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92 fractional electric charge and quark confinement Including the identity, this gives a discrete group of six elements, isomorphic to Z 6 . It is straightforward to verify that the transformations generated by (5.9) act identically on the Standard Model matter representations, 2 ( ) ( ) ( ) ν e e ( u d L , ) L , ν µ µ ( c s L ) , L ν τ τ L : (1, 2, −1) e R , µ R , τ R : (1, 1, −2) ( ) t , : (3, 2, 1 3 b ) (5.10) L u R , c R , t R : (3, 1, 4 3 ) d R , s R , b R : ( ) (3, 1, − 2 3 ) φ = φ + : φ 0 (1, 2, 1), where φ is the Higgs doublet and the fundamental SU(3) and SU(2) representations are labeled by their dimension. In this form the fractional charge of quarks is expressed as fractional hypercharge relative to the leptons and the Higgs. 5.2.1 Historical notes The discrete Z 6 symmetry is often overlooked when the Standard Model gauge group is written down, but it has played an important historical role. For one thing, the matter representations in Eq. (5.10) are almost completely fixed by demanding the cancellation of anomalies that otherwise spoil the consistent formulation of a quantum field theory [134, 135]. There are three sources of anomalies that must be taken care of. First of all, the perturbative triangular chiral anomaly must be avoided, which destroys gauge invariance and renormalizability. Then there is Witten’s SU(2) chiral gauge anomaly [136], which is a global anomaly that forces the path integral to vanish for an SU(2) gauge theory with an odd number of chiral fermion doublets. Finally there is the chiral gauge-gravity anomaly, which, like the triangle anomaly, is perturbative. It breaks general covariance. Canceling all three of these anomalies fixes the relative U(1) Y charges uniquely for N = 3 colors [135]. 3 Anomaly cancellation tightly constrains the charge assignments in the Standard Model. Grand unification is often invoked as a possible origin story. Placing the various matter fields in the same representation of a larger gauge group fixes their assignments after symmetry breaking. In fact, the Standard Model representations (5.10) would not easily fit in grand unified theories based on the prototypical gauge groups SU(5) ⊂ SO(10) without the Z 6 symmetry. A product of the color and electroweak gauge groups may only be accommodated after dividing out the Z 6 redundancy, SU(3) × SU(2) × U(1) Y /Z 6 ≃ S(U(3) × U(2)) ⊂ SU(5). (5.11) 2 We’ve left out right-handed neutrinos, which are present if a Dirac mass is at least partly responsible for neutrino oscillations. 3 It is more accurate to say that anomaly cancellation almost fixes the relative charge assignments. There is a so-called bizarre solution with zero-hypercharge for the leptons [134, 135].
5.2 a hidden global symmetry 93 The product SU(3)×SU(2)×U(1) Y is a cover of the ‘true’ minimal gauge symmetry in the same sense that SU(N) is a cover of the minimal gauge symmetry SU(N)/Z N in pure Yang-Mills theory. In SU(5) GUT [137], the Standard Model gauge symmetry is reproduced when an adjoint scalar field Φ acquires an expectation value at some energy scale above the electroweak transition, which leads to an effective reduction of the gauge theory, SU(5) M GUT −→ SU(3) × SU(2) × U(1) Y /Z 6 O(200) GeV −→ SU(3) × U(1) em /Z 3 . (5.12) A potential for the GUT Higgs Φ is chosen such that it is minimized in the unitary gauge by, 〈Φ〉 ∝ Y = diag (−2/3, −2/3, −2/3, 1, 1), (5.13) which generates a compact hypercharge U(1) Y and commutes with su(3) and su(2) generators that are arranged in blocks along the diagonal. Tracelessness of Y then quantizes the hypercharge of, for instance, matter in the fundamental 5 representation of SU(5) that decomposes into SU(3) × SU(2) × U(1) Y representations like, 5 M GUT −→ (3, 1, − 2 ) ⊕ (1, 2, 1). (5.14) 3 Placing the down quark SU(3) triplet and an SU(2) doublet (e + , ¯ν e ) in this representation fixes their quantum numbers. See Ref. [138] for a detailed treatment of the algebra of several GUTs, how they accomodate the Standard Model, and how they relate to one another. Note that it is usual to describe the electroweak and GUT transition via the Higgs mechanism as spontaneous symmetry breaking transitions, in spite of Elitzur’s theorem which states that local gauge symmetry cannot break spontaneously. This is rather misleading, and the subtleties require a careful treatment when one goes from a semi-classical to fully quantum description of a theory with the Higgs mechanism. Ref. [139] provides an up to date discussion. 5.2.1.1 Topological inspiration from GUTs This is where we make contact with the GUT monopoles considered in Chapter 4. Toplogical defects are a generic consequence of breaking a unified gauge group via the Higgs mechanism, provided that it that has a simply connected cover [132]. The latter point ensures that the effective low energy gauge theories inherits non-trivial topology. If the unifying gauge group is SU(N), then magnetic charge is quantized with respect to the residual U(1) N−1 gauge invariance that is generated by the diagonal Lie algebra elements that commute with the Higgs field in unitary gauge (cf. Section 4.1). For SU(5) GUT, the U(1) subgroups are embedded in the hypercharge U(1) Y and the enlarged residual SU(3) color and electroweak SU(2) gauge symmetries that follow from the submaximal symmetry breaking pattern in (5.12). Consider the residual U(1) subgroup generated by exponentiating the Lie algebra element, diag(0, 0, −1, 1, 0) ∈ su(5), (5.15)
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T O P O L O G Y, S Y M M E T RY, A
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To Angela.
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Z U S A M M E N FA S S U N G Wir un
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viii contents 4.6 Algebraic formula
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2 introduction While waiting on exp
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G A U G E T H E O R I E S 2 Before
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2.2 quantization 7 The non-abelian
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2.3 lattice field theory 9 Figure 2
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2.3 lattice field theory 11 the pro
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2.3 lattice field theory 13 picked
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16 universal aspects of confinement
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Page 89 and 90: 4.6 algebraic formulation 81 Figure
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ibliography 143 [60] J. Engels, E.
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ibliography 145 [89] A. S. Kronfeld
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ibliography 147 [120] Y. Aoki, G. E
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ibliography 149 [151] M. P. Nightin
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ibliography 151 [179] C. Korthals A
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C U R R I C U L U M Name: Samuel Ry
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