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In this paper, we investigate tree-level scattering amplitude relations in $U(N)$ non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24] both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell $U(1)$ identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy $U(1)$-decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total $2m$-point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.
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We describe the kink solitary waves of a massive non-linear sigma model with an ${mathbb S}^2$ sphere as the target manifold. Our solutions form a moduli space of non-relativistic solitary waves in the long wavelength limit of ferromagnetic linear spin chains.
We summarize recent progress on the symmetric subtraction of the Non-Linear Sigma Model in $D$ dimensions, based on the validity of a certain Local Functional Equation (LFE) encoding the invariance of the SU(2) Haar measure under local left transformations. The deformation of the classical non-linearly realized symmetry at the quantum level is analyzed by cohomological tools. It is shown that all the divergences of the one-particle irreducible (1-PI) amplitudes (both on-shell and off-shell) can be classified according to the solutions of the LFE. Applications to the non-linearly realized Yang-Mills theory and to the electroweak theory, which is directly relevant to the model-independent analysis of LHC data, are briefly addressed.
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We have introduced Faddeev-Niemi type variables for static SU(3) Yang-Mills theory. The variables suggest that a non-linear sigma model whose sigma fields take values in SU(3)/(U(1)xU(1)) and SU(3)/(SU(2)xU(1)) may be relevant to infrared limit of the theory. Shabanov showed that the energy functional of the non-linear sigma model is bounded from below by certain functional. However, the Shabanovs functional is not homotopy invariant, and its value can be an arbitrary real number -- therefore it is not a topological charge. Since the third homotopy group of SU(3)/(U(1)xU(1)) is isomorphic to the group of integer numbers, there is a non-trivial topological charge (given by the isomorphism). We apply Novikovs procedure to obtain integral expression for this charge. The resulting formula is analogous to the Whiteheads realization of the Hopf invariant.
The stability of the kinks of the non-linear ${mathbb S}^2$-sigma model discovered in Phys. Rev. Lett. 101(2008)131602 is discussed from several points of view. After a direct estimation of the spectra of the second-order fluctuation operators around topological kinks, first-order field equations are proposed to distinguish between BPS and non-BPS kinks. The one-loop mass shifts caused by quantum fluctuations around the topological kinks are computed using the Cahill-Comtet-Glauber formula proposed in Phys. Lett. 64B(1976)283. The (lack of) stability of the non-topological kinks is unveiled by application of the Morse index theorem. These kinks are identified as non-BPS states and the interplay between instability and supersymmetry is explored.
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