Topics in Quantum Field Theory and Gauge Theories pp 221-291 | Cite as

# Part A: An introduction to unification

(Mirror-symmetry, confinement versus liberation, unifying mass-scale etc.

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Yukawa Coupling Local Symmetry Spontaneous Symmetry Breaking Gauge Coupling Constant Spontaneous Breakdown## Preview

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## References and footnotes

- 1.C. N. Yang and R. C. Mills, Phys. Rev. D96, 191 (1954).Google Scholar
- 2.J. Schwinger, Ann. Phys. (N.Y.) 2, 407 (1957); S. L. Glashow, Nucl. Phys. 22, 57a (1961); Abdus Salam and J. C. Ward, Physics Letters 13, 168 (1964). In the present context, the model based on spontaneous breakdown of the local symmetry SU(2)
_{L}x U(1) is due to S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967) and Abdus Salam, in Elementary Particle Theory, Nobel Symposium, Ed. by N. Svartholm (Almgvist, Stockholm, 1968), p. 367.Google Scholar - 3.J. C. Pati and Abdus Salam, Phys. Rev. D8, 1240 (1973); Phys. Rev. Lett. 31, 661 (1973); Phys. Rev. D10, 275 (1974), and Physics Letters 58B, 333 (1975).Google Scholar
- 4.H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974); H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93, 193 (1975); F. Gürsey and P. Sikivie, Phys. Rev. Lett. 36, 775 (1976); P. Ramond, Nucl. Phys. B110, 214 (1976).Google Scholar
- 5.These attempts are admittedly incomplete, until gravity is in the picture.Google Scholar
- 6.In general, if G is a direct product of several groups, correspondingly several coupling constants would appear.Google Scholar
- 7.P. W. Higgs, Phys. Rev. Lett. 12, 132 (1964); P. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964); G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1965); P. W. Higgs, Phys. Rev. 145, 1156 (1966); T. W. B. Kibble, Phys. Rev. 155, 1554 (1967).Google Scholar
- 8.J. Goldstone, Nuovo Cim. 19, 15 (1961); Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961); J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962).Google Scholar
- 9.One still needs to examine if quantum-corrections could disturb the character of the ground state of the theory (see e.g. S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973)).Google Scholar
- 10.G' t Hooft, Nucl. Phys. B33, 173 (1971); B35, 167 (1971); B. W. Lee and J. Zinn-Justin, Phys. Rev. D5, 3121, 3137 (1972).Google Scholar
- 11.See for example, J. M. Cornwall, D. N. Levin and G. Tiktopoulos, Phys. Rev. Letters, 30, 1268 (1973); C. H. Llewellyn Smith, University of Hawaii Summer School Lecture Notes (1973) and references therein.Google Scholar
- 12.See for example R. N. Mohapatra, S. Sakakibara and J. Sucher, Phys. Rev. D10, 1844 (1974).Google Scholar
- 13.As to whether spontaneous symmetry breaking is entirely due to the presence of elementary Higgs-scalars, or whether it is partly or entirely induced dynamically is not yet clear.Google Scholar
- 14.If potential V(φ) has a global symmetry higher than the local symmetry, one needs to examine whether light pseudo goldstone bosons are generated in the theory.Google Scholar
- 15.D. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).Google Scholar
- 16.See for example H. D. Politzer “Testing of Scaling and QCD with muon Beams”, HUTP-77/A038 and references therein; G. Altarelli, G. Parisi and R. Petronzio, Physics Letters 63B, 183 (1976).Google Scholar
