Research-TPP

À la recherche de TPP                                             David Bailin
 

                                                 Left to Right: Gabriel Barton, David Bailin, Roger Blin-Stoyle, Norman Dombey

                                                     Picture by taken by Roger's former student, Peter Rosen, on July 12, 1973

Sometime in the summer of 1964, the Civil Rights summer in the USA,  I received a preprint request card from one "Roger Blin-Stoyle" of the "School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton. (In those days, long before the advent of the internet or the arXiv, papers were typed by secretaries on to "onion skins", one for each page. These were then wrapped around an ink drum and typically one to two hundred copies of each page were printed. After collation, usually by press-ganged graduate students, the preprints were posted to the main groups around the world and to selected individuals whom the author adjudged might have a particular interest in the topic.) That was my first contact with Roger; I thought what a funny name he had and wondered about its origin. I'd never previously heard of the University of Sussex. The following year I was a postdoc in Oxford and by then the new University of Sussex was infamous if not famous: the social antics of the Jay twins, the daughters of the Labour Government's  minister Douglas Jay, and their contemporaries meant that the University was often featured by the tabloid press, and deplored by the Daily Telegraph. With such credentials, clearly the place couldn't be all bad, so I had little hesitation in applying for one when I read an advertisement in The Guardian  for "Lectureships in Theoretical Physics" at Sussex; in any case, I needed a job.  There was no M25 in those days so the drive to Brighton for my interview was lengthy, but enjoyable; I was impressed to be given overnight accommodation in The Old Ship  hotel on the seafront which was, I was later assured by Gabriel Barton, if not the best, certainly the most distinguished  hotel in Brighton. The interview was conducted by Roger (apparently perfectly normal, notwithstanding the name), Ken Smith and Gabriel; the latter gave me a walking tour of the campus and the teacher training college (now Brighton University Falmer campus) then under construction on the other side of the railway tracks. A month or two after I had accepted the job offer, in the summer of 1965, I received a letter from Roger informing me that they had admitted a research student for me to supervise, a Miss Anjali Medhi, recently graduated from Imperial College's DIC programme. For these reasons, I have observed that Roger organised my whole life! He gave me my first permanent job and arranged my introduction to  the woman who later became my wife. In Oxford, Lady Peierls, the wife of Prof Peierls,  then Head of the Department of Theoretical Physics, was often ascribed similar powers, even telling young faculty members which house they should buy. Roger never did that, but he did volunteer to be the godfather of our first child, an offer that I now regret that we declined; I assumed that he was religious, and it was only many years later, at his humanist funeral in 2007, that I discovered he was not.

As Dean of MAPS, Roger had a nice office in Physics I, now Pevensey I  2C1 and currently used as a seminar room. There were then three of us who regarded ourselves as particle theorists: Gabriel, Norman Dombey and me. (We didn't use the abbreviation "TPP" in those days.)   We had offices in the "Terrapin", a prefabricated temporary building situated alongside the service road, now called the North-South Road, and overlooking what is now Sussex House. The windows were totally opaque due to the mud thrown up by the interminable stream of construction traffic.  There was little insulation, so we were freezing in winter and frying in summer; one could easily hear conversations, or worse, in the adjacent offices.
