An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem FP

Ward Identities, Renormalization of QED — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · decays-of-the-higgs-boson

Problem

Decays of the Higgs Boson. Consider the minimal form of the Glashow-Weinberg-Salam electroweak theory with one Higgs scalar field ϕ\phi. The physical Higgs boson h0h^0 of this theory was discussed in Section 20.2. You can use that information to compute the amplitudes for the various possible decays of the h0h^0 as a function of its mass mhm_h. For the other parameters of weak-interaction theory, use the following values: mb=5m_b = 5 GeV, mt=175m_t = 175 GeV, mW=80m_W = 80 GeV, mZ=91m_Z = 91 GeV, sin⁡2θw=0.23\sin^2\theta_w = 0.23, αs(mZ)=0.12\alpha_s(m_Z) = 0.12. (a) Compute first the rate for h0→ffˉh^0 \to f\bar{f}, where ff is a quark or lepton of the standard model. After a completely trivial computation, you should find Γ(h0→ffˉ)=(αmh8sin⁡2θw)⋅mf2mW2(1−4mf2mh2)3/2⋅Nc(f),\Gamma(h^0 \to f\bar{f}) = \left(\frac{\alpha m_h}{8\sin^2\theta_w}\right) \cdot \frac{m_f^2}{m_W^2} \left(1 - \frac{4m_f^2}{m_h^2}\right)^{3/2} \cdot N_c(f), where Nc(f)=1N_c(f) = 1 for leptons, 3 for quarks. If you have studied Chapter 18, you might improve this result for the case in which the fermion ff is a quark, by computing the leading-log QCD corrections for the case mh≫mqm_h \gg m_q. Remember that the quark mass mqm_q is determined at the quark threshold M2∼mq2M^2 \sim m_q^2. (b) If mh>2mWm_h > 2m_W, the Higgs boson can decay to W+W−W^+W^-; if it is just a bit heavier, it can also decay to Z0Z0Z^0Z^0. Compute the decay widths to these final states. You can check your result in the following way: If mh≫mWm_h \gg m_W, the dominant contribution to the decay comes from production of longitudinally polarized WW or ZZ bosons, and this contribution can be estimated as follows: Γ(h0→W+W−)≈Γ(h0→ϕ+ϕ−),Γ(h0→Z0Z0)≈Γ(h0→ϕ3ϕ3),\Gamma(h^0 \to W^+W^-) \approx \Gamma(h^0 \to \phi^+\phi^-), \quad \Gamma(h^0 \to Z^0Z^0) \approx \Gamma(h^0 \to \phi^3\phi^3), where ϕ±\phi^\pm, ϕ3\phi^3 are the Goldstone bosons of the Higgs sector and the quantities on the right-hand sides of these relations are computed in the ungauged Higgs theory. Explain why this statement should be true, and verify it explicitly. (c) The third important decay mode of the Higgs is the decay to 2 gluons. The amplitude for this decay is generated by diagrams involving quark loops. Compute these diagrams, using dimensional regularization. The diagrams will be finite, but nevertheless there is a subtle contribution which apparently depends on the regulator. Check that you have computed the amplitude correctly by verifying that it is gauge invariant. Then square the amplitude and construct the decay rate. You should find Γ(h0→2g)=(αmh8sin⁡2θw)⋅mh2mW2⋅αs29π2⋅∣∑qI ⁣(mh2mq2)∣2,\Gamma(h^0 \to 2g) = \left(\frac{\alpha m_h}{8\sin^2\theta_w}\right) \cdot \frac{m_h^2}{m_W^2} \cdot \frac{\alpha_s^2}{9\pi^2} \cdot \left|\sum_q I\!\left(\frac{m_h^2}{m_q^2}\right)\right|^2, where the sum runs over all quark species and I(mh2/mq2)I(m_h^2/m_q^2) is a form factor with the property that I(x)→1I(x) \to 1 as x→0x \to 0 and I(x)→0I(x) \to 0 as x→∞x \to \infty. This property implies that the dominant contribution to the decay rate comes from very heavy quarks. You need not evaluate I(x)I(x) explicitly at this stage; just leave it in the form of a Feynman parameter integral. (d) The existence of the process h0→2gh^0 \to 2g implies the existence of the inverse process g+g→h0g + g \to h^0, which is a mechanism for production of Higgs bosons in proton-proton collisions. Using the parton model, derive a relation between the partial width Γ(h0→2g)\Gamma(h^0 \to 2g) and the total cross section for pp→h0+Xpp \to h^0 + X. Compute this cross section numerically (in nanobarns) for a 30 GeV Higgs for pppp collisions of center of mass energy 1--40 TeV. (1 TeV =103= 10^3 GeV.) For the purposes of this problem set (though this is not actually a good approximation) you may ignore the Q2Q^2 dependence of the gluon distribution function and take simply fg(x)=8⋅1x(1−x)7.f_g(x) = 8 \cdot \frac{1}{x}(1-x)^7. You may also set I(mh2/mt2)=1I(m_h^2/m_t^2) = 1; this is correct to about 10\%. (e) The final decay mode that you should consider is h0→2γh^0 \to 2\gamma. Consider first the contribution from the loop diagrams involving quarks and leptons. Show that the result is simply expressed in terms of the form factor I(mh2/m2)I(m_h^2/m^2) that you derived in part (c). (f) Next, compute the contribution to this decay amplitude from the loop diagram involving WW bosons, and the various diagrams one must add to this to obtain a gauge-invariant result. It is easiest to work in Feynman-'t Hooft gauge. Add this contribution to that of very heavy quarks and leptons, each with electric charge QfQ_f. Your result should reduce to the following expression in the limit mh≪mWm_h \ll m_W: Γ(h0→2γ)=(αmh8sin⁡2θw)⋅mh2mW2⋅α218π2⋅∣∑fQf2Nc(f)−214∣2.\Gamma(h^0 \to 2\gamma) = \left(\frac{\alpha m_h}{8\sin^2\theta_w}\right) \cdot \frac{m_h^2}{m_W^2} \cdot \frac{\alpha^2}{18\pi^2} \cdot \left|\sum_f Q_f^2 N_c(f) - \frac{21}{4}\right|^2. (g) Now work out the detailed behavior of the form factor I(x)I(x) defined in part (c). Reduce your expression from part (c) to a one-parameter integral, then evaluate this integral numerically. Plot I(mh2/mt2)I(m_h^2/m_t^2) over the range 50 GeV <mh<< m_h < 500 GeV, and compute the decay width Γ(h0→2g)\Gamma(h^0 \to 2g) numerically (in keV) over this range. The computation of part (f) introduces an additional form factor; compute this function in the same way. (h) Finally, put together all the pieces. Find the branching fraction of the h0h^0 into each of its five major decay modes bbˉb\bar{b}, ttˉt\bar{t}, gggg, W+W−W^+W^-, Z0Z0Z^0Z^0, for Higgs bosons of mass 50 GeV -- 500 GeV.

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