Abstract
The soft bootstrap program aims to construct consistent effective field theories (EFT’s) by recursively imposing the desired soft limit on tree-level scattering amplitudes through on-shell recursion relations. A prime example is the leading two-derivative opera tor in the EFT of SU(N) x SU(N)/SU(N) nonlinear sigma model (NLSM), where \U0001d4aa(p2 ) amplitudes with an arbitrary multiplicity of external particles can be soft-bootstrapped. We extend the program to \U0001d4aa(p4) operators and introduce the “soft blocks,” which are the seeds for soft bootstrap. The number of soft blocks coincides with the number of independent operators at a given order in the derivative expansion and the incalculable Wilson coefficient emerges naturally. We also uncover a new soft-constructible EFT involving the “multi-trace” operator at the leading two-derivative order, which is matched to SO(N + 1) /SO(N) NLSM. In addition, we consider Wess-Zumino-Witten (WZW) terms, the existence of which, or the lack thereof, depends on the number of flavors in the EFT, after a novel application of Bose symmetry. Remarkably, we find agreements with group theoretic considerations on the existence of WZW terms in SU(N) NLSM for N ≥ 3 and the absence of WZW terms in SO(N) NLSM for N ≠ 5.
Highlights
We introduce the notion of a “soft block” here, which serves as the seed of soft bootstrap, and present a new EFT with a flavor structure that is different from the commonly studied SU(N ) adjoint representation
We will see that the new EFT corresponds to SO(N +1)/SO(N ) nonlinear sigma model (NLSM), where N massless scalars transform under the fundamental representation of SO(N ) group
We find 7 soft blocks in total, up to O(p4), which are summarized in table 1
Summary
The modern approach to the soft bootstrap of EFT’s was initiated in refs. [7, 8], which considered various kinds of one-parameter scalar effective theories. The soft bootstrap program is predictive only when the higher-pt amplitudes constructed using the soft recursion relation are independent of the arbitrary coefficients A(r) and B in the general solution of ai’s in eq (2.9). All operators with ρ = 0 can be soft-bootstrapped from this particular soft block This can be seen explicitly from eq (2.25): an arbitrary number of insertions of the two-derivative soft block will generate an amplitude with two powers of external momenta. The only free parameter is the unknown coefficient c0 in eq (2.17), which can be absorbed into the definition of f This agrees with the outcome of imposing shift symmetries in the Lagrangian to bootstrap two-derivative operators in NLSMs that are higher orders in the 1/f expansion [4, 5]. We will see that the new EFT corresponds to SO(N +1)/SO(N ) NLSM, where N massless scalars transform under the fundamental representation of SO(N ) group. Since the flavor-ordered partial amplitudes for multi-trace operators are not commonly encountered in the literature, in appendix A we provide a definition of multi-trace partial amplitudes, which is relevant at O(p4)
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