- 17.S. Abers and B. W. Lee, Physics Reports 19C, (1973); M.A.B. Bég and A. Sirlin, Ann. Rev. Nucl. Science, 24, 379 (1974). C. Llewellynsmith, Hawaii University Lecture Notes (1973). S. Weinberg, Rev. Mod. Phys. 46, No. 2, (1976).Google Scholar
- 18.Recent CDHS measurements yield sin
^{2}θ_{W}= 0.24 ± 0.02(M. Holder et. al. CERN Preprint, 1977). Earlier measurements, gave somewhat higher values for sin^{2}σ_{W}, see for example, J. Blietschau et.al., Nucl. Phys. B118, 218 (1977); B. Barish et. al., Proc. Internat. Neutrino Conf. Aachen, 1976 (p. 289); A. Benvenuti et. al., Proc. Internat. Neutrino Conf. Aachen, 1976 (P. 296).Google Scholar - 19.J. Blietschau et. al., Nucl. Phys. B118, 218 (1977); H. Faissner et. al., reports of Aachen-Padova results at Aachen, Tbilisi and Ben-Lee Memorial Conference (1977).Google Scholar
- 20.C. Bouchiat, J. Iliopoulos and P. Meyer, Phys. Lett. 38B, 519 (1972).Google Scholar
- 21.H. Georgi and S. L. Glashow, Phys. Rev. D6, 429 (1972).Google Scholar
- 22.D. J. Gross and R. Jackiw, Phys. Rev. D6, 477 (1972).Google Scholar
- 23.See for example the first paper of Ref. 3 where the relevant Yukawa interaction is written down.Google Scholar
- 24.S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2, 1285 (1970).Google Scholar
- 25.R. N. Mohapatra, J. C. Pati and P. Vinciarelli, Phys. Rev. D8, 3652 (1973); S. Weinberg, Phys. Rev. Lett. 31, 494 (1973).Google Scholar
- 26.A. I. Vainshtein and I. B. Khriplovich, JETP Letters 18, 141 (1973); M. G. Gaillard and B. W. Lee, Phys. Rev. D10, 897 (1974).Google Scholar
- 27.G. Goldhaber et. al., Phys. Rev. Lett. 37, 569 (1976). For a recent review, see the report by A. M. Litke, Invited Talk, European Physical Society Meeting, held at Budapest (July, 1977), SLAC-PUB-2024.Google Scholar
- 28.B. Wiik, Lectures presented at Cargese summer school, 1977.Google Scholar
- 29.See e.g. C. Baltay et. al., Phys. Rev. Lett. 39, 62 (1977) and references therein.Google Scholar
- 30.E. G. Cazzoli et. al., Phys. Rev. Lett. 34, 1125 (1975); G. Blietschau et. al., Phys. Lett. 60B, 207 (1975); J. Von Krogh et. al., Phys. Rev. Lett. 36, 710 (1976).Google Scholar
- 31.“Color” was first introduced implicitly through paraquarks of order three by 0. W. Greenberg, Phys. Rev. Letters, 13, 598 (1964), and subsequently explicitly through three triplets of quarks, by M. Han and Y. Nambu, Phys. Rev. 139, B1006 (1965) and P.G.O. Freund, Phys. Letters 15, 352 (1965). “Explicit” Color, which in general permits either solution for quark-charges (integral or fractional), appears to be needed for the generation of local SU(3)'-color-gauge interactions. [A straightforward formulation to generate Yang-Mills gauge interactions in the parafield-theory generates SO(3) rather than local color SU(3)-symmetry: see P. G. 0. Freund, Phys. Rev. D13, 2322 (1976), and also remarks by O. W. Greenberg and C. A. Nelson, Physics Reports 32C, Page 88 (1977)].Google Scholar
- 32.J. C. Pati and C. H. Woo, Phys. Rev. D3, 2920 (1971); K. Miura, T. Minamikawa, Prog. Theor. Phys. 38, 954 (1967). Plausible arguments for K-decays were given by C. A. Nelson and K. J. Sebastian, Phys. Rev. D8, 3144 (1973).Google Scholar
- 33.M. K. Gaillard and B. W. Lee, Phys. Rev. Letters 33, 108 (1974); G. Altarelli and L. Maiani, Phys. Letters 52B, 351 (1974).Google Scholar
- 34.The first suggestion of using SU(3)′-Color-gauges to generate superstrong interactions, with additional interactions generating strong (or medium strong) interactions, is due to Han and Nambu (Ref. 31).Google Scholar
