon generated some enthusiasm because it was considered a possible candidate for the description of weak and strong interactions. The equations of the Yang-Mills theory were much more complicated than those of QED because they were non-linear. From a physical point of view, this meant that the fields describing the interactions were themselves the source of the interaction. In other words, the quanta of the radiation fields interacted between themselves – a behaviour very different from that of photons that mediate electromagnetic interactions without being charged.

Although promising, the Yang-Mills theory had a serious problem. In order to maintain gauge invariance, the gauge bosons had to be massless. On the other hand, a massless particle needed to mediate a long-range force and have already been detected, while the only massless particle that had up to that point been observed was the photon. This issue came to be known as the “zero-mass problem” of the Yang-Mills theory.

In his 1961 electroweak theory, Glashow put in the mass of the intermediate vector bosons “by hand,” arguing for a “partial symmetry” of the theory. However, there were serious doubts that such a theory - like any Yang-Mills theory with massive gauge bosons - was renormalizable, because contemporary analyses pursued by Salam and others seemed to show that it was not. Many physicists did not find it useful to concentrate on gauge field theories because the difficulties seemed to be insurmountable. In any case, some tried to solve these issues. During the 1960s, various theoretical progressions - essentially based on the concept of spontaneous breaking of the underlying gauge symmetry - led to the formulation of the so called Higgs mechanism for the mass production of the gauge bosons and, later, on the electroweak unification of the interaction between leptons based on such a mechanism. The final outcome of these developments was the SU(2)xU(1) gauge scheme proposed by Weinberg in 1967 and, independently, by Salam in 1968.

While the Weinberg-Salam theory solved the mass problem of the gauge bosons of electroweak interactions by means of the Higgs mechanism, the issues concerning the renormalization of such a theory remained unaddressed. Although some believed that the theory was renormalizable, no formal proof existed, and the Weinberg-Salam theory was almost ignored until 1971. Veltman’s student ‘tHooft provided such a proof in 1971 and Veltman played an essential role in this achievement by pursuing an unusual and solitary path towards the renormalization of massive Yang-Mills theory.

**Towards the Renormalization of Massive Yang-Mills Theory**

Veltman’s interest in the Yang-Mills theory was sparked by some calculations related to neutrino experiments performed at CERN. Driven by the 1962 analyses made by Yang and Lee on the Feynman rules for the vector bosons, Veltman began working on calculations for the Feynman rules concerning the interactions between charged vector bosons and photons. The calculations were very complex involving more than 50 000 terms. In order to complete the computation without introducing errors, in December 1963 Veltman developed an algebraic program called Schoonschip - one of the first computer algebra systems ever developed. This program allowed Veltman to calculate the value of the vector bosons magnetic moment that minimised the divergences. As Veltman later recognised, the value was the one suggested by the Yang-Mills theory, but Veltman had no knowledge of such a theory in 1963. This early work was also important for a different reason: the evolution of Schoonship became a very efficient tool in the calculation of the integrals of higher order terms in perturbation theory.

In 1966, Veltman shifted his interest from phenomenological calculations to more fundamental topics. He began working on Gell-Mann’s current algebra of the V-A theory of weak interactions. Veltman tried to recover the commutation relations of current algebra by employing a field theoretical formalism. He demonstrated that he could recover all the sum-rules of the current algebra - including the Adler-Weisberger relation - starting from a couple of simple equations that had a structure similar to those of the Yang-Mills gauge theory. The works of other physicists, including that of his friend John Bell, convinced Veltman that the success of current algebra and, in particular, of the Adler-Weiberger sum rule was a consequence of gauge invariance of the underlying equations. Since he firmly believed in the existence of the intermediate vector bosons, as he later reported, he “interpreted the Adler-Weisberger relation as experimental evidence for the renormizability of a vector-boson theory of weak interactions.” Following this line of reasoning, Veltman began working on the renormalization of massive Yang-Mills theory in 1968 - a work that was eventually completed by his PhD student ‘t Hooft.

**The Renormalization of the Electroweak Theory**

In his renormalization programme, Veltman encountered many difficulties and extreme complications, but he noticed that by employing some simplification several infinities cancelled out. He then had the idea that it was necessary to modify the Feynman rules in order to incorporate the cancellation of infinities since the beginning. He made this change by introducing a free scalar field that did not interact with the vector bosons and by employing some manipulations in order to keep the physics unchanged. The new Feynman rules might be chosen in a way to minimise the divergences, but such a method had the inconvenience that a new free particle entered as a ghost. In any case, the important result of this approach was that the Yang-Mills theory with an explicit mass term could be made one-loop renormalizable. Veltman had decided to work on massive Yang-Mills gauge theory because he considered the massless theory to be void of physical significance. At the same time other theorists were working on this problem, and by 1969 Veltman became aware that works on Feynman rules had demonstrated the massless case to be renormalizable.