 
Roger regarded himself more as a nuclear theorist  interested in the manifestation of particle physics effects in nuclei, rather than a particle theorist. He was interested in things like the charge dependence of nuclear forces, and meson exchange effects. The observation by Madame Wu in 1956 of parity violation in the beta decays of polarised cobalt nuclei led to the reformulation in 1958 by Feynman and Gell-Mann of Fermi's 1934 current-current theory of the weak interactions. In the modern version,  the leptonic and hadronic (charged) weak currents both had vector and axial vector pieces; the vector and axial vector  components of the weak nucleon current lead respectively to Fermi and Gamow-Teller beta decay transitions. Further, the purely hadronic (weak) Hamiltonian is also parity-violating. Since the strong part of the Hamiltonian is parity-conserving, it follows that the nuclear eigenstates have small admixtures of the "wrong" parity. This in turn means that transitions that would otherwise be forbidden may actually occur but at a much slower rate than the allowed transitions. Roger was particularly active in persuading the experimentalists, Dennis Hamilton and Jim Byrne, to devise experiments to study these transitions. The discovery by Fitch and Cronin in 1964 of CP-violation (charge-conjugation C times parity reversal P) in the (weak) neutral kaon decays was strong evidence, later confirmed, of T-violation, i.e.violation of time-reversal T-symmetry, although the precise origin of the interaction responsible was not (and is still not) known.  If it is in the weak interactions, then, as for parity, the nuclear eigenstates have (presumably small) admixtures of the "wrong T-parity. So later on, Roger was again active in urging experimental searches for T-violation in nuclear transitions. I particularly recall the arrival in 1968 of Roger's student Peter Herczeg, a dear friend,  now retired from the Theory Division at Los Alamos National Laboratory. Peter  fled Czechoslovakia during the Prague Spring in which the Soviet army invaded the country to halt reforms introduced by Alexander Dubcek. Peter worked on the effects of "irregular" components of the weak hadronic current, i.e.  components having the "wrong  or opposite G-parity to the standard components in the Feynman and Gell-Mann theory. One day, at tea on the Bridge, now the Bridge Café, I joined Peter talking to Gabriel. While waiting to ascertain the topic of their discussion, I realised that they were conversing in a foreign tongue, Hungarian, in fact. Until that day, I had had no inkling that Gabriel was anything other than an archetypal English gentleman -  tweed sports jacket, woolen tie, Westminster and Christ Church: you can't get much more English establishment than that -  nor of his flight from the Nazis in Budapest.

In the early 1960s data from the new particle accelerators dominated the particle physics scene.The Bevatron, built at  Berkeley in 1954 but upgraded to about 25 GeV in 1960, the 33 GeV Alternating Gradient Synchrotron at Brookhaven, built in 1960, and the 25 GeV Proton Synchroton that began operating at  CERN in 1959, all produced large volumes of strong interaction data on the scattering of pions and kaons on  (fixed target) nucleons. Quantum field theory was not  much use for these processes at these energies; it still isn't, although "chiral perturbation theory, developed later, has had a measure of success.  Instead, theorists attempted to make quantitative predictions from general considerations. The scattering amplitude is constrained by the unitarity of the S-matrix, and, for 2-particle<math>\rightarrow</math> 2-particle processes,  it was assumed that it had certain analyticity and crossing properties when regarded as a function of the complexified energy and scattering angle. The analyticity means that the scattering amplitude satisfies a dispersion relation involving its imaginary part, which is in turn determined by unitarity. The golden boy  and arch evangelist of this era was Geoffrey Chew, a Californian based at Berkeley. He was alleged to have observed that "Every time