- 35.The first suggestion of a renormalisable gauge theory of weak, electromagnetic and strong interaction using flavor to generate basic weak and SU(3)′-Color to generate basic strong interactions (via the gauge structure SU(2)
_{L}x U(1) x SU(3)′_{Col}was made by J. C. Pati and A. Salam (Aug, 1972, unpublished), work reported by J. D. Bjorken, Proceedings of the 1972 Batavia Conference (Vol. 2, Page 304). The same local symmetry permits either solution for quark-charges depending upon the nature of spontaneous symmetry breaking (see text). This work is elaborated in the first paper of Ref. 3.Google Scholar - 36.aThe hypothesis that local color symmetry remains unbroken, and that color is confined was proposed in stages by H. Fritzsch, M. GellMann and H.Leutwyler, Physics Letters B47,365 (1973).Google Scholar
- 36b.S. Weinberg, Phys. Rev. Letters 31, 494 (1973) and D. J. Gross and F. Wilczek, Phys. Rev. D8, 3633 (1973). See also, C. Itoh, et. al. (Preprint, 1973, unpublished).Google Scholar
- 37.0. W. Greenberg and D. Zwanziger, Phys. Rev. 150, 1177 (1966); H. J. Lipkin, Phys. Lett. 45B, 267 (1973).Google Scholar
- 38.For a good review of the color-hypothesis and its various versions, see 0. W. Greenberg and C. A. Nelson. Physics Reports 32C (1977).Google Scholar
- 39.J. C. Pati, Phys. Rev. D4, 2143 (1971); J. C. Pati and A. Salam, Trieste Report IC/73/81, A. D. Dolgov, L. B. Okun, and V. I. Zacharov) Phys. Lett. 47B, 258 (1973). Breaking global SU(9) via Higgs mechanism would require a rather elaborate set of Higgs-scalars, such a large global symmetry would inevitably generate a large number of light pseudoGoldstone bosons. Thus spontaneous breakdown of global SU(9) is unlikely to remove the SU(9)-difficulty discussed in the text.Google Scholar
- 40.A number of such models were proposed in the early stages of model building. See the last two references in Ref. 39 for a listing of references on these models.Google Scholar
- 41.In such a theory, spontaneous symmetry breaking may still induce a mixing between flavor and color-gauge particles; such mixing in turn may generate a color-nonsinglet component in the weak-current.Such a theory, in which color is “brought” into weak interactions only through spontaneous symmetry breaking, differs drastically from the one in which the basic weak interactions use the color-symmetry, prior to spontaneous symmetry breaking. In particular, the former does not suffer from the SU(9)-difficulty. See discussions later for other characteristic features of such a theory pertaining to leptoproduction experiments.Google Scholar
- 42.In general, if there is an abelian U(l)-factor contributing to electric charge, SU(3)′-generators (F'
_{3}and F'_{8}) may contribute to electric charge with arbitrary coefficients associated with them. This arbitrariness is removed,once the symmetry (e.g. G_{o}) is embedded within a nonabelian local symmetry; and the constraint that an effective good global SU(3)-Color Symmetry must be preserved (despite the breaking of the local SU(3)-Color Symmetry) is imposed.(See discussions in Ref. 43).Google Scholar - 43.R. N. Mohapatra, J. C. Pati and A. Salam, Phys. Rev. D13, 1733 (1976).Google Scholar
- 44.See Ref. 35 for details.Google Scholar
- 45.J. C. Pati and A. Salam, Phys. Rev. Lett. 36, 11 (1976); G. Rajasekharan and P. Roy, Pramana 6, 303 (1975); V. Elias, J. C. Pati, A. Salam and J. Strathdee, Pramana 8, 303 (1977).Google Scholar
