From Yang-Mills to Asymptotic Freedom to Quantum Chromodynamics

1. From Yang-Mills to Asymptotic Freedom to Quantum Chromodynamics Jiří Chýla, Institute of Physics, Prague

2. From Yang-Mills to Asymptotic Freedom to Quantum Chromodynamics Jiří Chýla, Institute of Physics, Prague The story of the emergence of the concept of gauge invariance and its importance for the formulation of physical laws show that Dirac was right to expect that Physical laws should have mathematical beauty

3. From Yang-Mills to Asymptotic Freedom to Quantum Chromodynamics Jiří Chýla, Institute of Physics, Prague The story of the emergence of the concept of gauge invariance and its importance for the formulation of physical laws show that Dirac was right to expect that Physical laws should have mathematical beauty but the converse is not true as mathematical beauty does not necessarily imply physical relevance.

4. There are many excellent texts covering various aspects of the emergence and application of nonabelian gauge theories. My recommendations: D. Gross: Twenty five years of asymptotic freedom • C.N. Yang:Interview in The Mathematical Intelligencer 15/4 • N. Straumann: Early Histrory of Gauge Theories • S. Weinberg: The Making of the Standard Model • H. Lipkin: Quark model and quark phenomenology • O. Greenberg: From Wigner’s supermultiplet theory to QCD • G. ‘t Hooft: When was the asymptotic freedom discovered? • de Rujula: Fifty years of Yang-Mills theories: a phenomeno- • logical point of view • D. Gross: Oscar Klein and gauge theory

5. Premature burial From Nambu’s bookQuarks The renormalization procedure,developed by Dyson, Feynman, Schwinger and Tomanaga was spectacularlysuccessful in QED. The physicalmeaning of renormalization was, however, not truly understood and the renormalization was considered by most physicists, including Dirac and Wignera trick. The prevalent feeling was that renormalization simply swept the infinities under the rug, but that theywere still there.

6. In middle1950’sLandauandPomeranchukattempted to give the renormalization procedure in QED good physical meaning and mathematical sense.They put a finite “bare” electric charge e0=e(r0)on a sphere of radius r0, placed it in the QED vacuum and calculated how it appears at a finite distance r>r0. bare charge e0 must be a function of the radius r0! i.e. the QED vacuumscreensthe bare electriccharge! Sending the radius of bare electron to zero and keeping the bare charge e0 constant, theeffective chargee2(r)vanishes for any fixed distance r! This is the famous problem of “zero charge”, which for Landau implied that QED is incomplete: We reach the conclusion that within the limits of formalelectrodymics a point interaction is equivalenttonointeraction at all.

7. Landau pole in QED ... Turning the argument around, they could have asked how would the bare charge e0=e(r0) or rather α(r0) have to depend on r0 to yield a finite effective electric coupling α(r)at distance rwhen r0 vanishes. The second formula suggests that it would have to grow to infinity at finite distance rLdefining the so called “Landau pole”. In fact, the problem with the renormalization proce-dure in QED is not the fact that bare electric charge diverges, but that it does soat a finite (though very small) distance!

8. ... is absent in QCD! One can only wonder whether Landau and Pomeranchuk asked themselves this natural question.Had they done it, they might be led to the concept of asymptotic freedom becauseit suffices to change the sign ofβ0 for the bare as well as effective charges to be well-defined, and actually vanish, at small distances Modern, inherently nonperturbative, approach to the renorma- lization, which lies at the heart of lattice gauge theory, is just to construct the dependence α0=α(r0) in such a way to yield finite values of physical quantitiesin the limit of vanishing r0.

9. Dirac on renormalization of QED in 1974 Hence most physicists are very satisfied with the situation. They say: “Quantum electrodynamicsis a good theory, and we do not have to worry about it any more.” I must say that I amvery dissatisfied with the situation, because this so-called “good theory” does involve neglectinginfinities which appear in its equations, neglecting them in an arbitrary way. This is just notsensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out tobe small – not neglecting it just because it is infinitely great and you do not want it!