I hear of a young man [sic] working on quantum field theory my heart bleeds and I think: there goes another lost soul". His collaborator, Henry Stapp, in a similar vein, quipped that "The contribution of quantum field theory to particle physics is less than epsilon", a dig at the axiomatic field theorists led by Arthur Wightman at Princeton and Rudolf Haag in Zurich. It's not that this approach is now known to be wrong or misguided, it's just that for the most part it wasn't very fruitful when applied to purely hadronic processes. It was much more successful when the hadronic interactions were probed by electromagnetic or weak interactions. The combination of (perturbative) quantum field theory to describe the probes and dispersion relations for the hadronic interactions was quite productive. In 1966 Gabriel was working on one of the most spectacular predictions that had been obtained using this technique, viz the claim by Dashen & Frautschi to have explained the neutron-proton mass difference as being due to electromagnetic radiative corrections. In the absence of electromagnetic interactions, it was believed that the neutron and proton would have equal masses, and indeed that the SU(2) isospin symmetry would also be exact. The "democratic" zeitgeist averred that all hadrons are equal, so that, for dispersion relation purposes, and using just pions and nucleons, a neutron can be regarded as a bound state of a proton and a negative pion, whereas the proton  can be regarded as a bound state of a neutron and a positive pion. In the former, there is an attractive  (i.e. negative) Coulomb contribution to the self energy deriving from single photon exchange, but not in the latter; magnetic effects are small. It follows that the neutron is lighter than the proton, in contradiction to the result of Dashen & Frautschi and, unfortunately, to the experimental data; the neutron is heavier than the proton by about 1.29 MeV/c². Gabriel identified the error in their work - the incorrect treatment of an infrared divergence - but amazingly they declined to correspond and, so far as I know, never recanted. These days it is believed that the mass difference of the nucleons derives from the mass difference between the  up u and down d quarks (see below) which constitute the nucleons, proton (uud) and neutron (udd). The origin of this latter difference is unknown, but is not now thought to have an electromagnetic origin. Another of Gabriel's projects used sidewise dispersion relations to calculate the electric dipole moment (EDM) of the neutron, a quantity whose measurement has been an enduring and continuing theme of the EPP group's research here. Both the magnetic dipole moment μ_n and electric dipole moment d_n, if there is one,  must be proportional to the spin s_n of the neutron,  which is an axial vector, since at rest there is no other vector available. However the former is coupled to the magnetic flux B, also an axial vector, whereas the latter is coupled to the electric fieldE, a vector. The magnetic coupling  is therefore parity-conserving, whereas the latter is parity-violating. Further, the spin changes sign under time-reversal T, as does B, whereas E is invariant. Thus the magnetic coupling  is also T-conserving and the latter isT-violating. The known existence of both parity-violation and T-violation in the weak interactions indicates that there must Gabriel, with his student Eddy White, published an upper bound on the EDM <math>|{d}_n| \lesssim 10^{-23}</math> e cm, and this result was refined the following year  by another of Gabriel's students, David Broadhurst. Interestingly, although I was unaware of it until recently, Gabriel must have been thinking about this since at least 1965, barely one year after the discovery of CP-violation.  His student Saime Goksu wrote her MPhil thesis that year on the topic; it is in the list of DPhil and MPhil theses that appears on another page of this site.