- 46.See Ref. 45 and J. C. Pati, “Unification-Liberation of Quarks and Leptons”, Univ. of Md. Tech. Rep. 77-073; Proceedings of the 1976 Scotthish University Summer School Lecture Notes ed. by I. M. Barbour and A. T. Davies (Pages 89–145, 1976). An account of the “Philosophical” implications of partial versus absolute confinement may be found in J. C. Pati and Abdus Salam, “Lepton-Hadron-Unification”, Proceedings of the 1976 Aachen Neutrino Conference, ed. by H. Faissner, H. Reithler and P. Zerwas (Pages 589–629, see especially Sec. V).Google Scholar
- 47.See for example, B. Barish, Particles and Fields, 1974 (APS/DPF-Williamsburg), ed. C. E. Carlson (AIP, New York, 1975).Google Scholar
- 48.L. L. Lewis, et. al., Phys. Rev. Lett. 39, 795 (1977); P. E. G. Baird, et. al., Phys. Rev. Lett. 39, 798 (1977).Google Scholar
- 49.For a review on present experimental and theoretical status, including calculations of shielding effect, see P. Sandars, Invited Talk at Ben Lee Memorial Conference, held at Batavia, Illinois (Oct., 1977), To appear in the Proceedings.Google Scholar
- 50.The possibility of observable fractionally charged quarks, though logically permissible, is disfavored within quark-lepton
*unifying*symmetries (without abelian U(1)-factors), unless massless gluons are seen; as for such symmetries, color-gluons can acquire mass and yet an effective good SU(3)-color global symmetry can emerge, provided quarks acquire integer-charges (See Ref. 43 and discussions in Ch. IV).Google Scholar - 51.See for example H. D. Politzer (Ref. 16); H. Georgi and H. D. Politzer “Clean Tests of QCD in μp-Scattering (HUTP-77/A063). For tests pertaining to jet-structure, see H. Georgi and M. Machacek, Phys. Rev. Lett. 39, 1237 (1977); G. Steerman and S. Weinberg, Phys. Rev. Lett. 39, 1436 (1977), and E. Farhi, Phys. Rev. Lett. Vol. 39, 1587 (1977).Google Scholar
- 52.Abdus Salam and J. Strathdee propose that spin-2 multiplet should play a vital role in generating partial or absolute confinement (Trieste Preprints, 1977). The link between Yang-Mills spin-1 and Einstein-Weyl spin-2 gauge-theories remains to be understood, quite independently of the question of whether spin-2 exchange produces confinement or not.Google Scholar
- 53.The first such calculation utilising the Georgi-Glashow SU(5)-model (Ref. 4) for illustration, was carried out by H. Georgi, H. R. Quinn and S. Weinberg, Phys. Rev. Lett. 33, 451 (1974).Google Scholar
- 54.S. W. Herb et. al., Phys. Rev. Lett. 39, 252 (1977); W. R. Innis, et. al., Phys. Rev. Lett. 39, 1249 (1977).Google Scholar
- 55.F. Gürsey and P. Sikivie; and P. Ramond (Ref. 4).Google Scholar
- 56.The example is fictitious, because with p and n-quarks having charges +1 and 0, the proton acquires a charge +2. This feature is remedied yet the essential content of the illustration leading to a correlation between quark and lepton-charges is retained-once quarks are assigned color as well as flavor (see discussions in Ch. IV).Google Scholar
- 57.V. Elias, Phys. Rev. D14, 1896 (1976); ibid Phys. Rev. D16, 1586 (1977); ibid Phys. Rev. D (To be published); F. J. Yndurdin, Nucl. Phys. B115, 293 (1976). M. Chanowitz, J. Ellis and M. K. Gaillard, CERN preprint TH-2312 (1977), To appear in Nuclear Physics.Google Scholar
- 58.V. Elias, J. C. Pati and Abdus Salam, “Unification of The Basic Particle Forces at a Mass-Scale of order 1000 m