10. Dirac draw uncompromising conclusion: Of course, the proper inference from this work is thatthe basic equations are not right.... There must be some drastic change introduced into them so thatno infinities occur in the theory at all and so that we can carry out the solution of the equationssensibly, according to ordinaryrules. • Dirac criticism of the renormalization procedure • was justified for QED, but • does not applytoYang-Mills gauge theories. • For these theories Dirac was thus wrong!

11. The beginning of all starting point: isotopic dublet of nucleons: In the present paper we wish to exlore the possibility of requiring all interactions to be invariant under independent rotations of the isotopic spin at all space-time points,.. We then propose that all physical processes (not involving electromagnetic field) be invariant under the isotopic gau- ge transformation

12. this requirement lead them to the following Lagrangian density gauge bosons Three electrically charged gauge bosons and their selfcoupling ensued automatically The quanta of the b-field clearly have spin unity and iso- spin unity. We know their electric charge too because all the interactions that we propose must satisfy the law of conservation of the electric charge, which is exact.

13. but question remained about the mass of the b-quantum We next come to the question of the mass of the b-quantum, to which we do not have a satisfactory answer. One may argue that without a nucleon field the lagrangian would contain no quantity of the dimension of a mass and that therefore the mass of the b-quantum in such a case is zero. The argument is how- ever subject to the criticism that, like all field theories, the b-field is beset with divergences and dimensional arguments are not satisfactory. b b mass of the gauge boson to be determined by its full propagator YM considered seriously the possibility that their gauge bosons will eventually be massive: A conclusion about the mass of the b-quantum is of course very important in deciding whether the proposal of the existen- ce of the b-field is consistent with experimental information.

14. Under the spell of gauge principle Since around 1960Sakurai, Salam, Ward, Neeman and others started considering local gauge invarianceas guiding principle in constructing theories of strong, weak as well as electromagnetic interactions. Abdus Salam & John Wardin On a Gauge Theory of Elementary particles Nuovo Cimento 11 (1960), 165 Our basic postulate is that it should be possible to generate strong, weak and electromagnetic inter- action terms by making local gauge transformations on the kinetic terms in the free Lagrangian for all particles. This is the statement of ideal, which in this paper at least, is only very partially realized.

15. The straightforward generalization of the original Yang-Mills was proposed in 1961 bySalam and Ward who extended isospin symmetry toSU(3) version of theSakata model by gauging the fundamental tripletof baryons 8-parameter traceless hermitian matrix they got the octet of selfinteracting gauge vector mesons infinitesimal gauge transformation

16. kinetic term invariant full YM these break GI! Close to Eightfold way but different inbasic multiplet and no discussion of baryons beyond the fundamental triplet p,n,

17. As an alternative to the Sakata model based on the relation Y. Neeman and M. Gell-Mann proposed in early 1961 the Eightfold Way which starts with the product of three SU(3) triplets and leads to different set of multiplets. At the beginning of 1961 it was still not quite clear which scenario was correct. The stories of their discoveries are quite different as are their professional careers and whole lives.

18. Eightfold way according to Y. Neeman Derivation of Strong interactions from a Gauge invariance Y. Neeman, Nucl. Phys. 26 (1961), 222 contains a full-fledged Yang-Mills gauge theory of strong interactions extending the original YM theory to SU(3) unitary symmetry. Baryons are assigned to octets as are the pseudoscalar mesons. Octet of selfinteracting vector bosons is predicted, though no vector meson was known at the end of 1960. But no interpretation of the fundamental tripletattempted.

20. Discovery of vector mesons proceeding as quasi two-body process followed by ρ in May, Φ in July and ωin August 1961

21. Who was afraid of gauge theory? MGM’s preprint is truly fantastic for the straightforwardness with which the idea is presented.