The first semblance of order in the burgeoning hadronic zoo was brought by Murray Gell-Mann (Norman's supervisor) in 1961. He incorporated the SU(2) isospin group into the larger SU(3) symmetry group. (The word "flavour" was a later addition to the lexicon.) He observed that the nine pseudoscalar mesons <math>(\pi^{\pm,0}, K^\pm, K^{\pm}, \bar{K}^0 \eta, \eta')</math> all having spin J and parity P with J^P=0^-,  fitted neatly into the octet 8'plus singlet 1 representation of SU(3), as did the vector J^P=1^- mesons <math>(\rho ^{\pm,0}, {K^*}^{\pm},{K^*}^0, \bar{K^*}^0, \omega, \phi)</math>. Similarly, the eight J^P=(1/2)⁺ nucleons and hyperons  <math>(p,n,\Sigma ^{\pm,0}, \Xi ^{-.0},\Lambda)</math> also filled out an 8, However, the nine J^P=(3/2)⁺ baryons <math>(\Delta ^{++,+,0,-}, {\Sigma ^*}^{\pm,0},{\Xi ^*}^{0,-})</math> left a single unfilled slot in the decuplet  10 representation. A negatively charged isospin singlet state Ω ⁻ ; with strangeness -3 was missing. The source of the breaking of the SU(3) symmetry was unknown, although manifestly considerably larger than the presumed electromagnetic breaking of the isospin symmetry. However the breaking between Σ ^* and Δ isospin multiplets is <math>m_{\Sigma^*}-m_{\Delta} \sim 150</math> MeV/c², and that between the Ξ ^* and Σ ^* multiplets is also <math>m_{\Xi^*}-m_{\Sigma ^*} \sim 150 </math> MeV/c². Thus the predicted mass of the Ω ⁻ was <math>m_{\Omega ^-} \sim m_{\Xi^*}+150</math> MeV/c² ~1680 MeV/c². Its discovery with a mass a little below the predicted value, the first of a particle with -3 units of strangeness, at Brookhaven in 1964, was a welcome and rare triumph for theory. The above representations of SU(3) can of course all be constructed using the fundamental triplet3 representation and its complex conjugate <math>\bar{3}</math>. In particular <math>3 \times \bar{3}= 8 + 1</math>, so the nine pseudoscalar mesons could be regarded as bound states constructed from a baryon triplet (p,n,Λ) and their antiparticles. This was originally proposed by Sakata in 1956. However it does not explain the origin of the baryon octet and decuplet. In 1964 Gell-Mann made a further proposal. More generally the 3 representation must consist of an isodoublet of "quarks" (u,d) with electric charges (in units of ) of (z+1,z) together with an isosinglet quarks with charge z; the Sakata model corresponds to the choice z=0. (Presumably in deference to Sakata, the quarks were often denoted by (p,n,λ) rather than the much better notation that we now use.) If we also demand that the electric charge Q is a generator of SU(3), then traceQ=0 and z=-1/3. The quark charges are then fractional (Q_u,Q_d,Q_s)=(2/3,-1/3,-1/3), and the anti-quarks in the <math> \bar{3}</math> representation have the negatives of these. The baryon octet and decuplet, with the correct charges, then arise in the product <math>3 \times 3\times 3= 1+8+8+10</math>. In this picture a baryon is a bound state of three quarks (qqq) and a meson is a quark-anti-quark bound state <math>(q\bar{q})</math>. Clearly the quarks had to be fermions, but initially the view was that they were only "mathematical entities" used to construct the weak hadronic current, which was the real quantity of physical interest. It was recognised too that Pauli's exclusion principle would forbid the s-wave ground state baryons, e.g. Δ ^{+ +} = uuu, and this was resolved by the invention of three "colours": each quark flavour q=u,d,s exists in three colours q_i ( i=1,2,3) labelling the fundamental representation 3 of a (different) SU(3)_colour symmetry. (Americans often preferred to use i= `red', `white', and `blue'.) Then s-wave baryons are allowed in the totally antisymmetric, colour-singlet representation. The SU(3)<sub>colour</sub> symmetry was eventually promoted to a local, gauge symmetry, and QCD was born. Note that the one-third integral charges of the quarks derived from the three then known flavours. This in turn requires baryons to be made of three quarks, and hence the need for three colours. Since then, three more quark flavours, `charm' c, `top' t and `bottom' b, have been identified, but they all still have one-third integral charges. I have always found it remarkable that the discovery of the SU(3)_colour symmetry was based on what must be a fallacious argument.