_{W}”, Univ. of Md. Tech. Rep. 78-041 (Sept., 1977).Google Scholar - 59.A compulsive explanation would require an opriori understanding of the ratio of the unifying mass-scale M and themass of the weak boson m
_{W}(100 GeV).Google Scholar - 60.For weak/electromagnetic symmetries other than SU(2)
_{L}x U(1) (e.g. SU(2)_{L}x SU(2)_{R}x U(1)_{L+R}, see later), one can still define a parameter analogous to θ_{W}.Google Scholar - 61.R. N. Mohapatra and J. C. Pati, Phys. Rev. Dll, 566, 2558 (1975).Google Scholar
- 62.J. C. Pati and Abdus Salam, Phys. Letters, 58B, 333 (1975); J. C. Pati, Proc. Second Orbis Scientae, Coral Gables, Florida, Jan., 1975 (p. 253–256), ed. by A. Perlmutter ands. Widmayer.Google Scholar
- 63.H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93, 193 (1975).Google Scholar
- 64.The basic distinction between the patterns I and II arises under the presumption that Fermi mass-matrix, generated by spontaneous symmetry breaking would mix −F
_{L}with F_{R}and −F^{m}_{L}with F_{R}^{m}with little or (in the extreme case) no mixing between F and F^{m}. [See also remarks later about the difficulty of generating Fermi masses for Case II].Google Scholar - 65.The decays of mirror F
^{™}F^{m}-composites to normal hadrons made of −FF-components would in this case be governed by a super Zweig rule.Google Scholar - 66.J, C. Pati and Abdus Salam: “SU(8) x SU(8)“-unpublished notes; J. C. Pati, Abdus Salam and J. Strathdee, Nuovo Cimento 26, 72 (1975).Google Scholar
- 67.See for example G. Feinberg and J. Sucher, Phys. Rev. Letters, 13, 1740 (1975); J. D. Jackson, invited talk at the American Physical Society Meeting, Budapest (July, 1977), CERN-Preprint (1977), To appear in the Proceedings.Google Scholar
- 68.The consistency of this hypothesis and other general consequences of low-mass axial color-gluons are being examined in collaboration with J. Sucher.Google Scholar
- 69.V. Elias, J. C. Pati and Abdus Salam, In preparation (1977).Google Scholar
- 70.The symmetries [SU(5)]
^{4}and [SU(6)]^{4}belonging to this class have been considered (J. C. Pati and Abdus Salam, unpublished), see also Ref. 46.The [SU(5)]^{4}symmetry has been studied in some detail by S. Rajpoot (Ph.D. Thesis, Imperial College, 1977).Google Scholar - 71.M. L. Perl et. a1., Phys. Rev. Lett. 35, 1489 (1975). For a good review of the recent experimental status covering SPEAR and DORIS-experiments, see M. L. Perl, invited talk presented at the Photon-Lepton-Symposium, Hamburg, W. Germany (August, 1977), SLAC-PUB 2022 (1977). The review emphasises the compatibility of the data with the heavy lepton-interpretation; the equally viable quark-interpretation of the data is not considered in this review. (The quark versus heavy lepton origins for the (\(\bar \mu\)e)-events are considered in Ch. IV).Google Scholar
- 72.Barring large mixing of F and F
^{m}via Fermi mass-matrix, which would, in general, lead to “unnatural” violations of quantum numbers as well as of β-decay versus μ-decay and μ versus e universalities. Small mixing-e.g. of v^{o}_{e}and E^{o}-can be present. Such mixing may indeed account naturally for the masslessness or small mass of the physical neutrinos (ν_{e}and v_{μ}).