22. The vector mesons are introduced in a very natural way, by extension of the the gauge principle of Yang and Mills. Here we have a supermultiplet of eight mesons. In the limit of unitary symmetry we have completely gauge-invariant and minimal theory like electromagnetism. and on another place Now the vector mesons themselves carry F spin and there- fore contribute to the current which is their source. The prob- lem of constructing a nonlinear theory of this kind has been completely solved in the case of isotopis spin by Yang and Mills and by Shaw. We have only to generalize their result (for three vector mesons) to the case of F spin and eight vector mesons.

23. leptons played the role of quarks: gauge transformations on all particles involved:

24. full Yang-Mills Lagrangian written out unique coupling noting that “There are trilinear and quadrilinear interactions amongst the vector mesons, as usual ...” But this preprint has never been published!!

25. instead we read in “Symmetries of Baryons and Mesons” In Section VIII we propose, as an alternative to the symmetrical Sakata model, another scheme with the same group, which we call ``eightfold way''.Here the baryons, as well as mesons, can form octets and singlets, and the baryons N, ,  and  are supposed to constitute an approximately degenerate octet. Nowhere does our work conflict with the program of the Chew et al. of dynamical calculation of the S-matrix from strong interactions using dispersion relations. If there are no fundamental fields ….all baryons and mesons being bound or resonant states of one another, models like Sakata will fail;the symmetry propertieswe have abstracted can still be correct, however. Remarkably, this paper does not mention the gauge principle and does not refer to Yang-Mills paper at all!

26. S-matrix and bootstrap: Theory of everything? From G. Chew: S-Matrix Theory, (W.A. Benjamin Inc, 1963). I believethe conventional association of fields with strong interacting particles to be empty. It seems to me thatno aspect of strong interactions has been clarified by the field concept.Whatever success theory has achieved in this area is based on the unitarity of the analytically continued S-matrix plus symmetry principles. I do not wish to assert (as does Landau) thatconventional field theory isnecessarily wrong, but only that it issterilewith respect to the strong interactionsand that, like an old soldier, it is destined not to die but just to fade away…The notion, inherent in conventional Lagrangian field theory, that certain particles are fundamental while others are complex, is becoming less and less palatable …

27. For application of YM theories to strong interactions the identification of correct space to gauge was crucial. This sounds trivial, but was not. It took 20 years to come to the conclusion that the fundamental object of nonabelian theory of strong interactions are coloredquarks and that forces acting between them follow from gauging the color degree of freedom.

28. Quark model according to Zweig Zweig: Both mesons and baryons are constructed from a set of three fundamental particles, called aces.Each ace carries baryon number 1/3 and isfractionally charged. SU(3)is adopted as a higher symmetry for the strong inte- interactions.Extensive space-time and group theoretic structure is then predicted for both mesons and baryons … An experimental search for the aces is suggested.

29. Quark model according to Gell-Mann PL 8 (1964), 214 A formal mathematical model based on field theory can be built up for the quarks exactly as for p, n and Λ in the old Sakata model, for example with all strong interactions ascribed to a neutral vector meson field interacting symmetrically with the three particles. Within such a framework the electromagnetic currents is just

30. MGM’s view of the role of quarks (Physics 1 (1964), 63) In order to obtain such relations that we conjecture to be true, weuse the method of abstraction from a Lagrangian field theory model. In other words, we construct a mathematical theory of the stronglyinteracting particles, which may or may not have anything to do withreality, find suitable algebraic relations that hold in the model, postulatetheir validityand then throw away the model. We may compare thisprocess to a method some-timesemployed in French cuisine: a pieceof phea-sant meat is cooked between two slices of veal, which are thendiscarded.