After these encouraging results, Veltman made other attempts that did not meet immediate success. In collaboration with his student J. Reiff, Veltman investigated the behaviour of two-loop diagrams for massive Yang-Mills theory and discovered that the limit to zero mass of massive Yang-Mills theory did not correspond to the massless theory. By trying to understand this paradox, Veltman became an expert of various formalisms, including path integrals developed by Feynman and ‘source theory’ formulated by Schwinger. By employing these methods, Veltman elaborated on his previous work and made evident the problems affecting the renormalization of massive Yang-Mills field theories. This last part of Veltman’s research overlapped significantly with the entering in the field of his PhD student ‘t Hooft who in 1969 decided to write a dissertation on the same subject. As Veltman himself emphasised, the research on the Yang-Mills theory was not considered a central topic in that period and he did not encourage his students to commit themselves with this somewhat idiosyncratic subject. However, ‘t Hooft insisted and Veltman began supervising ‘t Hooft’s work starting a collaboration in which it is difficult to precisely identify the role held by the two scientists. ‘t Hooft was able to overcome the difficulties of Veltman’s approach and in a couple of years he demonstrated that massive Yang-Mills theory was indeed renormalizable when mass was acquired by means of spontaneous breaking of the original gauge symmetry. By this indirect route, ‘t Hooft independently derived the Higgs mechanism and the Weinberg-Salam theory of electroweak interactions, while at the same time showing that such a theory was renormalizable. When ‘tHooft made his proof public, the community of theoretical physicists was not familiar with the methods ‘tHooft was employing. After ‘t Hooft had published his method, other physicists - notably B. Lee and J. Zinn-Justin - published their proof of the renormalizability of the massive Yang-Mills theory employing path integrals. Veltman also made another fundamental contribution by developing with ‘tHooft the dimensional regulation method published in the paper “Regularization and Renormalization of Gauge Fields,” – a method that was independently put forward by South American physicists J. J. Giambiagi and C. Bollini.

These different approaches helped in spreading the belief that the Glashow-Weinberg-Salam theory was renormalizable. Several physicists began working on the improvements of the theory and on the calculation of its predictions. Consequently, a number of experiments were planned to test such predictions. The first successful series of experiments were performed in 1973. Some of them were interpreted as a demonstration that weak neutral currents exist, confirming one of the predictions of the SU(2)xU(1) gauge electroweak theory. In 1978, the observation of parity-violating neutral currents involving charged leptons persuaded many physicists of the validity of the theory. Between 1973 and 1976, the electroweak theory had further evolved in order to include the quarks. This evolution led to the prediction and experimental discovery of a fourth quark, called the charm quark. After these events, the gauge theory of strong interactions called quantum chromodynamics (QCD) gained consensus within the community of physicists. Many physicists began accepting the unifying model of particle interaction called the Standard Model of particle physics composed of the SU(2)xU(1) theory of electroweak interactions as well as of the SU(3) colour theory of strong interactions. In 1979, the consensus on such a model was so widespread that Glashow, Weinberg, and Salam were awarded the Nobel Prize in Physics “for their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current.” In 1983, two international teams working at the CERN proton-antiproton collider reported that their experiments – known as UA1 and UA2 – showed convincing evidence of the existence of the three intermediate vector bosons of the electroweak theory, which had been christened W and Z bosons. Weinberg and others have always given strong importance to ‘t Hooft’s proof as the event that modified physicists’ attitude towards quantum field theories of particle physics in the early 1970s. Before 1971, indeed, Weinberg’s 1967 paper was almost never quoted, while his theory became a respected candidate for the interpretation of particle interactions as soon as ‘t Hooft’s demonstration came to be accepted. In 1999, when electroweak theory and QCD had been broadly accepted as the Standard Model of particle physics for almost twenty years, Veltman and ‘t Hooft were jointly awarded the Nobel Prize in Physics “for elucidating the quantum structure of electroweak interactions in physics.”

**Bibliography**

Hoddeson, L., Brown, L. M., Riordan, M. & Dresden, M. (1995) The Rise of the Standard Model: Particle Physics in the 1960s and the 1970s. Cambridge University Press, Cambridge.

Pais, A. (1986) Inward Bound: Of Matter and Forces in the Physical World. Clarendon Press, Oxford.

Pickering, A. (1999). Constructing quarks: A sociological history of particle physics. University of Chicago Press, Chicago.

Veltman, M. J. (2000). From weak interactions to gravitation. International Journal of Modern Physics, A 15 (29), pp. 4557-4573.

Veltman M. J. (1995) The Path to Renormizability, in Hoddeson, L., Brown, L. M., Riordan, M. & Dresden, M. The Rise of the Standard Model: Particle Physics in the 1960s and the 1970s. Cambridge University Press, Cambridge, pp. 145-178.

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