[Consider the possibility that v^{o}_{e}— diagonal mass element α^{2}M_{E}o and the non-diagonal (ν_{e}^{o}E^{o})-mixing mass is αm_{E}o. Such a massmatrix would generate a massless eigenstate; the mixing angle would however be tiny*(0(a))*and thus compatible with the observed universalities. These considerations will be elaborated in a note under preparation.]Google Scholar - 73.The prequark-hypothesis has been considered in different forms. For gauge motivated considerations, see J. C. Pati and A. Salam, Phys. Rev. D10, 275 (1974), Footnote 7; 1975 Palermo Conference Proceedings, ed. by A. Zichichi (page); J. C. Pati, A. Salam and J. Strathdee, Phys. Lett. 59B, 265 (1975) and Supplement Trieste Preprint IC/75/139/ (unpublished). The hypothesis under other motivations has been considered by W. Krolikowski, Nuovo Cim. 72, 645 (1972); J. D. Bjorken (private communications, 1973, unpublished); C. H. Woo (private communications 1974, unpublished); K. Matumoto, Prog. Theor. Phys. 52, 1973 (1974);, and 0. W. Greenberg, Phys. Rev. Lett. 35, 1120 (1975). Certain
*Perturbative*dynamical calcualtions testing the compataibility of the hypothesis has recently been carried out by E. Nowak, J. Sucher and C. H. Woo (Phys. Rev. D. To be published).Google Scholar - 74.J. C. Pati (Ref. 46, Page 109 of the Proceedings). The same constraint follows by demanding that the physical weak currents satisfy left-right symmetry, which would require that the left and right Cabibbo-like angles be equal up to
*0(a)*-corrections (M.A.B. Beg, R. Budny, R. N. Mohapatra and A. Sirlin, Phys. Rev. Lett. 38, 1252 (1977)).Google Scholar - 75.M. Holder et al., Phys. Rev. Letters, 39, 433 (1977).Google Scholar
- 76.M.A.B. Beg, and A. Zee, Phys. Rev. Letters 30, 675 (1973). For a list of references on other vector-like models see R. M. Barnett, Review talk, Proceedings of the Brookhaven APS meeting (Sept. 1976).Google Scholar
- 77.Skewness angles, arising due to F-F' mixing, are assumed to be small for reasons mentioned before (see also Ref. 74).Google Scholar
- 78.A Bervenutti et al., Phys. Rev. Letters 37., 1039 (1976); J. Blietschau et al., Preprint CERN/EP/Phys 76–55; B. C. Barish et al. (CALT-68-544). The clearest distinction is shown by elastic scattering data (i.e. measurements of σ(\(\bar v\)p→\(\bar v\)p) versus σ(
*v*p→*v*p); D. Cline et al., Phys. Rev. Lett. 37, 252 648 (1976) and W. Lee et al., Phys. Rev. Lett. 37, 186 (1976).Google Scholar - 79.Precise limit would depend upon low-momentum effective X-coupling constant. This is being studied by V. Elias.Google Scholar
- 80.Technically, the scalar mass terms need not respect the discrete symmetries; since such terms, having dimension two, induce only finite corrections to coupling constants. However, aesthetically they should.Google Scholar
- 81.G. Senjanovic and R. N. Mohapatra, Phys. Rev. D12, 1502 (1975).Google Scholar
- 82.The discussion on the spontaneous breaking of [SU(4)]
^{4}follows a recent paper by V. Elias, J. C. Pati and A. Salam, “Light Neutral Z°-Boson In the PETRA and PEP Energy Range Within the Unifying Symmetry [SU(4)]^{4}”, Univ. of Md. Tech. Rep. No. 78-043 (Nov. 1977).Google Scholar - 83.As to whether an allowed minimum of the potential (subject to the parameters being in a given range) will permit such different patterns of VEV for fields transforming similarly (e.g. Li, Ei and M
_{i}), remains to be examined.Google Scholar - 84.Two multiplets, related by discrete symmetries, may well have unequal vacuum expectation values (see e.g. Ref. 81).Google Scholar