31. Confinement: consequence or source of nuclear democracy? M. Gell-Mannat 1992 ICHEP: I was reflecting that if those objects (i.e. quarks) could not emerge to be seen individually, then all observable hadrons could still have integral charge and also the principle of “nuclear democracy” could be preserved unchanged for observable hadrons. With this proviso, the scheme appealed to me. For MGM nuclear democracy was fundamental principle of strong interactions andconfinement its consequence: Since I was always convinced that quarks would not emerge to be observed as single particles (“real quarks”), I never paid much attention to the Hahn-Nambu model in which their emergence was supposed to be made possible by giving them integral charges.

32. The concept of colored quarks has been introduced in late 1964 primarily in order to explain the apparent problem of quark statistics implied by the success of SU(6) symmetric quark model. To reconcile this model with Pauli principle Greenberg proposed to interpret quarks as parafermions of rank 3. It soon became clear that this assumption is equivalent to assigning to each quark flavor another internal quantum number, which could take three different values and which, following Pais’ suggestion at 1965 Erice Summer School, has beencalled color.

33. While for most of theorists color was introduced to solve the quark statistics problem Nambu had used it since early 1965 as a dynamical variable generating the force between quarks, assuming furthermore that the force between colored quarks is due to the exchange of octet of colored gauge bosons, which induce the effective four quark coupling of the type and lead to (potentially infinite) gap between colorless and colored states. In this way his model contained all essential elements of QCD, except that it was not Quantum Field Theory.

34. Gell-Mann on quarks (summer 1967) The idea that mesons and baryons are made primarily of quarks is difficult tobelieve, since we know that, in the sense of dispersion theory, they are mostly, if not entirely,made up out of one another. The probability that a meson consists of a real quarkpair rather than two mesons or a baryon and antibaryon must be quite small. Thusit seems to me that whether or not real quarks exist, the q or q wehave been talking aboutare mathematical entities ...... If the mesons and baryons are made of mathematical quarks, then the quark model mayperfectly well be compatible with bootstrap hypothesis, thathadrons are made up outof one another.

35. Too much scaling may be misleading Bjorken derived scaling behavior observed at SLAC from current algebra considerations assuming that the nucleon structure functions stay finite in the limit of infinite momentum transfer. But we now know that in QCD the above assumption does not hold and, consequently, his paper is, indeed, empty!

36. Bardeen, Fritzsch, Gell–Mann in1972 (hep-ph/0211388) One is considering the abstraction of results that are true only formally, withcanonical manipulation of operators, and that fail, by powers of logarithmic factors, in eachorder of renormalized perturbation theory, in all barely renormalizable models. The reason for the recent trend is, of course, the tendency of thedeep inelastic electron scattering experiments at SLAC to encourage belief in Bjorken scaling, which fails to every orderof renormalized perturbation theoryin barelyrenormalizable models.There is also the availability of beautiful algebraic results, with Bjorken scaling as one of their predictions, ….

37. Why asymptotic freedom? Because only asymptotically free QFT could explain surprisingly good scaling behavior of nucleon structure functions observed since 1967 in deep inelastic electron-nucleon scattering at SLAC and reconcile it with experimental fact of quark confinement. & Because for asymptotically free quantum field theories the renormalization procedureas formulated by Landau & Pomeranchukcan be consistently carried through. In this sense asymptotically free Quantum Field Theories do not contain ultraviolet divergencies. For these theories Dirac was thus wrong!

38. In 1972 quarks were still not taken seriously In Summer 1972 Gell-Mann and Fritzsch presented their view at XVI ICHEP in Chicago in a contribution called Current Algebra: Quarks and WhatElse? We assume here thatquarks do not have real counterparts that are detectable in isolationin the laboratory – they are supposed to bepermanently bound inside mesonsand baryons.........It might be a convenience to abstract quark operators themselves, or other non–singlets with respect to color, …,but it is not a necessity. It may not even be much of aconvenience since we would .... be discussing a fictitiousspectrum for each fictitious sector of Hilbert space, and we probably don’t want to loadourselves with so much spurious information.