Gabriel and Norman both had students working on the quark model of hadrons. Ken Bowler, Norman's student, (now also retired(!) from Edinburgh) wrote his thesis in 1969 on electromagnetic transitions in the quark model. He was concerned with the radiative decays of the baryon resonances, and the possible necessity of including spin-orbit effects if the quarks have large magnetic dipole moments.  It was around this time, at SLAC, that the first evidence of  (Bjorken) "scaling" was observed in the deep inelastic scattering of electrons on protons in which <math>ep \rightarrow eX</math>; only the scattered electron in the final state was observed, X denotes everything else. The interpretation of the data is that the electrons "see" hard constituents of the proton, christened "partons" by Feynman. (This is a direct analogue of Rutherford's discovery in 1911 of the atomic nucleus by the scattering of alpha particles.) The immediate inference was that the partons are quarks. Indeed, since the electron is scattered by the charge of the quark, the cross-section is proportional to the sum of the squares of the  constituent quark charges (=5e²/9 for a proton), not by the square of the sum, viz the square of the proton's charge (=e²). The data support the former prediction. Stanley Brodsky from SLAC and Geoffrey West from Los Alamos were long-term visitors at this time and taught us about these matters. The deep inelastic scattering of neutrinos on nucleons is determined by the weak charges of the constituent quarks, and I later worked on this in collaboration with Alex Love and Dimitri Nanopoulos (see below).  Norman had a long-standing interest in photoproduction <math>\gamma N \rightarrow \pi N</math>, his first student, Chau, having written his thesis in 1966 on this and singular potential theory, another long-standing interest. Daresbury Nuclear Physics Laboratory in Cheshire started up in the late sixties with the construction of a new synchotron NINA accelerating electrons to 5 GeV, which was also used to produce a collimated photon beam with energy 4.6 GeV; Norman was a consultant there. It was therefore a natural development to study  electro-pion production <math>eN \rightarrow e \pi N</math>, which utilises a virtual photon rather than a real, on-shell one. This too became a long-term interest. Brian Read was the first of his students to work on this project, and there were several others. Using the electromagnetic current as a probe (but also the pion, related by PCAC to the axial current), they used dispersion relation techniques to make predictions both for the exclusive process above, and the inclusive process <math>eN \rightarrow e \pi X</math>. Norman, with his student Nigel McKenzie, also worked on low energy theorems for the elastic scattering of photons and neutrinos, using a novel technique in which the zero-mass limit of a massive vector particle is taken when it is coupled to a non-conserved current. The low energy theorems are readily converted to sum rules for the total cross sections using the optical theorem and analyticity.

The Princeton preprint that sparked Roger's interest in my work presented the (perturbative field theory) calculation of the electromagnetic radiative corrections to muon and beta decay, mediated by an intermediate vector boson, then, as now, called the W boson.  The  pointlike current-current theory of the weak interactions developed by Feynman & Gell-Mann is a non-renormalisable theory. Although the (lowest order) radiative corrections to muon decay are finite, those to beta decay are not; they diverge as the logarithm of the cut-off Λ. In those days it was believed that strong interactions would supply the cut-off so the choice Λ~m_p was customarily made, but  the results are pretty insensitive to precisely which value is used. As one of Chew's lost souls, my PhD project was to redo the calculation with a massive charged W mediating the current-current interaction, an idea originally proposed by Lee & Yang, I believe. The W had not then been discovered, and all that was known was that if it existed it was heavier than about 1 GeV/c². (There was a buzz of excitement when its discovery at Brookhaven in 1962 was announced in the  New York Times but, of course, it was wrong.) It was known that this theory, involving a massive vector particle coupled to a non-conserved (weak) current, is also non-renormalisable. Nevertheless, my calculations showed that in the ratio of the decay rates the  mass m_W of the  intermediate W replaced the  cut-off Λ  that arose in the case of the point-like interaction; there were other differences deriving from the assumed interaction of the W with the electromagnetic field, but