- 85.H. Fritzsch and P. Minkowski, Nucl. Phys. B103, 61 (1976); R. N. Mohapatra and D. P. Sidhu, Phys. Rev. Letters 38, 667 (1977); A. DeRujula, H. Georgi and S. L. Glashow, Preprint (1977); J. C. Pati, S. Rajpoot and Abdus Salam, Imperial College Preprint ICTP/76/11 (Phys. Rev. D To be published).Google Scholar
- 86.The possible importance of the group SU(2)
_{L}x SU(2)_{R}x U(1)_{L}x U(1)_{R}has been emphasized by J. C. Pati (Ref. 62), J. C. Pati, S. Rajpoot and Abdus Salam (Ref. 85, Footnote 15). Recently its consequences have been emphasized in some detail by Q. Shafi and Ch. Wetterich, Univ. of Freiburg Preprint (Oct., 1977), and by V. Elias, J. C. Pati and Abdus Salam (Ref. 82).Google Scholar - 87.J. C. Pati, S. Rajpoot and Abdus Salam, ICTP/76/11 (Phys. Rev. D to be published), ibid ICTP/76/15 (Physics Letters, To be published). H. S. Mani, J. C. Pati, S.' Rajpoot and Abdus Salam, Trieste Preprint IC/77/88 (Physics Letters To be published).Google Scholar
- 88.R. Cahn and F. Gilman (SLAC-PUB 1977). The analogous calculations for the left-right symmetric model allowing for its generality (see Ref. 87), has been recently carried out by Mr. A. Janah at Univ. of Md. (To be published).Google Scholar
- 89.J. Leveille, Imperial College Preprint (Dec. 1977).Google Scholar
- 90.This is proved in its full generality for the SU(2)
_{L}x SU(2)_{R}x U(1)_{L+R}-model by J. C. Pati, S. Rajpoot and A. Salam (Ref. 87); the equivalence for the special case of vanishing atomic parity violation was shown by H. Fritzsch and P. Minkowski (Ref. 85). Recently, the result has been further generalized to encompass bigger group structures by H. Georgi and S. Weinberg, Harvard Preprint (HUTP-77/AO52).Google Scholar - 91.K. Bardacki, M. B. Halpern, Phys. Rev. D6, 696 (1972). B. De Witt, Nucl. Phys. B51, 237 (1973).Google Scholar
- 92.R. N. Mohapatra and J. C. Pati, Physics Letters, 63B, 204 (1976).Google Scholar
- 93.Recently attempts have been made to give mass to color-gluons (with fractionally charged quarks), e.g. by A. De Rujula and R. Jaffe, Harvard Preprint (1977). Such a possibility, it seems, cannot however be entertained within a
*unified*theory of quarks and leptons for reasons mentioned in the text (See Ref. 92 for details). A number of interesting results obtained by these authors can be carried over, however, to the case of unconfined integer-charge quarks, developed in Ref. 3.Google Scholar - 94.J. C. Pati, Abdus Salam and S. Sakakibara, Phys. Rev. Lett. 36, 1229 (1976); J. C. Pati, S. Sakakibara and Abdus Salam, “The Missing Quark Mystery” Trieste Preprint IC/75/93 (unpublished). These two papers should be referred to for details on quark and proton-decay considerations.Google Scholar
- 95.The q → \(\ell \ell \bar \ell\)decay modes proceed through tree diagrams, which are damped by two heavy mass propagators, where as the loop-diagram Fig. 5(a) is damped by only one heavy mass-propagator m
_{x}^{−2}. The decay mode-such as q → e^{−}π^{+}involving emission of a charged lepton,proceed either via tree or via loop-diagrams with pion-emission from inside the loop. (Compare with Fig. 5(c)). Given that pion is a q−q-composite, such loopdiagrams would also be damped (compared to the loop-diagram shown in Fig. 5(c)), if the emission of composite pion from the quark-line is governed by a form-factor ∞ (1/(k^{2}−m^{2})). See Ref. 94 for details of these arguments.Google Scholar - 96.The ratio of the amplitudes of W
