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Scalar Quantum Electrodynamics

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Scalar Quantum Electrodynamics A complex system that works is invariably found to have evolved from a simple system that works. John Gall 17 Scalar Quantum Electrodynamics In nature, there exist scalar particles which are charged and are therefore coupled to the electromagnetic field. In three spatial dimensions, an important nonrelativistic example is provided by superconductors. The phenomenon of zero resistance at low temperature can be explained by the formation of so-called Cooper pairs of electrons of opposite momentum and spin. These behave like bosons of spin zero and charge q =2e, which are held together in some metals by the electron-phonon interaction. Many important predictions of experimental data can be derived from the Ginzburg- Landau theory of superconductivity [1]. The relativistic generalization of this theory to four spacetime dimensions is of great importance in elementary particle physics. In that form it is known as scalar quantum electrodynamics (scalar QED). 17.1 Action and Generating Functional The Ginzburg-Landau theory is a three-dimensional euclidean quantum field theory containing a complex scalar field ϕ(x)= ϕ1(x)+ iϕ2(x) (17.1) coupled to a magnetic vector potential A. The scalar field describes bound states of pairs of electrons, which arise in a superconductor at low temperatures due to an attraction coming from elastic forces. The detailed mechanism will not be of interest here; we only note that the pairs are bound in an s-wave and a spin singlet state of charge q =2e. Ignoring for a moment the magnetic interactions, the ensemble of these bound states may be described, in the neighborhood of the superconductive 4 transition temperature Tc, by a complex scalar field theory of the ϕ -type, by a euclidean action 2 3 1 m g 2 E = d x ∇ϕ∗∇ϕ + ϕ∗ϕ + (ϕ∗ϕ) . (17.2) A Z " 2 2 4 # The mass parameter depends on the temperature like m2 (T T )/T . Above ∝ − c c the transition temperature, the parameter m2 is positive, below it is negative. As 1077 1078 17 Scalar Quantum Electrodynamics a consequence, the system exhibits a spontaneous symmetry breakdown discussed in Chapter 16. The symmetry group that is broken, is given by the U(1) phase transformation iα ϕ(x) e− ϕ(x) (17.3) → under which is obviously invariant. Alternatively we may write AE ϕ cos αϕ + sin αϕ , 1 → 1 2 ϕ sin αϕ + cos αϕ , (17.4) 2 → − 1 2 so that we may equally well speak of an O(2)-symmetry. The action (17.2) is the euclidean version of a relativistic field theory in D = 2+1 dimensions, with an action 2 D 1 µ m g 2 E = d x ∂µϕ∗∂ ϕ ϕ∗ϕ (ϕ∗ϕ) . (17.5) A Z "2 − 2 − 4 # For the sake of generality, we shall discuss this theory for arbitrary D. Most formulas will be written down explicitly in D = 4 dimensions, to emphasize analogies with proper QED discussed in Chapter 12. The phenomenon of spontaneous symmetry breakdown in this system has been studied in detail in Chapter 16. For T > Tc, where the mass term is positive, small oscillations around ϕ = 0 consist of two degenerate modes carried by ϕ1,ϕ2, both 2 of mass m . For T < Tc, where the mass is negative, the energy is minimized by a field ϕ with a real non-vanishing vacuum expectation value, say ϕ(x) ϕ . (17.6) h i ≡ 0 Then the symmetry between ϕ1 and ϕ2 is broken and there are two different modes of small oscillations: one orthogonal to ϕ , which is the massless Nambu-Goldstone mode, and one parallel to ϕ , which hash i a positive mass 2m2 > 0. h i − In order to describe the phenomena of superconductivity with the euclidean version of (17.5), we must include electromagnetism. According to the minimal substitution rules described in Chapter 12, we simply replace ∂µϕ in (17.5) by the covariant derivative ∂ ϕ D ϕ (∂ iqA ) ϕ. (17.7) µ → µ ≡ µ − µ The extra vector potential Aµ(x) is assumed to be governed by Maxwell’s electro- magnetic action [see (12.3)]:1 1 D µν γ = d xFµν (x)F (x). (17.8) A −4 Z The combined action becomes 2 D 1 2 m 2 g 4 1 2 se[ϕ,ϕ∗, A]= d x Dµϕ ϕ ϕ Fµν . (17.9) A Z (2| | − 2 | | − 4| | − 4 ) 1In this chapter we use natural units with ¯h = c = 1. 