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Showing that gauge fields become massless and massive

  1. Dec 6, 2011 #1
    1. The problem statement, all variables and given/known data


    Consider a non-abelian gauge theory of SU(N) × SU(N) gauge fields coupled to N[itex]^{2}[/itex]
    complex scalars in the (N,N[itex]^{_}[/itex]) multiplet of the gauge group. In N × N matrix notations,
    the vector fields form two independent traceless hermitian matrices Bμ(x) =[itex]\Sigma[/itex][itex]_{a}[/itex] B[itex]^{a}_{\mu}[/itex](x) [itex]\frac{\lambda}^{a}{2}[/itex] and Cμ(x) = [itex]\Sigma[/itex][itex]_{a}[/itex] C[itex]^{a}_{\mu}[/itex](x) [itex]\frac{\lambda}^{a}{2}[/itex] , the scalar fields form a complex matrix [itex]\Phi[/itex], and the Lagrangian is
    L = −(1/2) tr(B[itex]_{\mu\nu}[/itex]B[itex]^{\mu\nu}[/itex] −(1/2) tr(C[itex]_{\mu\nu}[/itex]C[itex]^{\mu\nu}[/itex]) + tr(D[itex]_{\mu}[/itex][itex]\Phi[/itex][itex]^{+}[/itex]D[itex]^{\mu}[/itex][itex]\Phi[/itex]) - ([itex]\alpha[/itex]/2)(tr([itex]\Phi[/itex][itex]^{+}[/itex][itex]\Phi[/itex]))^2 - ([itex]\beta[/itex]/2)(tr([itex]\Phi[/itex][itex]^{+}[/itex][itex]\Phi[/itex][itex]\Phi[/itex][itex]^{+}[/itex][itex]\Phi[/itex])) - (m^2)*(tr([itex]\Phi[/itex][itex]^{+}[/itex][itex]\Phi[/itex])

    where

    B[itex]_{\mu\nu}[/itex] = [itex]\partial[/itex][itex]_{\mu}[/itex]B[itex]_{\nu}[/itex] - [itex]\partial[/itex][itex]_{\nu}[/itex]B[itex]_{\mu}[/itex] - ig[B[itex]_{\mu}[/itex], B[itex]_{\nu}[/itex]]
    C[itex]_{\mu\nu}[/itex] = [itex]\partial[/itex][itex]_{\mu}[/itex]C[itex]_{\nu}[/itex] - [itex]\partial[/itex][itex]_{\nu}[/itex]C[itex]_{\mu}[/itex] - ig[C[itex]_{\mu}[/itex], C[itex]_{\nu}[/itex]]
    D[itex]_{\mu}[/itex][itex]\Phi[/itex] = [itex]\partial[/itex][itex]_{\mu}[/itex][itex]\Phi[/itex]
    + igB[itex]_{\mu}[/itex][itex]\Phi[/itex] - ig[itex]\Phi[/itex]C[itex]_{\mu}[/itex]
    D[itex]_{\mu}[/itex][itex]\Phi[/itex][itex]^{+}[/itex] = [itex]\partial[/itex][itex]_{\mu}[/itex][itex]\Phi[/itex][itex]^{+}[/itex]
    + ig[itex]\Phi[/itex][itex]^{+}[/itex]B[itex]_{\mu}[/itex] + igC[itex]_{\mu}[/itex][itex]\Phi[/itex][itex]^{+}[/itex]

    (2)
    For simplicity, let both SU(N) factors of the gauge group have the same gauge coupling g.
    Besides the local SU(N)×SU(N) symmetries, the Lagrangian (1) has a global U(1) phase
    symmetry [itex]\Phi[/itex](x) → e[itex]^{i\theta}[/itex][itex]\Phi[/itex](x), but some of these symmetries become spontaneously broken
    for m[itex]^{2}[/itex] = −μ[itex]^{2}[/itex] < 0. Specifically, for m[itex]^{2}[/itex] = −μ[itex]^{2}[/itex] < 0 but α, β > 0, the scalar potential has
    a local maximum at [itex]\Phi[/itex] = 0 while the minima lie at
    [itex]\Phi[/itex] = [itex]\sqrt{\frac{\mu^{2}}{N\alpha + \beta}}[/itex] × a unitary matrix. (3)

    All such minima are related by SU(N) × SU(N) × U(1) symmetries to
    [itex]\Phi[/itex] = [itex]\sqrt{\frac{\mu^{2}}{N\alpha + \beta}}[/itex] × 1[itex]_{NxN}[/itex]


    Now let’s be more specific: Show that the vector fields A[itex]^{a}_{\mu}[/itex] = [itex]\frac{1}{\sqrt{2}}[/itex](B[itex]^{a}_{\mu}[/itex] + C[itex]^{a}_{\mu}[/itex]) remain massless while the orthogonal combinations X[itex]^{a}_{\mu}[/itex] = [itex]\frac{1}{\sqrt{2}}[/itex](B[itex]^{a}_{\mu}[/itex] - C[itex]^{a}_{\mu}[/itex]) become massive.
    Hint: Fix the unitary gauge in which the [itex]\Phi[/itex](x) matrix is hermitian up to an overall
    phase, [itex]\Phi[/itex][itex]^{+}[/itex](x) = [itex]\Phi[/itex](x) × e[itex]^{-2i\theta(x)}[/itex]. Explain why this gauge condition is non-singular for [itex]\Phi[/itex](x) near the minima (3).

    3. The attempt at a solution

    From what I've seen in textbooks, to see what new massive and massless field arise, you just substitute the new field (ie [itex]\phi[/itex] -> [itex]\phi[/itex]e[itex]^{-i\theta}[/itex])
    But I'm confused with this problem. Do I just substitute the A[itex]^{a}_{\mu}[/itex] in place of B and C in equation (1)? Or expand out (1) using the [itex]\Phi[/itex][itex]^{+}[/itex](x) = [itex]\Phi[/itex](x) × e[itex]^{-2i\theta(x)}[/itex], and then factor it out to get A[itex]^{a}_{\mu}[/itex] and X terms?

    I don't see how by doing the latter, we will see a mass term in front of A[itex]^{a}_{\mu}[/itex]? In equation (1), the only term the mass term is in front of is tr([itex]\Phi[/itex][itex]^{+}[/itex][itex]\Phi[/itex])
     
  2. jcsd
  3. Dec 6, 2011 #2
    The corrections in the typos in the OP are:
    B[itex]_{\mu}[/itex](x) = [itex]\Sigma[/itex][itex]_{a}[/itex]B[itex]^{a}_{\mu}[/itex](x)[itex]\frac{\lambda^{a}}{2}[/itex]

    C[itex]_{\mu}[/itex](x) = [itex]\Sigma[/itex][itex]_{a}[/itex]C[itex]^{a}_{\mu}[/itex](x)[itex]\frac{\lambda^{a}}{2}[/itex]
     
  4. Dec 6, 2011 #3

    micromass

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