_{14}versus W_{L}-loops (see Fig. 5(c) and 5(a)) is given by [c^{2}_{4}Rl*ln*(m_{X}/m_{W}_{14})][clc4kn(m_{X}/m_{W}_{L}+)]^{−l}, assuming that W_{L}^{+}mixes with X (rather than W_{R}+ of the GIM SU(2)_{R}group).[Noting that VEV of B and C-multiplets are interchangeable due to the L ~> R Symmetry in the theory (See Eq. (73)), W^{+}_{R}can mix with X rather than W_{L}^{+}. Decays of charged red quarks into charged leptons via the two step process (see section 4.6) would exhibit V + A or V-A coupling depending upon whether it is W_{L}^{+}or W_{R}^{+}, which mixes with X (see Ref. 94 for details)].Google Scholar - 97.Hadrons of 2–3 GeV mass are not expected to have
*pair*production crosssection at ISR-Fermilab energies much bigger than = 10^{−30}cm^{2}, as judged from the empirical observation of J/ϕ—production (e.g.).Google Scholar - 98.The best empirical “lower limit” on proton-lifetime at present is 2 × 10
^{30}years (F. Reines and M. Crouch, Phys. Rev. Lett. 32, 493 (1974) and see references therein).The above search is sensitive to energetic μ^{+}s, which may be expected to arise especially from an assumed two-body decay mode of the proton (p → μ^{+}π^{o}). For a suggestion on decay-mode independent search based on geochemical methods, see S. P. Rosen, Phys. Rev. Lett. 34, 774 (1975). Recent decay-mode independent searches, yielding proton lifetime limits ≥ 2.2 × 10^{26}yr years, have been made by R. I. Steinberg and J. C. Evans, Jr., Invited Paper, Neutrino-Conference, Elbrus, USSR (1977) and Science 197, 989 (1977). For an excellent up-to-date review, see M. Goldhaber, invited talk at Ben Lee Memorial Conference (Oct., 1977), To appear in the Proceedings.Google Scholar - 99.See a recent estimate by A. J. Buras, J. Ellis, M. K. Gaillard and D. V. Nanopoulos, Preprint CERN TH 2403 (1977).Google Scholar
- 100.For an up-to-date review on the tests of unconfined color, see a recent paper by J. C. Pati and Abdus Salam,“Design of Future Experiments: I-To Distinguish Between the Alternatives of Physical and Hidden Color; II-To Test if the Neutral Gauge Boson lies in the vicinity of PETRA-PEP-Region” Trieste Preprint IC/77/65. See also Ref. 46.Google Scholar
- 101.J. C. Pati, J. Sucher and C. H. Woo, Phys. Rev. D15, 147 (1977).Google Scholar
- 102.S. Sakakibara, B. Kayser, J. C. Pati and G. Zorn (unpublished).Google Scholar
- 103.This takes into account the photon-gluon-cancellation-effect in lepto-production of color (Ref. 45).Google Scholar
- 104.See the third paper in Ref. 45 for a derivation of
^{R}Color = (1/8) (m^{2}_{U}/μ^{2})^{2}, with m_{U}denoting the gluon-propagator-mass and μ the effective mass of the gluon-partons (μ S m_{N}).Google Scholar - 105.Recent PLUTO-data, reported at Photon-Lepton Symposium, held at Hamburg, W. Germany (August, 1977).Google Scholar
- 106.T. Appelquist and J. Carrazone, Phys. Rev. D11, 2856 (1975).Google Scholar
- 107.Ref. 58 for other cases (e.g. M° m
_{W}-100 GeV).Google Scholar - 108.For recent attempts in this direction see F. Wilczek and A. Zee (Preprint, Princeton University, 1977), H. Fritzsch, CERN preprint TH2358 (1977) and S. Weinberg, Harvard Preprint (1977).Google Scholar
- 109.S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973); E. Gildener, Phys. Rev. D14, 1667 (1976). For a suggestion in the recent context, see H. S. Mani, J. C. Pati and Abdus Salam, Trieste Preprint IC/77/80 and Ref. 58.Google Scholar

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