17.1 Action and Generating Functional 1079 2 µν where Fµν is short for FµνF . This action is invariant under local gauge transfor- mations ϕ(x) eiqΛ(x)ϕ(x), A (x) A (x)+ ∂ Λ(x). (17.10) → µ → µ µ A functional integration leads to the generating functional of scalar QED: ∗ µ ∗ ∗ µ µ i se[ϕ,ϕ ,A ] λ ϕ ϕ λ j Aµ Z [λ,λ∗, j] = ϕ ϕ∗ A e {A − x − x − x }, (17.11) D D phys. D Z Z R R R 4 where the symbol x abbreviates the volume integral d x, and λ(x), λ∗(x) are the µ sources of the complexR scalar fields ϕ∗(x) and ϕ(x). TheR vector j (x) is an external electromagnetic current coupled to the vector gauge field Aµ(x). To preserve the gauge invariance, the current is assumed to be a conserved quantity and satisfy ∂ jµ(x) = 0. The functional integration over the vector field Aµ assumes µ phys D that a Faddeev-Popov gauge fixing factor is inserted, together withR a compensating determinant (recall the general discussion in Section 14.16): µ µ 1 A A Φ− [A]F [A]. (17.12) Zphys D ≡ Z D For m2 > 0, we may choose [recall (14.353)] 1 µ 2 (∂µA ) F [A]= F4[A]= e− 2α x , (17.13) R and derive from (17.11) the Feynman rules for perturbation expansions. The gauge- fixed photon propagator was given in Eq. (14.373): D d k ik(x y) i kµkν − − (17.14) G0 µν(x y)= D e 2 gµν (1 α) 2 . − − Z (2π) k " − − k # We have seen in Chapter 12 that in QED all scattering amplitudes are independent of the gauge parameter α. In addition, the perturbation expansion derived from the generating functional (17.11) employs propagators of scalar particles represented by lines D ik(x y) d k ik(x y) i e− − G (x y)= e− − 0 − (2π)D k2 m2 (17.15) Z − and vertices . (17.16) Since the scalar field is complex, the particle carries a charge, and the charged particle lines in the Feynman diagrams carry an orientation. The coupling of the scalar field to the vector potential does not only give rise to a three-point vertex of the type (12.95) in QED [left-hand diagram in (17.16)], but also to a four-vertex in which two photons are absorbed or emitted by a scalar particle [right-hand diagram in (17.16)]. This diagram is commonly referred to as seagull diagram. 1080 17 Scalar Quantum Electrodynamics 17.2 Meissner-Ochsenfeld-Higgs Effect 2 The situation is drastically different for T < Tc where m < 0. At the classical level, the lowest energy state of the pure ϕ theory is reached at ϕ = ϕ0 with ϕ = φ = m2/g, i.e., | 0| 0 − q 2 iγ0 m iγ0 ϕ0 = φ0e− = − e− . (17.17) s g Now the photon part of the action (17.9) reads 2 2 D 1 2 q m 2 γ = d x Fµν Aµ . (17.18) A Z −4 − 2 g ! Extremization of this leads to the Euler-Lagrange equation q2m2 ∂2A ∂ ∂A A =0. (17.19) µ − µ − g µ Note that we could have chosen as well 2 iγ0(x) m iγ0(x) ϕ = φ0e− = − e− , (17.20) s q with an arbitrary spacetime-dependent phase γ0(x). Then the photon Lagrangian would read 2 2 D 1 2 q m 2 γ = d x Fµν (Aµ ∂µγ0) , (17.21) A Z "−4 − 2 g − # with the Euler-Lagrange equation m2 ∂2A ∂ (∂A) q2 (A ∂ γ )=0. (17.22) µ − µ − g µ − µ 0 The extremal field is reached at Aµ(x)= ∂µγ0(x). (17.23) Of course, the two results (17.19) and (17.22) are equivalent by a gauge transforma- tion A A + ∂ γ (x), and the physical content of (17.21) is independent of the µ → µ µ 0 particular choice of γ0(x). Consider now the fluctuation properties of this theory. The generating func- tional has the form (17.11) in which the measure sums over all ϕ and physical Aµ configurations orthogonal to the gauge degrees of freedom. If we decompose ϕ into radial and azimuthal parts iγ(x) ϕ(x)= ρ(x)e− , (17.24) 17.2 Meissner-Ochsenfeld-Higgs Effect 1081 the path integral takes the form ∗ µ ∗ ∗ µ µ i se[ϕ,ϕ ,A ] λ ϕ ϕ λ j Aµ Z [λ,λ∗, j]= ρ ρ γ A e {A − x − x − x }. (17.25) D D phys D Z Z R R R But from the structure of the covariant derivative it is obvious that no matter what the fluctuating γ(x) configuration is, it can be absorbed into A(x) as a longitudi- nal degree of freedom. Thus we have the option of considering complex fields ϕ(x) together with physical transverse Aµ fluctuations, or real ρ together with trans- verse and longitudinal Aµ(x)-fluctuations. This may be expressed by rewriting the functional integral (17.25) as µ ∗ ∗ µ µ i se[ρ,A ] λ ϕ ϕ λ j Aµ Z [λ,λ∗, j]= ρ ρ A e {A − x − x − x }, (17.26) D all D Z Z R R R with the action 2 2 µ D 1 2 q 2 2 m 2 g 4 1 2 [ρ, A ]= d x (∂µρ) + ρ Aµ ρ ρ + Fµν .
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