39. Their hope that We might eventually abstract from the quarkvector–gluon field theory model enoughalgebraic information about the color singlet operators in the model to describe all thedegrees of freedom that are present. and thus We would havea complete theory of the hadrons and their currents, and we need never mention anyoperators other than color singlets. • has not been born out by further theoretical developments and experimental results, in particular those on • heavy quarkonia spectra and • jet phenomena which require that we treat quarks and gluons in the same way as leptons and basically forget about confinement.

40. This paper is quoted as containing the suggestion that gluons could form the octet of Yang-Mills gauge bosons. In fact this option is mentioned in the following context Now the interesting question has been raised lately whether we should regard the gluonsas well as the quarks as being non–singlets with respect to color (private communication of J. Wess to B. Zumino). For example, theycould form a color octet ofneutral vector fields obeying the Yang–Mills equations. they, however, ignored this option: In the next three Sections we shall usually treat the vector gluon, forconvenience, as acolor singlet.

41. D. Gross: QFT must be destroyed! I decided, quite deliberately, to prove that local fieldtheory could not explain the experimental fact of scaling and thus wasnot an appropriate framework for the description of the stronginteractions.Thus, deep inelastic scattering would finally settle the issue as tothe validity of quantum field theory.The plan of the attack was twofold. First,I would prove that “ultravioletstability,” the vanishing of the efective coupling at short distances,later calledasymptotic freedom, was necessary to explain scaling.

42. Second, I would show that there existed no asymptotically free field theories.The latter was to be expected. After all the paradigm of quantumfield theory –QED- was infrared stable; inother words, the efective charge grew larger at short distances and noone had ever constructed a theory in which the opposite occurred Together with FrankWilczek they succeeded in the first step, but failed in the second because: Nonabelian gauge theories have turned out to be (under certain circumstances) asymptotically free! D. Gross:For me the discovery of asymptotic freedom was totally unexpected. …. Field theory wasnot wrong, insteadscaling must be explained by an asymptotically free gauge theory ofthe strong interactions.

44. Conversion of Saul of Tars to St. Paul

45. Asymptotic freedom had been discovered in these two papers

46. shortly followed by three papers containing complete formulation of QCD, together with elaboration of its application to DIS. All these papers existed as preprints by the date of their submissions and thus months before the submission of the paper which is often, but incorrectly, credited with the formulation of QCD.

47. A fourth apparent advantage of the color octetgluon scheme has recently been demonstrated……the behavior of light conecommutators comes closer to scaling behavior than inthe color singlet vector gluon case.However, actualBjorken scaling does not occur….. For us, the result that the color octet field theorymodel comes closer to asymptotic scaling than thecolor singlet model is interesting, but not necessarilyconclusive, since we conjecture that there may be amodification at high frequencies that produces trueasymptotic scaling.

48. 31 years after their work Gross, Wilczek and Politzer were awarded the 2004 Nobel Prize. Their theory provides the basic framework for reconciling the apparently conflicting facts that quarks do not exist as free particles but in some situations appear to behave as almost free. The key manifestation of their “existence” are jets

49. People knowing without understanding Perhaps the first who observed this behaviour of a coupling constant were V. S.Vanyashin and M. V. Terentev in 1965. Studying the effects of vacuum polari-zation due to loops of charged vector bosons on the renormalized electric charge they found the expression and noted that these loops give the opposite sign that those of fermion loops in standard QED! But they attributed this result to the fact that the theory with charged vector bosons coupled to photons is not renormalizable.

50. G. ‘t Hooft:When was asymptotic freedom discovered? hep-th/9808154 I knew about thebeautiful scaling behaviour of non-Abelian gauge theories. Suspecting that this featureshould be known by now by the experts on the subject of scaling, I did not speak uplouder.Veltman … warned me thatno-one would take such an idea seriously as long as it could not be explained why quarkscannot be isolated one from another. By 1972, I had calculated the scaling behavior, and I wrote it in the form (5.3) where Cj are Casimirs for VB, fermions and scalars