overall the effect was numerically very small. I continued to work on radiative corrections after my arrival in Sussex in 1965, to pion decay with Anjali, and with Richard Shann, and to muon capture in hydrogen with Martin Goldman, all research students. The divergence structure in higher orders of perturbation theory was studied with another student, Alex Bregman, with the ultimate aim, never actually achieved by me and my students, of identifying how these might be tamed by the introduction of new fields. One day in 1967 - it must have been late November - I wandered into the Physics Library, overlooking the Meeting House,  (in what is now 1A9 of Pevensey I, currently used for a PC cluster) to peruse recently arrived preprints and journals. In the new issue of Physical Review Letters I happened upon an article by Steven Weinberg entitled "A model of leptons". (I knew of other work by Weinberg and recalled his visit to Cambridge while I was a research student.) It considered just what we would now call the first generation of leptons, viz the electron e and its neutrino ν_e, arranged as a doublet of left-chiral fields <math>(e_L,{\nu_e}_L)</math> and the right-chiral singlet e_R - the neutrino was assumed to be massless and to have only its known left-chiral component. These interacted with the photon and (charged and neutral) intermediate vector bosons. The latter were constructed from the  Yang-Mills gauge boson fields for the group SU(2)xU(1), a generalisation of the familiar local U(1) gauge symmetry of the electromagnetic field. As in the more familiar case, the local symmetry requires that the gauge bosons are massless. (Having a Cambridge PhD, I am obliged at this juncture to record again that in 1954 another Cambridge graduate student, Ron Shaw, a contemporary of my future supervisor John C Taylor, addressed the problem of how to make the global SU(2) isospin symmetry of the hadrons into a local symmetry. He reached the same conclusion as Yang and Mills in the same year. However, his supervisor was unimpressed, observing that "Everyone knows that there are no charged photons". His supervisor was Abdus Salam!) Since the W bosons had to be massive the symmetry had to be broken, and Weinberg did this by "spontaneous" symmetry breaking. This required the introduction of a doublet of scalar fields coupled  to the leptons and gauge bosons and to themselves. The key bit was to break the SU(2) symmetry by giving the neutral component of the doublet a vacuum expectation value. (I was non-plussed by this, having always been taught that a quantum field is a fluctuation around the vacuum, in which, by definition, all fields are zero.) The upshot was a theory with the charged currents coupled correctly to charged, massive W bosons, the electromagnetic current coupled to a massless photon, but in addition a neutral current coupled to a massive neutral vector  boson Z. Tantalisingly, and prophetically, Weinberg observed that "The model may be renormalizable".  My immediate interest in the paper was that it offered a reasonable project for my MSc student, Gerry Kendall, to calculate the neutrino scattering cross sections predicted by the model. In 1968, after Gerry had left -  he subsequently joined the National Radiological Protection Board at Chilton -  I wrote up a paper on his results and submitted it for publication. Unfortunately, the model was adjudged to be of insufficient interest to merit publication of the paper, and indeed the absence of citations in those early years of what was to become a Nobel-prize winning paper fully justifies the referee's judgement. This was three years before 't Hooft  proved (in 1971) the renormalisability of spontaneously broken non-abelian gauge theories, a result that  triggered the  sea change towards quantum field theory that has lasted ever since. Roger did finally make use of my 1964 results in a paper he wrote in 1970 with an experimentalist, Joan Freeman, at AERE Harwell. The group at Harwell made precision measurements of the ft-values of various 0⁺<math>\rightarrow</math>0⁺nuclear transitions,  essentially the beta decay lifetimes. The theory was sufficiently well understood that electromagnetic radiative corrections had to be included. Even with their inclusion, though, the theoretical predictions following from the Feynman & Gell-Mann theory, with the cut-off Λ~m_p, disagreed with the experimental data. However, if instead the W-boson mediated the process, then there was accord provided that m_W<math>\gtrsim</math> 40 GeV/c². So far as I know, Roger and Joan's paper was the first to claim that the data required the existence of a W-boson, thirteen years before the discovery of the W and Z.

