Standard Model and Beyond · Problem 3.15

Gauge Theories

Physics — Particle Physics · su2-invariants-pion-self-interactions

Problem

Generalize the pion self-interaction term in (3.84) to the case of four distinct types of pions, πAi≡Ai\pi_{Ai} \equiv A_i, πBi≡Bi\pi_{Bi} \equiv B_i, πCi≡Ci\pi_{Ci} \equiv C_i, πDi≡Di\pi_{Di} \equiv D_i, each transforming as an adjoint under SU(2)SU(2). In general, there are cubic and quartic interactions such as LI=−cijklAiBjCkDl−dijkAiBjCk.\mathcal{L}_I = -c_{ijkl}A_i B_j C_k D_l - d_{ijk} A_i B_j C_k. (There are also other possible terms, such as A2BCA^2 BC or ABDABD, which we will not consider.) However, LI\mathcal{L}_I is only SU(2)SU(2) invariant for the combinations LI=−cαTr(AB)Tr(CD)−cβTr(AC)Tr(BD)−cγTr(AD)Tr(BC)−d Tr(ABC),\mathcal{L}_I = -c_\alpha \mathrm{Tr}(AB)\mathrm{Tr}(CD) - c_\beta \mathrm{Tr}(AC)\mathrm{Tr}(BD) - c_\gamma \mathrm{Tr}(AD)\mathrm{Tr}(BC) - d\,\mathrm{Tr}(ABC), where A=A⃗⋅τ⃗/2A = \vec{A} \cdot \vec{\tau}/\sqrt{2}, etc., so that Tr(AB)=A⃗⋅B⃗\mathrm{Tr}(AB) = \vec{A} \cdot \vec{B}. (If AA, BB, and CC are all pseudoscalars, then reflection invariance would require d=0d = 0.) (a) Show explicitly that invariants like Tr(ABCD)\mathrm{Tr}(ABCD) are not independent. (b) The existence of only 3 quartic invariants can be justified as follows: AA, BB, CC, and DD are all isospin-1 operators. Thus, the product ABAB is a direct sum of operators OABIAB\mathcal{O}^{I_{AB}}_{AB} with isospin IAB=0,1I_{AB} = 0, 1, or 22, and similarly for CDCD. However, the only overall singlets involve the pairing with IAB=ICDI_{AB} = I_{CD}. Applying similar reasoning to the cubic term, LI=−∑I=02cIOABIOCDI−d1OAB1C.\mathcal{L}_I = -\sum_{I=0}^{2} c_I \mathcal{O}^I_{AB} \mathcal{O}^I_{CD} - d_1 \mathcal{O}^1_{AB} C. Construct these invariants explicitly, and relate the coefficients cIc_I and d1d_1 to cα,β,γc_{\alpha,\beta,\gamma} and dd. (c) Show why there is only a single invariant in (3.84).

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