Symmetries and symmetry breaking

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Abstract

We have seen in the last chapter that the discrete Z 2 symmetry of our standard λφ 4 La-grangian could be hidden at low temperatures, if we choose a negative mass term in the zero temperature Lagrangian. Although such a choice seems at first sight unnatural, we will investigate this case in the following in more detail. Our main motivation is the expectation that hiding a symmetry by choosing a non-invariant ground-state retains the "good" properties of the symmetric Lagrangian. Coupling then such a scalar theory to a gauge theory, we hope to break gauge invariance in a "gentle" way which allows e.g. gauge boson masses without spoiling the renormalisability of the unbroken theory. As additional motivation we remind that couplings and masses are not constants but depend on the scale considered. Thus it might be that the parameters determining the Lagrangian of the Standard Model at low energies originate from a more complete theory at high scales, where the mass parameter µ 2 is originally still positive. In such a scenario, µ 2 (Q 2) may become negative only after running it down to the electroweak scale Q = m Z. 11.1 Symmetry breaking and Goldstone's theorem Let us start classifying the possible destinies of a symmetry: • Symmetries may be exact. In the case of local gauge symmetries as U(1) or SU(3) for the electromagnetic and strong interactions, we expect that this holds even in theories beyond the SM. In contrast, there is no good reason why global symmetries of the SM as B − L should be respected by higher-dimensional operators originating from a more complete theory valid at higher energy scales. • A classical symmetry may be broken by quantum effects. As a result, the corresponding Noether currents are non-zero and the Ward identities of the theory are violated. If the anomalous symmetry is a local gauge symmetry, the theory becomes thereby non-renormalisable. Moreover, we would expect e.g. in case of QED that the universality of the electric charge does not hold exactly. • The symmetry is explicitly broken by some "small" term in the Lagrangian. An example for such a case is isospin which is broken by the mass difference of the u and d quarks. • The Lagrangian contains an exact symmetry but the ground-state is not symmetric under the symmetry. In field theory, the ground-state corresponds to the mass spectrum of particles. As a result, the symmetry of the Lagrangian is not visible in the spectrum of physical particles. If the ground-state breaks the original symmetry because one or several scalar fields acquire a non-zero vacuum expectation value, one calls this spontaneous symmetry breaking (SSB). As the symmetry is not really broken on the Lagrangian level, a perhaps more appropriate name would be "hidden symmetry." 156

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van Oers, W. T. H. (2001). Symmetries and symmetry breaking. Nuclear Physics A, 684(1–4), 266–276. https://doi.org/10.1016/S0375-9474(01)00406-7

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