Srednicki — Quantum Field Theory · Problem 97.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · proton-decay-rate-su5

Problem

Consider the proton decay mode with the most stringent experimental bound, p→e+π0p \to e^+\pi^0. The terms in eq. (97.43) relevant for this mode are Leff=C1εαβγ(euγ)(dˉα†uˉβ†)+2C2εαβγ(eˉ†uˉγ†)(dαuβ)+h.c.(97.45)\mathcal{L}_{\text{eff}} = C_1 \varepsilon^{\alpha\beta\gamma}(eu_\gamma)(\bar{d}^\dagger_\alpha \bar{u}^\dagger_\beta) + 2C_2 \varepsilon^{\alpha\beta\gamma}(\bar{e}^\dagger \bar{u}^\dagger_\gamma)(d_\alpha u_\beta) + \text{h.c.} \tag{97.45} Note that, under the SU(2)L×SU(2)R\mathrm{SU}(2)_\mathrm{L} \times \mathrm{SU}(2)_\mathrm{R} global symmetry of QCD that we discussed in section 83, the operator u(dˉ†uˉ†)u(\bar{d}^\dagger \bar{u}^\dagger) transforms as the first component of a (2,1)(2,1) representation, while the operator uˉ†(du)\bar{u}^\dagger(du) is related by parity, and transforms as the first component of a (1,2)(1,2) representation. At low energies, we can replace these operators (up to an overall constant factor) with hadron fields with the same properties under Lorentz and SU(2)L×SU(2)R×U(1)V\mathrm{SU}(2)_\mathrm{L} \times \mathrm{SU}(2)_\mathrm{R} \times \mathrm{U}(1)_V transformations. a) Show that PL(uN)1P_\mathrm{L}(u\mathcal{N})_1 and PR(u†N)1P_\mathrm{R}(u^\dagger \mathcal{N})_1 transform appropriately. Here u=exp⁡[iπaTa/fπ]u = \exp[i\pi^a T^a/f_\pi], where πa\pi^a is the triplet of pion fields, and Ni\mathcal{N}_i is the Dirac field for the proton-neutron doublet. b) Show that the low-energy version of eq. (97.45) is then Leff=C1AEC‾PL(uN)1+2C2AEC‾PR(u†N)1+h.c.,(97.46)\mathcal{L}_{\text{eff}} = C_1 A \overline{\mathcal{E}^C} P_\mathrm{L}(u\mathcal{N})_1 + 2C_2 A \overline{\mathcal{E}^C} P_\mathrm{R}(u^\dagger \mathcal{N})_1 + \text{h.c.} , \tag{97.46} where EC\mathcal{E}^C is the charge conjugate of the Dirac field for the electron (in other words, EC\mathcal{E}^C is the Dirac field for the positron), and AA is a constant with dimensions of mass-cubed. Lattice calculations have yielded a value of A=0.0090 GeV3A = 0.0090\,\text{GeV}^3 for μ=2 GeV\mu = 2\,\text{GeV}. c) Write out the terms in eq. (97.46) that contain the proton field and either zero or one π0\pi^0 fields. d) Compute the amplitude for p→e+π0p \to e^+\pi^0. Note that there are two contributing Feynman diagrams: one where eq. (97.46) supplies the proton-positron-pion vertex, and one where the proton emits a pion via the interaction in eq. (83.30), and then converts to a positron via the no-pion terms in eq. (97.46). Neglect the positron mass. Hint: your result should be proportional to 1+gA1 + g_A. e) Compute the spin-averaged decay rate for p→e+π0p \to e^+\pi^0. Use the values of C1C_1 and C2C_2 for μ=2 GeV\mu = 2\,\text{GeV} that you computed in problem 97.3. How does your answer compare with the naive estimate we made earlier? Hint: your result should be proportional to C12+4C22C_1^2 + 4C_2^2.

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