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Particle # conservation in a spontaneously broken theory

  1. Aug 31, 2015 #1
    So consider a theory of a complex scalar field ##\phi## with a global U(1) symmetry ##\phi \rightarrow e^{i\theta} \phi##. The theory admits a conserved current
    $$ J^\mu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^\dagger)} \delta\phi^\dagger = -i \left( \phi^\dagger \partial^\mu \phi - \partial^\mu \phi^\dagger \phi \right) $$
    which gives us the conserved charge (or particle number in this case)
    $$ N = \int d^3x\, J^0 = -i \int d^3x\, \left( \phi^\dagger \dot{\phi} - \dot{\phi}^\dagger \phi \right) $$
    Now this is fine when the fields involved are complex (due to the U(1) symmetry) and die off exponentially as ## r \rightarrow \infty##, but what happens when there is spontaneous symmetry breaking? Is particle number still conserved? Does the concept of particle number still make sense?

    Consider now an abelian Higgs model:
    $$ \mathcal{L} = \partial_\mu H^\dagger \partial^\mu H - \lambda \left( |H|^2 - v_0^2/2 \right)^2 $$
    This clearly has a U(1) symmetry, so at the outset we might assume the particle number is conserved analogous to the ##\phi## model above. However, due to the potential, the Higgs takes on a vacuum expectation value (vev), spontaneously breaking the U(1) symmetry. If we parametrize the Higgs in the unitary gauge so as to expand around the new vacuum
    $$ H = \frac{1}{\sqrt{2}} e^{i\theta/v_0} (h + v_0)$$
    where ##h## and ##\theta## are real scalar fields, the lagrangian becomes (ignoring the massless Goldstone bosons)
    $$ \mathcal{L} = \frac{1}{2} \partial_\mu h \partial^\mu h - \frac{\lambda}{4} \left( h^2 + 2v_0 h \right)^2
    = \frac{1}{2} \partial_\mu h \partial^\mu h - \frac{1}{2} m_h^2 h^2 - \lambda v_0 h^3 - \frac{\lambda}{4} h^4$$
    where ## m_h^2 = 2 \lambda v_0^2##. The U(1) symmetry in this new parametrization is now equivalent to ## \theta \rightarrow \theta + \alpha ##, and you can see the broken lagrangian is invariant under this change (the terms I ignored were derivatives of ##\theta## and are clearly invariant under a constant shift). If we just substitute this into the particle number integral we derived before, we get
    $$ N = -i \int d^3x\, \left( H^\dagger \dot{H} - \dot{H}^\dagger H\right) = \int d^3x\, \dot{\theta} \left( h + v_0 \right)^2 $$
    However, this integral is now divergent because of the ##v_0^2 \dot{\theta}## term which (may or may not) die out at infinity. If ##\theta=\text{const}##, everything is fine, but if we say the ##\theta## field only has a simple linear time dependence ##\theta = \omega t## this becomes a constant. Does this mean the Higgs particle number cannot be defined? Or can this be saved by simply dropping the constant in the same way we drop infinite energy shifts from a Hamiltonian? The issue I have with the second one is that when we shift the Hamiltonian, it's by some universal constant that is the same throughout the theory. Here, the constant ##\omega## may depend on the system under consideration and is not universal to the theory. If there is any more complicated time dependence of ##\theta## than linear, this just compounds the issue. Is this suggesting that the argument of the ##H## field is fixed at infinity (which kind of makes sense) and I should just forget about this term altogether? Any insights?
     
  2. jcsd
  3. Sep 1, 2015 #2

    fzero

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    If you include the gauge field in your Lagrangian, you'll find that ##\theta## mixes with the gauge field via quadratic terms like ##A^\mu \partial_\mu \theta##. The usual interpretation of this is to note that the field redefinition (gauge transformation) ##A_\mu' = A_\mu + \partial_\mu \theta/q## completely removes ##\theta## from the Lagrangian. Here ##q## is the charge of the Higgs field, perhaps other numerical factors will appear depending on conventions. This is interpreted as ##\theta## being absorbed ("eaten") by the gauge field to give the 3rd polarization of the massive gauge field. So the 2 d.o.f. of the massless gauge field and 2 d.o.f. of the complex scalar become 3 d.o.f. of a massive gauge field and 1 d.o.f. of a real massive scalar. There are no non-trivial conserved charges in this theory because there is no longer any global ##U(1)## invariance. Explicitly, one finds ##N=0## because ##H## is real in this gauge.
     
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