Alex Love joined our group as a postdoc in 1972 and almost immediately we began a collaboration that has endured until today. Of course, we were interested in the new "unifiedelectroweak gauge theories. The Weinberg model, by then enlarged to include hadrons (i.e.the three quark flavours), was disfavoured  because it generated strangeness-changing neutral current processes, <math>K^0\rightarrow \mu^+ \mu^-</math> for example, at a level well above the experimental upper bounds. Instead the Georgi-Glashow model with gauge group O(3) was initially favoured. Only  the electromagnetic current was neutral, but the price paid was an unobserved new positively charged leptonE⁺ to join e and ν_e in the fundamental lepton triplet representation. In collaboration with Graham Ross, Alex's former collaborator then at RAL (now at Oxford), and Alex's research student,  Dimitri Nanopoulos (officially Norman's, now at Houston and the Academy of Athens), we studied  the consequences for the purely hadronic weak interactions, viz nonleptonic hyperon and kaon decays, and parity-violating nuclear forces. The discovery in 1973, by Politzer and by Gross & Wilczek,  that non-abelian gauge theories in general, and QCD in particular, are asymptotically free explained the scaling behaviour in the deep inelastic processes, and led to the calculation of the (slow) "running" effects induced by QCD. We looked at these effects in the deep inelastic scattering of  neutrinos, in QCD and in theories that were asymptotically "nearly" free.  Arguably CERN's first big discovery was the discovery in 1973 of weak neutral currents in the Gargamelle bubble chamber. Clearly falsifying the Georgi-Glashow model, it led to the instatement of the Weinberg(-Salam) model as the prime candidate for a realistic renormalisable electroweak theory. The problem of strangeness-changing weak neutral currents was solved by the Glashow-Iliopoulos-Maiani mechanism that required the existence of a fourth quark flavour having the same electric charge as the u. It was named "charm" c, since it is naturally paired with strangeness. In the autumn of 1974, while I was trying to finish my boo<span class="texhtml" />k on Weak Interactions, the extremely narrow J / ψ particle, with J^P = 1 ⁻ , was discovered at Brookhaven and SLAC.  It seemed like a tidal wave had broken over particle theory, that a new era had begun. There were feverish phone calls to people in Oxford who were in contact with (people who were in contact with)  the experimental groups. The popular press talked about the "new physics", but it wasn't really.Everyone wrote papers on the topic, and we were no exception. In collaboration with Tim Jones, then a newly-arrived postdoc from Oxford (now at Liverpool) we wrote an (entirely justifiably) long-forgotten paper arguing that the J / ψ was part of an  adjoint representation of a hypothetical SU(3) flavour symmetry. The correct explanation, actually published by Ben Lee before its discovery, is that  J / ψ is a <math>c\bar{c}</math> bound state, analogous to the <math>\phi=s\bar{s}</math> in the J^P = 1 ⁻ meson octet.  Confirmation of the fourth flavour followed in the following summer when the charmed isodoublet bound states <math>(c\bar{d},c\bar{u}) =(D^+,D^0)</math> were discovered in e⁺'e^-collisions at SLAC. However, whereas m_φ > 2m_K , so that <math>\phi \rightarrow K^+K^-</math> is the dominant decay mode, m_ψ < 2m_D so that the decay <math>\psi\rightarrow D^+D^-</math> is kinematically forbidden; the width is small because it can decay to lighter states only via the emission of three gluons. By then I was on sabbatical in India, and the foundations of what came to be called the "Standard Model" had been laid. My student, Wilson Angerson, and John C Taylor's student, Douglas Ross  in Oxford (now at Southampton), completed the "proper" calculation of the radiative corrections to muon and beta decay in the Weinberg model in 1974. This seems like a good place to conclude my account of the early years of TPP. There is, of course, much to be written about in the succeeding years. Maybe others will feel moved to do so, or maybe  I shall. But not now.

The forgoing paragraphs are my recollections, impressions and musings on our early years. I have tried to write a balanced account, but obviously I remember most about what I did.  I hope at least that everything I have said is true, but even if it is, it will certainly not be the whole truth. I therefore tender my apologies in advance to anyone who, after reading what I have written, feels that I have overlooked them and/or their contribution. However, you, dear reader, can do something about it too. Just over a year ago, when it was first suggested that we mark our fiftieth anniversary by writing some sort of history of Physics & Astronomy,  I was asked if I would like to contribute. I was reluctant myself to attempt writing a comprehensive and balanced account of our history, for all of the reasons just mentioned, which is why I proposed constructing a wiki. Errors and omissions can easily be corrected by the author, but also by you.  If, perchance, you think that there is a major topic that should be covered, in this period, or later, please just write about it and mount it. That's the wonder of the wiki!